Toward Massive Detection of Planets Around M Dwarfs Using the Radial Velocity Technique

Home , 13 Ceti, 14 Ceti, 49 Ceti, Gliese 581b, Gliese 682, HD 107148

TOWARD MASSIVE DETECTION OF PLANETS AROUND M DWARFS USING THE RADIAL VELOCITY TECHNIQUE

JI WANG

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2012 ⃝c 2012 Ji Wang

2 I dedicate it to my parents and my wife

3 ACKNOWLEDGMENTS

I would like to thank my parents, who have given all they have to raise and educate me.

They taught me how to become a good man, a man with skills, faith and integrity. I want to thank my advisor, Dr. Jian Ge, for his valuable advices and continuous support throughout my graduate study at the University of Florida. I learned from him the 3-R standard, i.e., responsive, reliable and responsible. The 3-R standard has become my guidance and will keep beneﬁting me in my future career. He taught me how to become a leader in the ﬁeld of Astronomy with vision and determination. I want to thank Dr. Eric Ford for his advices on academic life. During our collaboration on the eccentricity paper, I have learned the entire process of publishing a paper. It is his patience and meticulousness that set me an example of how to become a responsible advisor and an exceptional researcher. I want to thank Dr.

Xiaoke Wan for his help in my laboratory work. He is a precise and careful experimentalist.

He taught me basics on optics and hands-on experience in the lab. I also thank him for his advices on how to survive as a Chinese in the US.

I thank my wife, He Huang, for her unconditional support for me. If I have to give a reason for her support, that is love, the most amazing thing in the world. It is her encouragement that motivates me to strive for more and makes me to believe that I can do better. It is her sacriﬁcing support at home that lessens my burden as a family member and frees up more of my time that is devoted to research. She is the person that made me start to realize the power of love and the faith that together we can overcome every adversity and achieve one goal and then another in life.

There are numerous teachers that I want to express my gratitudes to. Mr. Zhenlong Ou, the math teacher in my junior school, he taught me to be more aggressive and hardworking when I naively thought that I can success just with my smartness. May he rest in peace! Mrs.

Xiaomei Tang is my Chinese teacher in junior school. I want to thank her for her voluntary tutoring and free dinners when I was preparing the entrance exam for high school. She is more like a mother to me than a teacher who is willing to take care of everything for me. Mr.

4 Zhifang Li is my high school math teacher. I want to thank her for motivating me to be the best student by letting me know that I am not her favorite student. She also taught me that study should not be everything for a person, there are other things that make my life colorful and diversiﬁed such as music, sports and so on. It became more valuable an advice after I was admitted in the University of Science and Technology of China (USTC), where I found myself living a happier life than most of my classmates. I would like to thank my advisors at

USTC, Dr. Tinggui Wang, Dr. Fuzhen Chen and Dr. Xu Kong. They are excellent professors in astronomy research and would be good examples for my upcoming path in astronomy.

The entire ET group led by Dr. Jian Ge has helped me a lot in ﬁnishing my dissertation.

I would like to thank former member Dr. Suvrath Mahadevan, who is now an assistant professor at PennState University. He is always there whenever I need help on M dwarf planet science no matter how busy he is. I thank Dr. Jullian van Eyken for his valuable discussion on DFDI theory. I want to specially thank Dr. Justin Crepp, who is going to start a new chapter of his life at the University of Notre Dame as an assistant professor. I thank him for the valuable experience I gained from our laboratory work for high-contrast imaging of exoplanet. I thank him for the good time we had at UF, Pasadena and Yellowstone.

I also want to thank him for his generous offer when I was looking for a job. I thank Dr.

Scott Fleming, Dr. Nathan De Lee and Dr. Brian Lee for his advices on my projects and observation proposals. I wish all the group members the best in their future researches and lives.

During my six years’ study at the University of Florida, I am so thankful to have many friends accompanied. I thank Pengcheng Guo, aka PC, for all his help when I ﬁrst came and settled down here and the good time playing pool, badminton and basket ball and swimming.

He taught me to be a gentleman and to be less emotional no matter what happens. Dr. Bo

Zhao came here two years ahead of me and treated me like his little brother. I thank Dan Li for being the dedicated ﬁshing partner. Dr. Peng Jiang is an elder alumni from USTC who keeps me motivated and I am always encouraged by him to think deeper into a problem. I

5 thank Liang Chang, aka, Liang ge, for all the time we spent together only trying to kill time.

I thank Dr. Jiwei Xie and his wife for their funny stories. I thank Bo Ma for always being the guy who is made fun of. We would be less happier without your existence. PC, me, Peng,

Jiwei and Bo made the old Astronomy Five, the basketball team that defeated other teams

”effortlessly”. With part of the old team leaving and new blood coming in, the new Astronomy

Five is formed with members being PC, me, Bo, Rui and Shuo. The new Astronomy Five is younger and more sophisticated in game execution and it is on its way to be the ”best” basketball team in the Chinese community within Gainesville area.

Besides my Chinese friends, I would like to thank my international friends (US friends included). I thank Soungchul and Mark for all the lunches we had together. We had such a good time in discussing recent projects, keeping each other motivated and exchanging ideas on topics that interested all of us. I thank Craig for keeping me ﬁt by pushing me to ﬁnish one more uphill stairs and then another in stadium running. I thank Catherine for helping me out of trouble in registration and departmental affairs. I thank Dr. Cullen Blake for valuable discussions and reference letters. I thank Dr. Robert Wittenmyer for his mental support from

Australia, the other end of the Earth. I thank Hali Jakeman for proofreading all my papers and for her help during Stephanie’s qualifying exam.

I would like thank my friends growing up with, Zhi Zeng, Li Zhou and Liqin Huang, for their generosity in treating meals every time I go back to China. I cannot thank enough because I am such a blessed person.

6 TABLE OF CONTENTS page

ACKNOWLEDGMENTS ...... 4

LIST OF TABLES ...... 11

LIST OF FIGURES ...... 12

ABSTRACT ...... 15

CHAPTER

1 BACKGROUND AND MOTIVATION ...... 17

1.1 The Chronicle of Exoplanets Search ...... 17 1.2 Detection Methods ...... 17 1.2.1 The Radial Velocity Technique ...... 17 1.2.2 The Transit Method ...... 18 1.2.3 Direct Imaging ...... 18 1.2.4 The Timing Technique ...... 18 1.2.5 Microlensing ...... 19 1.2.6 Astrometry ...... 19 1.3 Conventional Spectrograph and the Dispersed Fixed Delay Interferometer .. 19 1.4 Major Questions To Be Answered ...... 20 1.5 Planets Around M Dwarfs ...... 24 1.5.1 Essential Facts About M-dwarf ...... 24 1.5.2 M-dwarf Planets ...... 25 1.5.3 Surveys and Results ...... 25

2 FUNDAMENTAL PERFORMANCE OF THE DFDI METHOD ...... 28

2.1 Introduction ...... 28 2.2 Methodology of Calculating Photon-limited RV Measurement Uncertainty ... 33 2.2.1 Photon-limited RV Uncertainty of DE ...... 34 2.2.2 Photon-limited RV Uncertainty of DFDI ...... 35 2.3 Comparison between DE and Optimized DFDI ...... 36 2.3.1 Optimized DFDI ...... 36 2.3.2 Influence of Spectral Resolution R ...... 39 2.3.3 Influence of Detector Pixel Numbers ...... 44 2.3.4 Influence of Multi-Object Observations ...... 47 2.3.5 Influence of Projected Rotational Velocity V sin i ...... 50 2.4 Summary and Discussion ...... 50 2.4.1 Q Factors for DFDI, DE and FTS ...... 50 2.4.2 Application of DFDI ...... 54

7 3 COMPREHENSIVE SIMULATIONS FOR HABITABLE PLANET SEARCH IN THE NIR ...... 56

3.1 Introduction ...... 56 3.2 Simulation Methodology ...... 58 3.2.1 High Resolution Synthetic Spectra ...... 58 3.2.2 RV Calibration Sources ...... 60 3.2.3 Stellar Noise ...... 62 3.2.4 Telluric Lines Contamination ...... 64 3.3 Results ...... 66 3.3.1 RV Calibration Uncertainty ...... 66 3.3.2 Optimal Spectral Band For RV Measurements ...... 67 3.3.2.1 Stellar Spectral Quality ...... 68 3.3.2.2 Stellar Spectral Quality + Stellar Rotation ...... 69 3.3.2.3 Stellar Spectral Quality + RV Calibration Source ...... 72 3.3.2.4 Stellar Spectral Quality + RV Calibration Source + Atmosphere 74 3.3.2.5 Comparisons to Previous Work ...... 76 3.3.3 Current Precision vs. Signal of an Earth-like Planet in Habitable Zone 79 3.3.3.1 Stellar Spectral Quality ...... 80 3.3.3.2 Stellar Spectral Quality + RV Calibration Source + Atmosphere 82 3.3.3.3 Stellar Spectral Quality + RV Calibration Source + Stellar Noise ...... 84 3.4 Summary and Discussion ...... 87

4 PLANET SEARCH AROUND M DWARFS ...... 92

4.1 Introduction ...... 92 4.1.1 Current Status ...... 92 4.1.2 Challenges ...... 92 4.1.2.1 Atmophsere ...... 92 4.1.2.2 Wavelength Calibration Sources ...... 93 4.2 Tackling Adversities in NIR RV Measurement ...... 93 4.2.1 Software Advancement ...... 93 4.2.1.1 Precise Telluric Lines Removal ...... 93 4.2.1.2 Binary Mask Cross Correlation ...... 95 4.2.2 Hardware Advancement ...... 96 4.3 M-dwarf Planet Search and Characterization-Results ...... 99 4.3.1 Telluric Line RV Stability ...... 99 4.3.2 RV Measurements of a Reference Star-GJ 411 ...... 100 4.4 M-dwarf Planet Search and Characterization-Future Works ...... 101 4.4.1 Searching For Planets Around M Dwarfs with EXPERT ...... 101 4.4.2 Multi-Band Study of Radial Velocity Induced by Stellar Activity with EXPERT ...... 104 4.4.3 Mid-Late Type M Dwarf Planet Survey Using FIRST ...... 107 4.4.3.1 Science Justiﬁcation ...... 108 4.4.3.2 Target Selection ...... 109

8 4.4.3.3 Planet Yield Prediction ...... 110

5 ACCURATE GROUP DELAY MEASUREMENT FOR RV INSTRUMENTS USING THE DFDI METHOD ...... 115

5.1 Introduction ...... 115 5.2 GD Measurement Using White Light Combs ...... 119 5.2.1 Method ...... 119 5.2.2 Data Reduction ...... 120 5.2.3 GD Measurement Results ...... 121 5.2.4 GD Measurement Error Analysis ...... 124 5.3 GD Calibration: Observing an RV Reference Star ...... 127 5.3.1 Method ...... 127 5.3.2 GD Calibration Precision ...... 128 5.4 Implementation of Measured GD in Astronomical Observations ...... 128 5.5 Summaries and Discussions ...... 130 5.5.1 Summaries ...... 130 5.5.2 Discussions ...... 130 5.5.2.1 White Light Comb (WLC) Method ...... 130 5.5.2.2 Reference Star (RS) Method ...... 132 5.5.2.3 A Future M-Dwarf Survey With the DFDI Method ...... 132

6 ECCENTRICITY DISTRIBUTION FOR SHORT-PERIOD EXOPLANETS ...... 134

6.1 Introduction ...... 134 6.2 Method ...... 134 6.2.1 Bayesian Orbital Analysis of Individual Planet ...... 135 6.2.2 Γ Analysis of Individual Systems ...... 137 6.3 Results for Individual Planets ...... 140 6.3.1 Comparison: Standard MCMC and References ...... 141 6.3.2 Comparison: Standard MCMC and Γ Analysis ...... 143 6.3.3 Discussion of Γ Analysis ...... 144 6.4 Tidal Interaction Between Star and Planet ...... 147 6.5 Eccentricity Distribution ...... 154 6.6 Discussion ...... 162 6.7 Conclusion ...... 164

7 SUMMARY, CONCLUSION AND CONTRIBUTION ...... 169

7.1 Chapter 2 ...... 169 7.2 Chapter 3 ...... 170 7.3 Chapter 4 ...... 170 7.4 Chapter 5 ...... 171 7.5 Chapter 6 ...... 171

REFERENCES ...... 172

9 BIOGRAPHICAL SKETCH ...... 181

10 LIST OF TABLES Table page

2-1 OPD choice as a function of R and V sin i at different Teﬀ ...... 38

2-2 Power Law Index χ as a function of Spectral Resolution R (Teﬀ = 2400K) ..... 41

2-3 Spectral Resolution and wavelength coverage on a given detector ...... 46

2-4 Q′′ comparison of DFDI and DE as a function of V sin i ...... 50

3-1 Deﬁnition of observational bandpasses ...... 61

3-2 RV uncertainties caused by calibration sources at different spectral resolutions .. 68

3-3 Photon-limited RV uncertainties based on stellar spectral quality at different spectral resolutions for different spectral types ...... 70

3-4 Spectral Type, corresponding Teﬀ, and typical stellar rotation V sin i ...... 72

3-5 Comparison of Q factors from our results to Bouchy et al. (2001) ...... 77

3-6 Comparison of predicted RV precision between our results to Reiners et al. (2010) 78

3-7 Comparison of predicted RV precision between our results to Rodler et al. (2011) . 79

3-8 Required S/N for detection of an Earth-like Planet in the HZ as a function of spectral type ...... 81

3-9 Two examples of telluric contamination ...... 83

3-10 Prediction vs. HARPS observation ...... 84

5-1 GD measurement results as a function of spectrum number (GD(#) = C0 + C1 · # + 2 C2 · # ) and standard deviation (δGD) at different frequencies (ν) ...... 127

5-2 MARVELS predicted RV uncertainty (at an average S/N of 100) vs. Teﬀ ...... 128

5-3 Comparison between two methods of GD measurement and calibration ...... 130

6-1 Comparison of Eccentricities Calculated From Different Methods ...... 148

6-2 Two-sample K-S test result ...... 155

6-3 Bayesian analysis results ...... 160

6-4 Catalog of Short-Period Single-Planet Systems ...... 165

11 LIST OF FIGURES Figure page

1-1 Illustration of conventional spectrograph and the DFDI method ...... 21

1-2 Illustration of Doppler sensitivity for a conventional spectrograph ...... 22

1-3 Illustration of Doppler sensitivity for the DFDI method ...... 23

2-1 DFDI layout diagram ...... 30

2-2 DFDI illustration ...... 30

2-3 Examples of M-dwarf spectra ...... 31

2-4 Power spectrum of a M-dwarf spectrum ...... 32

2-5 Optimal OPD vs. V sin i and R ...... 37

2-6 Q factor gain vs. spectral resolution ...... 40

2-7 Q factor vs. spectral resolution ...... 42

2-8 Improvement of QDFDI over QDE vs. spectral resolution ...... 43

′ ′ 2-9 Comparison of QIRET and QDE at different spectral resolutions ...... 46 ′′ ′′ 2-10 Comparison of QDFDI and QDFDI,R=100,000 at different R ...... 48 ′′ ′′ 2-11 Comparison of QDE and QDE,R=100,000 at different spectral resolutions ...... 49

2-12 QDFDI vs. V sin i ...... 51

3-1 Comparisons between synthetic and observed M-dwarf spectra ...... 59

3-2 RV calibration uncertainties vs. spectral resolutions ...... 67

3-3 RV precision based on spectral quality factor ...... 69

3-4 RV precision based on spectral quality factor and typical stellar rotation ...... 71

3-5 RV precision based on spectral quality factor and RV calibration uncertainties ... 73

3-6 RV precision considering spectral quality factor, RV calibration uncertainties and telluric contamination ...... 75

3-7 The percentage contribution of RV uncertainty induced by telluric contamination .. 77

3-8 RV precisions considering spectral quality factor at a S/N of 425 ...... 81

3-9 RV precision (R=120,000) considering spectral quality factor (S/N=425), RV calibration uncertainties and telluric contamination ...... 83

12 3-10 RV precision considering spectral quality factor (S/N=425), RV calibration uncertainties and stellar noise ...... 85

3-11 RMS error of Keplerian orbit ﬁtting for planets detected by HARPS since 2004 ... 87

4-1 Comparison between two spectra before and after removing telluric lines ...... 94

4-2 Comparison between an observed stellar spectra (GJ 411, telluric lines removed) and a synthetic spectrum ...... 95

4-3 Telluric line removal residual is 2.7% ...... 96

4-4 An example of binary mask template ...... 97

4-5 Application of the sine source as an absorption cell ...... 99

4-6 Application of the sine source as an emission lamp for simultaneous wavelength calibration ...... 100

4-7 Sin source demonstration experiment ...... 101

4-8 Telluric lines RV stability ...... 102

4-9 RV measurements for GJ 411 ...... 103

4-10 RV measured in I band using DEM of EXPERT for KEP 11859158 ...... 106

4-11 RV measured in V band using DEM of EXPERT for KEP 11859158 ...... 107

4-12 V and J band distribution for FIRST survey targets ...... 110

4-13 Teﬀ distribution for FIRST survey targets ...... 111

4-14 Predicted RV measurement precision for the FIRST survey ...... 112

4-15 The predicted survey completeness contours based on observation strategy and RV precision for the pessimistic case ...... 113

4-16 The predicted survey completeness contours based on observation strategy and RV precision for the baseline case ...... 114

5-1 Illustration of the DFDI method ...... 117

5-2 Simulated WLCs of an interferometer ...... 120

5-3 Phase of simulated WLCs ...... 121

5-4 The normalized ﬂux and visibility (γ) as a function of frequency ...... 122

5-5 Top and side view of an individual ﬁber beam feeding of the MARVELS interferometer ...... 123

13 5-6 White light combs phase as a function of frequency ...... 124

5-7 Measured group delay as a function ﬁber number ...... 125

5-8 GD as a function of frequency at different ﬁber numbers ...... 126

5-9 RVs of HIP 14810 (barycentric velocity not corrected) over a period of 70 days ... 129

6-1 Examples of how credible intervals of standard MCMC analysis are calculated using posterior distribution of e ...... 137

6-2 Contours of posterior distribution in h and k space for HD 68988 ...... 139

6-3 Γ¯ as a function of eccentricity e for HD 68988 ...... 141

6-4 Comparison among standard MCMC analysis, Γ andlysis and previous references . 142

6-5 Cumulative distributions functions (CDFs) of eccentricities from different methods . 146

6-6 Distribution of short-period single-planet systems in (e,τage/τcirc ) space ...... 150

6-7 Cumulative distribution function of eccentricity ...... 154

6-8 Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and Rayleigh pdfs ...... 157

6-9 Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and uniform pdf ...... 158

6-10 Cumulative distributions functions (cdf) of eccentricities from different methods ... 159

6-11 Distribution of short-period single-planet systems in period-eccentricity space ... 162

14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

TOWARD MASSIVE DETECTION OF PLANETS AROUND M DWARFS USING THE RADIAL VELOCITY TECHNIQUE

Ji Wang

August 2012

Chair: Jian Ge Major: Astronomy

M dwarfs are the least massive but the most common type of stars in the solar neighborhood. Discoveries of M dwarf planets would lead to a complete understanding of planet formation and evolution around stars of different types. Radial velocity (RV) technique is one of the leading technologies that detect exoplanets and the RV technique favors detection of planet in the habitable zone of an M dwarf. The dispersed ﬁxed delay interferometer (DFDI) method is one branch of the RV technique that uses an interferometer to boost Doppler sensitivity of a spectrograph with a given resolution. I systematically studied the comparison between the DFDI method and the traditional high-resolution spectrograph method (the DE method). My work provides a guidance for future exoplanet survey: 1, a survey of a large sample of stars should adopt the DFDI method, which enables both adequate RV precision and high survey efﬁciency; 2, high precision low-mass exoplanet search should adopt the

DE method with a high resolution spectrograph. I concluded that NIR observation of mid-late type M dwarfs is the most realistic and likely approach in the search for a habitable Earth-like planet. Current M dwarf planet survey in the NIR faces two severe challenges, telluric contamination and lack of a precise wavelength calibration source. I developed the binary mask cross correlation technique and the telluric standard star method in the NIR to eliminate telluric contamination. I also developed an precise and stable wavelength calibration source, i.e., the Sine source, which provides a promising candidate for NIR wavelength calibrator. All these software and hardware developments will pave the way for the next wave of massive

15 detection of M dwarf planets using instruments such as the FIRST. The MARVELS project is multi-object planet survey with the DFDI method. Owing to my work in MARVELS interferometer group delay calibration, over 250 binary and a dozen of brown dwarfs have been discovered and they provide a valuable insight of formation and evolution of low-mass stellar companion and brown dwarf. I have envisioned an M-dwarf planet survey and provided a concept study of such survey.

16 CHAPTER 1 BACKGROUND AND MOTIVATION

1.1 The Chronicle of Exoplanets Search

The search for planets around other stars (i.e., exoplanets) was ﬁrst proposed by Struve

(1952). van de Kamp (1963) claimed a detection an astrometric wobble of Barnard’s star

which is later disputed by Gatewood & Eichhorn (1973). Grifﬁn (1973) demonstrated

∼30 m · s−1 stellar radial velocity measurement precision using telluric lines for wavelength

reference. It was not until 1992 that the ﬁrst exoplanet around a pulsar was detected by

using the pulsar timing technique (Wolszczan & Frail, 1992). The ﬁrst exoplanet around a

main sequence star was 51 Pegasi discovered by Mayor & Queloz (1995). More discoveries

of exoplanets quickly followed (Marcy & Butler, 1996), and the ﬁeld of exoplanet started to

gain tremendous attention since then. As of Feb 2012, there are more than 700 exoplanets

detected by a variety of methods1 .

1.2 Detection Methods

1.2.1 The Radial Velocity Technique

In a star-planet system, a star and a planet orbit around their common center of mass.

The wobble of the star can be detected by measuring the stellar velocity along line of sight,

i.e., the radial velocity (RV). RV is usually measured by monitoring tiny shifts of stellar

absorption lines with a high-resolution spectrograph. Most (more than 70%) of currently

known exoplanets are discovered by the RV technique2 . RV measurements precision

of ∼1 m · s−1 has been routinely obtained (Bouchy et al., 2009; Howard et al., 2010b)

with instruments such as HARPS (Mayor et al., 2003) and HIRES (Vogt et al., 1994). In

comparison, Jupiter causes 12.5 m · s−1 RV wobble while the Earth induces 10 cm · s−1 annual RV variation in our solar system. With a increasing time baseline, exoplanets with

1 http://exoplanet.eu/

2 http://exoplanets.org/

17 semi-major axis of ∼6 AU (orbital period∼5000 day) have been detected (Fischer et al., 2008;

Jones et al., 2010).

1.2.2 The Transit Method

Star brightness drops when an object blocks a portion of star light during a transit

event. A periodical star dimming event may be indicative of an orbiting exoplanet. The transit

method is powerful tool in studying properties of an exoplanet such as radius, absorption

and transmission spectrum of atmosphere and so on. Ground-based transiting observation

has yielded many discoveries including a super-Earth around a low mass star (Charbonneau

et al., 2009). A super-Earth refers to a planet with mass roughly 2-10 times of the Earth.

Space-based transiting planets survey such as CoRot (Baglin, 2003) and Kepler (Borucki

et al., 2011a,c) overcome the atmospheric turbulence-induced noise in precision photometric

measurements. Many interesting exoplanet systems are discovered including super-

Earths (Batalha et al., 2011; Leger´ et al., 2009), habitable planet (Borucki et al., 2012) and

Earth-like planets in a multiple system (Gautier et al., 2011).

1.2.3 Direct Imaging

Exoplanets can be directly imaged by both ground-based and space-based telescopes.

Because of the high brightness contrast between a star and an exoplanet, it is extremely

difﬁcult to directly image an close-in exoplanet. Therefore, most of currently known directly-

imaged exoplanets are more than 10 AU away host stars. The directly-imaged exoplanets

complement exoplanets discovered by the RV technique and with the transit method in the

discovery parameter space.

1.2.4 The Timing Technique

Some astronomical phenomena exhibit stringent repeatability in time domain. The

existence of an exoplanet may be betrayed by a small but detectable perturbation of the

repeatability. For example, Wolszczan & Frail (1992) discovered 3 planets around a pulsar

PSR-1257-12 by measuring the timing variation of its rotation period, Qian et al. (2012)

made a tentative discovery of a Jupiter-like planet around a binary star with the eclipsing

18 binary timing technique. Similar idea was proposed in the measurement of transit timing variation (Ford & Holman, 2007; Holman & Murray, 2005), and this technique has also resulted in several discoveies, e.g., Ballard et al. (2011).

1.2.5 Microlensing

Microlensing is predicted by Einstein with his general theory of relativity. When a field star is aligned with the line of sight between the Earth and a distant star, the brightness of the distant star experiences a sudden boost because of the field star (lensing star). More interestingly, if the lensing star has an orbiting planet, a finer spike due to the planet will be observed together with major spike due to the lensing star in a continuous photometry measurement. The microlensing method is capable of detecting an exoplanet at a great distance that is out of reach for other detection methods, which are typically limited within several hundred parsec away from the Earth. One latest detection by this method is reported by Batista et al. (2011)

1.2.6 Astrometry

Star position is periodically perturbed by a surrounding planet. A precise astrometry measurement of a nearby star would be a good candidate method for detecting an exoplanet.

Like the direct imaging method it complements the RV technique and the transit method in the discovery parameter space for its sensitivity to planets far away from host stars. Unlike the RV and the transit methods, astrometry can independently measure planet mass without the aid from other methods. However, no exoplanet has been detected by this method yet.

1.3 Conventional Spectrograph and the Dispersed Fixed Delay Interferometer

Two major methods exist for the RV technique for exoplanet detection, conventional spectrograph Marcy & Butler (1996); Mayor & Queloz (1995) and the Dispersed Fixed Delay

Interferometer (DFDI) method (Erskine, 2003; Erskine & Ge, 2000; Ge, 2002; Ge et al.,

2002). The RV is obtained by directly measuring the movement of stellar spectral lines along the dispersion direction for a conventional spectrograph (see bottom of Fig. 1-1).

In comparison, the DFDI measures the movement of vertical ﬂux distribution in order to

19 measure RV (see top right of Fig. 1-1). The vertical ﬂux distribution is created by stellar

absorption lines and white light combs generated by a ﬁxed delay interferometer.

Stellar absorption lines are usually very narrow (with a line width of ∼0.1 Å ), a high

resolution spectrograph with R ≥ 50, 000 is usually required to resolve those lines in order

to precisely determined the line movement. Such a high resolution spectrograph requires a

long light path and therefore large and expensive. It is difﬁcult to realize such design under

tight ﬁnancial budget and space constraint for a new equipment. In contrast, the ﬁxed delay

interferometer in a DFDI instrument provides an extra spectral resolving power and enables

an DFDI instrument with low spectral resolution to have a equivalent Doppler sensitivity

with a conventional spectrograph at a high resolution. Therefore, a DFDI instrument can be

made at a lower cost and more compact. What’s more, the multi-object capability of a DFDI

instrument makes it attractive for future large area Doppler planet surveys (Ge, 2002; Wang

et al., 2011).

Erskine (2003) used a simple example to illustrate the advantage of the DFDI method

over a conventional spectrograph. I will brieﬂy introduce the illustration and its main con-

clusion. Reader can refer to his original paper for more details. For a conventional spectro-

graph, as illustrated in Fig. 1-2, an intrinsic absorption line is blurred by the limited spectral

resolution. The reaction function to a small frequency shift due to Doppler shift is the slope

of the blurred profile. According to Erskine (2003), the S/N for a fixed Doppler shift of ∆νD is √ √ 3/2 2n∆νD Hi / Ai (Ao /Ai ) . In contrast, for the DFDI method with a fixed delay interferometer added into the optical path prior to the dispersing element (as illustrated in Fig. 1-3), the

dominant measurable becomes line depth change instead of line centroid movement. The

S/N for a ﬁxed Doppler shift for the DFDI method is, according to Equation 10 in Erskine √ √ 1/2 (2003), n∆νD Hi /2 Ai (Ao /Ai ) . Therefore, the ratio of S/N induced by a ﬁxed Doppler √ shift, (S/N)DE /(S/N)DFDI is 2 2/(Ao /Ai ). For example, Ao is ∼1 Å at 5000 Å at R=5,000,

(S/N)DFDI is a factor of 3.5 higher than (S/N)DE for a ﬁxed Doppler shift.

20 Figure 1-1. Top left: a sample spectrum of the DFDI method. The spectrum is composed of a stellar absorption line and white light combs generated by a ﬁxed delay interferometer. Top right: Flux distribution of vertical direction. Bottom: ﬂux distribution of horizontal direction, i.e., dispersion direction.

1.4 Major Questions To Be Answered

Facing the number and the diversity of current known exoplanets, we cannot help wondering many questions. Among them, two sequential questions may be the most frequent. How common are exoplanets and how common is life on other exoplanets? With continuously advancing technologies, we begin to have a reasonable handling of the ﬁrst question. For the second one, we may not be able to answer until a population of habitable exoplanets has been discovered. However, we start seeing the tip of iceberg after a couple of potential habitable worlds are detected (Borucki et al., 2012; Charbonneau et al., 2009).

21 Figure 1-2. Illustration of Doppler sensitivity for a conventional spectrograph (Fig. 6 in Erskine (2003)). a) Intrinsic absorption line. b) An absorption line after blurring due to limited spectral resolution. c) Reaction function to a small frequency shift due to Doppler effect, i.e., the slope of line proﬁle in b).

Planet occurrence rate is a complicated issue involving many dependences such as

stellar metallicity, planet and stellar mass. There is a well-established planet-metallicity corre-

lation indicating that occurrence rate rises from 3% for [Fe/H]≤0 to 25% for [Fe/H]≥+0.4 (Fis- cher & Valenti, 2005). As measurement precision keeps increasing, a population of exoplan-

ets, such as super-Earths and sub-Neptunes (∼ 2 ≤ M ≤∼ 20M⊕), starts to be probed.

This population together with other planets with lower mass are what is called low-mass ex-

oplanets. The study of planet occurrence has been focused on low-mass exoplanets around

solar-type stars. Scientists using HARPS (Mayor et al., 2003) estimated a 30-50% planet

occurrence for super-Earths (Lovis et al., 2009; Mayor et al., 2009a; Udry, 2010). Another

22 Figure 1-3. Illustration of Doppler sensitivity for the DFDI method (Fig. 6 in Erskine (2003)). a) Intrinsic absorption line. b) Interferometer combs. c) Product of intrinsic absorption line and interferometer combs. d) After blurring, combs are smoothed and become half continuum, ”Bite area” becomes a flux dip. e) Reaction function to a small frequency shift due Doppler effect. Unlike a conventional spectrograph, flux dip as shown in d) is a more dominant measurable than flux change along dispersion direction.

23 RV survey yielded a slightly lower occurrence rate ∼20%, which is reported by Howard

et al. (2010b). The occurrence rate is even lower, i.e., 13±0.8% (Howard et al., 2011) or

19% (Youdin, 2011), according to recently-released Kepler data (Borucki et al., 2011c).

Discrepancies among different surveys spurs speculations and efforts to explain (Wolfgang

& Laughlin, 2011), however, a complete understanding would not be obtained until we fully

understand the bias and completeness of each survey and strive for better measurement

precision in the future.

Johnson et al. (2010a) studied the correlation between planet occurrence and stellar

mass and found a positive correlation characterized as a rise from 3% at 0.5M⊙ to 14% at

2.0M⊙. Occurrence rate of low-mass exoplanets around stars other than solar-type stars is rarely mentioned until very recently. Bonﬁls et al. (2011b) found that super-Earths are

∼ +54 abundant around M dwarfs ( 35%) and the occurrence rate for habitable planets is 41−13% for a M-dwarf sample in their survey. In comparison, Howard et al. (2010b) found this

+16 number to be 23−10% for solar-type stars. The large error bars from their reports indicate the amount of effort required for constraining the low-mass planet occurrence rate as a function of stellar mass.

Other questions are as interesting as, if not more interesting than, those mentioned above. For example, what is the mass distribution of exoplanet? This question helps us to distinguish between planets and other objects such as brown dwarfs and stars and to explain the observed ”brown dwarf desert” (Marcy et al., 2005). What is the eccentricity

distribution of exoplanets and what can be inferred from it? What is the statistics of multiple

planetary systems? All these statistical informations help to constrain and reﬁne theoretical

models (Ida & Lin, 2005; Mordasini et al., 2009) that eventually provide complete and

accurate picture of planet formation and evolution.

1.5 Planets Around M Dwarfs

With hundreds of exoplanets discovered, searching for low-mass exoplanets has been

gaining increasing attention. A planet of given mass and semi-major axis would produce

24 larger RV signal around a M dwarf than around a solar-type star. The fact makes M dwarfs

interesting targets in RV surveys for low-mass exoplanets.

1.5.1 Essential Facts About M-dwarf

In astronomy, most stars are classiﬁed as O, B, A, F, G, K and M according to their

spectral features such as spectral energy distribution and spectral line characteristics. F, G

and K stars are usually referred to as solar-type stars. M stars have the lowest stellar mass

(M≤∼0.45 M⊙) and the lowest effective temperature (Teff ≤∼3700 K) among all classified stars. Because of low Teff and thus low luminosity (L≤∼0.08 L⊙), M stars are usually called

M dwarfs because of their positions on a H-R diagram (lower stellar mass end). M dwarfs spend a very long time on the main sequence, which is even longer compared to the age of the universe. Therefore, M stars on the main sequence have not evolved to an advanced evolutionary stages.

According to the law of black body radiation, M dwarfs are generally faint and emit the bulk of energy in the near infrared (NIR). There are few M dwarfs that can be seen by naked eyes despite the fact that more than 70% solar neighborhood stars are M dwarfs (Henry,

1998).

1.5.2 M-dwarf Planets

RV signal increases with planet mass and inversely with square root of stellar mass and

semi-major axis. The equation is given by Zechmeister et al. (2009): √ ( ) 1/2 G m sin i − m sin i M⊙ AU K = √ = . · 1 · , (1–1) 2 28 4 km s 1 − e (M + m)a MJup M a

where K is RV amplitude, G is the universal gravitational constant, e is eccentricity, m is planet mass, i is orbital inclination with i=90◦ when seen edge-on, M is stellar mass and a is semi-major axis.

