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CHARACTERIZATION OF CANDIDATE EXOPLANET

COMPANIONS FROM HUBBLE SPACE TELESCOPE

ASTROMETRY, GROUND-BASED RADIAL

VELOCITY, AND INFRARED INTERFEROMETRY

Eder Martioli

Thesis submitted in partial fulﬁllment of the requirements for the degree of Ph.D.

in Astrophysics, supervised by Drs. Francisco Jablonski, Barbara McArthur, and

Fritz Benedict, approved in June 9, 2010.

Original document registry:

<http://urlib.net/xxx>

INPE

São José dos Campos

2010

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CHARACTERIZATION OF CANDIDATE EXOPLANET

COMPANIONS FROM HUBBLE SPACE TELESCOPE

ASTROMETRY, GROUND-BASED RADIAL

VELOCITY, AND INFRARED INTERFEROMETRY

Eder Martioli

Thesis submitted in partial fulﬁllment of the requirements for the degree of Ph.D.

in Astrophysics, supervised by Drs. Francisco Jablonski, Barbara McArthur, and

Fritz Benedict, approved in June 9, 2010.

Original document registry:

<http://urlib.net/xxx>

INPE

São José dos Campos

2010

Cataloging in Publication Data

Martioli, Eder.

Cutter

Characterization of Candidate Exoplanet Companions From

Hubble Space Telescope Astrometry, Ground-Based Radial Veloc-

ity, and Infrared Interferometry / Eder Martioli. – São José dos

Campos : INPE, 2010.

?? + 166 p. ; (INPE-00000-TDI/0000)

Tese (Doutorado em Astrofísica) – Instituto Nacional de

Pesquisas Espaciais, São José dos Campos, 2010.

Orientadores : Francisco J. Jablonski; Barbara E. McArthur;

G. Fritz Benedict.

1. Exoplanets. 2. Binary Stars. 3. Astrometry. 4. Radial Ve-

locity. 5. Infrared Interferometry.

CDU 000.000

 2010 do MCT/INPE. Nenhuma parte desta publicação pode ser reproduzida, ar-

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recording, microﬁlming, or otherwise, without written permission from INPE, with the exception

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system, for exclusive use of the reader of the work.

ATENÇÃO! A FOLHA DE

APROVAÇÃO SERÁ INCLU-

IDA POSTERIORMENTE.

Doutorado em Astrofísica

iii

ACKNOWLEDGEMENTS

I would like to start by conveying my heartfelt gratitude to my parents, Aparecido

and Iraci, my sister Lia and her family, my brother Cid and his family, and my

dearest friend Carla, for their enormous support throughout my journey, and also

for their caring and invaluable aﬀection and friendship.

I am also very appreciative to my advisor Francisco Jablonski who stood by me

since the beginning of my graduate studies, and has made available his support in

a number of ways.

This thesis would not have been possible without the support of my co-advisors in

Texas, Fritz Benedict and Barbara McArthur, those of whom provided me with the

basic ideas and material for this thesis work. I am especially grateful for our fruitful

discussions and for the continuous academic support they provided me at the time

I was in Texas, and for their patience in keeping advising me through hundreds of

emails, even after I came back to Brazil.

I would also like to thank my colleagues Carla Gil and Ramarao Tata, who helped

me in developing the pioneering work on the infrared interferometry experiment.

They have also been decisive in the preparation of observing proposals, obtaining

data, and in the data reduction and analysis.

I am grateful for many of the personnel involved in my research work. Particularly,

I would like to thank the competent work of clerical assistants at DAS-INPE, and

at the University of Texas at Austin. Not to mention the always eﬃcient work of

clericals and technicians at the observatories.

I would also like to thank the Coordenação de Aperfeiçoamento de Pessoal de Nível

Superior (CAPES) agency for the ﬁnancial support, and all the Brazilian and inter-

national funds that I made use of for the development of my research. I am grateful

for the support provided by NASA through grant GO-10704-10989, and 11210 from

the Space Telescope Science Institute, for observations obtained from the Hobby-

Ebberly Telescope at McDonald Observatory, and from the Very Large Telescope

Array at European Southern Observatory. I would also like to thank the services

provided by SIMBAD database, operated at CDS, Strasbourg, France, and NASA’s

Astrophysics Data System Abstract Service.

vii

Finally, I should point up that I have not mentioned everyone who deserves being

here. For this reason I would like to express my general but sincere gratitude to

all of my friends, co-workers, and family members, who witnessed my quest and in

some way supported me and made this thesis possible. These people surely know

how important they were and still are to me.

viii

ABSTRACT

This work presents the development of observational techniques and data analy-

sis for the follow-up of RV-detected exoplanet candidates and low-mass compan-

ions. We present high-cadence radial velocity data obtained with the HRS/HET

combined with previously published data, and relative FGS/HST astrometry for

HD 136118 and HD 33636. We perform a simultaneous analysis of these data in order

to characterize the companion’s orbit thoroughly. This establishes the actual mass

of HD 136118 b, M

= 63

+22

−13

, in contrast to the minimum mass determined from

the radial velocity data only, M

sin i ∼ 12 M

. Therefore, the low-mass companion

to HD 136118 is now identiﬁed as a likely brown-dwarf residing in the “brown-dwarf

desert”. We have performed a similar analysis for the object HD 33636. For this ob-

ject we also present experimental AMBER/VLTI infrared interferometric data. The

latter provides the measurement of visibility variations, which are consistent with

an additional light that presents ﬂux ratio of 30%, for a system with a G0 V primary

star and an M-dwarf companion.

CARACTERIZAÇÃO DE CANDIDATOS A EXOPLANETAS VIA

ASTROMETRIA COM O TELESCÓPIO ESPACIAL, MEDIDAS EM

TERRA DE VELOCIDADES RADIAIS E INTERFEROMETRIA NO

INFRAVERMELHO

RESUMO

Neste trabalho desenvolvem-se técnicas observacionais e análise de dados para o

estudo de candidatos a exoplanetas e companheiras de baixa massa detectados via

velocidades radiais. Apresentamos medidas de alta-cadência de velocidades radiais

obtidas com o Espectrógrafo de Alta Resolução no Telescópio Hobby-Eberly, com-

binado com dados publicados anteriormente, e medidas astrométricas com o instru-

mento FGS-1r no Telescópio Espacial Hubble dos sistemas HD 136118 e HD 33636.

Realizamos a análise simultânea desses dados para caracterização completa da ór-

bita das companheiras. O trabalho resultou na determinação da massa verdadeira de

HD 136118 b, M

= 63

+22

−13

, em contraste com a massa mínima determinada ante-

riormente via velocidades radiais, M

sin i ∼ 12 M

. Portanto, HD 136118 b é identi-

ﬁcada como uma provável anã-marrom que reside no “deserto das anã-marrons”. Re-

alizamos uma análise semelhante para os dados do também candidato a exoplaneta

HD 33636. Para este objeto, apresentamos medidas interferométricas experimentais

no infravermelho com o instrumento AMBER no VLTI. Esse experimento resultou

na medida de variações na visibilidade consistentes com uma luz adicional que ap-

resenta razão de ﬂuxos de aproximadamente 30%, para um sistema binaário que

constitui-se de uma estrela primária do tipo G0 V e uma companheira anã do tipo

LIST OF FIGURES

Page

3.1 Ellipse geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Simple representation of the binary brightness distribution and the bi-

nary parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 HRS optical layout. Source: http://www.as.utexas.edu/mcdonald. . . . . 33

4.2 HRS System Parameters. Source: http://www.as.utexas.edu/mcdonald. 33

4.3 HRS design parameters. Source: http://www.as.utexas.edu/mcdonald. . 34

4.4 HRS performance parameters. Source:

http://www.as.utexas.edu/mcdonald. . . . . . . . . . . . . . . . . . . . . 34

4.5 FGS1r optical train schematic. Source: Nelan et al. (2010). . . . . . . . . 38

4.6 Great circle: FGS ﬁeld-of-view on the HST focal plane (projected onto

the sky). Right panel: FGS-1r S-curve response for a point-like source.

Source: Nelan et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.7 Constructive and destructive interference in the Koester prism. Source:

Nelan et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.8 Simpliﬁed optical setup of AMBER instrument. Source: Tatulli et al.

(2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.9 AMBER sample image of an exposure of a calibration source. DK is the

dark channel, P1, P2 and P3 are the ordinary chanels from each telescope

and IF indicates the interferometric channel, where the fringe pattern is

recorded. The wavelength dispersion is on the vertical direction. Source:

Tatulli et al. (2007). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 ASTROSPEC pipeline ﬂowchart. . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Pair of a single ﬂat-ﬁeld exposure of echelle HRS spectra. The images are

respectively the RED CCD (left) and BLUE CCD (right). . . . . . . . . 52

5.3 Directory tree of ASTROSPEC. . . . . . . . . . . . . . . . . . . . . . . . 53

5.4 Sample piece of a normalized ﬂat-ﬁeld. . . . . . . . . . . . . . . . . . . . 54

5.5 Example of the blaze function for CCD 1 (left) and CCD 2 (right). . . . 55

5.6 Extraction scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.7 A sample piece of an extracted spectrum where it has been adopted 5

diﬀerent values for the sample size, M

= 1, 3, 5, 8, and 12. Each spectrum

was shifted for the better visualization. . . . . . . . . . . . . . . . . . . . 57

xiii

5.8 A sample piece of an extracted spectrum where it has been adopted 5

diﬀerent values for the aperture size, N

= 7, 9, 11, 13, and 15. . . . . . . 58

5.9 Example of the normalization of a given spectral order. The full circles

represent the maximum points found by the normalization algorithm,

the green line represents the spline interpolation through the maximum

points. The spectrum is shown in red. . . . . . . . . . . . . . . . . . . . . 61

5.10 Normalized spectrum after running normspec_img. . . . . . . . . . . . . 62

5.11 OUTSPEC normalized spectral images of HD 136118 for both the RED

CCD (top) and the BLUE CCD (bottom). . . . . . . . . . . . . . . . . . 62

5.12 Th-Ar entire calibrated spectrum. . . . . . . . . . . . . . . . . . . . . . . 64

5.13 Th-Ar calibrated spectrum for order 115 (red line) and 116 (green line),

and the reference spectrum (blue line). . . . . . . . . . . . . . . . . . . . 65

5.14 Instrument Proﬁle (IP) model (red). We have used a parameterized model

comprising a central gaussian (green) plus 10 satellite gaussians (blue). . 68

5.15 Sample chunk 2 Angstrom wide showing the template I

spectra (solid

thin line), the observed Flat+I

spectra (ﬁlled circles) and the ﬁt model

(solid line connecting solid circles), and ten times the residuals (open

circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.16 Gasgano window showing the interferometric data. . . . . . . . . . . . . 78

5.17 Gasgano snapshot window showing the loaded maintenance ﬁles of

the following categories: AMBER_2P2V, AMBER_2WAVE, AMBER_-

3P2V, AMBER_3WAVE, AMBER_BADPIX, and AMBER_FLAT-

FIELD. These data are used by routine “amber_p2vm”. . . . . . . . . . 79

5.18 Uniform disk model for the squared visibility of calibrators listed in Table

5.1 and the Sci object, HD 33636. Dashed vertical line shows the cut-oﬀ

frequency for our experiment. . . . . . . . . . . . . . . . . . . . . . . . . 81

5.19 Baseline 1 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636. . . . . . . . . . . . 81

5.20 Baseline 2 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636. . . . . . . . . . . . 82

5.21 Baseline 3 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636. . . . . . . . . . . . 82

5.22 Solid lines show the mean diﬀerence between each point and the average,

∆V

. Dashed lines show the standard deviation. . . . . . . . . . . . . . . 84

5.23 System visibility, V

sys

(transfer function), for all calibrators as indicated

in the legend. The thick solid line is the average of all data sets. . . . . . 85

xiv

5.24 List of calibrator data sets that have been ﬁltered out for calibrating

interferometric data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.25 HD 19637 squared visibility data (ﬁlled circles) and the disk model ﬁt

(solid line). Data points with σ

> 1.5 have been cut oﬀ. The ﬁt angular

diameter for HD 19637 is Θ = 1.452 ± 0.035 mas. . . . . . . . . . . . . . . 86

5.26 HD 36134 squared visibility data (ﬁlled circles) and the disk model ﬁt

(solid line). Data points with σ

> 2.0 have been cut oﬀ. The ﬁt angular

diameter for HD 36134 is Θ = 1.362 ± 0.029 mas. . . . . . . . . . . . . . . 87

5.27 HD 33636 squared visibility data is represented by triangles (baseline 1),

circles (baseline 2), and diamonds (baseline 3). The disk model for the

derived angular size of HD 33636, Θ = 0.383 mas, is also shown (solid line). 88

6.1 HD 136118 radial velocities as function of time and the best-ﬁt model. . 91

6.2 HD 136118 radial velocities and the best-ﬁt model in the phase diagram.

Folding period is P = 1188.67 days. . . . . . . . . . . . . . . . . . . . . . 92

6.3 HD 33636 radial velocities as function of time and the best-ﬁt model. . . 93

6.4 HD 33636 radial velocities and the best-ﬁt model in the phase diagram.

Folding period is P = 2119.68 days. . . . . . . . . . . . . . . . . . . . . . 94

6.5 Histogram of HD 136118 RV residuals from 1-companion model for 3

datasets: all combined (ALL) (top panel), Lick (middle panel) and HET

(bottom panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.6 Lomb-Scargle periodogram for RV residuals from 1-companion model and

using all datasets combined (solid line). The thresholds for false alarm

probability of 1% and 10% are plotted in dotted lines. Above it is shown

the AIPD (see text) for the same dataset (dashed line). The AIPD here

is multiplied by a factor 20 and shifted for the sake of better visualization. 98

6.7 Lomb-Scargle periodogram for HET RV residuals from 1-companion

model (solid line). The thresholds for false alarm probability of 1% and

10% are plotted in dotted lines. Above it is shown the AIPD (see text)

for the same dataset (dashed line). The AIPD here is multiplied by a

factor 20 and shifted for the sake of better visualization. . . . . . . . . . 99

6.8 χ

map from a 2-companion ﬁt model for HET and Lick data. The grid

resolution is about 0.3 day (1100 × 90 points). Contour lines show four

diﬀerent levels of χ

. The best ﬁt solution has the lowest value at χ

76.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.9 HD 136118 radial velocities as function of time and the best-ﬁt two-

companion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.10 Phase diagram folded with period 1191.52 days. RV HET (ﬁlled circles)

and Lick (open marks) data subtracted the “nuisance orbit” model. Solid

line shows the best ﬁt HD 136118 b orbit model. Residuals from the 2-

companion model is plotted in the bottom panel. . . . . . . . . . . . . . 103

6.11 RV HET (ﬁlled circles) and Lick (open marks) residuals from

HD 131168 b orbit model and the best ﬁt “nuisance orbit” model plotted

in the phase diagram folded with period 254.06 days. Residuals from the

2-companion model is plotted in the bottom panel. . . . . . . . . . . . . 104

6.12 HD 136118 ﬁeld-of-view. Source: image from the Science and Engineer-

ing Research Council Survey (SERC), Space Telescope Science Institute

(STSI), digitized with the Plate Densitometer Scanner (PDS). This is

superimposed on the Naval Observatory Merged Astrometric Dataset

(NOMAD) catalog. These were obtained through the Aladin previewer

and the SIMBAD database. . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.13 HD 33636 ﬁeld-of-view. Source: image from the Palomar Observatory Sky

Survey (POSSII), Space Telescope Science Institute (STSI), digitized

with the Plate Densitometer Scanner (PDS). This is superimposed on

the Naval Observatory Merged Astrometric Dataset (NOMAD) catalog.

These were obtained through the Aladin previewer and the SIMBAD

database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.14 Predicted orbit perturbation size α as function of orbital inclination i

for HD 136118 b (solid line), HD 33636 b (dashed line) and the nuisance

orbit (dotted line). The latter is identiﬁed as a possible component ‘c’ of

HD 136118. The horizontal line represents a 1 mas detection threshold. . 112

6.15 Open circles are the reduced astrometry data for HD 136118 (top left

panel) and for the astrometric reference stars. Filled circles show the

median for clumps with 100 data points. Black lines show the ﬁt model

for the apparent path of each star. Each component of this model is also

shown separately: blue lines for the parallax, red lines for the proper

motion, and magenta line for the perturbation orbit. . . . . . . . . . . . 117

6.16 Open circles are the reduced astrometry data for HD 33636 (top left

panel) and for the astrometric reference stars. Filled circles show the

median for clumps with 40 data points. Black lines show the ﬁt model

for the apparent path of each star. Each component of this model is also

shown separately: blue lines for the parallax, red lines for the proper

motion, and magenta line for the perturbation orbit. . . . . . . . . . . . 118

xvi

6.17 Histogram of astrometric residual data for FGS X and Y positions for

HD 136118 (left panels) and HD 33636 (right panels). . . . . . . . . . . . 119

6.18 ∆ξ and ∆η components of HD 136118 perturbation orbit versus time. . 120

6.19 Filled circles with error bars are the median and standard deviation of

three groups of astrometric residuals of HD 136118, representing three

diﬀerent epochs. Solid line is the ﬁt model of the apparent perturbation

orbit of HD 136118. Open circles are the positions calculated from the ﬁt

model, each of which is connected by a thick solid line to its respective

observed epoch. The open square shows the predicted position of the

periastron passage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.20 Filled circles with error bars are the median and standard deviation of ﬁve

groups of astrometric residuals of HD 33636, representing ﬁve diﬀerent

epochs. Solid line is the ﬁt model of the apparent perturbation orbit of

HD 33636. Open circles are the positions calculated from the ﬁt model,

each of which is connected by a thick solid line to its respective observed

epoch. The open square shows the predicted position of the periastron

passage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.21 CDF (reﬂected vertically for values > 0.5) for the masses of HD 136118 b

(red curve) and HD 33636 B (blue curve). Filled in grey presents the re-

gion within which we consider the uncertainty on the mass measurement.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.22 CDF (reﬂected vertically for values > 0.5) for the semimajor axes of

HD 136118 b (red curve) and HD 33636 B (blue curve). Filled in grey

presents the region within which we consider the uncertainty on the semi-

major axis measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.23 Apparent orbit of HD 33636 A (solid line) and B (dashed line). Filled

circles show the predicted positions for each component in UT 2008-10-

14. The periastron is indicated with open squares. . . . . . . . . . . . . 128

6.24 Squared visibility reduced data (ﬁlled circles) for the three baselines and

the best ﬁt model (solid line). The model consists of a binary with a

disk-like source with size Θ

= 0.38 mas plus a point-like source. The ﬁt

parameters are the ﬂux ratio, f = 0.323 ± 0.025, the binary separation,

ρ = 169.52 ± 0.11 mas, and the position angle θ = 269

◦

.03 ±0.16. . . . . 129

6.25 Open circles are the reduced interferometric phases for HD 33636. Filled

circles show the median taken over clumps containing 15 points. . . . . . 130

xvii

6.26 Open circles are the reduced closure phases for HD 33636. Filled circles

show the median taken over clumps containing 15 points. . . . . . . . . 131

7.1 FGS-1r photometry of HD136118. Magnitude variation is relative to the

mean magnitude, V=6.93. Dashed lines show the amplitude of variation

possible from a (single) spot ﬁlling factor of 6%, the spot ﬁlling factor

required to produce the observed RV variation from HD136118 b (SAAR;

DONAHUE, 1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.2 Predicted emission spectrum of HD 136118 (dash-dotted line), emis-

sion/reﬂection spectrum of HD 136118 b and the ﬂux ratio between the

brown dwarf and the parent star for a 5 Gyr and 1 Gyr system age, as

indicated in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 Mass versus absolute magnitude in the K band for low-mass stars

(BARAFFE et al., 1998). The models correspond to an age of 5 Gyr. The

yellow ﬁlled circle shows the position of HD 33636 A. The red ﬁlled circle

shows the predicted position of HD 33636 B from its dynamical mass. The

red open circle shows the position of HD 33636 B considering both the

dynamical mass and the infrared ﬂux ratio obtained from interferometry. 139

7.4 Blackbody emission spectra of HD 33636 A (dash-dotted line) and

HD 33636 B (solid line). The ﬂux ratio between the two components and

the limit where the ﬂux ratio is 30% is shown in dotted lines. . . . . . . 140

xviii

LIST OF TABLES

Page

2.1 Properties of HD 136118. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Properties of HD 33636. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1 Log of astrometric observations. . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Log of interferometric observations. . . . . . . . . . . . . . . . . . . . . . 47

5.1 Information for the Interferometric Calibrators (MÉRAND et al., 2005). . 79

6.1 HD 136118: RV best-ﬁt parameters. . . . . . . . . . . . . . . . . . . . . 95

6.2 HD 33636: RV best-ﬁt parameters. . . . . . . . . . . . . . . . . . . . . . 96

6.3 HD 136118: RV best-ﬁt parameters for a two-companion model. . . . . . 101

6.4 HET relative radial velocities for HD 136118. . . . . . . . . . . . . . . . . 105

6.5 Lick relative radial velocities for HD 136118. . . . . . . . . . . . . . . . . 106

6.6 HET relative radial velocities for HD 33636. . . . . . . . . . . . . . . . . 107

6.7 Lick relative radial velocities for HD 33636. . . . . . . . . . . . . . . . . . 107

6.8 Keck relative radial velocities for HD 33636. . . . . . . . . . . . . . . . . 108

6.9 Elodie relative radial velocities for HD 33636. . . . . . . . . . . . . . . . 108

6.10 HD 136118: classiﬁcation spectra and photometric information for the

astrometric reference stars. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.11 HD 33636: classiﬁcation spectra and photometric information for the as-

trometric reference stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.12 HD 136118, HD 33636, and the astrometric reference stars information

from the NOMAD catalog (ZACHARIAS et al., 2005). . . . . . . . . . . . . 114

6.13 HD 136118, HD 33636, and the astrometric reference stars information

from the UCAC-3 catalog (ZACHARIAS et al., 2009). . . . . . . . . . . . . 115

6.14 HD 136118 and HD 33636 astrometric information from the Hipparcos

catalog (PERRYMAN et al., 1997; LEEUWEN, 2007). . . . . . . . . . . . . . 115

6.15 Resulting astrometric catalog for HD 136118 from the RV and astrometry

simultaneous ﬁt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.16 Resulting astrometric catalog for HD 33636 from the RV and astrometry

simultaneous ﬁt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.17 HD 136118: best-ﬁt parameters for a two-companion model from the si-

multaneous RV and astrometry data analysis. . . . . . . . . . . . . . . . 123

xix

6.18 HD 33636: best-ﬁt parameters from the simultaneous RV and astrometry

data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

CONTENTS

Page

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 RELATED WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Individual Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 The Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Physics of Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 N-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Experiment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5.1 Astrometry: Proper Motion and Parallax . . . . . . . . . . . . . . . . . 16

3.5.2 Astrometry: Apparent Orbital Motion . . . . . . . . . . . . . . . . . . 18

3.5.3 Spectroscopy: Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.4 Spectroscopy: Radial Velocity for Keplerian Orbits . . . . . . . . . . . 21

3.5.5 Spectroscopy: Multi-planet Keplerian RV Model . . . . . . . . . . . . . 22

3.5.6 Interferometry: Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5.7 Interferometry: Measuring Visibility Functions . . . . . . . . . . . . . . 24

3.5.8 Interferometry: Visibility Model . . . . . . . . . . . . . . . . . . . . . . 25

3.5.9 Interferometry: Binary Model . . . . . . . . . . . . . . . . . . . . . . . 26

4 EXPERIMENTS DESIGN . . . . . . . . . . . . . . . . . . . . . . 31

4.1 Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 The Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.3 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.3.1 HET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.3.2 HRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

xxi

4.1.4 Strategy and Observations . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.1 The Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.2 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.2.1 HST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.2.2 FGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3 Strategy and Observations . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Infrared Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 The Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.3 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.3.1 VLTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.3.2 AMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.3.3 FINITO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.4 Strategy and Observations . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 DATA REDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.2 Data Reduction for Obtaining Radial Velocities . . . . . . . . . . . . . 50

5.1.3 The Automatized Pipeline for Spectra Reduction (ASTROSPEC Pack-

age) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.3.1 Preparation of data . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1.3.2 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.3.3 Extraction of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.3.4 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.3.5 Wavelength Calibration (Th-Ar) . . . . . . . . . . . . . . . . . . . . 60

5.1.3.6 Measuring Radial Velocities (Iodine Method) . . . . . . . . . . . . . 63

5.2 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.1 FGS Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.2 Calibration of FGS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.3 Astrometric Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Infrared Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3.1 Reduction of the AMBER/VLTI Data . . . . . . . . . . . . . . . . . . 77

5.3.2 Calibrating Interferometry Visibility Data . . . . . . . . . . . . . . . . 79

5.3.3 Calibrating Interferometry Phase Data . . . . . . . . . . . . . . . . . . 86

xxii

6 DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1 Radial Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Limits on Additional Periodic Signals in the HD 136118 RV Data . . . 93

6.2 Astrometry Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.0.1 RV constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.0.2 Spectro-Photometric Parallaxes . . . . . . . . . . . . . . . . . . . . . 113

6.2.0.3 Proper Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.0.4 Hipparcos Data for the Targets . . . . . . . . . . . . . . . . . . . . . 114

6.3 Combining Astrometry, Radial Velocity, and Priors . . . . . . . . . . . . 115

6.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4 Derivation of the True Mass . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.5 Infrared Interferometry for HD 33636 . . . . . . . . . . . . . . . . . . . . 125

6.5.1 Interferometric Phase and Closure Phase . . . . . . . . . . . . . . . . . 127

7 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . 133

7.1 HD 136118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.1.1 Activity in HD 136118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.1.2 HD 136118 Spin Axis and the Companion’s Orbit Alignment . . . . . . 133

7.1.3 HD 136118 b: a Brown Dwarf Companion . . . . . . . . . . . . . . . . . 134

7.1.4 HD 136118 c: A Possible New Planet? . . . . . . . . . . . . . . . . . . . 136

7.2 HD 33636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.2.1 RV and Astrometry Results . . . . . . . . . . . . . . . . . . . . . . . . 137

7.2.2 Reliability of our Interferometric Results . . . . . . . . . . . . . . . . . 137

7.2.3 Flux Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.2.4 The Spectral Distribution of HD 33636 Components . . . . . . . . . . . 139

7.2.5 What would be the Proper Interferometer Set Up to Study HD 33636? 140

8 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . 143

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

10 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.1 Model for the Simultaneous Analysis of Astrometry and Radial Velocity

HD 136118 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xxiii

1 INTRODUCTION

Centuries ago philosophers and scientists like Giordano Bruno and Isaac Newton

have already assumed the possibility of planets existing around stars other than

the Sun. Although it has been conjectured long ago, the search for exoplanets only

became possible when the experiments in astronomy reached maturity. The search

for other worlds has started with astrometric measurements. Jacob (1855) announced

the discovery of a likely planetary body orbiting the binary star 70 Ophiuchi. He

asserted the discovery of a third unseen companion aﬀecting the orbit of the two

visible stars. His discovery has never been conﬁrmed. Later on, during the 1950s

and 1960s, Peter van de Kamp also claimed the astrometric detection of planetary

bodies orbiting the Barnard’s Star (KAMP, 1969), the largest proper motion star in

the sky. It was found later that his photographic plate technique had not attained

precision enough for the detections he was claiming. In a recent work, Benedict et

al. (1999), using the Fine Guidance Sensors (FGS) with the Hubble Space Telescope

(HST), have found no companion to Barnard’s star, with a companion detection

sensitivity less than or equal to one Jupiter mass, for periods longer than 150 days.

Although the ﬁrst attempts to detect exoplanets were astrometric, the ﬁrst discovery

of a planetary body outside the Solar System came along through a diﬀerent tech-

nique. In 1992, radio astronomers Aleksander Wolszczan and Dale Frail announced

the discovery of planets around a pulsar, PSR 1257+12 (WOLSZCZAN; FRAIL, 1992),

using pulsar timing variations. On October 6, 1995, Michel Mayor and Didier Queloz

of the University of Geneva announced the ﬁrst deﬁnitive detection of an exoplanet

orbiting an ordinary main-sequence star, 51 Pegasi (MAYOR; QUELOZ, 1995). The

latter was possible using the Radial Velocity (RV) method. The technological ad-

vances and the development of this technique have contributed to the detection of

the majority of exoplanets to date. The number of detections has been raising very

rapidly in the last few years and today over 400 of these objects are known, most of

which has been detected through the RV method. A full up-to-date list of exoplanets

can be found at http://exoplanet.eu.

Planetary systems and binary (or multiple) star systems are currently becoming part

of the same ﬁeld of study in astronomy. The reason for this is that these objects are

all of the same kind, a Multiple Orbiting Body System (MOBS). Although the

bodies involved in each kind of system may be very diﬀerent physically, on what con-

cerns the observing techniques they turn out to be very alike. The study of MOBS

has become highlighted in the past few years due to the discovery of exoplanets.

The great attention given to this ﬁeld is obviously motivated by the possibility of

ﬁnding bodies that might be similar to the Earth and perhaps host some kind of

extraterrestrial life. Although this would be the most exciting achievement for the

exoplanetary research, an important progress in this ﬁeld has already been done. For

example, the understanding of how planets, brown-dwarfs and low-mass stars are

formed. Few years ago the only known planets were those in the Solar System. To-

day, the hundreds of exoplanets present an entire “zoo” of planetary systems. Among

these, one can ﬁnd even unprecedented classes of planets, like the Hot-Jupiters and

Super-Earths. There are also planets in very unusual and unexpected conditions,

like those in very eccentric orbits, or those in extremely short period orbits. Multi-

ple systems are common, including some systems that resemble our own, like υ And

and 55 Cnc. Exoplanet candidates have been found in many diﬀerent environments,

like in binary stars systems, around pulsars, around brown dwarfs, among others.

