NEW INFORMATION from RADIAL VELOCITY DATA SETS Robert 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 solutions. 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 "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 9 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 ) 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 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 " 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 " 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 1 t 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 s1, ‘‘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. Using augmented uncertainties uk in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ equation (9), where pffiffiffiffiffiffiffiffiffiffiffiffiffi Tm m 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 p þ s mp sin i ¼ K 1 " ; ð5Þ 0 2 2 2G uk ¼ uk þ 7:2 ; ð10Þ

2 or algebraically, produces values of r 1, indicating reliable fits and good " understanding of sources of error. We have consistently used hiÀÁ 1=3 2 3 2 unaugmented uncertainties to generate synthetic data sets and mp; min ¼ 2 6ms 3 2 þ 18 ms 2 2 3 to compute r in portrayal and used augmented uncer- # tainties only when fitting data sets. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffi 1=3 Table 2 summarizes three fits to the HD 72469 data set, 2 3 3 2 4 þ 27ms þ 3 3 4 ms þ 27 ms which are all basically the same in the context of this paper and which we call the ‘‘accepted solution.’’ Marcy provided the observers’ fit. Our fits 1 and 2 explore the effect of aug- 3 2 2 þ 2 þ 18 ms þ 27ms menting the uncertainties. We find that augmentation does not qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hi significantly change the results of the fit to the HD 72469 data. pffiffiffi 1=3 1 3 3 2 4 1=3 þ 3 3 4 ms þ 27 ms ðÞ3 2 ; ð6Þ 3 See http://exoplanets.org/esp/hd72659/hd72659.shtml. No. 2, 2004 NEW INFORMATION FROM RADIAL VELOCITY DATA SETS 1081

TABLE 1 Radial Velocity Data for HD 72659

tk uk uk Index k (JD 2,440,000) (m s1) (m s1)

1...... 10,838.893 35.586 4.26 2...... 10,861.915 53.129 4.34 3...... 11,171.026 47.898 4.15 4...... 11,226.938 52.108 4.52 5...... 11,551.989 37.209 4.41 6...... 11,553.007 36.538 3.99 7...... 11,580.932 30.938 3.99 8...... 11,883.116 15.369 3.85 9...... 11,898.127 4.776 1 3.54 10...... 11,971.996 11.657 3.97 11...... 12,243.116 33.295 4.40 12...... 12,362.963 19.773 5.20 13...... 12,576.125 41.064 3.93 14...... 12,777.797 27.477 3.81 15...... 12,804.748 17.741 4.18 16...... 12,805.743 25.535 3.96

Note.—Data provided by G. Marcy, at whose advice we added 7.2 m s1 in quadrature to the tabulated values of uk before using them in least-squares ﬁts.

Compared with the observers’ fit, our fits have insignificantly The probability density plots in Figures 2–4 clarify that the different eccentricities and periods but significantly nonzero two solution types suggested by the histograms are related and offset velocities (see below). may or may not be distinguished from each other by an inter- We explore the range of Keplerian solutions permitted by vening gap in parameter space. Both possible types have T > the data in the first stage of Monte Carlo projection. We 2500 days (log10T > 3:4 log-days) and !s ’ 4:6rad,which generate 10,000 synthetic data sets for HD 72469 that repre- means that stellar periastron is located closer to the observer sent the real data set statistically. To accomplish this, we re- than the center of mass. Type A1 has period 2500 days T place the value of uk by a Gaussian random deviate with the 25; 000 days, with 0 " 0:8and4rad M0 6rad,and mean value f (tk), using the accepted solution, and the variance type A2 has period 25; 000 days T 250; 000 days, with 2 (uk) ; we retain the values of tk and uk in Table 1. We 0:8 "<0:95 and M0 2 rad. By counting the number of compute a new best-fit orbital solution for each synthetic data solutions in each type’s zone and computing the fraction of the set using augmented errors (eq. [10]). The histograms in sample, we estimate that the probabilities are 0.95 and 0.05 for Figures 1a–1h are based on the results. types A1 and A2, respectively, with two prior assumptions. The The histograms confirm the accepted solutions as most first is that exactly one companion is present. The second as- probable. They also show three minor but possibly significant sumption, discussed by Press et al. (1986, p. 530), is that the departures from the accepted solutions: secondary peaks at shape of the probability distribution of deviations of the syn- high eccentricities, long periods, and initial mean anomaly thetic solution parameter values from the approved solution near 2 rad (red arrows in Figs. 1a,1b,and1e). We find that parameter values is the same (or nearly the same) as the shape there may be different types of Keplerian solutions to radial of the probability distribution of the deviations of the synthetic velocity data sets. solution parameter values from the true parameter values.