As considerable interest has been focused on searching for planets in the habitable zone (HZ), M dwarfs become promising targets for several reasons:

25 • According to Equation 1–1, for give m and a, K is larger because of low stellar mass of a M dwarf

• Because of low luminosity of M dwarfs, HZ, deﬁned as a region around a star where liquid water may exist, is closer in compared to solar-type stars whose stronger radiation pushes HZ further away. For example, HZ around the Sun is ∼1 AU, where the Earth orbit is. In comparison, HZ is typically 0.03-0.4 AU for a M dwarfs (Zechmeister et al., 2009)

• Theoretical work has shown that low-mass planets should be common around M dwarfs (Ida & Lin, 2005)

1.5.3 Surveys and Results

There are several RV surveys targeting M-dwarf planets (Bean et al., 2010; Blake

et al., 2010; Bonﬁls et al., 2011b; Clubb et al., 2009; Endl et al., 2006; Zechmeister et al.,

2009). The results of these surveys indicate that, while gas giants are rare (Endl et al., 2006;

Zechmeister et al., 2009), low-mass planets may be abundant around M dwarfs (Bonﬁls

et al., 2011b). 21 planets around 15 M dwarfs have been detected by the RV technique.

The ﬁrst exoplanet around a M dwarf was GJ 876 b discovered in 1998 by Delfosse

et al. (1998); Marcy et al. (1998). The second planet GJ 876 c around the star was an-

nounced by Marcy et al. (2001), which is another gas giant in resonance with the one

previously discovered. This type of resonance, commonly found for small objects such as

moons and astroids, is known for gas giants for the ﬁrst time. In 2004, another gas giant

HD 41004B b was discovered by (Zucker et al., 2004). The HD 41004 binary star system is

unique because it has a brown dwarf orbiting the faint companion and a planet orbiting the

bright companion. The discoveries of gas giants around low-mass M dwarfs pose challenges

to planet formation and evolution theory and motivate theorists to upgrade and reﬁne models.

The ﬁrst Neptune-size exoplanet was found by Butler et al. (2004) with a minimum mass

m sin i of 21 M⊕. The third planet around GJ 876 d, which is a ∼7.5 M⊕ super-Earth, was

discovered by Rivera et al. (2005) in 2005. With the discovery of the fourth planet, GJ 876

e (Rivera et al., 2010), which is Neptune-sized, GJ 876 system becomes the ﬁrst known

Sun-analog with two outside gas giants shepherding an inside low-mass planet and the

26 outermost one to be a Neptune-size planet. GJ 581 b was found by Bonﬁls et al. (2005) in the same year, which is the prelude for a series of subsequent discoveries of other members c,d (Udry et al., 2007) and e (Mayor et al., 2009b) in the system. Bonﬁls et al. (2007) detected a planet with a minimum mass of 11 M⊕, GJ 674 b. According to their analysis, they found evidence of the existence of planet-metallicity correlation for M dwarfs. In the same year, Johnson et al. (2007a) found the third M-dwarf giant planet with a long period (P=692.9 d). They found giant planet occurrence rate is correlated with host star mass even after correcting for metallicity. Forveille et al. (2009) added one super-Earth in the discovery list after correcting the claim made by Endl et al. (2008). The planet found by Bailey et al. (2009),

GJ 832 b, hold the record for the longest orbital period and the lowest host star metallicity among current census of M-dwarf planets. GJ 1214 b is the first planet around a M dwarf discovered with the transit method and later confirmed with the RV technique (Charbonneau et al., 2009). Haghighipour et al. (2010) found a saturn-mass planet, HIP 57050 b. In 2010, three more giant planet discoveries, HIP 79431 b (Apps et al., 2010), GJ 649 b (Johnson et al., 2010b) and GJ 179 b (Howard et al., 2010a), were announced. Bonfils et al. (2011a) detected another short-period super-Earth GJ 3634 b.

The discovery list will keep being replenished as exoplanet scientists continue to pushing the detection limit for surveys using a variety of methods. For example, 1235 planetary candidates were announced after Kepler releases its ﬁrst 4 months of data (Borucki et al.,

2011c). There are M-dwarf planet candidates waiting to be conﬁrmed by other methods such as the RV technique.

27 CHAPTER 2 FUNDAMENTAL PERFORMANCE OF THE DFDI METHOD

2.1 Introduction

The popular Doppler instruments are based on the cross-dispersed echelle spectrogaph design, which we called the direct echelle (DE) method. In this method, the RV signals are extracted by directly measuring the centroid shift of stellar absorption lines. The fundamental photon-limited RV uncertainty using the DE method has been studied and reported by several research groups (e.g., Bouchy et al. (2001); Butler et al. (1996)). While DE is the

most widely adopted method in precision RV measurements, a totally different RV method

using a dispersed ﬁxed delay interferometer (DFDI) has also demonstrated its capability

in discovering exoplanets (Fleming et al., 2010; Ge et al., 2006b; Lee et al., 2011). In this

method, the RV signals are derived from phase shift of the interference fringes created by

passing stellar absorption spectra through a Michelson type interferometer with ﬁxed optical

path difference (OPD) between the two interferometer arms (Erskine, 2003; Erskine & Ge,

2000; Ge, 2002; Ge et al., 2002). The stellar fringes are separated by a post-disperser, which

is typically a medium-resolution spectrograph. Doppler sensitivity of DFDI can be optimized

by carefully choosing the optical path difference of the interferometer. The DFDI method is

promising for its low cost, compact size and potential for multi-object capability (Ge, 2002).

van Eyken et al. (2010) discussed the theory and application of DFDI in details. However,

its fundamental limit for Doppler measurements has not been well studied before. In this

chapter, I will introduce a method to calculate photon-noise limited Doppler measurement

uncertainty in the near infrared (NIR) wavelength region, where we plan to apply the DFDI

method for launching a Doppler planet survey around M dwarfs.

NIR Doppler planet surveys are very important to address planet characteristics around

low mass stars, especially M dwarfs. M dwarfs emit most of their photons in the NIR region.

Due to the lack of NIR Doppler techniques, only a few hundreds bright M dwarfs have been

searched for exoplanets using optical DE instruments (Blake et al., 2010; Clubb et al., 2009;

28 Endl et al., 2006; Zechmeister et al., 2009). To date, only about 20 exoplanets around M

dwarfs have been discovered compared to more than 700 exoplanets discovered around

solar type stars (i.e., FGK stars) despite of the fact that M dwarfs account for 70% stars in

local universe. Nonetheless, searching for planets around M dwarfs is essential to answer

questions such as the dependence of planetary properties on the spectral type of host

stars. In addition, the smaller stellar mass of M dwarfs favors detection of rocky planets in

habitable zone (HZ) using the RV technique. However, the stellar absorption lines in NIR

are not as sharp as those in the visible band. Recent study by Reiners et al. (2010) shows

that precision RV measurements can only reach better Doppler precision in the NIR than in

visible wavelength for M dwarfs with stellar types later than M4. In this chapter, I will report

results from our study on fundamental limits with the NIR Doppler technique using the DFDI

method.

The theory of DFDI has been discussed by several papers (Erskine, 2003; Ge, 2002;

van Eyken et al., 2010; Wang et al., 2011). Readers may refer to previous references for

more detailed discussion. DFDI is realized by coupling a ﬁxed delay interferometer with a

post-disperser (Fig. 5-1). The resulting fringing spectrum is recorded on a 2-D detector. The formation of the ﬁnal fringing spectrum is illustrated in Fig. 2-2. B(ν, y) is a mathematical

representation of the ﬁnal image formed at the 2-D detector and it is described by the

following equation: [ ] S (ν) B(ν, y) = 0 × IT (ν, y) ⊗ LSF (ν, R), (2–1) hν

where S0(ν) is the intrinsic stellar spectrum and ν is optical frequency. S0 is divided by hν

to convert energy ﬂux into photon ﬂux. IT is the intensity transmission function (Equation

2–2), y is the coordinate along the slit direction which is transverse to dispersion direction, ⊗

represents convolution and LSF is the line spread function of the post-disperser which is a

function of ν and spectral resolution R. In Equation 2–2: γ is visibility for a given frequency

channel, the ratio of half of the peak-valley amplitude and the DC offset, which is determined

by stellar ﬂux S0(ν); c is the speed of light; and τ is the optical path difference (OPD) of the

29 !"#$%

.'($%/$%01$($%

&'(%)'*$+,-"(

,5$*(%08%)59

203(+4"35$%3$%

667

Figure 2-1. A schematic layout of an RV instrument using the DFDI method.

Q !!"#

"#$"$Q # $

%$"$Q # Q

Figure 2-2. DFDI Illustration. S0(ν) is a stellar spectrum; IT (ν, y) is interferometer transmission; B(ν, y) is the image taken at a 2-D detector. ν is optical frequency and y is coordinate of slit direction. DFDI measures RV by monitoring phase shift of stellar absorption line fringes in the y direction (the slit direction).

30 interferometer which is designed to be tilted along the slit direction such that several fringes are formed along each ν channel (Middle, Fig. 2-2). We assume the LSF is a gaussian function (Equation 2–3), ∆ν = ν/R/2.35 because we assume that one resolution element is equal to the FWHM of a spectral line. [ ] 2πντ(ν, y) IT = 1 + γ(ν) · cos , (2–2) c [ ] 1 (ν − ν )2 LSF (ν , ∆ν) = exp − 0 . (2–3) 0 2πν2 2∆ν2

− Figure 2-3. Examples of synthetic M dwarf stellar spectra (V sin i=0 km · s 1), which are generated by PHOENIX (Allard et al., 2001; Hauschildt et al., 1999). The top panel shows the spectra between 0.8-1.35 µm, the bottom panel shows an enlarged spectral region around 1177 nm showing stellar line proﬁles.

Figure 2-3 shows high-resolution (0.005 Å spacing) synthetic spectra of M dwarfs with solar metallicity (Allard et al., 2001; Hauschildt et al., 1999). Teﬀ ranges from 2400K to

31 Figure 2-4. Top: power spectrum of derivatives of stellar spectrum, F [dS0/dν], with SRF (ρ) = F [LSF (ν, R)] for different R overplotted, where LSF is line spread function; Bottom: F [dS0/dν] shifted by ∆ρ = 20mm using a ﬁxed-delay Michelson interferometer. The SRF s for different R are overplotted.

3100K, and log g is 4.5. No rotational broadening is added in the spectrum. Most absorption lines are shallow with FWHMs of several tenths of an Å. Since RV information is embedded in the slope of an absorption line, sharp and deep lines contain more RV information than broad and shallow lines. Mathematically, the slope is the derivative of ﬂux as a function of optical frequency, i.e., dS0/dν. The power spectrum of dS0/dν is obtained by Fourier transform. According to properties of Fourier transform, F [dS0/dν] = (iρ) ·F [S0], where

F manifests Fourier transform, i is the unit of imaginary number and ρ is the representation of ν/c in Fourier space. We plot F ([dS0/dν] in Fig. 2-4 as well as the spectral response function (SRF), which is F [LSF ]. SRF at R = 5, 000 drops drastically toward high spatial

32 frequency (high ρ value) such that it misses most of the RV information contained in stellar

spectrum. As R increases, SRF gradually increases toward high ρ where the bulk of RV

information is stored. A spectrograph with R of ∼100,000 is capable of nearly completely

extracting RV information. Unlike DE, DFDI can shift F [dS0/dν] by an amount determined by

the OPD of the interferometer (Erskine, 2003). For example, Fig. 2-4 also shows the power

spectrum of F ([dS0/dν] of a fringing spectrum obtained with a DFDI instrument with a 20

mm optical delay, which shifts F [dS0/ν] by 20 mm. In this case, RV information has been shifted from the original high spatial frequencies to low spatial frequencies which can be resolved by a spectrograph with a low or medium R in DFDI.

2.2 Methodology of Calculating Photon-limited RV Measurement Uncertainty

In the DE method, an efﬁcient way based on a spectral quality factor (Q) was introduced by Bouchy et al. (2001) to calculate the fundamental uncertainty in the Doppler measure-

ments. The Q factor is a measure of spectral proﬁle information within a given wavelength

region considered for Doppler measurements. Here we develop a similar method to calculate

Q values for the DFDI method. Instead of representing the spetral line proﬁle information in

the DE method, the Q factor in our DFDI method represents stellar fringe proﬁle information.

We use high resolution (0.005 Å spacing) synthetic stellar spectra generated by PHOENIX

code(Allard et al., 2001; Hauschildt et al., 1999) because observed spectra of low mass

stars do not have high enough resolution and broad effective temperature coverage. Reiners

et al. (2010) have conducted several comparisons between synthetic spectra generated by

PHOENIX and the observed spectra. They concluded that the synthetic spectra are accurate

enough for RV measurement uncertainty calculation. We used synthetic stellar spectra of

solar abundance with Teﬀ ranging from 2400K to 3100K (corresponding spectral type from

M9V to M4V) and a surface gravity log g of 4.5. The Q factor is calculated for a series of

10 nm spectral slices from 800 nm to 1350 nm. We artiﬁcially broaden spectra with V sin i

− − from 0 km · s 1 to 10 km · s 1 assuming a limb darkening index of 0.6, which is a typical value

for an M dwarf. We convolve the rotational broadening proﬁle with each spectral slice of

33 10 nm to obtain a rotationally-broadened spectrum. We assume a Gaussian LSF which is

determined by spectral resolution R (Equation 2–3). After artiﬁcial rotational broadening and

LSF convolution, we rebin each spectral slice according to 4.2 pixels per resolution element

(according to the optical design of IRET by Zhao et al. (2010)) to generate the ﬁnal 2D image

on a detector based on which we compute the Q factor.

2.2.1 Photon-limited RV Uncertainty of DE

Bouchy et al. (2001) described a method of calculating the Q factor for the DE method.

We brieﬂy introduce the method here and the reader can refer to Bouchy et al. (2001) for

more details. Let S0(ν) designate an intrinsic stellar spectrum. A0, a digitalized and calibrated spectrum, is considered as a noise-free template spectrum for differential RV measurement, which is related to S0(ν) via the following equation:

S (ν) A (i) = 0 ⊗ LSF (ν), (2–4) 0 hν

where i is pixel number and S0 is divided by hν to convert energy ﬂux into photon ﬂux.

Another spectrum A is taken at a different time with a tiny Doppler shift, which is small relative to the typical line width of an intrinsic stellar absorption. Assuming that the two spectra have the same continuum level, Doppler shift is given by:

δv δν = , (2–5) c ν

where c is speed of light and ν is optical frequency. The overall RV uncertainty for the entire

spectral range is given by (Bouchy et al., 2001):

 − / δ ∑  1 2 vrms −1   1 = Q ·  A0(i) = √ , (2–6) c − i Q Ne

where Q is deﬁned as: ∑  1/2  W (i)    i  Q ≡  ∑  ,   (2–7)  A0(i)  i

34 and W (i) is expressed as: [ ] ∂ 2 A0(i) ν 2 ∂ν(i) (i) W (i) = . (2–8) A(i) The Q factor is independent of photon ﬂux and represents extractable Doppler information

given an intrinsic stellar spectrum and instrument spectral resolution R. According to

Equation 2–6, we can calculate photon-limited RV uncertainty given the Q factor and photon ∑ ﬂux Ne− = A0(i) within the spectral range. i 2.2.2 Photon-limited RV Uncertainty of DFDI

A new method of calculating the Q factor for DFDI is developed and discussed here.

After a digitalization process, a 2-D ﬂux distribution expressed by Equation 2–1 is recorded

on a 2-D detector in DFDI. The digitalization process involves distributing photon ﬂux into

each pixel according to: 1) pixels per resolution element (RE); 2) spectral resolution; 3)

number of fringes along slit. B0(i, j), which is a noise-free template, is then calculated.

B(i, j) is a frame taken at a different time with a tiny Doppler shift. i is the pixel number along

the dispersion direction, and j is the pixel number along the slit direction. The observable

intensity change at a given pixel (i, j) in DFDI is expressed by:

∂B (i, j) B(i, j) − B (i, j) = 0 δν(i) 0 ∂ν(i) ∂B (i, j) δv = 0 · · ν(i). (2–9) ∂ν(i) c

The Doppler shift is measured by monitoring the intensity change at a given pixel in the

equation: δv B i, j − B i, j = ( ) 0( ) ∂ . (2–10) c B0(i,j) · ν ∂ν(i) (i)

Frame B0 is assumed to be a noise-free template and the noise of frame B is the quadratic sum of the photon noise and the detector noise σD : √ = + σ2 Brms (i, j) B(i, j) D . (2–11)

35 √ Equation 2–11 is approximated under photon-limited conditions as Brms (i, j) = B(i, j).

Therefore, the RV uncertainty at pixel (i, j) is given by: √ δv i, j B i, j rms ( ) = ( ) ∂ . (2–12) c B0(i,j) · ν ∂ν(i) (i)

The overall RV uncertainty for the entire spectral range is given by:

 [ ] − / ∑ −2 1 2 δvrms  δvrms (i, j)  =   c  c  i,j   ∑ −1/2   ≡  W (i, j) i,j   −1/2 ∑  ≡ −1 ·   Q  B0(i, j) i,j 1 = √ , (2–13) Q Ne−

where ( )2 ∂ B0(i,j) ν 2 ∂ν(i) (i) W (i, j) ≡ , (2–14) B(i, j) and ∑ W (i, j) [ ]1/2 i,j Q ≡ ∑ . (2–15) B0(i, j) i,j Equation 2–15 calculates the Q factor for the DFDI method, which is also independent of

ﬂux and represents the Doppler information that can be extracted with the DFDI method.

According to Equation 2–13, we can calculate photon-limited RV uncertainty given the Q ∑ factor and photon ﬂux Ne− = B0(i, j) within the spectral range. i,j 2.3 Comparison between DE and Optimized DFDI

2.3.1 Optimized DFDI

Optical Path Difference (OPD) of a ﬁxed delay interferometer is a crucial parameter that

affects the Doppler sensitivity of a DFDI instruments (Ge, 2002). An optimized OPD can

36 help increase the instrument Doppler sensitivity. We calculate the optimal OPD for spectra of

various Teﬀ and V sin i at different spectral resolutions (Table 2-1). We assume a wavelength

range from 800 nm to 1350 nm and an OPD range from 10mm to 41mm with a step size of

1mm in the calculation as described in §2.2.2. Optimal OPD is selected as the one which

results in the highest Q factor value. Increasing V sin i or decreasing R naturally broadens absorption lines, decreasing the coherence length of each stellar absorption line (Ge,

2002). Consequently, our simulations show in general that the optimal OPD decreases with

increasing V sin i or decreasing R values (Fig. 2-5). We also note that Teﬀ inﬂuence on

optimal OPD is not signiﬁcant.

Figure 2-5. Optimal OPD correlation with V sin i (left) and spectral resolution R (right). Optimal OPD for R=20,000 (solid), 50,000 (dotted), 80,000 (dashed) are used on the left panel. V sin i=2 (solid), 5 (dotted), 10 (dashed) km · s−1 are assumed on the right panel. Complete results of optimal OPD can be found in Table 2-1. Teff influence on optimal OPD is not significant, Teff=2800 K is adopted in the plot.

37 Table 2-1. OPD choice as a function of R and V sin i at different Teﬀ − R V sin i [km · s 1] 0 1 2 3 4 5 6 7 8 9 10 5,000 19,27,29 19,19,28 19,19,19 15,15,17 15,15,15 13,13,13 11,12,12 10,11,11 10,10,10 10,10,10 10,10,10 10,000 21,23,27 19,21,26 19,19,20 17,17,17 15,15,15 13,13,13 11,12,12 11,11,11 10,10,10 10,10,10 10,10,10 15,000 21,23,26 21,22,23 19,20,20 17,17,17 15,15,15 14,14,14 13,13,13 12,11,11 11,11,11 11,10,10 10,10,10 20,000 22,23,26 21,23,24 20,21,21 19,19,19 17,17,17 15,15,15 14,14,14 13,13,13 12,12,12 11,11,11 11,11,11 25,000 23,24,26 23,24,25 21,22,22 19,20,20 19,18,18 17,17,16 16,15,15 14,14,14 14,13,13 14,13,12 12,12,12 30,000 24,26,26 24,25,26 23,23,24 21,21,21 19,19,19 18,17,17 17,16,16 16,15,15 14,14,14 14,14,14 14,14,14 35,000 26,26,28 26,26,26 24,24,25 22,22,23 21,21,20 19,19,19 19,18,17 17,17,17 16,16,16 16,16,16 16,16,15 40,000 27,28,28 27,27,28 26,26,26 24,24,24 22,22,22 21,20,20 19,19,19 19,18,18 19,18,18 19,18,18 19,18,16 45,000 29,29,30 28,28,29 27,27,28 25,25,26 24,23,23 22,22,21 22,20,20 22,20,20 22,20,18 22,18,18 22,18,18 50,000 30,30,31 29,30,30 28,28,28 27,26,26 24,24,24 24,22,23 22,22,22 22,22,22 22,22,22 22,22,22 24,22,22 55,000 32,32,32 31,31,32 30,30,30 27,28,28 27,26,26 24,24,24 24,24,24 24,24,22 24,24,22 24,24,22 24,24,24 60,000 32,32,34 32,32,33 32,31,32 29,28,28 27,27,26 27,26,26 24,24,24 24,24,24 24,24,24 24,24,24 24,24,24

38 65,000 34,34,35 34,34,34 32,32,32 32,30,30 29,28,28 27,27,26 27,26,26 27,24,26 24,24,26 24,24,24 24,24,24 70,000 35,35,36 35,35,36 34,32,34 32,32,32 32,30,30 32,28,28 27,28,28 27,28,26 27,24,26 24,24,26 24,24,26 75,000 37,36,37 37,36,37 35,35,35 32,32,32 32,32,32 32,32,32 32,32,28 32,32,28 32,32,32 32,32,32 32,24,32 80,000 37,37,39 37,37,37 37,36,36 35,34,35 35,32,32 32,32,32 32,32,32 32,32,32 35,32,32 35,32,32 35,32,32 85,000 40,39,41 40,39,39 37,37,37 37,36,36 35,32,32 35,32,32 35,32,32 35,32,32 35,32,32 35,32,32 35,32,32 90,000 40,41,41 40,41,41 40,37,39 37,36,36 37,36,36 37,36,36 35,32,36 35,32,36 37,37,36 37,37,36 37,37,36 95,000 41,41,41 41,41,41 40,41,41 40,37,37 37,37,36 37,37,36 37,37,36 37,37,36 37,37,37 37,37,37 37,37,37 100,000 41,41,41 41,41,41 40,41,41 40,40,41 37,37,37 37,37,36 37,37,36 37,37,37 37,37,37 37,37,37 37,37,37 105,000 41,41,41 41,41,41 41,41,41 40,41,41 40,40,41 37,37,37 37,37,37 37,37,37 37,37,37 37,37,37 37,37,37 110,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 37,37,37 115,000 41,41,41 41,41,41 41,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 120,000 41,41,41 41,41,41 41,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 125,000 41,41,41 41,41,41 41,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 130,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 135,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 37,37,37 140,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,41,41 37,37,41 37,37,37 37,37,37 145,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,41,41 37,41,41 37,41,41 150,000 41,41,41 41,41,41 40,41,41 40,41,41 40,41,41 40,41,41 37,41,41 37,41,41 37,41,41 37,41,41 37,41,41 We investigate how the Q factor is affected if OPD is deviated from the optimal value.

We calculate the Q factor when the actual OPD is deviated from the optimal OPD by 5mm.

We choose the lower value of the two Qs from the 5 mm deviation from the optimal delay

(both positive and negative sides) as Qdeviated. We plot the ratio of Qoptimal and Qdeviated as a function of spectral resolution R in Fig. 2-6. We found that deviating OPD by 5mm does

not result in severe degradation of the Q factor. The maximum degradation is 1.115 and

−1 occurs at R of 5,000 and V sin i of 5 km · s for a star with Teﬀ of 2800 K. The degradation

can be compensated by increasing the integration time by 24% (1.1152) to reach the same

photon-limited Doppler precision according to Equation 2–13. The degradation becomes

smaller as R increases. As shown in Fig. 2-4, DFDI shifts the power spectrum of dS0/dν by an amount determined by the interferometer OPD so that the SRF has a reasonable response at a region where most of the RV information is contained. The SRF broadens as

R increases. Therefore, it can still recover most of the RV information in a stellar spectrum even if OPD is deviated from the optimal value. At low and medium resolutions (5,000 to 20,000), Q factor degradation becomes larger as V sin i increases. This is because rotational broadening removes the high frequency signal from a stellar spectrum, which makes the region containing most of the RV information more sensitive to the choice of OPD as the power spectrum distribution becomes narrower due loss of high ρ component.

2.3.2 Inﬂuence of Spectral Resolution R

In theory, a spectrograph with an inﬁnitely high resolution would be able to extract all

the RV information contained in a stellar spectrum. However, in practice, it is impossible to

completely recover the RV information with a spectrograph with a ﬁnite spectral resolution

whose spectral response function drops at the high spatial frequency end. Although the

power spectrum of the derivative of the stellar spectrum is shifted to the low frequency

region where most of the RV information is carried, the power spectrum is still broad in the

spatial frequency (ρ) domain (see Fig. 2-4). Therefore, high R can help to extract more RV information. In a wavelength coverage from 800 nm to 1350 nm, we calculate Q values for

39 Figure 2-6. Ratio of Qoptimal and Qdeviated as a function of spectral resolution, where Qoptimal is Q factor at optimal OPD and Qdeviated is Q factor when OPD is deviated from Qoptimal by 5 mm. Different line styles represent different V sin i while colors indicate different Teﬀ.

− stellar spectra with V sin i of 0 ,2 ,5 and 10 km · s 1 at different R (5,000 to 150,000 with

a step of 5,000) in order to investigate the dependence of Q on R (Fig. 2-7). We ﬁnd that

more RV information (higher Q factor) can be extracted as R increases. Q factors for DFDI

and DE converge at high R because the spectral response function is wide enough in the ρ domain to cover the region rich in RV information, not affected by the power spectrum shifting involved in DFDI. In addition, the Q factor at a given R increases as Teﬀ drops from 3100K to

2400K, which is largely due to stronger molecular absorption features in the I, Y and J bands

(see Fig. 2-3).

40 Table 2-2. Power Law Index χ as a function of Spectral Resolution R (Teﬀ = 2400K) DFDI DE V sin i 5,000 20,000 50,000 5,000 20,000 50,000 − R [km · s 1] -20,000 -50,000 -150,000 -20,000 -50,000 -150,000

0 0.62 0.59 0.31 1.08 0.93 0.44 2 χ 0.63 0.56 0.27 1.07 0.89 0.38 5 0.62 0.45 0.16 1.01 0.69 0.21 10 0.58 0.28 0.09 0.87 0.39 0.10

We divide R into three regions, low resolution (5,000 to 20,000), medium resolution

(20,000 to 50,000) and high resolution (50,000 to 150,000). We use a power law to ﬁt Q for

both DFDI and DE as a function of R. The power indices χ of three regions for Teﬀ = 2400K − are presented in Table 2-2. At low R region, χ remains roughly a constant for 0 km · s 1

− − ≤ V sin i ≤ 5 km · s 1, but it drops for stars with V sin i of 10 km · s 1 indicating stellar absorption lines begin to be resolved even at low R. At higher R regions, χ decreases as

V sin i increases, a reduced value of χ implies diminishing beneﬁt brought by increasing R.

Stellar absorption lines are broadened by stellar rotation, and they are resolved at a certain

R beyond which increasing R does not signiﬁcantly gain Doppler sensitivity. Overall, χ for

DE is larger than that of DFDI, especially for low and medium R. In other words, QDFDI is less sensitive to a change of R, and the DFDI instrument can extract relatively more Doppler information at low or medium spectral resolution than the DE method. For example, for

−1 0.63 slow rotators (V sin i=2 km · s ) at the low R region (R=5,000-20,000), QDFDI ∝ R . √ Doppler sensitivity δvrms is inversely proportional to two factors: Q and Ne− according to

Equation 2–6 and 2–13, where Ne− is the total photon count collected by the CCD detector.

2 Ne− ∝ (S/N) · Npixel, where S/N is the average signal to noise ratio per pixel, and Npixel is total number of pixels. Note that Ne− ∝ R if the wavelength coverage, S/N per pixel and the

−0.63−0.5 −1.13 resolution sampling are ﬁxed. Therefore, δvrms ∝ R = R for DFDI. In comparison,

−1.57 δvrms ∝ R for DE given the same wavelength coverage and S/N per pixel. The power law is consistent with the previous theoretical work by Ge (2002) and Erskine (2003).

41 Figure 2-7. Q factor as a function of spectral resolution. (left: Teff = 2400K; right: Teff = 3100K. Open circles represent QDFDI; crosses represent QDE; solid lines are best power-law fits for QDFDI; dashed lines are best power-law fits for QDE)

We compare Q factors for both DFDI and DE at given R values, and the results are

− − shown in Fig. 2-8. For very slow rotators (0 km · s 1 ≤ V sin i ≤ 2 km · s 1), the advantage of DFDI over DE is obvious at low and medium R (5,000 to 20,000) because the center of the power spectrum of the derivative of the stellar spectrum is at a high frequency domain which cannot be covered in DE due to the limited frequency response range of its SRF at low and medium R. The improvement of DFDI is ∼3.1 times (R=5,000), ∼2.4 times (R=10,000) and ∼1.7 times (R=20,000) respectively. In other words, optimized DFDI with R of 5,000,

10,000 and 20,000 is equivalent to DE with R of 16,000, 24,000 and 34,000 respectively in terms of Doppler sensitivity for the same wavelength coverage, S/N per pixel and spectral sampling (otherwise, see more discussions in §2.3.3, the gain with the DFDI would be more

42 signiﬁcant for a ﬁxed detector size and exposure time). Overall, DE with the same spectral

resolution as DFDI at R=5,000-20,000 requires ∼3-9 times longer exposure time to reach the same Doppler sensitivity as DFDI if both instruments have the same wavelength coverage and same detection efﬁciency (i.e., Ne− is the same). The improvement of DFDI at R=20,000-

50,000 is not as noticeable as at the low R range. The difference between DFDI and DE

becomes negligible when R is over 100,000. In other words, the advantage of the DFDI over

DE gradually disappears as R reaches high resolution domain (R > 50, 000). In addition, the

− − improvement for relatively faster rotators (5 km · s 1 ≤ V sin i ≤ 10 km · s 1) with DFDI is less signiﬁcant than it is for slow rotators.

Figure 2-8. Improvement of QDFDI over QDE as a function of spectral resolution.

43 2.3.3 Inﬂuence of Detector Pixel Numbers

In the NIR, detector pixel number is typically smaller than the optical detector. Further-

more, the total cost for an NIR array is much higher than an optical detector with the same

pixel number. In the foreseeable future, detector size may be one of the major limitations

for Doppler sensitivity improvement. We study the impact of the limited detector resource

on the Doppler measurement sensitivity. Using the same detector resource, we ﬁnd that it

is fair to compare their Doppler performance for the same target with the same exposure

to understand strength and weakness for each method although DFDI and DE are totally

different Doppler techniques. According to Equation 2–6 and 2–13, we deﬁne a new merit

function, √ ′ Q = Q · Ne− , (2–16) to study photon-limited Doppler performance for both methods with the same detector size.

Note that the newly deﬁned merit function is directly related to photon-limited RV uncertainty, i.e., inverse proportionality. Ne− is calculated by Equation 2–17:

F∗ · η · Stel · texp Ne− = , (2–17) 2.512mJ

in which F∗ is the photon ﬂux in the wavelength coverage region ∆λ of an mJ = 0 star with

−1 −2 the unit of photons · s · cm ; η is instrument total throughput; Stel is the effective surface area

of the telescope; texp is the time of exposure; and mJ is the J band magnitude.

Here we use IRET (Zhao et al., 2010) as an example to illustrate strengths of the DFDI

method for Doppler measurements. IRET adopts the DFDI method and has a wavelength

coverage from 800 nm to 1350 nm and a spectral resolution of 22,000. For a ﬁxed detec-

tor size (i.e., total number of detector pixels) and ﬁxed number of pixels to sample each

resolution element, the total wavelength coverage of a Doppler instrument, ∆λ, is

Npix λc ∆λ = · , (2–18) Porder R · NS

44 where Npix is total number of pixels available on a CCD detector and NS is the number of

pixels per resolution element, λc is the central wavelength and Porder is the number of pixels

sampling each pixel channel between spectral orders1 . Equation 2–18 shows that ∆λ is inversely proportional to R. Table 2-3 gives the relation of R and ∆λ assuming Npixel, Porder

and NS as constants. λc is set to be 1000 nm because it is approximately the center of the

′ ′ Y band. We calculated the ratio of QDFDI and QDE in which we use the photon ﬂux of a star ′ ′ with a Teﬀ of 2400K (Fig. 2-9). QIRET is consistently higher than QDE regardless of R of the DE instrument. In other words, IRET is able to achieve lower photon-limited RV uncertainty

compared to a DE instrument with the same detector. The result seems to be different from

the conclusion we drew in §2.3.2, in which we compare Q factors of the same R and ∆λ and

reached a conclusion that DFDI with an R of 22,000 is equivalent to DE with an R of 35,000

(a factor of 1.6 gain) for the same wavelength coverage and the same detection efﬁciency

(Fig. 2-7). The key difference between this case and the earlier case is the ﬁxed detector

resource instead of ﬁxed total collected photon numbers. Since lower spectral resolution

allows to cover more wavelengths, more photons will be collected for the same instrument

detection efﬁciency for both DFDI and DEM. Note that Q′ consists of two components, Q

and Ne− . For a given number of pixels on the detector, Ne−,DFDI is higher than Ne−,DE due

to the larger wavelength coverage. In addition, QDFDI(∆λDFDI) is more than QDE(∆λDE).

