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

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

2 c 2012 Ji Wang 2

3 I dedicate it to my parents and my wife 3

4 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

5 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

6 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

7 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 BACKGROUND AND MOTIVATION The Chronicle of Exoplanets Search Detection Methods The Radial Velocity Technique The Transit Method Direct Imaging The Timing Technique Microlensing Astrometry Conventional Spectrograph and the Dispersed Fixed Delay Interferometer Major Questions To Be Answered Planets Around M Dwarfs Essential Facts About M-dwarf M-dwarf Planets Surveys and Results FUNDAMENTAL PERFORMANCE OF THE DFDI METHOD Introduction Methodology of Calculating Photon-limited RV Measurement Uncertainty Photon-limited RV Uncertainty of DE Photon-limited RV Uncertainty of DFDI Comparison between DE and Optimized DFDI Optimized DFDI Influence of Spectral Resolution R Influence of Detector Pixel Numbers Influence of Multi-Object Observations Influence of Projected Rotational Velocity V sin i Summary and Discussion Q Factors for DFDI, DE and FTS Application of DFDI

8 3 COMPREHENSIVE SIMULATIONS FOR HABITABLE PLANET SEARCH IN THE NIR Introduction Simulation Methodology High Resolution Synthetic Spectra RV Calibration Sources Stellar Noise Telluric Lines Contamination Results RV Calibration Uncertainty Optimal Spectral Band For RV Measurements Stellar Spectral Quality Stellar Spectral Quality + Stellar Rotation Stellar Spectral Quality + RV Calibration Source Stellar Spectral Quality + RV Calibration Source + Atmosphere Comparisons to Previous Work Current Precision vs. Signal of an Earth-like Planet in Habitable Zone Stellar Spectral Quality Stellar Spectral Quality + RV Calibration Source + Atmosphere Stellar Spectral Quality + RV Calibration Source + Stellar Noise Summary and Discussion PLANET SEARCH AROUND M DWARFS Introduction Current Status Challenges Atmophsere Wavelength Calibration Sources Tackling Adversities in NIR RV Measurement Software Advancement Precise Telluric Lines Removal Binary Mask Cross Correlation Hardware Advancement M-dwarf Planet Search and Characterization-Results Telluric Line RV Stability RV Measurements of a Reference Star-GJ M-dwarf Planet Search and Characterization-Future Works Searching For Planets Around M Dwarfs with EXPERT Multi-Band Study of Radial Velocity Induced by Stellar Activity with EXPERT Mid-Late Type M Dwarf Planet Survey Using FIRST Science Justification Target Selection

9 Planet Yield Prediction ACCURATE GROUP DELAY MEASUREMENT FOR RV INSTRUMENTS USING THE DFDI METHOD Introduction GD Measurement Using White Light Combs Method Data Reduction GD Measurement Results GD Measurement Error Analysis GD Calibration: Observing an RV Reference Star Method GD Calibration Precision Implementation of Measured GD in Astronomical Observations Summaries and Discussions Summaries Discussions White Light Comb (WLC) Method Reference Star (RS) Method A Future M-Dwarf Survey With the DFDI Method ECCENTRICITY DISTRIBUTION FOR SHORT-PERIOD EXOPLANETS Introduction Method Bayesian Orbital Analysis of Individual Planet Γ Analysis of Individual Systems Results for Individual Planets Comparison: Standard MCMC and References Comparison: Standard MCMC and Γ Analysis Discussion of Γ Analysis Tidal Interaction Between Star and Planet Eccentricity Distribution Discussion Conclusion SUMMARY, CONCLUSION AND CONTRIBUTION Chapter Chapter Chapter Chapter Chapter REFERENCES