Exoplanetary systems also present themselves as excellent laboratories to test theo-

ries in orbital dynamics. This has provided important contributions to the study of

planet formation in general as well as to the study of our own Solar System. These

discoveries are certainly just the tip of the iceberg, since the techniques employed

are still limited and under growing technological development. The outstanding pre-

cision required for the detection of exoplanets in part motivates these advances and

as a beneﬁt many other areas in astrophysics may be privileged. In particular those

that make use of the same techniques, like the MOBS.

A good illustration to show the limitation of methods for detecting exoplanets is the

fact that for most objects detected to date only a few of them have their actual mass

known. The widely used Doppler spectroscopy technique (same as Radial Velocity)

yields the radial component of the stellar perturbation velocity only. Consequently,

the inclination of the orbital plane is unknown and only the minimum mass of the

companion may be determined. To fully determine the orbit and obtain a compan-

ion’s true mass it is necessary to make use of additional techniques. Note that the

mass is a crucial parameter, since it is deﬁnitive to classify a body as a planet. The

ﬁrst precise determination of an exoplanet mass was made for a transiting system

(HENRY et al., 2000). However, transits are observed to occur only for systems that

are oriented edge-on, and they have only a small probability of occurrence for close-

in planets (semimajor axes less than about 0.1 AU). Another way to determine the

orbital inclination of an unseen companion is by measuring the stellar reﬂex mo-

tion astrometrically. The ﬁrst astrometrically determined mass of an exoplanet by

Benedict et al. (2002) was possible thanks to the high precision of the FGS instru-

ment on the HST. The FGS provides per observation precisions of better than 1 mas

for small angle relative astrometry. This unique capability enables the detection of

stellar perturbations due to planetary mass companions in wide orbits.

For this thesis work we were granted observing time with the HST to measure the

perturbation and determine the true mass of HD 136118 b, which is an exoplanet can-

didate found by radial velocity measurements (FISCHER et al., 2002). We have also

been granted observing time with the Hobby-Eberly Telescope (HET) to obtain

high-cadence radial velocity measurements. This will supplement previously pub-

lished data and provide better constrain on the companion’s spectroscopic orbital

parameters. A similar experiment has been performed for the exoplanet candidate

HD 33636 b, for which the results are published in Bean et al. (2007). The compan-

ion to HD 33636 is an M-dwarf star. We have selected this object as a test-bed to

perform interferometric measurements as an additional and alternative technique to

the HST astrometry. We were granted VLTI time in a short science veriﬁcation run

for testing the new instrument FINITO working with AMBER in order to measure

the binary parameters of HD 33636.

In summary, the primary goal of this work is the development of methods for com-

bining the two techniques, Doppler spectroscopy and relative astrometry, and if

possible, include infrared interferometry data. These are meant to be analyzed si-

multaneously, in order to provide a thorough characterization of exoplanet candi-

dates and MOBS in general. The advantage of combining multiple techniques is that

one may constrain parameters in the model and determine them more precisely. It

means that it provides more information about bodies involved in the system. Some

parameters are critical and without this information it is almost impossible to deﬁne

what kind of body is involved in the MOBS.

Below we outline the contents of this work.

• Related Work - brief description of works that motivated us and that

present results obtained in a similar way as the experiments developed

in this thesis. We also include the previous results from each individual

object in study.

• Background - overview on the background theory for the astrophysical

models and experiments.

• Experiment Design - detailed description of the experiments and the ob-

servational techniques employed.

• Data Reduction - detailed description of the methods used to calibrate

data and extraction of observables from each experiment.

• Data Analysis - analysis of the reduced data in order to obtain the relevant

physical information from each observed system.

• Results and Discussion - discussion on the results and their main astro-

physical implications.

• Summary and Conclusions - summary of achievements obtained with this

work and concluding remarks.

2 RELATED WORK

2.1 Motivation

The main motivation for this thesis work is that the Doppler spectroscopy technique

is insuﬃcient to characterize the systems thoroughly. Therefore, we make use of HST

astrometry in order to better understand these systems. We also propose infrared in-

terferometry as an additional tool for obtaining further information from the system.

Below we describe some related works found in the literature that have motivated

us and that provided us with the necessary background for the development of these

experiments.

2.2 Radial Velocity

The RV technique consists of measuring the star’s velocity through the shift of spec-

tral lines due to the Doppler eﬀect. In order to obtain precise radial velocities, it is

important to perform a careful treatment of the spectral data. Baranne et al. (1996)

present a method to reduce the ELODIE spectrograph data, which was the respon-

sible for the ﬁrst, and many other detections of extrasolar planet candidates around

Sun-like stars. At about the same time, another eﬃcient technique was developed by

Butler et al. (1996), who have attained comparable radial velocity precision using

an echelle spectrograph with the iodine method. This technique has also led to the

discovery of many exoplanet candidates. We have employed the same methodology

for our spectroscopic measurements. This powerful technique together with the large

collecting area of the Hobby-Eberly Telescope (HET), and the quality of its High-

Resolution Spectrograph (HRS) allowed us to obtain relative velocity precision of

about ∼ 3 m/s.

2.3 Astrometry

One of our main results is to show the importance of using complementary techniques

to the RV method in the characterization of low-mass companions. The relative

astrometry with FGS/HST is a potential technique to explore the entire orbit of the

star due to unseen companions. This method is shown to be eﬃcient for variations

due to companions within a range of minimum masses, 5 < M sin i < 17 M

. In

our measurements, we were able to estimate the key orbital elements of the RV-

detected companion to HD 136118, like the perturbation orbit size α, the longitude

of the ascending node, Ω, and the inclination, i, thus turning M sin i into true mass,

M. A similar approach has already been performed in a previous work, where HST

astrometry and ground-based radial velocities have been combined to determine the

orbit and the true mass of exoplanet candidates. Gliese 876 b, a planetary companion

to the M dwarf star Gl 876 (BENEDICT et al., 2002), is the case where HST astrometry

allowed the determination of the true mass of an exoplanet candidate companion,

= 1.89 ± 0.34 M

, which is now unequivocally deﬁned as a Jupiter-like giant

planet. Only the transiting star-planet systems can provide a mass estimate with

smaller error. In contrast to that short-period system, the same techniques have been

used to determine the inclination for the outermost known planet 55 Cnc d in the

55 Cnc system (MCARTHUR et al., 2004), a companion with a period, P = 4517 days.

In this case seventeen data sets over 380 days resulted in an inclination with a

13% error. The perturbation had a semi-amplitude α = 1.9 ± 0.4 mas. As it will

be shown in our results, we found similar levels of precision for HD 136118 with

observations spread over about a similar time span, because shorter periods result

in great curvature in the perturbation. In a recent work McArthur et al. (2010)

have determined the true mass of two exoplanet candidate companions to the υ And

system, M

= 13.98

+2.3

−5.3

and M

= 10.25

+0.7

−3.3

, which is the ﬁrst multi-planet

system with accurate mass determination for two components. This is also the ﬁrst

determination of the mutual inclination between objects in an extrasolar system.

Notice that υ And is a wide binary star system, for which the component υ And A

is a solar type star with three planetary companions.

The FGS/HST astrometric observations of the G4 IV star HD 38529 have also pro-

vided the measurement of the mass of the outermost of two previously known com-

panions (BENEDICT et al., 2010). The mass of HD 38529 c is M

= 17.6

+1.5

−1.2

, which

is 3σ above a 13 M

deuterium burning, brown dwarf lower limit.

In contrast to the exoplanet candidates Gl876 b, 55 Cnc d, and υ And c and d, which

have been identiﬁed as truly planetary bodies, and HD 38529 c, as a brown dwarf

companion, the other exoplanet candidate observed by FGS/HST, HD 33636 b, has

been identiﬁed as an M dwarf stellar companion, with true mass M = 142 ± 11 M

(BEAN et al., 2007). HD 33636 b has ﬁrstly been detected through RV (VOGT et

al., 2002; PERRIER et al., 2003; BUTLER et al., 2006), showing a minimum mass of

M sin i = 9.3 M

, which is consistent with planetary mass. This discrepancy occurs

due to the nearly face-on orbit orientation. This proves the eﬃciency of astrometry

in complementing the RV method to access the real nature of a planetary system.

Given the high astrometric signal from HD 33636 we have selected this system as an

experimental case of study to validate the methods used in the analysis of HD 136118

data. The companion to HD 33636 is likely to provide a larger ﬂux contribution to

the system, therefore it is a suitable target to perform an interferometric experiment

using the AMBER instrument at VLTI. This sort of experiment has already been

performed as explained below.

2.4 Interferometry

The interferometric experiment we present in this work is not an attempt of direct

imaging, rather it is just an attempt of detecting any interferometric signature in

the infrared originated from the unseen companion. The contrast between a hot

parent star and its cooler companion is much favorable in the infrared region of

spectra. Besides, interferometry provides two main features that favors the detection

of exoplanets and low mass companions. First it combines the light collected from

many optical telescopes, therefore resulting in larger collecting area than in single

instruments. Second, an interferometer can probe the light distribution at very small

angular scales. These characteristics are essential for the detection of any weak

luminous feature lying close to the stellar disk, just like planets. The theory relating

interferometric observables to the binary parameters is explained in Chelli et al.

(2009). Duvert et al. (2010) have recently published the results of an interferometric

experiment for the detection of a close faint companion using AMBER/VLTI, thus

very similar to the experiment presented here.

2.5 Individual Objects

In this section we provide a compilation of facts from the targets in study. The

targets, HD 136118 and HD 33636, are solar type main-sequence stars with RV-

detected low mass companions, both of which present M sin i below the planetary

limit,  13 M

. HD 136118 (=HIP 74948) is a V = 6.93, F9 V star with roughly

solar photospheric abundances (GONZALEZ; LAWS, 2007). Table 2.1 summarizes its

observed properties given in the literature. HD 33636 (=HIP 24206) is a high-proper

motion, V = 7.0 G0 V solar-type star at a distance of 30 pc. Table 2.2 summarizes

its observed properties.

TABLE 2.1 - Properties of HD 136118.

ID HD 136118 Unit Reference

RA(2000) 15 : 18 : 55.4719 (8.18) h:m:s Perryman et al. (1997)

Dec(2000) −01 : 35 : 32.590 (5.37) d:m:s Perryman et al. (1997)

−124.1 ±0.9 mas yr

−1

Martioli et al. (2010)

23.5 ±0.7 mas yr

−1

Martioli et al. (2010)

abs

19.1 ±0.8 mas Martioli et al. (2010)

Γ −3.6 ±0.1 km s

−1

Perryman et al. (1997)

Spc type F9V - Martioli et al. (2010)

Age 4.8

+0.7

−1.9

Gyr Saﬀe et al. (2005)*

[Fe/H] −0.010 ±0.053 dex Gonzalez e Laws (2007)

[C/H] 0.049 ±0.081 dex Gonzalez e Laws (2007)

[O/H] 0.112 ±0.045 dex Gonzalez e Laws (2007)

[Si/H] −0.042 ±0.058 dex Gonzalez e Laws (2007)

[Ca/H] −0.057 ±0.062 dex Gonzalez e Laws (2007)

d 52.3 ±0.6 pc Martioli et al. (2010)

v sin i 7.33 ±0.5 m s

−1

Butler et al. (2006)

rot

12.2 day Fischer et al. (2002)

eff

6097 ±44 K Butler et al. (2006)

log g 4.16 ±0.09 cm s

−2

Prieto e Lambert (1999)

∗

1.24 ±0.07 M



Fischer et al. (2002)

∗

1.58 ±0.11 R



Prieto e Lambert (1999)

BC 0.01 ±0.03 mag Prieto e Lambert (1999)

3.34 mag Zacharias et al. (2005)

B 7.432 mag Zacharias et al. (2005)

V 6.945 mag Zacharias et al. (2005)

R 6.630 mag Zacharias et al. (2005)

J 5.934 mag Zacharias et al. (2005)

H 5.693 mag Zacharias et al. (2005)

K 5.599 mag Zacharias et al. (2005)

*Age value and limits derived from isochrone method

TABLE 2.2 - Properties of HD 33636.

ID HD 33636 Unit Reference

RA(2000) 05 : 11 : 46.449 h:m:s Zacharias et al. (2005)

Dec(2000) +04 : 24 : 12.74 d:m:s Zacharias et al. (2005)

180.28 ±0.35 mas yr

−1

this work

−137.81 ±0.30 mas yr

−1

this work

abs

34.98 ±0.28 mas this work

d 28.59 ±0.23 pc this work

Γ 5.3 ±0.2 km s

−1

Nordstrom et al. (2004)

Spc type G0 V - Bean et al. (2007)

Age 5.0 ± 1.9 Gyr Fischer e Valenti (2005)*

[Fe/H] -0.13 dex Vogt et al. (2002)

[Na/H] -0.22 dex Fischer e Valenti (2005)

[Si/H] -0.09 dex Fischer e Valenti (2005)

v sin i 3.08 m s

−1

Butler et al. (2006)

rot

13 day Vogt et al. (2002)

eff

5904 K Butler et al. (2006)

log g 4.37 ± 0.04 cm s

−2

Prieto e Lambert (1999)

∗

1.02 ±0.03 M



Takeda et al. (2007)

∗

1.26 ±0.03 R



Prieto e Lambert (1999)

BC 0.07 mag Prieto e Lambert (1999)

4.71 ±0.17 mag Prieto e Lambert (1999)

B 7.559 mag Zacharias et al. (2005)

V 6.990 mag Zacharias et al. (2005)

R 6.600 mag Zacharias et al. (2005)

J 5.931 mag Zacharias et al. (2005)

H 5.633 mag Zacharias et al. (2005)

K 5.572 mag Zacharias et al. (2005)

*Age value and limits derived from isochrone method

3 BACKGROUND

This chapter provides the basic background theory concerned in the experiments

and data analysis.

3.1 The Kepler’s Laws

Johannes Kepler deduced empirically three ‘laws’ of planetary orbital motion. Al-

though these were obtained speciﬁcally for the orbits of the planets around the Sun

they can be generalized for any unperturbed orbit involving two bodies. For this

reason it is widely used in the analysis of binary stars and exoplanets. The Kepler’s

laws and their generalization are given below:

1st Law - The planet moves along elliptical paths with the Sun at one focus. We

can express the planet’s distance from the Sun as:

r =

a(1 −e

)

1 + e cos f

, (3.1)

with a the semimajor axis, e the eccentricity and f the true anomaly. This

expression is only true if the central body is much more massive than

the planet. For the general case of two orbiting bodies with comparable

masses (hereafter binary), both bodies move along elliptical paths with the

barycenter at one focus. Nevertheless, Equation 3.1 still holds for the dis-

tance of each body to the barycenter. Note that each body has a particular

semimajor axis, which can be computed through the following equation:

= a

, (3.2)

where a

and a

are the semimajor axes, and M

and M

are the masses.

By convention the true anomaly f is the angle measured from the peri-

astron passage counterclockwise. This makes f from each component be

delayed by 180

◦

2nd Law - A line connecting the planet and the Sun sweeps out an area, A, at

constant rate dA/dt. For binaries, the line connecting the two bodies passes

through the shared focus and both bodies sweep out areas at constant rate.

3rd Law - The square of planet’s orbital period (in years) is equal to the cube of its

semimajor axis (in AU):

= a

. (3.3)

Equation 3.3 only holds for Solar System planets. The general expression

that may be used for binaries must include the masses of each body ex-

plicitly, and is given as follow:



2π



G(M

+ M

)

, (3.4)

where G is the gravitational constant, and a is the semimajor axis of the

relative motion between the two bodies, also given by a = a

+ a

. By

making use of Equation 3.2 we can rewrite Equation 3.4 for the orbit of a

particular body around the barycenter:



2π



+ M

)

, (3.5)

where we have chosen the body A. In order to obtain this expression for

body B one just needs to switch the index numbers.

3.2 Physics of Orbital Motion

The Kepler’s laws were empirically and work well only when the dominant force is

the one between the two bodies. However, the basic theory to explain orbits for the

more general case with multiple interacting bodies and additional external forces is

based on Newton’s Universal Law of Gravitation,

F = G

, (3.6)

where G is the gravitational constant, F is the attractive force between two particles

with masses M

and M

, which are separated by the distance r. The total gravita-

tional force applied to a given particle in a system of N particles may be written as

the resultant of all forces:

= Gm



j=i

. (3.7)

where it has been introduced the vectorial force, F

, and the diﬀerence between the

position vectors, r

≡ r

− r

. The positions are relative to an arbitrary inertial

system of coordinates. It has also been used the deﬁnition r ≡ rˆr.

In order to obtain the equations of motion for a given particle, one should make use

of Newton’s Second Law in Equation 3.7 for all particles, and therefore solve the

following system of diﬀerential equations:











= ...

...

= G



j=i

...

= ...

(3.8)

This expression is valid when no external force is present. Any additional force

should be included as an additional term on the right side of Equation 3.8 divided

by the particle mass m

3.3 The Two-Body Problem

For the case where N = 2, the two-body problem, Equation 3.8 leads to the following

system of equations:











= G

(3.9)

Note that r

= r

and ˆr

= −ˆr

. Subtracting the ﬁrst equation from the second,

and letting the relative position vector be r ≡ r

− r

, we have the following:

r +

r = 0, (3.10)

where it has been deﬁned the gravitational coeﬃcient µ ≡ G(m

+ m

). Equation

3.10 is a second order ordinary diﬀerential equation that has to be solved in order to

obtain the equation of motion for the relative position r(t) of particle 2 with respect

to the particle 1.

The general solution for Equation 3.10 is a conic section, which depending on the

total energy of each body can be either an ellipse, a hyperbola or a parabola. The

particular solution that leads to an ellipse is the same as Equation 3.1, which was

obtained empirically by Kepler.

Since Equation 3.10 is of vectorial type, it is in fact a system of three independent

diﬀerential equations of second order, each of which requires two initial values in

order to provide a unique solution. Therefore r(0) and

r(0) must be provided. The

positions and velocities are not the only set of initial values that are needed. Instead,

one could use a diﬀerent set of six parameters, the orbital elements. Although there

are diﬀerent deﬁnitions of orbital elements, we adopt here the Keplerian elements:

eccentricity (e), semimajor axis (a), inclination (i), longitude of the ascending node

(Ω), argument of periastron (ω) and mean anomaly at epoch (M).

A proper choice of the system of coordinates may provide a handful of relationships

between the orbital elements and the cartesian coordinates. It will be shown that this

is also important when one wants to relate these elements with the observables in

the experiments. Figure 3.3 shows the geometry of an ellipse that has its focus lying

closer to the periastron centered in a cartesian system of coordinates. By setting up

this system, one is led to the deﬁnitions shown in Figure 3.3.

Note that parameters e and a alone already deﬁne the shape of the orbit in the orbital

plane. The mean anomaly M is the parameter that contains the time dependence

and deﬁnes the orbital phase of the object. Although M is mathematically deﬁned

as an angle, it is not a real angle in the ellipse, it is rather the angle that varies

linearly with time and is also deﬁned as function of the orbital period (P ), and of

the epoch of periastron passage (T

) through the following expression:

M =

2π

(t −T

). (3.11)

The angle E indicated on Figure 3.3 is the eccentric anomaly, which is related to

FIGURE 3.1 - Ellipse geometry.

the mean anomaly by the following expression:

M = E − e sin E. (3.12)

Equation 3.12 is known as the Kepler equation. The eccentric anomaly is related

geometrically to the true anomaly (f), which is the argument of the position vector

(r) in polar coordinates (see Figure 3.3). Thus the following expression holds:

tan



1 + e

1 −e

tan

. (3.13)

The remaining orbital elements, i, Ω, and ω are the orientation angles. They rep-

resent the Euler angles to describe the orientation of an orbit in 3-dimensional

Euclidean space. Therefore, any Keplerian orbit can be represented by its natural

shape rotated by the orientation angles in space.

3.4 N-Body Problem

The N-body problem consists of solving the equations of motion (Equation 3.8) for

a given body that is part of a system of N gravitationally interacting bodies. An

analytic exact solution only exists for trivial problems, like the two-body problem.

Therefore, any attempt to solve a general N-body problem is likely to be an approx-

imation of the reality. In fact, to be more comfortable with non-exact solutions, we

remind the reader that even for a two-body problem the solution presented above

is an approximation, since the more correct theory to calculate orbits is the Gen-

eral Theory of Relativity, developed by Albert Einstein. Given the precision of our

experiments, the classical theory is suitable. However, although the interactions be-

tween bodies in a given observed system may be small enough to perturb the orbits

at a level that is not detected in the experiments, when this small perturbations

are propagated through many years, the outcome solution may diverge from what

is expected. This means that Keplerian orbits may represent a suitable model for a

short-term experiment, but it might be necessary to make use of a full treatment of

the N-body problem for experiments that last more than a couple of orbits. More-

over, performing long-term simulations of a system with more than two components

also requires N-body numerical integrator, and in some cases it is even necessary

to make use of General Relativity (ADAMS; LAUGHLIN, 2006). In our analysis we

have used the symplectic numerical integrator MERCURY which is fully described

in Chambers (1999).

3.5 Experiment Models

3.5.1 Astrometry: Proper Motion and Parallax

In the analysis of astrometry data there are two main apparent intrinsic movements

for stars that should be taken into account. First the relative displacement of stars

due to their galactic motions, also called the proper motion, µ, which is usually

written in terms of its equatorial components, µ

and µ

. These are angular velocities

measured in units of milliseconds of arc per year (mas yr

−1

). Stars appear to move

slowly due to their large distances, therefore, for most of the cases, the displacement

in each direction can be approximated by an uniform linear motion:

∆ξ = µ

∆t, (3.14)

∆η = µ

∆t, (3.15)

where ξ and η are the standard coordinates.

Besides the intrinsic movement of each star, there is also an apparent movement

resulting from the orbital motion of the Earth around the Sun, the parallactic dis-

placement. This presents an annual parallactic motion with shape deﬁned by the

position of the target in the sky and size deﬁned by the intrinsic distance of the star

to the Sun. The parallactic orbit displacement is given by the following expressions:

∆ξ = P

abs

, (3.16)

∆η = P

abs

, (3.17)

where π

abs

is the amplitude of the parallactic ellipse, also called the absolute parallax.

This relates to the physical distance to the object, d = 1/π

abs

, where d is expressed

in parsec and π

abs

in seconds of arc. The indices α and δ represent the equatorial

coordinates, Right Ascension (RA) and Declination (DEC), respectively. The terms

and P

are the parallax factors, deﬁned in Kamp (1967),

= R

⊕

(cos  cos α sin  −sin α cos ), (3.18)

= R

⊕

[(sin  cos δ − cos  sin α sin δ) sin  − cos α sin δ cos ], (3.19)

where R

⊕

is the Earth distance to the Sun expressed in astronomical units,

  23

◦

.4392794, is the obliquity of the ecliptic, and  is the longitude of the

Sun in the ecliptic system, measured from the vernal equinox, in the direction of

increasing right ascensions. All of these quantities vary on time. The most accu-

rate data to calculate these quantities can be found in the JPL Planetary and Lu-

nar Ephemerides (STANDISH JR., 1998), which can be accessed through the address

http://ftpssd.jpl.nasa.gov.

The proper motion and parallax can be modeled together and removed from the

astrometric data using the following expressions:

ξ = ξ

− P

abs

− µ

∆t, (3.20)

η = η

− P

abs

− µ

∆t, (3.21)

where ξ

and η

are the measured standard coordinates for a given star in a given

epoch, and ξ and η are the reduced standard coordinates. The latter provide the

star catalogue, constituted by star coordinates for a reference frame, which is chosen

arbitrarily. These coordinates must match for all observations, and for this reason

Equations 3.20 and 3.21 are called equations of condition.

3.5.2 Astrometry: Apparent Orbital Motion

The high precision of FGS allows us to measure the apparent reﬂex orbital mo-

tion of the star due to the presence of a low mass unseen companion. This orbital

displacement may also be removed from the equations of conditions similarly to

the parallactic orbit. The model for the apparent orbital displacements of the star

around the barycenter between the star and a given low mass unseen companion is

given below.

First we write the elliptical rectangular coordinates, x and y, in the unit orbit:

x = (cos E − e), (3.22)

y =

√

1 −e

sin E, (3.23)

where e is the eccentricity and E is the eccentric anomaly. Remember that E car-

ries the dependence on time through the Kepler’s equation (Equation 3.12). The

projection of this true orbit onto the plane tangent to the sky gives the coordinates

∆x,∆y. This projection can be expressed mathematically by:

∆x = Bx + Gy, (3.24)

∆y = Ax + F y, (3.25)

where B, A, G, F are the Thiele-Innes constants, given by:

B = a(cos ω sin Ω + sin ω cos Ω cos i), (3.26)

A = a(cos ω cos Ω − sin ω sin Ω cos i), (3.27)

G = a(−sin ω sin Ω + cos ω cos Ω cos i), (3.28)

F = a(−sin ω cos Ω − cos ω sin Ω cos i), (3.29)

where a is the semimajor axis, Ω is the longitude of the ascending node, i is the

inclination of the orbit plane to the plane tangent to the sky and ω is the argument

of periastron. It should be noted that this can be the orbit coordinates for either the

parent star or its companion around the barycenter, depending on which semi-major

axis is taken; the ω in the respective orbits diﬀer by 180

◦

. We measure the star orbit,

so the coordinates of interest are ∆x

, ∆y

, obtained by taking a = a

A more practical expression for the apparent orbit displacements is given by

∆ξ = αQ

, (3.30)

∆η = αQ

, (3.31)

where α is the semimajor axis expressed in milliseconds of arc (mas), and Q

and

are the orbit factors, which are analogous to the parallax factors (KAMP, 1967).

These can be calculated through the following:

= B



x + G



y, (3.32)

= A



x + F



y, (3.33)

where the factors indicated with the prime symbol are related to the Thiele-Innes

constants as follows:



= −B/a, G



= −G/a, (3.34)



= −A/a, F



= −F/a. (3.35)

The semimajor axis in angular measure and in linear measure are related through

the following expression:

a(AU) =

α(mas)

abs

(mas)

. (3.36)

If the observed star is a binary system, then the astrometric model of Equations 3.20

and 3.21 may incorporate Equations 3.30 and 3.31, providing the following model:

ξ = ξ

− π

abs

− µ

∆t −αQ

, (3.37)

η = η

− π

abs

− µ

∆t −αQ

. (3.38)

3.5.3 Spectroscopy: Doppler Shift

In the spectroscopy experiment the quantity obtained from the measurements is the

Radial Velocity (RV), which is the velocity of the star in the direction of the line

of sight. The actual quantity measured is the Doppler shift of the spectra, which is

related to the RV by the following expression:

v = c

∆λ

, (3.39)

with c the speed of light, ∆λ the shift in wavelength and λ the central wavelength

at which the shift has been observed.

3.5.4 Spectroscopy: Radial Velocity for Keplerian Orbits

For systems containing only one companion, the radial velocity of the parent star’s

movement around the barycenter of the system is given by the ﬁrst derivative of

the radial component of its position. This is given by the projection of a Keplerian

orbital velocity to observer’s line of sight plus a constant systemic velocity Γ,

v = Γ + K[cos (f + ω) + e cos ω], (3.40)

where ω is the argument of periastron, e is the eccentricity, K is the velocity semi-

amplitude and f is the true anomaly.

The velocity semi-amplitude can also be written in terms of orbital elements,

K =

2π

sin i

√

1 −e

, (3.41)

where i is the orbital inclination and a

is the semi-major axis of the orbit of the star.

Using the proportionality between the masses and semi-major axes, a

= a

and Kepler’s third law, we can rewrite Eq 3.41 as

sin i

+ M

)

2/3



2πG



1/3

√

1 −e

. (3.42)

Note that we have introduced the indices s and b to distinguish between stellar and

companion’s parameters. Eq 3.42 provides a way to calculate the projected minimum

mass M

sin i of the companion with the assumption that there is a measurement of

the stellar mass by other means (e.g. stellar atmospheric models). It is to be noted

that this is a lower limit to the mass with the uncertainties mostly dominated by

the determination of the stellar mass.

3.5.5 Spectroscopy: Multi-planet Keplerian RV Model

A system containing more than one companion results in a more complex orbital

movement of the parent star around the barycenter. Strictly, this should be modeled

using a full consideration of the N-body interactions in the system. However, for

weak interacting companions, this orbital motion can be approximated by a linear

sum of the Keplerian contribution from each companion j, producing a total RV

perturbation given by

= Γ +



j=1

[cos (f

+ ω

) + e

cos ω

], (3.43)

where N is the number of companions in the system and the remaining parameters

follow the same notation used in Equation 3.40.

3.5.6 Interferometry: Basics

In this section we discuss basic concepts to understand the principles behind an in-

terferometric experiment. This discussion is based on Haniﬀ (2007a), Haniﬀ (2007b).

In observations of astronomical sources we are usually scarce of photons, hence we

need to collect light using reﬂector or refractive lenses. The light is gathered and

focused in order to obtain images. If one examines the resulting image of a point-like

source, as seen by an unaberrated telescope with circular pupil, it is observed that

light does not simply sum up, instead it presents a pattern of concentric rings, the

Airy disk. This phenomenon occurs due to the interference between coherent electro-

magnetic waves (light) coming from the diﬀerent points of the mirror/lens surface.

This pattern can be well described by the so-called spatial coherence function.