′ ′ Consequently, we see in Fig. 2-9 that QIRET is higher than QDE at all R of a DE instrument. ′ / ′ Figure 2-9 also shows that the minimum of QDFDI QDE is dependent of V sin i. The ratio of ′ ′ Q reaches a minimum (QDE reaches a maximum) around an R of 50,000 for slow rotators − (V sin i ≤ 5 km · s 1). It increases at the low R end because the spectrograph has not yet

1 In principle, each frequency channel for both DFDI and DE instruments can be designed identical. In practice, the DFDI instrument tends to use ∼ 20 pixels to sample fringes in the slit direction, which is only for measurement convenience, not a requirement. In fact, Muir- head et al. (2011) has demonstrated a phase-stepping method which does not require sample fringes in the slit direction.)

45 resolved stellar absorption lines. On the other hand, the ratio increases at the high R end

− because of fewer photons (see Table 2-3). For fast rotators (V sin i=10 km · s 1), the ratio reaches a minimum around R of 30,000.

Table 2-3. Spectral Resolution and wavelength coverage on a given detector

R ∆λ λmin − λmax (nm) (nm) 25,000 480 800−1280 30,000 400 800−1200 40,000 300 850−1150 50,000 240 880−1120 60,000 200 900−1110 70,000 170 910−1080 80,000 150 920−1070

√ ′ ′ ′ = · − Figure 2-9. Comparison of QIRET and QDE at different R. Note that Q Q Ne . Different color represents different rotational velocity.

46 −1 0.63 For a slow rotator (V sin i=2 km · s ) at low R region (R=5,000-20,000), QDFDI ∝ R . √ Since Doppler sensitivity δvrms is inversely proportional to two factors: Q and Ne− according to Equation 2–6 and 2–13, the Doppler sensitivity becomes nearly independent of spectral resolution for the DFDI method (∝ R−0.13) if the detection size (or total number of pixels) is

ﬁxed. This indicates that we can use quite moderate resolution spectrograph to disperse the stellar fringes produced by the interferometer in a DFDI instrument while maintaining high

Doppler sensitivity. This opens a major door for multi-object Doppler measurements using the DFDI method as proposed by Ge (2002). In comparison, the Doppler sensitivity for the

DE method still strongly depends on spectral resolution for a ﬁxed number of detector pixels

(∝ R−0.57), indicating that higher spectral resolution will offer better Doppler sensitivity.

2.3.4 Inﬂuence of Multi-Object Observations

As discussed in §2.3.2 and 2.3.3, the DFDI instrument can be designed to have a moderate resolution spectrograph coupled with a Michelson type interferometer. Moderate spectral resolution allows a single order spectrum or a few order spectra to cover a broad wavelength region in the NIR region while keeping the Doppler sensivitiy similar to a high resolution DE design which requires a large detector array to cover spectra from a single target. This indicates that the DFDI method has much greater potential for accommodating multiple targets on the same detector as proposed by Ge (2002) than a DE instrument. In order to evaluate the potential impact of multi-object DFDI instruments, we redeﬁne the merit function Q′′ as: √ ′′ α = · − · Q Q Ne Nobj , (2–19) where Nobj is the number of objects that can be monitored simultaneously, and α is the index of importance for multi-object observations. From the perspective of photon count and S/N, multi-object observations are equivalent to an increase of Ne− , and thus α is 0.5. However,

′′ from an observational efﬁciency point of view, Q should be proportional to Nobj because the more objects are observed simultaneously, the quicker the survey is accomplished, and α is therefore equal to 1.

47 Q′′ Q′′ R Figure 2-10. Comparison√ of DFDI and DFDI,R=100,000 at different . Note that ′′ α = · − · Q Q Ne Nobj . The maximum of each curve is indicated by ﬁlled circle. Different color represents different rotational velocity, the same as Fig. 2-9.

We assume a detector that covers from 800 nm to 1350 nm at R=100,000 so that we can use the Q factors obtained in §2.3.2. Ne− is a constant since we assume identical ∆λ.

Nobj is inversely proportional to the number of pixels per object which is proportional to spectral resolution R (Equation 2–18). Note that we do not require Nobj to be an integer because we can, in principle, ﬁt a fraction of spectrum on a detector to make full use of the detector. Q′′s for both DFDI and DE are calculated. Figure 2-10 shows the ratio of Q′′ and ′′ α α QR=100,000 for DFDI under two different assumptions of . For =0.5, i.e., increase of Nobj is equivalent to photon gain, only a slight improvement is achieved if the detector is used for multi-object observations at lower resolution than 100,000. In comparison, from a survey efﬁciency point of view (i.e., α=1), we see a factor of ∼4-6 times boost of Q′′ in multi-object

48 Q′′ Q′′ R Figure 2-11. Comparison√ of DE and DE,R=100,000 at different . Note that ′′ α = · − · Q Q Ne Nobj . The maximum of each curve is indicated by ﬁlled circle. Different color represents different rotational velocity, the same as Fig. 2-9. observations. The truncation at R=5,000 is due to a practical reason that a lower resolution than 5,000 is rarely used in planet survey using RV techniques. On the other hand, similar calculation is also conducted for the DE method (Fig. 2-11), in which we ﬁnd that high resolution single object observation is an optimal operation mode for DE from a perspective of photon gain (α=0.5). At α=1, the increase of Q′′ is a factor of ∼3 at the most.

We compare the maximum of Q′′ for both DFDI and DE at different V sin i in Table 2-4.

At α=0.5, the advantage of DFDI over DE is ∼1.1 for a wide range of V sin i. In other words, from the photon gain point of view, there is no signiﬁcant difference between DFDI and DE in multi-object RV instruments. However, from the survey efﬁciency point of view (α=1), we see a factor of 3 boost of Q′′ in DFDI for slow rotators (V sin i ≤2km · s−1), suggesting 9

49 Table 2-4. Q′′ comparison of DFDI and DE as a function of V sin i DFDI DE · −1 ′′ ′′ ′′ ′′ V sin i [km s ] Roptimal QDFDI Roptimal QDE QDFDI/QDE 0 50,000 7502 110,000 6623 1.065 2 50,000 6806 75,000 6001 1.134 α=0.5 5 25,000 4979 50,000 4424 1.125 10 15,000 3394 25,000 2996 1.133

0 5,000 26384 30,000 8705 3.031 2 5,000 24297 25,000 8490 2.862 α=1.0 5 5,000 18929 15,000 7884 2.401 10 5,000 12877 10,000 7239 1.779

times faster in terms of survey speed. For fast rotators (i.e., V sin i=10km · s−1), the boost

drops to 1.78. Our study conﬁrms that the DFDI method has an advantage for multi-object

RV measurements over the DE method as suggested by Ge (2002).

2.3.5 Inﬂuence of Projected Rotational Velocity V sin i

Projected rotational velocity V sin i broadens stellar absorption lines and thus reduces

− the Q factor. We carry out simulations calculating Q factors of different V sin i(0 km · s 1 ≤

−1 V sin i ≤10 km · s ) at R=150,000 and various Teﬀ. We assume a wavelength range from

800 nm to 1350 nm. The results are shown in Fig. 2-12. The Q factor decreases as V sin i

increases. It is clear that slow rotators would be better targets to reach higher photon-limited

RV precision because the spectrum of a slow rotator contains more Doppler information.

2.4 Summary and Discussion

2.4.1 Q Factors for DFDI, DE and FTS

We develop a new method of calculating photon-limited Doppler sensitivity of an

instrument adopting the DFDI method. We conduct a series of simulations based on

high resolution synthetic stellar spectra generated by PHOENIX code(Allard et al., 2001;

Hauschildt et al., 1999). In simulations, we investigate the correlations of Q and other

parameters such as OPD of the interferometer, spectral resolution R and stellar projected

rotational velocity V sin i. We ﬁnd that optimal OPD increases with increasing R and

decreasing V sin i. Empirically, the optimal OPD is chosen such that the density of the

50 Figure 2-12. QDFDI as a function of V sin i. interference combs matches with the line density of the stellar spectrum. Based on the simulation results, the optimal OPD is determined as the one that maximizes the Q factor. In fact, optimal OPDs found from empirical way and from numerical simulation are consistent with each other. For example, for V sin i=0 km·s−1 and R=50,000, simulation gives an optimal

OPD of 30 mm. The interference comb density of an interferometer with OPD of 30 mm is

∼0.3 Å at 1000 nm, which indeed matches the width of a typical absorption line after spectral blurring with R of 50,000. An independent method to calculate photon-noise limited Doppler measurement uncertainty in the optical is being developed, and the results will be reported in a separate paper (Jiang et al., 2011). We have compared results from both methods and conﬁrmed that both independent methods produce essentially the same results for both optical and NIR Doppler measurements.

51 We investigate how the Q factor is affected if OPD is deviated from the optimal value and ﬁnd that a deviated OPD (5mm) does not result in a signiﬁcant Q factor degradation,

which is mitigated as R increases. We ﬁnd that the Q factor increases with R for both DFDI

and DE, and eventually converge at very high R (R ≥100,000). The convergence of DFDI

and DE methods is a natural consequence because the measurement method does not

make a difference after the spectral resolution becomes extremely high. In addition, Q

factors at a given R increase as Teﬀ drops from 3100K to 2400K, which is due to stronger

molecular absorption features in NIR (see Fig. 2-3). The Q factor decreases as V sin i

increases because stellar rotation broadens the absorption lines, leading to less sensitive

measurement.

− We compare Q factors for both DFDI and DE at a given R. For slow rotators (0 km · s 1 ≤

− V sin i ≤ 2 km · s 1), DFDI is more advantageous over DE at low and medium R (5,000 to

20,000) for the same wavelength coverage ∆λ. The improvement of DFDI compared to DE is ∼3.1 (R=5,000), ∼2.4 (R=10,000) and ∼1.7 (R=20,000), respectively. In other words, optimized DFDI with R of 5,000, 10,000 and 20,000 are equivalent in Doppler sensitivity to DE with R of 16,000, 24,000 and 34,000, respectively. The improvement of DFDI at R 20,000 to 50,000 is not as noticeable as at low R range. The difference between

DFDI and DE becomes negligible when R is over 100,000. For relatively faster rotators

− − (5 km · s 1 ≤ V sin i ≤ 10 km · s 1), the improvement with DFDI is less obvious than it is for very slow rotators. DFDI has strength when the spectral lines in a stellar spectrum are not resolved by a spectrograph, which is the case for low and medium resolution spectrograph.

Under such conditions, the ﬁxed delay interferometer provides additional resolving powers for the system. After the lines are fully resolved by the spectrograph itself, the interferometer in the system becomes dispensable, which is the reason why we see the convergence of DFDI and DE at very high spectral resolution.

Fundamental performance of a Fourier-transform spectrometer (FTS) in the application of Doppler measurements has been discussed by Maillard (1996). There are similarities

52 between the FTS and the DFDI method, for example: 1, both methods use the interferom-

eter as a ﬁne spectral resolving element; 2, RV is measured by monitoring the temporal

phase change at a ﬁxed OPD of the interferometer. In DFDI method, OPD is scanned

in each frequency channel because of two relatively tilted mirrors, and the resolution of

the post-disperser in DFDI is chosen to ensure a reasonable fringe visibility. Therefore,

the DFDI method is a extended version of the FTS method with a low-medium resolution

post-disperser. However, one major difference between these two methods is that the in-

terferometer itself is used as a spectrometer by OPD scanning in the FTS method while an

additional spectrograph is employed in the DFDI method. The advantage of introducing an

additional spectrograph into the system is that the visibility (or fringe contrast) is no longer

limited by the bandpass as in the FTS case, which is the reason that the DFDI method

can be applied in broad-band Doppler measurements. Mosser et al. (2003) discussed the

possibility of an FTS working in broad band by introducing a low resolution post-disperser

and concluded that the FTS method is inferior (by a factor between 1 and 2) to DE method

even after employing an post-disperser. This conclusion should be accepted with cautions

because they compared an FTS with a post-disperser (R=1200) with a DE instrument with a

much higher spectral resolution (R=84,000), which is not necessarily a fair comparison.

We deﬁne new merit functions (Equations 2–16 and 2–19) to objectively evaluate

Doppler performance for both DFDI and DE methods. For Q′, the merit function for single

′ ′ object observation, we ﬁnd that QDFDI is consistently higher than QDE regardless of the R of the DE instrument under the constraint of total number of pixels, i.e., both the DFDI and DE

instrument adopt the same NIR detector. The DE instrument requires using a larger detector

in order to reach the same wavelength coverage as the DFDI instrument. Note that the above

conclusion is based on the assumption that the number of pixels per spectral order are the

same for DFDI and DE. In practice, a DFDI instrument uses ∼20 pixels to sample spatial

direction, i.e., the direction transverse to dispersion direction, while ∼5 pixels are usually

used to sample spatial direction in a DE instrument. However, the ∼20 pixels sampling is

53 not a requirement for DFDI but rather for the convenience of data reduction. Normally, ∼7

pixels sample one spatial period of a stellar fringe, which in principle are adequate based on

a phase-stepping algorithm provided by Erskine (2003).

If the same detector is used, the spare part of the detector in DFDI can be used

for multi-object observations. Consequently, in addition to single-object instrument, we

also investigate Q′′, a merit function for multi-object RV measurement for both DFDI and

DE. Different conclusions are reached depending on different value of α, an index of the

importance of multi-object observation. From a pure photon gain point of view, DFDI and ′′ ′′ ′′ ∼ DE instruments have similar Q values with QDFDI slightly better than QDE (a factor of 1.1). From a survey efﬁciency point of view, a DFDI multi-object instrument is 9 times faster than

− − its counterpart using DE for slow rotating stars (0 km · s 1 ≤ V sin i ≤ 2 km · s 1) and ∼4 times

− faster for fast rotators (V sin i ≥ 10 km · s 1).

2.4.2 Application of DFDI

There may be other practical concerns about the instrument using the DFDI method,

most of them are due to the relative low spectral resolution compared to current DE instru-

ments. First of all, an absolute wavelength calibration for a DFDI instrument is not as precise

as a DE instrument with a higher spectral resolution. For example, at a spectral resolution

of 22,000 for IRET, a line proﬁle with a FWHM of ∼0.45 Åin Y band is expected. Following

the method described in Butler et al. (1996), it corresponds to 136.4 m · s−1 RV uncertainty at a S/N of 100 if only one spectral line is used. Ramsey et al. (2010) proposed to use a

U-Ne emission lamp as a wavelength calibration source and it has approximately ∼500

lines in Y band according to their measurement. Therefore, after all the lines in Y band are

considered, ∼6 m · s−1 RV uncertainty is introduced in the process of absolute wavelength

calibration. In comparison, a DE instrument at R of 110,000 causes ∼1.2 m · s−1 RV uncertainty in an absolute wavelength calibration. However, an absolute wavelength solution is only required for the DE method in order to measure RV drift due to instrument instability, which is measured in a different method in a DFDI instrument. It is similar to a stellar RV

54 measurement, the difference is that the object is switched from a star to an wavelength cali-

bration source. Vertical fringe movement of absorption or emission lines of an RV calibration

source is measured instead of centroid movement measurement in a DE instrument. In this

case, a DFDI instrument (e.g., IRET, R=22,000) is equivalent to a DE instrument with R of

37,000 in terms of Doppler measurement precision (see §2.3.2). Therefore, instrument RV

drift calibration process introduces an RV uncertainty of ∼3.5 m · s−1 for IRET in the example of a U-Ne lamp calibration source. In addition, the RV uncertainty can be further reduced by increasing S/N and number of measurement. Secondly, at a low spectral resolution, it is challenging to perform spectral line profile analysis and thus it requires high-resolution follow- up in order to confirm or exclude a possible detection. Last but not least, in a binary case in which the observed spectrum is blended, two approaches can be used for identification:

1, from measured RV, if the ﬂux ratio is small, similar to the planet companion case, then the lower mass companion can be identiﬁed in the measured RV curve even though small

flux contamination exists; if the flux ratio is about unity, indicating strong flux contamination, a large RV scattering is expected because this case is not considered and modeled in the current data reduction pipeline; 2, from measured spectrum, even the observed spectrum is a 2-D fringing spectrum in DFDI, we can still de-fringe the spectrum into a 1-D traditional spectrum, on which special treatment can be performed to quantify the blending such as

TODCOR (Zucker et al., 2003).

55 CHAPTER 3 COMPREHENSIVE SIMULATIONS FOR HABITABLE PLANET SEARCH IN THE NIR

Discovering an Earth-like exoplanet in habitable zone is an important milestone for

astronomers in search of extra-terrestrial life. While the radial velocity (RV) technique re-

mains one the most powerful tools in detecting and characterizing exo-planetary systems,

we calculate the uncertainties in precision RV measurements considering stellar spectral

quality factors, RV calibration sources, stellar noise and telluric contamination in different ob-

servational bandpasses and for different spectral types. We predict the optimal observational

bandpass for different spectral types using the RV technique under a variety of conditions.

We compare the RV signal of an Earth-like planet in the habitable zone (HZ) to the near

future state of the art RV precision and attempt to answer the question: How close are we to

detecting Earth-like planet in the HZ using the RV technique?

3.1 Introduction

The fundamental photon-noise RV uncertainties have been discussed in several previous papers (Bouchy et al., 2001; Butler et al., 1996). However, only intrinsic properties of

stellar spectra are discussed in their works while no detailed calculation of RV uncertainties

introduced by the calibration sources. In a recent paper, Reiners et al. (2010) considered

the uncertainties in the NIR caused by RV calibration sources, i.e., a Th-Ar lamp and an

Ammonia gas absorption cell. However, these two RV calibration sources can not be com-

pletely representative of the calibration sources used and proposed in current and planned

Doppler planet survey in the NIR. For example, there are other emission lamps available in

the NIR for RV calibration such as a U-Ne lamp as proposed by Mahadevan et al. (2010).

In addition, other gas absorption cells besides the Ammonia cell have been proposed in

the NIR (Mahadevan & Ge, 2009; Valdivielso et al., 2010). Futhermore, in Reiners et al.

(2010), the calculation of RV calibration uncertainty of a gas absorption cell assumes a 50

nm band width in K band, and then the uncertainty was applied to other NIR bands, which is

purely hypothetical. Therefore, a more comprehensive and detailed study of RV calibration

56 uncertainties is necessary at different observational bandpasses in the NIR in the search

of planets around cool stars. On the other hand, in the visible, even though the current RV

precision is not limited by the RV calibration source such as a Th-Ar lamp or an Iodine ab-

sorption cell, a better understanding of their performances under the photon-limited condition

helps us discern a stage in which the RV calibration source becomes the bottle neck as RV

precision keeps improving. Rodler et al. (2011) recently investigated RV precision achiev-

able for M and L dwarfs, but did not quantitatively discussed the inﬂuence of RV calibration

sources and stellar noise on precision Doppler measurement.

Stellar noise is a signiﬁcant contributor to RV uncertainty budget, which falls into three

categories: p-mode oscillation, spots and plagues, and granulations. P-mode oscillation

usually produces an RV signature with a period of several minutes. The oscillation mode

has been relatively well studied by previous work (e.g., Carrier & Bourban (2003); Kjeldsen

et al. (2005)). Exposure time of 10-15 min is proposed in order to smooth the RV signature

induced by p-mode oscillation (Dumusque et al., 2011). Spots and plagues induced RV

signal has been discussed by several papers (e.g., Desort et al. (2007); Lagrange et al.

(2010); Meunier et al. (2010); Reiners et al. (2010)). Meunier et al. (2010) concluded that

the photometric contribution of plages and spots should not prevent detection of Earth-mass

planets in the HZ given a very good temporal sampling and signal-to-noise ratio. Granulation

is considered to be the major obstacle in detection of Earth planets in the HZ because

it produces an RV signal with an amplitude of 8∼10 m · s−1 based on observation on the

Sun (Meunier et al., 2010). In addition, there is by far no good method of removing the RV noise from this phenomenon. Dumusque et al. (2011) provided a model of noise contribution

in RV measurements based on precision RV observation on stars of different spectral type

and at different evolution stages.

The telluric lines from the Earth’s atmosphere are usually masked out in calculations

of the photon-limited RV uncertainties in NIR (Reiners et al., 2010; Rodler et al., 2011).

Although Wang et al. (2011) proposed a method to quantitatively estimate the inﬂuence of

57 atmosphere removal residual on precision Doppler measurement using the Dispersed Fixed

Delay Interferometer (DFDI) method (Erskine, 2003; Ge, 2002; van Eyken et al., 2010), no attempt has ever been made for precision Doppler measurements using a high-resolution

Echelle spectrograph. In practice, the telluric lines are not masked out, but instead modeled and removed. Therefore, a quantitative way of estimating the RV uncertainties produced by the residual of telluric line removal is necessary before we fully understand the performance of an RV instrument. In the visible band, the estimation of telluric line contamination is equally important as higher RV precision is required in the search of lower-mass planets around solar type stars.

We address two basic questions in this chaper after considering a variety of factors including stellar spectrum quality, RV calibration precision, stellar noise and atmosphere contamination: 1, which observational bandpass is optimal to conduct precision Doppler measurements for stars of different spectral types; 2, is current RV precision adequate for detecting Earth-like planets in the HZ in the most optimistic scenario, in which the star is the least active and telluric lines are perfectly modeled and removed. The methods and ﬁndings of this study will provide insights to the design and optimization of a planned or ongoing precision Doppler planet survey. In addition, it also helps us to access at what stage we are in the search of Earth-like planets in the HZ.

3.2 Simulation Methodology

3.2.1 High Resolution Synthetic Spectra

Because observed stellar spectra do not have high enough spectral resolution and broad effective temperature coverage, we decide to use high resolution synthetic stellar spectra in the calculation of photon-limited RV uncertainty. For solar type stars, i.e., FGK type stars (3750 K≤ Teﬀ ≤7000 K), we adopt the spectra with a 0.02 Å sampling from Coelho et al. (2005). For M dwarfs, (2400 K≤ Teﬀ ≤3500 K), we use high-resolution (0.005 Å sampling) synthetic stellar spectra generated by PHOENIX code(Allard et al., 2001; Hauschildt

58 et al., 1999). Reiners et al. (2010) conducted several comparisons between synthetic spec-

tra generated by PHOENIX and observed spectra in NIR. They concluded that the synthetic

spectra are accurate enough for the purpose of simulations. For more massive stars (7000

K< Teﬀ ≤9600 K), we also use the synthetic spectra with 0.005 Å sampling generated

by PHOENIX. Throughout the chapter, we assume a metallicity of solar abundance and a

surface gravity log g of 4.5 for main sequence stars. We assume a Gaussian line spreading function (LSF) which is determined by spectral resolution(R). After an artiﬁcial rotational

line broadening using a kernel provided by Gray (1992) and an LSF convolution, we rebin

each spectral slice according to 4.0 pixels per resolution element (RE) to generate the a one-

dimensional spectrum. In comparison, the sampling rate is 3.2 pixel/RE for HARPS (Mayor

et al., 2003) and 3.5 pixel/RE for HIRES (Vogt et al., 1994).

We compare the synthetic spectra to the observed ones in the visible to ensure that

the synthetic spectra are good approximations of observed stellar spectra (Bagnulo et al.,

2003). Comparison in NIR requires carefully removing telluric lines from the observed

stellar spectra, which is beyond the scope of my work. Figure 3-1 shows comparisons of

synthetic spectra and the observed high resolution (R=80,000) stellar spectra from Bagnulo

et al. (2003). The comparison spans a wide range of spectral types from M6V to A5V.

The synthetic spectrum of an A5V star matches well the an observed one with an RMS of

0.02. As the features in a stellar spectrum increases due to cooler Teﬀ and slower stellar

rotation, the RMS increases due to an increasing complexity of comparison and imprecise

spectral line modeling. The RMS of difference is 0.05 and 0.04 for an F8V and a G2V star.

It get worse in the comparison for a K5V star, in which the RMS is 0.10. And the RMS of

difference is 0.05 for an M6V star. The results from the comparisons between synthetic

and observed spectra indicate the difﬁculty in modeling the spectra of cool stellar objects.

Although not perfect, the synthetic spectra are able to reproduce majority of the features in

the observed spectra. Therefore, we decide to use the synthetic spectra in our calculation

59 Figure 3-1. Comparisons between synthetic and observed spectra. Black lines represent observed spectra and red lines are synthetic spectra after rotational line broadening and LSF convolution at R=80,000. The Teff and V sin i are chosen according to the spectral type and line width empirically, they are not necessarily the best-fit parameters for the observed spectra. The chosen Teff and V sin i are, from top to bottom, 9000 K and 80.0 km · s−1 for HD 39060 (A5V), 6250 K and 4.5 km · s−1 for HD 30562 (F8V), 5750 K and 6.0 km · s−1 for HD 14802 (G2V), 4750 K and 4.0 km · s−1 for HD 10361 (K5V), 2900 K and 10.0 km · s−1 for HD 34055 (M6V). The difference between observed and synthetic spectrum is also plotted at the bottom of each panel with RMS of difference.

60 of RV uncertainty. Photon-limited RV measurement uncertainty is calculated based on the

method discussed in §2.2.1.

3.2.2 RV Calibration Sources

RV calibration sources are important in precision Doppler measurements because

they not only provide wavelength solutions but also help track drift due to instrument in-

stabilities. The RV uncertainties due to calibrations must be considered if we want to fully

understand the performance of an RV instrument. We consider the photon-limited uncer-

tainties introduced by RV calibration sources based on their spectral quality factors. Two

types of calibration sources have been successfully applied in RV measurements in the

visible bands: 1), a Th-Ar emission lamp (Lovis & Pepe, 2007); 2), an Iodine gas absorption

cell (Butler et al., 1996). Searching for planets in NIR using the RV technique has already

been conducted by several groups (Bean et al., 2010; Blake et al., 2010; Figueira et al.,

2010b; Mahadevan et al., 2010; Muirhead et al., 2011) and several high resolution NIR spec-

trographs will be put into use in the foreseeable future (Ge et al., 2006a; Quirrenbach et al.,

2010). We limit the discussions in the emission lamps and gas absorption cells although

there are other candidates for RV calibration sources, for examples, laser combs (Li et al.,

2008; Steinmetz et al., 2008), which are unfortunately very expensive and not yet readily

available, and interferometer calibration sources as proposed by Wildi et al. (2010) and Wan

& Ge (2010)

In the following discussions, the RV calibration sources are categorized by the observational bandpass in which they are applied. The corresponding wavelength range for each observational bandpass is given in Table 3-1.

In B band, a Th-Ar lamp is a suitable calibration source. The lines list of a Th-Ar lamp from Lovis & Pepe (2007) is adopted in my work. Only Thorium lines are used in

the calculation because the instability of Argon lines is at the order of ∼10 m · s−1, which

is not stable enough for high precision Doppler measurements. A Iodine absorption cell

is assumed in V band for RV calibration, a Th-Ar lamp is also considered in this band

61 Table 3-1. Deﬁnition of observational bandpasses used in this study: center wavelength and wavelength range

Band λ0 λmin − λmax (nm) (nm) B 450 400−500 V 545 500−590 R 660 590−730 Y 1020 960−1080 J 1220 1110−1330 H 1580 1480−1680 K 2275 2170−2380 for comparison. We obtained a high resolution spectrum (R ≥200,000) using the Coude

Spectrograph at Kitt peak for a Iodine cell with a 6-inch light path at 60 ◦C. Note that an iodine cell spectrum is superimposed on a stellar spectrum (Butler et al., 1996), the S/N of

RV calibration is thus determined by the S/N of the continuum of a stellar spectrum. This case is called Superimposing in this chapter. Howerver, for very stable instruments, there are other ways of calibrating the non-stellar drift including spatial (Mayor et al., 2003) and temporal approaches (Lee et al., 2011). In a spatial approach, the light from a star and a

Th-Ar lamp is fed onto nearby but different parts of CCD by two separate ﬁbers (We call this case Non-Common Path in the chapter). In Bracketing method, on the other hand, RV calibrations are conducted right before and after a stellar exposure in a temporal approach.

In both cases, the S/N of RV calibration is not dependent on stellar ﬂux. The disadvantage is, however, the light from a calibration source does not pass through the instrument in exactly the same path or at the same time as the light from a star. We choose a Th-Ar lamp as the

RV calibration source in R band, where strong Argon lines exist that saturate the CCD. Since we exclude Argon lines in RV calibration uncertainty calculation, a more practical result when

Argon lines are considered is expected to be worse unless a CCD with higher dynamic range is used.

In Y and J band, a U-Ne emission lamp is proposed by Mahadevan et al. (2010), we use a lines list of Uranium provided by Stephen Redman (Redman et al., 2011). In H band,

62 a series of absorption cells is proposed by Mahadevan & Ge (2009), in which a mixture of

13 14 12 12 13 gas cells including H C N, C2H2, CO, and CO creates a series of absorption lines

that spans over 120 nm of the H band. Bean et al. (2010) demonstrated that an Ammonia

absorption cell is a good candidate for calibration source in K band. Therefore, we assume

an Ammonia cell in the calculation of RV calibration uncertainty in the K band. Valdivielso

et al. (2010) proposed a gas absorption cell with the mixture of acetylene, nitrous oxide,

ammonia, chloromethanes, and hydrocarbons covering most of the H and K bands. We do

not consider this cell in this study since a detailed lines list of the cell is not available.

3.2.3 Stellar Noise

Stellar noise is a signiﬁcant contributor to RV uncertainty budget, therefore we devote

the following part to discuss a method of quantifying its inﬂuence on precision Doppler

measurement. Granulation is considered to be the major obstacle in detection of Earth

planets in the HZ because it produces an RV signal with an amplitude of 8∼10 m · s−1 based on observation on the Sun (Meunier et al., 2010). In addition, there is by far no good method

of removing the RV noise from this phenomenon. Dumusque et al. (2011) provided a model

of noise contribution in RV measurements based on precision RV observation on stars of

different spectral type and at different evolution stages. We adopt this model and quantify

the RV uncertainty contribution of granulation based their measurement of three stars, i.e.,

α Cen A (G2V), τ Ceti (G8V), and α Cen B (K1V). The sum of three exponentially decaying

functions represents a power spectrum density function with contributions from granulation,

meso-granulation and super-granulation, using the values given in Table 2 from Dumusque

et al. (2011). An RV RMS error due to granulation is then calculation based on Equation

(6) in their paper assuming a 100-day (300-day) consecutive observation for K (G) type

star with an optimal strategy found in the paper, i.e., three measurements per night of 10

min exposure each, 2 h apart. The total length of consecutive observation is roughly in

accordance with the orbital period of a planet in the HZ. We ﬁnd that the RV RMS error due

63 to granulation is 0.55, 1.05 and 1.05 m · s−1 for a K1V, G8V and G2V star respectively. These number are going to be used later in this study to estimate a total RV uncertainty.

Detailed study of RV uncertainty induced by stellar noise has so far been limited in K and G type stars due to practical concerns such as stellar photon ﬂux and stellar activity.

Despite their intrinsic faintness and relative higher level of stellar activity due to fast rotation and deep convection zone, M dwarfs are among primary targets in search of planets in the

HZ. The RMS fitting error of orbit of GJ 674 b (Bonfils et al., 2007) is 0.82 m · s−1 after RV noise due to a stellar spot is modeled and removed, providing an good target for Earth-like planet search with an upper limit of other stellar noise contribution of 0.82 m · s−1, if we interpret the RMS fitting error is due to an combination of instrument instability, photon- noise and other source of stellar noise. We adopt the model proposed by Dumusque et al.

(2011) to estimate the RV RMS error due granulation phenomenon for an M dwarf using

the parameters for a K or G star. Aware of the caveat of different stellar type, we ﬁnd the

RMS error is 0.52, 1.07 and 1.04 m · s−1 using the parameters for a K1V, G8V or G2V star.

50-day consecutive observation is assumed with an optimal strategy described in Dumusque

et al. (2011). The theoretical calculation of granulation-induced RV RMS error is worse than

observation of GJ 674 b using parameters for G stars, suggesting the parameters for G

stars are not representative of optimistic scenario in observation of M dwarfs. Therefore, we

use 0.52 m · s−1, which is a result of using the parameters for K stars, as an estimation of

granulation-induced RV RMS error for an M dwarf in an optimistic case.

3.2.4 Telluric Lines Contamination

Ground-based observations are prone to contamination by telluric lines. Precision

Doppler measurements in the NIR requires a signiﬁcant level of disentanglement of stellar

absorption lines and telluric lines. As RV precision keeps improving, Doppler measurements

in the visible band such as B, V and R band should also consider telluric lines, because the contamination of them will no longer be negligible. The quantiﬁcation of telluric line contamination has been discussed by Wang et al. (2011) in the context of Dispersed Fixed

64 Delay Interferometer method (Erskine, 2003; Ge, 2002; van Eyken et al., 2010), here we

present a generalization of the method for the case of Echelle spectrograph, which is a more

conventional application. Atmospheric transmission (AT) is calculated by a service provided

by spectralcalc.com based on a method described in Gordley et al. (1994).

The following equation describes the ﬂux distribution on a CCD detector if telluric absorption lines are considered: [ ] S (ν) F ν = 0 × AT ν ⊗ LSF ( ) ν ( ) [ h ] S (ν) = 0 × − α × AA ν ⊗ LSF ν (1 ( )) [ h ] [ ] S (ν) S (ν) = 0 ⊗ LSF + − 0 × α × AA(ν) ⊗ LSF hν hν

= FS (ν) + FN (ν), (3–1)

where S0(ν) is stellar energy ﬂux, which is converted into photon ﬂux by being divided by hν,

AT is the atmospheric transmission function, AA is the atmospheric absorption function, and

α is a parameter describing the level of telluric line removal as a ﬁrst-order estimation.