10 BIOGRAPHICAL SKETCH

11 Table LIST OF TABLES page 2-1 OPD choice as a function of R and V sin i at different T eff Power Law Index χ as a function of Spectral Resolution R (T eff = 2400K) Spectral Resolution and wavelength coverage on a given detector Q comparison of DFDI and DE as a function of V sin i Definition of observational bandpasses RV uncertainties caused by calibration sources at different spectral resolutions Photon-limited RV uncertainties based on stellar spectral quality at different spectral resolutions for different spectral types Spectral Type, corresponding T eff, and typical stellar rotation V sin i Comparison of Q factors from our results to Bouchy et al. (2001) Comparison of predicted RV precision between our results to Reiners et al. (2010) Comparison of predicted RV precision between our results to Rodler et al. (2011) Required S/N for detection of an Earth-like Planet in the HZ as a function of spectral type Two examples of telluric contamination Prediction vs. HARPS observation GD measurement results as a function of spectrum number (GD(#) = C 0 + C 1 # + C 2 # 2 ) and standard deviation (δgd) at different frequencies (ν) MARVELS predicted RV uncertainty (at an average S/N of 100) vs. T eff Comparison between two methods of GD measurement and calibration Comparison of Eccentricities Calculated From Different Methods Two-sample K-S test result Bayesian analysis results Catalog of Short-Period Single-Planet Systems

12 Figure LIST OF FIGURES page 1-1 Illustration of conventional spectrograph and the DFDI method Illustration of Doppler sensitivity for a conventional spectrograph Illustration of Doppler sensitivity for the DFDI method DFDI layout diagram DFDI illustration Examples of M-dwarf spectra Power spectrum of a M-dwarf spectrum Optimal OPD vs. V sin i and R Q factor gain vs. spectral resolution Q factor vs. spectral resolution Improvement of Q DFDI over Q DE vs. spectral resolution Comparison of Q IRET and Q DE at different spectral resolutions Comparison of Q DFDI and Q DFDI,R=100,000 at different R Comparison of Q DE and Q DE,R=100,000 at different spectral resolutions Q DFDI vs. V sin i Comparisons between synthetic and observed M-dwarf spectra RV calibration uncertainties vs. spectral resolutions RV precision based on spectral quality factor RV precision based on spectral quality factor and typical stellar rotation RV precision based on spectral quality factor and RV calibration uncertainties RV precision considering spectral quality factor, RV calibration uncertainties and telluric contamination The percentage contribution of RV uncertainty induced by telluric contamination RV precisions considering spectral quality factor at a S/N of RV precision (R=120,000) considering spectral quality factor (S/N=425), RV calibration uncertainties and telluric contamination

13 3-10 RV precision considering spectral quality factor (S/N=425), RV calibration uncertainties and stellar noise RMS error of Keplerian orbit fitting for planets detected by HARPS since Comparison between two spectra before and after removing telluric lines Comparison between an observed stellar spectra (GJ 411, telluric lines removed) and a synthetic spectrum Telluric line removal residual is 2.7% An example of binary mask template Application of the sine source as an absorption cell Application of the sine source as an emission lamp for simultaneous wavelength calibration Sin source demonstration experiment Telluric lines RV stability RV measurements for GJ RV measured in I band using DEM of EXPERT for KEP RV measured in V band using DEM of EXPERT for KEP V and J band distribution for FIRST survey targets T eff distribution for FIRST survey targets Predicted RV measurement precision for the FIRST survey The predicted survey completeness contours based on observation strategy and RV precision for the pessimistic case The predicted survey completeness contours based on observation strategy and RV precision for the baseline case Illustration of the DFDI method Simulated WLCs of an interferometer Phase of simulated WLCs The normalized flux and visibility (γ) as a function of frequency Top and side view of an individual fiber beam feeding of the MARVELS interferometer

14 5-6 White light combs phase as a function of frequency Measured group delay as a function fiber number GD as a function of frequency at different fiber numbers RVs of HIP (barycentric velocity not corrected) over a period of 70 days Examples of how credible intervals of standard MCMC analysis are calculated using posterior distribution of e Contours of posterior distribution in h and k space for HD Γ as a function of eccentricity e for HD Comparison among standard MCMC analysis, Γ andlysis and previous references Cumulative distributions functions (CDFs) of eccentricities from different methods Distribution of short-period single-planet systems in (e,τ age /τ circ ) space Cumulative distribution function of eccentricity Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and Rayleigh pdfs Marginalized probability density functions of parameters for analytical eccentricity distribution with a mixture of exponential and uniform pdf Cumulative distributions functions (cdf) of eccentricities from different methods Distribution of short-period single-planet systems in period-eccentricity space