Now if one uses an optical interferometer, which is composed of two (or more) optical

telescopes that are far apart to collect light from the same source, and combine

these two beams together, the result is a wave-like pattern, similar to that formed

by a single dish optics. The diﬀerence here is that Airy disks are formed due to the

circular shape of telescope mirrors, and the shape of an interferometer image pattern

is determined by the slit. This technique known as interferometry is not in principle

diﬀerent than regular imaging, but on what concerns the observing techniques, data

reduction and analysis it turns out to be very diﬀerent.

The fundamental concept of interferometry for astronomy lies in answering the fol-

lowing question: what is the interferometric response (visibility) for a given source

morphology? The source morphology is usually expressed by the source brightness

distribution function, I(α, β), on the sky. The link between the brightness distribu-

tion and the response of an interferometer is the Cittert-Zernike theorem (HANIFF,

2007a), which states that for sources in the far ﬁeld the normalized value of the

spatial coherence or “visibility” function is equal to the Fourier transform of the

normalized sky brightness distribution. This can be expressed mathematically as

follows:

r,norm

(u, v) =

 

I(α, β)e

−2πi(uα+vβ)

dαdβ

 

I(α, β)dαdβ

, (3.44)

where α and β are angular coordinates on the sky, u and v are spatial frequencies, or

the coordinates in the reciprocal plane. Notice that spatial frequency (u, v) coordi-

nates can be condensed into a vector,



f = uˆu + vˆv, where by convention component

ˆu points East and ˆv points North. For each spatial frequency there is a correspond-

ing baseline vector in the pupil,



B, those of which can be related in terms of the

wavelength, λ, as follows



B =

0.648



f, (3.45)

where the spatial frequency is in arcsec

−1

, wavelength in µm and baseline in meters.

Equation 3.44 provides a way to obtain the coherence function (or visibility) from

the source brightness distribution. The reciprocal is also true, i.e. one can measure

the coherence function in order to infer the brightness distribution, which is obtained

by the inverse Fourier transform of Equation 3.44,

norm

(α, β) =



∞

−∞



∞

−∞

V (u, v)e

2πi(αu+βu)

dudv. (3.46)

where V (u, v) is the complex visibility function, which is normalized, thus V (u, v) =

ν(u, v)/ν(0, 0), for ν(u, v) and ν(0, 0) being complex visibility functions correspond-

ing to I(α, β) and I(0, 0), respectively. In the next section we explain how to measure

coherence functions in practice.

3.5.7 Interferometry: Measuring Visibility Functions

The way for measuring complex visibility functions in practice depends on which

type of radiation is under study. For example, in the radio domain it is possible

to measure directly the amplitudes and phases of the complex visibility function.

However, for the optical domain (including infrared), the coherence time is extremely

short, therefore it is still impossible to access these quantities directly. Instead the

quantity to be measured is the squared visibility, which allows us to obtain the

amplitude and phases indirectly. To grasp the idea of measuring visibilities in optical

interferometry we refer to the Young’s double slit experiment, where the intensity

of a given point on the image plane is given by the modulus squared summation of

the electric ﬁelds, E

and E

, arriving from the two slits. This intensity is given by:

I = |E

 + |E

 + 2|E

|cos φ, (3.47)

where φ is the phase diﬀerence between the electric ﬁeld components E

and E

, and

the angle brackets refers to the time average. The ﬁrst two terms of Equation 3.47

are directly proportional to the square of visibility and can be directly measured

from the intensity of the fringe pattern. The last term carries the oscillatory nature

of the fringes. Therefore one can access phases of complex visibility functions from

the light-dark modulation of fringe patterns. Notice that these phases are measured

with respect to an oﬀset of a whole fringe period.

The amplitude of visibility function can be obtained from the fringe contrast, also

known as “Michelson Visibility”, which is given by:

V =

max

− I

min

max

+ I

min

. (3.48)

Therefore it is feasible to measure both amplitude and phases of the visibility (or

coherence) functions with square-law detectors, like optical interferometers.

3.5.8 Interferometry: Visibility Model

From Equation 3.44 one can obtain the visibility from a brightness distribution of an

observed source. The latter may be rather complex, thus resulting in a complex form

for the visibility function. In practice interferometry presents a limitation where the

(u, v) plane is not well sampled. The current optical interferometers have only a few

elements, namely 2 to 3 telescopes. This implies in a very poor coverage of the (u, v)

plane, therefore making it diﬃcult to recover the true image of the source. There

are techniques to increase the (u, v) coverage, such as by observing the same target

at diﬀerent sky positions (as the Earth rotates), or by observing within a range of

wavelengths, but even though the coverage remains poor. Given this limitation, one

usually needs to rely on very simple models that may provide conclusive results.

Below we present a description of these models based on Berger e Segransan (2007).

The simplest model one can think of is a point-like source, for which the Fourier

pair image/visibility model is given by:

I(α, β) = δ(α − α

, β − β

), (3.49)

V (u, v) = e

−2πi(uα

+vβ

)

. (3.50)

This is an important model to use in the calibration of an interferometer. The re-

sponse for an unresolved source, which is known to be punctual, is given by Equation

3.50. The model of a more complex brightness distribution may also be represented

by a composition of point-like sources.

The other important models to consider here are the uniform and gaussian disk.

These become useful when a given source is resolved by the interferometer. It is

common for calibrators since they are usually required to be bright sources in the

visible, thus these are likely to be nearby stars. The gaussian disk model is given by:

I(α, β) =



4 ln 2

−

4 ln 2ρ

, (3.51)

V (u, v) = e

−

(πΘ

√

)

4 ln 2

, (3.52)

where Θ is the FWHM, and ρ =

√

+ β

. The uniform disk model is given by:

I(α, β) =











πΘ

if ρ ≤ Θ/2,

0 if ρ > Θ/2,

(3.53)

V (u, v) = 2

(πΘr)

πΘr

, (3.54)

where Θ is deﬁned here as the disk diameter, r =

√

+ v

, and J

is the ﬁrst order

Bessel function.

3.5.9 Interferometry: Binary Model

Figure 3.5.9 shows a simple representation of the brightness distribution of a binary,

like an exoplanet and its parent star. It also shows the astrometric binary parameters

that we want to include in the visibility model.

star

companion

FIGURE 3.2 - Simple representation of the binary brightness distribution and the binary

parameters.

The primary diﬀerence between the binary model and those shown in the last section

is that the brightness distribution is no longer symmetric, what makes the visibility

to be a complex function. This has another important consequence, the phase of

the complex visibility function is no longer constant (zero or π). This allows us to

obtain information about the source from models for either the phases φ or, if three

or more baselines are used, the closure phase Φ. We discuss these models below.

The visibility model for a binary can be described as a composition of two sources,

each represented by either a point-like or a disk model, as shown in the latter section.

This choice depends on the characteristics of the system in study.

The brightness distribution for a binary with two point-like sources is given by:

I(α, β) = F

δ(α − α

, β − β

) + F

δ(α − α

, β − β

), (3.55)

where (α

, β

) are the coordinates of component ‘j’, and F

is the ﬂux. In this case

the visibility amplitude for each source is constant, but the corresponding visibility

amplitude of the pair is not constant. This is given by the Fourier transform of

Equation 3.55, which provides the unnormalized complex visibility,

ν(u, v) = F

2πi(uα

+vβ

)

+ F

2πi(uα

+vβ

])

. (3.56)

The normalized squared visibility is then given by

|V (u, v)|

ν(u, v)ν(u, v)

∗

|ν(0, 0)|

, (3.57)

which can be expanded to deliver the expression for the normalized squared visibility

amplitude of a resolved binary pair with both unresolved components,

|V (u, v)|

1 + f

+ 2f cos [(2π/λ)



B.ρ]

(1 + f)

. (3.58)

where f = F

/F 1 is the ﬂux ratio, for the brightest component ‘1’ at the center, ρ is

the position vector of component ‘2’ with respect to component ‘1’ (primary). The

projection of the baseline onto the position vector is given by



B.ρ = Bρ cos (θ − θ

This introduces the astrometric binary parameters shown in Figure 3.5.9 into the

visibility equation.

Equation 3.55 may not reﬂect the true brightness of the pair if any of the components

are resolved. The more general expression for the complex visibility of a binary pair

is given by:

V (u, v) =

(u, v)e

2πi(uα

+vβ

)

+ F

(u, v)e

2πi(uα

+vβ

)

+ F

, (3.59)

where V

(u, v) is the normalized visibility for each individual component ‘j’. One

may introduce any visibility function for the components, such as the disk model.

For a system containing a faint companion (e.g. exoplanet or brown dwarf) around

a star, for which the star visibility is V

∗

, the companion visibility is V

, and the ﬂux

ratio is f = F

∗

, the squared visibility is given by:

|V (u, v)|

∗

+ f

+ 2f|V

∗

||V

|cos [(2π/λ)



B.ρ]

1 + f

. (3.60)

From Equation 3.60 we notice that if f is small, which is the case for a faint compan-

ion, the visibility reduces to the star’s visibility. The precision required to measure

such small ﬂux ratios is what limits the current interferometers in detecting very

faint companions. However, phases may present a detectable signal. The phase φ of

a complex visibility function V (u, v) is deﬁned as follows

tan φ ≡

ImV (u, v)

ReV (u, v)

. (3.61)

Therefore, one can write the phase model for a binary using Equation 3.59, which

provides:

tan φ =

sin [(2π/λ)



B.ρ]

∗

+ fV

cos [(2π/λ)



B.ρ]

. (3.62)

For small ﬂux ratios and except around the nulls one can approximate the above

equation by:

φ =

sin [(2π/λ)



B.ρ]

∗

+ nπ, (3.63)

where n = 0 if V

∗

> 0, and n = 1 if V

∗

< 0. We notice from Equation 3.63 that when

the ﬂux ratio is small, the phase is either equal to 0 or π, depending on the sign

of the visibility of the star. However, at spatial frequencies where visibility becomes

close to the ﬂux ratio V

∗

∼ f, the phase assumes very diﬀerent values. This happens

at interferometric nulls of the star visibility function. If the baseline and wavelength

are chosen adequately, then one could probe these regions and obtain information

from the companion, even for very low values of ﬂux ratio. The closure phase (Φ) for

a three elements interferometric array is deﬁned as the sum of phases between each

pair of baselines. Therefore measuring Φ is equivalent to measuring three phases at

the same time. The amplitude of the signal in this case depends on a combination

of the three baseline orientations.

4 EXPERIMENTS DESIGN

In this chapter we describe the experimental set up and observing strategies for each

of the following three experiments. 1. Doppler spectroscopy to obtain the Radial

Velocities (RV). 2. Astrometry. 3. Infrared interferometry.

4.1 Optical Spectroscopy

4.1.1 The Experiment Overview

Spectroscopic measurements were carried out with the High Resolution Spectrograph

(HRS) on the Hobby-Eberly Telescope (HET), McDonald Observatory, Texas, USA.

Each target spectrum provides a measurement of the velocity of the star in the radial

direction, which is obtained from the Doppler shift of the whole stellar spectrum with

respect to some zero reference spectrum generated by a source situated at an inertial

frame. We have used ThAr lamp emission spectrum produced in the laboratory as

a ﬁrst wavelength calibration reference. In order to improve the precision we have

also used the iodine absorption cell method Butler et al. (1996). In this method a

low-pressure, temperature controlled cell with I

gas, is positioned in the light path.

The absorption of light by the I

gas produces a set of well known spectral features

imprinted at the same time as the stellar spectrum. However they are produced with

no wavelength shift with respect to the observatory frame. This provides a much

better reference for wavelength calibration and also permit us to characterize the

instrumental proﬁle with great accuracy.

4.1.2 Goals

This experiment is aimed to provide a time-series of the HD 136118 radial velocity

with higher cadence and better precision ( 3 m/s) than previous work (FISCHER

et al., 2002). This will enlarge the time coverage and improve the determination

of the companion’s orbit parameters. Also, it will permit us to look for additional

companions in the system. Apart from the results for HD 136118 we also aim to

improve the reduction and analysis procedures, all of which are detailed in Chapters

5 and 6.

4.1.3 Instruments

The following two sections provide some technical details from the instruments used

in our experiments. All of this information has been obtained in the McDonald

Observatory webpage (http://www.as.utexas.edu/mcdonald).

4.1.3.1 HET

The Hobby-Eberly Telescope (HET) has been designed for spectroscopic survey

work. It is a segmented mirror telescope with 91 Zerodur panels that provide a

collecting area of 77.6 m

and a ﬁlling factor of 98% (spherical), thus an eﬀective

aperture of 9 m. Focal length is 13.1 m. Its mounting has a ﬁxed elevation axis at

◦

zenith distance, prime focus tracker will follow objects when passing through

the beam up to 1.5 hours. Azimuth is changeable by 360

◦

. HET site is at Ft. Davis

(Latitude: 30

◦



N. Longitude: 104

◦



W. Elevation: 2240 m), therefore all objects

between declinations −10

◦

and 71

◦

(70% of the sky) can be observed.

4.1.3.2 HRS

The HRS is an optical echelle spectrograph, which is detailed in Tull (1998). Figure

4.1 shows the HRS optical layout assembly. This instrument uses an R-4 (blaze angle

of 75

◦

) echelle mosaic with cross-dispersing gratings to separate spectral orders. An

all-refracting camera images onto a mosaic of two thinned and anti-reﬂection coated

2K x 4K Marconi CCDs with 15 µm pixels. The CCDs are abutted along their 4K

side with a ∼ 72 pixel dead space between them. This dead space is approximately

parallel to the spectral orders. The HRS is a “white pupil” spectrograph using the

2-mirror collimator system pioneered by Hans Dekker and Bernard Delabre at ESO:

Mirror M1, the main collimator, is an oﬀ-axis paraboloid used in auto-collimation,

with the entrance slit at its focus. After the dispersed light is reﬂected from M1 the

beam comes to an intermediate focus, oﬀset from the slit by an amount controlled

by the oﬀ-plane tilt of the echelle, which is 0.8

◦

. Mirror M2 has identical ﬁgure and

focal length but is displaced oﬀ-axis by a distance equal to the separation of slit and

intermediate focus. Resolving powers of R ∼ 15,000, 30,000, 60,000, and 120,000 are

available by means of four eﬀective slit widths. Spectral coverage is 420 - 1100 nm.

The parameters of the system are summarized in Figure 4.2, the HRS design pa-

rameters in Figure 4.3 and the HRS performance parameters in Figure 4.4.

FIGURE 4.1 - HRS optical layout. Source: http://www.as.utexas.edu/mcdonald.

FIGURE 4.2 - HRS System Parameters. Source: http://www.as.utexas.edu/mcdonald.

4.1.4 Strategy and Observations

Our spectroscopic observations of HD 136118 include a total of 168 high resolution

spectra, which were obtained between UT dates 2005 December 4 and 2008 May

20. Multiple observations were taken most nights, so the velocities obtained on the

FIGURE 4.3 - HRS design parameters. Source: http://www.as.utexas.edu/mcdonald.

FIGURE 4.4 - HRS performance parameters. Source: http://www.as.utexas.edu/mcdonald.

same night may be combined, producing individual measurements of the stellar

Radial Velocity (RV) at 61 diﬀerent epochs.

The spectrograph was used in the R = 60, 000 mode with a 316 line/mm echelle

grating. The position of the cross dispersion grating was chosen so that the central

wavelength of the order that fell in the break between the two CCD chips was 5936 Å.

The temperature controlled cell containing low pressure iodine (I

) gas was placed

in front of the spectrograph slit entrance during all the exposures. The exposure

times were nominally 120 s, but were increased on a few nights due to bad seeing

conditions. In addition to the program spectra we have also obtained HD 136118

template spectra. For these we removed the I

cell, the resolution was set to R =

120, 000, and the exposure times were 230 s.

Our RV data combined with previously published velocities from Lick Observatory

(FISCHER et al., 2002) produces a total data set that spans 10.3 yr. This corresponds

to about 3 times the orbital period of the known companion, P ∼ 1190 d.

4.2 Astrometry

4.2.1 The Experiment Overview

Astrometric measurements were obtained with the Fine Guidance Sensor 1r (FGS-

1r) instrument aboard the Hubble Space Telescope (HST). As an interferometer

working with a 2.4 m telescope in space, the FGS has a potential capability for

measuring relative stellar positions with sub-millisecond of arc precision, with a

dynamical range up to 12 magnitudes. The primary goal of this experiment is to

obtain FGS-1r relative astrometry of stars with known radial velocity variations due

to unseen low-mass companions. The FGS has the capability of measuring the reﬂex

motion due to companions with planetary and brown dwarf masses at relatively

wide orbits, with perturbation orbit size of order of α  0.5 mas. Below we give an

overview of the instrumental set up and strategies.

4.2.2 Instruments

4.2.2.1 HST

The HST is an optical space telescope that was deployed in low-Earth orbit by the

crew of the space shuttle Discovery (STS-31) on 25 April 1990. Since then it has

passed through four servicing missions for repairing and upgrading. These eﬀorts

are hoped to keep the telescope fully functioning at least until 2014 and perhaps

longer.

The HST is a 2.4 m reﬂecting aplanatic Cassegrain telescope of Ritchey-Chrètien de-

sign. The collecting area is 4.5 m

, and the focal length is 57.6 m. It weighs 11,110 kg,

and is currently in a near-circular low-Earth orbit, at height 559 km, orbital period

of 96 - 97 minutes, orbit velocity of 7,500 m/s, and acceleration due to gravity of

8.169 m/s

It is equipped with six science instruments, which operate in ultraviolet, optical and

infrared wavelengths. The instruments are the following:

NICMOS - infrared camera/spectrometer

ACS - optical survey camera (partially failed)

WFC3 - wide ﬁeld optical camera

COS - ultraviolet spectrograph

STIS - optical spectrometer/camera

FGS - three ﬁne guidance sensors

We are particularly interested in the FGSs, which we describe in more detail in the

following section.

4.2.2.2 FGS

The FGS is a two-axis white-light optical interferometer aboard HST. A schematic

view of its optical train is shown on Figure 4.5. This instrument has been designed

to work as a guider for the Space Telescope, requiring a remarkable astrometric

precision, which makes it suitable for scientiﬁc research as well.

The whole instrument has three FGSs operating at the same time, of which two

(FGS2 and FGS3) are used for guiding. FGS1r is currently being used for science

(before SM-4). A detailed description about the FGSs can be found in Nelan et al.

(2010). Figure 4.6 shows a scheme of the ﬁeld-of-view (FOV) of the three FGSs on

the focal plane of the telescope.

Figure 4.7 shows the Koester prism, the device responsible for the interference of

light collected by the telescope. The light coming outward from each side of the

prism is focused by a lens into photomultiplier tubes (PMT-A and PMT-B). Their

measured photon counts (A and B) can be combined by the following expression

S = (A −B)/(A + B), which gives the response of the interferometer. The read po-

sition can be changed by performing slight rotations on a secondary mirror, which

is read by an encoder. Then the angles are translated to the telescope focal plane

(pickle) coordinates (X or Y ). This gives the position in one direction. The light

beam is split into two identical prisms rotated by 90

◦

, providing positions in two

orthogonal directions. Astrometric measurements are made by scanning S over a

range of positions. For example, the response S(X or Y ) for a point-like source is

given by the curve shown in Figure 4.6. This is the FGS interferometric fringe. Ex-

tended sources like resolved stellar disks or binary stars show diﬀerent patterns for

the S-curve. These can be modeled giving direct measurements of the light distribu-

tion of these objects. In our work we are particularly interested in highly accurate

measurements of relative stellar positions in a ﬁeld which only requires the measure-

FIGURE 4.5 - FGS1r optical train schematic. Source: Nelan et al. (2010).

ment of the photo-center of each star (POSITION mode) and not the morphology

of the entire S-curve (TRANSFER mode).

4.2.3 Strategy and Observations

HD136118 data were obtained with the FGS in fringe-tracking POSITION mode.

A neutral density ﬁlter (F5ND) was applied when observing HD 136118 due to its

brightness. For the reference stars we used the F583W ﬁlter. All observations were

secured under 2-gyro guiding, an operational mode dictated by gyro failures on HST.

This mode results in major constraints on HST roll angle and observation dates.

This restriction no longer applies since in the last servicing mission (SM-4) all the

six gyros have been replaced by brand new ones.

FIGURE 4.6 - Great circle: FGS ﬁeld-of-view on the HST focal plane (projected onto the

sky). Right panel: FGS-1r S-curve response for a point-like source. Source:

Nelan et al. (2010).

The proposal was to obtain a group of observations (a group is six data sets = six

orbits) at two epochs in Cycle 14 and one epoch in Cycle 15 to space the observations

over a time span of about a year. Each group would be secured within two weeks.

The HD 136118 ﬁeld has a suﬃcient number of guide stars, and these guide stars

are bright (V > 14.5). Thus, this target is relatively bullet-proof as regards two-

gyro operations. Two-gyro operations had one scheduling consequence. Rather than

acquire all three epochs within one Cycle, the two-gyro visibility windows required

that we waited until a year after the ﬁrst epoch to secure that last data sets (i.e., wait

for the window of opportunity to roll around once again). Obviously the three-gyro

mode would be preferable, because the parallactic ellipse would be better sampled.

But, because we had an HIPPARCOS parallax in hand to use as Bayesian prior,

two-gyro pointing control was adequate.

FIGURE 4.7 - Constructive and destructive interference in the Koester prism. Source:

Nelan et al. (2010).

Table 4.1 shows the dates of observation, the number of measurements, N

obs

, and

the HST Roll angle for each visit to the HD 136118 ﬁeld. Our data sets span 1.8 yr,

covering about 55% of the HD 136118 b’s orbital period.

TABLE 4.1 - Log of astrometric observations.

Epoch Date N

obs

HST Roll

1 2005/Jun/15 4 58.00

2 2005/Jun/16 4 58.00

3 2005/Jun/17 4 58.00

4 2005/Jun/18 4 58.00

5 2005/Jun/19 4 58.00

6 2005/Jun/24 4 59.10

7 2006/Mar/02 4 261.00

8 2006/Mar/10 4 264.17

9 2006/Mar/15 4 266.00

10 2006/Mar/22 4 269.15

11 2006/Apr/03 4 274.00

12 2006/Apr/07 4 280.41

13 2007/Mar/03 4 261.00

14 2007/Mar/09 4 263.67

15 2007/Mar/15 4 266.00

16 2007/Mar/24 4 269.74

17 2007/Apr/01 4 274.00

18 2007/Apr/08 4 280.70

4.3 Infrared Interferometry

4.3.1 The Experiment Overview

This experiment is meant to be a preliminary study of what we consider a promis-

ing technique to obtain information of high-contrast binaries, like brown dwarfs and

exoplanet companions. Infrared interferometry in this context is still under develop-

ment. For this reason, as it will be shown, it still presents a lot of open issues to be

explored in future work.

Our interferometric measurements were carried out with the Astronomical Multi-

BEam combineR (AMBER), instrument fed by the three 8.2 m Unit Telescopes

(UTs) at the Very Large Telescope Interferometer (VLTI), Chile. In this experi-

ment we obtained the measurements of infrared low-resolution spectrally dispersed

interferometric fringes of HD 33636, which is a solar type star with an RV-detected

exoplanet candidate companion (see Chapter 2). HD 33636 has been observed in

the same fashion as HD 136118, as part of the FGS/HST and HRS/HET observ-

ing programs. We have selected HD 33636 for the interferometric experience due

to its larger companion mass, thus resulting in a lower contrast, easier to detect

interferometrically.

4.3.2 Goals

The primary goal of this experiment is to obtain spectrally dispersed squared visi-

bilities and interferometric phases for HD 33636 in a single quick observing run. The

more speciﬁc objectives are outlined below.

• Conﬁrm the mass and the orbital parameters of the lower mass companion

to HD 33636.

• Investigate the visibility and diﬀerential phase signal in order to look for

any spectral signatures of the companion.

• Demonstrate and evaluate the use of AMBER+FINITO+UTs to better

characterize RV-detected low-mass companions.

• Validate our FGS astrometry results.

Some more general projected objectives of this experiment are:

• Once the use of AMBER+FINITO+UTs/ATs is established to obtain/es-

timate true mass, this method can be used to eﬀectively ﬁlter out "true" ex-

oplanet candidates for future astrometric programs like PRIMA and SIM.

• Some of these exoplanet companions may turn out to be brown dwarfs

(true mass between 13M

Jup

and 72M

Jup

). This will help us in a better

understanding of the brown dwarf desert (if real).

4.3.3 Instruments

Below we provide a brief description of the instrumentation used in this experiment.

4.3.3.1 VLTI

The VLTI is one of the largest optical facilities for astronomy. Located on Cerro

Paranal, Chile, it is a complex of four 8.2 m ﬁxed Unit Telescopes (UTs) and another

four movable 1.8 m Auxiliary Telescopes (ATs), all of which can be used in single

mode observations or can be combined into two or three elements of an optical

interferometer array. The six baselines spanned by the UTs ranges from 47 m to

130 m, while for the movable ATs it spans from 8 m to 202 m. The interferometric

science instruments work primarily in the near- and mid-infrared wavelength. The

four UTs are all equipped with the adaptive optics system MACAO (Arsenault et

al. 2004).

4.3.3.2 AMBER

AMBER is a near-infrared, interferometric instrument, used to combine simultane-

ously the beam coming from up to 3 telescopes. A documentation of this instrument

is found in the regularly updated user manuals published in the ESO/AMBER home-

page, http://www.eso.org/sci/facilities/paranal/instruments/amber/doc,

latest version by Merand et al. (2010). A much more detailed description of AMBER

instrumental set up, data products and data reduction can be found in Petrov et al.

(2007), Robbe-Dubois et al. (2007), and Tatulli et al. (2007).

To understand the basic idea of AMBER we present Figure 4.8, which shows a

sketch of the AMBER instrument showing how the beam-combination is performed.

Figure 4.9 presents an AMBER sample reconstituted image, using three telescopes,

of a standard calibrator. Both Figures 4.8 and 4.9 have been obtained from Tatulli

et al. (2007).

The entire light path inside AMBER is rather complex, nevertheless we summarize

this process as follows. The beam coming from each telescope is split into the near

infrared J, H and K fractions of the light. This is done with dichroic mirrors. Each

beam is fed separately into a set of J, H and K single-mode optical ﬁbers for the

spatial ﬁltering. The output light of each ﬁber set, corresponding to each telescope,

is collimated and split into two portions, one for the ordinary channels (P1, P2 and

P3) and another for the interferometric channel (IF), see Figures 4.8 and 4.9. The

interferometric portions from all telescopes are focused into a common point, forming

an Airy pattern that contains the interferometric fringes. The interferometrically

combined beams, as well as the ordinary portions of each separate beam, enter a

standard long-slit spectrograph that disperses the light vertically. The spectrograph

can be set into three spectral resolutions: R = 30 (LR), R = 1500 (MR) and

R = 12000 (HR). The wavelength coverage is 1 −2.5 µm. Images are recorded by an

infrared CCD detector. This description is, of course, over-simpliﬁed. We refer the

reader to the literature for further details.

FIGURE 4.8 - Simpliﬁed optical setup of AMBER instrument. Source: Tatulli et al.

(2007).

4.3.3.3 FINITO

The 2008 Science Veriﬁcation (SV) run of AMBER was created in order to validate

the Fringe-tracking Instrument of NIce and Torino (FINITO). FINITO is the ﬁrst

FIGURE 4.9 - AMBER sample image of an exposure of a calibration source. DK is the

dark channel, P1, P2 and P3 are the ordinary chanels from each telescope

and IF indicates the interferometric channel, where the fringe pattern is

recorded. The wavelength dispersion is on the vertical direction. Source:

Tatulli et al. (2007).

generation VLTI fringe sensor. The basic idea behind FINITO is that interferometric

fringes are strongly aﬀected by atmospheric variability. This causes blurring and

loss of fringe information for very short integration times, limiting interferometric

observations for only a few bright targets. Without FINITO any exposure time

longer than the atmospheric coherence time (∼ 10 ms) becomes impractical. The

task of FINITO is to operate as a fringe tracking unit that measures the optical

path diﬀerence variation along time, caused by atmospheric turbulence. This unit

uses interferograms produced by two combined telescope beams. Information is then

delivered to the delay line control loop, which performs slight mechanical corrections

that counteract atmospheric variations. FINITO increases the practical integration

time to a few hundred seconds, improving instrumental sensitivity by as much as

5 magnitudes. A more detailed description of FINITO is found in Corcione et al.

(2003), Gai et al. (2004).

4.3.4 Strategy and Observations

The interferometric observations of HD 33636 were obtained with U1, U2 and U4,

using AMBER+FINITO in the Low Resolution (LR) mode. The observations have

been carried out in UT 2008 Oct 14. The largest baseline used was the U1-U4, with

116 m. The seeing was around 1” at an airmass of ∼ 1.2.

Our observing strategy was as simple as it is required for a typical SV run. We

proposed to use the shortest practical telescope time that would provide any de-

tectable information from the companion to HD 33636, or at least to set a limit

of detection. We requested a total of 3 hours and 30 minutes of observations with

AMBER+FINITO+UTs. According to the latest AMBER User Manual, it would

take about 70 minutes to observe one calibrated visibility point with the UTs. We

proposed 3 calibrated points, then it would take 3 x 70 min = 210 min = 3h30min. If

three measurements were not possible we proposed to have a minimum of two pairs

of Cal/Sci observations. Notice that one calibrated point means a pair of interfer-

ometric observations, comprising one point for the Science (Sci) object bracketed

by two Calibrators (Cal). The actual observations provided us with 5 Sci sequential

points and ﬁve useful sets, each containing 5 Cal sequential points in one night.