In Equation 3–1, photon ﬂux distribution on the detector, F , is comprised of a signal component FS and a noise component FN . Ideally, we require that the detector ﬂux change,

δF , is entirely due to the stellar RV change δvS . However, δF is also partly induced by telluric line shift δvN resulting from random atmospheric motions. Therefore, both δvS and δvN contribute to δF . We have two sets of RV measurements, δvS + σ(0, δvrms,S ) for stellar RV and δvN + σ(0, δvrms,N ) for RV induced by the Earth’s atmosphere, where σ(0, δ) represents random numbers following a gaussian distribution with a mean of 0 and a standard deviation of δ. δvrms,S is the photon-limited measurement error for component FS and δvrms,N is the

photon-limited measurement error for component FN . We weigh the ﬁnal RV measurement

with the inverse square of photon-limited RV uncertainties of these two components, which is

expressed by the following equation:

δ + σ δ · δ −2 + δ + σ δ · δ −2 ( vS (0, vrms,S )) vrms,S ( vN (0, vrms,N )) vrms,N δv = , (3–2) δ −2 + δ −2 vrms,S vrms,N

65 In practical Doppler measurements, δvS consists two components, stellar RV and Earth’s

barycentric RV. Depending on the position of the Earth in its orbit, there is an offset between

δvS and δvN , which is the Earth’s barycentric velocity. The Earth’s barycentric motion has a

semi-amplitude of 30 km · s−1. Statistically, observed star has an annually-varying RV with a semi-amplitude of on-average 21.21 km · s−1. We artiﬁcially shift a stellar spectrum by an amount less than 21.21 km · s−1 in order to generate a offset between stellar spectrum and AA spectrum. δvrms,S and δvrms,N are then calculated for FS and FN . We choose the median of δvrms,N to represent a typical δvrms,N value from calculations based on different input barycentric velocities. We further assume that observed star has a constant RV (i.e., no differential RV), and FN has an RV ﬂuctuation with an RMS of δvN,ATM because of the Earth’s turbulent atmosphere. The measured RV uncertainty δv is equal to:

δ · δ −2 + δ 2 + δ 2 1/2 · δ −2 ( vrms,S ) vrms,S ( vN,ATM vrms,N ) vrms,N δv = , (3–3) rms δ −2 + δ −2 vrms,S vrms,N

In reality, RV uncertainty of FN is not dominated by photon-noise, instead, it is dominated by

atmospheric behaviors such as wind, molecular column density change, etc. Figueira et al.

−1 (2010a) used HARPS archive data and found that O2 lines are stable to a 10 m · s level over 6 years. However, long term stability of telluric lines (over years) becomes worse if we take into consideration other gas molecules such as H2O and CO2. The uncertainty induced by atmospheric telluric lines is transferred to δvrms via Equation 3–3. In order to calculate the

ﬁnal RV uncertainty, δvrms , we need to calculate photon-limited RV uncertainty δvS,rms and

δvN,rms according to Equation 2–6, in which two terms need to be calculated: Q and Ne− . The

spectral quality factors (QS and QN ) for the two components (FS and FN ) from Equation 3–1

are calculated based on Equation 2–7. Ne−,S and Ne−,N , the photon ﬂux of FS and FN are

calculated based on stellar type, magnitude, exposure time, instrument speciﬁcations and

telluric absorption properties. Note the ratio of Ne−,S and Ne−,N remains constant as long as

atmospheric absorption stays unchanged because telluric line absorption is imprinted on the

stellar spectrum.

66 The method described above provides a quantitative way of answering the questions

such as: 1), how the RV uncertainty is correlated with different levels of residual of telluric

line removal; 2), what the contribution of RV uncertainty due to telluric contamination is in the

ﬁnal RV error budget in each different observational bandpass.

3.3 Results

3.3.1 RV Calibration Uncertainty

RV calibration sources are used to track the drift that is not caused by the stellar reﬂex

motion due to an unseen companion. A emission lamp or a gas absorption cell is usually

used for such purpose. We calculate the RV uncertainties brought by the calibration sources

themselves based on their spectral properties. The RV calibration sources in different

observational bandpasses are discussed in §3.2.2. For gas absorption cells, we assume

a continuum level of 30,000 ADU (within the typical linear range) on a CCD with a 16-bit

dynamic range, which corresponds to a S/N of 425 if the gain is at 6 electron/ADU. For a

emission lamp, we assume that the strongest line in the spectral region has a peak ﬂux of

30,000 ADU. Note that 4 pixels per resolution element is assumed throughout the paper.

Figure 3-2 shows the RV calibration uncertainties as a function of observational bandpass

at different spectral resolutions. Note that there are currently two successful calibration

sources in V band, i.e., a Th-Ar lamp (asterisk) and an Iodine cell (square). Therefore, both are considered and plotted for V band in Fig. 3-2. The results shown in Fig. 3-2 are also

summarized in Table 3-2. Gas absorption cells usually offer higher calibration precision than

emission lamps because of denser lines distribution and on-average higher S/N. However,

this conclusion depends on the calibration methods, it is usually the case for Non-Common

Path and Bracketing method while it is not always true in Superimposing scheme, which will

be discussed later in this section.

3.3.2 Optimal Spectral Band For RV Measurements

The optimal observational bandpass for precision RV measurements depends on the

quality of a stellar spectrum (Q factor), photon ﬂux (S/N), RV calibration uncertainty, the

67 Figure 3-2. RV calibration uncertainties as a function of observational bandpass at different spectral resolutions (color-coded). Squares in V band represent Iodine cell method and asterisks in V band represent Th-Ar lamp method. Refer to Table 3-2 for RV calibration sources in different observational bandpasses.

Table 3-2. RV uncertainties caused by calibration sources at different spectral resolutions. R B V R Y J H K Th-Ara Th-Ar, Iodineb Th-Ar U-Nec U-Ne Mixed celld Ammoniae (m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1) 20,000 0.78 0.90, 0.33 1.0 3.0 2.4 1.4 0.57 40,000 0.65 0.74, 0.21 0.81 2.0 1.6 0.75 0.30 60,000 0.54 0.60, 0.15 0.63 1.4 1.1 0.50 0.21 80,000 0.46 0.50, 0.12 0.51 1.1 0.87 0.38 0.17 100,000 0.38 0.41, 0.10 0.43 0.90 0.70 0.31 0.15 120,000 0.33 0.36, 0.08 0.36 0.76 0.59 0.27 0.14

Note. — a: Lovis & Pepe (2007); b: Butler et al. (1996); c: Redman et al. (2011); d: Mahadevan & Ge (2009); e: Bean et al. (2010).

68 severity of telluric line contamination and other factors. We will consider different situations in the following discussion. We assume a S/N (per pixel) of 100 at the center of Y band

(i.e., λ=1020 nm) at R=60,000, the S/N in other observational bandpass varies with stellar spectral energy distribution (SED) and spectral resolution accordingly. The S/N reported in this paper is at the center of each observational bandpass (see Table 3-1) unless otherwise speciﬁed. We will investigate the optimal observational bandpass for precision Doppler measurements given the same exposure time, the same telescope aperture and the same instrument throughput (independent of wavelength).

3.3.2.1 Stellar Spectral Quality

We start with the simplest case in which the RV uncertainty is only determined by the stellar spectral quality factor and the SED of a star. In other words, the RV calibration source is perfect and no uncertainty is introduced when calibrating out the non-stellar drift.

In addition, telluric lines are perfectly removed from the observed stellar spectrum. Table 3-3 summarizes the obtainable RV precisions and the S/Ns at three different spectral resolutions, i.e., 20,000, 60,000 and 120,000. An example of R = 120, 000 is plotted in Fig. 3-3. We

ﬁnd the optimal observational bandpass is B band for a wide range of spectral types from K to A. The optimal observational bandpass for an M dwarf is either in R band or in K band.

More speciﬁcally, R band is optimal for an early-to-mid-type M dwarf while K band for an late-type M dwarf. The ﬁnding remains the same for a wide range of spectral resolutions from

20,000 to 120,000. The RV uncertainty for another spectral type or at a different S/N can be obtained by either interpolating or scaling based on the results in Table 3-3.

3.3.2.2 Stellar Spectral Quality + Stellar Rotation

Stellar rotation broadens the absorption lines in a stellar spectrum, resulting in less

Doppler sensitivity. It is therefore necessary to consider the stellar rotation in the discussion of photon-limited RV uncertainty. Typical values of rotation velocities of different spectral types are obtained based on the measurement results from Jenkins et al. (2009) for M dwarfs and Valenti & Fischer (2005) for FGK stars. In addition, typical rotational velocities

69 Table 3-3. Photon-limited RV uncertainties based on stellar spectral quality at different spectral resolutions for different spectral types, average S/N per pixel is reported in perentheses Spec. Type B V R Y J H K (m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1) R=20,000 A5V 4.7(211.7) 8.3(209.0) 21.5(198.3) 23.9(173.2) 22.6(155.1) 62.9(126.8) ...(...) F5V 3.8(166.9) 6.3(177.1) 12.8(182.1) 23.3(173.2) 20.9(164.2) 27.2(148.9) ...(...) G5V 3.3(126.4) 5.1(145.6) 8.6(163.3) 20.8(173.2) 15.9(171.9) 14.9(168.0) ...(...) K5V 3.7(88.8) 4.8(110.9) 7.5(141.0) 17.3(173.2) 13.5(181.8) 11.2(199.8) ...(...) M5V 14.7(26.8) 11.7(44.0) 9.0(66.2) 12.8(173.2) 12.0(203.0) 9.5(205.6) 6.8(191.6) M9V 28.2(9.3) 18.4(17.9) 13.8(26.6) 8.2(173.2) 4.6(250.5) 5.4(246.3) 2.7(250.9)

R=60,000 70 A5V 1.5(122.2) 2.7(120.7) 6.9(114.5) 10.0(100.0) 9.6(89.5) 24.6(73.2) ...(...) F5V 1.1(96.4) 1.8(102.3) 3.6(105.1) 8.8(100.0) 7.9(94.8) 9.4(86.0) ...(...) G5V 1.0(73.0) 1.6(84.0) 2.5(94.3) 6.9(100.0) 5.7(99.3) 5.6(97.0) ...(...) K5V 1.2(51.3) 1.5(64.0) 2.3(81.4) 5.9(100.0) 5.1(105.0) 4.6(115.3) ...(...) M5V 4.6(15.5) 3.5(25.4) 2.5(38.2) 4.8(100.0) 4.4(117.2) 3.8(118.7) 2.4(110.6) M9V 9.3(5.4) 5.9(10.4) 4.2(15.3) 3.3(100.0) 1.8(144.6) 2.0(142.2) 1.0(144.9)

R=120,000 A5V 1.1(86.4) 2.0(85.3) 5.1(81.0) 8.0(70.7) 7.9(63.3) 18.9(51.8) ...(...) F5V 0.7(68.2) 1.2(72.3) 2.4(74.3) 6.4(70.7) 5.9(67.0) 6.5(60.8) ...(...) G5V 0.6(51.6) 1.1(59.4) 1.6(66.7) 4.6(70.7) 3.9(70.2) 4.0(68.6) ...(...) K5V 0.8(36.3) 1.0(45.3) 1.5(57.6) 4.0(70.7) 3.5(74.2) 3.0(81.6) ...(...) M5V 3.1(11.0) 2.3(18.0) 1.5(27.0) 3.3(70.7) 3.1(82.9) 2.8(83.9) 1.8(78.2) M9V 6.4(3.8) 4.1(7.3) 2.8(10.8) 2.3(70.7) 1.3(102.3) 1.5(100.6) 0.8(102.4) Figure 3-3. RV precision (R=120,000) based on spectral quality factor as a function of observational bandpass for different spectral types from M9V to A5V (color-coded). Average S/N per pixel is also shown in the plot, see Table 3-3 for results at other spectral resolutions. K band RV uncertainties are not calculated for stars with Teﬀ higher than 3500 K because they are usually observed in the visible band at current stage.

for early type stars such as A stars are extrapolated from values of solar type stars. Table

3-4 summarizes the spectral types and the corresponding Teﬀ and Vsin i used in the paper.

A trend of increasing Vsin i is seen as spectral type moves either to early type end (F and A) or late type end (M). After considering typical stellar rotation velocities for different spectral types (as shown in Fig. 3-4, R=120,000), F and A stars are not suitable targets for

precision Doppler measurements because of their intrinsic high stellar rotation. M dwarfs

RV uncertainties are getting worse than non-rotating case, but 2−5 m · s−1 RV precision are

expected in optimal cases, i.e., R band for M5V and K band for M9V. For K and G stars, sub

71 m · s−1 precision is reached under photon-limited condition even after considering typical stellar rotation.

Figure 3-4. RV precision (R=120,000) based on spectral quality factor and typical stellar rotation as a function of observational bandpass for different spectral types from M9V to A5V (color-coded). Average S/N per pixel is the same as shown in Fig. 3-3.

3.3.2.3 Stellar Spectral Quality + RV Calibration Source

The above RV precisions considering typical stellar rotation broadening are good indicators for RV planet surveys in which a population of stars is observed with a distribution of stellar rotations. However, in the search for an Earth-like planet, a different approach is taken in which stars with favorable properties for Doppler measurements are assigned higher priority in observation. The properties usually include slow stellar rotation and low stellar activity. Therefore, we will reduce stellar rotation in the following discussion since we

72 Table 3-4. Spectral Type, corresponding Teﬀ, and typical stellar rotation V sin i

Spectral Type Teﬀ Vsin i (K) (km · s−1) A0V 9600 131.0 A2V 9000 108.0 A5V 8400 85.5 A8V 7800 62.5 F0V 7400 47.5 F2V 7000 32.0 F5V 6750 23.0 F8V 6250 6.5 G0V 6000 3.5 G2V 5750 2.2 G5V 5500 1.7 G8V 5250 1.8 K0V 5000 1.9 K2V 4750 1.8 K5V 4500 2.0 K8V 4000 2.5 M0V 3750 2.8 M2V 3500 2.8 M5V 3100 3.9 M8V 2600 6.8 M9V 2400 8.0 emphasize discovery of an Earth-like planet. After considering the uncertainties brought by an RV calibration source, RV uncertainties in Fig. 3-3 degrade to those in Fig. 3-5

(R = 120, 000). Two scenarios of calibration are considered: Superimposing (Dotted), and

Non-Common Path and Bracketing (Solid). The difference between these two is whether the

S/N depends on the stellar flux. In the Superimposing case, because the absorption cell is in the light path of the stellar flux, the continuum of the resulting Iodine absorption spectrum is determined by the the continuum flux of a star. Consequently, the RV calibration uncertainty is strongly dependent on the incoming stellar flux. In the comparison of the two cases in

Fig. 3-5, we see the Non-Common Path and the Bracketing methods always introduce less uncertainty in RV calibration than the Superimposing method. The major reason for that is the S/N in the former case may be optimized by adjusting the source intensity (Non-Common

73 Path) or the exposure time (Bracketing). The main conclusion about optimal observational bandpass from §3.3.2.1 remains unchanged.

Figure 3-5. RV precision (R=120,000) based on spectral quality factor and RV calibration uncertainties as a function of observational bandpass for different spectral types from M9V to A5V (color-coded). Average S/N per pixel is the same as shown in Fig. 3-3. Squares in V band represent Iodine cell method and asterisks in V band represent Th-Ar lamp method. Dotted lines show result from Superimposing cases and solid lines for Non-Common Path and Bracketing cases. Refer to Table 3-2 for RV calibration sources in different observational bandpasses.

3.3.2.4 Stellar Spectral Quality + RV Calibration Source + Atmosphere

The optimal band for Doppler measurements is in the NIR (K band in particular) for stars with spectral types later than M5 from previous discussions in this paper. However, one important element is missing in the discussion, which is the contamination from the telluric lines in the Earth’s atmosphere, which is a severe problem in the NIR observation. The

74 quantitative analysis of telluric line contamination is introduced in §3.2.4 and we apply that

method in estimating the RV uncertainty brought by the telluric contamination. We conﬁne

our discussions for late-type M dwarfs since NIR observation does not gain advantage for

other spectral types earlier than M5V.

Figure 3-6 shows an example of how RV uncertainty for an M9V star changes with

observational bandpass under different values of α (i.e., level of telluric line removal, see

Equation 3–1). 1 indicates no telluric line removal and 0 indicates complete removal of

telluric lines (see Equation 3–1). RV ﬂuctuation of 10 m · s−1 due to random atmospheric movement is assumed in the calculation. Bracketing RV calibration is assumed in the calculation. There are several points worth noting in this plot: 1), different observational bandpasses are affected differently by telluric lines, the signiﬁcance of telluric line contamination is indicated by the span of RV uncertainties at different α values. For example, B band is the least affected by telluric lines because the RV uncertainties in B band at different

levels of telluric line removal remain roughly the same, while J, H and K bands suffer severe

telluric line contamination because any small change of α results in signiﬁcant change of RV

uncertainty. 2), If there is no attempt of removing telluric lines from observed stellar spectrum

(purple in Fig. 3-6), there is no advantage in observing late-type M dwarfs in NIR, RV uncer-

tainty is dominated by Earth’s atmosphere behavior in the NIR. In this case, the optimal band

is V and R band. Only when α ≤0.01, i.e., more than 99% telluric line strength is removed, the advantage of observing late-type M dwarfs in the NIR becomes obvious, at a factor of ∼3

improvement.

In practice, there have been several examples in which telluric line modeling and

removal is demonstrated to be successful. Vacca et al. (2003) achieved maximum deviations

of less than 1.5% and RMS deviations of less than 0.75% with R=2000 and S/N≥100 using

a telluric standard star nearby the science target star. Bean et al. (2010) has shown that

the RMS deviation is as low as 0.7% after using a 3-component model (Stellar spectrum,

telluric absorption and Ammonia absorption) to ﬁt an observed NIR spectrum. In both cases,

75 Figure 3-6. RV precision (R=120,000) considering spectral quality factor, RV calibration uncertainties and telluric contamination for an M9V star as a function of α, i.e., telluric line removal level (color-coded). 1 indicates no telluric line removal and 0 indicates complete removal of telluric lines. Bracketing RV calibration is assumed for the results shown in the plot. Squares in V band represent Iodine cell method and asterisks in V band represent Th-Ar lamp method. Refer to Table 3-2 for RV calibration sources in different observational bandpasses.

an α value of better than 0.01 has been demonstrated showing great potential of precision

Doppler measurement in the NIR band.

Figure 3-7 shows the percentage contribution of RV uncertainty introduced by telluric

contamination at different α values. If no telluric line removal is performed, the RV uncer-

tainty in the NIR is dominated by those caused by telluric contamination, i.e., the percentage

contributions are more than 87.9% in Y , J, H and K band. In comparison, the percentage

contribution of telluric contamination induced RV uncertainty is 2.9%, 3.1% and 53.0% in B,

76 Y and R band respectively. As α decreases, i.e., more strength of telluric lines is removed, less RV uncertainty is contributed to the ﬁnal RV uncertainty budget. However, there is still a signiﬁcant fraction (more than 70%) of RV uncertainty contributed by telluric contamination in J, H and K band even after 90% of telluric line strength is removed. The percentage contribution drops below 10% throughout considered observational bandpasses when more than 99.9% strength is removed. To sum up the discussion, RV uncertainty is dominated by telluric contamination in the NIR band. Therefore, telluric line removal in the NIR is a necessary step to reduce the telluric contamination and extract more of Doppler information intrinsically carried by a stellar spectrum.

Figure 3-7. The percentage contribution of RV uncertainty induced by telluric contamination as a function of observational bandpass. Different α values are indicated by colors. 1 indicates no telluric line removal and 0.001 indicates 99.9% effective removal of telluric lines.

77 3.3.2.5 Comparisons to Previous Work

There are works that have been previously done in attempts to understand the funda-

mental photon-limited RV uncertainties based on high resolution synthetic stellar spectra.

Bouchy et al. (2001) calculated Q factors for a set of synthetic stellar spectra for solar type dwarf stars. We restrict the comparison to spectra with the same turbulence velocity (Vt ).

−1 Since the spectra for solar type stars in our study have a Vt of 1.0 km · s , we only compare

−1 the results from spectra with Vt of 1.0 km · s in Bouchy et al. (2001). Table 3-5 summarizes

a comparison of our results to those from Bouchy et al. (2001). The Q factors from our study

are generally 10−15% lower if no stellar rotation is considered, i.e., V sin i=0 km · s−1. It may be due a different sampling rate in the synthetic spectra, 0.005 Å in Bouchy et al. (2001)

and 0.02 Å in our paper. More ﬁne features are seen in a spectrum with higher sampling

rate and thus more Doppler information is contained. At low stellar rotation rate (V sin i=4 and 8 km · s−1), our results agree with theirs within 6%, which is improved compared to non-rotating case because the ﬁne features are smoothed out by stellar rotation. For fast rotators, i.e., V sin i=12 km · s−1, 10% difference is seen in the worst case, for which a different

limb-darkening value might be responsible.

Table 3-5. Comparison of Q factors from our results to Bouchy et al. (2001) (numbers in parenthathes).

Teﬀ logg Vt Vsin i (K) (cm · s−1)(km · s−1) 0 km · s−1 4 km · s−1 8 km · s−1 12 km · s−1 4500 4.5 1.0 30238(34940) 17235(17080) 8700(8440) 5793(5380) 5000 4.5 1.0 30001(33405) 16607(16140) 8305(7815) 5432(4930) 5500 4.5 1.0 26892(30375) 14858(14700) 7397(7020) 4700(4385)

Reiners et al. (2010) investigated the precision that can be reached in RV measure-

ments for stellar objects cooler than solar type stars in the NIR. The treatment of telluric lines

in their calculations was to block the regions where the telluric absorption is over 2% and

30 km · s−1 in the vicinity. Following the method described in their paper, we calculated the

fraction of the wavelength range affected by telluric contamination in V , Y , J and H band,

the results are 2.4%, 22.7%, 60.0% and 50.6%. In comparison, the results are 2%, 19%,

78 55% and 46% for V , Y , J and H band in their paper. The difference may be caused by a

different atmospheric absorption used in calculation.

Both Reiners et al. (2010) and we reach the same conclusion that NIR RV measure-

ments start to gain advantage over visible bands for mid-to-late-type M dwarfs. However, we

predict that Y and H band are similar in terms of giving the highest RV precision among V ,

Y , J and H bands, while it is found in their paper that Y band is the optimal band considering stellar spectrum quality and telluric line masking. Note that we adopt the deﬁnition of V ,

Y , J and H bands according to Reiners et al. (2010) in comparisons. Table 3-6 summarizes

our calculations of RV precision can be reached for an M9V star (Teﬀ = 2400K) in compari-

son to the results in their paper as well as the S/N obtained in each observational bandpass.

In further examination, we compare our results in J and H band and ﬁnd that RV precisions

in H band are in general better than those in J band. It is explained by the wider wavelength

coverage and richer absorption features in H band for an M9V star. On the contrast, the

improvement of RV precision in H band is not seen in the comparison of J and H band in the

results from Reiners et al. (2010). In addition, we report better RV precisions in V band by a

factor of 1.5.

Table 3-6. Comparison of predicted RV precision (in the unit of m · s−1) between our results to −1 Reiners et al. (2010) for an M9 dwarf (Teﬀ=2400 K, V sin i=0 km · s ). The Following results are calculated based on a telluric masking treatment in which telluric lines with more than 2% absorption depth and 30 km · s−1 within its vicinity are masked out. R S/N This study Reiners et al. (2010) VYJHVYJHVYJH 60000 12 100 134 128 5.1 3.0 4.9 2.7 8.0 2.2 4.6 4.0 80000 10 86 116 111 4.2 2.5 4.1 2.2 6.2 1.7 3.5 3.5 100000 9 77 104 99 3.9 2.4 3.8 2.0 5.3 1.5 2.9 3.3

We have also conducted similar calculation to Rodler et al. (2011) for an M9.5 dwarf

−1 (Teﬀ=2200 K, V sin i=5 km · s ) and the comparison is presented in Table 3-7. Instead of

ﬁnding Y band, we ﬁnd K band gives the highest RV precision in telluric-contamination-free

case, while H band gives the highest RV precision in the case where telluric lines with an

79 absorption depth more than 3% are masked out in RV calculation. We also notice that our predicted RV precisions are less sensitive to spectral resolution. Note that stellar absorption lines typically become resolved by spectrograph after spectral resolution goes over 50,000, we do not expect RV precision increases steeply as spectral resolution goes well beyond

50,000.

Table 3-7. Comparison of predicted RV precision (in the unit of m · s−1) between our results to −1 Rodler et al. (2011) for an M9.5 dwarf (Teﬀ=2200 K, V sin i=5 km · s ). Case A is for complete and perfect removal of telluric contamination; Case B is for the case in which telluric lines with absorption depth of ≥3% were masked out. R S/N This study Rodler et al. (2011) YJHKYJHKYJHK Case A

20000 139 180 171 152 16.4 12.2 8.7 5.1 22.2 25.5 22.8 27.9 40000 98 127 121 108 9.4 6.9 5.4 3.2 6.9 8.7 7.8 10.8 60000 80 104 99 88 7.7 5.5 4.6 2.8 4.2 5.7 5.1 3.8 80000 70 90 85 76 6.8 5.1 4.4 2.6 3.3 4.0 3.8 5.1

Case B 20000 139 180 171 152 18.8 19.8 13.9 16.4 24.2 29.7 29.1 39.3 40000 98 127 121 108 10.5 10.9 8.5 9.5 8.7 12.2 11.9 17.3 60000 80 104 99 88 8.6 8.7 7.0 7.8 5.4 7.1 6.8 10.7 80000 70 90 85 76 7.6 8.0 6.7 7.2 3.8 5.2 5.1 7.9

3.3.3 Current Precision vs. Signal of an Earth-like Planet in Habitable Zone

One of the most intriguing tasks in exoplanet science is to search and characterize

Earth-like planets in the HZ. Over the past two decades, great advances have been seen but we have not yet discovered another Earth. We are trying to answer several questions in the following discussion: 1), is it possible to detect an Earth-like planet in the HZ using the

RV technique? 2), If so, at what S/N in which observational bandpass and for which spectral type? 3), Based on the current available RV calibration sources and knowledge of stellar noise, is it practical to detect an Earth-like planet in the HZ in the most optimistic case?

80 3.3.3.1 Stellar Spectral Quality

We ﬁrst consider an ideal situation in which the RV precision is only determined by the

Q factor of a stellar spectrum. The highest S/N per pixel obtainable for a single exposure

is 425 (assuming 30,000 ADU and a gain of 6 electron/ADU). Figure 3-8 shows the RV

precisions obtainable at spectral resolution of 120,000 for different spectral types in different

observational bandpasses. Overplotted are RV signals of an Earth-like planet in the inner

(dashed) and outer edge (dash-dotted) of the HZ of a star with a certain spectral type (color

coded). The position of the HZ is calculated based on Kasting et al. (1993). The position

of the HZ gets closer to the host star as stellar temperature and luminosity drops. The RV

signal is enhanced by both the decreasing distance to the star and the decreasing stellar

mass. Earth-like planet is detectable in every observational bandpass at a S/N as high as

425 for M dwarfs. B and V band bear the highest probability for K stars and B band is the

sweet sopt for G stars. Predicted RV precisions are not adequate to detect the signal of an

Earth-like planet in the HZ around F and A stars with single exposure on a current typical

CCD with 16-bit dynamic range. Table 3-8 summarizes the S/N required for detection of an

Earth-like planet in the HZ as a function of spectral type, in which we assume that a detection

is possible when the RV precision is equal to the signal. Even though it only require a S/N

of 17 for an M9V star in B band to detect an habitable Earth-like planet, the exposure time

could be as long as 1 hour even at the Keck telescope for a J=6 M9V star and there is no

such bright late M type star in the sky. In addition, only 25 M stars are available with J band

magnitude less than 6 (Lepine´ & Shara, 2005). In comparison, ∼1 min exposure time at Keck will obtain a S/N of 175 for a B=8 star, which is adequate for detecting habitable Earth-like planet around a K5V star. 10% instrument throughput is assumed in the above calculations.

3.3.3.2 Stellar Spectral Quality + RV Calibration Source + Atmosphere

Current RV precision is not only restricted by the photon-limited uncertainty determined

by a stellar spectrum, but also by the uncertainties brought by an RV calibration source and

the telluric contamination from the Earth’s atmosphere. Figure 3-9 shows the RV precisions

81 Figure 3-8. RV precisions (R=120,000) considering spectral quality factor at a S/N of 425 as a function of observational bandpass for different spectral types from M9V to A5V (color-coded). Overplotted are RV signals of an Earth-like planet in the inner (dashed) and outer edge (dash-dotted) of the HZ.

Table 3-8. Required S/N for detection of an Earth-like Planet in the HZ as a function of spectral type Spectral Band HZ properties BVR Y J HK m ain aout vin vout −1 −1 (M⊙) (AU) (AU) (m · s )(m · s ) A5V 3483 6087 14724 20389 17827 35053 ... 1.82 2.893 4.167 0.03 0.03 F5V 837 1547 3088 8042 7003 6976 ... 1.30 1.614 2.325 0.06 0.05 G5V 318 625 1053 3149 2660 2648 ... 0.87 0.720 1.037 0.11 0.09 K5V 175 280 535 1710 1570 1496 ... 0.64 0.387 0.558 0.18 0.15 M5V 47 58 58 328 359 330 193 0.19 0.068 0.098 0.78 0.65 M9V 17 22 22 118 99 108 58 0.10 0.034 0.049 1.51 1.26

82 taking into consideration of Q factors, RV calibration uncertainties and telluric contamination.

Bracketing calibration (at a S/N of 425) is considered in the calculations in which the S/N of calibration is not determined by the continuum of the observed star. Two cases are discussed for telluric contamination, in one case no telluric removal is attempted (solid) while in the other case 99.9% of telluric line strength is removed (dotted). For the visible bands (i.e., B, V and R band), RV precision in B band is barely affected by telluric contamination (10 m · s−1 random RV of telluric lines is assumed in the calculations) but limited by the RV calibration uncertainty due to a Th-Ar lamp. There are two RV calibration sources considered in the V band, a Th-Ar lamp and a Iodine absorption cell, the latter one provides higher calibration precision in the Bracketing case.

Even though only 2.4% of the wavelength range is affected by telluric line contamination

(§3.3.2.5), handling telluric lines is still very important. The RV uncertainty budget of a K5V star in Table 3-9 shows an example in which the RV precision is worse than detection limit if no telluric line removal is involved while it is below the detection limit in the 99.9% removal case (α=0.001). This example address the importance of telluric line removal in the search of an Earth-like planet even in the visible band where telluric contamination is less severe than the NIR band. After 99.9% of telluric line strength is removed, the RV uncertainty of a K5V star is dominated by the spectral quality of a K5V star and the RV calibration source.

Table 3-9. Two examples of telluric contamination

Spectral Type Bandpass α δvS,rms δvATM,rms δvcal δvrms δvHZ (m · s−1)(m · s−1)(m · s−1)(m · s−1)(m · s−1) 1.0 0.19 0.24 K5V V 0.11 0.08 0.18 0.001 0.01 0.14

1.0 7.97 7.98 M5V K 0.33 0.14 0.78 0.001 0.17 0.39

In comparison, in the NIR (Y , J, H and K band), the RV uncertainties are dominated by telluric contamination, resulting RV precisions at 5−10 m · s−1 that are not adequate in habitable Earth-like planet detections. A similar example of how telluric contamination

83 Figure 3-9. RV precision (R=120,000) considering spectral quality factor (S/N=425), RV calibration uncertainties and telluric contamination as a function of observational bandpass for different spectral types from M9V to G5V (color-coded). Overplotted are RV signals of an Earth-like planet in the inner (dashed) and outer edge (dash-dotted) of the HZ. Solid lines represent non-telluric-removal cases while dotted lines represent cases in which 99.9% of the strength of telluric lines is removed. Squares in V band represent Iodine cell method and asterisks in V band represent Th-Ar lamp method. Refer to Table 3-2 for RV calibration sources in different observational bandpasses. raises the ﬂoor of RV uncertainty is also given for an M5V star in K band in Table 3-9. In the

99.9% removal case, RV uncertainties in the NIR are no longer mainly dominated by telluric contamination, but by spectral quality factor. To sum up, telluric line removal is an important and indispensable step toward the discovery of an Earth-like planet even in the visible band.

After telluric lines are successfully removed from observed stellar spectrum, the RV precision is limited by the uncertainty caused by stellar spectral quality and RV calibration sources.

84 After completely removing the telluric contamination, we compare our prediction of

RV uncertainties and what is reported from HARPS instrument (Mayor et al., 2003). An

example of HD 47186 (Bouchy et al., 2009) is given in Table 3-10. HD 47186 is a G5V star

with a V sin i of 2.2 km · s−1, the best achievable RV precision for this star is 0.3 m · s−1 at a

S/N of 250 according to Bouchy et al. (2009). Our prediction indicates that an RV precision

of 0.24 m · s−1 is possible to achieve at the same S/N for the same wavelength coverage.

The difference may come from those uncounted factors in our calculation, for example, stellar noise. However, our prediction of RV precision is within 20% to precision from real observation.

Table 3-10. Prediction vs. HARPS observation. a: HARPS observation of HD 47186 from Bouchy et al. (2009), the best achievable RV precision is at a S/N of 250 for this G5V star with V sin i of 2.2 km · s−1; b: our RV uncertainties prediction for this star assuming the same S/N, spectral type, observation bandpasses and stellar rotation.

Bandpass δvS,rms δvcal δvrms (m · s−1)(m · s−1)(m · s−1) HD47186a B+V+R ...... 0.30

B 0.14 0.33 0.36 V 0.29 0.36 0.46 G5Vb R 0.48 0.36 0.60 B+V+R 0.12 0.20 0.24

3.3.3.3 Stellar Spectral Quality + RV Calibration Source + Stellar Noise

Assuming the telluric lines are perfectly measured and removed, we consider the

obtainable RV precision based on stellar spectral quality, RV calibration precision and stellar

noise. Stellar noise of different spectral type is estimated in §3.2.3 based on Dumusque

et al. (2011). Figure 3-10 shows predicted RV precision, it is clear that the RV precision for

G and K type stars is not adequate for detecting Earth-like planet in the HZ after stellar noise

is taken into consideration. However, it is still possible to detect Earth-like planets around

M dwarfs because of relatively larger RV signal. For an M5V star, visible band and K band

85 provide adequate precision for an Earth-like planet detection while all bandpasses allow an

Earth-like planet detection for an M9V star.

Figure 3-10. RV precision (R=120,000) considering spectral quality factor (S/N=425), RV calibration uncertainties and stellar noise as a function of observational bandpass for different spectral types from M9V to G5V (color-coded). Overplotted are RV signals of an Earth-like planet in the inner (dashed) and outer edge (dash-dotted) of the HZ. HZs of G and K type stars are not plotted because they are out of reach based on the predicted RV precision. Squares in V band represent Iodine cell method and asterisks in V band represent Th-Ar lamp method. Refer to Table 3-2 for RV calibration sources in different observational bandpasses.