15 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 Chair: Jian Ge Major: Astronomy By Ji Wang August 2012 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

16 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

17 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 methods The Radial Velocity Technique 1.2 Detection Methods 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 technique 2. 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

18 semi-major axis of 6 AU (orbital period 5000 day) have been detected (Fischer et al., 2008; Jones et al., 2010) 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; Léger et al., 2009), habitable planet (Borucki et al., 2012) and Earth-like planets in a multiple system (Gautier et al., 2011) 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 directlyimaged 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 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 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

19 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) 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) 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

20 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 conclusion. Reader can refer to his original paper for more details. For a conventional spectrograph, 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 ν D is 2n νd H i / A i (A o /A i ) 3/2. 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 (2003), n ν D H i /2 A i (A o /A i ) 1/2. Therefore, the ratio of S/N induced by a fixed Doppler shift, (S/N) DE /(S/N) DFDI is 2 2/(A o /A i ). For example, A o 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

21 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

22 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 correlation indicating that occurrence rate rises from 3% for [Fe/H] 0 to 25% for [Fe/H] +0.4 (Fischer & Valenti, 2005). As measurement precision keeps increasing, a population of exoplanets, 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 exoplanets. 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

23 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

24 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 abundant around M dwarfs ( 35%) and the occurrence rate for habitable planets is % for a M-dwarf sample in their survey. In comparison, Howard et al. (2010b) found this number to be % 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

25 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 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 (T eff 3700 K) among all classified stars. Because of low T eff 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) 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): G m sin i K = = 28.4 km s 1 m sin i ( M 1 e 2 (M + m)a M Jup M ) 1/2 AU, (1 1) 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

26 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 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) 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 announced 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 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 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

27 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 b. In 2010, three more giant planet discoveries, HIP 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

28 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

29 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 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: [ S0 (ν) B(ν, y) = hν ] IT (ν, y) LSF (ν, R), (2 1) where S 0 (ν) is the intrinsic stellar spectrum and ν is optical frequency. S 0 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 S 0 (ν); c is the speed of light; and τ is the optical path difference (OPD) of the 29

30 Figure 2-1. A schematic layout of an RV instrument using the DFDI method. Figure 2-2. DFDI Illustration. S 0 (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

31 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 c LSF (ν 0, ν) = 1 [ 2πν exp (ν ν 0) ν 2 ], (2 2) ]. (2 3) 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 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). T eff ranges from 2400K to 31

32 Figure 2-4. Top: power spectrum of derivatives of stellar spectrum, F [ds 0 /dν], with SRF (ρ) = F [LSF (ν, R)] for different R overplotted, where LSF is line spread function; Bottom: F [ds 0 /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., ds 0 /dν. The power spectrum of ds 0 /dν is obtained by Fourier transform. According to properties of Fourier transform, F [ds 0 /dν] = (iρ) F [S 0 ], 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 ([ds 0 /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

33 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 [ds 0 /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 ([ds 0 /dν] of a fringing spectrum obtained with a DFDI instrument with a 20 mm optical delay, which shifts F [ds 0 /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 measurements. 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 T eff 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

34 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 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 S 0 (ν) designate an intrinsic stellar spectrum. A 0, a digitalized and calibrated spectrum, is considered as a noise-free template spectrum for differential RV measurement, which is related to S 0 (ν) via the following equation: A 0 (i) = S 0(ν) hν LSF (ν), (2 4) where i is pixel number and S 0 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 c = δν ν, (2 5) 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): δv rms c = Q 1 A 0 (i) i 1/2 = 1 Q, (2 6) N e where Q is defined as: W (i) i Q A 0 (i) i 1/2, (2 7) 34

35 and W (i) is expressed as: W (i) = [ A0 ] (i) 2 ν(i) ν(i) 2. (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 N e = A 0 (i) within the spectral range. i 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. B 0 (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 0 (i, j) = B 0(i, j) δν(i) ν(i) = B 0(i, j) δv 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 c = B(i, j) B 0(i, j). (2 10) B 0 (i,j) ν(i) ν(i) Frame B 0 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 σ D : B rms (i, j) = B(i, j) + σ 2 D. (2 11) 35