Table 4.2 shows the log of interferometric observations.

Since our main goal was to measure absolute visibilities and perhaps absolute closure

phases, we proposed to use the smallest Detector Integration Time (DIT) possible.

In interferometry it is a major concern to have Cal observations as close as possible in

space and time to the Sci observations. This requirement is important to guarantee

that a minimum systematic error from the Optical Path Delay (OPD) is introduced

due to the Earth rotation or to the diﬀerence in the observing direction. We discuss

better this topic in the data analysis section.

TABLE 4.2 - Log of interferometric observations.

Set Object UT Date Category

1 HD 19637 2008-10-14 05:42:14.224 Cal

1 HD 19637 2008-10-14 05:43:45.093 Cal

1 HD 19637 2008-10-14 05:45:15.374 Cal

1 HD 19637 2008-10-14 05:46:45.983 Cal

1 HD 19637 2008-10-14 05:48:16.336 Cal

2 HD 19637 2008-10-14 06:44:55.493 Cal

2 HD 19637 2008-10-14 06:46:26.107 Cal

2 HD 19637 2008-10-14 06:47:57.475 Cal

2 HD 19637 2008-10-14 06:49:27.920 Cal

2 HD 19637 2008-10-14 06:50:59.078 Cal

3 HD 33636 2008-10-14 07:11:28.056 Sci

3 HD 33636 2008-10-14 07:13:47.387 Sci

3 HD 33636 2008-10-14 07:16:06.996 Sci

3 HD 33636 2008-10-14 07:18:26.313 Sci

3 HD 33636 2008-10-14 07:20:46.033 Sci

4 HD 34137 2008-10-14 07:35:31.496 Cal

4 HD 34137 2008-10-14 07:37:53.126 Cal

4 HD 34137 2008-10-14 07:40:13.257 Cal

4 HD 34137 2008-10-14 07:42:33.701 Cal

4 HD 34137 2008-10-14 07:44:54.276 Cal

5 HD 34137 2008-10-14 08:00:32.489 Cal

5 HD 34137 2008-10-14 08:02:02.980 Cal

5 HD 34137 2008-10-14 08:03:33.305 Cal

5 HD 34137 2008-10-14 08:05:04.751 Cal

5 HD 34137 2008-10-14 08:06:35.237 Cal

6 HD 36134 2008-10-14 09:26:40.496 Cal

6 HD 36134 2008-10-14 09:28:10.942 Cal

6 HD 36134 2008-10-14 09:29:41.273 Cal

6 HD 36134 2008-10-14 09:31:11.962 Cal

6 HD 36134 2008-10-14 09:32:42.283 Cal

5 DATA REDUCTION

As in Section 4 we have divided this section in three parts, each containing the

description of the data reduction procedures for each experiment.

5.1 Optical Spectroscopy

5.1.1 Raw Data

The spectroscopic data arrive in standard FITS format. In each FITS ﬁle there are

two images, one for each CCD chip. The longer wavelength part of the spectrum

falls in CCD 1, which we also call the RED CCD. The shorter wavelength part falls

in CCD 2, the BLUE CCD.

Below one can ﬁnd a list of raw image ﬁles for a typical date of observation:

README.hrs hrs620040.ﬁts.Z hrs620041.ﬁts.Z

hrs620042.ﬁts.Z hrs620043.ﬁts.Z hrs620044.ﬁts.Z

hrs620051.ﬁts.Z hrs620053.ﬁts.Z hrs620054.ﬁts.Z

hrs620055.ﬁts.Z hrs620057.ﬁts.Z hrs620059.ﬁts.Z

hrs620085.ﬁts.Z hrs620086.ﬁts.Z hrs620087.ﬁts.Z

hrs620090.ﬁts.Z hrs620091.ﬁts.Z hrs620092.ﬁts.Z

The ﬁles above are spectra of either the science target or the calibration, all of

which are discriminated by their “OBSTYPE” keyword in the header. There can be

basically six diﬀerent types of ﬁles:

• ZERO - spectra taken with the shutter closed. There are normally a few

of these images, so they can be combined in order to estimate the average

zero-level in the CCD.

• FLAT - spectra of an incandescent light source. It is also usual to ﬁnd

many ﬂat exposures for a date. These consist of bright and continuous

spectra, which can be used in many steps of the reduction. For example,

for calibrating the diﬀerences in pixel sensitivity, for ﬁnding the position of

each spectral echelle order, and for measuring the global blaze illumination

function.

• FLAT+I2 - spectrum of an incandescent light source with the iodine cell

in the light path. There is usually just one exposure of this type. This is

in fact an absorption spectra of the iodine gas, which is used for obtaining

the instrumental proﬁle.

• ThAr - spectra of a Thorium-Argon lamp. This is an emission spectra

that produces many well known features that can be used as wavelength

calibrators.

• OBJECT - spectra of the science target. This type of image is not present

in all dates. The reason for this is that the object is usually observed with

the iodine cell in. Observations like this are made for obtaining object

template spectra only. These are normally higher signal-to-noise images,

and in some cases higher resolution.

• OBJECT+I2 - spectra of the science target with the iodine cell in the

light path. There are usually a couple of images of this type, namely 1 to

3, depending on the weather conditions and the telescope time allocated for

the project. These are typically lower signal-to-noise images, which provide

the main scientiﬁc results, the wavelength shift of the stellar spectra with

respect to the iodine spectra.

5.1.2 Data Reduction for Obtaining Radial Velocities

The HD 136118 and HD 33636 RV data presented in our analysis have been reduced

using a pipeline written in IDL, which is detailed in Bean et al. (2007), hereafter

Bean’s Pipeline (BP). BP performs the CCD reduction and the optimal order extrac-

tion for all individual spectra using the REDUCE package (PISKUNOV; VALENTI,

2002). For measuring the radial velocities from the target spectra it uses an inde-

pendent adaptation of the techniques described by Valenti et al. (1995) and Butler

et al. (1996).

In order to understand the procedures involved in the reduction of these echelle

spectral data, that aims to obtain radial velocities at the level of precision for de-

tecting exoplanets, we have worked in a thorough and independent investigation of

every step of this reduction. We have assembled the procedures in our own pipeline,

which we called ASTROSPEC. The ASTROSPEC is under development and still

lacks computational eﬃciency for reducing large amounts of data, as it is required in

our observing program. For this reason we have not yet been able to perform tests

to compare ASTROSPEC and BP results. Therefore, the RV data presented in our

analysis has been reduced only by BP, in the same fashion as in the previous work

(BEAN et al., 2007; MARTIOLI et al., 2010). Consequently our reduced data are consis-

tent with these works, thus we are able to compare the physical results obtained in

our analysis, which are discussed in Section 6. Below we present the ASTROSPEC

package and its preliminary results.

5.1.3 The Automatized Pipeline for Spectra Reduction (ASTROSPEC

Package)

As an attempt to develop a more eﬃcient tool and also for learning the procedures

and caveats involved in the reduction of spectroscopic data, we have developed an

independent pipeline, the ASTROSPEC package, for the reduction of HRS data to

obtain high-precision radial velocities. The main frame of this streamline is built in

a C-shell script, which can be visualized from the chart in Figure 5.1.3.

Preparation of Data

Pre-Processing

Optimal Extraction

Normalization

λ-Calibration

Measure RV

(Iodine method)

Pipeline

./HET_prep.sh

Scripts Codes

split_hrs_list, makelists, ccdproc,

imcombine

./prep_flat.sh

./prep_data.sh

./extract.sh

./normalize.sh

./calibrate_wl.sh

./get_rv.sh

Required Files

clean_bright_strip, extract_flat,

extract_blaze, imarith

extract_sum

wlcal_img

get_rv

normspec_img, cleanspc, imcombine,

imarith

Raw data, myprocfile, myprocfile_1,

myprocfile_2

Images split into CCD1 and CCD2:

Master flats, bias-subtracted

images, mask_1.fits, mask_2.fits

Reduced echelle spectra images

(-bias, /flat, *mask)

OUTSPEC FITS images

thar_ref.dat, guess.dat,

thar_1.fits, thar_2.fits

NORM_OUTSPEC images,

I2_template.dat, obj_template.dat

ord.cal files

./pipeline.sh

FIGURE 5.1 - ASTROSPEC pipeline ﬂowchart.

Below we provide the description of each step followed by ASTROSPEC.

5.1.3.1 Preparation of data

The preparation of data is the very ﬁrst step in the reduction. The script for this

stage is HET_prep.sh. This employs routines to organize images and give them the

ﬁrst treatment, like trimming and bias subtraction. Below we outline the main steps

followed by HET_prep.sh.

• Split a raw image into two new images holding the same rootname and

adding the suﬃxes ‘_1.fits’ and ‘_2.fits’, one for each CCD chip.

Figure 5.1.3.1 shows a pair of images for a typical ﬂat-ﬁeld exposure.

FIGURE 5.2 - Pair of a single ﬂat-ﬁeld exposure of echelle HRS spectra. The images are

respectively the RED CCD (left) and BLUE CCD (right).

• Create a list of ﬁles for each observation type (OBSTYPE).

• Combine zero images (delete individual images) (imcombine).

• Subtract zero from all images (CCDProc).

• Combine ﬂat-ﬁeld images into a master frame, but do not discard the

individual frames. (imcombine).

• Organize images into new directories. Figure 5.1.3.1 presents a scheme of

the directory tree where ASTROSPEC is mounted.

ASTROSPEC

CODES

DATABASE

DOCS

EXE

LAB

SPEC-DATA

PIPELINE

FLAT

FILES

FLAT+I2

OBJECT+I2

RAW

MASK

OBJECT

EMPTY_TREE

ThAr

WORK

ZERO

TEMP

EXTRACT

CLEANSPC

GET_RV

NORMSPEC

CCDProc

PREP_DATA

WLCAL

...

FIGURE 5.3 - Directory tree of ASTROSPEC.

5.1.3.2 Pre-processing

In the pre-processing phase we make use of ﬂat-ﬁeld images to obtain important

calibrations, and these are used to apply the proper corrections to the data.

The three initial steps for the preparation of ﬂat-ﬁeld reducing tasks are performed

within the script prep_flat.sh and are explained below.

• First the routine clean_bright_strip performs the cleaning of a bright

nuisance feature appearing in the bottom of BLUE CCD ﬂat images. This

step is particularly important to avoid malfunctioning of the algorithm

that automatically detects echelle orders.

• The routine extract_flat detects spectral orders and the regions within

which signal is stronger than a given threshold. For this regions, the routine

calculates the normalized pixel-by-pixel sensitivity variations. The space

between orders with negligible signal is set to unit. This routine returns an

output date_flat_CCDNUM.fits image. Figure 5.1.3.2 presents a sample

piece of the normalized ﬂat-ﬁeld extracted with extract_flat.

FIGURE 5.4 - Sample piece of a normalized ﬂat-ﬁeld.

• Finally the routine extract_blaze selects a set of points that represents

the overall illumination (blaze) function. It ﬁts a two-dimensional poly-

nomial to these points and returns an output date_blaze_CCDNUM.fits

image. Figure 5.1.3.2 presents a pair of blaze function images.

Once obtained the ‘ﬂat’ and the ‘blaze’ functions, one needs to apply these correc-

tions to the images. This is performed within the script prep_data.sh. It simply

divides each object image by the normalized ﬂat-ﬁeld and by the blaze function. At

the end of this stage it still permits one to apply masks to the data. Note that there

is a directory “MASK” where one can ﬁnd the mask images named mask_1.fits

and mask_2.fits.

FIGURE 5.5 - Example of the blaze function for CCD 1 (left) and CCD 2 (right).

5.1.3.3 Extraction of Spectra

This is an important stage of the reduction process. In this stage the spectral echelle

orders are extracted into N

1-D spectra, where N

is the number of orders. The

pipeline script for this is extract.sh, which runs the routine extract_sum. This

routine detects order positions, enumerates orders, ﬁnds the useful area for extrac-

tion, and performs an optimal extraction of spectra. The output is saved in the

OUTSPEC image ﬁles.

The extraction algorithm calculates the sum of mean counts of pixels within a given

interval along the dispersion direction. Namely, each ﬂux point in the output spec-

trum, f

, for a given row j, is calculated by the following expression:



i=i

−N





j+M





=j−M







, (5.1)

where S

is the counts of a pixel at the detector position (i, j), with i being the

column number and j the row number. i

is the column pixel at the photocenter of

an order, N

is the slit width (extraction aperture) in pixels, and M

is the sample

size, i.e. the number of neighbor pixels within which the average count is calculated.

This operation is performed for each row, providing one point in the spectra. Figure

5.1.3.3 illustrates how this operation is performed.

-N

-1 i

, j i

j +1

j + M

j -1

- M

slit size (N

)

sample size (M

)

f(i

-N

) f(i

-1)

f(i

)

+.. +

+ +

average counts

output

ﬂux f

orders

101 102 103

λ dispersion

ref

row j

.. ..

+.. +

....

f(i

)f(i

+1)

FIGURE 5.6 - Extraction scheme.

As an example of the eﬀects of parameter M

in the extraction, in Figure 5.1.3.3 we

present a piece of the spectrum of a given order. We show the extraction performed

for 5 diﬀerent values of M

. Notice that larger values of M

smooth out the spectrum

but also make it less noisy. It seems that a low value of M

, such as M

= 3, is a

reasonable choice.

We have also inspected the eﬀects of N

in the quality of the extraction. Figure 5.1.3.3

presents a region of a given order, in which it has been performed the extraction

using 5 diﬀerent aperture sizes, N

= 7, 9, 11, 13, and 15. Notice that the depth of

features increases for larger values of N

, therefore the signal-to-noise ratio (SNR)

is increased. On the other hand, the depth of features does not change signiﬁcantly

from N

= 13 to 15. This means that after a certain limit value, as N

increases,

more pixels in the region between orders are included, thus adding more noise to

80000

100000

120000

140000

160000

180000

200000

220000

240000

260000

1900 1950 2000 2050 2100 2150 2200

counts

pixel

= 1

= 3

= 5

= 8

= 12

FIGURE 5.7 - A sample piece of an extracted spectrum where it has been adopted 5

diﬀerent values for the sample size, M

= 1, 3, 5, 8, and 12. Each spectrum

was shifted for the better visualization.

the overall ﬂux. Based on this we chose the value N

= 15 for the slit aperture size.

A couple of other approaches have been tested for the extraction. The one pre-

sented here seems to be the most reliable so far. However, further improvements

with alternative approaches may provide better results, specially on what concerns

the detection and exclusion of bad pixels. In our approach the bad pixels are blindly

included in the summation, and therefore they add noise to the spectra, or as in

most of the cases they spoil the entire point evaluation, even though the majority of

points bears valid information. One could use, for example, the deviation from the

mean to estimate whether a pixel is an outlier, and then it could be replaced by a

simple interpolation. This approach has not been tested yet, but may be explored

in future research.

5.1.3.4 Normalization

Each one-dimensional spectrum needs to be calibrated both in ﬂux and wavelength.

This section deals with the ﬂux calibration. For the scientiﬁc goals of our experiment

50000

100000

150000

200000

1400 1600 1800 2000 2200 2400 2600

counts

pixel

= 7

= 9

= 11

= 13

= 15

FIGURE 5.8 - A sample piece of an extracted spectrum where it has been adopted 5

diﬀerent values for the aperture size, N

= 7, 9, 11, 13, and 15.

it is not important to perform an absolute calibration of the ﬂux. The quantity to be

measured, the wavelength shift, depends exclusively on the measurement of narrow

absorption features in the spectra. Therefore the entire continuum ﬂux emission may

be identiﬁed and removed. The identiﬁcation of this continuum envelope is the bulk

idea of the normalization algorithm, which will be explained below.

The script to perform the normalization is normalize.sh, which runs two routines,

cleanspc and normspec_img, iteratively.

The cleanspc identiﬁes and cleans out spurious features like cosmic rays and hot

pixels. The outliers are replaced by a local interpolation.

The normspec_img reads a spectrum row from the OUTSPEC ﬁle, then searches for

maximum points that better describe the overall shape of the continuum. Either a

spline or a polynomial function is ﬁt to these points. All data points are normalized

by the ﬁtted curve. This routine has ﬁrstly been designed to be used on absorption

spectra, however it has already been adapted to normalize emission spectra. The

latter may be used to extract scattered light and to change the scale of an emission

spectrum.

The automatic search for maximum points is the key point of this program. This

step is particularly important because it provides the source of data points to feed

the ﬁtting algorithms. Therefore this should represent very well the shape of the

continuum function. This continuum is not necessarily the continuum emission, but

rather a combination of any (may be many) low frequency information imprinted

along with the spectra. We link below the steps followed by the program to search

for the maximum points:

1. Read spectra data.

2. Set a box with n

data points, where n

is an input parameter of

normspec_img.

3. Perform a robust linear ﬁt in a box containing 3×n

points. Move n

points

forward and perform another ﬁtting. Move again and so on until the end of

data points. The last box may contain less than 3 ×n

points, therefore it

takes all remaining data points plus the necessary amount of points before

them to complete 3 ×n

points.

4. Calculate the standard deviation (σ) of residuals from the ﬁt line inside

each box of n

points. The ﬁrst and second boxes make use of the same ﬁt

line. The last box and the one before also use the same ﬁt line.

5. Select points lying at less than 1σ away from the ﬁt line. Outlier points have

their values replaced by a simulated data following a normal distribution

with mean lying on the ﬁt line and with dispersion σ. Then perform again

a robust ﬁt of a line to these points.

6. Find ﬁve maximum positive deviations from the ﬁt line inside each box

containing n

points.

7. Choose the maximum that better represents the upper level to normalize

the spectrum. The ﬁve maximum points found in step 6 are evaluated. If

the higher maximum is lower than 3 ×σ, then assume this and discard the

other four. However if it is greater than 3 ×σ, tests the second maximum,

and so on until the ﬁfth point. Finally if the ﬁfth point did not pass the

condition, then it assumes the level as equal to 3 × σ above the ﬁt line.

8. Reset all ﬁt lines to their respective upper level (ﬁt line + maxpoint).

Where maxpoint is chosen in step 7.

9. Calculate the average of each ﬁt line inside each box. This provides one

global maximum for each box.

10. Export max dataset to the ﬁtting routines. The ﬁt can be either polyno-

mial using a Levenberg-Marquardt minimization algorithm or a third order

spline interpolation.

Figure 5.1.3.4 shows an example of a spectral order, the continuum envelope found

by the normspec_img algorithm and a spline interpolation through these points.

Figure 5.1.3.4 presents the normalized spectrum, now with relative ﬂuxes scaled

between [0,1].

Figure 5.1.3.4 presents the HD 136118 OUTSPEC images containing the calibrated

spectra for both CCD chips.

5.1.3.5 Wavelength Calibration (Th-Ar)

The wavelength calibration is of chief importance for the determination of radial

velocities. This section deals with the initial calibration using Th-Ar emission lines as

the reference standards. Further reﬁnement in the calibration will be made using the

Iodine method, however the results in the latter strongly depend on a ﬁne previous

calibration. For this reason we have been careful in this stage. Below we describe

the methods employed for the wavelength calibration.

The script for this stage is calibrate_wl.sh, which runs the routine wlcal_img.

This will delivery a set of calibration ﬁles, one for each order. Each of these ﬁles

is named with the order number and the extension ‘.cal’, for example the order

101 will delivery the ﬁle ‘101.cal’. The algorithm to perform the calibration is

described below:

1. Read the following data ﬁles: a) Th-Ar OUTSPEC normalized spectra ﬁle

obtained at the same conditions as the program spectra. b) Program OUT-

SPEC normalized spectra ﬁle. c) File guess.dat, which contains a list of

50000

60000

70000

80000

90000

100000

110000

120000

130000

0 500 1000 1500 2000 2500 3000 3500 4000

counts

pixel

FIGURE 5.9 - Example of the normalization of a given spectral order. The full circles

represent the maximum points found by the normalization algorithm, the

green line represents the spline interpolation through the maximum points.

The spectrum is shown in red.

positions of Th-Ar lines in units of pixel and the correspondent wavelength.

d) A full catalogue of Th-Ar lines, thar_ref.dat (LOVIS; PEPE, 2007).

2. Get rid of saturated spectral lines. It ﬁrstly uses the trace_ord algorithm

(which is the same used to ﬁnd echelle spectral orders). The threshold

must be very high, so it will only detect very broad and strong features,

which are characteristic of saturated lines. Then it ﬁts a gaussian proﬁle

to the line. This is obviously not the best model, but is enough to place a

scale within which the feature will be deleted. We have used 10 times the

FWHM as the radius around the center, which places a boundary within

which the feature is cut oﬀ. The cut itself is done by replacing the ﬂux

with zero.

3. GUESS FIT - In this step it reads the line positions in the guess.dat ﬁle

and performs a gaussian ﬁt to each of these lines in the observed Th-Ar

spectrum. The ﬁt positions (in pixels) against the predicted positions (in

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1.05

0 500 1000 1500 2000 2500 3000 3500 4000

relative flux

pixel

FIGURE 5.10 - Normalized spectrum after running normspec_img.

FIGURE 5.11 - OUTSPEC normalized spectral images of HD 136118 for both the RED

CCD (top) and the BLUE CCD (bottom).

wavelength) provide the data for the ﬁrst calibration model.

4. POLY FIT - Fits a polynomial function to the line positions. The ﬁtting is

tested with several polynomial orders, varying from 1 to 7. The one with

lowest chi-square is selected.

5. Apply calibration to the observed Th-Ar spectrum.

6. With new calibrated data, a new ﬁtting of line positions is performed. Now

all features in guess.dat plus the features in the catalog ﬁle are taken.

7. The new ﬁtted positions are stored and the statistical moments are calcu-

lated. Since some bad ﬁtted lines are expected, an algorithm to ﬁlter out

all features presenting parameters with high deviations (> 3σ) from the

median is applied. It is worth to mention here that we do not use the reg-

ular deﬁnition of RMS. The reason is that outliers (always present) brings

the RMS to an unrealistic value of the quality of our ﬁtted lines. Instead

we use the Root Median of Squared Residuals (RMSR), deﬁned as follows;

RMSR =



Median(residuals

8. Re-iterate from step 4 to reﬁne the calibration.

Although the wlcal_img routine already has one built-in iteration it may be run iter-

atively with the same dataset. Previous calibration in the ﬁle ord.cal are searched,

and additional calibration is added to the end of these ﬁles. Notice that the new

calibration will only be added if it improves the ﬁt. Below is an example of the

contents of a typical calibration ﬁle for order 101, 101.cal:

3 5993.87109375 0.03058147 -0.00000142

5 0.53082484 -0.02510870 0.00041834 -0.00000295 0.00000000746

The ﬁrst integer number in each line tells the polynomial order and the remaining

numbers in the line are the coeﬃcients derived in the ﬁt.

Figure 5.1.3.5 shows the entire Th-Ar observed spectrum displaying all orders to-

gether after calibrated using wlcal_img. Figure 5.1.3.5 presents a piece of the cali-

brated Th-Ar spectrum showing two orders, 115 and 116, and the reference Th-Ar

spectrum (LOVIS; PEPE, 2007) for comparison. Notice that each order has been cal-

ibrated independently and their overlay regions present a good agreement in the

positions of Th-Ar lines.

5.1.3.6 Measuring Radial Velocities (Iodine Method)

The traditional way for obtaining radial velocities from stellar spectra is performed

by ﬁrst identifying spectral lines and then measuring their positional shifts, in units

0.2

0.4

0.6

0.8

4000 4500 5000 5500 6000 6500 7000 7500 8000

relative flux

! (A)

FIGURE 5.12 - Th-Ar entire calibrated spectrum.

of wavelength. Since each spectrum may present many lines, the shifts are averaged

and converted to velocities through Equation 3.39. We use a diﬀerent approach, the

Iodine method (BUTLER et al., 1996), which is described below.

The main idea of this method is to introduce a known source of spectral features

that permits one to precisely calibrate the spectra. This idea was introduced by

Campbell e Walker (1979), where the source of calibrating features used was the

hydrogen ﬂuoride (HF) gas. The application of this method is better explained in

Campbell et al. (1988). The HF presents the drawback of being a hazardous gas.

We use the Iodine (I

) gas instead, which is introduced in a thermally controlled low

pressure cell, positioned in the beam path of light collected by the telescope. The

main characteristics of these sources is that they should contain a large quantity

of well known absorption lines, and it is desirable that the gas is not polluted by

any external product that might produce additional nuisance features. It should be

noted that by introducing any substance in front of the beam path, a considerable

amount of light is absorbed and wasted, thus more photons are needed in order to

obtain the usual signal-to-noise ratio. This implies in limiting the method for bright

-0.1

0.1

0.2

0.3

0.4

5290 5295 5300 5305

relative flux

! (A)

Order 115

Order 116

Reference Spectrum

FIGURE 5.13 - Th-Ar calibrated spectrum for order 115 (red line) and 116 (green line),

and the reference spectrum (blue line).

sources, or the need for larger telescope apertures.

Below we explain how to implement this method to obtain the radial velocities based

on the discussion in Butler et al. (1996).

First we deﬁne each spectrum to be used in the calibration process. The spectrum

of an incandescent source of light, I

, the same ﬂat-ﬁeld spectrum observed through

the iodine cell, I

F +I

, the stellar spectrum through the iodine cell, I

S+I

, the tem-

plate transmission spectrum of the iodine gas, T

, and ﬁnally the template stellar

spectrum, I

. Each of these quantities can be related through the following expres-

sions:

F +I

(λ) = k

(λ)I

(λ)] ∗IP + C

eq : i2cal1 (5.2)

S+I

(λ) = k

(λ)I

(λ + ∆λ)] ∗ IP + C

, (5.3)

where we have introduced normalization and oﬀset constants, k

, k

, C

, and

the Instrumental Proﬁle, IP . The sign “∗” represents convolution.

Notice that I

, I

F +I

and I

S+I

can be directly observed every night, and T

can

be obtained from a catalogue. We have used the iodine template available in Salami

e Ross (2005). The stellar template, I

, is somewhat more diﬃcult to obtain in the

literature, therefore, in order to obtain our own template, we performed additional

higher resolution and better signal-to-noise observations of the target star without

the iodine cell. The generation of this template is performed in the same fashion,

i.e. using the same iodine template to calibrate it in wavelength. This has the ad-

ditional beneﬁt that our template will have similar wavelength calibration as any

other observed spectra.

Thus, the method itself consists of modeling an IP function that better ﬁts the

observed data involved in Equation ??. Then apply this IP in Equation 5.3 to look

for a wavelength shift, ∆λ, in the template stellar spectra, that better reproduces

the observed spectra, I

S+I

(λ).

Let us focus now on the determination of the IP function, which is a vital part

of this method. The IP is a transfer function that speciﬁes how the instrument

degrades sharp features. The blurring caused by the IP is a combination of many

factors that we do not necessarily need to know in order to measure and use it.

The only assumption we make here is that it remains constant over the same region

of the detector and over time during the same night. The IP can essentially be

measured from Equation ??, but this is not as simple as it seems since it involves an

operation of deconvolution, which may be performed with many diﬀerent methods,

each presenting pros and cons. Below we discuss and explore some of these methods.

Valenti et al. (1995) present two methods, a NonLinear Least Squares (NLLS), and

the Singular Value Decomposition (SVD) method. They also mention a Fourier

transform method, which is not explored. Endl et al. (2000) presents a Maximum

Entropy Deconvolution (MEM) method. We have chosen to our pipeline the NLLS

method due to simplicity. Below we discuss how this method is implemented and

the issues raised in the implementation.

The convolution in its integral form is given by:

I  F =



∞

−∞

I(λ −λ



)F (λ



)dλ



. (5.4)

In practice it is more convenient to apply this expression in the pixel space, repre-

sented here by the variable x, as presented in Valenti et al. (1995):

g(x) =



∞

−∞

φ(x −x



)f(x



)dx



, (5.5)

where g(x) represents any observed spectrum, f(x



) the internal known function and

φ(x −x



) describes the IP. The normalization of f (x



) is preserved by requiring that



∞

−∞

φ(x −x



)dx



= 1. (5.6)

Moreover Equation 5.7 can be rewritten in its discrete form

q(i+1)−1



j=qi





j+p





=j−p

j−j







, (5.7)

where p is the number of pixels within which the function φ is considered, and

q is the number of sub-pixels that ﬁt inside an oversampled detector pixel. This

oversampling operation is arbitrary, but is highly recommended, since the resolution

of a template is better than an observed spectra. Therefore by limiting the sampling

to the detector’s sampling one would be losing template information for measuring

the IP function.

The radial velocities are obtained by ﬁrst running the routine gen_template, which

generates a template spectrum for a high signal-to-noise and higher resolution object

spectrum. This is used by the routine get_rv, which will obtain the velocities as

explained below.

The routine get_rv splits the spectra into chunks two Angstroms wide. For each

chunk it takes a ﬂat+i2 spectra to measure the instrumental proﬁle. The IP is then

used to deconvolve the stellar spectra from its template. Then it builds a model of

obj+i2, for which we calculate the cross-correlation with the observed obj+i2. The

latter operation is repeated for many wavelength shifts between obj and i2 spectra in

the model spectra. It ﬁnally returns the shift for the maximum correlation. Finally,

the wavelength shift is converted to radial velocity through Equation 3.39. The whole

useful part of the spectrum provides hundreds of chunks, each of which provides a

measurement for the radial velocity. A robust statistics should then be applied in

order to obtain the ﬁnal velocity for a single image.

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

-15 -10 -5 0 5 10 15

Pixel

FIGURE 5.14 - Instrument Proﬁle (IP) model (red). We have used a parameterized model

comprising a central gaussian (green) plus 10 satellite gaussians (blue).

