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Science Metric for RV Exoplanet Characterization Robert A. Brown DRAFT May 16, 2013 Introduction Known RV exoplanets are prime targets for a probe-class, direct-detection mission, which could spectroscopically characterize their atmospheres or surfaces in reflected starlight. To help differentiate mission concepts, and to help develop reasonable scientific expectations, we propose a science metric for RV exoplanets. This metric is the estimated number of such planets detected and characterized during the mission, and its standard deviation. In this report, we derive the metric and estimate its value as (XXX + YYY) ± UUU. We use simplifying assumptions, which could be relaxed in the future to improve the systematic accuracy of the metric. We drew our sample of known RV exoplanets from www.exoplanets.org on May 10, 2013. It includes all 423 objects satisfying the search term “PLANETDISCMETH == 'RV'.” Because it is not constrained by the RV technique, the orbital inclination angle (i) is unknown for the 410 of these objects with no additional information. For those exoplanets, we can only estimate the probability of detection, based on the natural probability density of i. For the other 13 objects, with known i, we can compute detectability with certainty: yes or no. Therefore, the metric is the number (XXX) of detectable planets with known i plus the expectation value (YYY ± UUU) of the sum of 410 Bernoulli random variables with a unique probability of detection. We define the detection probability of an RV exoplanet with unknown i to be the maximum probability of satisfying three criteria simultaneously, sometime during the mission duration. The three criteria are #1 #2 #3 permitted pointing: angle between the host star and the sun ( g ) greater than the solar avoidance angle, g > g 1 and less than the angle at which the star shade appears illuminated (star-shade glint), g < g 2 object resolved: apparent separation s > IWA, the inner working angle adequate contrast: delta magnitude less than the systematic limit, Dmag < Dmag0 In this report, we adopt these mission parameters relevant to the metric: (a) (b) (c) (d) mission duration = three years: January 1, 2020, to December 31, 2025 IWA = 0.165 arcsec Dmag0 = 26 g 1 = 45° (coronagraph), g 1 = 45° and g 2 = 80° (star shade) www.exoplanets.org provides values of these parameters for each RV exoplanet,: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) semimajor axis (a) in AU orbital eccentricity ( e ) orbital period (T) in days time of periapsis (T0) in JD argument of periapsis of the star ( w s ) in degrees mass of the star (ms) in solar masses minimum planetary mass (mp sini) in Jupiter masses stellar distance (d) in pc stellar coordinates, right ascension ( a ) and declination ( d ) in decimal hours and degrees, respectively From these parameters, with the caveat that we must know or assume a value of i, we can compute the three-dimensional position of the exoplanet, in space, relative to the star, and from this position, we can compute s, the radial distance (r) from star to the planet, and the phase angle ( b ), which is the planetocentric angle between the star and earth. We can compute the position for any RV exoplanet at any time (t) in JD, following the procedure in §3 of Brown (2004). To compute Dmag , we must make additional assumptions about the photometric parameters: (x) (xi) (xii) geometric albedo of the planet (p) planetary phase function ( F(b ) ) planetary radius (Rp) In this report, we choose p = 0.5, F = F L , the Lambert phase function, and Rp = RJupiter . We compute Dmag from the parameter vector {r, b , p, F , Rp } using Eq. (19) in Brown (2004). To compute q , we compute the unit vectors from earth to the star and to the sun, and take the dot product, which is cosq . For XXX of the 13 RV exoplanets with known i, we compute s, Dmag , and q for each mission day, and we find that the three detection criteria are satisfied at least once during the mission. In the case of the 410 RV exoplanets with unknown i, we proceeded probabilistically, as follows. For each exoplanet, for each day of the mission, we compute we compute s, Dmag for a large number of random values of i drawn from a uniform density of orbital poles on the celestial sphere. This density is achieved by drawing from the random deviate cos-1 (1- 2R), where R is a uniform random deviate on the interval 0–1. The detection probability for the jth exoplanet on the kth day (Pj,k) is defined as the fraction of the random values of i for which the three detection criteria are satisfied. We define the mission detection