36 Equation 2 11 is approximated under photon-limited conditions as B rms (i, j) = B(i, j). Therefore, the RV uncertainty at pixel (i, j) is given by: δv rms (i, j) c = B(i, j) B 0 (i,j) ν(i). (2 12) ν(i) The overall RV uncertainty for the entire spectral range is given by: δv rms c = [ ] 2 δvrms (i, j) c 1/2 i,j 1/2 W (i, j) i,j Q 1 B 0 (i, j) = i,j 1/2 1 Q, (2 13) N e where W (i, j) ( ) 2 B 0 (i,j) ν(i) 2 ν(i), (2 14) B(i, j) and [ W (i, j) ] 1/2 i,j Q. (2 15) B 0 (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 N e = B 0 (i, j) within the spectral range. i,j 2.3 Comparison between DE and Optimized DFDI 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

37 help increase the instrument Doppler sensitivity. We calculate the optimal OPD for spectra of various T eff 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 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 T eff 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. T eff influence on optimal OPD is not significant, T eff =2800 K is adopted in the plot. 37

38 Table 2-1. OPD choice as a function of R and V sin i at different Teff R V sin i [km s 1 ] ,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 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 38

39 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 Q deviated. We plot the ratio of Q optimal and Q deviated as a function of spectral resolution R in Fig We found that deviating OPD by 5mm does not result in severe degradation of the Q factor. The maximum degradation is and occurs at R of 5,000 and V sin i of 5 km s 1 for a star with T eff of 2800 K. The degradation can be compensated by increasing the integration time by 24% ( ) to reach the same photon-limited Doppler precision according to Equation The degradation becomes smaller as R increases. As shown in Fig. 2-4, DFDI shifts the power spectrum of ds 0 /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 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

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

41 Table 2-2. Power Law Index χ as a function of Spectral Resolution R (T eff = 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, ,000-20,000-50, , χ 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 T eff = 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, Q DFDI 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 slow rotators (V sin i=2 km s 1 ) at the low R region (R=5,000-20,000), Q DFDI R Doppler sensitivity δv rms is inversely proportional to two factors: Q and N e according to Equation 2 6 and 2 13, where N e is the total photon count collected by the CCD detector. N e (S/N) 2 N pixel, where S/N is the average signal to noise ratio per pixel, and N pixel is total number of pixels. Note that N e R if the wavelength coverage, S/N per pixel and the resolution sampling are fixed. Therefore, δv rms R = R 1.13 for DFDI. In comparison, δv rms R 1.57 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

42 Figure 2-7. Q factor as a function of spectral resolution. (left: T eff = 2400K ; right: T eff = 3100K. Open circles represent Q DFDI ; crosses represent Q DE ; solid lines are best power-law fits for Q DFDI ; dashed lines are best power-law fits for Q DE ) We compare Q factors for both DFDI and DE at given R values, and the results are shown in Fig 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

43 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., N e 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 Q DFDI over Q DE as a function of spectral resolution. 43

44 2.3.3 Influence of Detector Pixel Numbers In the NIR, detector pixel number is typically smaller than the optical detector. Furthermore, 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 N e, (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. N e is calculated by Equation 2 17: N e = F η S tel t exp, (2 17) m J in which F is the photon flux in the wavelength coverage region λ of an m J = 0 star with the unit of photons s 1 cm 2 ; η is instrument total throughput; S tel is the effective surface area of the telescope; t exp is the time of exposure; and m J 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 detector 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 λ = N pix λ c, (2 18) P order R N S 44

