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Analytical Determination of Orbital Elements Using Fourier Analysis. I Astronomy & Astrophysics manuscript no. DSBA c ESO 2018 May 15, 2018 Analytical determination of orbital elements using Fourier analysis I. The radial velocity case J.-B. Delisle1; 2, D. Ségransan1, N. Buchschacher1, and F. Alesina1 1 Observatoire de l’Université de Genève, 51 chemin des Maillettes, 1290, Sauverny, Switzerland e-mail: [email protected] 2 ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau, 75014 Paris, France May 15, 2018 ABSTRACT We describe an analytical method for computing the orbital parameters of a planet from the periodogram of a radial velocity signal. The method is very efficient and provides a good approximation of the orbital parameters. The accuracy is mainly limited by the accuracy of the computation of the Fourier decomposition of the signal which is sensitive to sampling and noise. Our method is complementary with more accurate (and more expensive in computer time) numerical algorithms (e.g. Levenberg-Marquardt, Markov chain Monte Carlo, genetic algorithms). Indeed, the analytical approximation can be used as an initial condition to accelerate the convergence of these numerical methods. Our method can be applied iteratively to search for multiple planets in the same system. Key words. celestial mechanics – planets and satellites: general – methods: analytical – techniques: radial velocities 1. Introduction centric planets). These algorithms are implemented in the DACE web-platform (see Buchschacher et al. 2015)1. In Sect.2 we derive an analytical decomposition of a Kep- In this article we are interested in retrieving planetary orbital lerian radial velocity signal in Fourier series. In Sect.3, we de- parameters from a radial velocity time series. These parameters scribe an analytical method to compute orbital parameters from are usually obtained using numerical least-squares minimization the Fourier coefficients of the fundamental and first harmon- methods (Levenberg-Marquardt, Markov chain Monte Carlo, ge- ics of a Keplerian signal. In Sect.4, we show how to find the netic algorithms, etc.). These methods enable one to explore the fundamental frequency and compute the Fourier coefficients. In parameter space and find the best fitting solution as well as es- Sect.5, we illustrate the performances and limitations of our timates of the error made in the parameters. However, numer- methods on observed planetary systems. In Sect.6, we summa- ical methods need to be initialized with starting values for the rize and discuss our results. parameters. When initial values are far from the solution, these algorithms can become very expensive in terms of computation time, or even unable to converge (non-linear fit can have sev- 2. Fourier decomposition of a Keplerian radial eral local χ2 minima, etc.). On the contrary, using a good guess velocity signal of the parameters as initial conditions significantly improves the efficiency of numerical methods. We assume here that the observed system is composed of a star and a planet (no perturbation). The motion of the star with re- spect to the center of mass is Keplerian. The radial velocity of Here we describe an analytical method that allows a very ef- the star in the reference frame of the center of mass reads ficient determination of the orbital parameters. This method is based on the Fourier decomposition of the radial velocity data, V(t) = K (cos(ν + !) + e cos(!)) ; (1) which is obtained using linear least-squares spectral analysis. arXiv:1512.03298v2 [astro-ph.EP] 8 Apr 2016 This idea of recovering the orbital parameters of the planet from with the Fourier decomposition of the radial velocity signal has al- ready been proposed by Correia(2008) and used in the analy- mp 2πa sin i K = ; (2) sis of different planetary systems (see Correia et al. 2005, 2008, ms + mp P p1 e2 2009, 2010). A similar idea was also developed in the case of − interferometric astrometry by Konacki et al.