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.1

5778 5778.5 5779 5779.5

rms = 0.2%

FIGURE 5.15 - Sample chunk 2 Angstrom wide showing the template I

spectra (solid thin

line), the observed Flat+I

spectra (ﬁlled circles) and the ﬁt model (solid

line connecting solid circles), and ten times the residuals (open circles).

5.2 Astrometry

5.2.1 FGS Raw Data

FGS astrometry data obtained from the HST archive arrive in FITS format. These

must be converted to GEIS format before processing. As it was mentioned in Section

4, the FGS instrument is part of HST’s pointing control system. Thus the contents of

the FGS data are determined solely by the engineering telemetry format. Diﬀerently

from other instruments, the FGS is linked to the HST engineering stream and not

to the science instruments stream.

The data ﬁle rootnames follow the HST convention as appear in Shaw et al. (2009).

For example in the name “fpppss01m”, ‘f’ stands for the FGS instrument, ‘ppp’

and ‘ss’ correspond to the HST program ID and visit ID, ‘01’ corresponds to the

exposure number and ‘m’ corresponds to the source of transmission, which is in this

case “merged real time and tape recorded”. As an example, we have the following

ﬁles that were generated by an exposure of a typical FGS observation:

• f9d2320am_a1f.fits, f9d2320am_a2f.fits, f9d2320am_a3f.fits -

FITS data ﬁles, one for each FGS.

• f9d2320am_cvt.dmh - File containing scheduling and support data relevant

to the observation.

• f9d2320am.a1h, f9d2320am.a2h, f9d2320am.a3h, f9d2320am.a1d,

f9d2320am.a2d, f9d2320am.a3d - Data ﬁles converted to GEIS format.

Thus for each FGS observation a support ﬁle (extension .dmh) and three sets of

GEIS ﬁles, one for each FGS (2,3 and 1r), are generated. Header ﬁles (ending with

‘h’) contain keywords that will help to interpret data ﬁles and the data ﬁles (ending

with ‘d’) contain an instantaneous record of the following 19 quantities:

• PMTXA, PMTXB, PMTYA and PMTYB. These are the photon counts

from the four photomultiplier tubes. They are stored in the groups 1

through 4.

• SSENCA and SSENCB. These are the values of the two star selector servo

angles, stored in the groups 5 and 6.

• There are 13 status ﬂags extracted from the engineering telemetry. Each

of these ﬂags can only assume binary values (1=ON and 0=OFF) and are

stored in the same group 7.

Each group of data is not a single record of a quantity, rather it has a given number

of samples of the observed quantity. The number of samples for each quantity is

determined by the exposure time and by the sample frequency. The sample frequency

is nominally 40 Hz for quantities in groups 1 through 6 and 6.67 Hz for quantities in

group 7.

A more detailed and complete description of data formats and contents of FGS data

ﬁles can be found in the FGS Data Handbook (NELAN et al., 2010) and in the HST

Data Handbook (SHAW et al., 2009).

5.2.2 Calibration of FGS Data

In order to obtain high astrometric precision (∼ 1 mas) with FGS it is absolutely

necessary to perform a careful treatment of the data. In this section we brieﬂy

describe the standard data reduction procedures and give a bit more details on

the non-standard procedures we used to calibrate our data. Note that the following

procedures are only valid for observations in the Position Mode.

We can identify and correct for errors that arise in FGS Position Mode observations

at three diﬀerent levels:

1. Observation Level - errors associated with each individual FineLock

acquisition and tracking sequence.

2. Visit Level - errors involved in constructing a virtual plate for a given

FGS astrometric visit.

3. Field Level - errors that arise when comparing virtual plates of the same

ﬁeld taken during diﬀerent visits.

For level 3, we use GaussFit (JEFFERYS et al., 1988) models, which will be described

in Chapter 6. For levels 1 and 2 there are two pipeline processors, ﬁrst calfgsa and

then calfgsb, which are used to process the Position Mode data. Below we describe

each of these processing tools.

1. calfgsa

This is an IRAF task that performs the low level processing (conversion)

to extract information of a single observation in a stand-alone fashion, i.e.

ignoring other observations belonging to the same HST visit.

calfgsa ﬁrst reads raw GEIS ﬁles and inspect the ﬂags and status bits

(group 7) to determine if the observation was successful. Then it reads the

encoder angles (groups 5 and 6) and the PMT counts (groups 1 - 4). These

are used to compute the median centroid of each star in the Instantaneous

Field of View (IFOV), for data obtained during the FineLock/DataValid

(FL/DV) interval (when the FGS was tracking the star’s interferometric

fringes). Data collected during the slew to the target is used to estimate

the background contribution, which is subtracted from the PMT counts

obtained for the target.

Therefore, calfgsa’s task is basically to convert the raw telemetry encoder

positions to instantaneous detector coordinates, (x, y), using a static set of

parameters. The product from calfgsa is a ﬁle named as rootname.tab.

This ﬁle contains (x, y) FGS coordinates of the centroid of the star’s po-

sition in an exposure, along with data needed for further processing by

calfgsb.

All of these procedures are repeated for the identical time interval in all

three FGSs (the astrometer and the two guiders).

2. calfgsb

The processor calfgsb ﬁrst reads TAB ﬁles from calfgsa. Then it ap-

plies two observation-level corrections, the LTSTAB and OFAD, which are

better explained below.

The conversion of raw telemetry encoder positions to instantaneous de-

tector coordinates makes use of several parameters, such as the lever arm

length, and oﬀset angles, which are known to vary in time. These changes

are monitored and updated multiple times a year by the Long Term Sta-

bility Monitor (LTSTAB) program. The corrections due to these changes

are performed by the LTSTAB calibration, which is part of the calfgsb

processor.

The Optical Field of Angle Distortions (OFAD) are responsible for the

largest source of error in reducing star positions from observations with

the FGS. This is an aberration of the optical telescope assembly and can

be modeled by a two dimensional ﬁfth order polynomial (JEFFERYS et

al., 1994). The coeﬃcients for this model are obtained through a self-

calibration performed with observations of a selected star ﬁeld in M35.

These have been systematically measured and updated for many years.

The description of the OFAD calibration and its previous analysis can

be found in Jeﬀerys et al. (1994), Whipple et al. (1995), McArthur et al.

(1997), McArthur et al. (2002), and McArthur et al. (2005).

OFAD calibrations have been made upon observations in 1993, 1994, 1995

(for FGS 3), 2000, 2008 (for FGS-1r), and the LTSTAB have been per-

formed between the OFAD calibrations to maintain the calibration. This

is basically done by measuring scale-like changes. Notice that for this work

we have used a non-published (2008) up to date OFAD calibration.

After performing the corrections above for all exposures, calfgsb starts

the visit level processing. In this stage it will perform essentially two cor-

rections, one based on the guide stars, which are those observed by the

guiding FGSs, and another based on the check star, which is the one with

more exposures observed by the astrometer FGS (in our case FGS-1r). The

ﬁrst is referred to as dejittering and the second as drift corrections.

– De-jittering. During a nominal visit, while the astrometer FGS se-

quentially measures the positions of the targets, the other two FGSs

guide the telescope. They track the same guide star in FineLock.

Therefore calfgsb uses the guide star data to remove eﬀects due to

spacecraft translation and roll diﬀerences between each exposure. The

dejittering correction is normally very small (∼ 1 mas), but sometimes

can get as high as ∼ 5 mas.

– Drift Corrections. The astrometer FGS (in our case the FGS-1r)

must observe at least two check stars (usually 4). The check stars are

observed many times during the visit. These eventually drift across

the FGS ﬁeld-of-view. To remove this drift, calfgsb ﬁts a low-order

polynomial (linear, quadratic, or quadratic with rotation) to the check

stars path along the visit. Since the exposures are executed sequen-

tially, the drift correction may be removed from all exposures.

The output ﬁle produced by calfgsb, the “AST” ﬁle, will have the suﬃx ‘_ast’.

This ﬁle is in GaussFit ASCII table format, where the reduced astrometric positions

and related information for all exposures are recorded.

Finally, the ﬁeld-level corrections will take place. These are discussed in the following

section. The analysis that involves ﬁeld-level correction also involves the determina-

tion of some physical parameters, like the proper motion and parallax, therefore it

will be treated in more detail only in Chapter 6.

5.2.3 Astrometric Reduction

We call this section as “Astrometric Reduction” because most of the procedures

adopted here are the same used in classical astrometric reduction of conventional

images. In fact the AST ﬁle permits to construct a virtual plate.

Three main calibrations should be taken into account in the ﬁeld-level corrections:

• Lateral Color - color-dependent error in the measured positions.

• Cross Filter Eﬀect - apparent change in the measured position as func-

tion of the ﬁlter selected for the observation.

• Plate Constants - scaling, rotation and oﬀset constants to adjust all visit

frames to the reference frame.

Besides these three corrections one should also model simultaneously the parallax

and proper motion components for each object, as explained in Chapter 3. Therefore,

given the positions (x



, y



) measured by FGS-1r we build a model that accounts for

positional changes occurring systematically in all reference stars from date-to-date.

This is accomplished by solving an overlapping plate model which includes scaling-

rotation (C

, C

) and oﬀset (D

) constants, which are constrained to

an arbitrary frame adopted as the reference (the constrained plate). This is the 6-

parameter model, however in some cases, where the number of reference stars is not

suﬃcient, one should opt for a 4-parameter model, with two scaling-rotation and

two oﬀset constants. Therefore the model is given by the standard coordinates of

each visit:

= C



+ C



+ D

, (5.8)

= C



+ C



+ D

. (5.9)

Notice that the lateral color and cross ﬁlter eﬀects have not been included yet.

These corrections depend on the color and the diﬀerence in magnitude between the

reference stars. The model that includes the lateral color and cross ﬁlter eﬀects is

also given by Equations 5.8 and 5.9, but with x



and y



given by



= x + lc

(B − V ) −∆XF

, (5.10)



= y + lc

(B − V ) −∆XF

, (5.11)

where x and y are the astrometer FGS coordinates, (B − V ) represents the color

of each star, lc

and lc

are the lateral color corrections, ∆XF

and ∆XF

are

the cross ﬁlter corrections in x and y. There are currently only two ﬁlters available

that are supported by the OFAD calibration, the F5ND and the F583W. For fainter

targets (V > 8), F583W is the recommended ﬁlter. For brighter targets (V < 8),

F5ND ﬁlter is more appropriate. Therefore, whenever both ﬁlters are used in the

same observing program, the cross ﬁlter correction should be performed.

Equations 5.8 and 5.9 can be replaced in Equations 3.37 and 3.38, forming the

complete equations of condition. These comprise a set of two equations for each star

and for each visit, all of which must be solved simultaneously. Since the ﬁeld-level

corrections have to be modeled together with some physical parameters, like the

parallax, proper motion, and the perturbation orbit, we leave this discussion to the

following chapter.

Note on the HST Roll Angle:

The plate constants should account for eventual ﬁeld rotations among visits with

respect to one reference plate. However, the HST has to spin around its own axis in

order to provide the orientation for which the solar panels will collect the Sun light

more eﬀectively. Therefore, in some cases, the orientation of plates may become so

diﬀerent that it messes up the minimization algorithms, which might get lost easily.

In order to avoid this, one needs to rotate all plates to roughly the same orientation.

The expression to rotate counterclockwise any given position (X, Y ) in the FGS-1r

frame to the (ξ, η) standard coordinates in the sky frame, is given by:

ξ = X cos (R) − Y sin (R), (5.12)

η = X sin (R) + Y cos (R), (5.13)

where R is the relative angle between the FGS internal orientation and the spacecraft

roll angle, Roll, given by R

F GS1r

= (Roll −90

◦

) (for FGS-1r) and R

F GS3

= (Roll −

270

◦

) (for FGS-3). In order to rotate from a given frame i of a plate with roll angle

to the reference frame with roll angle R

, one has to apply Equations 5.12 and

5.13, with R = (R

− R

5.3 Infrared Interferometry

The processing of interferometric AMBER data is rather complex and it is not the

scope of this work to deal with this here. We make use of available tools, all of

which are well supported by the team of specialists from the European Southern

Observatory (ESO) and aﬃliated organizations. The principles of interferometric

reduction with AMBER/VLTI is fully described in Tatulli et al. (2007). Below we

discuss in summary the pathway for obtaining the interferometric observables from

AMBER raw data.

5.3.1 Reduction of the AMBER/VLTI Data

Raw interferometric data downloaded from the ESO server

(http://archive.eso.org) are in FITS format. In order to reduce AMBER

data one needs both the SCIENCE and CALIB category ﬁles. The SCIENCE ﬁles

comprise the OBJECT (Sci), DARK, and SKY types. The CALIB ﬁles comprise

OBJECT (Cal), DARK, SKY, and the maintenance ﬁles, which are given by the

WAVE and 3P2V (or 2P2V for two-telescopes mode) types. These ﬁles together with

the AMBER_BADPIX and AMBER_FLATFIELD ﬁles contain all information to

allow AMDLIB processors to extract uncalibrated interferometric observables from

scientiﬁc data. The latter two ﬁles are supported by the ESO engineering team and

can be downloaded from the ESO server.

AMDLIB is a package of C programs developed speciﬁcally to process AMBER

data. The front-end program called GASGANO permits one to access and run the

pipeline recipes from the AMDLIB library. Once GASGANO is set up it practically

processes data itself. One just needs to load FITS ﬁles in GASGANO, inform which

routine is to be used, and run the processor. More information about the AMDLIB

recipes can be found in Duvert et al. (2008).

Figure 5.3.1 presents a snapshot of GASGANO window showing the HD 33636 Sci

and Cal loaded data. The Cal data are interferometric exposures for the stars

HD 19637, HD 34137, and HD 36134. Notice that besides the Sci/Cal (OBJECT)

data, there are also DARK and SKY data, which are exposures taken right before

and after every set of program exposures, which are needed for calibration.

To run the reduction pipeline one needs to select ﬁles used by the recipe, and go to

menu Select Files ⇒ to recipe ⇒ Load recipe ⇒ “recipe name”.

FIGURE 5.16 - Gasgano window showing the interferometric data.

The ﬁrst important AMBER pipeline recipe to be run is the “amber_p2vm”, which

is used to create a pixel to visibility matrix (TATULLI et al., 2007). To run this routine

one needs to load AMBER ﬁles of the following six categories: AMBER_BADPIX,

AMBER_FLATFIELD, AMBER_2P2V, AMBER_2WAVE, AMBER_3P2V, and

AMBER_3WAVE. These are loaded in the “maintenance” group, as shown in Figure

5.3.1. The bad-pixel map and ﬂat-ﬁeld ﬁles are provided by ESO engineering support.

The product of the amber_p2vm recipe is the P2VM ﬁle, which is used by the

“amber_SciCal” routine.

The AMBER pipeline recipe “amber_SciCal” is used to calculate raw (uncalibrated)

visibilities from interferometric observations. The ﬁles needed by this routine are the

ones listed in Figure 5.3.1 (SKY, DARK, BADPIXEL, FLATFIELD, SCI, or CAL)

plus the P2VM ﬁle. Notice that amber_SciCal must be run separately for each

target, and for each date. The products are written in ﬁles with names “amber_-

xxxx.ﬁts”, each of which is in the OI-FITS format (PAULS et al., 2005). For each

observation there will be three of these ﬁles, one for each infrared band: J, H, and

K. These ﬁles carry the primitive header information, reduction parameters, and the

FIGURE 5.17 - Gasgano snapshot window showing the loaded maintenance ﬁles of

the following categories: AMBER_2P2V, AMBER_2WAVE, AMBER_-

3P2V, AMBER_3WAVE, AMBER_BADPIX, and AMBER_FLAT-

FIELD. These data are used by routine “amber_p2vm”.

resulting uncalibrated interferometric observables. We have developed independent

tools to extract and handle these data from OI-FITS ﬁles. The resulting visibilities

and phases, as well as the calibration process are described below.

5.3.2 Calibrating Interferometry Visibility Data

An important feature of interferometry is that the observation of calibrators is as im-

portant as the science object observations. The instrumental eﬀects can be condensed

in a transfer function, which gives the point-source response of an interferometer,

and can be obtained through observations of calibrators. This is only possible when

the interferometric model for calibrators is well understood. The calibrators are usu-

ally well known previously studied targets. Our calibrators have been provided by

the ESO “search calibrator tool” (SearchCal) (BONNEAU et al., 2006). Table 5.1 lists

information from our interferometric calibrators, all of which has been used in the

analysis of HD 33636 data. Information listed in Table 5.1 has been obtained from

Mérand et al. (2005).

TABLE 5.1 - Information for the Interferometric Calibrators (MÉRAND et al., 2005).

Object RA Dec π

abs

Spec V J H K Θ

ID (2000) (2000) mas Type mag mag mag mag mas mas mas mas mas

HD 34137 05:15:11.9 +01:33:22 3.29 ± 0.95 K2III 7.20 4.89 4.36 4.18 0.825 0.786 0.798 0.802 ±0.011

HD 19637 03:10:27.0 +26:53:46 8.03 ± 0.97 K3III 6.04 3.86 3.24 3.04 1.333 1.274 1.292 1.298 ±0.017

HD 36134 05:29:23.7 -03:26:47 6.98 ± 0.83 K1III 5.81 3.90 3.21 3.12 1.201 1.148 1.164 1.170 ±0.016

Prior to the reduction we investigate whether Sci and Cal sources are resolved or

not by the interferometer. The transmitted spatial frequencies observed in our ex-

periment range from 50 arcsec

−1

to 550 arcsec

−1

approximately. Therefore there is

a cut-oﬀ at the maximum transmitted frequency f

max

, which limits the detection

of structures with size inferior than f

max

, i.e. the star apparent diameter will be

detected by the interferometer if Θ  1.818 mas. From Table 5.1 we notice that all

calibrators have apparent sizes relatively close to this value, therefore one should

consider a model that accounts for the size of calibrator sources, like the uniform

disk model (Equation 3.54). For unresolved sources one should use Equation 3.50,

which provides a constant value for the visibilities. Figure 5.3.2 presents the uniform

disk model for the squared visibility (V

) of calibrators listed in Table 5.1. We have

also included in Figure 5.3.2 a plot of the V

model for HD 33636. The apparent

angular diameter Θ (in seconds of arc) of HD 33636 has been estimated through the

following expression:

Θ =

107.5 d

, (5.14)

where R is the physical radius in units of R



, and d = 1/π

abs

is the distance in

parsecs. Both radius and distance for HD 33636 are presented in Table 2.2.

In order to calibrate our interferometric data we have followed the procedure de-

scribed in Boden (2007), which is shortly discussed below.

Figures 5.3.2, 5.3.2, and 5.3.2 present the data for the three baselines of the raw

squared visibilities for the two Cal objects and for HD 33636. We can clearly notice

that visibility for all targets follow basically the same pattern. This common pattern

is due to the instrumental transfer function, which is also known as the ‘system

visibility’. In fact the visibilities shown in Figures 5.3.2, 5.3.2, and 5.3.2 contain

the signal for each target multiplied by the system visibility. The latter is assumed

to be constant within the same baseline, within the same spectral band, and over

observations performed within a short time interval.

The transfer function, V

sys

(system visibility) can be obtained from Cal objects

through the following equation:

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 200 400 600 800 1000 1200 1400 1600

Squared Visibility

Spatial Frequency (arcsec

-1

)

HD 33636

HD 34137

HD 19637

HD 36134

FIGURE 5.18 - Uniform disk model for the squared visibility of calibrators listed in Table

5.1 and the Sci object, HD 33636. Dashed vertical line shows the cut-oﬀ

frequency for our experiment.

-0.05

0.05

0.1

0.15

0.2

0.25

0.3

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Raw Squared Visibility

! (µm)

BASELINE 1 (f = 100 to 200 arcsec

-1

)

J Band H Band K Band

HD 33636

HD 34137 Set A

HD 34137 Set B

HD 19637 Set A

HD 19637 Set B

FIGURE 5.19 - Baseline 1 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636.

-0.05

0.05

0.1

0.15

0.2

0.25

0.3

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Raw Squared Visibility

! (µm)

BASELINE 2 (f = 170 to 350 arcsec

-1

)

J Band H Band K Band

HD 33636

HD 34137 Set A

HD 34137 Set B

HD 19637 Set A

HD 19637 Set B

FIGURE 5.20 - Baseline 2 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636.

-0.05

0.05

0.1

0.15

0.2

0.25

0.3

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Raw Squared Visibility

! (µm)

BASELINE 3 (f = 240 to 480 arcsec

-1

)

J Band H Band K Band

HD 33636

HD 34137 Set A

HD 34137 Set B

HD 19637 Set A

HD 19637 Set B

FIGURE 5.21 - Baseline 3 raw squared visibility data for the two calibrators, HD 34137

and HD 19637, and for the science target, HD 33636.

sys

m−cal

i−cal

, (5.15)

where V

i−cal

is the intrinsic target visibility, V

m−cal

is the measured target visibility.

Notice that the intrinsic visibility of calibrators is assumed to be known, which are

given by the curves provided in Figure 5.3.2.

Finally, the Sci target calibrated visibility can be obtained through the following

expression:

c−sci

m−sci

sys

, (5.16)

where V

m−sci

is the measured Sci object visibility.

Before performing the calibration we have to evaluate the data quality. By making an

eyeball inspection of Figures 5.3.2, 5.3.2, and 5.3.2 we could notice that data labeled

as ‘HD 34137 set B’ and ‘HD 19637 set A’ (open marks), for some reason, at least

for the baseline 1, present systematic disagreement with respect to the remaining

curves. Observations usually bracket the Sci target with two calibrators, and these

two data sets are the ones more distant in time, which may cause some discrepancy

in the system visibility. The two data sets taken closest in time to the Sci data,

‘HD 34137 set A’ and ‘HD 19637 set B’, shows better agreement. The calibrator

HD 36134, which has been observed further in time, is a bright target and presents

an excellent agreement with the other calibrators. Therefore we have also included

this in our analysis.

These systematic eﬀects can be inspected by looking at the behavior of the system

visibility of each data set with respect to the mean system visibility. Figure 5.3.2

presents a plot of the mean deviation from the average (solid line) and their standard

deviation (dashed line). We notice that for the H and K bands, these deviations

increase signiﬁcantly. This may introduce larger errors in the determination of the

system visibility and consequently in the Sci calibrated visibility points. In order

to check whether any speciﬁc data set is causing these eﬀects, we have plotted in

Figure 5.3.2 the system visibility for each set (discontinued lines) and the average

of all data sets (thick solid line). It is clear that some sets are exceedingly deviant

from the mean. For this reason we have ﬁltered out the calibrator data sets listed

in Figure 5.3.2.

0.06

0.05

0.04

0.03

0.02

0.01

1.2 1.4 1.6 1.8 2 2.2 2.4

! (µm)

Baseline 1

0.05

0.04

0.03

0.02

0.01

Squared Visibility

Baseline 2

0.1

0.08

0.06

0.04

0.02

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Baseline 3

Mean deviation

Standard deviation

FIGURE 5.22 - Solid lines show the mean diﬀerence between each point and the average,

∆V

. Dashed lines show the standard deviation.

Finally we employ Equations 5.15 and 5.16 to obtain the calibrated visibilities.

In order to validate our results we have calibrated two calibrator’s data, HD 19637

and HD 36134. This procedure turns the calibrator into Sci target and automatically

excludes it from the calibrator list. The last cosmetic operation performed in our data

is a σ-cut, where we have excluded points with error larger than a certain threshold.

For HD 19637 we have selected points with σ

< 1.5 and for HD 36134 with σ

2.0. Figures 5.3.2 and 5.3.2 present the median taken every 15 calibrated squared

visibilities points (solid circles with error bars), and the uniform disk ﬁt model

for HD 19637 and HD 36134, respectively. We have obtained the ﬁt angular sizes,

HD19637

= 1.452 ± 0.035mas and Θ

HD36134

= 1.362 ± 0.029mas, which we consider

in good agreement with the values listed in Table 5.1, given that we have not used

a λ-dependent model. Figure 5.3.2 presents HD 33636 data, which we have reduced

in exactly the same way as the two calibrators listed above. The disk model is also

shown. We can clearly notice a disagreement between data and the predicted model

for an isolated disk-like source. This implies the need of an additional component

0.1

0.08

0.06

0.04

0.02

-0.02

-0.04

1.2 1.4 1.6 1.8 2 2.2 2.4

! (µm)

Baseline 1

0.1

0.08

0.06

0.04

0.02

-0.02

-0.04

1.2 1.4 1.6 1.8 2 2.2 2.4

! (µm)

Baseline 1

0.08

0.06

0.04

0.02

-0.02

Squared Visibility

Baseline 2 0.08

0.06

0.04

0.02

-0.02

Squared Visibility

Baseline 2

0.1

0.08

0.06

0.04

0.02

-0.02

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Baseline 3

Mean Vsys

HD 19637 Set A

HD 19637 Set B

HD 34137 Set A

HD 34137 Set B

HD 36134

FIGURE 5.23 - System visibility, V

sys

(transfer function), for all calibrators as indicated

in the legend. The thick solid line is the average of all data sets.

Baseline 1

Baseline 2

Baseline 3

J-band H-band K-band

HD 19637 Set 1 HD 19637 Set 1

HD 19637 Set 1

HD 19637 Set 1HD 19637 Set 1

HD 34137 Set 1 HD 34137 Set 1

HD 34137 Set 1HD 34137 Set 1

HD 34137 Set 2HD 34137 Set 2

Filtered Out Data Sets

FIGURE 5.24 - List of calibrator data sets that have been ﬁltered out for calibrating

interferometric data.

to explain the brightness morphology, like the binary. This will be explored in the

data analysis chapter.

0.2

0.4

0.6

0.8

1.2

1.4

0 200 400 600 800 1000 1200 1400 1600

Squared Visibility

Spatial Frequency (arcsec

-1

)

Uniform Disk Model

HD19637 VLTI Data

FIGURE 5.25 - HD 19637 squared visibility data (ﬁlled circles) and the disk model ﬁt

(solid line). Data points with σ

> 1.5 have been cut oﬀ. The ﬁt angular

diameter for HD 19637 is Θ = 1.452 ± 0.035 mas.

5.3.3 Calibrating Interferometry Phase Data

The calibration of interferometric diﬀerential phases (φ) and closure phases (Φ) is

conceptually similar to the calibration of visibilities. The brightness distribution of

calibrators is symmetric, thus the complex visibility function is real. From Equation

3.61 we notice that either the phases or closure phases for calibrators follow a step-

like function, where it has constant values, 0 or π, depending on the range of spatial

frequency. The change from 0 to π, or vice-versa, occurs at every null of the visibility

function. As one can see from Figure 5.3.2 the maximum spatial frequency sampled

by our experiment falls before the ﬁrst interferometric visibility null of all calibrators.

Thus we do not need to concern about the phase inversion in this calibration.

Therefore, in order to obtain the relative calibrated phases for the target φ

cal−trg

one just needs to divide the target raw phase φ

raw−trg

by the calibrator raw phase

raw−sys

, so

0.2

0.4

0.6

0.8

1.2

0 200 400 600 800 1000 1200 1400 1600

Squared Visibility

Spatial Frequency (arcsec

-1

)

Uniform Disk Model

HD36134 VLTI Data

FIGURE 5.26 - HD 36134 squared visibility data (ﬁlled circles) and the disk model ﬁt

(solid line). Data points with σ

> 2.0 have been cut oﬀ. The ﬁt angular

diameter for HD 36134 is Θ = 1.362 ± 0.029 mas.

cal−trg

raw−trg

raw−sys

. (5.17)

Since we have observations from three calibrators, we calculate φ

raw−sys

through the

average of raw phases from all calibrators at a given spatial frequency. The OI-FITS

ﬁles also provides the closure phase data, so one does not need to combine phases

from the three baselines. Instead one can directly obtain the closure phase value and

perform the calibration in exactly the same way as for the phases. The calibrated

phases and closure phases for HD 33636 are presented in Chapter 6.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 200 400 600 800 1000 1200 1400 1600

Squared Visibility

Spatial Frequency (arcsec

-1

)

HD 33636 VLTI Data

Uniform Disk Model

Baseline 1

Baseline 2

Baseline 3

FIGURE 5.27 - HD 33636 squared visibility data is represented by triangles (baseline 1),

circles (baseline 2), and diamonds (baseline 3). The disk model for the

derived angular size of HD 33636, Θ = 0.383 mas, is also shown (solid

line).

6 DATA ANALYSIS

This section is a description of the data analysis to obtain the physical parameters

of the systems in study, and comprises the primary results of this work. The main

scientiﬁc target in our analysis is HD 136118, but we also present a reassessment of

HD 33636 previously published radial velocity and astrometry data, for which we

have applied the same methodology as for HD 136118. We obtained interferometric

data only for HD 33636, which will be included in our analysis and discussed at the

end of this chapter.

The spectroscopic RV data obtained with HRS/HET are combined with previously

published data, all of which are used to model the system’s parameters simultane-

ously along with FGS astrometry, spectro-photometric parallaxes, and all available

information in the literature. This is made in such a way that parameters are con-

strained and therefore consistent with all datasets. The uncertainties play an impor-

tant role since they deﬁne the degree of inﬂuence of each data set in the ﬁnal results.

This will become more clear as we present our results. The data sets with better

precision and/or better sampling can be, in some cases, analyzed alone in order to

provide a ﬁrst solution, which is used as prior in the search for a more complete

solution, including all data sets. This procedure restricts some parameters ranges

within which minimization algorithms will search for the best solution.