45 where N pix is total number of pixels available on a CCD detector and N S is the number of pixels per resolution element, λ c is the central wavelength and P order is the number of pixels sampling each pixel channel between spectral orders 1. Equation 2 18 shows that λ is inversely proportional to R. Table 2-3 gives the relation of R and λ assuming N pixel, P order and N S as constants. λ c is set to be 1000 nm because it is approximately the center of the Y band. We calculated the ratio of Q DFDI and Q DE in which we use the photon flux of a star with a T eff of 2400K (Fig. 2-9). Q IRET is consistently higher than Q DE 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 N e. For a given number of pixels on the detector, N e,dfdi is higher than N e,de due to the larger wavelength coverage. In addition, Q DFDI ( λ DFDI ) is more than Q DE ( λ DE ). Consequently, we see in Fig. 2-9 that Q IRET is higher than Q DE at all R of a DE instrument. Figure 2-9 also shows that the minimum of Q DFDI /Q DE is dependent of V sin i. The ratio of Q reaches a minimum (Q DE 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, Muirhead et al. (2011) has demonstrated a phase-stepping method which does not require sample fringes in the slit direction.) 45

46 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 λ min λ max (nm) (nm) 25, , , , , , , Figure 2-9. Comparison of Q IRET and Q DE at different R. Note that Q = Q N e. Different color represents different rotational velocity. 46

47 For a slow rotator (V sin i=2 km s 1 ) at low R region (R=5,000-20,000), Q DFDI R Since Doppler sensitivity δv rms is inversely proportional to two factors: Q and N e 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 Influence of Multi-Object Observations As discussed in 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: Q = Q N e N α obj, (2 19) where N obj 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 N e, and thus α is 0.5. However, from an observational efficiency point of view, Q should be proportional to N obj because the more objects are observed simultaneously, the quicker the survey is accomplished, and α is therefore equal to 1. 47

48 Figure Comparison of Q DFDI and Q DFDI,R=100,000 at different R. Note that Q = Q N e N α obj. The maximum of each curve is indicated by filled circle. Different color represents different rotational velocity, the same as Fig 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 N e is a constant since we assume identical λ. N obj 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 N obj 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 Q R=100,000 for DFDI under two different assumptions of α. For α=0.5, i.e., increase of N obj 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

49 Figure Comparison of Q DE and Q DE,R=100,000 at different R. Note that Q = Q N e N α obj. The maximum of each curve is indicated by filled circle. Different color represents different rotational velocity, the same as Fig 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

50 Table 2-4. Q comparison of DFDI and DE as a function of V sin i α=0.5 DFDI DE V sin i [km s 1 ] R optimal Q DFDI R optimal Q DE Q DFDI /Q DE 0 50, , , , , , , , α= , , , , , , , , times faster in terms of survey speed. For fast rotators (i.e., V sin i=10km s 1 ), the boost drops to Our study confirms that the DFDI method has an advantage for multi-object RV measurements over the DE method as suggested by Ge (2002) 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 V sin i 10 km s 1 ) at R=150,000 and various T eff. We assume a wavelength range from 800 nm to 1350 nm. The results are shown in Fig 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 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

51 Figure Q DFDI 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

52 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 T eff 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

53 between the FTS and the DFDI method, for example: 1, both methods use the interferometer 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 interferometer 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 Q DFDI is consistently higher than Q DE 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

54 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 Q DFDI slightly better than Q DE (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 ) 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 instruments. 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 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

55 measurement, the difference is that the object is switched from a star to an wavelength calibration 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 followup 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

56 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 remains 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 observational 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 completely 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

57 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 absorption 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 achievable 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 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

58 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 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 T eff 7000 K), we adopt the spectra with a 0.02 Å sampling from Coelho et al. (2005). For M dwarfs, (2400 K T eff 3500 K), we use high-resolution (0.005 Å sampling) synthetic stellar spectra generated by PHOENIX code(allard et al., 2001; Hauschildt 58

59 et al., 1999). Reiners et al. (2010) conducted several comparisons between synthetic spectra 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< T eff 9600 K), we also use the synthetic spectra with 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 onedimensional 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 As the features in a stellar spectrum increases due to cooler T eff 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 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

60 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 T eff 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 T eff and V sin i are, from top to bottom, 9000 K and 80.0 km s 1 for HD (A5V), 6250 K and 4.5 km s 1 for HD (F8V), 5750 K and 6.0 km s 1 for HD (G2V), 4750 K and 4.0 km s 1 for HD (K5V), 2900 K and 10.0 km s 1 for HD (M6V). The difference between observed and synthetic spectrum is also plotted at the bottom of each panel with RMS of difference. 60