(2002). In this ar- and a is the semi-major axis of the planet with respect to the ticle, we complete the sketch proposed by Correia(2008) and star, e the eccentricity, i the inclination, ν the true anomaly, ! provide a fully analytical method to retrieve the orbital parame- the argument of periastron, P the orbital period, mp the planet ters. In addition to this Fourier method, we propose an alterna- tive algorithm that is based on information contained in the ex- 1 The DACE platform is available at http://dace.unige.ch. The algo- trema of the radial velocity curve and that can be used when the rithms described in this article are part of the project “Observations” of Fourier decomposition is unreliable (e.g. ill-sampled, very ec- DACE. Article number, page 1 of 12 A&A proofs: manuscript no. DSBA mass, ms the star mass. We denote by n the mean-motion of the we only need the expansion of X 1 and X 2. We have (see Ap- planet: pendixA) ± ± r 2 4 2π µ X1 = 1 e + O e (10) n = = ; (3) − P a3 5 X = e e3 + O e5 (11) 2 − 4 with 1 2 1 4 6 X 1 = e + e + O e (12) µ = (m + m ); (4) − −8 48 G s p 1 3 1 5 7 X 2 = e + e + O e : (13) and is the gravitational constant. The radial velocity signal is − −12 48 P-periodicG and can be decomposed in discrete Fourier series: X iknt 3.2. Eccentricity and phase V(t) = Vke ; (5) k Z Since we assume that the complex coefficients V , V are known, 2 1 2 the ratio ρ V2=V1 is also known. From Eq. (9), we have with ≡ 2i! ! V2 iM X2 1 + X 2=X2e− Z P ρ = = 0 − : 1 iknt e 2i! (14) Vk = V(t)e− dt V1 X1 1 + X 1=X1e− P 0 − Z 2π From Eqs. (10)-(13) we obtain 1 ik(M M0) = V(t)e− − dM; (6) 3 2π X2 e 0 = e + O e5 (15) X1 − 4 where M is the mean anomaly and M0 is the mean anomaly at the 2 X 1 e 4 reference time t = 0. The coefficients Vk are complex numbers. − = + O e (16) Since V(t) is real, we have V k = Vk (where Vk is the complex X1 − 8 − l;q 2 conjugate of V ). We denote by X the Hansen coefficients X 2 e 4 k k − = + O e : (17) X2 −12 Z 2π l l;q 1 r iqν ikM X (e) = e e− dM: (7) We thus have k π 2 0 a 2i! 2 1 + X 2=X2e− e 2i! 4 − = 1 + e− + O e ; (18) ffi 2i! These coe cients are real numbers that only depend on the ec- 1 + X 1=X1e− 24 l; q l;q − centricity (e), and we have X − = X . The Fourier expansion k k and of the radial velocity can be rewritten− using Hansen coefficients 3 3 ! iM e e 2i! 5 ρ = e 0 e + e− + O e : (19) V0 = 0 (8) − 4 24 ikM0 Ke 0;1 i! 0; 1 i! V = X (e)e + X − (e)e− We introduce the complex coefficient k 2 k k 2i! ! KeikM0 1 e− 0;1 i! 0;1 i! C(!) = 1 ; (20) = Xk (e)e + X k (e)e− (k Z∗): (9) 2 − 2 4 − 6 We observe that the radial velocity only depends on Hansen co- and use, in the following, the approximation 0;1 efficients of the form X (k Z∗). In the following, we drop the k 2 ρ eiM0 e C(!)e3 ; (21) exponents and use the notation X X0;1. ≈ − k ≡ k We note that V0 = 0 is only valid if the observer is fixed in where order 5 (and more) terms are neglected. We first suppose the reference frame of the center of mass. If we account for a that C(!) is known and show how to determine the eccentricity constant motion of the observed system with respect to the solar and the angle M0. Using approximation (21), we only have to system, we have V0 , 0 but all other coefficients are unaffected. solve a third order polynomial equation to obtain the eccentricity ρ e (C)e3; (22) 3. From Fourier coefficients to orbital parameters j j ≈ − < where (C) denotes the real part of C, Let us assume that we are able to determine the period of the < ! " # planet as well as the values of V1 (fundamental) and V2 (first har- 1 cos(2!) 5 7 monics) from an observed radial velocity signal. We thus have (C) = 1 ; : (23) < 4 − 6 2 24 24 access to five observables (because Vk are complex numbers). Therefore, this information should be sufficient to determine the Since we always have (C) < 1=3, there is at most one solution < five parameters: P, K, e, !, M0. for e in the interval [0; 1], and only one, provided that ρ 1 (C). One can easily verify that this solution is given byj j ≤ − < 0 !1 3.1. Expanding Hansen coefficients 3 p3 (C) Bπ + < ρ C B arccos 2 C Hansen coefficients are functions of the eccentricity alone and 2 B j j C eˆ = p cos B C ; (24) can be expanded in power series of the eccentricity. Since we 3 (C) B 3 C restrict our study to the fundamental and the first harmonics, < @B AC Article number, page 2 of 12 J.-B. Delisle et al.: Analytical determination of orbital elements using Fourier analysis where the hat denotes an estimation of the considered parame- is the only important angle. Our analytical method provides an ter (here the eccentricity e).
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