TABLE 2 ‘‘‘‘Ac ce pte d’’’’Keplerian Solutions for the Radial Velocity of HD 72659

Parameter Observers’ Fit Our Fit 1 with Measurement Uncertainties Our Fit 2 with Augmented Uncertainties

Period, T (days) ...... 3530 3519 3938 Eccentricity, "...... 0.26 0.26 0.33 Periastron passage (JD) ...... 2,451,696.60 2,451,697.23 2,451,682.61 Initial mean anomaly, M0 (rad) ...... 4.757 4.751 4.937 Argument of periastron, !s (rad)...... 4.515 4.517 4.489 Argument of periastron, !p (rad)...... 1.373 1.375 1.347 Velocity amplitude, K (m s1)...... 42.3 42.3 41.3 CM or offset velocity, V (m s1) ...... 7.2 7.1 Minimum mass, mp;min ...... 3.1MJ 3.1MJ 3.0MJ Semimajor axis, a (AU) ...... 4.54 4.53 4.88 2 2 Reduced , r ...... 3.28/8.17/2.15 3.26 0.84

Notes 2 2 .—The observers’ fit was provided by G. Marcy, who reported r ¼ 3:28 using unaugmented measurement uncertainties. We find r ¼ 8:17 and 2.15 for the observers’ fit (with V ¼ 0) without and with augmented uncertainties, respectively. We derive the ‘‘observers’ value’’ of M0 using the provided period and epoch of periastron passage. Our fits permit V to be a formal free parameter. The observers carry no information about velocities in the solar system barycentric frame. We derive all values of mp; mini and a from eqs. (6), (7), and (8). Fig. 1.—(a–h) Histograms of 10,000 Keplerian solutions compatible with the HD 72659 data set. The red arrows in (a), (b), and (e) indicate type A2 solutions; (c)and(d ) include periods less than 25,000 days only. ( j, k) Histograms of the implied minimum mass and semimajor axis of the companion. The ordinate is the fraction of solutions in a bin.

1082 Fig. 2.—Probability density plot against T and " for the 10,000 Keplerian solutions for the radial velocity of HD 72659, based on synthetic data sets consistent with the real data set. We sorted the solutions into 200 bins vertically and 323 bins horizontally. The color reports the probability density. Two distinctive types of solutions are present. Type A1 has period 2500–25,000 days and eccentricity 0–0.8, and type A2 has period 25,000–250,000 days and eccentricity 0.8–0.95. The orange curve is the expected variation for constant K. The white circles are particular solutions summarized in Table 3.

Fig. 3.—Probability density plot against T and M0. The orange curve, part of which is displaced downward for ease of viewing the data, is the expected variation of initial phase to preserve the date of periastron as the period varies. The sprinkling near T 2500 days reﬂects the indeterminate periastra of near-circular orbits. 1083 1084 BROWN Vol. 610

Fig. 4.—Probability density plot against T and !s . The sprinkling near T 2500 days reﬂects the indeterminate periastra of near-circular orbits.