61 of RV uncertainty. Photon-limited RV measurement uncertainty is calculated based on the method discussed in 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 instabilities. 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 uncertainties 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 spectrographs 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

62 Table 3-1. Definition of observational bandpasses used in this study: center wavelength and wavelength range Band λ 0 λ min λ max (nm) (nm) B V R Y J H K 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

63 a series of absorption cells is proposed by Mahadevan & Ge (2009), in which a mixture of gas cells including H 13 C 14 N, 12 C 2 H 2, 12 CO, and 13 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 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

64 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, photonnoise 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 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

65 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: F (ν) = = = [ S0 (ν) ] hν AT (ν) LSF [ S0 (ν) ] hν (1 α AA(ν)) LSF [ S0 (ν) ] [ LSF + S ] 0(ν) hν hν α AA(ν) LSF = F S (ν) + F N (ν), (3 1) where S 0 (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 F S and a noise component F N. Ideally, we require that the detector flux change, δf, is entirely due to the stellar RV change δv S. However, δf is also partly induced by telluric line shift δv N resulting from random atmospheric motions. Therefore, both δv S and δv N contribute to δf. We have two sets of RV measurements, δv S + σ(0, δv rms,s ) for stellar RV and δv N + σ(0, δv rms,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. δv rms,s is the photon-limited measurement error for component F S and δv rms,n is the photon-limited measurement error for component F N. 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: δv = (δv S + σ(0, δv rms,s )) δv 2 rms,s + (δv N + σ(0, δv rms,n )) δv 2 δv 2 rms,s + δv 2 rms,n rms,n, (3 2) 65

66 In practical Doppler measurements, δv S 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 δv S and δv N, 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 km s 1. We artificially shift a stellar spectrum by an amount less than km s 1 in order to generate a offset between stellar spectrum and AA spectrum. δv rms,s and δv rms,n are then calculated for F S and F N. We choose the median of δv rms,n to represent a typical δv rms,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 F N has an RV fluctuation with an RMS of δv N,ATM because of the Earth s turbulent atmosphere. The measured RV uncertainty δv is equal to: δv rms = (δv rms,s) δv 2 rms,s + (δv 2 N,ATM + δv 2 rms,n )1/2 δv 2 δv 2 rms,s + δv 2 rms,n rms,n, (3 3) In reality, RV uncertainty of F N is not dominated by photon-noise, instead, it is dominated by atmospheric behaviors such as wind, molecular column density change, etc. Figueira et al. (2010a) used HARPS archive data and found that O 2 lines are stable to a 10 m s 1 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 H 2 O and CO 2. The uncertainty induced by atmospheric telluric lines is transferred to δv rms via Equation 3 3. In order to calculate the final RV uncertainty, δv rms, we need to calculate photon-limited RV uncertainty δv S,rms and δv N,rms according to Equation 2 6, in which two terms need to be calculated: Q and N e. The spectral quality factors (Q S and Q N ) for the two components (F S and F N ) from Equation 3 1 are calculated based on Equation 2 7. N e,s and N e,n, the photon flux of F S and F N are calculated based on stellar type, magnitude, exposure time, instrument specifications and telluric absorption properties. Note the ratio of N e,s and N e,n remains constant as long as atmospheric absorption stays unchanged because telluric line absorption is imprinted on the stellar spectrum. 66

67 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 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 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 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 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

68 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-Ar a Th-Ar, Iodine b Th-Ar U-Ne c U-Ne Mixed cell d Ammonia e (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) 20, , , , , , , , , , , , 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

69 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) 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 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 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

70 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 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) 70

71 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 T eff 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 T eff 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

72 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 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

73 Table 3-4. Spectral Type, corresponding T eff, and typical stellar rotation V sin i Spectral Type T eff Vsin i (K) (km s 1 ) A0V A2V A5V A8V F0V F2V F5V F8V G0V G2V G5V G8V K0V K2V K5V K8V M0V M2V M5V M8V M9V 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