In order to compare our predicted RV precision to observations, we choose 69 planets detected by HARPS since 2004 after an instrument upgrade (Mayor et al., 2003) and plot the RMS errors of Keplerian orbit fitting as a function Teff (Fig. 3-11). The minima of three subsets (corresponding to G, K and M stype stars) are found based on Teff. In comparison,

86 the predicted RV predictions (after combining results from B, V and R bands) for a G5V,

K5V and M5V star are plotted as open diamonds. Since the predicted RV precision is based

on an optimistic case, we compare the predictions with the RMS minima we ﬁnd in the

observation. For M dwarfs, the minimum of RMS errors is found at 0.8 m · s−1 with GJ 674

b (Bonﬁls et al., 2007). In comparison, our prediction is 0.62 m · s−1 considering stellar spectral quality factor, RV calibration error and stellar noise. If ∼0.5 m · s−1 instrumental uncertainty as mentioned in Bonﬁls et al. (2007) is added in quadrature, our prediction is

well matched with HARPS M dwarfs observation in the best case scenario. Lovis et al. (2006)

reported 0.64 m · s−1 RMS errors for a planet system of a K0V star (i.e., HD 69830), which

−1 is consistent with our prediction of 0.65 m · s . It is plotted in the bin with Teﬀ between

5000 K and 6000 K because reported Teﬀ of 5385 K. For G type stars, Bouchy et al. (2009)

reported 0.91 m · s−1 RMS error for HD 47186 b and c, a planetary system around a G5V

star. In comparison, we predict a total RV uncertainty of 1.1 m · s−1. The overestimation

of RV measurement uncertainty is possibly due to an overestimation of stellar noise or an

increasing S/N because of multiple measurements in real observation.

We predict a total RV measurement uncertainty of 0.62, 0.65 and 1.1 m · s−1 for spectral type M5V, K5V and G5V considering stellar spectral quality, RV calibration and stellar noise.

According to the calculations in §3.2.3, RV uncertainty due to stellar noise is 0.52, 0.55 and

1.05 m·s−1 for the above three types of stars, accounting for 70.3%, 71.6% and 91.1% of total

RV measurement uncertainty. Based on comparisons of our predictions and observation,

we therefore conclude that stellar noise is one major contributor in error budget of precision

Doppler measurement. M dwarfs should be the primary targets in search of Earth-like

planets in the HZ. Unlike G and K stars, the RV signal of Earth-like planets in the HZ is not

overwhelmed by stellar noise for M dwarfs in the most optimistic case.

3.4 Summary and Discussion

We provide a method of practically estimating the photon-limited RV precision based

on the spectral quality factor, stellar rotation, RV calibration uncertainty, stellar noise and

87 Figure 3-11. RMS error of Keplerian orbit fitting for planets detected by HARPS since 2004. Predicted RV precisions considering stellar noise for different stellar types are overplotted as open diamonds. telluric line contamination. The methodology described and the results presented in this paper can be used for design and optimization of planned and ongoing precision Doppler planet surveys. For pure consideration of stellar spectral quality without artificial rotationally broadening the absorption line profile, the optimal band for RV planet search is B band for a wide range of spectral types from K to A, while it is R or K band for mid-late type M dwarfs.

Nevertheless, the above conclusion remains unchanged after considering typical stellar rotation of each spectral type. However, F and A stars become unsuitable for precision RV measurements because of typically fast stellar rotation. We conﬁrm the ﬁnding in Reiners et al. (2010) that the NIR Doppler measurements gain advantage for mid-late M dwarfs.

However, instead of ﬁnding Y band as the optimal band considering stellar spectrum quality

88 and telluric masking, we ﬁnd that both Y and H bands give the highest RV precision among

V , Y , J and H bands. In a comparison to Rodler et al. (2011), we ﬁnd K band is the optimal

band for precision Doppler measurement in a telluric-free case and H band is optimal in a

telluric-masking case, while they found Y band gives the highest RV precision in both cases.

Fundamental photon-limited RV precision for evolved stars has been discussed by Jiang

et al. (2011), which is valuable for ongoing RV planet search around retired stars discussed

in Johnson et al. (2007b).

We also consider the uncertainties brought by current available RV calibration sources

at different spectral resolutions (Fig. 3-2). Sub m · s−1 calibration precision can be reached for each observational bandpass. Note that the Q factors may change as gas pressure, length of light path and temperature changes. The precision also depends on the methods used in the

RV calibration. We categorized the calibration methods into several cases: Superimposing, in which the calibration spectrum is imprinted onto a stellar spectrum; Non-Common Path and

Bracketing, in which the calibration is conducted either spatially or temporally. The former method depends on stellar ﬂux while the latter one can only be applicable for very stable instruments. There are other calibration sources we have not included into the discussions in this study, for example, laser combs (Li et al., 2008; Steinmetz et al., 2008), the Fabry-Perot

calibrator (Wildi et al., 2010) and the Monolithic Michelson Interferometer (Wan & Ge, 2010).

Once they become more economically affordable or more technically ready, the RV precision

will be greatly improved in the future.

For the ﬁrst time we have quantitatively estimated the uncertainty caused by the residual

of telluric contamination removal for high resolution echelle spectroscopy method. Depending

on the telluric absorption, different observational bandpasses are affected differently. B

band is the least sensitive to telluric contamination because there are barely any telluric

absorption features in B band. However, the NIR bands suffer the most in precision RV

measurements because the stellar absorption lines and telluric lines are mixed together

severely in this spectral region. Only when α ≤0.01, i.e., more than 99% of strength of telluric

89 lines is removed, the advantage of NIR observation of mid-late type M dwarfs begins to show,

which is a factor of 3 improvement. This quantitative method in estimating the RV uncertainty

induced by telluric contamination can be easily adapted to other problems, for example,

estimating the moon light contamination.

Besides telluric line removal, telluric line masking has also been discussed in several

of previous studies (Reiners et al., 2010; Rodler et al., 2011; Wang et al., 2011). In Reiners

et al. (2010), telluric absorption with depth more than 2% and 30 km · s−1 in the vicinity is

blocked out when measuring RV. Based on this blocking criterium, the photon-limited RV

uncertainty, δvrms,S (refer to Equation 3–3) , for an M9V star at R=100,000 is 3.9, 2.2, 3.9, 2.2

−1 m · s in V , Y , J and H band respectively (see Table 3-6). In comprison, δvrms,N is 71.3,

−1 5.8, 6.5, 3.7 m · s in V , Y , J and H band respectively. Except for V band, δvrms,S and

δvrms,N are at the same order of magnitude, and the uncertainty caused by telluric absorption cannot be neglected even though that the spectral region with any telluric absorption of more than 2% is blocked. If more strict criterium of telluric line masking is applied, fewer photons are considered in measuring the RV, which effectively increases the photon-limited

RV uncertainty. In order to reach photon-limited RV precision predicted by pure consideration of spectral Q factor, telluric removal should be applied in which telluric contamination is measured or modeled and then removed from measured stellar spectrum.

RV uncertainty due to stellar granulation is taken into consideration in this paper. High frequency (∼min) stellar noise such as p-mode oscillations usually have a RV amplitude of

0.1 to 4.0 m · s−1 (Schrijver & Zwaan, 2000) and they can be averaged out within typical

10−15 exposure time. RV uncertainties due to low frequency (10−100 day) stellar noise such as stellar spots have been discussed in recent papers, for example, Desort et al. (2007)

and Reiners et al. (2010). The amplitudes of spot-induced RV range from one to several

hundred m · s−1. Since stellar spot induced RV uncertainties are periodic and therefore can

be modeled and removed, however, the amplitude of residual is unknown at this stage.

90 We compare the RV precision based on stellar spectral quality and the signal of an

Earth-like planet in the HZ of a star with a certain spectral type. We ﬁnd that it is likely to

detect a habitable Earth-like planet around G,K and M stars while it is too demanding to

detect one around F and A stars. B band is the optimal band for G and K stars and K band

for M dwarfs. After considering practical issues such as telluric contamination, we ﬁnd that,

except for B band, every observational bandpass is affected by telluric contamination to some extent. The major RV measurement uncertainty comes from telluric contamination, which overwhelms the RV signal of an habitable Earth-like planet around G and K stars.

Surprisingly, telluric contamination becomes an issue in V band even there is only 2.4%

of spectral region affected by telluric lines. After telluric lines are removed at a very high

level, i.e., α ≤0.001, the error from RV calibration becomes the major contributor of Doppler

measurement uncertainty. After stellar noise (granulation only) is taken into consideration,

which is dominant contributor to RV uncertainty, M dwarfs become the only type of star that is

suitable for the search for Earth-like planets in the HZ.

The RV precision in the discussion of habitable Earth detectability considers four factors:

stellar spectral quality, RV calibration uncertainty, stellar noise and telluric contamination.

However, the discussion of stellar noise should be treated with great caution for several

reasons: 1), stellar noise is not very well understood and characterized at this stage; 2), it

is different from case to case and therefore it is difﬁcult to draw a general conclusion; 3), a

habitable planet search is different from a planet survey, the targets are chosen in favor of

detection at the best case scenario, for example, high stellar ﬂux, slow stellar rotation, low

stellar activity and low stellar noise and so on. Therefore, the stellar noise assumed in this

study is in the best case scenario according to current theory and observation. In addition,

we assume the highest signal within linear range (30,000 ADU) for current typical CCD

(16-bit dynamic range) in single exposure in the discussion, note that the S/N can also be

improved by multiple independent measurements. The S/Ns required for Earth-like planet

detections are provided in Table 3-8 based on stellar spectral quality. Please note that the HZ

91 changes over time as the luminosity of the host star changes. It also depends on properties of a planet such as atmosphere composition, albedo and orbit. The purpose of discussion in this paper is to provide a basic idea of the comparison of current best obtainable RV precision to a typical RV signal of an habitable Earth-like planet.

92 CHAPTER 4 PLANET SEARCH AROUND M DWARFS

4.1 Introduction

4.1.1 Current Status

As of May 2011, there are 35 planets in 28 planetary systems of M dwarfs. Radial

velocity technique is the most productive method in M-dwarf planet search with discoveries

of 21 planets in 15 systems. Microlensing ranks the second with 12 planet detections in 11

systems. Transiting method has by far detected 2 planets around M dwarfs.

Giant planet occurrence rate for M dwarfs is generally thought to be lower than that for

solar-type stars. Bonﬁls et al. (2011b) found a low frequency (f ) of giant planet around M

≤ = +3 dwarfs, f 1% for P=1-10 day and f 2−1% for P=10-100 day, P denotes orbit period. In comparison, Cumming et al. (2008) found that the frequency is ∼10% for solar-type stars.

On the other hand, low-mass planets are frequently detected despite of an adverse detection

bias. Bonﬁls et al. (2011b) found that super-Earths (m sin i = 1 − 10M⊕) are abundant around

M dwarfs with a frequency of ∼35%. Given the frequency of super-Earth around M dwarfs, there are many planets awaiting for discoveries.

4.1.2 Challenges

M dwarfs emit the bulk their energy in the near infrared (NIR), they are thus much more

brighter in the NIR than in the optical wavelengths. NIR observation provides a promising

way of detecting planets around M dwarfs. However, there are several obstacles that prevent

us from making discoveries.

4.1.2.1 Atmophsere

Ground-based NIR RV measurement is severely affected by the Earth’s atmosphere,

which consists absorption lines of many species including H2O, O2, CO, CO2 and so on

(telluric lines). The RV of these species is correlated with the motion of the atmosphere such

as wind and turbulence. Measured RV of an M dwarf is therefore affected if telluric lines are

not properly removed from observed spectrum.

93 4.1.2.2 Wavelength Calibration Sources

Another fact that has limited the improvement of RV precision in the NIR is the lack of a stable and precise wavelength calibration source. A wavelength calibration source provide a caliber (absolute wavelength calibration) for those obtained stellar spectra and enables exclusion of instrument instability and measurement of stellar RV. Unlike those matured wavelength calibration sources in the visible bands such as an Iodine cell (Butler et al.,

1996) and a Th-Ar emission lamp (Lovis & Pepe, 2007), the quest for a suitable wavelength calibration source in the NIR remains a wide open question. Please refer to §3.2.2 for a more detailed and complete review of the ﬁeld of wavelength calibration sources.

4.2 Tackling Adversities in NIR RV Measurement

4.2.1 Software Advancement

4.2.1.1 Precise Telluric Lines Removal

Telluric lines exit and some times populate in the NIR part of an observed stellar spectrum. Stellar RV will not be precisely measured unless telluric lines are carefully removed from an observed stellar spectrum. Several ways of telluric removing scheme have been proposed and practiced including telluric line forward modeling (Bean et al., 2010) and observing a telluric standard star (Vacca et al., 2003). The former method relies on a synthetic telluric absorption spectrum to forward model an observed spectrum together with spectra of an stellar template and an absorption cell that provides an absolute wavelength solution. It is computationally intense and the cross talk between different components in the model is difﬁcult to fully understand. The latter method relies on an observation of a telluric standard star, usually a fast-rotating early type star whose intrinsic spectrum is almost featureless. Telluric line spectrum can be obtained based on the observed spectrum of an telluric standard star. This method requires more observational time but reduces the complexity in data reduction process. I adopted the telluric standard star method to remove telluric lines from an observed spectrum.

94 Figure 4-1 show an example of an observed stellar spectrum (GJ 411) contaminated with telluric lines (mostly water lines) in the NIR region between 8130 Å and 8270 Å . Most of the observed lines are not associated with the star but formed by the Earth’s atmosphere.

The comparison between a telluric-line-removed stellar spectrum and a synthetic stellar spectrum can be seen in Fig. 4-2. At this stage, a precise measurement of stellar RV becomes possible. To further investigate the telluric removal level, I divide these two spectra and remove outliers due to mismatches between an observed spectrum and a synthetic spectrum. I found the residual after telluric line removal has an rms of 0.027 indicating ∼97% of telluric line strength has been successfully removed.

Figure 4-1. Comparison between two spectra before (black) and after (red) removing telluric lines.

95 Figure 4-2. Comparison between an observed stellar spectra (black GJ 411, telluric lines removed) and a synthetic spectrum (red).

4.2.1.2 Binary Mask Cross Correlation

Stellar RV can be measured with the so-called cross correlation function (CCF) method in which a fully wavelength-calibrated stellar spectrum is multiplied with a series of Doppler- shifted template spectra. The maximum of the CCF corresponds the most likely stellar RV.

The template spectra can be obtained either from observation or from theoretical high- resolution synthetic spectra. I adopted the binary mask cross correlation technique (Baranne et al., 1996; Pepe et al., 2002; Queloz, 1995). The template spectra are generated from synthetic stellar spectra and each absorption line has a boxed shape whose width and depth are determined by the actual line width and line depth. The advantage of the binary mask cross correlation technique is that it makes it possible to select speciﬁc lines of interests

96 Figure 4-3. Telluric line removal residual is 2.7% after ejecting points that are caused by mismatches between an observed spectrum and a synthetic spectrum (marked by red asteriks). while discarding potential contaminating spectral region. It is particularly useful when dealing with spectra taken in the NIR which is severely contaminated by the telluric lines. This technique is also useful to eliminate noise source contributed by weak absorption lines in the low S/N case. Figure 4-4 shows an example of a wavelength-calibrated stellar spectrum in the NIR (black) and a binary mask template (red) used in the cross correlation process.

4.2.2 Hardware Advancement

There is a lack of precise and stable wavelength calibration source in the NIR although many candidates have been proposed (Bean et al., 2010; Li et al., 2008; Mahadevan & Ge,

2009; Redman et al., 2011; Steinmetz et al., 2008; Wildi et al., 2010). Suitable gas cells such as an Iodine cell in the optical wavelengths are hard to ﬁnd in the NIR and the precision is

97 Figure 4-4. Red: an example of binary mask template. Black: wavelength-calibrated stellar spectrum.

limited by temperature and species contamination. Sources such as emission lamps are

facing aging problem. What’s more, their spectra are irregular and non-uniform, which poses

challenges in data reduction process.

Etalon can be used for precise wavelength calibration (Wildi et al., 2010). However, the

high ﬁnesse mirror limits its operating band to be relatively narrow. Laser frequency combs

technique (Li et al., 2008; Steinmetz et al., 2008) is widely believed to be the next generation wavelength calibration source in the NIR, but it is still too expensive and inmatured to be incorporated with an astronomical Doppler instrument.

98 We are developing a Michelson interferometer (namely the sine source) that can be used as a precise and stable wavelength calibration source. When compared to previously- mentioned candidate sources, it has many advantages:

1. It is precise enough to provide a calibration precision of better than 10 cm · s−1which would enable detection of an Earth-like planet in the habitable zone of a solar-type star.

2. Its spectral stability is solely dependent upon its thermal stability. This feature reduces the complexity when designing the interferometer and its enclosure.

3. It has a very broad wavelength coverage from optical wavelengths to the NIR with a operation bandwidth of more than 800 nm. The wide wavelength coverage allows us to simultaneously include more spectral region to increase S/N.

4. The spectral feature is uniformed and periodical which enables fast and precise data processing.

5. It is compact with a dimension of 2 × 2 × 2 inch for its optics components. When equipped with a thermal enclosure, its dimension is 4 × 4 × 4 inch.

The applications of the sine source is ﬂexible. It can be used either as an absorption cell (Fig. ??) or as an emission lamp source (Fig. 4-6). In the former case the star light goes through the sine source before recorded by a CCD. This application is similar to the Iodine absorption cell method currently being used at the HIRES for the Keck telescope, but the wavelength region with calibration is dramatically increased because of the broad wavelength coverage of the sine source. However, the Earths telluric lines have to be considered in the

NIR. In the latter case where the sine source is used as an emission lamp source, star light and the emission spectrum of the sine source are directed to the detector by two nearby but separated ﬁbers. This case is similar to the calibration scheme adopted by HARPS using a

Th-Ar emission lamp, but data reduction is expected to be simpliﬁed because of the simple spectrum output of the sine source.

We have conducted demonstration experiments both in the lab and at the observatory using the EXPERT instrument at the KPNO 2.1m telescope. Figure 4-7 shows the comparison between two difference wavelength calibration sources. One is the sine source, the other one is a Th-Ar lamp. In a 2-day experiment, we have demonstrated that the measured

99 Figure 4-5. Application of the sine source as an absorption cell. Obtained spectrum is a multiplication of stellar spectrum, atmosphere absorption and sine source spectrum.

instrument drift (in m·s−1) from the two sources over time is consistent with each other at 10.7

m · s−1 level which is at the level of predicted photon-noise limited measurement error. The

sine source provides an overall more precise measurement and thus a better wavelength

calibrator.

4.3 M-dwarf Planet Search and Characterization-Results

4.3.1 Telluric Line RV Stability

It has long been wondered how stable the Earth’s telluric lines are and different RV

stabilities of different species. Figueira et al. (2010a) has studied O2 lines RV stability using

−1 HARPS data and concluded that O2 lines are as stable as 10 m · s over 6 years and the

−1 intrinsic stability of O2 lines is even higher (2-3 m · s ) when a simple physical atmosphere

100 Figure 4-6. Application of the sine source as an emission lamp for simultaneous wavelength calibration. Stellar spectrum and sine source spectrum are arranged next to each other.

model is considered. Bean et al. (2010) mentioned that H2O lines is as stable as 20

m · s−1 over half year. The small wavelength coverage (one spectral order, 8130-8270Å for

H2O and 6870-6935Å for O2) and unknown systematic error prevent us from studying the

absolute RV stability of these two species. However, the relative stability can be studied by

subtracting one RV measurement result from the other. Figure 4-8 shows the relative RV

−1 stability between H2O and O2 lines and the RV scatter rms is 18.3 m · s over 10 days. The result is consistent with previous studies by Figueira et al. (2010a) and Bean et al. (2010).

The implication is that telluric lines can be used as a wavelength calibration source at ∼10-20

m · s−1 precision level.

101 Figure 4-7. Top: Monitored RV drift over 2-day period. Results from Sin Source are plotted in black with error bars, and the results from Th-Ar lamp are in red. The median of measurement error is 3.8 m · s−1 for Sin Source and 18.4 m · s−1 for Th-Ar lamp. Bottom: The difference between results from Sin Source and Th-Ar lamp. The two methods track each other with an RV RMS of 10.7 m · s−1.

4.3.2 RV Measurements of a Reference Star-GJ 411

GJ 411 is a known RV stable star with an RV scatter rms less than 7 m · s−1 according

to Endl et al. (2006). We have shown in Fig. 4-9 that the RV rms is 24.7 m · s−1 with a Th-Ar lamp wavelength calibration and 40.6 m · s−1 with telluric lines as a wavelength calibration source. Both results are 1.5 times worse than photon-noise limited prediction. The results are expected to be improved signiﬁcantly with larger wavelength coverage.

102 Figure 4-8. Telluric water lines RV stability compared to O2 lines RV stability.

4.4 M-dwarf Planet Search and Characterization-Future Works

4.4.1 Searching For Planets Around M Dwarfs with EXPERT

The endeavor of searching for exoplanets has led to discoveries of over 400 planets around stars of spectral type A-M1 . Only 16 (4.0%) planets has been found around M type stars despite of the fact that they make up more than 70% of the galaxy including solar neighbors. As of today, planets search programs around M type stars have resulted in relative low detection rate compared to solar type stars (Cumming et al., 2008; Endl et al.,

2006; Zechmeister et al., 2009). Part of the reason is that RV measurement precision is

1 http://exoplanet.eu/

103 Figure 4-9. RV measurements for GJ 411. Black: results using a Th-Ar lamp as a wavelength calibration source. Red: results using telluric lines as a wavelength calibration source. relatively worse for M type stars, making it difﬁcult to detect Neptune-like or lower mass planets. On the other hand, the RV measurement is precise enough to detect close-in

Jupiter-like planets. The relative low detection rate indicates the low frequency of gas giant around M type stars compared to solar type stars. Endl et al. (2006) gives 1σ upper limit on frequency of gas giant around M type stars of < 1.27%. Butler et al. (2006) estimates planets fraction of 1.8% ± 1.0% for planets masses over 0.4 MJ . Johnson et al. (2007a) ﬁnds this fraction 1.8% for stellar mass range from 0.1-0.7 M⊙, and Cumming et al. (2008) found that M dwarfs are 10 times unlikely to harbor a gas giant within a 2000-day orbit compared to solar type stars. In contrast, Bonﬁls et al. (2007)found that planets less massive than ∼25

M⊕ are signiﬁcantly more frequent around M dwarfs which supports the prediction that the

104 frequency of Neptune-like planets are higher around M type stars than G type stars (Ida &

Lin, 2005; Kennedy & Kenyon, 2008; Kennedy et al., 2006). Among 16 planets discovered

around M type stars, 3 (18.8%) has been found more massive than 1 MJ and another 3

(18.8%) planets between 0.6 MJ and 1 MJ . These Jupiter-like exoplanets pose challenges

to traditional core-accretion model in the way that classical core accretion model has severe

problem with forming gas giant planets due to less massive protoplanetary disk (Laughlin

et al., 2004), while competing gravitational instability model can effectively form Jupiter-like planets around M type stars (Boss, 2006). More observations are needed to constrain planetary formation theory and discoveries of planet around M type stars will help us address the question from statistical perspective.

We are proposing to use EXPERT(EXtremely high Precision ExtrasolaR planet Tracker)

DEM(Direct Echelle Mode) at KPNO 2.1m telescope to conduct precise RV(radial Velocity) measurements of 41 mid-late M type stars of spectral range between M3.5 and M6. We have increased the mid-late M type stars sample by a factor of 1.5 compared to similar RV surveys combined in the past (Cumming et al., 2008; Endl et al., 2006; Zechmeister et al., 2009). It

will provide robust statistical constraints on the frequency of close-in Jupiter-like planets and

Neptune-like planets around mid-late M type stars. Simulations show that we will be able to

obtain better than 3 m/s RV precision in photon-noise limit case in 10 min exposure for J=7

target. In comparison, a 20 M⊕ planet in a edge-on orbit with a of 0.1 AU (p∼20 day) around

a star of 0.3 M⊙ produces RV signal with semi-amplitude of 10 m/s.

M type stars emit the bulk of energy around 1 µm, so it may be advantageous observing

them in NIR(near infrared). Practically, we are inevitably facing two major obstacles going

into NIR: (1) telluric lines contamination; (2) lack of source for absolute wavelength cali-

bration. We will observe bright fast rotating early type stars(mostly A type) as telluric lines

standard star and try to remove the contamination of telluric lines from M type star spectrum.

Telluric lines contaminating portion in spectrum will be masked during data reduction if no

telluric standard star is available.

105 4.4.2 Multi-Band Study of Radial Velocity Induced by Stellar Activity with EXPERT

Searching for earth-like habitable exoplanet has long been pursued by planet hunters.

However, it is extremely difﬁcult to achieve due to the low RV signal (0.1 m · s−1) for solar type

star in contrast with current RV precision in visible band (∼ 1 m · s−1). On the other hand, planets around late-type stars (i.e. M dwarfs) induce relatively larger stellar reflex motion due to lower stellar mass, thus higher RV signal. Therefore, M dwarfs become favorable targets in the search of habitable planets. However, visible band observation of M dwarfs is difficult due to the intrinsic faintness of the objects. Therefore, RV measurements in near infrared (NIR) is becoming an increasingly interesting field. Meanwhile, it is claimed that RV jitters due to stellar activities are reduced in NIR, which becomes anther advantage of NIR

RV measurements. The argument appears to be observationally conﬁrmed in a number of cases (Bean et al., 2010; Huelamo´ et al., 2008; Mart´ın et al., 2006; Prato et al., 2008).

However, in above observations, the measurements in visible band and in NIR were not

conducted simultaneously, in which we cannot rule out the possibility that the stars observed

were experiencing a less active period during the NIR observation. Therefore, simultaneous

measurements are required to conﬁrm the trend of decreasing RV jitter toward longer

wavelength, i.e., NIR.

Desort et al. (2007) studied the RV induced by a star spot and gave an empirical correlation between RV amplitude in visible and other parameters such as spectral type, spot size and V sin i. Reiners et al. (2010) carried it further to compare the RV amplitude induced

by a star spot in the visible and NIR band. According to the simulation, they found that the

RV amplitude in Y band is at least twice smaller than that in V band for hot star spot. It is

an theoretical support for the trend of decreasing RV in NIR which needs conﬁrmation from

multi-band simultaneous observation.

We have demonstrated the short-term RV precision in I band (λ0=806 nm, ∆λ=149 nm)

by observing a photometric stable star, KEP 11859158. The RMS is 57.6 m · s−1 for 7 days’

continuous observation (Fig. 4-10). We used Telluric lines as wavelength and RV calibration

106 reference since the lines of Th-Ar lamp are very sparse in the same wavelength region.

We also removed the telluric lines from stellar spectra using a nearby telluric line standard star which is observed at almost the same time as the observation of KEP 11859158. We suspect that the scattering of the RV measurements is due to the intrinsic instability of telluric lines (i.e., wind, water vapor density, etc.). We have developed a new RV calibration source, i.e., an RV-calibration interferometer which produces dense lines over large wavelength coverage including B, V, R, I and Y band. In-lab demonstration has shown that RV calibrations in

V band using Th-Ar lamp and with the RV-calibration interferometer track each other and the

RV calibration with the interferometer shows much smaller RMS scattering than Th-Ar lamp

(3.8 m · s−1 VS. 18.4 m · s−1, Fig. 4-7). After the recent installation of the RV-calibration interferometer in Feb 2011, the RV precision beyond 0.7 µm is expected to be greatly improved.

21.9 m · s−1 RV RMS scatter with the same star has been reached in the V band using DEM of EXPERT (Fig. 4-11).

Most objects with RV measured both in V and NIR band show RV jitter or amplitude more than 300 m · s−1 in V band. Plus, spectral types of the objects span from brown dwarfs (Mart´ın et al., 2006) to active young stars (Prato et al., 2008). If the RV jitter is indeed smaller in NIR than in V band by a factor of at leasst two as predicted by Reiners et al.

(2010), then it is observable using EXPERT. Furthermore, the question of multi-band RV jitter difference dependence on spectral type will for the ﬁrst time be answered if the sample of targets is carefully chosen. The ﬁnding of the proposal will provide insights for future habitable earth-like exoplanet searching mission in NIR, helping understand the RV jitter of stars of different spectral types.

4.4.3 Mid-Late Type M Dwarf Planet Survey Using FIRST

We proposes to conduct a pilot survey of 50 J≤8 nearby M dwarfs for exoplanets with the 2-m Automatic Spectroscopic Telescope (AST) in 2013-2015 with a new generation cryogenic high-resolution (R=60,000, 0.8-1.35 micron) near IR (NIR) cross-dispersed echelle spectrograph. This instrument, called FIRST, is scheduled to see ﬁrst light at the AST 2m

107 Figure 4-10. RV measured in I band using DEM of EXPERT for KEP 11859158, a photometric stable star. The RV RMS is 57.6 m · s−1. telescope in June 2013. This pilot survey is designed to tune the new instrument and the data pipeline to get ready for launching a NIR high precision Doppler exoplanet survey of 215

M dwarfs.

4.4.3.1 Science Justiﬁcation

Historically, exoplanet searches have focused on stars with masses similar to that of our Sun. Lower mass stars have received less attention in large part because they are intrinsically faint and cool, emitting most of their light at NIR wavelengths where our observational techniques are less well developed. Radial velocity searches have probed the early M dwarfs (mass≥0.4 Solar mass) and have clearly indicated that short-period giant planet companions to early M dwarfs are rare (Endl et al., 2006; Johnson et al.,

108 Figure 4-11. RV measured in V band using DEM of EXPERT for KEP 11859158, a photometric stable star. The RV RMS is 21.9 m · s−1.

2007a). However, planetary companions to stars at the peak of the stellar mass function and

below (mass≤0.4 Solar mass) remain essentially unexplored. These small stars represent perhaps our best opportunity to detect Earth-mass planets, including those orbiting in the

Habitable Zone (HZ), given current levels of RV precision. At the same time, these planets will necessarily be in the solar neighborhood, making them amongst the most important targets for future space-based efforts to directly image Earth-like planets and to study their atmospheres.

There is mounting evidence that sub-Neptune mass planets, including the Super-Earths, may be very common in orbit around low-mass stars. While Kepler observes only a small number of low-mass stars, it provides evidence that the planet mass function increases

109 toward smaller planetary and stellar mass. An analysis by Howard et al. (2011) indicates that up to 30% of early M dwarfs have Super- Earth sized planets with orbital periods less than

50 days. Analysis of the HARPS RV data by Bonfils et al. (2011b) finds similarly high values for the rate of occurrence of super-Earths orbiting early M dwarfs with periods between 10 and 100 days (35%) as well evidence for a significant population with orbital periods less than 10 days (36%). The detection of a system of Mars-size companions orbiting a late M dwarf in Kepler data by Muirhead et al. (2012), one of only a handful of late-M dwarfs in the

Kepler ﬁeld, provides further circumstantial evidence that these types of companions are very common.

M dwarfs later than M4 are of great scientiﬁc interest. For these stars, the mass, size, and temperature of the stars begin to rapidly decrease. To date, most exoplanet searches targeting M dwarfs have been conducted at visible wavelengths (Bonﬁls et al., 2011b; Endl et al., 2006; Johnson et al., 2007a) for stars with spectral type earlier than M4 because of intrinsic faintness of mid-late type M dwarfs. There are only 12 M4 or later type stars with

V≤12 north of -30 degrees (Reid & Gizis, 1997). For comparison, there are about 300 nearby stars M4 or later with J≤9 (Lepine´ & Shara, 2005). Therefore, for the latest types of stars, an observing program must operate in the NIR.

The current state-of-the-art for NIR RV detection of planets around late M dwarfs has been demonstrated with the VLTs CRIRES with moderate simultaneous wavelength coverage (364 ) using an ammonia gas cell for calibration (Bean et al., 2010). Long-term

( 6 months) RV precisions of 5 m/s have been demonstrated with this system. With this precision, and the observing time available at this facility, searches for giant planets orbiting late-M dwarfs can be carried out. The precision that will be delivered by FIRST (better than 4 m/s) and the cadence enabled by TSUs AST make this system a logical next step for survey of low-mass planets around mid-late type M dwarfs.

110 4.4.3.2 Target Selection

Our targets were selected from the following catalogs: Gliese Catalog of Nearby

Stars; Gliese Catalog of Nearby Stars cross identiﬁed with 2MASS (Stauffer et al., 2010);

ROSAT All-Sky Survey: Nearby Stars (Hunsch¨ et al., 1999). The selection was based on the following criteria:

• J≤10 and dec≥-20◦

• MV ≤8.7 and V-K≥3.5

• Ratio between X ray luminosity and bolometric luminosity, RX ≤-3.0

215 M dwarfs were selected with the above criteria (Note: before we launch the survey, we will use the 2MASS catalog to reject additional M dwarfs with a J≤14 stellar companion within 5 arcsec and replace them with slightly fainter M dwarfs. During the survey, we will reject spectroscopic binaries after 3 RV measurements from our targets and replace them with new survey targets). Based on the empirical equation of rotation velocity vs. RX in Kiraga & Stepien (2007), we expect 87% of our M dwarfs with rotational velocity less than

5 km/s. Therefore, most of them are inactive stars, which can help to minimize the RV jitters caused by stellar activities although the jitter level is signiﬁcantly reduced in NIR (Ma & Ge,

2012; Reiners et al., 2010). Figure 4-12 and Figure 4-13 show the number distribution in the

J and V bands of the FIRST M dwarf survey targets and the effective temperature distribution of the survey targets.

4.4.3.3 Planet Yield Prediction

A Monte-Carlo simulation is used to estimate planet yield of the survey with FIRST.

For each selected star, an RV measurement precision is calculated based on its apparent magnitude and spectral type. Figure 4-14 shows RV measurement precision in two cases.

In the baseline case, we assume total measurement error consists of 1.5 times photon- noise measurement uncertainty, 0.5 m/s wavelength calibration error, 98% telluric line masking error. The baseline case represents a reasonable scenario for the FIRST survey.

In the pessimistic case, we assume total measurement error consists of a 3 m/s unknown

111 Figure 4-12. V and J band distribution for FIRST survey targets. systematic error in addition to error sources assumed in the baseline case. The pessimistic case represents a worst case scenario for the FIRST survey. Stellar mass is estimated from its absolute magnitude. Once RV precision and stellar mass is known, we can generate a detectability plot on mass-period space. More speciﬁcally, for a given planet mass and orbital period, we generate a RV curve of 100 days from which 24 RV points are randomly drawn to form a RV data set, eccentricity distribution follows that from (Wang & Ford, 2011).