Table 3 gives 10 representative solutions, including the drawn for the values of T and M0 of the accepted solution. accepted solution. Figure 5 shows them plotted against the The sprinkling of M0 and !s near T 2500 days reﬂects the data, and Figures 2–4 indicate their positions with white indeterminate periastra of nearly circular orbits. circles. The orange curve in Figure 2 holds (1 "2)1=2T 1=3 approxi- Figures 2–5 illustrate how solutions adapt to preserve the mately constant to maintain the relationship between mp sin i two essential facts in the data set: the timing and magnitude of and K in equation (5). We can explain the departures from that the radial velocity swing near JD 2,451,800. simple theory as follows. When we extrapolate type A2 (long The orange curve in Figure 3 shows how M0 adjusts to period, high eccentricity) solutions beyond the data, the curves preserve the periastron for a range of periods: must be nearly ﬂat exiting the data, which means a preference for lower K to avoid large residuals from the earliest and latest T data points. Conversely, for the type A1 (intermediate period, M 0 ¼ 2 ðÞ2 M ; ð11Þ 0 T 0 0 low to moderate eccentricity) solutions, the lower period curves

TABLE 3 Ten Particular Solutions to the Radial Velocity Data Set for HD 72659

2 2 Number ID T " M0 !s KVr (ﬁt) r (data)

1967...... A 3155 0.22 4.80 4.25 41.9 8.4 0.17 0.96 ‘‘Accepted’’..... B 3938 0.33 4.94 4.49 41.3 7.1 0.00 0.84 1832...... C 4963 0.44 5.20 4.51 42.7 7.7 0.74 0.97 7710...... D 6354 0.53 5.40 4.60 41.7 5.8 0.11 1.00 7894...... E 7958 0.57 5.60 4.49 39.5 5.3 0.32 0.92 8325...... F 50242 0.88 6.20 4.09 40.0 6.9 0.15 1.14 3782...... G 62480 0.90 6.20 4.30 39.3 3.7 0.16 0.96 8269...... H 79699 0.91 6.22 4.39 38.6 0.7 0.17 0.95 6170...... J 100038 0.93 6.22 4.77 38.0 9.9 0.08 1.00 8399...... K 124720 0.94 6.25 4.24 38.5 5.25 0.21 1.02

Notes.—Solution B is our fit 2, an accepted solution. We selected the others to represent solution types A1 and A2. The white circles in Figs. 2–4 identify these solutions in the correlation plots. Fig. 5 shows these solutions plotted against the 2 2 data. We computed r (fit) and r (data) with respect to the fitted and measured radial velocity values, respectively, using augmented uncertainties. No. 2, 2004 NEW INFORMATION FROM RADIAL VELOCITY DATA SETS 1085

