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Modeling Radial Velocity Signals for Exoplanet Search Applications MODELING RADIAL VELOCITY SIGNALS FOR EXOPLANET SEARCH APPLICATIONS Prabhu Babu∗, Petre Stoica Division of Systems and Control, Department of Information Technology Uppsala University, P.O. Box 337, SE-75 105, Uppsala, Sweden {prabhu.babu, ps}@it.uu.se Jian Li Department of Electrical and Computer Engineering, University of Florida, Gainesville, 32611, FL, U.S.A. [email protected]fl.edu Keywords: Radial velocity method, Exoplanet search, Kepler model, IAA, Periodogram, RELAX, GLRT. Abstract: In this paper, we introduce an estimation technique for analyzing radial velocity data commonly encountered in extrasolar planet detection. We discuss the Keplerian model for radial velocity data measurements and estimate the 3D spectrum (power vs. eccentricity, orbital period and periastron passage time) of the radial velocity data by using a relaxation maximum likelihood algorithm (RELAX). We then establish the significance of the spectral peaks by using a generalized likelihood ratio test (GLRT). Numerical experiments are carried out on a real life data set to evaluate the performance of our method. 1 INTRODUCTION on iterative deconvolution in the frequency domain to obtain a clean spectrum from an initial dirty one. A Extrasolar planet (or shortly exoplanet) detection is periodogram related method is the least squares peri- a fascinating and challenging area of research in the odogram (also called the Lomb-Scargle periodogram) field of astrophysics. Till mid 2009, 353 exoplanets (Lomb, 1976; Scargle, 1982) which estimates the si- have been discovered. Some of the techniques avail- nusoidal components by fitting them to the observed able in the astrophysics literature to detect exoplanets data. Most recently, (Yardibi et al., 2010; Stoica et al., are astrometry, the radial velocity method, pulsar tim- 2009) introduced a new method called the Iterative ing, the transit method and gravitational microlens- Adaptive Approach (IAA), which relies on solving an ing. Among these methods, the radial velocity analy- iterative weighted least squares problem. sis is the most commonlyused technique, in which the In this paper, we analyze the radial velocity data Doppler shift in the spectral lines and hence the radial by using a relaxation maximum likelihood algorithm velocity of the parent star is measured. The spectrum (RELAX) initialized with IAA estimates. The signif- of the measured Doppler shifts is then analyzed to de- icance of the spectral peaks is then established via a tect the exoplanet(s) revolving around the star. generalized likelihood ratio test (GLRT). Numerical Most often the radial velocity measurements are ob- experiments are carried out on a real life radial veloc- tained at nonuniformly spaced time intervals due to ity data set. hardware and practical constraints, which limits the In Section 2, we describe the model used in this pa- application of commonly used spectral analysis meth- per for the radial velocity data. Section 3 presents the ods. The most straightforward way to deal with this RELAX and GLRT methods, and Section 4 contains problem is to use the standard periodogram by ignor- the results for a real life example. Finally, the paper ing the nonuniformity of data samples, which results is concluded in Section 5. in an inaccurate spectrum. In (Roberts et al., 1987), a method named CLEAN was proposed, which is based ∗Corresponding author. This work was supported in part by the Swedish Research Council (VR). 155 ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics 2 DATA MODEL enough (to reduce the computation time), then IAA might miss some true peaks (see (Babu et al., 2010) N Let {y(tn)}n=1 denote the radial velocity of a star for an elaborate discussion on IAA for radial veloc- measured at a set of possibly nonuniform time in- ity data). In that case, applying RELAX (Li and N stants {tn}n=1. Based on the Keplerian model of plan- Stoica, 1996), a parametric iterative estimation al- etary motion (Zechmeister and Kurster, 2009) (Cum- gorithm, can refine the IAA estimates. Algorithm 1 ming, 2004), the radial velocity data are modeled as briefly describes the steps involved in RELAX. The follows: P largest peaks picked from IAA are used as initial M estimates for RELAX, which has a beneficial effect β ω ν ω y(tn)= C0 + ∑ m[cos( m + m(tn)) + em cos( m)], on the convergence of RELAX compared with using m=1 n = 1,··· ,N other more arbitrary initial estimates. In the case of (1) radial velocity data, the choice of P = 5 peaks ap- pears to be reasonable for most applications. RELAX where C0 is the constant radial velocity, and generally converges within a few iterations (typically 1+em tan(νm(tn)) = tan(Em(tn)) in less than 10 iterations). 