The data set is then analyzed by a detection code based on periodogram, if the peak of the periodogram agrees with the input period and the false alarm probability (FAP) is less than

1/1000, then we mark it as a detection. This test is repeated 100 times for each given planet mass and period. Therefore, there is a detectability/completeness plot for each star. The

112 Figure 4-13. Teﬀ distribution for FIRST survey targets. 3180 K is used to divide early and mid-late type M dwarfs in the sample.

survey completeness plot (Fig. 4-15 and Fig. 4-16) is the average of completeness plots of all selected stars.

To estimate planet yield, for example, estimated number of detected super Earths (1-10

M⊕), the planet yield is calculated by x · η · N, where x is the median of completeness in the region with 1≤Msini≤10 M⊕ and 1≤Period≤100 day, η=0.35 (Bonﬁls et al., 2011b) is the

frequency of Super-Earth around M dwarfs with 1-100 day period, and N is the sample size.

In total, 23 planets are expected to be detected for the pessimistic case including 5 super-

Earths, 2 giant planet (m sin i ≥ 100M⊕) and 16 intermediate-mass planets (10-100 M⊕). For

the baseline case, 30 planets are expected to be detected including 10 super-Earths, 2 giant

planets and 18 intermediate-mass planets.

113 Figure 4-14. Predicted RV measurement precision for the FIRST survey. Black dots represent baseline case (1.5 times photon-noise+calibration error+98% telluric masking error) and red dots represent pessimistic case (1.5 times photon-noise+calibration error+98% telluric masking error+3.0 m/s unknown systematic error). Dashed line represents the best precision achieved by HARPS M dwarf planet survey.

114 Figure 4-15. The predicted survey completeness contours based on observation strategy and RV precision for the pessimistic case.

115 Figure 4-16. The predicted survey completeness contours based on observation strategy and RV precision for the baseline case.

116 CHAPTER 5 ACCURATE GROUP DELAY MEASUREMENT FOR RV INSTRUMENTS USING THE DFDI METHOD

5.1 Introduction

As of Apr 2012, there are over 700 discovered exoplanets, and most of them are detected by the radial velocity (RV) technique1 . RV precision of 1 m · s−1 has been routinely achieved (Bouchy et al., 2009; Howard et al., 2010b) with instruments such as

HARPS (Mayor et al., 2003) and HIRES (Vogt et al., 1994), which are cross-dispersed

echelle spectrographs. While cross-dispersed echelle spectrographs are commonly used in

instruments for precision RV measurements, a method using a dispersed ﬁxed delay inter-

ferometer (DFDI) has offered an alternative method (Fleming et al., 2010; Ge et al., 2006b;

Lee et al., 2011). In this method, a Michelson-type interferometer is used in combination with

a moderate resolution spectrograph, RV signals are then extracted from phase shift of inter-

ference fringes of stellar absorption lines (Erskine, 2003; Erskine & Ge, 2000; Ge, 2002; Ge et al., 2002). The details about the DFDI theory and applications are discussed in van Eyken

et al. (2010) and Wang et al. (2011). Instrument adopting the DFDI method has demon-

strated advantages such as low cost, compact size and multi-object capability (Fleming et al.,

2010; Ge, 2002; Ge et al., 2006b; Lee et al., 2011; Wisniewski et al., 2012).

The MARVELS (Multi-object APO Radial Velocity Exoplanet Large-area Survey) (Ge

et al., 2009) is a ground-based Doppler survey with the main goal of obtaining a large-scale, statistically well-deﬁned sample of giant planets. It has operated since 2008 until 2014. It will search for gaseous planets around 11,000 stars that have orbital periods ranging from hours to 2 years, and are between 0.5 and 10 Jupiter masses. It has completed observation of 3,300 stars with over 94,000 RV data points, i.e., on average 28 data points per star. Over

250 binaries and a dozen of brown dwarfs have been detected from the survey.

1 http://exoplanet.eu/; http://exoplanets.org/

117 In the DFDI method, a ﬁxed delay interferometer (Wan et al., 2011, 2009) plays a crucial role in creating stellar spectral fringes for high precision RV measurements (Erskine,

2003; Ge, 2002). The Doppler sensitivity can be optimized by carefully choosing the group delay (GD) of the interferometer (Wang et al., 2011). More specifically, GD of an interferometer should be chosen such that the spatial frequency of white light combs (WLCs) matches with that of a stellar spectrum after rotational broadening. GD is defined by the following equation: 1 dϕ GD(ν) = − · , (5–1) 2π dν where ϕ is phase shift and ν is optical frequency. The interferometer in a DFDI instrument is usually designed to be field-compensated to minimize the influence of input beam instability (Wan et al., 2009; Wang et al., 2010). It is realized by carefully selecting glass materials and thicknesses of two second surface mirrors such that their virtual images are overlapped.

Because glasses are used in the optical paths, ϕ does no longer linearly change with frequency, therefore GD is dependent on optical frequency. An inaccurate GD measurement may signiﬁcantly limit the RV measurement accuracy (Barker & Schuler, 1974; van Eyken et al., 2010).

In practice, there may be several methods of measuring GD:

1. Calculate GD based on glass refractive indices using Sellmeier equation and thicknesses from manufacturer speciﬁcation.

2. Forward model the spectrum of a known spectral source, such as an Iodine cell or a Th-Ar lamp.

3. Measure phase and frequency using a while light source, such as a tungsten lamp.

4. Calibrate GD using a source with known velocity.

Method 1 is straightforward but may lack of adequate precision because of uncertainty in parameters in Sellmeier equation and manufacturer tolerance for glass thickness. Method

2 holds great promise for accurately determining GD but there some current practical issues

118 preventing us from adopting this method (see more detailed discussion in §5.5.2.1). We will use Method 3 and 4 to measure GD of an interferometer in this paper.

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! !%

Figure 5-1. Illustration of the DFDI method. Tilted lines represent interference combs generated by an interferometer. Vertical line represents an stellar absorption line (solid: original position with a frequency of ν0; dashed: shifted position with a frequency of ν).

In the DFDI method, GD determines the phase-to-velocity (PV) scale, the proportionality between the measured phase shift and the velocity shift. Since the DFDI method is realized by coupling a ﬁxed delay interferometer with a post-disperser, the resulting fringing spectrum−stellar absorption lines superimposing on the WLCs−is recorded on a CCD detector (illustrated in Fig. 5-1). The fringe phase is expressed by the following equation:

2π · τ(ν, y) · ν ϕ(ν, y) = , (5–2) c

119 where y is the coordinate along slit direction, which is transverse to dispersion direction,

τ is the optical path difference (OPD) of an interferometer and c is the speed of light. Two mirrors (arms) of the interferometer are designed to be tilted towards each other along the slit direction such that several fringes are formed along each ν channel. The intersection of a stellar absorption line and a WLC moves (from Po to P in Fig. 5-1) if there is a shift of an absorption line due to a change of stellar RV. Consequently, a small change of ϕ in the dispersion direction, ∆ϕx , is induced:

dϕ dϕ dν ∆ϕ = · ∆v = · · ∆v x dv dν dv dϕ ν = · · ∆v = Γ · ∆v, (5–3) dν c where Γ is deﬁned as phase-to-velocity scale (PV scale). It is determined by the GD of an interferometer, which becomes explicit if Equation 5–1 and 5–3 are combined:

ν Γ = −2π · GD · . (5–4) c

At resolutions typically adopted by the DFDI method (5, 000 ≤ R ≤ 20, 000), stellar lines

(line width∼0.1) are not resolved and a measurement of ∆ϕx is extremely difﬁcult. Instead,

∆ϕy , phase shift along y direction can be measured, which is equal to ∆ϕx if the combs generated by an interferometer are parallel to each other. This is a good approximation at very high orders of interference. The advantage of measuring ∆ϕy instead of ∆ϕx is seen from Fig. 5-1, in which the physical shift in the ν direction is amplified in y direction, the amplification rate is determined by the relative angle between the interferometer combs and a stellar absorption line. Therefore, ∆ϕy is relatively easier to measure compared to ϕx and it is measured by fitting a well-sampled periodical flux signal along the y direction in the DFDI method. Compared to conventional high-resolution Echelle method, the number of freedom for the DFDI method in the fitting process is much less and small Doppler phase shift can be relatively easier detected with a simple functional form, i.e., a sinusoidal function. However, we want to point out that while the DFDI method provides a boost in instrument Doppler

120 sensitivity, the Doppler sensitivity is not strongly dependent on the ampliﬁcation rate because

ﬂux slope decreases as ampliﬁcation rate increases, which negates the gain of phase slope.

5.2 GD Measurement Using White Light Combs

5.2.1 Method

MARVELS (Multi-object Apache Point Observatory Radial Velocity Exoplanet Large-

area Survey) instrument covers a wavelength range from 500 nm to 570 nm and uses

a post-dispersive grating with a spectral resolution of 11,000 after a ﬁxed delay interfer-

ometer (Ge et al., 2009). A Th-Ar emission lamp and an iodine absorption cell serve as

wavelength calibration sources. The instrument setup of MARVELS (Ge et al., 2009; Wan

et al., 2009) is similar to the equipments that measure GD as described in Kovacs´ et al.

(1995) and Amotchkina et al. (2009), in which a white light interferometer (WLI) is combined

with a post-disperser. However, the OPD is scanned by a moving picomotor in Amotchk-

ina et al. (2009) while it is realized by two relatively tilted arms in the WLC method using

MARVELS instrument. WLCs are generated by the interferometer when fed by a white light

source (e.g., a tungsten lamp). ϕ(ν), the phase of each frequency channel ν, is measured

and then unwrapped to remove ambiguity of 2π. GD is then derived by taking the derivative of ϕ(ν) according to Equation 5–1.

Fig. 5-2 shows an example WLCs created by an interferometer with a ﬁxed delay of

4 mm. The combs is created by an input continuum modulated with frequency due to con-

structive and destructive interference. The phase is determined by Equation 5–2. The phase

can be measured with a Fourier-transform-based algorithm described by (Rochford & Dyer,

1999): the signal H(ν) is obtained by ﬁrstly removing the negative Fourier components of

F (ν)−the ﬂux distribution with frequency−and then conducting an inverse Fourier transform.

The phases ϕ(ν) are obtained by calculating and unwrapping the arguments of H(ν). Fig. 5-3

shows the unwrapped phase measured from ﬂux distribution in Fig. 5-2. GD can be deter-

mined by measuring the derivative of unwrapped phase with respect to frequency according

to GD deﬁnition (Equation 5–1).

121 Figure 5-2. Simulated WLCs of an interferometer with a ﬁxed delay of 4mm. Flux is modulated with frequency due to constructive and destructive interference.

5.2.2 Data Reduction

Standard spectroscopy reduction procedures are performed with an IDL data reduction

pipeline dedicated to MARVELS. Figure 5-4 shows an example of normalized ﬂux as a function of frequency for a processed spectrum. A zoom-in sub-plot shows the WLCs produced by frequency modulation of the interferometer. Visibility, deﬁned as the ratio of half of peak-valley value to the DC offset, increases with frequency in the red part of the spectrum. The increasing visibility in the blue end of the spectrum is not physical but caused by an increasing photon noise and our algorithm of visibility calculation.

The fringe phases as a function of ν are calculated by the Hilbert transform technique (Rochford & Dyer, 1999) described in §5.2.1. We ﬁnd that the phase change between

122 Figure 5-3. Phase of simulated WLCs. Phase can be calculated by Fourier-transform-based algorithm described in §5.2.2.

pixels exceeds π in the blue part and therefore the phase unwrap cannot be successfully applied, so we decide to use only part of the spectrum with a pixel range from 1800 to 3800 for phase unwrapping. A third-order polynomial is used to ﬁt ϕ as a function ν. GD is then calculated according to its deﬁnition (Equation 5–1).

5.2.3 GD Measurement Results

The top view and side view of the MARVELS interferometer are shown in Fig. 5-5. 60

ﬁbers are mounted and each creates two spectra, one is picked from the forwarding beam

and the other one is from the returning beam (see Fig. 5-5). In total, 120 spectra are formed,

allowing us to measure GD at 60 positions on the interferometer along vertical (slit) direction.

Each position corresponds to a ﬁber number. There are 24 pixels along the slit direction

123 Figure 5-4. The normalized flux and visibility (γ) as a function of frequency of a tungsten spectrum taken with MARVELS. The solid line is the normalized flux and filled circles represent visibilities in different frequency channels. for each spectrum. We chose 15 rows in the middle to measure GD because of relatively higher photon flux, and thus smaller photon noise in the middle region of the spectrum.

The top panel of Fig. 5-6 shows phase measurement results for center row as a function of frequency at different ﬁber numbers. Phase ﬁtting residual (shown on the bottom panel of Fig. 5-6, RMS=0.9 rad) is consistent with photon-noise limited measurement error (see

§5.2.4 for details). GD for a particular ﬁber number is obtained by averaging the results of

GD measurements for those rows associated with the fiber. Figure 5-7 shows the results at ν=550 THz as a function of fiber number. Note that the two arms of the interferometer are intentionally tilted to each other and the 60 fibers are evenly mounted along the slit

124 direction. The measured GDs should gradually vary with ﬁber number. We use a second-

order polynomial to fit the GD variation with the fiber number. The fitting residual has an

RMS of 0.0046 ps. Figure 5-8 shows ﬁtted GD as a function of frequency for different ﬁbers.

GD varies 0.15 ps (0.6%) across measurement range from 540 to 565 THz. Ignoring GD

dependence of frequency would result in 180 m · s−1 measurement offset between two ends of measurement range (assuming a true RV of 30,000 m · s−1, which is a typical stellar

RV value due to the Earth’s barycentric motion). Table 5-1 provides the polynomial ﬁtting

coefﬁcients of GD vs. ﬁber number at different frequencies within measurement range.

Figure 5-5. Top: top view of an individual fiber beam feeding of the MARVELS interferometer. Two spectra are formed by one fiber. One (Slit A) is from the returning beam arm while the other one (Slit B) is from the forwarding beam arm. Bottom: side view of the fiber array beam feeding of the MARVELS interferometer. There are 60 fibers yielding 120 spectra. Note the exaggerated wedge angle of the shown second surface mirror, GD gradually changes along the vertical direction.

125 Figure 5-6. Top: white light combs phase as a function of frequency at different ﬁber locations. Bottom: phase residual after third-order polynomial ﬁtting.

5.2.4 GD Measurement Error Analysis

Two physical parameters, ϕ and ν, are measured in the experiment. The uncertainty of the ϕ measurement is ∼0.8 rad under photon-noise limited condition assuming a S/N of

120 and a typical fringe visibility of 1.5%. The uncertainty due to the wavelength calibration is ∼0.002 THz (0.02Å). We conduct a bootstrapping process to investigate the uncertainty of GD caused by the measurement uncertainties of ϕ and ν. We add gaussian noises with

standard deviation of measurement errors to both ϕ and ν and calculate the group delay.

We run 1000 iterations for bootstrapping in order to estimate the uncertainty of GD. The

median of the relative error of GD measurements, δGD/GD, is 4.4 × 10−5. In comparison,

the median of the relative GD measurement error is ∼ 1.8 × 10−4 after smoothing by ﬁtting

126 Figure 5-7. Measured group delay as a function fiber number. Filled circles are measured results, solid line represents the best second-order polynomial fitting with an RMS fitting error of 0.0046 ps. Measurement results can be found in Table 5-1 at other frequencies. a polynomial to GD variation with the fiber number. This number does not agree with the relative GD error predicted by the bootstrapping experiment. We suspect that the uncounted error in the bootstrapping simulation comes from image distortion due to optics which the data pipeline has not fully corrected for, e.g., spectrum curvature, spectral line slant, etc.

In a 2-D spectrum as illustrated in Fig. 5-1, the phase shift between adjacent pixels along slit direction is ∼0.6 rad, and the phase shift between each wavelength chanel is ∼2.5 rad.

An imperfect spectrum curvature tracing tends to shift pixel in the slit direction while an imperfect slant correction can affect pixel shifting in both slit and dispersion directions. The range of unwrapped phase is ∼4000 rad. For one ﬁber, if a gradually-changing phase error

127 Figure 5-8. GD as a function of frequency at different fiber numbers. GD measurement results vs. frequency and fiber numbers can be found in Table 5-1. is introduced by the data pipeline when correcting for the optical distortion, for example, 0.4 rad deviation from true value at one end while no deviation at the other end, a relative error of GD would be caused with an estimation of 0.4/4000 = 1 × 10−4. If different fibers are treated independently, which is the case for the MARVELS data reduction pipeline, then this gradually-changing phase error, introduced by imperfect optical distortion correction, may explain the standard deviation error we see after the polynomial fitting for GDs as a function of fiber number.

128 Table 5-1. GD measurement results as a function of spectrum number 2 (GD(#) = C0 + C1 · # + C2 · # ) and standard deviation (δGD) at different frequencies (ν)

ν[THz] C0 C1 C2 δGD[ps] 540.0000 -2.5195975233e+01 4.1224184409e-03 -1.3537701123e-05 0.0066 542.0000 -2.5216854974e+01 4.8064043246e-03 -2.3438315106e-05 0.0070 544.0000 -2.5219400268e+01 3.8524896642e-03 -8.1355543995e-06 0.0057 546.0000 -2.5232735338e+01 3.5346857088e-03 -2.6247780042e-06 0.0055 548.0000 -2.5246276599e+01 3.5171101494e-03 -3.3394045215e-06 0.0047 550.0000 -2.5259435353e+01 3.3877963986e-03 -1.7477210448e-06 0.0046 552.0000 -2.5270775315e+01 2.8029235840e-03 9.6338400722e-06 0.0066 554.0000 -2.5286851225e+01 3.5335302125e-03 -4.0412480867e-06 0.0093 556.0000 -2.5295872014e+01 3.4000221425e-03 -2.5813725398e-07 0.0067 558.0000 -2.5303654859e+01 2.3432563708e-03 1.8389109388e-05 0.0087 560.0000 -2.5319347252e+01 2.6464319719e-03 1.2493738694e-05 0.0075

5.3 GD Calibration: Observing an RV Reference Star

5.3.1 Method

A deviated PV scale would result in an inaccurate velocity measurement given the same

amount of fringe phase shift:

′ ′ ∆ϕ = ∆v · Γ = ∆v · Γ, (5–5) where ∆v ′ represents a measured velocity shift while ∆v represents a true velocity shift.

Combining Equation 5–4 and 5–5, we obtain the following equation from which GD can be

calculated by using the measured velocity shift of an object with a known velocity.

GD′ · ∆v′ GD = . (5–6) ∆v

This approach is similar to that of Barker & Schuler (1974), but the difference is that the latter

applied correction for discrete laser frequencies while we seek corrections for a continuous

frequency distribution.

In order to realize the method of GD calibration using an RV reference star, we need to:

1), assume a GD′ that is close to the true value of GD; 2), measure velocity shift ∆v ′ based

on an assumed GD′; 3), know the true value of the velocity shift of an RV reference star.

129 Table 5-2. MARVELS predicted RV uncertainty (at an average S/N of 100) vs. Teﬀ

Teﬀ[K] 4500 5000 5500 6000 6500 7000 δv ′[m · s−1] 1.9 2.3 2.7 3.1 3.5 4.0

5.3.2 GD Calibration Precision

The calibration precision using an RV reference star is determined by measurement

error of ∆v ′: GD′ · δv′ δGD = , (5–7) ∆v where δv ′ is RV measurement uncertainty. Wang et al. (2011) provided a method of calculat-

ing photon-limited RV uncertainty for the DFDI method. Under photon-noise limited condition,

we expect the GD calibration error to be determined by the photon-limited RV uncertainty

within the instrument band width, which is provided in Table 5-2. In Equation 5–7, GD′ is usu-

ally estimated to be within a few percent of true GD, ∆v is statistically ∼10,000 m · s−1 given

a uniform reference star distribution and a quarter year observational availability. Therefore,

−4 relative error of GD measurement is ∼ 2 × 10 for an RV reference star with a Teﬀ of 4500 K.

5.4 Implementation of Measured GD in Astronomical Observations

We use an RV reference star, HIP 14810 (V=8.5), as an example to show the RV

measurement results after implementation of the newly measured GD using the WLC

method. HIP 14810 is a star known to harbor 3 planets and its RV jitter is estimated to be 2

m · s−1 (Wright et al., 2009). After RV changes due to instrument drift, the Earth’s barycentric

motion and orbiting planets are removed, RV RMS error is 17.13 m · s−1 but has not reached the predicted photon-limited RV uncertainty (4.8 m · s−1, S/N=80 with a half wavelength coverage from 535 to 565 nm). RV RMS error is expected to be further reduced after the data pipeline is improved in the future.

We also examine the reference star GD calibration method. We use one spectral block within measurement range centering at 550 THz (540-560 THz) and set GD′ to be an arbitrary value of -23.873 ps. The measured RVs (barycentric velocity not corrected) are shown in Fig. 5-9. After applying correction according to Equation 5–6, we ﬁnd that the

130 GD is -25.107±0.027 ps. In comparison, GD measurement result of ﬁber number 51 (the

fiber for HIP 14810) using the WLC method gives -25.091±0.005 ps (refer to Table 5-1). We confirm that the GDs measured by these two methods are consistent with each other at 68% significance level.

Figure 5-9. Top: measured (∆v ′) and true (∆v) RVs of HIP 14810 (barycentric velocity not corrected) over a period of 70 days. Bottom: the ratio of ∆v ′ and ∆v as a function of time.

Many binary and brown dwarf discoveries are made owing to the well calibrated interferometer GD. As of June 2012, more than 250 binaries have been discovered and a dozen of brown dwarfs have been discovered by MARVELS and later conﬁrmed by follow- up observations conducted at other observatories (Fleming et al., 2010; Lee et al., 2011;

Wisniewski et al., 2012).

131 Table 5-3. Comparison between two methods of GD measurement and calibration WLC RS Spectrum Coverage Half Full S/N ∼15 100 for V∼8a Current precision 4.6 × 10−3 ∼ 9.3 × 10−3 ps 0.027 ps Current RV errorb ∼ 2 m · s−1 10.8 m · s−1 Potential precision ∼ 3.1 × 10−4 ps ∼ 5 × 10−3 ps Potential RV errorb ∼ 0.1 m · s−1 ∼ 2 m · s−1 Dependence on observation × X Dependence on pipeline X X

Note. — a: assuming MARVELS throughput; b: RV error is calculated assuming a true velocity shift (∆v) of 10,000 m · s−1 according to Equation 5–6.

5.5 Summaries and Discussions

5.5.1 Summaries

The PV scale is an important parameter in the DFDI method that translates a measured phase shift to an RV shift, and is determined by the group delay (GD) of an interferometer.

We have provided and discussed two methods of GD measurement and calibration: 1),

GD measurement using white light combs (WLCs) generated by the interferometer in a

DFDI Doppler instrument; 2), GD calibration using an RV reference star (RS). Table 5-3 summarizes the main results and the comparison between these two methods. The accuracy of GD measurement is sufﬁcient for current RV precision achieved with instruments using the

DFDI method (Fleming et al., 2010; Lee et al., 2011; Muirhead et al., 2011). However, higher measurement and calibration precision is required in the near future as higher RV precision is achieved by DFDI instruments in search for exoplanets. RS and WLC methods can serve as complementary methods of GD measurement and calibration for DFDI instruments.

5.5.2 Discussions

5.5.2.1 White Light Comb (WLC) Method

The GD measurement using WLCs created by the interferometer provides a direct way of calibrating the PV scale. In the region where combs are visible, effective S/N is relatively

132 low (∼15) because of low comb visibility (1.5%). In addition, GD cannot be measured in the

region where combs are not visible, which limits the application of this method. We are able

to measure GD in a region that accounts for half of the spectrum coverage. Extrapolation

beyond the measurement range may result in large uncertainties. The major issue facing

the method is that the data reduction pipeline may have introduced unknown errors while

correcting optical distortions such as spectrum curvature and slant.

In principle, we can use a tungsten lamp with an iodine cell or a Th-Ar lamp instead of

a tungsten lamp in order to increase the fringe visibility. However, there are some practical

concerns that hinder us from applying the above solutions: 1) line blending, because of low

spectral resolution, many spectral lines can not be resolved and it is not certain at this stage

how line blending affects phase measurement; 2) illumination correction, which is required

to correct for illumination proﬁle in slit direction in order to properly measure the phase. We

adopt a self-illumination correction procedure in the pipeline which requires a certain contin-

uum level to be successfully achieved. Th-Ar is less affected by line blending if a careful line

selection process is involved, but it does not have enough continuum level for self-illumination

correction. An experiment is being conducted in which a second interferometer is used to

improve the visibility of WLCs so that GD is more precisely measured at a higher effective

S/N for a wider frequency coverage.

When compared to previous work in the ﬁeld of GD measurement, Amotchkina et al.

(2009) achieved a measurement precision of 1 × 10−4 ps. Our measurement of GD has a typical accuracy of ∼ 6 × 10−3 ps (limited by effective S/N and systematic errors), which is more than an order of magnitude lower. However, it is shown in §5.2.4 that substantial

improvement would be able to be achieved once the data reduction pipeline has a better

handle of optical distortion. Wan et al. (2010) measured GD for the MARVELS interferometer

using a scanning WLI method and achieved a precision of 0.6 × 10−5 ∼ 2.4 × 10−5 ps,

which is more than two order of magnitude better than the results in this paper. However,

there are practical concerns using GD measurement results from the scanning WLI method

133 because they are not measured in situ, therefore it is not easy to associate a position in a

WLI measurement to a ﬁber position.

5.5.2.2 Reference Star (RS) Method

The GD calibration using an RV reference star (RS) is a self-calibrating process and has

the potential of achieving a high calibration precision if the following requirements are met:

1), the RV reference star has a large velocity shift during a observation window; 2), the RV

reference star is bright; 3), the data reduction pipeline is able to produce the photon-limited

RV precision. In addition, GD is practically measured within a certain band width: ∫ ν ω ν ν ∆ν GD( ) ( )d GD(ν) = ∫ , (5–8) ω ν ν ∆ν ( )d

where ω(ν) is weight function. The band width, ∆ν, should be small such that the dispersion effect is negligible. The limitations stated above prevent us from precisely determining the PV scale at the position of each ﬁber because not every ﬁber has a continuous observation on a bright known RV reference star. For MARVELS, the brightest RV reference star available has V mag of 8 and the resulting S/N is ∼100 per pixel. However, the RS method is a very

promising approach for a single-object DFDI instrument because only one bright reference

star is required in the ﬁeld. We are planning to apply this method in calibrating GD of another

DFDI instrument (EXPERT) at KPNO 2.1m telescope (Ge et al., 2010). Note that the S/N

can be further increased by conducting multiple independent measurements and increasing

instrument throughput.

5.5.2.3 A Future M-Dwarf Survey With the DFDI Method

The MARVELS survey concept can be adopted by a future M-dwarf planet survey using

the DFDI method. As the planets around M dwarfs are gaining increasingly more attention, a

survey of nearby M dwarfs is important in order to completely understand planet occurrence

rate for M dwarfs and its dependency on stellar metallicity, stellar mass and activity and so

on. This type survey is ideal for large area spectroscopic telescopes such the SDSS 2.5m

telescope and the LAMOST telescope. There are about 15,000 mid-late type M dwarfs

134 with J magnitude less than 12 in the northern sky (Wang et al., 2011). Considering about 6 square degree ﬁeld of view for these telescopes, there are 5 such stars in the ﬁeld of view.

According to previous study on the multi-object case, a resolution of 5,000 would be optimal for a multi-object DFDI survey (see §2.3.4). At such resolution, we expect 125 m/s RV precision for a J=12 star and 8 m/s for a J=6 star. With this design concept and instrument speciﬁcation, a 1 K by 1 K near infrared detector if sufﬁcient for 800 to 1350 nm wavelength coverage.

135 CHAPTER 6 ECCENTRICITY DISTRIBUTION FOR SHORT-PERIOD EXOPLANETS

6.1 Introduction

The discovery of exoplanets has signiﬁcantly advanced our understanding of formation and evolution of planetary system (Marcy & Butler, 1996; Mayor & Queloz, 1995; Wolszczan,

1994). As of February 2011, over 500 exoplanets have been discovered including 410 sys-

tems detected by radial velocity (RV) technique1 . The eccentricity distribution of exoplanets

is very different from that of solar system. For sufﬁciently short-period planets, it is expected

that tidal circularization would lead to nearly circular orbits. Yet, several short-period planets

appear to have eccentric orbits. Several mechanisms (e.g. planet scattering, Kozai effect)

have been proposed to explain the observed eccentricity distribution (Ford & Rasio, 2008;

Juric´ & Tremaine, 2008; Takeda & Rasio, 2005; Zhou & Lin, 2007). This chapter aims to

improve our understanding of the true eccentricity distribution and its implications for orbital

evolution.

The Bayesian approach offers a rigorous basis for determining the posterior eccentricity

distribution for individual system. The Bayesian method is particularly advantageous relative

to traditional bootstrap method when the orbital eccentricity is poorly constrained by RV

data (Ford, 2006). Ford (2006) discussed eccentricity estimation using Markov Chain

Monte Carlo (MCMC) simulation in the framework of Bayesian inference theory and found

a parameter set that accelerates convergence of MCMC for low eccentricity orbit. For a

population of planets on nearly circular orbits, eccentricity estimates for planets on circular

orbit are biased resulting in overestimation of orbital eccentricities (Zakamska et al., 2011).

Further complicating matters, the population of known exoplanets is not homogeneous, and

the observed eccentricity distribution is affected by the discovery method, selection effects

and data analysis technique.

1 http://exoplanet.eu/; http://exoplanets.org/

136 6.2 Method

We select all the systems with: 1) a single known planet discovered with the radial velocity technique as of April 2010; 2) an orbital period of less than 50 days; and 3) a publicly available radial velocity data set. We exclude planets discovered by the transit technique in order to avoid complications due to selection effects (Gaudi et al., 2005). We perform an orbital analysis on each system in our sample using: 1) a standard MCMC analysis (§2.1) and 2) a new method, Γ analysis (described in §2.2). We focus on the eccentricity estimation for each planet since the eccentricity is an important indication of orbital evolution and tidal interaction.

6.2.1 Bayesian Orbital Analysis of Individual Planet

We performed a Bayesian analysis of the published radial velocity observation using a model consisting of one low-mass companion following a Keplerian orbit. If a long-term RV trend is included in the original paper reporting the RV data or if a linear trend of more than 1 m · s−1 · yr−1 is apparent, then we add to the model a constant long-term acceleration due to distant planetary or stellar companion.

We calculate a posterior sample using the Markov Chain Monte Carlo (MCMC) technique as described in Ford (2006). Each state in the Markov chain is described by the ⃗ parameter set θ = {P, K, e, ω, M0, Ci , D, σj }, where P is orbital period, K is the velocity semi-amplitude, e is the orbital eccentricity, ω is the argument of periastron, M0 is the mean anomaly at chosen epoch τ, Ci is constant velocity offset (subscript i indicates constant for different observatory), D is the slope of a long-term linear velocity trend, and σj is the ”jitter” parameter. The jitter parameter describes any additional noise including both astrophysical noises, e.g., stellar oscillation, stellar spots (Wright et al., 2005) and any instrument noise not accounted for in the reported measurement uncertainties. The RV perturbation of a host star at time tk due to a planet on Keplerian orbit and possible perturbation is given by

= · ω + + · ω + · − τ vk,⃗θ K [cos( T ) e cos( )] D (tk ), (6–1)

137 where T is the true anomaly which is related to eccentric anomaly E via the relation,

( ) √ ( ) T 1 + e E tan = tan . (6–2) 2 1 − e 2 The eccentric anomaly is related to the mean anomaly M via Kepler’s equation,

2π E(t) − e · sin[E(t)] = M(t) − M = (t − τ). (6–3) 0 P We choose priors of each parameter as described in Ford & Holman (2007). The prior

is uniform in logarithm of orbital period. For K and σj we use a modiﬁed Jefferys prior in the

−1 −1 form of p(x) ∝ (x + xo ) (Gregory, 2005) with Kmin = σj,min = 0.1 m · s . Priors are uniform ⃗ for: e (0 ≤ e ≤ 1), ω and M (0 ≤ ω, M < 2π), Ci and D. We veriﬁed that the parameters in θ

did not approach the limiting values. We assume each observation results in a measurement

drawn from normal distribution centered at the true velocity, resulting in a likelihood (i.e.,

conditional probability of making the speciﬁed measurements given a particular set of model

parameters) of

∏ 2 2 exp[−(v ⃗θ − vk ) /2σ ] p ⃗v|⃗θ, M ∝ k, k , ( ) σ (6–4) k k

where vk is radial velocity at time tk , and vk,θ is the model velocity at time tk given the model ⃗ parameters θ. Noise σk consists of two parts. One component is from the observation uncertainty σk,obs reported in the radial velocity data, and the other is the jitter, σj , which accounts for any unforseen additional noise including instrument instability and stellar jitter.

The two parts are added in quadrature in order to generate σk . We calculate the Gelman-

Rubin statistic, Rˆ , to test for nonconvergence of Markov chains.

We perform a MCMC analysis for RV data set of each system in the sample and obtain posterior samples of h and k, where h = e cos ω and k = e sin ω. This parameterization

has been shown to be more effective in description of the eccentricity distribution for low

eccentric orbits (Ford, 2006). We take steps in h and k and adjust the acceptance rate

138 according to the Jacobian of the coordiante transformation, so as to maintain a prior that is uniform in e and ω. Mean values, h¯ and k¯, from posterior samples of h and k are adopted √ 2 2 to calculate eMCMC using the equation eMCMC = h¯ + k¯ . The posterior distribution of e is not always Gaussian distribution especially near e ∼0. Therefore, it is not appropriate to calculate the uncertainty of e using the equation of error propagation in which gaussian noise is assumed. We use posterior distribution of e to infer the credible interval of e. The boundaries of the region where 68% posterior samples populate are adopted as elower and eupper (Table 6-1). Figure 6-1 illustrates two examples of how the credible intervals are inferred for HD 68988 (eccentric orbit) and HD 330075 (circular orbit).

