The Astrophysical Journal, 610:1079–1092, 2004 August 1 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A. NEW INFORMATION FROM RADIAL VELOCITY DATA SETS Robert A. Brown Space Telescope Science Institute,1 3700 San Martin Drive, Baltimore, MD 21218; [email protected] Receivved 2004 January 7; accepted 2004 April 9 ABSTRACT Radial velocity data sets hold information about the direct observability (e.g., separation and flux) of inferred companions. They also contain information about the types of Keplerian solution compatible with the data. ‘‘Monte Carlo projection’’ and ‘‘2 portrayal’’ are two techniques for discovering and pursuing the implications of this information. The first (projection) involves random solutions consistent with the data set, from which we can estimate (1) the probability distribution of the true solution in the six-dimensional space of the Keplerian parameters and (2) the probability distribution of the companion’s position in space at future times, in order to predict observability. The second technique (portrayal) involves the distribution in parameter space of values of the numerical 2 function, from which we can estimate the regions that contain the true solution at various levels of confidence. We study the case of HD 72659, a Sun-like star at 51 pc with a radial velocity companion inferred from 16 data points. We find at least two types of Keplerian solutions present in the data set: (1) periods 2500– 25,000 days and eccentricity 0–0.8 (type A1), (2) periods 25,000–250,000 days and eccentricity 0.8–0.95 (type A2), and (3) periods 2000–2500 days and eccentricity 0–0.5 (type B). (Types A1 and A2 may not be distinct.) Pursuing direct observability, we randomize the inclination angle and compute the apparent separation, true separation, and phase angle of the companion. We compute a minimum flux ratio to the star assuming no self- luminosity and that the companion is Jupiter sized and has Jupiter’s albedo and the phase function of a Lambert sphere. We plot the probability distribution of direct observability at specific epochs. Subject headinggs: binaries: spectroscopic — instrumentation: high angular resolution — planetary systems — stars: individual (HD 72659) — techniques: radial velocities 1. INTRODUCTION the direct observability at any epoch. We proceed by trans- lating the sample of compatible solutions into companion Radial velocity observers monitor nearby stars for evidence orbits, which we use to compute positions in space at a future of Jupiter-class companions. As of 2003 December, they had epoch, which we use to compute apparent separations and published discoveries of 104 such systems, consisting of 119 2 delta magnitudes, which we use to compute the fraction of the companions and including 13 multiple-companion systems. The scientific community, as well as the public, is interested in sample that is directly observable, which is our estimate of the probability of direct observability at that epoch. the nature of these objects, many of which may be planets The art in Monte Carlo projection lies in creating the initial in the solar system sense, while others may be brown dwarfs or sample of compatible solutions, for the subsequent calcu- even objects not yet categorized. Only by resolving, detecting, lations are strictly mechanical. In this paper we pursue Monte and analyzing their light can we learn what these objects are Carlo projection based on the techniques described by Press in themselves. Engineers and scientists are designing optical et al. (1986, pp. 529–532). We generate random samples of systems to make such observations possible. Therefore, it is synthetic data sets statistically consistent with the real data useful to understand what the existing data sets say about the set, which we fit to find the sample of compatible solutions. direct observability of radial velocity companions, particu- To provide a pertinent example, G. Marcy has kindly pro- larly if they have the photometric properties of planets. Here we treat the case of direct detection in visible light, vided a radial velocity data set for HD 72659, which evidences a Jupiter-class companion (Butler et al. 2003). While HD which is starlight scattered by the object toward the observer. 72659 is not particularly favorable for direct detection, being at The instrument implied but not discussed is an optimized coronagraph, which ideally offers an annular field of view about 51 pc distance, it nevertheless illustrates our techniques. Monte Carlo projection invites us to conceptualize the with uniform sensitivity. The key metrics of a companion’s domain of all Keplerian solutions compatible with a radial direct observability are its apparent separation and flux ratio to velocity data set. This domain is enclosed by surfaces in the the star (Brown & Burrows 1990; Brown et al. 2003). hyperspace spanned by the six Keplerian parameters (‘‘pa- We call our technique for predicting direct observability rameter space’’). An improved representation of this domain ‘‘Monte Carlo projection.’’ We generate large numbers of is an opportunity to deepen our understanding of the possible Keplerian solutions for the star’s radial velocity that are sta- orbits. As we show below, the sample of solutions obtained tistically compatible with the data set. Each solution corre- from synthetic data sets provides a striking view of parameter sponds to a possible companion, for which we can compute space in terms of the probability density of compatible so- lutions. To pursue the implications, we introduce a second 2 1 technique, called ‘‘ portrayal,’’ based on pp. 532–535 in The Space Telescope Science Institute is operated by the Association Press et al. (1986). Here, without the intermediary of a fitting of Universities for Research in Astronomy, Inc., under NASA contract 2 NAS 5-26555. routine, we use a probe to directly locate in parameter 2 See http://www.obspm.fr/encycl/catalog.html. space confidence regions that contain the true solution at 1079 1080 BROWN Vol. 610 various levels of confidence. These results complement, con- where firm, and extend the insights from fitting synthetic data sets. 1:5 Thus, two new techniques for understanding the orbits of K3ðÞ1 À "2 T : ð7Þ spectroscopic binaries are an unexpected result of our interest 2G in direct observability. To demonstrate the practicality of predicting the direct ob- To derive equation (5), we use the semimajor axis of the servability of radial velocity companions using Monte Carlo orbit of the companion around the star, which we obtain from projection, we pose and answer the question, ‘‘What is the Kepler’s Third Law informed by Newton’s Law of Gravity: probability that the radial velocity companion of HD 72659, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À9 ÀÁ if it is a planet, will be brighter than 10 times the star and Gm þ m T 2 have a separation greater than 0B05 on 2010 January 10?’’ (That 3 p s a ¼ 2 ; ð8Þ date is the quadricentennial of Galileo’s discovery of the 4 satellites of Jupiter.) to eliminate the semimajor axis of the star around the center 2. KEPLERIAN SOLUTIONS FOR THE of mass from Smart’s equation (58), p. 359. Here G is the STELLAR RADIAL VELOCITY gravitational constant. A ‘‘Keplerian solution’’ for the radial velocity of a star A radial velocity data set consists of N triplets: the epoch of consists of six values for the parameters of the Keplerian the observation (tk), the measured radial velocity (uk), and the theory. We choose for these parameters the period (T ), ec- estimated measurement error in the radial velocity (uk), for k centricity ("), mean anomaly at the epoch of the first data point from 1 to N. The best estimates for the Keplerian parameters are those that minimize the reduced 2 statistic: (‘‘initial mean anomaly,’’ M0), argument of periastron (!s), radial velocity amplitude (K ), and radial velocity of the center P N fg½f (t ) À u =u 2 of mass or ‘‘offset velocity’’ (V ). Following Smart (1962), the 2 k¼1 k k k ; ð9Þ theoretical value of radial velocity at any time t is the function r N À 6 f (t) ¼ V þ K½cosðÞþ þ !s " cos !s ; ð1Þ which is a numerical function of the data set and the six Keplerian parameters. (To find our best estimates, we use the where the true anomaly () is the root of the equation, Levenberg-Marquardt option of FindMinimum in Mathemat- rffiffiffiffiffiffiffiffiffiffiffi ica ver. 5.0.) 1 þ " E tan ¼ tan ; ð2Þ 2 1 À " 2 2.1. HD 72659: The Rangge of Keplerian Solutions Table 1 gives the radial velocity data for HD 72659 provided in which the eccentric anomaly (E) is the root of Kepler’s by G. Marcy. According to SIMBAD, this star has spectral type Equation, G0 V, visual magnitude 7.5, and parallax 19.5 mas; it is a 1 M , E À " sin E ¼ M; ð3Þ main-sequence star 51 pc away. The observers have published a preliminary orbital solution and make updated information 3 and the mean anomaly (M )is available at their Web site. For purposes of the least-squares fit, Marcy advised us to À1 t À t1 add 7.2 m s in quadrature to the tabulated values of uk M ¼ M0 þ 2 ; ð4Þ 2 T before using them to compute r in equation (9) to fit the data. His team has calibrated the velocity jitter of HD 72659 using where t1 is the epoch of the first data point. The inclination chromospheric and photospheric diagnostics and estimate the angle (i) is neither observed nor determined. Using astro- value of 7.2 m sÀ1, ‘‘typical of G0 V stars.’’ According to physical characteristics to estimate the stellar mass (ms), the Marcy, this estimate was computed ab initio, using only the minimum mass of the companion (mp; min)isthemp root of this team’s stellar information, independent of the residuals in their 0 equation when sin i ¼ 1: radial velocity data set.