1−em (2) q 2π(tn−Tm) Next we note that, under the assumption the noise Em(tn) − em sin(Em(tn)) = , Pm in the data is Gaussian distributed, the RELAX esti- M: Number of exoplanets revolving around the mates are optimal in the maximum likelihood sense star. The number of planets M is usually unknown. (Li and Stoica, 1996). We can then use the general- th em: Eccentricity of the orbit of the m planet. ized likelihood ratio test (GLRT) to establish the sta- th ωm: Longitudeof the periastron for the m planet. tistical significance of the estimated planet parame- th Pm: Orbital period of the m planet; Pm is related ters. We first apply RELAX to the largest IAA peak to orbital frequency f by f = 1 . m m Pm and use GLRT to test the null hypothesisthat there are th Tm: Periastron passage time of the m planet. no planets (or, in other words, that the data is made th βm: Radial velocity amplitude of the m planet. only of white noise) against the hypothesis that there νm(tn), Em(tn): True and eccentric anomaly of the is at least one exoplanet. If the test rejects the null th m planet, with tn denoting their time dependence. hypothesis then we will proceed and apply RELAX We divide the entire 2D space G, defined as G = to the two largest peaks and subsequently test the hy- fmax fmax pothesis that there is one exoplanet in the data against {(e, f ), 0 ≤ e < emax, − 2 < f < 2 }, into a grid of prespecified size K. We point out here that the grid the hypothesis that there are at least two exoplanets; G does not include the parameter T , and that T is and so on. As an example, for the following hypothe- taken to be zero for the time being but will be esti- ses H : There are no planets. mated as described later on. The choices of emax and 0 f in G depend on the sampling pattern: to deter- H1: There is at least one exoplanet with eccentricity max ˆ mine them, we calculate the spectral window defined eˆ1, orbital frequency f1 and periastron passage time ˆ as: T1. the log-likelihood (LL) functions are given by: N 2 1 W (e, f ) = ∑ exp( jν(tn)) , 0 ≤ e < 1,−∞ ≤ f ≤ ∞. N LL(H ) = n=1 0 (3) N N ∑ 2 ν − 2 ln |y(tn)| +C, For any choice of (e, f ) and tn, there exists a (tn) ob- n=1 tained via (2). Following (Eyer and Bartholdi, 1999), LL(H1) = N the parameters emax and fmax are chosen such that the N ∑ ν ν 2 − 2 ln |y(tn) − rˆ1 cos( (tn)) − qˆ1 sin( (tn))| +C region G leads to an unambiguous W(e, f ) (see (Babu n=1 et al., 2010) for more details). (4) where C is an additive constant, ν(tn) is calculated from the RELAX estimates (ˆe1, fˆ1, Tˆ1), andr ˆ1,qˆ1 3 PARAMETER ESTIMATION are the least square estimates of r,q corresponding to (ˆe1, fˆ1, Tˆ1), see Algorithm 1. Under the assumption AND STATISTICAL that hypothesis H0 is true, the log-likelihood-ratio, SIGNIFICANCE TESTING: defined as 2(LL(H1) − LL(H0)), is asymptotically a RELAX AND GLRT random variable with a chi-square distribution. Then the GLRT is given by The estimates obtained from IAA are usually fairly H1 2(LL(H ) − LL(H )) ≷ Λ (5) accurate. However, if the grid ( ) is not chosen fine 1 0 G H0 156 MODELING RADIAL VELOCITY SIGNALS FOR EXOPLANET SEARCH APPLICATIONS where Λ denotes a fixed threshold. The threshold is of spectral peaks (planets) in radial velocity data and usually chosen such that prob(X ≤ Λ)= ξ, where X ∼ accurately identifies both their frequencies and eccen- χ2 5 denotes a chi-square distributed random variable tricities as well as their periastron passage times. The with 5 degreesof freedom(becauseof the 5 unknowns example used here is typical of cases usually encoun- per planet in the data model, namely e, f , T , r and tered in exoplanet search and hence the proposed al- q), and ξ determines the significance level of the test. gorithm is believed to be an effective and useful tool. Choosing ξ = 0.99 gives a false alarm probability of 0.01 and the corresponding threshold is Λ = 15. REFERENCES 4 A REAL LIFE EXAMPLE: Babu, P., Stoica, P., Li, J., Chen, Z., and Ge, J. (2010). Analysis of radial velocity data by a novel adaptive HD 208487 approach. The Astronomical Journal, 139:783–793. Cumming, A. (2004). Detectability of extrasolar planets in In this section, we consider the application of the al- radial velocity surveys. Monthly Notices of the Royal gorithm introduced in the previous section to a real Astronomical Society, 354(4):1165–1176. life radial velocity data set. Our goal is to detect Eyer, L. and Bartholdi, P. (1999). Variable stars: Which the exoplanets present in a star system and estimate Nyquist frequency? Astronomy and Astrophysics, their eccentricities, frequencies and periastron pas- Supplement Series, 135:1–3.
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