Our results were obtained using GaussFit (JEFFERYS et al., 1988) programs, which

is an eﬃcient system for solving least-square problems. GaussFit allows one either

to employ a regular Levenberg-Marquardt (LM) method or, if desirable, a robust

method (HOLLAND; WELSCH, 1977; HUBER, 1981; REY, 1983). The robust method

is preferable when the data set is contaminated with outliers. However, there is an

issue when one wants to compare the goodness of ﬁt between the two methods. The

deﬁnition of χ

is diﬀerent in each approach. For this reason we will mostly present

the χ

obtained from both LM and robust solutions, but the results had ultimately

been obtained through the robust method. It will be made explicit in the text when

using a diﬀerent approach.

6.1 Radial Velocity Analysis

The RV data alone already provides important information about the system. This

analysis is usually employed for most of the exoplanet candidates found by the

RV method. We present below an analysis of RV data alone for HD 136118 and

HD 33636.

The ﬁrst step in the RV analysis is to separate reduced velocities by their instru-

ment, telescope, and/or site of observation. This segmentation of data must be done

because each data set provides a diﬀerent value for the zero level, which is an oﬀset

parameter, Γ (Equation 3.40). Eventually, one should also separate data that has

been obtained with the same instrument. This should be done whenever there are

reasons to believe that something could have introduced a diﬀerent oﬀset in part

of the data. This might happen, for example, in observations performed after some

critical maintenance operation in the spectrograph. A situation like this may not

interfere in the instrument’s relative precision but may aﬀect the zero level, therefore

one should take it into account by introducing a diﬀerent Γ for each portion of the

data. Of course there is a drawback in which as more pieces you split your data as

more relative information you lose.

Tables 6.4 and 6.5 present HD 136118 RV data obtained from HET/HRS (MARTIOLI

et al., 2010) and from the Lick Observatory (FISCHER et al., 2002). Tables 6.6, 6.7,

6.8, and 6.9 present HD 33636 RV data obtained from HET/HRS (BEAN et al., 2007),

Lick Observatory (BUTLER et al., 2006), Keck Observatory (BUTLER et al., 2006), and

from the Elodie spectrograph at Observatoire de Haute-Provence (PERRIER et al.,

2003), respectively.

The model used to ﬁt RV data is given by Equation 3.40. As we mentioned, we use

GaussFit programs to perform this ﬁt. However, if you just input the data with a

model given by Equation 3.40, and try to run GaussFit without a good set of initial

parameter values, the search for the minimum χ

will probably fail. The reason for

this is that minimization algorithms get lost easily. It may sometimes either get

stuck in a local minimum or enter an inﬁnite loop when doing an outbound search

for parameter values. In particular, we found that angular quantities often present

problems for minimization algorithms like LM.

HD 136118 and HD 33636 are two systems with RV-detected exoplanet candidate

companions. A previous analysis of these companion’s orbits made by Fischer et al.

(2002) and Bean et al. (2007) provide a solution for the RV orbital parameters of

HD 136118 b and HD 33636 b. We adopt these solutions as our initial set of param-

eters. Figures 6.1 and 6.3 present the Γ-subtracted radial velocities as function of

time, and the best-ﬁt models for the RV orbit of each companion. Figures 6.2 and

6.4 present the same plots folded with the companion’s period. The residuals are

shown in the bottom panel of each ﬁgure.

-200

-150

-100

-50

100

150

200

250

300

1000 1500 2000 2500 3000 3500 4000 4500 5000

Radial Velocity (m/s)

HJD - 2450000

Model

Lick1.dat

Lick2.dat

HET.dat

-30

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

UT date

FIGURE 6.1 - HD 136118 radial velocities as function of time and the best-ﬁt model.

Tables 6.1 and 6.2 present the orbital parameters obtained from the ﬁt, the derived

minimum masses and semimajor axes, where we have assumed the parent star masses

HD136118

= 1.24 ± 0.07 M



(FISCHER et al., 2002) and M

HD33636

= 1.02 ± 0.03 M



(TAKEDA et al., 2007), and the statistical quantities from the residuals.

Notice that in this analysis we have not considered the possibility of additional

companions to these systems. In fact Bean et al. (2007) have investigated this and

found no additional companions to HD 33636. Below we discuss the HD 136118 data.

From Table 6.1 we note that the RMS of HET residuals for HD 136118 is about 2.5

times larger than for HD 33636 (Table 6.1). Given that both data sets have been

obtained under similar conditions, and that both stars are relatively similar on what

-200

-150

-100

-50

100

150

200

250

300

Radial Velocity (m/s)

Model

Lick1.dat

Lick2.dat

HET.dat

-50

0 0.5 1 1.5 2

Orbital phase (P = 1188.67 days)

FIGURE 6.2 - HD 136118 radial velocities and the best-ﬁt model in the phase diagram.

Folding period is P = 1188.67 days.

concerns the velocity errors, it is likely that an unaccounted source of systematic

error is increasing the dispersion of the residuals of HD 136118 above the expected

level of usual HET noise. The reduced chi-square, χ

, for HD 136118 HET data

only is 1.58 and for Lick data only is 0.94. A χ

close to unit indicates satisfactory

agreement between the dispersion of residuals and the individual errors. Therefore

this agreement seems not to be as deﬁnitive for the HET as it is for Lick data.

We ﬁrst check whether the residuals follow a Gaussian distribution and inspect the

errors involved. Figure 6.5 shows the histogram of distribution of residuals for the two

individual datasets separately, Lick and HET, and also for both datasets combined

(hereafter “ALL”). We can see the diﬀerent dispersion for each dataset. We note that

each individual dataset, either HET and Lick, are not following exactly a normal

distribution. We call attention to the fact that the dispersion on HET residuals is

about three times larger than the error (∼ 3 m/s) estimated from previous work

(BEAN et al., 2007, e.g.). This discrepancy may be identiﬁed with an unaccounted for

-150

-100

-50

100

150

200

250

500 1000 1500 2000 2500 3000 3500 4000 4500

Radial Velocity (m/s)

HJD - 2450000

Model

Lick.dat

Keck.dat

HET.dat

Elodie.dat

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

UT date

FIGURE 6.3 - HD 33636 radial velocities as function of time and the best-ﬁt model.

systematic eﬀect. Below we investigate the detection limits for any further periodic

signal that could still be present in the data.

6.1.1 Limits on Additional Periodic Signals in the HD 136118 RV Data

The customary method for searching periodic signals in unevenly spaced data is by

means of the Lomb-Scargle Periodogram (LSP) (SCARGLE, 1982). Figures 6.6 and

6.7 show the LSP of residual RV data for two diﬀerent datasets respectively: ALL

and HET. The sets are analyzed separately because they have diﬀerent errors (see

Table 6.1). The power in the LSP is weighted by the overall variance, therefore if one

mixes two sets with diﬀerent variances it would result in an overestimated power for

higher levels of noise. The downside of analyzing datasets separately is that sampling

becomes diﬀerent as you have diﬀerent time coverage and it may aﬀect the signal

detectability for some frequencies.

Fischer et al. (2002) show that the LSP for Lick data does not seem to present any

-150

-100

-50

100

150

200

250

Radial Velocity (m/s)

Model

Lick.dat

Keck.dat

HET.dat

Elodie.dat

0 0.5 1 1.5 2

Orbital phase (P = 2119.68 days)

FIGURE 6.4 - HD 33636 radial velocities and the best-ﬁt model in the phase diagram.

Folding period is P = 2119.68 days.

expressive power. The LSP for HET dataset (Figure 6.7) shows some peaks at the

limit where False Alarm Probability (FAP) is as low as about 1%. The combined

dataset also shows some power below the level of 1% FAP. This indicates that either

there is still an unaccounted periodic signal or the sampling for those frequencies is

poor. The latter may be analyzed by the method we describe below.

We introduce a quantity to evaluate how much we can trust some high power found

at a given frequency based on sampling for that frequency. We call this quantity the

Amount of Information in the Phase Diagram (AIPD). It is deﬁned by the following

expression,

I = −



i=1

ln p

/ ln N

, (6.1)

where N

is the number of bins in the phase diagram, p

is the probability of ﬁnding

TABLE 6.1 - HD 136118: RV best-ﬁt parameters.

Orbital parameters:

P (days) = 1188.7 ± 2.1

T (JD) = 2450613.9 ± 6.6

e = 0.33 ± 0.01

ω (

◦

) = 317.4 ± 1.0

K (m s

−1

) = 211.4 ± 1.5

Derived parameters:

M sin i (M

) = 12.06 ± 0.46 (3.82%)

M sin i (M



) = 0.0115 ± 0.0004 (3.82%)

a sin i (AU) = 2.35 ± 0.16 (6.90%)

Systemic velocities (m s

−1

Lick1

= 0.6 ± 4.0

Lick2

= -10 ± 13

HET

= 486.3 ± 1.4

Statistical quantities:

= 116.74 DOF = 88

= 1.33

RMS

Lick

= 16.58 m s

−1

RMS

HET

= 8.26 m s

−1

a data point within a given bin i, and may be calculated by p

= n

/N, where n

is the number of points inside the bin i and N is the total number of data points.

Note that the dependence on the period arises from the construction of the phase

diagram. This quantity is normalized and therefore it varies from 0 to 1. When

I = 0 it means that all data points are found within a single bin in the phase

diagram and hence sampling for that frequency is very poor. From the other hand

when I = 1 it means that probability p

is the same for every bin and data is equally

distributed along all bins. This gives you an ideal coverage of the phase diagram.

We call attention to the fact that AIPD does not measure the statistical signiﬁcance

of the number of data points but only the signiﬁcance of how well distributed are

these points in the phase diagram. Therefore an issue of concern is the choice of an

adequate N

. We suggest the choice is made in the same fashion as when you build

a traditional histogram for inspecting probability distributions. Our choice of N

that of N/N

> 30 data-points-per-bin. If N is small such that makes N

< 10 then

we assume N

= 10.

Figures 6.6 and 6.7 also show a plot of AIPD (in the plot it is multiplied by 20 and

TABLE 6.2 - HD 33636: RV best-ﬁt parameters.

Orbital parameters:

P (days) = 2119.7 ± 7.8

T (JD) = 2451195.5 ± 5.8

e = 0.477 ± 0.007

ω (

◦

) = 337.0 ± 1.4

K (m s

−1

) = 161.4 ± 1.8

Derived parameters:

M sin i (M

) = 9.13 ± 0.21 (2.30%)

M sin i (M



) = 0.0087 ± 0.0002 (2.30%)

a sin i (AU) = 3.23 ± 0.13 (4.02%)

Systemic velocities (m s

−1

Lick

= -38.1 ± 3.4

Keck

= -32.8 ± 1.4

HET

= -0.1 ± 2.7

Elodie

= 155.1 ± 1.8

Statistical quantities:

= 184.17 DOF = 137

= 1.34

RMS

Lick

= 13.06 m s

−1

RMS

Keck

= 6.03 m s

−1

RMS

HET

= 3.35 m s

−1

RMS

Elodie

= 11.87 m s

−1

shifted for the sake of visualization). We notice that some of the high power periods

in the LSP also present a decreasing on the AIPD, which means a deﬁcit of sampling

for those periods. This is evident for the 1 yr period where there is always lack of

data for some part of the phase diagram. If you fold the 1 yr phase diagram twice

there will still be some lack of data coming from the 1 yr sampling problem. This

can be seen from the smaller decreases at half year period. Although it is smaller

it may still aﬀect the LSP. If one disregards the powers at periods which are close

integer fractions of a year there will be no signiﬁcant power left on the LSPs.

However, the existence of an additional signal cannot be ruled out by looking only

at the LSP for the following reason. If the signal comes from an orbit which follows

the RV model (Equation 3.40), then it may be considered solutions for eccentric

orbits instead of strict sine and cosines as in the LSP. An alternative periodogram

using the orbit solution is explored in Gregory (2007). We propose a strategy to ﬁnd

a possible hidden signal in our particular case, although it could be expanded and

0.3

0.2

0.1

! = 10.34 m/s

ALL

0.3

0.2

0.1

Prob

! = 13.71 m/s

Lick

0.2

0.1

-40 -30 -20 -10 0 10 20 30 40

RV Residuals (m/s)

! = 8.67 m/s

HET

FIGURE 6.5 - Histogram of HD 136118 RV residuals from 1-companion model for 3

datasets: all combined (ALL) (top panel), Lick (middle panel) and HET

(bottom panel).

applied for any other system.

Our strategy consists in making multiple attempts to model the RV, using a 2-

companion model (linear superposition of two Keplerian orbits), and keeping the

trial periods as constants in the ﬁtting process. This process forces the minimization

algorithm to search for the best solutions for each chosen period. This approach could

result in a lower χ

when including a hidden periodic component. Figure 6.8 shows

a χ

map over a range of periods for a 2-companion model ﬁtting RV HET and Lick

data simultaneously. The grid point resolution is 1100 ×90, which means a step on

the trial periods of about ∼ 0.3 day for both components. From Figure 6.8 we can

see a region around 255 days where we found an island of lower χ

(darker regions),

which indicates the presence of an additional signal. We note that the relatively

high power peak at ∼ 95 days in the LSP (see Figure 6.6) is now ruled out, because

any attempt of ﬁtting a secondary orbit with this period results in larger χ

. This

1 10 100 1000

Power

Period (days)

FAP

10%

190

125

LSP

AIPD

FIGURE 6.6 - Lomb-Scargle periodogram for RV residuals from 1-companion model and

using all datasets combined (solid line). The thresholds for false alarm

probability of 1% and 10% are plotted in dotted lines. Above it is shown

the AIPD (see text) for the same dataset (dashed line). The AIPD here is

multiplied by a factor 20 and shifted for the sake of better visualization.

approach does not prove the existence of another companion in the system. Rather it

shows an eﬀective way of ﬁnding solutions that are considerably improved by adding

a weak periodic signal that could not be detected in the LSP. Moreover, our model

uses Keplerian orbits, which look not only for a periodic signal but for a signal with

the shape of an orbit.

Then, we performed a reﬁned ﬁt using the LM method and then a robust ﬁt method

for a 2-companion model with an additional signal with period of about 255 days.

This solution presents a signiﬁcant improvement of 30% on the χ

if compared to

the 1-companion model (see Tables 6.1 and 6.3). The parameters for a 2-companion

model are shown on Table 6.3. The σ dispersion for HET data is now in agreement

with that we expected. Figures 6.10 and 6.11 show the phase diagram of RV data

and the respective component orbit model. The reality of this additional companion

1 10 100 1000

Power

Period (days)

FAP

10%

180

125

235

LSP

AIPD

FIGURE 6.7 - Lomb-Scargle periodogram for HET RV residuals from 1-companion model

(solid line). The thresholds for false alarm probability of 1% and 10% are

plotted in dotted lines. Above it is shown the AIPD (see text) for the same

dataset (dashed line). The AIPD here is multiplied by a factor 20 and

shifted for the sake of better visualization.

to the system will be discussed in Chapter 7. The parameters adopted in the fol-

lowing sections are those from a 2-companion model. The modelling of the second

component in the RV has a very marginal eﬀect on the parameters of the astrometric

detection, so it is not “polluting” our result.

FIGURE 6.8 - χ

map from a 2-companion ﬁt model for HET and Lick data. The grid

resolution is about 0.3 day (1100 × 90 points). Contour lines show four

diﬀerent levels of χ

. The best ﬁt solution has the lowest value at χ

= 76.6.

100

TABLE 6.3 - HD 136118: RV best-ﬁt parameters for a two-companion model.

HD 136118 b Nuisance orbit

Orbital parameters:

P (days) 1191.5 ± 1.9 254.1 ±1.7

T (JD) 2450608.9 ±5.8 2453758.9 ± 9.5

e 0.348 ±0.009 0.41 ±0.13

ω (

◦

) 316.4 ±1.1 190 ±19

K (m s

−1

) 216.7 ± 1.5 10.9 ±1.8

Derived parameters:

M sin i (M

) 12.31 ± 0.47 (3.82%) 0.36 ± 0.06 (18.01%)

M sin i (M



) 0.0117 ± 0.0004 (3.82%) 0.00034 ± 0.00006 (18.01%)

a sin i (AU) 2.35 ± 0.16 (6.89%) 0.84 ±0.24 (28.01%)

Systemic velocities (m s

−1

Lick1

= 0.14 ± 3.66

Lick2

= −9.7 ± 10.9

HET

= 486.21 ± 1.17

Statistical quantities:

= 74.44 DOF = 83

= 0.90

RMS

Lick

= 15.60 m s

−1

RMS

HET

= 5.60 m s

−1

101

-200

-150

-100

-50

100

150

200

250

300

1000 1500 2000 2500 3000 3500 4000 4500 5000

Radial Velocity (m/s)

HJD - 2450000

Model

Lick1.dat

Lick2.dat

HET.dat

-50

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

UT date

FIGURE 6.9 - HD 136118 radial velocities as function of time and the best-ﬁt two-

companion model.

102

-200

-150

-100

-50

100

150

200

250

300

Radial Velocity (m/s)

Model

Lick1.dat

Lick2.dat

HET.dat

-50

0 0.5 1 1.5 2

Orbital phase (P = 1191.52 days)

FIGURE 6.10 - Phase diagram folded with period 1191.52 days. RV HET (ﬁlled circles)

and Lick (open marks) data subtracted the “nuisance orbit” model. Solid

line shows the best ﬁt HD 136118 b orbit model. Residuals from the 2-

companion model is plotted in the bottom panel.

103

-80

-60

-40

-20

Radial Velocity (m/s)

Model

Lick1.dat

Lick2.dat

HET.dat

-50

0 0.5 1 1.5 2

Orbital phase (P = 254.06 days)

FIGURE 6.11 - RV HET (ﬁlled circles) and Lick (open marks) residuals from HD 131168 b

orbit model and the best ﬁt “nuisance orbit” model plotted in the phase

diagram folded with period 254.06 days. Residuals from the 2-companion

model is plotted in the bottom panel.

104

TABLE 6.4 - HET relative radial velocities for HD 136118.

HJD - 2450000 RV (m/s) ±σ (m/s)

3472.831 432.6 4.1

3482.881 421.4 3.5

3527.763 407.8 4.8

3544.727 392.4 4.1

3575.630 394.6 4.6

3755.051 319.0 10.6

3757.041 320.5 8.4

3765.026 312.2 8.8

3766.026 313.0 8.8

3767.020 321.9 7.7

3769.011 321.9 8.2

3787.982 329.2 7.5

3808.904 319.1 6.9

3809.909 322.5 7.8

3815.886 342.9 7.3

3816.898 334.4 7.3

3816.965 338.7 7.7

3818.873 324.5 8.0

3820.897 337.5 9.2

3832.840 331.2 6.9

3835.853 333.6 6.9

3836.858 334.6 10.2

3840.895 321.5 6.1

3844.909 328.9 5.6

3866.774 335.1 5.4

3867.754 332.1 4.3

3877.724 328.3 4.1

3880.810 339.0 5.0

3883.778 329.3 3.7

3888.700 330.7 4.2

3890.679 333.3 4.9

3891.682 333.7 4.6

3892.689 329.5 4.7

3893.768 325.2 4.4

3895.744 341.4 4.4

3897.749 332.9 4.6

3898.678 339.1 4.5

3901.740 336.0 4.3

3905.734 341.8 5.7

3911.730 333.9 5.8

3917.689 345.5 4.8

3938.639 341.2 16.4

3937.648 350.4 4.6

3939.631 338.5 4.7

4129.036 598.0 9.3

4131.023 587.5 9.0

4135.035 601.4 10.3

4144.998 619.0 8.0

4164.019 660.3 7.7

4176.992 696.4 7.3

4180.889 698.6 6.1

4186.887 711.6 5.5

4190.869 711.2 6.5

4191.864 713.7 5.9

4211.816 734.9 5.8

4221.789 744.9 5.7

4253.699 745.3 4.0

4282.631 735.8 4.9

4556.884 513.6 9.6

4565.914 510.6 8.7

4574.895 509.8 7.2

4580.893 491.0 7.2

4606.803 474.3 6.7

105

TABLE 6.5 - Lick relative radial velocities for HD 136118.

set HJD - 2450000 RV (m/s) ±σ (m/s)

1 1026.698 -26 26

1 1027.688 -19 22

1 1243.013 -108 19

1 1298.905 -127 22

1 1299.820 -102 18

1 1303.868 -126 18

1 1304.818 -133 18

1 1305.833 -120 17

1 1337.757 -167 21

1 1362.751 -140 20

1 1541.094 -160 19

1 1608.003 -109 21

1 1627.902 -117 17

1 1629.891 -100 19

1 1733.725 75 18

1 1751.729 94 16

1 1752.682 109 18

1 1914.096 251 20

1 1928.092 230 17

1 1946.039 239 15

1 1976.046 176 18

1 1998.971 184 17

1 1999.933 163 17

1 2000.978 125 21

1 2033.907 135 17

1 2040.850 131 15

1 2056.871 135 20

1 2103.736 69 19

1 2121.686 88 16

1 2157.646 42 20

2 2316.050 -42 22

2 2448.778 -153 18

2 2449.751 -137 18

106

TABLE 6.6 - HET relative radial velocities for HD 33636.

HJD - 2450000 RV (m/s) ±σ (m/s)

3633.938 85.6 3.2

3646.908 64.7 3.3

3653.901 74.5 3.5

3663.872 65.8 2.9

3666.841 66.9 3.2

3668.833 62.5 3.4

3676.839 61.8 3.5

3678.810 58.0 2.9

3680.805 56.1 3.0

3682.797 55.4 3.2

3683.812 48.8 3.3

3689.922 44.3 3.2

3691.790 51.9 3.0

3692.789 47.8 2.9

3696.771 47.4 3.0

3697.768 50.3 2.9

3700.760 47.3 2.6

3703.752 43.0 4.0

3708.862 43.2 3.3

3709.879 40.5 3.4

3713.724 48.2 3.9

3714.870 43.7 3.9

3719.698 41.3 4.1

3719.844 43.1 3.9

3724.819 41.2 3.4

3724.822 39.7 3.6

3730.667 31.5 3.7

3731.674 40.8 3.5

3732.665 41.4 3.5

3738.661 38.7 3.1

3739.646 32.5 3.4

3746.623 37.7 4.7

3748.633 30.4 3.7

3751.752 25.9 3.4

3753.748 28.9 3.7

3754.612 28.6 3.7

3755.602 28.7 3.6

3757.751 24.6 3.9

3762.592 26.2 4.4

3985.980 -21.3 3.4

3987.965 -31.4 2.7

3988.970 -30.3 2.7

3989.969 -31.3 2.7

3990.963 -32.3 2.7

3997.952 -27.1 2.7

4007.922 -33.6 2.8

4008.905 -32.9 3.1

4014.901 -35.0 2.7

4015.906 -33.7 3.3

4018.887 -35.4 3.0

4019.878 -36.6 2.8

4020.875 -38.4 3.0

4021.873 -37.5 3.0

4031.847 -38.2 3.1

4072.738 -45.0 3.4

4073.736 -43.6 3.0

4075.863 -46.1 3.0

4076.728 -44.4 3.1

4079.719 -37.6 3.3

4080.844 -45.4 3.0

4081.859 -48.9 3.2

4105.656 -48.5 4.0

4106.773 -50.6 3.7

4108.781 -50.5 3.9

4109.775 -51.8 3.7

4110.787 -48.8 4.3

4121.610 -51.6 4.1

TABLE 6.7 - Lick relative radial velocities for HD 33636.

HJD - 2450000 RV (m/s) ±σ (m/s)

831.736 -87.6 30.1

1154.792 150.9 11.3

1447.032 101.4 10.5

1607.684 8.5 12.8

1628.629 22.9 12.0

1859.945 -65.5 9.3

1860.907 -64.6 10.2

1913.782 -86.2 6.1

1914.844 -87.5 7.9

1915.801 -82.7 8.0

1945.718 -80.6 5.7

107

TABLE 6.8 - Keck relative radial velocities for HD 33636.

HJD - 2450000 RV (m/s) ±σ (m/s)

838.759 -82.1 4.5

1051.103 38.1 3.9

1073.040 61.9 3.6

1171.845 167.4 3.3

1228.803 200.0 3.9

1412.107 112.1 4.3

1543.900 28.5 4.2

1550.886 19.7 2.9

1580.836 15.2 4.7

1581.868 11.6 4.0

1582.785 10.3 4.1

1793.120 -38.1 4.6

1882.934 -69.1 4.3

1884.085 -67.0 4.0

1898.032 -73.2 3.9

1899.045 -71.6 3.6

1900.065 -63.7 3.7

1901.014 -60.9 3.4

1973.749 -90.4 5.8

2003.746 -83.0 4.1

2188.139 -91.2 4.4

TABLE 6.9 - Elodie relative radial velocities for HD 33636.

HJD - 2450000 RV (m/s) ±σ (m/s)

863.326 112 8

1156.524 327 10

1240.334 387 12

1509.583 235 9

1541.427 226 10

1542.398 240 11

1556.390 220 9

1559.405 210 9

1561.436 201 11

1588.345 206 13

1590.350 211 9

1804.677 159 11

1835.654 130 10

1836.669 135 10

1852.600 139 11

1856.583 141 13

1882.499 130 10

1901.483 125 10

1906.538 130 14

1953.349 110 12

1955.316 115 11

1980.304 104 14

2197.668 81 10

2198.655 84 9

2214.610 105 13

2220.600 100 11

2248.517 89 11

2248.529 80 12

2278.421 76 11

2280.447 60 9

2308.439 76 10

2310.396 67 10

2495.887 72 12

2496.927 63 6

2497.924 61 6

2498.909 61 6

2499.928 67 6

2533.656 57 10

2559.661 43 14

2565.653 75 10

2597.588 77 11

2616.528 35 9

2637.481 67 10

2638.454 52 10

2649.414 70 10

2681.431 34 12

2723.309 58 11

108

6.2 Astrometry Analysis

Figure 6.12 and 6.13 provide sample images of the HD 136118 and HD 33636 ﬁeld-

of-view. The reference stars used for astrometry are indicated.

REF-14

HD136118

REF-16

REF-17

FIGURE 6.12 - HD 136118 ﬁeld-of-view. Source: image from the Science and Engineer-

ing Research Council Survey (SERC), Space Telescope Science Institute

(STSI), digitized with the Plate Densitometer Scanner (PDS). This is su-

perimposed on the Naval Observatory Merged Astrometric Dataset (NO-

MAD) catalog. These were obtained through the Aladin previewer and

the SIMBAD database.

The astrometry analysis consists of solving the overlapping plate model through

Equations 5.8 and 5.9. By setting one plate ﬁxed, the reference frame, we are forcing

the minimization algorithm to ﬁnd the best rotation, scaling and oﬀset constants

that brings all plates to the same reference frame. Additionally to the plate model we

109

HD33636

REF-60

REF-58

REF-57

REF-55

REF-47

FIGURE 6.13 - HD 33636 ﬁeld-of-view. Source: image from the Palomar Observatory Sky

Survey (POSSII), Space Telescope Science Institute (STSI), digitized with

the Plate Densitometer Scanner (PDS). This is superimposed on the Naval

Observatory Merged Astrometric Dataset (NOMAD) catalog. These were

obtained through the Aladin previewer and the SIMBAD database.

also have to solve simultaneously the astrometric model, which is given by Equations

3.37 and 3.38. The latter provides each star’s coordinates in the reference frame

(ξ, η), parallaxes and proper motion components for each reference star, and the

perturbation orbit.

A particular interesting feature of GaussFit, which is important for this problem,

is that it permits the inclusion of constraints in the model. The constraints in

GaussFit can be included in two diﬀerent ways. First by using the in-built function

exportconstraint[u], to ﬁx a given condition ‘u’, or by setting a given variable

parameter as observation with error. The latter may be used whenever there is a

110

previous measurement of the quantity, which may be improved by the addition of

new data. The use of a prior information to obtain a posterior updated solution

resembles the Bayesian methods. For this reason we sometimes call this a quasi-

Bayesian approach. In the sections below we describe the constraints we have used

in order to obtain our global solution.

6.2.0.1 RV constraint

The analysis of astrometry data can not be made independently as we have done

for RV data. This is due to the poor sampling and to the exceeding number of free

parameters in the astrometric model. Nevertheless we can use the RV parameters

in order to estimate the amplitude of the astrometric signal due to the RV-detected

companions to HD 136118 and HD 33636. Equations 3.41 and 3.36 can be combined

to give the following (POURBAIX; JORISSEN, 2000),

abs

(9.192 ×10

−8

)KP

√

1 −e

sin i

, (6.2)

where α is the semimajor axis and π

abs

is the absolute parallax both in units of

milliseconds of arc (mas). K, P , and e are the orbital elements obtained from

RV analysis, where the velocity is in m s

−1

and the period in days. i is the in-

clination angle, which is also unknown. In Figure 6.14 we plot the semimajor

axis (perturbation size) as function of inclination, adopting the parameters ob-

tained from RV analysis only and also the parallaxes from Hipparcos previous

determination, π

abs

[HD 136118] = 19.13 ± 0.85 mas (PERRYMAN et al., 1997) and

abs

[HD 33636] = 34.9 ± 1.3 mas (PERRYMAN et al., 1997).

From Figure 6.14 we notice that HD 136118 b, for inclinations below the limit i  25

◦

(similarly i  155

◦

), would produce a detectable astrometric signal at the limit of

1 mas. We can also notice that even if the nuisance orbit, found in the HD 136118 RV

data is conﬁrmed as a companion to the system, it would produce an undetectable

astrometric signal, therefore we do not expect to conﬁrm this detection with our

FGS astrometry data. Figure 6.14 also shows the perturbation size of HD 33636 b,

which presents a detectable astrometric signal for almost all possible inclinations.