Fig. 5.—Ten particular solutions to the radial velocity data set for HD 72659. The solutions are listed in Table 3. Solution B is the accepted solution. We selected the others from the 10,000 solutions to synthetic data sets representing the statistical variation of the data. The circles identify these solutions in the correlation plots, Figs. 2–4. Both measurement error bars (red ) and error bars augmented for chromospheric and photospheric effects (gray) are given. prefer higher K to pass more vertically through the end data the other two parameters held fixed at the coordinate values. points. At a given period, lower (higher) K means a departure Where the coverage is sparse (e.g., in Fig. 10 for log10T > 3:8, toward smaller (larger) values of ". where the data poorly constrain the offset velocity), the opti- The 2 portrayal provides a different view of compatible mization is poor or nonexistent, and we are looking through the solution types for HD 72659, as shown in Figures 6–10. We missing top layer of best fits into ignorance. In the interpretation generate hundreds of millions of random points in parameter that follows, we refer to the regions of Figures 6–10 where 2 space and at each compute the value of r using unaugmented coverage is sufficiently intense that the displayed ‘‘best available uncertainties, uk . At first, we cover a broad region, the entire fit’’ can be assumed to well approximate optimized values. 2 displayed range of parameter space, generating the coordinates Given the minimum value of r from optimizing four 2 of points using uniform random deviates over the ranges parameters with two held fixed (we call this value r;2), we 3 log-days log10T 5:5 log-days, 0 " 1, 0 rad M0 can estimate the local level of confidence using Theorem D of 1 1 2 rad, 0 rad !s 2 rad, 0 m s K 100 m s ,and Press et al. (1986, p. 535). By ‘‘local level of confidence’’ we 30 m s1 V 30 m s1. We follow up with more intense mean the probability that the true solution lies within the 2 2 2 coverage on areas producing relatively low values of r , regions of parameter space with r r;2. According to sometimes ‘‘painting in’’ smaller regions of parameter space, Theorem D, the statistic sometimes seeking to find low values close to promising points 2 2 2 already found, and always by generating new coordinates (N 6) r;2 r; min ; ð12Þ randomly in some tailored range. In plotting the 2 portrayal results, we superpose points with 2 2 where r; min is the global minimum of r , is distributed as a 2 2 lower values of r (better fits) over those with higher values distribution with 2 degrees of freedom. That is, the prob- (worse fits), which has the effect of presenting to the eye the best ability that 2 should be less than the observed value (if the available fit at any coordinates in the two-dimensional plots, theory is correct, the values of the parameters are the true ‘‘along the line of sight’’into the other four parametric dimensions. values, and the errors are normally distributed) is P(1, 2/2) , 2 We expect the patchy appearance of the five-frame portrait a regularized incomplete gamma function, where in Figures 6–10 because the cardinality of grid points is so high: Z 12 greater than 10 grid points for 1% resolution or better along six 1 t 1 parametric axes. Where the coverage is intense (e.g., near the P(; ) e t dt; ð13Þ () 0 accepted solutions), the values of the four undisplayed parameters of the plotted points are close to their optimized values for where () is the gamma function. Fig. 6.—Confidence regions in the T-" plane for the radial velocity solution for HD 72659. This is the first of five 2 portraits of the radial velocity data set for HD 72659. The colors show confidence regions, containing the true solution with 1–6 confidence in the sense of the normal distribution. (Compare with Fig. 2, which shows probability density.) The accepted solutions are well within the 1 confidence region, which extends at least over the range from 3:45 < log10T < 3:8and 0:1 <"<0:5 and which contains only type A solutions. The 2 confidence region, also all type A, extends over virtually the whole range of eccentricities, 0 <"<0:95, and a broad range of periods, 3:4 < log10T < 5:1, and perhaps beyond. Type B solutions, with 2000 days < T < 2500 days and 0 <"<0:5, are unlikely, being located entirely in confidence regions labeled 3 or higher. The patchy appearance of the 2 portraits is due to incomplete coverage of the grid points, which have a cardinality greater than 1012.

2 Fig. 7.—The portrait of conﬁdence regions in the T-M0 plane. (Compare with Fig. 3, which shows probability density.) Type B solutions occur with M0 0or2. 1086 2 Fig. 8.—The portrait of conﬁdence regions in the T-!s plane. (Compare with Fig. 3, which shows probability density.) Type B solutions occur with !s 1:7rad.

Fig. 9.—The 2 portrait of conﬁdence regions in the T-K plane. Toward lower T, type B solutions call for a near-vertical dive at the ends of the data record, leading to higher K, to compensate in eq. (5) for both lower T and higher ". 1087 1088 BROWN Vol. 610

Fig. 10.—The 2 portrait of conﬁdence regions in the T-V plane. The value V ¼ 0 for the accepted period is compatible with the data only at the 5 or 6 level.