74 Path) or the exposure time (Bracketing). The main conclusion about optimal observational bandpass from 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 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 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

75 quantitative analysis of telluric line contamination is introduced in 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 uncertainty 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

76 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 uncertainty 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

77 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 indicates 99.9% effective removal of telluric lines. 77

78 Comparisons to Previous Work There are works that have been previously done in attempts to understand the fundamental 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 (V t ). Since the spectra for solar type stars in our study have a V t of 1.0 km s 1, we only compare the results from spectra with V t of 1.0 km s 1 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, 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). T eff logg V t Vsin i (K) (cm s 1 ) (km s 1 ) 0 km s 1 4 km s 1 8 km s 1 12 km s (34940) 17235(17080) 8700(8440) 5793(5380) (33405) 16607(16140) 8305(7815) 5432(4930) (30375) 14858(14700) 7397(7020) 4700(4385) Reiners et al. (2010) investigated the precision that can be reached in RV measurements 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

79 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 measurements 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 (T eff = 2400K ) in comparison 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 Reiners et al. (2010) for an M9 dwarf (T eff =2400 K, V sin i=0 km s 1 ). 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) V Y J H V Y J H V Y J H We have also conducted similar calculation to Rodler et al. (2011) for an M9.5 dwarf (T eff =2200 K, V sin i=5 km s 1 ) 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

80 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 Rodler et al. (2011) for an M9.5 dwarf (T eff =2200 K, V sin i=5 km s 1 ). 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) Y J H K Y J H K Y J H K Case A Case B 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

81 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 (Lépine & 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 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

82 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 B V R Y J H K m a in a out v in v out (M ) (AU) (AU) (m s 1 ) (m s 1 ) A5V F5V G5V K5V M5V M9V

83 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 ( ), 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 δv S,rms δv ATM,rms δv cal δv rms δv HZ (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) (m s 1 ) K5V V M5V K 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

84 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

85 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 (Bouchy et al., 2009) is given in Table HD 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 Prediction vs. HARPS observation. a: HARPS observation of HD 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 δv S,rms δv cal δv rms (m s 1 ) (m s 1 ) (m s 1 ) HD47186 a B+V+R G5V b V R B B+V+R 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 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

86 provide adequate precision for an Earth-like planet detection while all bandpasses allow an Earth-like planet detection for an M9V star. Figure 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 T eff (Fig. 3-11). The minima of three subsets (corresponding to G, K and M stype stars) are found based on T eff. In comparison, 86

87 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 is consistent with our prediction of 0.65 m s 1. It is plotted in the bin with T eff between 5000 K and 6000 K because reported T eff of 5385 K. For G type stars, Bouchy et al. (2009) reported 0.91 m s 1 RMS error for HD 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

88 Figure RMS error of Keplerian orbit fitting for planets detected by HARPS since 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

89 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

90 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, δv rms,s (refer to Equation 3 3), for an M9V star at R=100,000 is 3.9, 2.2, 3.9, 2.2 m s 1 in V, Y, J and H band respectively (see Table 3-6). In comprison, δv rms,n is 71.3, 5.8, 6.5, 3.7 m s 1 in V, Y, J and H band respectively. Except for V band, δv rms,s and δv rms,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 exposure time. RV uncertainties due to low frequency ( 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

91 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

92 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

93 CHAPTER 4 PLANET SEARCH AROUND M DWARFS 4.1 Introduction 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 dwarfs, f 1% for P=1-10 day and f = % for P= 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 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 Atmophsere Ground-based NIR RV measurement is severely affected by the Earth s atmosphere, which consists absorption lines of many species including H 2 O, O 2, CO, CO 2 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

94 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 for a more detailed and complete review of the field of wavelength calibration sources. 4.2 Tackling Adversities in NIR RV Measurement Software Advancement 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

95 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 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 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

96 Figure 4-2. Comparison between an observed stellar spectra (black GJ 411, telluric lines removed) and a synthetic spectrum (red) 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 Dopplershifted 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 highresolution 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

97 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 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