Figure 6-1. Examples of how credible intervals of standard MCMC analysis are calculated using posterior distribution of e. Grey region contains 68% of total number of posterior samples of e.

139 6.2.2 Γ Analysis of Individual Systems

A fully Bayesian analysis of the population of exoplanet eccentricities would be computationally prohibitive due to the large number of dimensions. Therefore, we develop a hybrid Bayesian-frequentist method to evaluate the signiﬁcance of a non-zero eccentricity measurement. We combine a bootstrap style approach of generating and analyzing synthetic data sets with MCMC analysis of each synthetic data set to obtain a frequentist conﬁdence level for each eccentricity that accounts for biases. First, we perform the standard MCMC analysis described in §2.1 on the real RV data set and adopt the mean value of each orbital parameter in ⃗θ except e. We generate a series of simulated radial velocity data sets at different values of e. The adopted K is scaled accordingly to K ∝ (1 − e2)−0.5. The simulated radial velocity data has the same number of observations, and each simulated observation takes place at exactly the same time and the same mean anomaly as the real observation.

Gaussian noise with standard deviation of σk (§2.1) are added to simulated radial velocity data sets at different eccentricities. Each simulated RV data set has the same reported RV measurement uncertainties as the real RV observations. Standard MCMC analysis is then performed on each of the simulated RV data sets.

For both real and simulated data sets, we construct a two-dimensional histogram using the posterior samples in (h, k) space to approximate a two-dimensional posterior distribution for h and k, dr (hi , kj ) and ds (hi , kj ), where i and j denote bin indices, and the subscripts r and s denote the real and simulated data set. We compare the distribution for each simulated data set ds (hi , kj ) to the distribution for real radial velocity data set dr (hi , kj ).

To quantify the similarity between ds (hi , kj ) and dr (hi , kj ), we calculate the statistic deﬁned ∑N ∑N 2 as Γ = [ (ds (hi , kj ) − µs ) · (dr (hi , kj ) − µr )]/[σs σr (N − 1)], where N is the number i=1 j=1 of bins in h or k dimension, µ and σ represent mean and standard deviation. In other words, the Γ statistic is obtained by cross-correlating two posterior distributions in h and k space. Figure 6-2 illustrates the process by which we obtain Γ for the case of HD 68988.

If ds (hi , kj ) matches dr (hi , kj ), we expect to obtain a Γ value that approaches unity (blue

140 and red contours). If the samples differ signiﬁcantly then Γ decreases towards zero (red

and green contours). For each eccentricity, we simulated 21 radial velocity data sets and

compare ds (hi , kj ) with dr (hi , kj ) to obtain 21 Γ statistics between simulated and real RV data.

We choose the median value Γ¯ as an indicator of overall similarity at given eccentricity.

Figure 6-2. Contours of posterior distribution in h and k space for HD 68988 (Solid-68% of sample points included; dashed-95% of sample points included; dotted-99% of sample points included). Red contours are posterior distribution for real observation, green contours are for simulated RV data set with e=0.00, and blue contours are for simulated RV data set with e=0.13. Eccentricity of HD 68988 is 0.1250±0.0087 according to Butler et al. (2006).

Based on above analysis of simulated radial velocity data with different input eccentric-

ities e, we obtain a relationship between Γ¯ and e, i.e. Γ¯(e). We use a high-order polynomial

to interpolate for Γ¯(e). We deﬁne e¯ to be the eccentricity at which Γ¯(e) reaches maximum

and we interpret e¯ as an estimator of eccentricity. For HD 68988 (Fig. 6-3), input eccentricity

141 ranges from 0.00 to 0.29 with step size of 0.01. Γ¯(e) reaches maximum at e = 0.134. We

estimate statistical conﬁdence level of e¯ using every pair of posterior eccentricity samples

calculated from the data sets that are generated assuming the same eccentricity. Consider

the example of HD 68988 again: 21 sets of posterior distribution in h and k space, ds (hi , kj ),

are obtained. Comparison between each pair gives a Γ statistic between simulated RV data.

20 = Σi=1 210 Γ statistics in total for simulated RV data sets are calculated at eccentricity of 0.13 and 68.1% of pairs (143 out of 210) have a Γ statistic greater than 0.3714. We deﬁne

this value, Γc,0.68, as the critical Γ value for HD 68988 at 68% conﬁdence level for e = 0.13.

Therefore, if Γ statistic obtained in comparison between ds (hi , kj ) and dr (hi , kj ) is less than

Γc,0.68, we argue that the eccentricity inferred from simulated RV data set is not consistent

with the observed eccentricity of the system at 68% conﬁdence level. In the case where e¯ is

located between grids of simulated e values, we calculate Γc,0.68 at e¯ using interpolation of

Γc,0.68 at nearby e values. We use a high-order polynomial to approximate the discrete data

Γ¯(e). The polynomial is later used to infer e¯, lower and upper limit of eccentricity. For HD

68988, Γc,0.68 is 0.3663 at e¯ = 0.134 after interpolation. Using the relationship between Γ¯ and

e (Fig. 6-3), we look for the e values corresponding to Γc,0.68 as estimators of the lower and upper limit for eccentricity of the planet system at a 68% conﬁdence level. In HD 68988, we

= +0.040 = +0.025 obtained e 0.134−0.040 using Γ analysis. In comparison, we have obtained e 0.119−0.022 using a standard MCMC analysis and Butler et al. (2006) reported e = 0.125 ± 0.009.

6.3 Results for Individual Planets

In our sample of 50 short-period single-planet systems, we successfully analyzed 42

systems using Γ analysis, and 46 systems using standard MCMC analysis (Table 6-1).

All the error are based on a 68% conﬁdence level (Γ analysis) or a 68% credible interval

(MCMC). The unsuccessful cases in standard MCMC analysis are HD 189733, HD 219828,

HD 102195 and GJ 176. In HD 189733, most of the RV data points were taken during

observation of Rossiter-McLaughlin effect, which is not modeled here. MCMC analysis fails

−1 for HD 219828 and GJ 176 because of low signal to noise ratio (K=7 m · s and nobs =20

142 Figure 6-3. Γ¯ as a function of eccentricity e for HD 68988. Open circles are results from simulations, solid line is the result of polynomial ﬁtting. The long dashed line is the critical threshold, Γc,0.68 at the 68% conﬁdence level.

−1 for HD 219828 and K=4.1 m · s and nobs =57 for GJ 176). RV data points of HD 102195 were taken at 3 observatories and MCMC analysis is complicated using different observatory offsets. In addition to the above, Γ analysis was unsuccessful for HD 162020, GJ 86, HD

17156 and HD 6434. Since Γ analysis involves generating simulated RV data, the limited number of observations and partial phase coverage for these systems can cause poor convergence for some simulated data sets. These limitations become more severe for systems with high eccentricity (e.g. HD 17156, e = 0.684) since phase coverage is more important for eccentric orbits.

143 Figure 6-4. Top: comparison between standard MCMC analysis (blue) and previous references (red); Bottom: comparison between standard MCMC analysis (blue) and Γ analysis (green). The systems where disagreements take place are marked with a number: 1, HD 149026; 2, τ Boo; 3, HD 195019.

6.3.1 Comparison: Standard MCMC and References

In Fig. 6-4 (top), we compare results from two sources, standard MCMC analysis and previous references. In most cases the two methods provide similar results. We found there are 3 systems for which the eccentricity estimates are not consistent, i.e., the published eccentricity error bar does not overlap the 68% credible interval from our MCMC analysis.

They are HD 149026, τ Boo, HD 195019. For HD 149026, Sato et al. (2005) set the

eccentricity to be zero when ﬁtting the orbit. In contrast, we treat eccentricity as a variable

and the standard MCMC method found that the 68% credible interval for the systems

mentioned above does not include zero. In addition, HD 149026 b is a known transiting

144 planet and transit photometry provides additional constraints on eccentricity which we

= +7.2 have not included (Charbonneau, 2003). Knutson et al. (2009) measured ∆tΠ 20.9−6.2 minute(2.9σ) for HD 149026 which is inconsistent with zero eccentricity, because e ≥ ω ≃ π e cos 2P ∆tΠ, where P is period and ∆tΠ is the deviation of secondary eclipse from midpoint of primary transits. Standard MCMC results for other two systems (τ Boo and HD

195019) are not consistent with those from previous references even though eccentricity was

treated as variable in previous references. Butler et al. (2006) report e = 0.023 ± 0.015 for τ

Boo and e = 0.014 ± 0.004 for HD 195019. On the contrary, standard MCMC analysis gives

+0.0382 τ +0.0049 e=0.0787−0.0246 for Boo and e=0.0017−0.0017 for HD 195019 (See Table 6-1). 6.3.2 Comparison: Standard MCMC and Γ Analysis

Figure 6-4 (bottom) compares the results from standard MCMC analysis and Γ analysis.

We ﬁnd that the 68% credible/conﬁdence intervals for the two methods overlap in the cases

where there are discrepancies between standard MCMC analysis and previous references

(see § 6.3.1). The conﬁdence interval from the Γ analysis is generally larger than the credible interval from a standard MCMC analysis. The larger uncertainty from Γ analysis is likely due

to the analysis accounting for the uncertainty in each velocity observation twice, ﬁrst when

generating synthetic data sets and a second time when analyzing the simulated data. Thus,

the Γ analysis results in slightly larger uncertainty in eccentricity estimation.

In order to understand the behavior of standard MCMC analysis and Γ analysis for

planets on nearly circular orbits, we conduct an additional experiment generating many

synthetic data sets where each system has a single planet on a circular orbit. We assume

that they are observed at the same times and with the same RV measurement precisions

as actual RV data sets. In order to understand the bias of each method for nearly circular

systems, we compare the output eccentricities and their uncertainties. Using standard

MCMC analysis, we ﬁnd that 76.4 ± 2.9% of the simulated data sets are consistent with zero

using a 68% credible interval, and 23.6 ± 1.6% of the simulated data sets have 68% credible

intervals that do not include zero. In contrast, for 16.8 ± 1.4% of the simulated data sets, the

145 Γ analysis does not result in a 68% conﬁdence interval that includes zero. In both cases,

more than 68% of data sets are consistent with a circular orbit at a 68% level using either

method. Using the Γ analysis, 6.8 ± 3.0% more simulated data sets are consistent with a circular orbit than based on the standard MCMC analysis. This conﬁrms our intuition that the

Γ analysis is a less biased method for analyzing systems at very small eccentricity. Thus, the

Γ analysis may be a useful tool in assessing the signiﬁcance of a measurement of a small non-zero eccentricity. In particular, we ﬁnd 5 cases (11.4%) in which elower =0 for Γ analysis

even though elower for MCMC is greater than zero (e.g., HD 46375, HD 76700, HD 7924, HD

168746, HD 102117).

A similar experiment is conducted except that an eccentricity of 0.2 is assigned to each

system instead of zero eccentricity. Again, we assess the accuracy of the two methods

by comparing the input and output eccentricities. When using the MCMC method, we ﬁnd

that the 68% credible interval for the eccentricity does not include the input eccentricity

for 26.0 ± 2.1% of simulated data sets. When using the Γ method, we ﬁnd that the 68%

conﬁdence interval for the eccentricity does not include the input eccentricity for 18.9 ± 1.8%

of simulated data sets. Again, there is a larger fraction of results from the MCMC method

that are not consistent with the input at a sizable eccentricity, indicating Γ analysis is less

likely to reject the correct eccentricity. We also investigate the bias of the two methods at a

signiﬁcant eccentricity (i.e. e=0.2). In the cases where the output eccentricities are consistent with the input, we ﬁnd that 47.9 ± 3.3% of the output eccentricities are below 0.2 while

52.1 ± 3.5% of outputs are over 0.2 for MCMC method indicating the MCMC method is not biased at a sizable eccentricity, which agrees with the ﬁnding from Zakamska et al. (2011).

In comparison, Γ analysis is also an unbiased analysis with 49.9 ± 3.3% below input and

50.1 ± 3.3% exceeding input. Therefore, we ﬁnd no evidence for signiﬁcant bias of either

method for data sets with a signiﬁcant eccentricity.

146 6.3.3 Discussion of Γ Analysis

The different methods for analyzing Doppler observations are complimentary. Bayesian methods and MCMC in particular are routinely used to sample from the posterior distribution for the Keplerian orbital parameters for a given system. However, the analysis of an exoplanet population is more complicated than simply performing a Bayesian analysis of each system. To illustrate this point, consider a population of planets that all have exactly circular orbits. Due to measurement uncertainties and finite sampling, the ”best-fit” eccentricity for each system will be non-zero. Similarly, since eccentricity is a positive-definite quantity, the analysis of each system will result in a posterior distribution that has significant support for e > 0. This property remains even if one combines many point estimates (e.g., ”best-fit”), frequentist confidence intervals or Bayesian posterior distributions. While the posterior distribution for the orbital parameters represents the best possible analysis of an individual system, the inevitable bias for nearly circular orbits is a potential concern for population analyses. Therefore, it is important to apply different methods for population analyses (e.g.,

Hogg et al. (2010); Zakamska et al. (2011)).

Since we intend to investigate the potential role of tidal effects on the eccentricity distribution of short-period planets, we developed a hybrid technique to assess the sensitivity of our results to bias in the posterior distribution for planets with nearly circular orbits. This hybrid technique (Γ analysis) involves performing Bayesian analyses of each individual planetary system along with several simulated data sets, each of varying eccentricity.

The MCMC analysis of each data set allows us to account for the varying precision of eccentricity measurements depending on the velocity amplitude, measurement precision, number of observations and phase coverage. We assess the extent of the eccentricity biases by performing the same analysis on simulated data sets with known eccentricity.

We compare the posterior densities for the actual data set to the posterior density for each of the simulated data sets to determine which input model parameters are consistent with the observations. We can construct frequentist conﬁdence intervals based on Monte Carlo

147 simulations (i.e., comparing the posterior distributions for the synthetic data sets to each other).

The basic approach of the Γ analysis is similar to likelihood-free methods more commonly used in approximate Bayesian computation. In this case, we do have a likelihood which allows us to sample from the posterior probability distribution using standard MCMC.

We compare the posterior densities calculated for several simulated data sets to the posterior density of the actual observations, so as to assess the accuracy and bias of the standard

MCMC analysis.

Figure 6-5. Cumulative distributions functions (CDFs) of eccentricities from different methods. The solid red line is for eMCMC , adopted from MCMC method (Table 6-1). The dotted red line is similar to MCMC, but any eccentricity with elower of 0 for a 95% credible interval is assigned to 0. The blue lines are for Γ analysis, where the solid line is for eΓ from the Γ analysis (Table 6-1) and the dotted line is similar, but any eccentricity with elower of 0 for a 95% conﬁdence interval is assigned to 0.

148 The problem of biased eccentricity estimators for nearly circular orbits is familiar from previous studies of binary stars. In particular, Lucy & Sweeney (1971) investigated the possibility of mistakenly assigning an eccentric orbit to a circular spectroscopic binary due to inevitable measurement errors. As many spectroscopic binaries may have been affected by tidal circularization, they suggested assigning a circular orbit to any system for which the eccentricity credible interval contained 0. When studying a population of systems for which circular orbits are common, this approach signiﬁcantly reduces the chance of erroneously concluding the system has a non-zero eccentricity. One obvious disadvantage of this approach is that it would lead to a negative bias for systems where the eccentricity is of order

σ/K, where σ is the typical measurement precision and K is the velocity amplitude. For binary stars, σ/K may be small enough that this is not a signiﬁcant concern. For exoplanets, where σ/K may be as small as ∼ 2 − 3, such a procedure would result in a signiﬁcant negative bias for many systems. The Γ analysis offers an alternative approach, which may be particularly useful when analyzing the eccentricity distribution of a population of planetary systems.

For the sake of comparison, we consider a modernized version of the Lucy & Sweeney

(1971) approach which is based on the posterior distribution from a standard MCMC analysis or the conﬁdence interval from our Γ analysis. We construct a histogram or cumulative distribution of the eccentricities for a population of systems, using a single summary statistic for each system: the posterior mean for the standard MCMC analysis or the eccentricity that maximizes the Γ statistic. Following Lucy & Sweeney (1971), we adopt an eccentricity of zero for any system for which the 95% signiﬁcance level (Γ method) or the 95% credible interval (MCMC) includes e = 0. The cumulative distribution functions of the eccentricities using different methods are plotted in Fig. 6-5. Based on the generalized Lucy & Sweeney

(1971) approach, ∼81% (70%) of the short-period planet systems in our sample are consistent with circular orbits using the Γ analysis (standard MCMC analysis). Clearly, the Lucy &

Sweeney (1971) approach results in a large fraction being assigned a circular orbit, largely

149 due to the choice of a 95% threshold. The fraction assigned a circular orbit is sensitive to

the size of the credible interval used when deciding whether to set each eccentricity to zero.

There is no strong justiﬁcation for the choice of the 95% threshold (as opposed to 68% or

99.9% threshold) and tuning the threshold to agree with other methods negates the primary

advantage of the Lucy & Sweeney (1971) method, that it requires no additional computa-

tions. Therefore, we do not recommend using the Lucy & Sweeney (1971) approach to learn

about the eccentricity distribution for a population when σ/K is not large.

6.4 Tidal Interaction Between Star and Planet

Several factors affect the eccentricity distribution of short-period planets including

tidal interaction between host star and planet and possible perturbation of an undetected

companion. We will discuss how these factors affect the eccentricity distribution and whether

the effect is observable.

In order to understand the inﬂuence of tidal interaction on eccentricity distribution, we

ﬁrst divide our sample into two subsets, one subset contains systems with a long-term RV

trend while the other subset contains systems that do not show a long-term RV trend (Fig.

6-6). The systems that are noted with a long-term velocity trend include 51 PEG, BD -10

3166, GJ 436, GJ 86, HD 107148, HD 118203, HD 149143, HD 68988, HD 7924, HD 99492

and τ Boo. We further divide the no-trend subset into two groups, one group is distinguished

by τage/τcirc ≥ 1, and the other group is distinguished by τage/τcirc < 1, where τcirc is tidal circularization time scale and τage is the age of the host star. We investigate whether there is a signiﬁcant difference in the eccentricity distribution between these two groups as expected if tidal interaction is an important factor in shaping eccentricity distribution.

Following Matsumura et al. (2008), we estimate τcirc using:

′ ( )5[ ′ ( )2( )5 ]−1 Q M a Q M R∗ τ = 2 p p p p + circ ′ F∗ Fp , (6–5) 81 n M∗ Rp Q∗ M∗ Rp where the subscripts p and ∗ denote planet and star, M is mass, R is radius, a is semi-major

′ 3 1/2 axis, Q is modiﬁed tidal quality factor and n = [G(M∗ + Mp)/a ] is the mean motion.

150 Table 6-1. Comparison of Eccentricities Calculated From Different Methods Name Ref. MCMC Γ eref δeref eMCMC elower eupper eΓ elower eupper HD 41004 B 0.081 0.012 0.058 0.000 0.109 0.000 0.000 0.085 HD 86081 0.008 0.004 0.058 0.008 0.166 0.013 0.001 0.098 HD 189733 0.000 0.000 ...... HD 212301 0.000 ... 0.015 0.000 0.051 0.034 0.000 0.091 GJ 436 0.159 0.052 0.191 0.146 0.248 0.223 0.128 0.295 HD 63454 0.000 ... 0.018 0.000 0.038 0.003 0.000 0.017 HD 149026 0.000 ... 0.192 0.118 0.270 0.182 0.050 0.344 HD 83443 0.012 0.023 0.007 0.000 0.020 0.006 0.000 0.037 HD 46375 0.063 0.026 0.052 0.030 0.084 0.065 0.000 0.121 HD 179949 0.022 0.015 0.014 0.000 0.024 0.013 0.000 0.042 τ Boo 0.023 0.015 0.079 0.054 0.117 0.086 0.034 0.139 HD 330075 0.000 ... 0.019 0.000 0.087 0.008 0.000 0.095 HD 88133 0.133 0.072 0.076 0.000 0.127 0.086 0.000 0.175 HD 2638 0.000 ... 0.041 0.000 0.076 0.005 0.000 0.042 BD -10 3166 0.019 0.023 0.010 0.000 0.030 0.019 0.000 0.064 HD 75289 0.034 0.029 0.021 0.000 0.043 0.000 0.000 0.063 HD 209458 0.000 ... 0.008 0.000 0.016 0.005 0.000 0.018 HD 2198281 0.000 ...... HD 76700 0.095 0.075 0.062 0.003 0.104 0.045 0.000 0.104 HD 149143 0.000 ... 0.012 0.000 0.022 0.009 0.000 0.014 HD 102195 0.000 ...... 51 Peg 0.013 0.012 0.007 0.000 0.014 0.006 0.000 0.020 GJ 6742 0.100 0.020 0.070 0.000 0.147 0.047 0.000 0.131 HD 49674 0.087 0.095 0.050 0.000 0.119 0.000 0.000 0.073 HD 109749 0.000 ... 0.045 0.000 0.070 0.042 0.000 0.143 HD 7924 0.170 0.160 0.119 0.022 0.224 0.054 0.000 0.297 HD 118203 0.309 0.014 0.293 0.264 0.328 0.297 0.218 0.367 HD 68988 0.125 0.009 0.118 0.096 0.143 0.134 0.094 0.174 HD 168746 0.107 0.080 0.079 0.025 0.139 0.086 0.000 0.155 HD 1852693 0.276 0.037 0.276 0.242 0.314 0.279 0.175 0.389 HD 162020 0.277 0.002 0.277 0.274 0.279 ...... GJ 1764 0.000 ...... HD 130322 0.011 0.020 0.007 0.000 0.052 0.031 0.000 0.086 HD 108147 0.530 0.120 0.526 0.429 0.624 0.556 0.302 0.698 HD 4308 0.000 0.010 0.068 0.000 0.123 0.060 0.000 0.111 GJ 86 0.042 0.007 0.042 0.034 0.051 ...... HD 99492 0.050 0.120 0.036 0.000 0.125 0.056 0.000 0.115 HD 27894 0.049 0.008 0.024 0.000 0.045 0.029 0.000 0.093 HD 33283 0.480 0.050 0.458 0.401 0.523 0.483 0.386 0.544 HD 195019 0.014 0.004 0.002 0.000 0.007 0.006 0.000 0.014 HD 102117 0.121 0.082 0.068 0.038 0.121 0.052 0.000 0.119

151 Table 6-1. Continued Name Ref. MCMC Γ eref δeref eMCMC elower eupper eΓ elower eupper HD 171565 0.684 0.013 0.683 0.672 0.691 ...... HD 6434 0.170 0.030 0.159 0.124 0.202 ...... HD 192263 0.055 0.039 0.026 0.000 0.055 0.015 0.000 0.118 HD 117618 0.420 0.170 0.352 0.233 0.511 0.381 0.212 0.538 HD 224693 0.050 0.030 0.031 0.000 0.055 0.019 0.000 0.089 HD 436916 0.140 0.020 0.090 0.055 0.133 0.104 0.021 0.182 ρ Crb 0.057 0.028 0.048 0.000 0.069 0.055 0.000 0.123 HD 456527 0.380 0.060 0.434 0.371 0.496 0.443 0.299 0.588 HD 107148 0.050 0.170 0.028 0.000 0.141 0.073 0.000 0.221 References: 1 Melo et al. (2007); 2 Bonﬁls et al. (2007); 3 Johnson et al. (2006); 4 Forveille et al. (2009); 5 Barbieri et al. (2009); 6 da Silva et al. (2007); 7 Santos et al. (2008); eref and δeref are from Butler et al. (2006) if otherwise noted.

152 Figure 6-6. Distribution of short-period single-planet systems in (e,τage/τcirc ) space. Filled circles are systems showing no linear RV trend and open circles are systems showing long-term linear RV trends. Different colors indicate different ′ ′ combinations of Q∗ and Qp

6 ′ Matsumura et al. (2008) adopted 10 as a typical value for Q∗ for short period planetary ′ 5 9 ′ = 6 7 system host stars and considered Qp ranging from 10 to 10 . We use Q∗ [10 , 10 ] and ′ = 5 7 9 Qp [10 , 10 , 10 ] in our analysis. The factors F∗ and Fp are deﬁned in the following two equations:

2 11 2 Ω∗,rot F∗ = f (e ) − f (e ) , (6–6) 1 18 2 n

11 Ωp,rot F = f (e2) − f (e2) , (6–7) p 1 18 2 n

153 where Ω is rotational frequency. For short-period planets one could set Ωp,rot /n = 1

based on the assumption that all the planets in our sample have been synchronized since

−3 τsynch ∼ 10 τcirc (Rasio et al., 1996). In order to check whether our conclusion is sensitive to

the choice of Ω∗,rot /n, we conduct calculations with other Ω∗,rot /n values in which we choose

stellar rotation period to be 3, 30, and 70 days for all the stars. We ﬁnd that this range for

Ω∗,rot /n does not change the conclusions in the chapter. Therefore, for future discussion, we adopt Ω∗,rot /n = 0.67, which results in stellar rotation periods consistent with typical values from 3 to 70 days (Matsumura et al., 2008). The uncertainties in Ω∗,rot /n are accounted for

by our subsequent data analysis (Equation. 6–10). And f1 and f2 are approximated by the equations:

15 15 5 / f (e2) = (1 + e2 + e4 + e6)/(1 − e2)13 2, (6–8) 1 4 8 64

3 1 f (e2) = (1 + e2 + e4)/(1 − e2)5. (6–9) 2 2 8

Planet radius Rp is estimated based on Fortney et al. (2007). We assume that the

planet and host star are formed at the same epoch. We assume that planet structure is

similar to Jupiter with a core mass fraction of 25M⊕/MJ = 7.86%. Radii of GJ 436 b and

HD 149026 b are adopted from reference papers because there is a factor of ≥2 difference

between observed values (Torres et al., 2008) and theoretically calculated values. Stellar

radius and age estimations are obtained from the following sources with descending priority:

1, Takeda et al. (2007); 2, nsted.ipac.caltech.edu; 3, exoplanet.eu. The calculated τcirc

values are presented in Table 6-4 in addition to the results of MCMC analysis of individual

system and other stellar and planetary properties.

We use the eccentricity posterior samples for each system for which the standard

MCMC analysis was successful (i.e., results of §2.1) to construct the eccentricity samples

of two groups separated by τage/τcirc . We note that there are considerable uncertainties in

the estimation of τage and τcirc , so τage/τcirc > 1 does not necessarily mean that the actual

154 system age is larger than the actual circularization time. We consider the sensitivity of our results to these uncertainties by adopting a probability function:   [ ]  τage τage τ  1 − 0.5 exp − η · ( − 1) if ≥ 1 age  τcirc τcirc ρ( ) =  [ ] (6–10) τ  τ τ circ  0.5 exp − η · ( circ − 1) if age < 1 τage τcirc where η is a parameter tuning the conﬁdence of τage and τcirc estimation. For example, if

τage/τcirc = 2 and η = 1, then ρ(τage/τcirc ) = 0.816, meaning there is 81.6% chance that the system is from the group of τage /τcirc ≥ 1 because of the uncertainties in τage and τcirc estimation. Therefore, we take 81.6% of the eccentricity posterior samples of the system to construct eccentricity sample for the group of τage/τcirc ≥ 1 and the remaining 18.4% eccentricity posterior samples to construct eccentricity sample for group of τage/τcirc < 1. The

η parameter reflects our confidence in τage and τcirc estimation. If we are not very confident in the estimation of τage and τcirc , then we set η to a small value approaching zero, so half of the eccentricity posterior samples for each system are assigned to the group with τage/τcirc ≥ 1 and the the other half are assigned to the group with τage/τcirc < 1. After constructing the eccentricity sample for the two groups, we use two-sample K-S test to test the null hypothesis that these two samples from two groups were drawn from the same parent distribution.

The results (Table 6-2) show that we are unable to exclude the null hypnosis at a low p value (statistic of two-sample K-S test) because of the small effective sample size (N′∼8).

If ∆max = 0.2, where ∆max is the maximum difference between cumulative distribution functions of two groups, we can exclude the null hypnosis at p = 0.05 only if N′ is more than

44. In comparison, our current sample size is inadequate to draw a statistically signiﬁcant conclusion on whether or not the the groups are from the same parent distribution. However, we do see a hint of a difference between cumulative distribution functions of two groups (Fig.

6-7 left), there are more systems with low-eccentricity for the group with τage/τcirc < 1, which is a consequence of tidal circularization. We also ﬁnd that the conclusion is unchanged for a η ′ ′ wide range of , Q∗, and Qp values.

155 Figure 6-7. The left panel compares the cumulative distribution function of eccentricity for two groups of planets: 1) τage/τcirc ≥ 1 (solid line) and 2) τage/τcirc < 1 (dotted line). The right panel compares cumulative distribution functions of eccentricity for two subsets of planets: planets without a long-term RV slope (dashed line) and planets with a long-term RV slope (dash-dotted line).

We conduct similar test for two subsets distinguished by whether or not a long-term

velocity trend is recognized and ﬁnd the similar result that our current sample size is inade-

quate to draw a statistically signiﬁcant conclusion on whether or not the the groups are from

the same parent distribution. Again, we see a hint of a difference between cumulative distri-

bution functions of two subsets (Fig. 6-7 right) although it is not statistically signiﬁcant. There

are more systems with low-eccentricity for subsets showing no sign of external perturbation.

′ The maximum difference between the cumulative distribution functions ∆max is 0.123, N is

8.37 and K-S statistic is 0.999. In that case, we need an effective sample size of 119 in order

156 Table 6-2. Two-sample K-S test result. ∆max is the maximum difference between cumulative ′ distribution functions of eccentricity for two groups separated by τage/τcirc = 1. N is the effective sample size, calculated by (N1 · N2)/(N1 + N2), p is the signiﬁcance level at which two-sample K-S test rejects the null hypothesis that the two eccentricity samples are from the same parent distribution; η is a parameter tuning the conﬁdence of τage and τcirc estimation. ′ 5 7 9 Qp 10 10 10 ∆max 0.088 0.175 0.086 η=1000 N′ 7.57 8.76 7.57 p 1.00 0.93 1.00

′ 6 Q∗ = 10 ∆max 0.103 0.160 0.093 η=1 N′ 7.72 8.72 7.02 p 1.00 0.97 1.00

∆max 0.043 0.027 0.061 η=0.001 N′ 8.59 8.50 8.08 p 1.00 1.00 1.00

∆max 0.088 0.286 0.190 η=1000 N′ 7.57 8.42 2.80 p 1.00 0.43 1.00

′ 7 Q∗ = 10 ∆max 0.103 0.259 0.168 η=1 N′ 7.72 8.36 2.81 p 1.00 0.56 1.00

∆max 0.040 0.027 0.065 η=0.001 N′ 8.61 8.39 7.67 p 1.00 1.00 1.00 to make a statistically signiﬁcant conclusion (p=0.05). In other cases, larger sample size is required since ∆max is less.

We have shown that any difference in eccentricity distribution depending on expected time scale for tidal circularization or the presence of additional bodies capable of exciting inner planet’s eccentricity is not statistically signiﬁcant, although this may be a consequence of small effective sample size. The data are also consistent with the argument that both factors play roles in affecting the eccentricity distribution.

157 6.5 Eccentricity Distribution

We seek an analytical function that is able to approximate the observed eccentricity

distribution for short period single planetary systems in the framework of Bayesian inference.

For this purpose, we ﬁrst exclude systems showing long-term RV trends to reduce the

effect of perturbation on the estimated eccentricity distribution. We also assume that the

distribution of τage/τcirc in our sample is representative of short-period single-planet systems.

Using the posterior samples of eccentricity from standard MCMC analysis, we obtain an observed eccentricity probability density function (pdf) f (e) by summing the posterior

distributions together. While not statistically rigorous, this provides a simple summary of

our results. Logarithmic binning is adopted because the shape of f (e) at low eccentricity

is of particular interest. The uncertainty σ(e) for each bin is set by assuming a Poisson

distribution. Then, we use a brute-force Bayesian analysis to ﬁnd the most probable values

of parameters for the candidate eccentricity pdf f ′(e) that approximates the observed

eccentricity pdf. In the observed eccentricity pdf, there is a pile-up in small eccentricities near

zero and a scatter of nonzero eccentricity less than 0.8. Therefore, we use a mixture of two

distributions: an exponential pdf fexpo (e, λ) = (1/λ) · exp(−e/λ) to represent the pile-up of

small eccentricity near zero and either a uniform distribution or a Rayleigh distribution (Juric´

& Tremaine, 2008) to represent the population with sizable eccentricities. We assume a

uniform distribution for parameters in prior Σ(⃗θ), where ⃗θ is vector containing the parameters

for f ′(e). Our results are not sensitive to the choice of priors. We adopt Poisson likelihood ( n ⃗ ′ ⃗ for each bin in the form of fPoisson(n; ν) = (ν · exp(−ν))/n!, i.e., L(ei |θ) = fPoisson f (ei |θ) · ) ′ ⃗ N; f (ei ) · N , where N is total number of posterior samples. The value of f (ei |θ) · N is

rounded if it is not an integer. The posterior distribution of ⃗θ is calculated as p(⃗θ|⃗e) = ∏M ∫ ∏M [ ] 1 / [ ] 1 ⃗ ⃗ M ⃗ ⃗ M ⃗ Σ(θ)L(ei |θ) Σ(θ)L(ei |θ) dθ , where M is the number of bins. We have i i explored a range of bin size from 10 to 40 bins in logarithmal space and conclude that the results of Bayesian analysis do not change signiﬁcantly with choice of bin size. Since

158 the plausible values of parameters for f ′(e) are limited, we do not have to explore a large parameter space. Therefore, a brute-force Bayesian analysis is practical.

We apply Bayesian analysis to three different planet populations for different η values

η ′ = 7 ′ = 7 ( =0.001,1,1000) assuming Q∗ 10 and QP 10 : 1) systems without a long-term RV slope and τage/τcirc ≥ 1; 2) systems without long-term RV slope and τage/τcirc < 1; 3) the union of 1) and 2). The results of Bayesian analysis are presented in Table 6-3.

Figure 6-8. Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and Rayleigh pdfs. Uncircularized systems (group 2) are represented by colored solid lines while circularized systems (group 1) are represented by colored dashed lines. Different colors indicate different η values, red: η = 0.001; green: η = 1; blue: η = 1000. Solid black lines are for union of group 1 and 2.