In principle, the integral of the probability density, as de- We turn to the companion of HD 72659. termined by Monte Carlo projection, over a confidence region in parameter space, as determined by 2 portrayal, equals the 3. THE RANGE OF COMPANIONS level of confidence of that region. To establish an analytical framework to discuss the com- In Figures 6–10 we label levels of confidence by the prob- panion’s orbit, we set the star at the origin of a Cartesian (xˆ, yˆ,ˆz) ability of integer standard deviations of the normal distribu- coordinate system and define (or repeat from above) the tion, given in the color key. For the current case of N ¼ 16 and quantities: 2 r; min ¼ 3:26,thecutsfor1,2,3,4,5,and6 confidence are at 2 3:49, 3.88, 4.44, 5.19, 6.13, and 7.27, respectively. r;2 ¼ ms mass of star; Figures 6–9 show that the accepted solutions are close together, well within the 1 zone of confidence, as expected, mp mass of companion; which justifies the earlier statement that their periods and d distance to system; eccentricities are not significantly different. However, as a semimajor axis of companion0sorbit; shown in Figure 10, a value of zero for the offset velocity, with 0 the accepted period, is compatible with the data only at the 5 " eccentricity of companion s orbitðÞ same as for star ; or 6 level. r distance from companion to star; The 2 portraits show a new type solution, which we call type s apparent separation between companion and star; B, with 2000 days < T < 2500 days, 0 <"<0:5, M0 0or 2,and!s 1:7 rad. For type B solutions, stellar periastron is M mean anomaly of companionðÞ same as for star ; located beyond the plane of the sky through the center of mass, M0 mean anomaly at JD 2; 450; 838:893ðÞ same as for star ; and the initial phase is near periastron passage. These short- period, low- to moderate-eccentricity solutions call for a near- E eccentric anomaly of companionðÞ same as for star ; vertical dive at the ends of the data record, leading to higher K, true anomaly of companionðÞ same as for star ; to compensate in equation (5) for both lower Tand higher ".The T period of companion0s orbitðÞ same as for star ; type B solutions are entirely 3 or higher, meaning that this solution type has no higher probability than 0.003. !s argument periastron of star; 0 The 1 confidence region, all type A, extends in the T-" plane !p argument of periastron of companion sorbitðÞ¼!s þ ; at least over the range from 3:45 < log10T < 3:8and0:1 <"< i inclination of companion0sorbit; 0:5. The 2 confidence region, also all type A, extends over 0 virtually the whole range of eccentricities, 0 <"<0:95, and a position angle of ascending node of companion sorbit: broad range of periods, 3:4 < log10T < 5:1, and perhaps beyond. The 2 portraits fail to confirm that the type A solutions are The observer is located at (0, 0, d) with line of sight in the divided into two components. +ˆz direction; xˆ is east, and yˆ is north. The ascending node is No. 2, 2004 NEW INFORMATION FROM RADIAL VELOCITY DATA SETS 1089 the point at which the companion passes through the plane object. This obscuration has the practical purpose of limiting of the sky with z ¼ 0, headed away from the observer, moving the amount of unwanted light from the primary object that is in the +ˆz direction. detected with the planet image and improving the signal-to- We introduce a base orbit of the companion, which is an ellipse noise ratio for detection. In an optical coronagraph, which is defined by a and " that lies in the plane of the sky with z ¼ 0. The our main interest here, the obscuration is a field mask, typi- periastron of the base orbit is on the +y or north axis, and the cally with angular radius a few times k/D,wherek is the companion revolves in the direction from north to east. We can wavelength of light and D is the diameter of the telescope derive all other a-" orbits from the base orbit by rotations. aperture. For this reason, the separation, The x and y coordinates of the companion’s position in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi base orbit are described by the parametric equations: s ¼ x2 þ y2; ð18Þ

xb ¼ r()sin; ð14aÞ is one of two prime metrics of a companion’s direct observ- y r()cos; 14b b ¼ ð Þ ability. Brown (2004) analyzes the selection effects caused by where a central field mask on direct-imaging searches for companions (‘‘obscurational completeness’’). aðÞ1 "2 At this point, we assume that the companion is a planet and r() ¼ : ð15Þ 1 þ " cos invest it with the photometric properties of a typical gas giant planet. All objects of one to tens of times Jupiter’s mass have In addition to a and ", we adopt M0, T, and the three (ordered) approximately Jupiter’s size (Hubbard 1984). At optical wave- rotations !p, i,and to form a complete set of elements for lengths, we must detect the companion in reflected starlight specifying the general orbit and determining the position of the because self-luminosity does not add significantly until the companion at any epoch. The angle !p is a left-hand rotation mass exceeds a hundred times Jupiter’s mass. In any event, around the z axis in the range 0 !p 2. The angle i aleft- the companion’s minimum flux is the starlight it scatters in the hand rotation around the y axis in the range 0 i .Theangle direction of Earth. We introduce the following quantities for is a left-hand rotation around the z axis in the range discussing the photometry: 0 !p 2. We apply these rotations, in order, to the base vector (xb , yb , 0) to get the position vector position of the R radius of companion; companion, (x, y, z). Fp spectral Cux of companion integrated over Four characteristics of the companion come directly from the wavelength range of observation; radial velocity solution: the orbital elements T, ",andM0 are C identical, and !p ¼ !s þ . We can compute two other char- Fs spectral ux of star integrated over acteristics of the companion from the radial velocity solution: wavelength range of observation; the minimum mass from equations (6) and (7) and the semimajor axis from equation (8). Figures 1j and 1k show histo- mag magnitude of companion minus magnitude of star; grams of mp; min and a computed in that manner, based on the p geometric albedo of companion; first stage of Monte Carlo projection of the HD 72659 data set. phase angle (angle at companion between In the second stage of Monte Carlo projection, we randomize the missing orbital elements of the companion, which star and observer); are rotations on the celestial sphere: ðÞ phase function of companion:

¼ 2R ð16Þ Following Brown & Burrows (1990) and Brown et al. (2003), the second prime metric for direct observability is the and flux ratio, Fp /Fs, expressed in magnitudes: "# i ¼ cos1(1 2R); ð17Þ F R 2 mag ¼2:5log p ¼2:5log p() : ð19Þ where R is an available uniform random deviate in the range Fs r 0–1. (The value of has no effect on our detectability anal- ysis because the position angle of the companion is not in- We introduce the phase function of a Lambert sphere (surface volved.) Then we can compute at any epoch. We compute scatters equally in all outward directions) and Jupiter’s radius 8 r() from equation (15) and apply the rotations !p, i,and to and approximate geometric albedo near 5000 : find (x, y, z). This logic establishes that we can start with a sin þ ( )cos randomized orbital solution for the stellar radial velocity based L() ¼ ; ð20Þ on synthetic data sets, use the solution to compute the companion’s orbit with randomized orientation, and then, for any R ¼ 0:00048 AU; ð21Þ epoch, compute the companion’s position in space. Conceptu- ally, we are projecting the radial velocity data set into the four- p ¼ 0:5; ð22Þ space of the companion’s position in time. In the next section we translate this information into the separation and flux of the which permits us to write equation (19) as companion (that is, into its direct observability) at any epoch. sin þ ( )cos mag ¼ 17:4 2:5log þ 5logr; 3.1. Metrics of Direct Observvability Observing systems designed to image faint companions directly obscure a region of the field centered on the primary ð23Þ Fig. 11.—Direct observability of the companion of HD 72659 at nine epochs. Each dot is one of 10,000 possible companions consistent with the real data set. The colored bands are the particular solutions in Table 3 produced with 5000 random inclination angles. The green curve is the maximum flux ratio (minimum delta magnitude) vs. separation for a Lambertian sphere of Jupiter’s size and albedo. NEW INFORMATION FROM RADIAL VELOCITY DATA SETS 1091

Fig. 12.—Direct observability of the companion of HD 72659 on 2010 January 10. Possible companions to the right of the vertical blue line have apparent separations greater than 0B05. Those below the horizontal blue line have a flux ratio to the star greater than 109. The black dots are the companions that satisfy both conditions. They would be detected with an instrument with a central obscuration of radius 0B05 and with a limiting sensitivity of 22.5 delta magnitudes. The fraction of 10,000 trial solutions that satisfy those conditions on this day is 0.38, which is the direct observability or detection probability of thecompanionof HD 72659 on the quadricentennial of Galileo’s discovery of astrophysical companionship. where r isnowmeasuredinAU.Because For the Lambertian phase function, equation (25) has one root in the range 0 <<: z ¼ cos1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð24Þ s2 þ z2 max; L ¼ 1:1047 rad (or 63N30); ð26Þ we can now compute both metrics of direct observability, s and mag, for a Jupiter-class companion at any epoch from which is the phase angle of the brightest possible planet along its position in space. Conceptually, this reprojects the data set any line of sight (i.e., any given apparent separation, s). This is 2 into direct observability. a fundamental number, akin to the value of 3 for the geometric In the next section we examine the direct observability of albedo of the Lambertian sphere. This angle depends only on the companion of HD 72659. the inverse square law of illumination and isotropic scattering. Expressed as a minimum delta magnitude, this maximum 3.2. HD 72659: The Rangge of Direct Observvabilities brightness as a function of s in AU is In the preceding sections we concatenate three stages of