Equation 6.2 provides not only a way to estimate the orbit size, rather it provides an

important constraint between RV and astrometry data. By imposing this equation

111

0.001

0.01

0.1

100

10 20 30 40 50 60 70 80 90

! (mas)

i (degree)

HD 136118 b

HD 136118 c

HD 33636 b

1 mas limit

FIGURE 6.14 - Predicted orbit perturbation size α as function of orbital inclination i

for HD 136118 b (solid line), HD 33636 b (dashed line) and the nuisance

orbit (dotted line). The latter is identiﬁed as a possible component ‘c’ of

HD 136118. The horizontal line represents a 1 mas detection threshold.

we guarantee, for a certain degree, the agreement between RV and astrometry orbital

solutions.

As we notice from Figure 6.14 the orbit contribution is extremely small, and if

detectable, it will lie close to the FGS detection limit. Therefore, we ﬁrst perform

a ﬁt to the FGS data without the orbit, and after we include the orbit with all

possible constraints, so our search will be restricted to a very speciﬁc orbit signal

and therefore it will not result in doubtful solutions. In fact, although our FGS data

may be precise enough to detect such small signals, the observations do not cover

the entire companion’s orbit neither the parallactic orbit. This situation requires the

use of “a priori” knowledge of the parallax through either a previous determination

(e.g. from Hipparcos) or spectro-photometric independent determination.

112

6.2.0.2 Spectro-Photometric Parallaxes

We ﬁrst call attention to the fact that the target’s parallax obtained from relative

astrometry is a relative measure with respect to the reference stars, each of which has

its own parallax. For this reason, in order to obtain absolute, not relative parallax,

we must either apply a correction from relative to absolute parallax or estimate the

absolute parallaxes of reference stars by other means. We have already mentioned

that absolute parallaxes can be determined independently through the photomet-

ric and spectral information of each star, from the deﬁnition of distance modulus.

The so-called spectro-photometric absolute parallax can be calculated through the

following expression,

abs

= 10

−(V −M

−A

+5)/5

, (6.3)

where V is the magnitude in the V -band, A

is the extinction, and M

is the abso-

lute magnitude. In principle, M

can be obtained from the color, spectral type and

luminosity class of each star. Because this is an additional and important constraint

to be used as prior in our models, we have also carried out independent photometric

and spectroscopic observations of FGS reference-frame stars.

The spectra were obtained at the KPNO 4 m Telescope and the photometric data at

the NMSU 1 m telescope in 2006 May. Tables 6.10 and 6.11 summarize spectral and

photometric information for HD 136118 and HD 33636 astrometric reference stars.

Tables 6.10 and 6.11 also show the derived spectro-photometric parallaxes.

TABLE 6.10 - HD 136118: classiﬁcation spectra and photometric information for the as-

trometric reference stars.

Star Sp Ty V B −V M

abs

(mas)

HD 136118 F9V 6.93 0.55 3.34 0.0 19.14 ± 3.8

REF-14 K0V 13.95 0.86 5.88 0.12 2.57 ±0.8

REF-16 G0V 12.46 0.73 4.2 0.45 2.74 ± 0.8

REF-17 K0.5III 13.55 1.13 0.65 0.21 0.29 ± 1.0

The spectro-photometric parallaxes in Tables 6.10 and 6.11 are adopted as priors in

our models unless there is any better determination of these quantities available, as

113

TABLE 6.11 - HD 33636: classiﬁcation spectra and photometric information for the astro-

metric reference stars.

Star Sp Ty V B −V M

abs

(mas)

HD 33636 G0V 7.06 0.58 (4.77) (0.0) 34.85 ± 1.33

REF-47 F6V 15.23 0.64 3.56 0.50 0.572 ± 0.1

REF-55 F6V 14.08 0.59 3.56 0.37 0.933 ± 0.2

REF-57 K6V 15.31 1.25 7.28 0.06 2.590 ± 0.5

REF-58 G2V 13.06 0.61 4.56 0.05 2.035 ± 0.4

REF-60 K3III 9.92 1.30 0.27 0.05 1.207 ±0.2

is the case of our targets, which have Hipparcos measurements (see Section 6.2.0.4).

6.2.0.3 Proper Motion

Proper motions are obtained from the NOMAD catalog (ZACHARIAS et al., 2005) and

the UCAC-3 catalog (ZACHARIAS et al., 2009), for which the values and respective

positions and additional information are listed in Tables 6.12 and 6.13. The proper

motions are also included as observations with errors, but FGS measurements are

more precise than any catalog based on ground-based observations, therefore we

expect to improve these quantities.

TABLE 6.12 - HD 136118, HD 33636, and the astrometric reference stars information from

the NOMAD catalog (ZACHARIAS et al., 2005).

Star NOMAD ID RA(2000) DEC(2000) µ

(mas yr

−1

) µ

(mas yr

−1

) B V R J H K

HD136118 0884-0265533 15:18:55.4718 -01:35:32.591 −124.0 ± 0.8 23.5 ± 0.6 7.432 6.945 6.630 5.934 5.693 5.599

REF-14 0883-0281420 15:18:57.4353 -01:37:40.726 −8.9 ± 5.8 6.3 ± 5.8 14.640 13.970 13.030 12.266 11.918 11.782

REF-16 0884-0265573 15:19:04.3421 -01:35:25.289 −1.3 ± 5.8 −6.5 ± 5.8 13.100 12.630 12.000 10.963 10.654 10.532

REF-17 0883-0281398 15:18:53.6195 -01:36:31.821 −4.9 ± 5.8 −4.8 ± 5.8 14.480 13.640 12.980 11.442 10.896 10.808

HD33636 0944-0053995 05:11:46.449 +04:24:12.74 180.8 ± 1.0 −137.3 ± 0.8 7.559 6.990 6.600 5.931 5.633 5.572

REF-47 0943-0054186 05:11:51.175 +04:23:45.95 10.7 ± 6.3 −13.8 ± 6.3 14.780 14.620 14.720 13.995 13.701 13.595

REF-55 0944-0054026 05:11:51.041 +04:24:13.84 −12.8 ± 6.3 −8.0 ± 6.3 14.340 13.360

REF-57 0944-0054038 05:11:52.954 +04:27:24.00 −5.2 ± 6.3 4.3 ± 6.4 15.700 14.920 14.460 12.910 12.249 12.136

REF-58 0944-0054066 05:11:58.555 +04:26:17.03 −0.8 ± 2.0 −0.7 ± 2.0 13.400 13.000 12.550 11.873 11.553 11.526

REF-60 0944-0054097 05:12:02.307 +04:26:32.02 1.1 ± 1.6 −7.1 ± 1.3 11.335 9.964 9.100 7.650 6.988 6.813

6.2.0.4 Hipparcos Data for the Targets

Perryman et al. (1997) present a catalog with the reduction of Hipparcos data. This

contains parallaxes and proper motion values for both of our targets, HD 136118

and HD 33636. This determination is more accurate than any other work. Leeuwen

(2007) has recently presented a new reduction of Hipparcos data that presents an

overall improvement. The parallaxes and proper motions for our targets from both

114

TABLE 6.13 - HD 136118, HD 33636, and the astrometric reference stars information from

the UCAC-3 catalog (ZACHARIAS et al., 2009).

Star UCAC-3 ID RA(2000) DEC(2000) µ

(mas yr

−1

) µ

(mas yr

−1

) J H K

HD 136118 177-135412 229.7311074 -01.5922470 −125.3 ± 13.2 30.2 ± 13.2 5.935 5.694 5.599

REF-14 177-135414 229.7393086 -01.6279823 −16.4 ± 6.8 6.0 ± 6.9 12.266 11.918 11.782

REF-16 177-135423 229.7680912 -01.5903600 7.4 ± 6.8 13.3 ± 7.1 10.963 10.654 10.532

REF-17 177-135407 229.7234153 -01.6088389 −18.4 ± 6.6 −10.9 ± 6.8 11.442 10.896 10.808

HD 33636 189-021417 077.9435289 +04.4034978 180.0 ± 1.0 −138.0 ± 1.0 5.931 5.633 5.572

REF-47 189-021432 077.9632321 +04.3961034 7.9 ± 9.6 −5.4 ± 9.6 13.995 13.701 13.595

REF-55 189-021431 077.9626727 +04.4038487 16.9 ± 8.5 −11.2 ± 8.5 - - -

REF-57 189-021437 077.9706595 +04.4566664 4.1 ± 10.8 −0.7 ± 9.7 12.910 12.249 12.136

REF-58 189-021464 077.9939842 +04.4380659 −1.0 ± 2.2 −0.3 ± 2.1 11.873 11.553 11.526

REF-60 189-021479 078.0096071 +04.4422239 −0.7 ± 0.7 −9.1 ± 0.9 7.650 6.988 6.813

catalogs are shown in Table 6.14.

TABLE 6.14 - HD 136118 and HD 33636 astrometric information from the Hipparcos cat-

alog (PERRYMAN et al., 1997; LEEUWEN, 2007).

Star Reference π

abs

(mas) µ

(mas yr

−1

) µ

(mas yr

−1

) B −V

HD 136118 Perryman et al. (1997) 19.13 ± 0.85 −124.05 ± 0.89 23.50 ± 0.66 0.553

HD 136118 Leeuwen (2007) 21.47 ± 0.54 −122.69 ± 0.62 23.72 ± 0.46 0.553

HD 33636 Perryman et al. (1997) 34.85 ± 1.33 180.83 ± 1.08 −137.32 ± 0.84 0.588

HD 33636 Leeuwen (2007) 35.25 ± 1.02 179.69 ± 0.92 −138.40 ± 0.63 0.588

6.3 Combining Astrometry, Radial Velocity, and Priors

The framework for obtaining an overall solution simultaneously from RV and as-

trometry data is summarized below.

1. Obtain an independent ﬁt for the RV parameters.

2. Set priors, which comprise the observed spectro-photometric parallaxes or,

if available, any previous determination of parallaxes, and proper motions

from catalogs.

3. Constrain one plate by setting its position as zero, with no scaling and no

rotation.

4. Set the constraint relationship between RV and astrometry orbit parame-

ters (Equation 6.2).

5. Given the astrometric priors one can search for an astrometric independent

solution in order to obtain an initial set of plate constant values. Notice

that the perturbation orbit parameters should not be included in this step.

115

6. Finally, run GaussFit including all these constraints, priors and observed

data, with all parameters free, in order to obtain the best ﬁt model for

astrometry and radial velocity simultaneously. The GaussFit model for

this analysis is available in the Appendix in Chapter 10.

6.3.1 Results

Notice that we have listed two catalogs, NOMAD and UCAC-3, each of which pro-

vides proper motions for the reference stars (Tables 6.12 and 6.13). We have also

listed two Hipparcos reductions (Perryman et al. (1997) and Leeuwen (2007)), which

provide target parallaxes and proper motion (Table 6.14). FGS astrometry data is

more precise than any of these catalog data, however our sampling is not good

enough to obtain an independent value neither for the parallaxes nor the proper

motion. For this reason we use the catalog information as priors in our analysis.

Notice that the proper motion values for the HD 33636 reference stars in the two

catalogs contradict each other. We prefer in this case to leave only the UCAC-3

values.

Figure 6.15 and 6.16 show the reduced astrometric data. The data are shown sepa-

rately for each star plotted together with the resulting parallax and proper motion

models. We also show the proper motion and the parallactic orbit components of

the model separately. The parallaxes, proper motion and statistics of residuals are

shown in Tables 6.15 and 6.16. Figure 6.17 shows the histogram of astrometric resid-

uals in X and Y FGS positions from all reference stars. We also show a Gaussian ﬁt

to each residual distribution, each of which indicates the corresponding FWHM (σ)

and the mode.

116

HD136118 REF-14

REF-17REF-16

-0.01

0.01

0.02

0.03

0.04

0.05

0.06

-0.25 -0.2 -0.15 -0.1 -0.05 0

!!" (arc sec)

!!# (arc sec)

0.005

0.01

0.015

-0.025 -0.02 -0.015 -0.01 -0.005 0

!!" (arc sec)

!!# (arc sec)

-0.01

-0.008

-0.006

-0.004

-0.002

0.002

0.004

0.006

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01

!!" (arc sec)

!!# (arc sec)

-0.025

-0.02

-0.015

-0.01

-0.005

0.005

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005

!!" (arc sec)

!!# (arc sec)

FIGURE 6.15 - Open circles are the reduced astrometry data for HD 136118 (top left

panel) and for the astrometric reference stars. Filled circles show the me-

dian for clumps with 100 data points. Black lines show the ﬁt model for the

apparent path of each star. Each component of this model is also shown

separately: blue lines for the parallax, red lines for the proper motion, and

magenta line for the perturbation orbit.

TABLE 6.15 - Resulting astrometric catalog for HD 136118 from the RV and astrometry

simultaneous ﬁt.

Star ξ (arcsec) η (arcsec ) µ

(mas yr

−1

) µ

(mas yr

−1

) π

abs

(mas)

HD 136118 59.7255 ± 0.0018 659.3267 ± 0.0014 −123.57 ± 0.36 23.59 ± 0.19 19.86 ± 0.59

Ref-14 16.4100 ± 0.0023 783.7946 ± 0.0017 −12.00 ±2.97 7.02 ± 2.21 2.34 ± 0.42

Ref-16 −72.6607 ± 0.0022 638.4372 ± 0.0029 0.87 ± 3.07 −2.96 ± 3.85 2.87 ± 0.37

Ref-17 80.4887 ± 0.0014 721.3350 ± 0.0015 −10.92 ±1.97 −10.55 ± 2.00 6.81 ± 1.52

Statistical Quantities:

Algorithm χ

DOF χ

RMS

LM 313.17 635 0.49 1.07 mas 1.40 mas

Robust 80.38 631 0.13 1.10 mas 1.36 mas

117

HD33636 REF-47

REF-55 REF-57

REF-60REF-58

-0.2

-0.15

-0.1

-0.05

0.05

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

! (arc sec)

" (arc sec)

-0.01

-0.008

-0.006

-0.004

-0.002

0.002

0.004

0.006

-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

! (arc sec)

" (arc sec)

-0.01

-0.005

0.005

-0.01 -0.005 0 0.005 0.01 0.015 0.02

! (arc sec)

" (arc sec)

-0.005

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 0.012

! (arc sec)

" (arc sec)

-0.008

-0.006

-0.004

-0.002

0.002

0.004

-0.004 -0.002 0 0.002 0.004 0.006

! (arc sec)

" (arc sec)

-0.02

-0.015

-0.01

-0.005

0.005

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

! (arc sec)

" (arc sec)

FIGURE 6.16 - Open circles are the reduced astrometry data for HD 33636 (top left panel)

and for the astrometric reference stars. Filled circles show the median for

clumps with 40 data points. Black lines show the ﬁt model for the apparent

path of each star. Each component of this model is also shown separately:

blue lines for the parallax, red lines for the proper motion, and magenta

line for the perturbation orbit.

118

TABLE 6.16 - Resulting astrometric catalog for HD 33636 from the RV and astrometry

simultaneous ﬁt.

Star ξ (arcsec) η (arcsec ) µ

(mas yr

−1

) µ

(mas yr

−1

) π

abs

(mas)

HD 33636 13.7722 ± 0.0041 789.7908 ± 0.0038 180.28 ± 0.35 −137.81 ± 0.30 34.98 ±0.28

Ref-47 −49.7357 ± 0.0009 828.4899 ± 0.0006 11.81 ± 4.40 −1.58 ±4.41 5.70 ± 0.03

Ref-57 −117.0620 ± 0.0009 619.2937 ± 0.0007 3.68 ± 5.23 13.12 ± 8.37 2.60 ± 0.15

Ref-60 −244.5390 ± 0.0004 696.8457 ± 0.0003 −0.70 ± 0.23 −9.04 ± 0.32 1.20 ± 0.06

Ref-55 −53.0216 ± 0.0006 800.7207 ±0.0004 3.64 ± 3.52 −2.74 ± 3.35 9.20 ± 0.06

Ref-58 −186.6731 ± 0.0005 700.8105 ± 0.0004 −0.93 ± 0.92 −1.93 ± 1.51 2.01 ± 0.12

Statistical Quantities:

Algorithm χ

DOF χ

RMS

LM 480.44 711 0.68 1.62 mas 1.19 mas

Robust 61.33 711 0.086 1.67 mas 1.21 mas

0.6

0.4

0.2

4 3 2 1 0-1-2-3-4

!x (mas)

3 2 1 0-1-2-3

!y (mas)

0.6

0.4

0.2

2 1 0-1-2-3

!x (mas)

3 2 1 0-1-2-3

!y (mas)

HD 136118 HD 33636

! = 0.1 mas

= -0.3 mas

! = 0.2 mas

= 0.4 mas

! = 0.2 mas

= -0.6 mas

! = 0.2 mas

= -0.4 mas

FIGURE 6.17 - Histogram of astrometric residual data for FGS X and Y positions for

HD 136118 (left panels) and HD 33636 (right panels).

119

Fig 6.18 shows the reduced star positions ∆ξ and ∆η, versus time for HD 136118.

Although our solutions were obtained considering each data point individually, in

Fig 6.18, in order to provide the reader a better visualization to show how the ﬁt

works, we also plot normal points which are the median and respective standard

deviation of the mean of each clump of data, representing 3 diﬀerent epochs. These

collapsed points are also shown on Fig 6.19, where we plotted ∆ξ versus ∆η and the

apparent orbit ﬁt model.

-0.004

-0.002

0.002

0.004

0.006

3400 3600 3800 4000 4200 4400 4600 4800

!!" (arc sec)

JD - 2450000

-0.003

-0.002

-0.001

0.001

0.002

0.003

3400 3600 3800 4000 4200 4400 4600 4800

!!" (arc sec)

JD - 2450000

FIGURE 6.18 - ∆ξ and ∆η components of HD 136118 perturbation orbit versus time.

For HD 136118 we obtain a semimajor axis of the perturbation orbit α

= 2.20 ±

0.77 mas, an inclination i = 168.5

◦

± 4.1

◦

and a longitude of the ascending node

120

-3

-2

-1

-3 -2 -1 0 1 2 3 4

!" (mas)

!# (mas)

Apparent Orbit of HD136118 A

HD136118 A

Median points

FIGURE 6.19 - Filled circles with error bars are the median and standard deviation of

three groups of astrometric residuals of HD 136118, representing three

diﬀerent epochs. Solid line is the ﬁt model of the apparent perturbation

orbit of HD 136118. Open circles are the positions calculated from the

ﬁt model, each of which is connected by a thick solid line to its respec-

tive observed epoch. The open square shows the predicted position of the

periastron passage.

Ω = 227

◦

± 39

◦

. A summary containing all HD 136118 b and the nuisance orbit

parameters derived from the simultaneous RV and astrometry solution is shown on

Table 6.17. Notice that all RV parameters have been improved. The statistics of RV

residual data is similar to that shown in Table 6.3 and the ﬁnal χ

is the one shown

in Table 6.15.

For HD 33636 we have performed a similar analysis except that we consider a one-

companion model. The results are presented in Table 6.18. The resulting apparent

orbit model and the astrometric residual data are shown in Figure 6.20.

121

-10

-5

-10 -5 0 5 10 15 20

!" (mas)

!# (mas)

Apparent Orbit of HD33636 A

HD33636 A

Median points

FIGURE 6.20 - Filled circles with error bars are the median and standard deviation of

ﬁve groups of astrometric residuals of HD 33636, representing ﬁve diﬀerent

epochs. Solid line is the ﬁt model of the apparent perturbation orbit of

HD 33636. Open circles are the positions calculated from the ﬁt model,

each of which is connected by a thick solid line to its respective observed

epoch. The open square shows the predicted position of the periastron

passage.

122

TABLE 6.17 - HD 136118: best-ﬁt parameters for a two-companion model from the simul-

taneous RV and astrometry data analysis.

HD 136118 b Nuisance orbit

Orbital parameters:

P (days) 1190.8 ±1.2 255.3 ± 1.1

T (JD) 2450610.7 ± 3.5 2453761.117109 ± 4.6

e 0.3534 ±0.0054 0.505 ±0.075

ω (

◦

) 316.45 ±0.57 198 ± 11

K (m s

−1

) 215.83 ± 0.88 11.3 ±1.3

Derived parameters from RV:

M sin i (M

) 12.23 ± 0.46 (3.77%) 0.350 ± 0.045 (12.82%)

M sin i (M



) 0.01167 ± 0.00044 (3.77%) 0.000334 ± 0.000043 (12.82%)

a sin i (AU) 2.35 ± 0.16 (6.81%) 0.85 ±0.17 (20.60%)

Astrometric quantities:

(mas) 2.20 ±0.77 –

i (

◦

) 168.5 ±4.1 –

Ω (

◦

) 227 ±39 –

(AU) 0.111 ± 0.035 –

Derived parameters from Astrometry:

) 63

+22

−13

–



) 0.060

+0.021

−0.012

–

(AU) 2.286

+0.036

−0.040

–

Systemic velocities (m s

−1

Lick1

= −1.5 ± 2.2

Lick2

= −11.5 ± 6.5

HET

= 485.48 ± 0.75

123

TABLE 6.18 - HD 33636: best-ﬁt parameters from the simultaneous RV and astrometry

data analysis.

HD 33636 B

Orbital parameters:

P (days) 2118.2 ±5.1

T (JD) 2451197.5 ± 3.5

e 0.4739 ±0.0040

ω (

◦

) 337.74 ±0.82

K (m s

−1

) 161.71 ± 0.97

Derived parameters from RV:

M sin i (M

) 9.17 ± 0.19 (2.06%)

M sin i (M



) 0.00875 ± 0.00018 (2.06%)

a sin i (AU) 3.23 ± 0.12 (3.68%)

Astrometric quantities:

(mas) 12.9 ±4.2

i (

◦

) 4.3 ± 1.4

Ω (

◦

) 250 ±20

(AU) 0.37 ±0.12

Derived parameters from Astrometry:

) 131

+43

−26



) 0.125

+0.041

−0.025

(AU) 3.007

+0.051

−0.073

Systemic velocities (m s

−1

Lick

= −36.8 ± 1.7

Keck

= −33.13 ± 0.85

HET

= 0.9 ± 1.6

Elodie

= 156.5 ± 1.0

124

6.4 Derivation of the True Mass

By determining the inclination we are able to remove the previous degeneracy on the

mass of the companions. However the true mass should not be calculated simply by

replacing the inclination into the M sin i expression. Instead, one should calculate

the actual mass by iterating Equation 3.42. Moreover one should take into account

the uncertainties from all parameters in Equation 3.42. In order to do this we have

calculated the mass 100,000 times, where the input parameters are chosen by a

normal probability distribution with dispersion given by their uncertainties. This

leads to an asymmetric probability distribution for the mass. Figure 6.21 presents

the Cumulative Distribution Function (CDF)

for the masses of HD 136118 b and

HD 33636 B. The abscissa corresponding to the 0.5 value in the CDF is the median.

The 1-σ uncertainties are obtained from the 34.13% percentile areas. This yields

the following true masses: M

= 63

+22

−13

for HD 136118 b and M

= 131

+43

−26

for

HD 33636 B.

Analogously to the calculation of the mass, the physical semimajor axis may also

be obtained through Equation 3.4. Figure 6.22 shows the CDF for the semimajor

axes of both systems. This yields a

= 2.286

+0.036

−0.040

AU for HD 136118 b and a

3.007

+0.051

−0.073

AU for HD 33636 B.

6.5 Infrared Interferometry for HD 33636

The interpretation of interferometric observations depends strongly on the model

adopted for the analysis. This model can be obtained from prior knowledge of the

system. In our case we expect to be able to test the binary model (see Chapter 3)

for HD 33636 A and its M-dwarf companion, HD 33636 B.

By making use of HD 33636 parameters obtained from RV and astrometry analy-

sis (Table 6.18) we are able to estimate the expected signal in the interferometry

experiment. The squared visibility model for a binary is given by Equation 3.60,

which requires the assumption of a visibility function for each individual target.

Figure 5.3.2 presents the squared visibility model for HD 33636 A, which is modeled

as a disk-like source with diameter Θ

∼ 0.38 mas. According to the evolutionary

model for very-low-mass stars with dusty atmospheres of Chabrier et al. (2000), the

companion HD 33636 B, with mass M

= 0.125

+0.041

−0.025



, has a physical diameter

The CDF is reﬂected vertically for values > 0.5

125

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200 250 300 350 400

CDF

Mass (M

)

FIGURE 6.21 - CDF (reﬂected vertically for values > 0.5) for the masses of HD 136118 b

(red curve) and HD 33636 B (blue curve). Filled in grey presents the region

within which we consider the uncertainty on the mass measurement.

D < 0.3 R



, which implies an apparent angular size of Θ

< 0.1 mas. The latter

produces a negligible signal in the visibility for the spatial frequencies considered in

our experiment. Therefore we use a binary model consisted of an uniform disk for

the central star and a point-like source for the companion.

The parameters involved in the interferometric model are the ﬂux ratio between

components A and B, f = F

, and the binary parameters, ρ and θ, which can

be estimated from our previous orbital results. Figure 6.23 shows the orbit of the

two components in the system. The angular separation between the two bodies is

ρ = 157.22 mas and the position angle is θ = 255

◦

.98. These values are input as

initial guesses to ﬁt the interferometric data.

We performed a ﬁt using GaussFit, which provided the ﬂux ratio f = 0.323 ±0.025,

the binary separation, ρ = 169.52 ± 0.11 mas, and the position angle θ = 269

◦

.03 ±

0.16. The squared visibility data points and the ﬁt model are shown in Figure 6.24.

126

0.1

0.2

0.3

0.4

0.5

2 2.2 2.4 2.6 2.8 3 3.2

CDF

Semimajor axis (AU)

FIGURE 6.22 - CDF (reﬂected vertically for values > 0.5) for the semimajor axes of

HD 136118 b (red curve) and HD 33636 B (blue curve). Filled in grey

presents the region within which we consider the uncertainty on the semi-

major axis measurement.

6.5.1 Interferometric Phase and Closure Phase

Figure 6.25 shows the reduced interferometric phases and Figure 6.26 shows the re-

duced closure phases for HD 33636. One can notice that the phase and closure phase

variations for HD 33636 are consistent with a symmetrical brightness distribution,

i.e. constant phase. The ﬁlled circles show the median and respective deviations for

clumps of 15 points.

127

-150

-100

-50

-150 -100 -50 0 50

!" (mas)

!# (mas)

Apparent Orbit of HD33636 A and B

2008-10-14

$ = 157.22

% = 255.98

HD33636 A

HD33636 B

FIGURE 6.23 - Apparent orbit of HD 33636 A (solid line) and B (dashed line). Filled

circles show the predicted positions for each component in UT 2008-10-

14. The periastron is indicated with open squares.

128

0.5

50 100 150 200 250 300 350 400 450

Spatial Frequency (arcsec

-1

)

Baseline 1

0.5

50 100 150 200 250 300 350 400 450

Spatial Frequency (arcsec

-1

)

Baseline 1

0.5

Squared Visibility

Baseline 2

0.5

Squared Visibility

Baseline 2

0.5

0 50 100 150 200 250 300 350 400 450 500

Baseline 3

0.5

0 50 100 150 200 250 300 350 400 450 500

Baseline 3

FIGURE 6.24 - Squared visibility reduced data (ﬁlled circles) for the three baselines and

the best ﬁt model (solid line). The model consists of a binary with a

disk-like source with size Θ

= 0.38 mas plus a point-like source. The ﬁt

parameters are the ﬂux ratio, f = 0.323 ± 0.025, the binary separation,

ρ = 169.52 ±0.11 mas, and the position angle θ = 269

◦

.03 ± 0.16.

129

-15

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

-15

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

-15

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

FIGURE 6.25 - Open circles are the reduced interferometric phases for HD 33636. Filled

circles show the median taken over clumps containing 15 points.

130

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

-10

-5

50 100 150 200 250 300 350 400 450 500

! (degree)

Spatial Frequency (arcsec

-1

)

FIGURE 6.26 - Open circles are the reduced closure phases for HD 33636. Filled circles

show the median taken over clumps containing 15 points.

131

7 RESULTS AND DISCUSSIONS

7.1 HD 136118

We start with the HD 136118 system, which presents RV-detected companion, con-

ﬁrmed from our HRS/HET RV data. We have also detected an additional signal

in our RV data with amplitude near the limit of our instrument sensitivity. Our

FGS/HST astrometry allowed us to determine the complete set of orbital parame-

ters of the known companion, which permit us to calculate the actual mass of this

companion. This has been identiﬁed as a likely brown dwarf companion.

7.1.1 Activity in HD 136118

The stellar chromospheric activity can produce radial velocity variations. These

could add either noise or periodic signals to the RV data. Observations of Ca II

H and K lines (FISCHER et al., 2002) indicate modest chromospheric activity for

HD 136118, therefore not many spots should be expected. Using the Saar e Donahue

(1997) relationship for the spot radial velocity amplitude versus ﬁlling factor: A

6.5f

0.9

v sin i, where f

is spot ﬁlling factor in percent, v sin i is the projected velocity

in km/s and A

is the spot radial velocity amplitude in m/s. For the radial velocity

amplitude of 216 m/s, and the measured velocity v sin i = 7 ± 0.5 km/s (BUTLER

et al., 2006), we obtain a spot ﬁlling factor of about 6%, i.e. about 60 millimag

variations. As shown in Benedict et al. (1998) and Nelan et al. (2010) the Fine

Guidance Sensor itself is a millimag precision photometer. The variations we see in

HD 136118 over 700 days are of the order of 4 parts per 1000, about 4 millimags, as

shown on Fig 7.1. This implies only small variations in spectral line shapes, which

typically introduces noise of the order of 5 − 10 m/s in the velocities. Notice that

the 255 d signal detected in the HET RV data has amplitude of 11.3 m/s, and its

period is not correlated with any known star cycle.