98 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

99 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 previouslymentioned candidate sources, it has many advantages: 1. It is precise enough to provide a calibration precision of better than 10 cm s 1 which 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 inch for its optics components. When equipped with a thermal enclosure, its dimension is 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

100 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 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 O 2 lines RV stability using HARPS data and concluded that O 2 lines are as stable as 10 m s 1 over 6 years and the intrinsic stability of O 2 lines is even higher (2-3 m s 1 ) when a simple physical atmosphere 100

101 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 H 2 O lines is as stable as 20 m s 1 over half year. The small wavelength coverage (one spectral order, Å for H 2 O and Å for O 2 ) 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 stability between H 2 O and O 2 lines and the RV scatter rms is 18.3 m s 1 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 m s 1 precision level. 101

102 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 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

103 Figure 4-8. Telluric water lines RV stability compared to O 2 lines RV stability. 4.4 M-dwarf Planet Search and Characterization-Future Works 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-M 1. 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

104 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 M J. Johnson et al. (2007a) finds this fraction 1.8% for stellar mass range from 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

105 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 M J and another 3 (18.8%) planets between 0.6 M J and 1 M J. 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 calibration. 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

106 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; Huélamo et al., 2008; Martín 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 (λ 0 =806 nm, λ=149 nm) by observing a photometric stable star, KEP 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

107 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 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 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 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 (Martín 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 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 with a new generation cryogenic high-resolution (R=60,000, micron) near IR (NIR) cross-dispersed echelle spectrograph. This instrument, called FIRST, is scheduled to see first light at the AST 2m 107

108 Figure RV measured in I band using DEM of EXPERT for KEP , a photometric stable star. The RV RMS is 57.6 m s 1. telescope in June 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 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

109 Figure RV measured in V band using DEM of EXPERT for KEP , a photometric stable star. The RV RMS is 21.9 m s a). 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

110 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 (Lépine & 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

111 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 (Hünsch et al., 1999). The selection was based on the following criteria: J 10 and dec -20 M V 8.7 and V-K 3.5 Ratio between X ray luminosity and bolometric luminosity, R X 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. R X 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 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 photonnoise 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

112 Figure 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

113 Figure T eff distribution for FIRST survey targets K is used to divide early and mid-late type M dwarfs in the sample. survey completeness plot (Fig 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 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 ( 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

114 Figure 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

115 Figure The predicted survey completeness contours based on observation strategy and RV precision for the pessimistic case. 115

116 Figure The predicted survey completeness contours based on observation strategy and RV precision for the baseline case. 116

117 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) technique 1. 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 interferometer (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 interference 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 demonstrated 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 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

118 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: GD(ν) = 1 2π dϕ dν, (5 1) 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

119 preventing us from adopting this method (see more detailed discussion in ). We will use Method 3 and 4 to measure GD of an interferometer in this paper. 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 ν 0 ; 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: ϕ(ν, y) = 2π τ(ν, y) ν, (5 2) c 119

120 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 P o 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, ϕ x, is induced: ϕ x = dϕ dv v = dϕ dν dν dv v = dϕ dν ν v = Γ v, (5 3) 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: Γ = 2π GD ν c. (5 4) 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 ϕ x is extremely difficult. Instead, ϕ y, phase shift along y direction can be measured, which is equal to ϕ x 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 ϕ y instead of ϕ x 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, ϕ y is relatively easier to measure compared to ϕ x 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

121 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 Method MARVELS (Multi-object Apache Point Observatory Radial Velocity Exoplanet Largearea 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 interferometer (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 Kovács 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 Amotchkina 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 constructive 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 GD can be determined by measuring the derivative of unwrapped phase with respect to frequency according to GD definition (Equation 5 1). 121

122 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 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 We find that the phase change between 122

123 Figure 5-3. Phase of simulated WLCs. Phase can be calculated by Fourier-transform-based algorithm described in 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) GD Measurement Results The top view and side view of the MARVELS interferometer are shown in Fig 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

124 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 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

125 direction. The measured GDs should gradually vary with fiber number. We use a secondorder polynomial to fit the GD variation with the fiber number. The fitting residual has an RMS of 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