Monte Carlo projections to translate the original HD 72659 mmin(s) ¼ 18:24 þ 5logs; ð27Þ data set into the direct observability of the companion as a Jupiter-like planet. In this section we examine the results. Figure 11 shows the 10,000 planets plotted on the s-mag which is the green curve. plane for nine epochs spread over 15,000 days on either side of Each of the 10 planets associated with the solutions in Table 3 the observed periastron. The density of dots is proportional to is a black dot in Figure 11. To expose the role played by the the probability that the companion to HD 72659 would be randomly selected inclination angle, we generate 5000 addi- observed at epoch at a given apparent separation and delta tional planets from each solution in Table 3, using equation (17) magnitude assuming that it is similar to Jupiter photometrically. repeatedly to randomize i, and we color-code the new dots to The green curve in Figure 11 is the maximum brightness form the colored curves (except green) in Figure 11. versus apparent separation. For any phase function, local ex- The simplest practical application of direct observability is to explore hard limits for the metrics: minimum separation a trema of Fp /Fs occur for phase angles that are roots of the 0 equation: and limiting delta magnitude mag0, such that detection is a certainty if @() 2cos() þ sin ¼ 0: ð25Þ @ s > a0 ð28aÞ 1092 BROWN and orbits for terrestrial planets around stars with radial velocity companions. mag < mag0; ð28bÞ We have demonstrated the ability to predict the direct observability of companions inferred from radial velocity data and impossible otherwise. The fraction of trials in the Monte sets. (They can only be brighter, as a result of self-luminosity, Carlo sample that satisfy equations (28a) and (28b) at any if the inclination angle is low and the companion’s true mass is epoch is the probability that the inferred companion would be high.) Even though these predictions of direct observability observable with an instrument operating with the performance are probabilistic, they are useful for planning real observations parameters a0 and mag0. with real instruments. We are now in a position to answer the question posed in x 1, In this paper we have pursued Monte Carlo projection– asking with what probability the companion of HD 72659 based random samples of synthetic data sets, which we fit to would be observable on a special date with an instrument find the sample of compatible solutions. This approach will B characterized by a0 ¼ (0 05)(51 pc) ¼ 2:55 AU and mag0 ¼ not work when the time extent of the data record is short 2:5log109 ¼ 22:5, assuming that it is similar to Jupiter compared with the period, for it will not be possible to locate 2 photometrically. Figure 12 is the direct-observability projection with confidence a global minimum of r or a best fit on which for the date in question. The fraction of 10,000 planets satis- to base the syntheses. In this case, it may prove possible to use fying the detection criteria is 0.38. an alternative technique to explore the future observability of an inferred companion: generating the random sample of 4. EPILOGUE compatible solutions directly from within a confidence region We find that there are different types and broad ranges of of a 2 portrait. Keplerian solutions compatible with a radial velocity data set, to which we can assign probabilities. Monte Carlo projection and 2 portrayal are available techniques to better understand these effects, reduce the risk of a low-probability solution type We are most grateful to G. Marcy and his team for pro- with a high-probability fit, and determine the probability dis- viding the HD 72659 radial velocity data set and for their tributions of orbital parameters of radial velocity companions helpful clarifications. We thank Christopher Burrows and as a class, perhaps developing a valuable new taxonomy of Wesley Traub for their critical reading of the manuscript. We spectroscopic binary systems based on the ‘‘fingerprints’’ of a thank Christian Lallo for his expertise and craftsmanship in data set in the hyperspace of Keplerian parameters. aiding the computations and rendering the results in graphical Monte Carlo projection and 2 portrayal could be used to form. JPL contract 1254081 with the Space Telescope Science determine the probability distribution of dynamically allowed Institute provided support for this research.

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