7.1.2 HD 136118 Spin Axis and the Companion’s Orbit Alignment

Fischer et al. (2002) have provided evidence that the stellar rotation period of

HD 136118 is about 12.2 d. Given a stellar radius of R = 1.58 ± 0.11 R



(PRIETO;

LAMBERT, 1999) we calculate the maximum rotation speed at the stellar equator

max

= 6.5 ± 0.2 km/s. The measured projected velocity v sin i = 7 ± 0.5 km/s is

then consistent with the maximum speed. This suggests a very high inclination of

the spin axis. Therefore, if the whole system follows the same angular momentum

133

-30

-20

-10

53500 53600 53700 53800 53900 54000 54100 54200

! V (milimag)

JD - 2400000

Expected variation for

spot filling factor of 6%

FIGURE 7.1 - FGS-1r photometry of HD136118. Magnitude variation is relative to the

mean magnitude, V=6.93. Dashed lines show the amplitude of variation

possible from a (single) spot ﬁlling factor of 6%, the spot ﬁlling factor

required to produce the observed RV variation from HD136118 b (SAAR;

DONAHUE, 1997).

orientation as that of the star, then the companion’s orbit would be close to an

edge-on orientation with respect to our line of sight.

Conversely, we found that HD 136118 b has an orbital inclination of i = 168.5

◦

4.1

◦

, nearly perpendicular to the inferred inclination of the stellar spin axis. This

misalignment is a puzzling result since conservation of angular momentum would

favor alignment between stellar spin and companion orbital axes, assuming both

were formed in the same primordial cloud.

7.1.3 HD 136118 b: a Brown Dwarf Companion

HD 136118 b is likely a brown dwarf companion orbiting at 2.3 AU that falls in the

so called ‘brown dwarf desert’ (GRETHER; LINEWEAVER, 2006). They showed that

the frequency of companions in the stellar mass range follows a slope with gradient

−9.1 ± 2.9, while in the planetary mass region, the gradient is 24.1 ± 4.7. These

two separate linear ﬁts intersect below the abscissa at M = 43

+14

−23

. HD 136118 b

mass is M

= 63

+22

−13

. Reﬀert e Quirrenbach (2006) measured astrometric masses

134

for two exoplanet candidates HD 38529 c (M = 37

+36

−19

) and HD 168443 c (M =

34 ± 12 M

). Benedict et al. (2010) presented a more accurate measurement for

the mass of HD 38529 c, M = 17.6

+1.5

−1.2

. Both objects are likely brown dwarf

companions around solar type stars like HD 136118 b. These are important cases for

studying the mass function and evolutionary models at the brown dwarf mass range,

as we explain below.

Most of brown dwarfs known to date are those detected through photometric surveys

(DELFOSSE et al., 1999, e.g.). The masses of these objects are determined based on

evolutionary models (BARAFFE et al., 1998) . These objects are also the main source

of data for the construction of mass-functions and evolutionary tracks in the HR

diagram for the brown dwarfs range (BARAFFE et al., 2002). However, there are a

lot of uncertainties in these models, some of which raising, for example, from the

age/mass degeneracy. Brown dwarfs in multiple systems with dynamical masses, like

HD 136118 b, may have their ages calibrated with the age of the primary, assuming

coevality. Therefore, these are important calibrators to improve the determination

of evolutionary tracks. However, due to the high contrast between the companion

and the primary, the luminosity of these companions are usually unknown. We have

described in this thesis a method for this purpose, the infrared interferometry, which

is shown to be a promising technique for the detection of the companion’s ﬂux

signature in high contrast binaries. Below we make use of the current models to

estimate the ﬂux contribution from HD 136118 b.

According to Chabrier et al. (2000), Baraﬀe et al. (2002) evolutionary dusty model

for brown dwarfs, assuming M

= 0.060

+0.021

−0.012



and the age of the brown dwarf

as 5 Gyr, HD 136118 b has a temperature of about T

= 1100 K and a radius of

= 0.084 R



. If one considers the uncertainty in the age of the system (see Table

2.1), this brown dwarf may be much younger, therefore considering the age as 1 Gyr,

HD 13118 b has a temperature of about T

= 1675 K and a radius of R

= 0.094 R



These characteristics classiﬁes HD 136118 b as either a L-dwarf, or if one considers

the inferior mass limit, a T-dwarf. Using these values we calculate the emission and

reﬂection spectra, and the ﬂux ratio between the brown dwarf and the parent star

as show in Fig 7.2. The ﬂux ratio increases toward the far infrared (L, M and N

bands), where it can get as high as 10

−4

135

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

100

0.1 1 10 100

arbitrary units

! (µm)

HD136118

HD136118b (5 Gyr)

Flux ratio (5 Gyr)

HD136118b (1 Gyr)

Flux ratio (1 Gyr)

FIGURE 7.2 - Predicted emission spectrum of HD 136118 (dash-dotted line), emission/re-

ﬂection spectrum of HD 136118 b and the ﬂux ratio between the brown

dwarf and the parent star for a 5 Gyr and 1 Gyr system age, as indicated

in the legend.

7.1.4 HD 136118 c: A Possible New Planet?

We have detected a low-amplitude 255 days signal with orbit shape in the HD 136118

RV data. Such solution suggests the presence of an additional lower mass companion

to the system. If conﬁrmed, this is identiﬁed as a likely planetary companion with

minimum mass M

sin i = 0.350 ± 0.045 M

and orbital distance a sin i = 0.85 ±

0.17 UA. Furthermore, if this companion’s orbit follows the same orientation as that

of the brown dwarf companion, the true mass would be M

∼ 1.75 M

Although it seems evident that this signal comes from an additional massive compo-

nent, we call attention to the fact that this is a very eccentric orbit and considerably

large M sin i planet, which makes us believe that such body could hardly coexist

with a brown dwarf orbiting the same system at the distances they seem to be. In

fact we have performed a dynamical stability analysis using the MERCURY pack-

136

age (CHAMBERS, 1999). We have input the two companions around HD 136118, and

considered the minimum mass, which minimizes interactions. We also explored the

full range of inclinations for the lower-mass component. The system always becomes

unstable for very short time scales.

Besides the stability constraint, this detection is at the limit of our instrument, and

also if we look at Fig 6.11 we note that only a few data points are contributing to

form the orbit we have obtained. For these reasons we prefer to be cautious and take

this as a “nuisance orbit” to ﬁx an unknown source of systematic error present in the

HET data, although the possibility of a second companion is not out of the question.

Further re-reductions and observations are advised to investigate the origin of this

signal.

7.2 HD 33636

7.2.1 RV and Astrometry Results

For HD 33636 we have used more constraints in our analysis than those used in

Bean et al. (2007). This was possible because some of the priors we used have

been published more recently than Bean’s work. For this reason we have obtained

slightly diﬀerent values for the system parameters. These are diﬀerent but consistent

with each other. The mass we obtained for HD 33636 B is M

= 131

+43

−26

, and

the previous determination is M

= 142 ± 11 M

. The disagreement between the

uncertainties is due to the fact that they have not been calculated in the same

fashion. Although our mass is smaller than that obtained by Bean et al. (2007), the

classiﬁcation of this companion remains an M-dwarf star.

7.2.2 Reliability of our Interferometric Results

Notice that the minimum baseline used in our interferometry experiment provides

the measurement at spatial frequency of about 100 arcsec

−1

. This limits the inter-

ferometric ﬁeld-of-view up to ∼ 10 mas. As one can see from Figure 6.23 the binary

separation is ρ ∼ 160 mas, thus larger than the ﬁeld-of-view. For this reason we

consider that the ﬁt shown in Figure 6.24 is not as deﬁnitive, thus we can not trust

the ﬁt parameters obtained from our analysis. In fact the visibility models for these

spatial frequencies are oscillating functions of high frequency, which can easily pro-

duce aliases, reproducing the positions of those data points for basically any phase

adopted in the model. From Equation 3.60 we notice that the angular variable, i.e.

137

the term inside the cosine, is only aﬀected by two parameters, the position angle, θ,

and the binary angular separation, ρ. The conclusion here is that unfortunately the

interferometer baselines of our experiment do not provide an adequate sampling to

provide information about these binary parameters.

However the variations in the observed visibility are clear. From Equation 3.60 we

can also notice that the visibility amplitude is aﬀected only by one parameter, the

ﬂux ratio. Variations in the visibility with this amplitude can only be reproduced

by a binary brightness distribution with ﬂux ratio of order of 30%.

Although we are positive about the detection of an additional light other than the

primary star, we can not draw any conclusion about the morphology of this source

only from the interferometric results. Therefore, we suggest three possible situations

to explain the origin of this signal.

First we recall that even though the interferometer baseline separations do not sam-

ple larger spatial scales in the ﬁeld-of-view, each individual telescope has an 8 m

mirror, each of which is, in principle, able to sample smaller spatial frequencies.

This fact may introduce contributions from the HD 33636 B luminosity to the de-

tected signal.

Another possible explanation could be the presence of an additional light that comes

either from a disk, a ﬁeld-star or an additional inner companion. The disk hypothesis

is supported by the fact that HD 33636 shows an excess emission at 70 µm (BEICH-

MAN et al., 2005), which is an evidence for a debris disk, possibly located at wider

distances. The inner companion hypothesis is contradicted by the fact that radial

velocity data do not present any expressive signal. This would limit the size of the

companion to a very low-mass, thus less luminous source. The ﬁeld-star hypothesis

is a possibility that may be ruled out using future observations. HD 33636 is a high

proper motion star (over 100 mas yr

−1

), thus it would get away very rapidly from

any “polluting” ﬁeld-star.

7.2.3 Flux Ratio

Let us consider that the ﬂux ratio between components A and B is given by the

one measured in Chapter 6 from interferometry, f ∼ 30% . As for comparison

we can calculate the same ﬂux ratio from other means. Figure 7.3 presents the

Baraﬀe et al. (1998) mass-magnitude diagram for low-mass stars, which provides

138

the relationship between mass and absolute magnitude in the K-band. The models

correspond to an age of 5 Gyr, which is consistent with the age of HD 33636 (see Table

2.2). This relationship allows us to obtain the magnitude of the companion from its

dynamical mass, and consequently we are able to calculate the ﬂux ratio. Figure

7.3 also presents the position of HD 33636 B in this diagram, where the absolute

magnitude has been calculated from the interferometric ﬂux ratio.

FIGURE 7.3 - Mass versus absolute magnitude in the K band for low-mass stars

(BARAFFE et al., 1998). The models correspond to an age of 5 Gyr. The

yellow ﬁlled circle shows the position of HD 33636 A. The red ﬁlled cir-

cle shows the predicted position of HD 33636 B from its dynamical mass.

The red open circle shows the position of HD 33636 B considering both the

dynamical mass and the infrared ﬂux ratio obtained from interferometry.

Although the luminosity of the companion determined by each method seems to

disagree, we call attention to the fact that the interferometric experiment has not

been performed in ideal conditions. This suggests that further observations with

more careful consideration of the spatial scales should be done. Below we discuss

the experimental conditions that should provide more deﬁnitive results for HD 33636.

7.2.4 The Spectral Distribution of HD 33636 Components

Figure 7.4 presents the blackbody emission spectra of the two components of the

HD 33636 system. The spectrum for the primary has been calculated from the pa-

139

rameters listed in Table 2.2, and the model spectrum for the companion has been

calculated with the aid of the evolutionary models for low-mass stars of Baraﬀe et al.

(1998), Baraﬀe et al. (2002). We notice from Figure 7.4 that the ﬂux ratio between

components can reach up to only 1%, which is far from the 30% value obtained from

interferometry.

1e-10

1e-08

1e-06

0.0001

0.01

100

0.1 1 10 100

arbitrary units

! (µm)

HD33636 A

HD33636 B

Flux ratio

f = 30 %

FIGURE 7.4 - Blackbody emission spectra of HD 33636 A (dash-dotted line) and

HD 33636 B (solid line). The ﬂux ratio between the two components and

the limit where the ﬂux ratio is 30% is shown in dotted lines.

7.2.5 What would be the Proper Interferometer Set Up to Study

HD 33636?

We saw that our interferometry experiment was not adequate to investigate

HD 33636. However, infrared interferometry is still a promising technique to study

high-contrast binaries, like HD 33636. As an example we cite the Duvert et al. (2010)

work, where they have used AMBER/VLTI to detect the 5-mag fainter companion

of HD 59717 at a distance of 4 stellar radii. They present measurements for the

140

squared visibilities that cover spatial frequencies between 50 and 250 arcsec

−1

. This

is not very diﬀerent from the range we observed. However, the diﬀerence is that for

HD 59717, the ﬁrst interferometric null of the visibility function of the primary star

falls within the observed range of spatial frequencies. This enables one to inspect

small deviations on the visibility function where the relative contribution from the

companion is more expressive, i.e. around the null. The phase of the complex vis-

ibility function of the primary star also changes abruptly from 0 to π at the ﬁrst

null. This change is a step-like function for a single disk-like visibility function (see

Equation 3.54). According to Equation 3.63 the eﬀect of an additional light from

the faint companion is the “smoothing out” of the phase function at the zero visibil-

ity crossing, i.e. around the null. The smaller the ﬂux ratio, the smaller the spatial

frequency range within which this “smoothing” happens. Notice that the alteration

in the phase function is not dictated by the ﬂux ratio, therefore the capacity of de-

tection of a companion’s signature depends only on how ﬁne you can sample spatial

frequencies around the null. We refer the reader to the work of Chelli et al. (2009),

where they explore the detectability of faint companions in a broader context. Below

we apply the concepts developed in Chelli et al. (2009) to provide a brief discussion

on the interferometer set up needed to detect HD 33636 B.

The spatial frequency at the ﬁrst null for a disk-like visibility function given by

Equation 3.54 is

0.61



, (7.1)

where R



is the star radius in angular units. Therefore, for the radius of HD 33636 A

that we have estimated in Chapter 5, Θ = 0.383 mas, the ﬁrst null falls at frequency

∼ 1600 arcsec

−1

. From Equation 3.45 we calculate that the baseline required to

observe frequencies around this value is B = 330λ, where λ is the wavelength in µm.

Thus, for λ = 1 µm one needs a baseline of 330 m to probe the ﬁrst visibility null of

HD 33636 A. The maximum baseline supported by VLTI is only 200 m and the Keck

Interferometer (KI) baseline is 85 m.

Therefore, with these instruments one can not reach spatial frequency scales large

enough to probe the ﬁrst null of the visibility function of HD 33636 A. Let us check

now whether one can sample the smaller frequencies to probe the binary signature.

The binary separation is ρ ∼ 160 mas, therefore one needs to sample the visibility

141

function at spatial frequencies of about 5 arcsec

−1

in order to provide a ﬁeld-of-view

that allows the measurement of the binary pair without alias. This frequency is

reached for a baseline of about 1.3 m. This means that we do not necessarily need

an interferometer to measure the pair. Instead one can use a monolithic instru-

ment with aperture larger than ∼ 2 m. For ground-based observations one would

need an Adaptive Optics (AO) system in order to correct for atmospheric blurring.

The instrument would also need a coronagraph, so that the high contrast could be

minimized.

142

8 SUMMARY AND CONCLUSIONS

Our spectroscopic data provide radial velocities for HD 136118 with precision better

than 5 m/s. These velocities have been combined with archival data to conﬁrm the

presence of the exoplanet candidate companion HD 136118 b. Our analysis revealed

an additional low amplitude 255 d signal with a typical orbital shape. We conclude

that this is not a positive detection of an additional companion to the system, but

we strongly suggest further observations and/or improvement on data reduction

methods to test the reality of this signal. If this companion is conﬁrmed, according

to the orbit ﬁt we obtained the minimum mass M

sin i = 0.350 ± 0.045 M

, thus a

likely planetary companion.

Our FGS/HST astrometry combined with radial velocity data allows the thorough

characterization of the perturbation orbit of HD 136118 due to its known companion

HD 136118 b. This allowed us to determine the actual mass of this object, M

+22

−13

, in contrast to the minimum mass obtained from the radial velocity data

alone, M

sin i ∼ 12 M

. Therefore, given that its mass is above the 13 M

limit, it

is not a planetary but a likely brown dwarf companion. This makes it an important

object for studying the initial mass function for brown dwarfs and low-mass stars,

since it provides the dynamical mass of an object that seems to be residing in

the “brown dwarf desert”. The conﬁrmation of an additional planetary body to

this system would make it an even more interesting case of study. Our long-term

dynamical stability analysis that includes a third body in the system has presented

severe constraints due to the strong interactions between the brown dwarf and the

possible planetary companions.

We have accessed the published RV and FGS/HST data from HD 33636, for which

we have used newer determinations for the astrometric priors in order to perform the

analysis of these data using the same methods as those we used to analyze HD 136118

data. From the HD 33636 results we were able to compare with previously published

results and conclude that our analysis is reliable. The new set of priors provided

slightly diﬀerent values that agree with previous results. This has also permitted us

to improve the proper motion and parallaxes of astrometric reference stars (Tables

6.15 and 6.16).

The new set of parameters for HD 33636 has been used to complement the analysis

of interferometric data. For the interferometry experiment we conclude that our

143

instrumental set up was not adequate to validate the binary parameters obtained

from RV and astrometry analysis, however we have detected visibility variations

that can not be explained by a symmetrical brightness distribution, like a disk

or point-like individual source. Although we have not concluded that this signal

comes from the companion, we estimated a ﬂux ratio of the order of 30% for the

observed variations. This is too high to be explained by the ﬂux of the companion.

We conclude that an ideal interferometer set-up to study HD 33636 would have a

baseline of 330 m, which is not available presently neither in the VLTI nor in the

Keck Interferometer.

The astrometric determination of the masses is deﬁnitive to establish the nature of

low mass companions. It can decisively characterize a companion as a planet. A good

illustration of this can be seen from the results for three objects that were previously

listed as exoplanet candidates: Gliese 876 b, HD 136118 b and HD 33636 b. Each ob-

ject has been found to belong to a diﬀerent class: a giant planet, a brown dwarf

and an M-dwarf star, respectively, Benedict et al. (2002), Martioli et al. (2010) (and

this work), and Bean et al. (2007) (and this work). These results demonstrate the

importance of the application of complementary techniques in observing extrasolar

planetary systems.

The HST observing time is precious and therefore scarce. There are prospec-

tive projects under development, like the Space Interferometric Mission (SIM)

(http://sim.jpl.nasa.gov) and PRIMA (BELLE et al., 2008), that aim to follow-up

nearby targets astrometrically with µarcsec precision. The analysis of these data will

certainly make use of similar methods as those presented in this thesis. Therefore we

conclude that this work is not only important for the analysis of FGS/HST data, but

rather it is an experiment that provides important background for future involve-

ment in worldwide astrometric missions that aim the detection and characterization

of low-mass star, brown dwarfs and exoplanets companions.

Infrared interferometry is a promising technique for following up high contrast bina-

ries with low mass companions. Although we were not able to determine the binary

parameters for HD 33636, interferometry potentially allows one to determine the

position angle and separation of binaries. This presents as a cheaper and faster al-

ternative over long term space astrometry. In ideal cases, interferometry allows one

to obtain the companion’s spectrum. This feature turns interferometry of utmost

importance, since it complements the other two methods with luminous information

144

from the low mass objects in study.

We ﬁnally conclude by saying that these techniques, when combined, may lead

to the determination of accurate masses, orbits and spectra of exoplanets, brown

dwarfs and low-mass star companions. These objects comprise the most informative

material from the bottom of the H-R diagram. Consequently, the implementation of

these techniques, using state-of-the-art instrumentation, will soon provide us with a

better understanding on the nature of these bodies and consequently a construction

of better evolutionary models for the star-planet transition range. Moreover, these

techniques may be improved and hence employed extensively in the characterization

of Neptunes, Super-Earths and Earth-like exoplanets. The Neptune-like exoplanets

are known to be more abundant than those with the size of Jupiter (SUMI et al.,

2010). Therefore, less massive planets tend to be even more abundant, which makes

us believe that in the near future these techniques will be part of astronomers’

quotidian for the exploration of extrasolar planetary systems. Ultimately, this may

soon lead to unearthing planetary formation processes, or to the ﬁrst detection of

an exoplanetary biological activity.

145

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10 APPENDIX

This appendix provides the GaussFit model for the simultaneous analysis of RV and

astrometry data.

10.1 Model for the Simultaneous Analysis of Astrometry and Radial

Velocity HD 136118 Data

parameter P01;

parameter T01;

parameter ecc01;

parameter wrv01;

parameter K01;

parameter P02;

parameter T02;

parameter ecc02;

parameter wrv02;

parameter K02;

parameter a01;

parameter inc01;

parameter omg01;

parameter gamma01;

parameter gamma02;

parameter gamma03;

parameter xi[star], eta[star];

parameter mux[star], muy[star];

parameter par[star];

parameter a[set], b[set], d[set] , e[set];

parameter f[set], c[set];

observation RV;

data jd_rv;

data obstime, P_alpha, P_delta, rollV3, pra, pdec;

observation X, Y;

159

observation muxabs, muyabs, Parabs;

data s tar,set,file;

variabl e u, v, i, pxt, pyt, tc, pix, piy, mx, my, R, OFF, DtoR;

variabl e obstime1 = 0;

variabl e par_asec, mu x_spd, muy_spd;

variabl e lx, ly, xx, yy, ww, xi0, eta0, Rback;

variabl e llcx, llcy, xxfx, xx fy, BmV;

variabl e orbx,o rby;

variabl e obs = 1, mu = 2, pis = 3;

variabl e sinE01, cosE01, ecos01, cosv01, sinv01;

variabl e param01, cosvw01, c oswrv01, sinwrv01, m u01, wwrv01, E01;

variabl e sinE02, cosE02, ecos02, cosv02, sinv02

variabl e param02, cosvw02, c oswrv02, sinwrv02, m u02, wwrv02, E02;

variabl e orbx01,orby01,x_orb01,y_orb01,rvatmcnstr01 ;

variabl e ag amma;

variabl e DegToRad;

variabl e pi = 3.1415926 54;

variabl e vp;

variabl e vorb;

variabl e ct;

variabl e EE;

main(){

DegToRa d = pi/180 .0;

doconstrain ts();

domoreconstrain ts();

while(import())

{

if (file == obs)

{

atmmode l();

}

else i f (file == pis)

{

pimodel();

160

}

else i f (file == mu)

{

mumod();

}

else {

rvmodel();

}

doconstrain ts() {

exportconstraint(a[8]-1);

exportconstraint(b[8]);

exportconstraint(c[8]);

exportconstraint(d[8]);

exportconstraint(e[8]-1);

exportconstraint(f[8]);

}

domoreconstrain ts() {

rvatmcnstr0 1 = (a01*(1.49598e11)*sin(DegToRad*inc 01)/par[3]) -

((P01*86400)*K01*sqrt(1 - ecc01*ecc01)/(2*pi));

exportconstraint(rvatmcnstr 01);

}

atmmode l(){

if (obstime1 == 0)

obstime 1 = obstime;

DtoR = 3.141592654/1 80;

R = Dt oR * (rollV3 - 90.0);

tc = obstime - obstime1;

par_ase c = par*1e-3;

mux_spd = mux*1e-3/365.25;

muy_spd = muy*1e-3/365.25;

pxt = P_alpha * cos(DtoR * pdec) ;

161

pyt = P_delta;

pix = (+pxt*cos(R) + pyt*sin(R)) * par_asec;

piy = (-pxt*sin(R) + pyt*cos(R)) * par_asec;

mx = (+mux_spd*cos(R) + muy_spd*si n(R));

my = (-mux_spd*sin(R) + muy_spd*co s(R));

if(star == 3)

{

x_orb01 = orbitx(obstime, P01, T01, ecc01);

y_orb01 = orbity(obstime, P01, T01, ecc01);

orbx01 = a01*1e-3*(

+ orbfac_xi(x_orb01,y_orb01,inc01,wrv01,omg01)*cos(R)

+ orbfac_eta (x_orb01,y_orb01,inc0 1,wrv01,omg01)*sin(R));

orby01 = a01*1e-3*(

- orbfac_xi(x_orb 01, y_orb01,inc01,wrv 01, omg01)*sin(R)

+ orbfac_eta (x_orb01,y_orb01,inc0 1,wrv01,omg01)*cos(R));

orbx = orbx0 1;

orby = orby0 1;

} else {

orbx = 0;

orby = 0;

}

llcx = 0;

llcy = 0;

xxfx = 0;

xxfy = 0;

xx = X - tc*mx - pix - orbx + llc x - xxfx;

yy = Y - tc*my - piy - orby + llc y - xxfy;

162

Rback = DtoR * (264.17 4900 - rollV3);

lx = xx*cos(Rback) + yy*si n(Rback);

ly = - xx*sin(Rback) + yy*cos(Rback);

xi0 = a*lx + b*ly + c;

eta0 = d*lx + e*ly + f;

u = xi0 - xi;

v = et a0 - eta;

export2(u,v); /* Export equations of c ondition */

}

pimodel() /* model for ref star parallaxes */

{

yy = Parabs - par;

export (yy); /*Export equations of condition*/

}

mumod()

{

ww = muxabs - mux;

yy = muyabs - muy;

export2 (ww, yy); /*Export e quations of condition

}

rvmodel()

{

ct = jd_rv;

make_RVorbi t();

vp = RV - vorb;

export(vp);

}

orbitx(time, period, T_0, ecce) {

variabl e x0, EA,MA;

MA = (2*pi/period)*(time-T _0);

163

EA = kepler(ecce,MA);

x0 = cos(EA) - ecc e;

return x0;

}

orbity(time, period, T_0, ecce) {

variabl e y0, EA, MA;

MA = (2*pi/period)*(time-T _0);

EA = kepler(ecce,MA);

y0 = sin(EA)*s qrt(1. - ecce*ecce);

return y0;

}

orbfac_xi(x_orb, y_orb,i_deg, w_ deg, o_deg) {

variabl e orbfacx;

variabl e i_rad, w_rad, o_rad;

variabl e bb,aa,gg,ff;

i_rad = DegToRad*i_deg;

w_rad = DegToRad*w_deg;

o_rad = DegToRad*o_deg;

bb = + cos(w_rad)*sin(o_rad)

+ sin(w_rad) *cos(o_rad)*cos(i_rad);

aa = + cos(w_rad)*cos(o_rad)

- sin(w_rad)*sin(o_rad)*cos(i_rad);

gg = - sin(w_rad)*sin(o_rad)

+ cos(w_rad) *cos(o_rad)*cos(i_rad);

ff = - sin(w_rad)*cos(o_rad)

- cos(w_rad)*sin(o_rad)*cos(i_rad);

orbfacx = bb*x_orb + gg*y_orb;

return orbfacx;

}

orbfac_eta(x_orb, y_orb,i_deg, w_deg, o_deg) {

variabl e orbfacy;

variabl e i_rad, w_rad, o_rad;

variabl e bb,aa,gg,ff;

i_rad = DegToRad*i_deg;

w_rad = DegToRad*w_deg;

164

o_rad = DegToRad*o_deg;

bb = + cos(w_rad)*sin(o_rad)

+ sin(w_rad) *cos(o_rad)*cos(i_rad);

aa = + cos(w_rad)*cos(o_rad)

- sin(w_rad)*sin(o_rad)*cos(i_rad);

gg = - sin(w_rad)*sin(o_rad)

+ cos(w_rad) *cos(o_rad)*cos(i_rad);

ff = - sin(w_rad)*cos(o_rad)

- cos(w_rad)*sin(o_rad)*cos(i_rad);

orbfacy = aa*x_orb + ff*y_orb;

return orbfacy;

}

make_RVorbi t() {

if (file == 04)

agamma = g amma01;

if (file == 05)

agamma = g amma02;

if (file == 06)

agamma = g amma03;

mu01 = 2*pi/P01;

wwrv01 = wrv01*DegToRad;

coswrv0 1 = cos(wwrv01);

sinwrv0 1 = sin(wwrv01);

param01 = sqrt(1 -ecc01*ecc01 );

kepler(ecc01,mu01*(ct-T01));

E01 = EE;

sinE01 = sin(E01);

cosE01 = cos(E01);

ecos01 = 1 - ecc01 * cosE0 1;

cosv01 = (cosE01 - ecc01)/ecos01;

sinv01 = p aram01*sinE01/ecos01;

cosvw01 = cosv01 * coswrv01 - si nv01 * sinwr v01;

mu02 = 2*pi/P02;

165

wwrv02 = wrv02*DegToRad;

coswrv0 2 = cos(wwrv02);

sinwrv0 2 = sin(wwrv02);

param02 = sqrt(1 -ecc02*ecc02 );

kepler(ecc02,mu02*(ct-T02));

E02 = EE;

sinE02 = sin(E02);

cosE02 = cos(E02);

ecos02 = 1 - ecc02 * cosE0 2;

cosv02 = (cosE02 - ecc02)/ecos02;

sinv02 = p aram02*sinE02/ecos02;

cosvw02 = cosv02 * coswrv02 - si nv02 * sinwr v02;

vorb = agamm a +

+ K01*(ecc01 *coswrv01+co svw01)

+ K02*(ecc02 *coswrv02+co svw02);

}

kepler(ecc,M)

{

variabl e n;

EE = M;

for(n = 0; n < 100; n=n+1)

{

EE = M + ecc*sin(EE);

}

return EE;

}

166

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