Figure 6-8 shows marginalized probability density of different parameters for analytical eccentricity distribution with a mixture of exponential and Rayleigh distributions (f ′(e) =

159 α · fexpo (e, λ) + (1 − α) · frayl (e, σ)). α is the fraction of exponential distribution component and

2 2 2 frayl (e, σ) represents Rayleigh distribution with the form of frayl (e, σ) = (e/σ ) · exp (−e /2σ ).

Group 1 (potentially circularized system) is represented by dashed lines of different colors

indicating different values of η while group 2 (systems unlikely to have been circularized) is

represented by colored solid lines. Group 1 and group 2 are well mixed if η=0.001, i.e., a

loose constraint is applied on the boundary of τage /τcirc = 1. When adopting η=0.001, the

marginalized posterior pdfs for both groups 1 and 2 approach the marginalized pdf for group

3 (solid black lines; the union of group 1 and 2, including all planets without a velocity slope).

When we adopt larger η values (η=1 or 1000), α, the fraction of the exponential component

of the pdf is greater for group 1 (Fig. 6-8 top: blue and green dashed lines) than group 2 (Fig

6-8 top: blue and green solid lines). This is consistent with the hypothesis that signiﬁcant tidal

circularization affected group 1. For group 1, the fraction of the exponential component of the

pdf is consistent with unity, implying that the eccentricity distribution for planets from group

1 can be described by an exponential pdf. In comparison, there is a substantial fraction of

Rayleigh component (40%) for the analytical function describing eccentricity distribution for

uncircularized systems (i.e., group 2). Similar conclusions are also drawn for the analytical

eccentricity distribution with a mixture of exponential and uniform distribution (Fig. 6-9) in the

′ form of f (e) = α · fexpo (e, λ) + (1 − α) · funif (e, β), where funif (e, β) is uniform distribution with

lower boundary of 0 and upper boundary of β.

Figure 6-10 shows the cumulative distribution of the eccentricities sample based on summing the posterior eccentricity samples of each system (green solid line) and the CDFs of the two analytic functions (blue dotted and dashed) with the parameters that maximize the posterior probability (see Table 6-3, subset 3). For comparison, the cumulative distribution of

eMCMC from standard MCMC analysis (Table 6-1) is shown in red. The difference between

the red and the green line cannot be distinguished at 0.05 signiﬁcant level in a K-S test

(N′=46).

160 Figure 6-9. Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and uniform pdf. Uncircularized systems (group 2) are represented by colored solid lines while circularized systems (group 1) are represented by colored dashed lines, red: η = 0.001; green: η = 1; blue: η = 1000. Different colors indicate different η values. Solid black lines are for union of group 1 and 2.

161 Figure 6-10. Cumulative distributions functions (cdf) of eccentricities from different methods. MCMC: most probable eccentricities, eMCMC , adopted from MCMC method (Table 6-1); MCMC, sum: eccentricities by summing up posterior distribution samples of each system; AF1: cdf of the analytical function with the most ′ probable parameters, f (e) = α · fexpo (e, λ) + (1 − α) · frayl (e, σ); AF2: cdf of the analytical function with the most probable parameters, ′ f (e) = α · fexpo (e, λ) + (1 − α) · funif (e, β).

162 Table 6-3. Bayesian analysis results. Group 1: systems without long-term RV slope and τage/τcirc ≥ 1; group 2: systems without long-term RV slope and τage/τcirc < 1; group 3: union of 1 and 2. Numbers in bracket are uncertainties of the last two digits. The fraction column reports the percentage of trials in which eccentricity testing samples generated by analytical function can not be differentiated from the observed eccentricity sample at 0.05 conﬁdence level. f ′(e) = α · f (e, λ) + (1 − α) · f (e, σ) f ′(e) = α · f (e, λ) + (1 − α) · f (e, β) η subset expo rayl expo unif λ[×10−2] α[×10−1] σ[×10−1] fraction λ[×10−2] α[×10−1] β[×10−1] fraction 1 6.83(84) 7.61(40) 3.38(36) 100.0% 7.08(82) 7.35(38) 7.56(41) 100.0%

163 0.001 2 6.67(59) 7.56(32) 3.15(22) 100.0% 6.86(59) 7.32(33) 7.06(23) 100.0%

1 7.71(58) 9.78(20) 0.83(54) 99.6% 7.96(59) 10.0(12) 1.8(11) 99.9% 1.000 2 5.55(53) 6.10(31) 3.19(16) 100.0% 5.55(57) 5.66(34) 7.12(18) 100.0%

1 7.41(72) 8.9(12) 0.95(24) 99.9% 7.62(56) 9.2(11) 2.10(57) 100.0% 1000. 2 5.24(49) 5.95(30) 3.27(16) 100.0% 5.11(53) 5.44(33) 7.20(19) 100.0%

3 6.67(47) 7.54(25) 3.22(19) 100.0% 6.89(47) 7.33(26) 7.22(20) 100.0% In order to check whether the analytical function with the most probable parameters is

an acceptable approximation to the eccentricity distribution for short-period single-planet

systems, we generate test samples from the analytical functions and compare the resulting

eccentricity samples to observations. For each test sample there are N′ eccentricities

following the distribution of the analytical function f ′(e|θˆ), where N′ is the effective sample

size. Each eccentricity is perturbed by an simulated measurement error that follows the

distribution of posterior samples for our analysis of the actual observations. The test sample

is then compared to the observed eccentricity samples obtained by standard MCMC

analysis using two-sample K-S test. We report in Table 6-3 the percentage of trials in which

eccentricity samples generated by our analytical function can not be differentiated from the

observed eccentricity sample at 0.05 conﬁdence level. All the candidate analytical functions

we have tested are able to reproduce the observed eccentricity distribution in more than

99.6% of the trials at 0.05 conﬁdence level. We conclude that the best-ﬁt analytical function

is an adequate approximation to observed eccentricity distribution. We also compare to

analytical eccentricity distribution used in Shen & Turner (2008) f ′(e) ∼ [1/(1 + e)a − e/2a]

in which a=4, although it is not speciﬁcally for short period single planetary systems. Similar

to what we did in previous test, we found that in 39.0% of the tests, the eccentricity samples

generated by analytical functions can not be differentiated from the observed eccentricity

sample at 0.05 conﬁdence level.

From the results of Bayesian analysis, there is a clear difference in α, the fraction of exponential distribution, between group 1 and 2, suggesting the role played by tidal circularization. Group 1 with τage/τcirc ≥ 1 shows more planets with near-zero eccentricities

(α ∼ 75%) as compared to group 2 (α ∼ 55%) with τage/τcirc < 1 (Table 6-3). Since the

eccentricity samples tested were perturbed by measurement errors, the analytical function

we found can be interpreted as an approximation of the underlying eccentricity distribution of

short-period single-planet systems. However, the actual parameter values and uncertainties

in the analytical function are dependent upon the quality of observation and the number of

164 systems in the sample of short-period single planets. As the measurement precision and the sample size improve, we will be able to better constrain the values of parameters in the analytical function which approximates the underlying eccentricity distribution.

6.6 Discussion

Figure 6-11. Distribution of short-period single-planet systems in our sample in period-eccentricity space (top: p ≤ 10 day; bottom: p > 10 day). Systems that are not consistent with zero eccentricity according to Γ analysis (ﬁlled circles) or MCMC analysis (open circles) are marked with corresponding names. We also include transiting planets (marked as cross) for comparison. Transiting systems not consistent with zero eccentricity are marked with numbers: 1, WASP-18; 2, WASP-12; 3, WASP-14; 4, HAT-P-13; 5, WASP-10; 6, XO-3; 7, WASP-6; 8,WASP-17; 9, CoRoT-5; 10, HAT-P-11; 11, HAT-P-2.

The median eccentricity of short-period single-planet systems in our sample is 0.088.

When compared to median eccentricity for all the detected exoplanets 0.15, it suggests that the population may be affected by tidal circularization. We use eccentricity estimated by

165 Γ analysis in the following discussion since it is a less biased method for accessing small

eccentricity. Figure 6-11 shows the period-eccentricity correlation. We would expect planets

with sufﬁcient short period to be tidally circularized. While this is generally true, there are 3

(17.6%) planets with P < 4d and non-circular orbits: GJ 436, HD 149026 and τ Boo. For

GJ 436, it is suspected that an outer companion may be pumping the eccentricity (Maness

et al., 2007; Ribas et al., 2008). Similarly, observations of τ Boo b are inconsistent with circular orbit, but might be explained by the perturbation of an unseen companion indicated by a long-term RV linear trend (Butler et al., 2006). Secondary eclipse timing indicates that

the eccentricity of HD 149026 is quite small but inconsistent with zero (Knutson et al., 2009).

Considering planets with orbital periods up to 10 days, there are 4 additional systems that are not circularized. HD 118203, HD 68988 and HD 185269 might be due to perturbations by additional bodies in the system (Butler et al., 2006) while the non-zero eccentricity of

HD 162020 b may be attributed to a different formation mechanism (Udry et al., 2002).

With period longer than 10 days, there are 8 (44.4%) planets with τage/τcirc > 1 which

have non-zero eccentricity and no detected long-term linear trend. In comparison, there

are 8 (28.6%) eccentric planets with orbital periods less than 10 days. The increasing

fraction of recognizably eccentric orbits as period increases is suggestive of decreasing

tidal circularization effect, but large sample of planets is required to draw ﬁrm conclusion.

The discovery of over 705 planets candidates by the Kepler mission presents an excellent

opportunity to analyze the eccentricity distribution of short-period planets (Borucki et al.,

2011b; Ford et al., 2008). We brieﬂy consider transiting planets. We note that 11 of 58

(19.0%) transiting systems as of June 2010 are not consistent with zero eccentricity, and

the orbital periods for all transiting planets but 3 (i.e. CoRoT-9b, HAT-P-13c HD 80606b)

are less than 10 days. We infer that the tidal circularization process might be effective for

isolated planets with orbital period of less than 10 days. Alternatively, the planet formation

and migration processes for short-period giant planet may naturally lead to a signiﬁcant

fraction of nearly circular orbits, even before tidal effect takes place.

166 It is worth noting that HD 17156 b (Fischer et al., 2007) has one of the most eccentric orbits among short-period planet systems in spite the fact that τage/τcirc could be well over 10

(Fig. 6-6, red). However, Barbieri et al. (2009) found no indications of additional companions based on observations of directing imaging, RV and astrometry measurement. Anglada-

Escude´ et al. (2010) investigated the possibility that a 2:1 resonant orbit can be hidden by an eccentric orbital solution. It is interesting to explore such possibility on this particular system to solve the discrepancy of high eccentricity and τage to τcirc ratio. Another possibility is that the system is in the process of circularization that began well after the star and planet formed

(e.g., due to planets scattering). However, we are cautious in drawing conclusions since

τage/τcirc could be less than unity (Fig. 6-6, green and blue).

6.7 Conclusion

We apply standard MCMC analysis for 50 short-period single-planet systems and construct a catalog of orbital solutions for these planetary systems (Table 6-4). We find general agreement between MCMC analysis and previous references with the primary exception being cases where eccentricity was held fixed in previous analysis. We develop a new method to test the significance of non-circular orbits (Γ analysis), which is better suited to performing population analysis. We find the eccentricity estimations from Γ analysis are consistent with results from both standard MCMC analysis and previous references.

Our results suggest that both tidal interactions and external perturbations may play roles in shaping the eccentricity distribution of short-period single-planet systems but large sample sizes are needed to provide sufficient sensitivity to make these trends statistically significant. We identify two analytical functions that approximate the underlying eccentricity distribution: 1) mixture of an exponential distribution and a uniform distribution and 2) a mixture of an exponential distribution and a Rayleigh distribution. We use Bayesian analysis to find the most probable values of parameters for the analytical functions given the observed eccentricities (Table 6-3). The analytical functions can be interpreted as the underlying

167 distribution of eccentricities for short-period single-planet systems. Thus, the analytical functions can be used in the future theoretical works or as priors for eccentricity distribution.

168 Table 6-4. Catalog of Short-Period Single-Planet Systems Name P K e ω M0 (day) (m · s−1) (deg) (deg)

± ± +0.0511 ± ± HD 41004B 1.32363 8.91594e-05 4599.21 337.57 0.0580−0.0580 149.0 72.2 119.5 72.1 ± ± +0.1080 ± ± HD 86081 1.99809 0.00677822 189.65 12.09 0.0575−0.0501 -27.4 71.5 64.8 74.5 HD 1897331 2.2186±0.0005 205±6 0±0.0002 90 270 ± ± +0.0365 ± ± HD 212301 2.24571 0.000147699 57.26 3.01 0.0147−0.0147 172.8 85.6 56.0 85.6 ± ± +0.0571 ± ± GJ 436 2.64394 9.85041e-05 18.07 1.03 0.1912−0.0449 -5.6 15.3 -66.6 14.5 ± ± +0.0203 ± ± HD 63454 2.81747 0.000382247 63.19 1.82 0.0177−0.0177 -122.9 76.2 25.7 76.3 ± ± +0.0777 ± ± HD 149026 2.87807 0.00146571 54.63 11.90 0.1918−0.0743 114.2 25.9 10.9 22.5 ± ± +0.0135 ± ± HD 83443 2.98572 5.30373e-05 56.00 1.05 0.0070−0.0070 117.3 82.2 134.0 82.0 ± ± +0.0320 ± ± HD 46375 3.02358 6.44902e-05 33.67 0.81 0.0524−0.0229 113.7 34.3 -53.2 34.3 ± ± +0.0099 ± ± HD 179949 3.0925 3.30046e-05 112.62 1.77 0.0104−0.0104 -170.7 63.4 42.4 63.3 τ ± ± +0.0382 ± ± Boo 3.31249 3.12595e-05 469.59 14.86 0.0787−0.0246 -141.6 25.0 24.4 24.9 ± ± +0.0508 ± ± HD 88133 3.41566 0.000841134 34.13 3.57 0.0761−0.0761 -2.8 60.4 -39.5 60.1 ± ± +0.0351 ± ± HD 2638 3.43752 0.00823876 66.26 2.83 0.0407−0.0407 126.9 78.6 -123.4 77.7 ± ± +0.0192 ± ± BD -10 3166 3.4878 0.000104858 60.53 1.44 0.0104−0.0104 -14.6 83.9 36.6 83.7 ± ± +0.0217 ± ± HD 75289 3.50928 7.2946e-05 54.84 1.87 0.0211−0.0211 136.3 73.9 -162.2 74.1 ± ± +0.0078 ± ± HD 209458 3.52472 2.81699e-05 84.30 0.88 0.0082−0.0082 43.8 68.4 92.5 68.5 ± ± +0.0684 ± ± HD 330075 3.6413 0.00187111 97.84 8.79 0.0187−0.0187 38.2 91.6 35.2 91.7 HD 2198282 3.833±0.0013 7±0.5 0 0 0 ± ± +0.0426 ± ± HD 76700 3.97101 0.000203194 27.24 1.31 0.0616−0.0587 12.3 54.3 45.6 53.8 ± ± +0.0093 ± ± HD 149143 4.07206 0.000320041 149.28 1.65 0.0123−0.0115 -155.2 55.9 -150.9 55.9 HD 1021953 4.1138±0.000557 63±2 0 0 0 ± ± +0.0066 ± ± 51 PEG 4.2308 3.72905e-05 55.65 0.53 0.0069−0.0069 54.1 72.3 85.1 72.3 ± ± +0.0766 ± ± GJ 674 4.6944 0.00182591 9.46 1.09 0.0700−0.0700 0.5 71.2 77.6 71.1 ± ± +0.0691 ± ± HD 49674 4.94739 0.000974925 11.78 1.18 0.0495−0.0495 -96.1 79.3 82.4 79.3 ± ± +0.0250 ± ± HD 109749 5.23921 0.000935533 28.49 1.12 0.0451−0.0451 72.3 55.8 -155.1 55.8 ± ± +0.1050 ± ± HD 7924 5.39785 0.00096697 3.74 0.44 0.1186−0.0970 25.0 55.6 62.1 54.8 ± ± +0.0342 ± ± HD 118203 6.13322 0.00129898 213.75 6.67 0.2943−0.0298 -27.3 5.5 -2.9 4.8 ± ± +0.0246 ± ± HD 68988 6.27699 0.0002195 184.63 4.69 0.1187−0.0216 32.4 11.2 61.3 10.8 ± ± +0.0595 ± ± HD 168746 6.40398 0.000979461 28.41 1.38 0.0791−0.0541 14.8 47.2 -147.2 47.3 ± ± +0.0386 ± ± HD 185269 6.83796 0.00119146 89.39 4.20 0.2758−0.0334 173.3 6.4 -99.4 5.9 ± ± +0.0023 ± ± HD 162020 8.42826 7.76104e-05 1808.84 5.26 0.2765−0.0025 -151.2 0.3 68.3 1.0 GJ 1764 8.783±0.0054 4.1±0.52 0 0 0 ± ± +0.0455 ± ± HD 130322 10.7086 0.00184045 108.10 7.31 0.0068−0.0068 128.3 93.9 -10.4 93.8 ± ± +0.0979 ± ± HD 108147 10.8984 0.00316767 24.60 3.62 0.5161−0.0966 -54.4 14.0 -68.0 9.9 ± ± +0.0547 ± ± HD 4308 15.5646 0.0213556 4.20 0.34 0.0682−0.0682 -166.9 60.6 -13.0 59.5 ± ± +0.0092 ± ± GJ 86 15.765 0.000382114 376.64 2.79 0.0416−0.0073 -93.7 12.4 -76.8 12.0 ± ± +0.0891 ± ± HD 99492 17.0495 0.00525091 8.39 0.98 0.0364−0.0364 -138.6 93.6 -137.6 93.5 ± ± +0.0211 ± ± HD 27894 18.0059 0.0163248 56.88 1.75 0.0240−0.0240 131.3 66.5 -60.0288 66.5 ± ± +0.0656 ± ± HD 33283 18.1801 0.00830584 24.48 2.41 0.4576−0.0569 156.0 9.6 -60.0 7.4 ± ± +0.0049 ± ± HD 195019 18.2018 0.000595221 269.70 1.60 0.0017−0.0017 -127.4 47.4 -175.5 47.3 ± ± +0.0529 ± ± HD 102117 20.8210 0.01006465 10.20 0.91 0.0685−0.0647 137.6 72.0 -9.5 72.5 ± ± +0.0080 ± ± HD 17156 21.2178 0.00371004 279.88 8.43 0.6829−0.0106 121.3 1.1 -156.7 1.2 ± ± +0.0434 ± ± HD 6434 21.9975 0.0127604 34.30 1.52 0.1589−0.0345 155.4 15.8 -14.2 15.5 ± ± +0.0297 ± ± HD 192263 24.3546 0.00507675 51.13 2.62 0.0256−0.0256 -147.0 78.9 -56.7 78.9 ± ± +0.1583 ± ± HD 117618 25.8221 0.0155045 12.25 1.70 0.3524−0.1192 -105.6 22.9 -123.3 21.1 ± ± +0.0236 ± ± HD 224693 26.732 0.0245934 39.73 1.54 0.0313−0.0313 11.6 68.6 162.5 68.9 ± ± +0.0432 ± ± HD 43691 36.9916 0.0350715 125.98 4.06 0.0897−0.0346 91.3 26.2 16.7 24.9 ρ ± ± +0.0218 ± ± Crb 39.8459 0.00917865 65.25 2.20 0.0476−0.0476 -62.1 42.2 -172.6 42.5 ± ± +0.0625 ± ± HD 45652 43.6896 0.105825 35.76 2.84 0.4339−0.0632 83.2 12.8 77.7 10.2 ± ± +0.1135 ± ± HD 107148 48.6168 3.59204 10.01 4.15 0.0279−0.0279 126.5 90.8 -50.1 88.8

169 Table 6-4. Continued Name τ trend Jitter Nobs M∗ R∗ −1 −1 −1 (day) (m · s · d ) (m · s ) (MS )(RS ) HD 41004B 2452532.699 ...±... 2761.45±168.90 149 0.40 0.40 HD 86081 2453753.2 ...±... 32.13±5.74 26 1.21 1.22 HD 1897331 2454037.612 ...±... 15 86 ...... HD 212301 2453388.9 ...±... 8.53±1.78 23 1.05 1.19 GJ 436 2452992.1 0.0037±0.0015 0.41±1.46 55 0.41 0.46 HD 63454 2453238.057 ...±... 5.70±1.22 26 0.80 0.78 HD 149026 2453545.35 ...±... 2.41±3.26 17 1.30 1.50 HD 83443 2452248.9 ...±... 3.12±1.67 51 1.00 1.02 HD 46375 2451920.7 ...±... 3.28±0.60 50 0.92 0.94 HD 179949 2452419.1 ...±... 9.44±1.06 88 1.21 1.22 τ Boo 2450529.2 -0.051±0.0036 94.30±8.13 98 1.35 1.33 HD 88133 2453180.0 ...±... 5.67±1.61 21 1.20 1.93 HD 2638 2453323.282 ...±... 5.48±4.87 28 0.93 1.01 BD -10 3166 2451844.7 0.005±0.0026 4.00±1.74 31 1.01 0.84 HD 75289 2452593.9 ...±... 4.73±1.73 30 1.21 1.28 HD 209458 2452499.3 ...±... 3.27±0.86 64 1.14 1.14 HD 330075 2452968.399 ...±... 24.59±5.00 21 0.70 0.90 HD 2198282 2453898.63 ...±... 1.7 27 ...... HD 76700 2452655.1 ...±... 1.35±4.21 35 1.13 1.34 HD 149143 2453413.1 0.027±0.0056 0.48±1.96 17 1.20 1.61 HD 1021953 2453895.96 ...±... 6.1 59 ...... 51 PEG 2450404.4 -0.0045±0.00046 0.27±0.91 256 1.09 1.18 GJ 674 2453823.784 ...±... 3.55±0.55 32 0.35 0.46 HD 49674 2452308.9 ...±... 3.56±0.82 39 1.06 0.95 HD 109749 2453426.3 ...±... 0.31±1.10 20 1.21 1.28 HD 7924 2454096.65 0.35±0.07 2.59±0.22 93 0.83 0.78 HD 118203 2453351.2 0.14±0.0166 23.08±3.96 43 1.23 2.15 HD 68988 2452441.3 -0.065±0.0056 13.30±2.34 28 1.18 1.14 HD 168746 2452510.8 ...±... 0.41±1.62 16 0.93 1.04 HD 185269 2453795.0 ...±... 7.70±2.10 30 1.28 1.88 HD 162020 2451672.02 ...±... 11.02±3.43 46 0.78 0.74 GJ 1764 2454399.8 ...±... 2.5 57 ...... HD 130322 2452430.2 ...±... 10.25±3.79 12 0.88 0.85 HD 108147 2452407.0 ...±... 8.65±1.49 54 1.19 1.25 HD 4308 2453338.121 ...±... 0.58±0.78 41 0.90 0.92 GJ 86 2452199.4 -0.260±0.0029 10.67±1.62 42 0.77 0.80 HD 99492 2452523.85 0.0035±0.00066 3.96±0.51 86 0.86 0.76 HD 27894 2453344.278 ...±... 4.48±1.07 20 0.75 0.90 HD 33283 2453560.0 ...±... 0.27±0.94 24 1.24 1.20 HD 195019 2451844.0 ...±... 10.50±1.22 154 1.07 1.38 HD 102117 2452931.7 ...±... 0.57±1.73 44 1.11 1.26 HD 17156 2454111.21 ...±... 3.68±0.64 34 1.24 1.63 HD 6434 2451753.933 ...±... 7.39±1.15 130 0.79 0.57 HD 192263 2451867.6 ...±... 6.99±1.31 31 0.81 0.77 HD 117618 2452838.0 ...±... 3.30±1.20 57 1.09 1.20 HD 224693 2453607.2 ...±... 1.00±2.78 23 1.33 1.70 HD 43691 2454046.7 ...±... 10.35±2.58 36 1.38 1.92 ρ Crb 2451181.1 ...±... 0.65±2.79 26 1.00 1.28 HD 45652 2453692.66 ...±... 8.38±2.20 45 0.83 1.04 HD 107148 2452799.9 0.003±0.0012 3.53±1.04 35 1.14 1.12

170 Table 6-4. Continued Name MP RP τcirc τage RV (MJ )(RJ ) (Gyr) (Gyr) ref. HD 41004B 18.40 1.06 0.26 6.32 7 HD 86081 1.50 1.08 0.56 6.21 8 HD 1897331 ...... 9 HD 212301 0.40 1.07 0.28 5.90 11 GJ 436 0.07 0.38 5.57 6.00 6 HD 63454 0.39 1.06 0.67 1.00 12 HD 149026 0.36 0.65 3.46 2.00 6 HD 83443 0.40 1.04 1.10 11.68 6 HD 46375 0.23 1.02 0.67 11.88 6 HD 179949 0.92 1.05 2.90 2.56 6 τ Boo 4.13 1.06 4.12 1.64 6 HD 88133 0.30 1.00 1.49 9.56 6 HD 2638 0.48 1.04 2.38 3.00 12 BD -10 3166 0.46 1.03 2.70 1.84 6 HD 75289 0.47 1.03 3.03 3.28 6 HD 209458 0.69 1.05 4.07 2.44 6 HD 330075 0.62 1.06 1.97 6.21 13 HD 2198282 ...... 2 HD 76700 0.23 0.99 2.80 9.84 6 HD 149143 1.33 1.05 7.01 7.60 14 HD 1021953 ...... 3 51 PEG 0.47 1.03 6.61 6.76 6 GJ 674 0.04 1.13 0.21 0.55 15 HD 49674 0.10 0.98 3.25 3.56 6 HD 109749 0.28 0.99 12.04 10.30 14 HD 7924 0.03 1.05 0.61 0.88 16 HD 118203 2.14 1.05 3.68 4.60 17 HD 68988 1.86 1.05 80.15 3.40 6 HD 168746 0.25 0.99 20.31 12.40 6 HD 185269 0.94 1.04 19.34 4.20 8 HD 162020 15.00 0.98 148.00 0.76 19 GJ 1764 ...... 4 HD 130322 1.09 1.04 670.08 10.80 6 HD 108147 0.26 0.98 8.43 3.20 6 HD 4308 0.05 1.00 169.24 8.68 20 GJ 86 3.91 1.05 6701.62 8.48 6 HD 99492 0.11 0.96 692.89 1.80 21 HD 27894 0.62 1.02 3620.33 3.90 12 HD 33283 0.33 0.99 203.12 3.20 8 HD 195019 3.69 1.05 4051.07 9.32 6 HD 102117 0.17 0.95 3205.49 9.40 6 HD 17156 3.20 1.04 19.10 8.00 10 HD 6434 0.40 1.00 4393.14 6.85 22 HD 192263 0.64 1.02 14630.88 2.56 6 HD 117618 0.18 0.95 1435.34 5.68 6 HD 224693 0.71 1.03 23506.41 2.00 8 HD 43691 2.49 1.04 42743.17 2.80 18 ρ Crb 1.09 1.03 166705.71 7.64 6 HD 45652 0.47 1.01 12199.36 5.00 23 HD 107148 0.21 0.96 122900.60 5.60 6

171 Table 6-4. Continued References: 1 Bouchy et al. (2005); 2 Melo et al. (2007); 3 Ge et al. (2006b); 4 Forveille et al. (2009); 5 Maness et al. (2007); 6 Butler et al. (2006); 7 Zucker et al. (2004); 8 Johnson et al. (2006); 9 Winn et al. (2006); 10 Winn et al. (2009); 11 Lo Curto et al. (2006);12 Moutou et al. (2005); 13 Pepe et al. (2004); 14 Fischer et al. (2006); 15 Ge et al. (2006b); 16 Bonﬁls et al. (2007); 17 Howard et al. (2009); 18 da Silva et al. (2006); 19 da Silva et al. (2007); 20 Udry et al. (2002); 21 Udry et al. (2006); 22 Marcy et al. (2005);23 Mayor et al. (2004); 24 Santos et al. (2008) Stellar radius and age estimations are obtained from the following sources with descending priority: 1, Takeda et al. (2007); 2, nsted.ipac.caltech.edu; 3, exoplanet.eu. τ ′ = 7 ′ = 7 circ is calculated assuming Q∗ 10 and Qp 10

172 CHAPTER 7 SUMMARY, CONCLUSION AND CONTRIBUTION

My dissertation works pave the way for massive detections of M-dwarf planets. More specifically, I compare two existing RV techniques, i.e., the DE method and the DFDI method, and find their own strength in exoplanet search. I answer the question whether and how to detect a habitable Earth-like planet. I take effort to overcome challenges in NIR RV precision measurement. I help design and understand the scientific deliverable of a M-dwarf planet survey. I take part into a multi-object planet survey project using the DFDI method and help solve instrumentation problem which leads to fruitful discoveries. I envision a multi- object M-dwarf planet survey to complete our understand of planet formation around stars with a broad mass spectrum. In addition to my main work on M dwarf planet, I present my collaboration work with Dr. Eric Ford on the eccentricity distribution of short-period planets.

7.1 Chapter 2

I conducted comprehensive comparison between the DE and the DFDI method. In order to ensure a fair and quantitative comparison, I invented a method of calculating the Q factor for the DFDI method. This method is a natural extension of the work of Bouchy et al. (2001) in which they developed a method of calculating the Q factor for the DE method. Comparison between the DE method and the DFDI method has been done by several previous work such as Erskine (2003); Erskine & Ge (2000); Ge (2002); Ge et al. (2002); van Eyken et al.

(2010), but it has never been done as systematic and complete as my work described in this dissertation. I have considered many cases such as the same wavelength coverage case, the same detector case and the multi-object case. My work provides a guidance for future exoplanet survey: 1, a survey of a large sample of stars should adopt the DFDI method, which enables both adequate RV precision and high survey efﬁciency; 2, high precision low-mass exoplanet search should adopt the DE method with a high resolution spectrograph.

173 7.2 Chapter 3

Following the conclusion of Chapter 2, I investigated into the high precision case for

which the DE method with a high resolution spectrograph is suitable. I was trying to answer

the question if we are able to detect a habitable Earth-like planet given spectral properties

and state-of-the-art RV technique. The theoretical framework is given by Bouchy et al.

(2001) that provides the method to estimate RV precision for a given spectrum at a given

S/N. There are several previous papers on this topic such as Reiners et al. (2010); Rodler

et al. (2011), but my work is the ﬁrst of its kind in which I have considered all the dominant

factors that contribute to RV precision errors. They include stellar properties such as spectral

SED, line properties and rotational velocity, wavelength calibration error, stellar noise and

telluric contamination. I concluded that NIR observation of mid-late type M dwarfs is the most

realistic and likely approach in the search for a habitable Earth-like planet.

7.3 Chapter 4

There are several challenges in NIR precision RV measurement. I have conducted

pioneering work using the EXPERT spectrograph (R=27,000) at the KNNO 2.1m telescope.

My work includes observation, data analysis and hardware development. For observation, I has worked with other members in the ET group for proposal writing and nightly observation.

I have two proposals as PI accepted for which a total of 14 nights were awarded. For data analysis, I have used a telluric standard star method described by Vacca et al. (2003) to

remove telluric contamination from observed stellar spectrum and demonstrated 2.7%

removal residual. I have for the ﬁrst time introduced the binary mask cross correlation

technique into the NIR RV measurement data analysis. This technique has only been used

for the optical wavelengths before by Baranne et al. (1996). For hardware development, I

was mainly working with Dr. Xiaoke Wan on the development of the sine source. We have

together conducted feasibility demonstration in lab and at the KPNO 2.1m telescope, I was

partially involved in experiment and data acquisition and mainly in charge of data analysis.

The sine source has been demonstrated a stable and precise wavelength calibration source

174 and further demonstration experiment is required. I was heavily involved in the FIRST

survey proposal. My contribution is mainly on target selection and planet yield simulation. I

predicted that 30 planets will be detected including 10 super-Earths, 2 giant planets and 18

intermediate-mass planets.

7.4 Chapter 5

The other major conclusion from Chapter 2 is that a survey of large sample stars should

adopt the DFDI method at a low-medium spectral resolution. The MARVELS project is planet

survey that follows this conclusion. This project takes enormous effort in design, instrument

building, commissioning, operating and data analysis. My major contribution to this project is

instrumentation and interferometer calibration. The RV error due to interferometer is limited

within 2 m · s−1. Owing to my work in MARVELS interferometer group delay calibration, over

250 binary and a dozen of brown dwarfs have been discovered and they provide a valuable insight of formation and evolution of low-mass stellar companion and brown dwarf. I was awarded the Architect status for the MARVELS project in 2012 for my contribution to this project and will be guaranteed data access. I have envisioned an M-dwarf planet survey and provided a concept study of such survey.

7.5 Chapter 6

This chapter reports my collaboration work with Dr. Eric Ford on the eccentricity dis-

tribution of short-period planet. We apply standard MCMC analysis for 50 short-period

single-planet systems and construct a catalog of orbital solutions for these planetary sys-

tems. We develop a new method to test the signiﬁcance of non-circular orbits (Γ analysis),

which is better suited to performing population analysis. Our results suggest that both tidal

interactions and external perturbations may play roles in shaping the eccentricity distribution

of short-period single-planet systems but large sample sizes are needed to provide sufﬁcient

sensitivity to make these trends statistically signiﬁcant.

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184 BIOGRAPHICAL SKETCH

Ji Wang was born in Guilin, a beautiful city located in the mountainous area in Southern

China. 18 years after happily living in Guilin, he moved to the University of Science and

Technology of China (USTC) in Hefei to pursue his dream of being a scientist and studying abroad. At USTC, the best university in China, he started to learn and to appreciate the concept of hard work and smart thinking. He built a solid foundation for study and research at USTC for the coming years before being admitted into the Astronomy program at the

University of Florida. He spent 6 good years in Florida, where he studied hard but exercised harder. Thanks to all the sport facilities on campus, he became an all-around athletic who is good at basket ball, tennis, golf, swimming, long distance running. Now he is ready for his next stop, Yale University where he will work as a post-doc in the Department of Astronomy.

Hopefully in a few years he will ﬁnd a tenure track position somewhere either in the US or back in China.

185