probability for the jth exoplanet—Pj—to be the maximum value of Pj,k over k, in other words, over the whole mission. As explained in Brown & Soummer (2010) §4.1, the density of the number successful detections for the whole mission is the convolution of the 410 Bernoulli densities with probabilities Pj. This density is shown in Figure 1. The expectation value and standard deviation of the density is YYY ± UUU. Criterion #1: Permitted Pointing Figure 1 shows the pointing restrictions on a day chosen at random, June 20, 2009. Figure 1. The celestial sphere on June 20, 2009, showing the positions of the ecliptic equator (black ring), vernal equinox, sun, and host stars of the 423 RV planets in our sample (blue dots). As an example, the host star HD 2952 is shown as a larger dot. Left: star shade ( g 1 = 45° , g 2 = 80° ); right: coronagraph ( g 1 = 45° , g 2 = 180° ). If a star lies in the green region, it can be observed, but not if it lies in the red. In this figure, as time passes, the sun, coordinate grid, and red/green zones remain fixed, while the vernal equinox and stars revolve at constant ecliptic latitude—counterclockwise in this view—at the rate of one revolution per year. In computing our metric, we implement the pointing restrictions of Criterion #1 by creating 423 “validity” lists of days, one list for each host star in our sample. Each validity list contains all—and only—the days during the mission when it is permitted to point at that star. During the mission, there are 1097 observing times separated by one day. If there were no pointing restrictions ( g 1 = 0° , g 2 = 180° ), there would be a total of 1097 ´ 423 = 464,031 valid observing times. With the actual pointing restrictions, the sum of the cardinalities of the validity lists for the star-shade mission is 122,027, and for the coronagraphic mission, the number of valid observing times is 393,689. We compute Criteria #2 and #3 only at times on the validity lists. References R. A. Brown 2004, “New information from radial-velocity data sets,” ApJ, 610, 1079. R. A. Brown & R. Soummer 2010, “New completeness methods for estimating exoplanet discoveries by direct detection,” ApJ, 715, 122 Workpoints Assumptions could be relaxed—or new paramters tried—in future studies, which would increase the accuracy of the metric. Many of these assumptions are optimistic or neutral, which means net optimistic. No attempt has been made to prioritize these workpoints 1. Exposure times are not taken into account—neither for the limiting search nor for the charactering spectroscopy. An exposure time calculator and a set of observational overheads could be introduced. 2. Compute solar avoidance for the actual position of the spacecraft, rather than the position of the earth. 3. Even though exoplanets with mass above Jupiter all have about 1 Jupiter radius, we could introduce a variable planet radius for smaller masses. It could be computed from the planet mass by some acceptable mass-radius relation. The planet mass m is well determined from m sin i for any known or assumed value of i. 4. Shaklan on solar avoidance and star-shade glint: “For solar avoidance, 45 deg is probably the minimum for a starshade. We are investigating a solar diffraction term that may limit us to larger values and we are considering 50 deg for now. For a coronagraph 45 deg is reasonable.” Lisman: “The maximum allowable off-sun angle for observations is 80 or 85 degrees. You cannot let sun strike the telescope facing side of the occulter.” Also from Lisman: “I do not agree with the 45 degree number [Shaklan’s 45° for the star shade]. We have been using 30 degrees as the lower limit and Stuart has just recently suggested a change but we have not settled on the exact number.” 5. Shaklan on limiting delta magnitude Dmag0 : We think that a systematic detection limit of delta_mag = 26 is reasonable for a starshade (tolerances aren’t that tough to achieve). I suggest the same value for a coronagraph for working angles about >3 l/D, and dmag=25 for < 3 l/D. 6. Shaklan on IWA: Since we’re talking about probes, a telescope diameter of about 1.5 m is the maximum. A coronagraph at 2 l/D in the visible would have IWA = 165 mas. A 32 m starshade at 36,000 km would have IWA = 90 mas. The starshade and a 1.5 m telescope could be co-launched. 7. Shaklan on mission duration: Mission duration is a cost driver. 3 years is reasonable for the baseline mission. The starshade could carry 5 years worth of fuel. 8. The longest wavelength is 1 micron, to ensure the ability to detect oxygen and methane. 9. The star is observed on the day that satisfies all three criteria with maximum probability.