A Science Metric for Direct Characterization of Known Radial-Velocity Exoplanets

Home , HD 131664, HD 192310, HD 197037, HD 204313, HD 217107, HD 220773

Robert A. Brown Space Telescope Science Institute [email protected]

June 10, 2013

Abstract

Known RV exoplanets are viewed as prime targets for a future high-contrast imaging mission, which may be able to detect and characterize these enigmatic objects in reflected starlight. To help define and differentiate the candidates for such a mission on the basis of scientific performance, and to help set realistic expectations, we develop a science metric,

NRV, defined as the estimated number of planets that would be detected and characterized by such a mission. In this report, we estimate upper limits to NRV for missions with apertures in the range 1–2.4 m. We treat both star-shade and coronagraphic missions. DRMs,

1. Introduction

We define the science metric NRV to be the number of known RV exoplanets that satisfy four criteria:

#1 permitted pointing: during observations, the angle between the host star and the

sun (γ ) must be greater than the solar avoidance angle, γ > γ 1 and, for a star- shade mission, it must also be less than the angle at which the star shade appears

illuminated, γ < γ 2 #2 systematic limit: the flux ratio of the planet to the host star, expressed in delta magnitudes ( Δmag ), must be better than the systematic limit in delta magnitudes

( Δmag0 ), i.e. Δmag < Δmag0 #3 wavelength: criteria #1–2 must be satisfied at all wavelengths deemed critical for planet detection and characterization, and particularly for the coronagraph, at the

longest critical wavelength (λlc) #4 time: criteria #1–3 must be satisfied at some time during the mission for at least as long as the exposure time required for detection or characterization of a planet as faint as the systematic limit.

The input catalog of RV exoplanets—plus working assumptions about the photometric properties of exoplanets—provide the natural parameters for evaluating criteria #1–4. The mission design and science operations provide the technical parameters. In this report, we develop the methods for computing NRV from these parameters. The algorithms and methods were developed earlier by Brown (2004a, 2004b, 2005) and Brown & Soummer (2010) for estimating direct-imaging search completeness and for studying design reference missions (DRMs).

2. Natural Parameters: The Input Catalog

On May 10, 2013, we drew 423 known RV exoplanets from www.exoplanets.org, comprising all planets satisfying the search term “PLANETDISCMETH == 'RV'.” We reduced the list to 419 planets by dropping three host stars with unknown distances (HD 13189, BD+48 738, and HD 240237) and one planet with missing orbital information (Kepler-68). The final catalog includes 406 planets with unknown orbital inclination angle (i) and 13 planets with known i. The input catalog includes 347 unique host stars.

The input catalog includes 13 planets with known i, only one of which—HD 192310 c— achieves sufficient separation to be observed by the optical systems considered in this report. For current purposes, we have simply carried HD 192310 c along as if we did not know its inclination.

The input catalog contains the following parameters for each RV exoplanet:

(i) stellar right ascension (α in hours) and declination (δ in degrees) (ii) stellar distance (d) in pc

(iii) mass of the star (ms) in solar masses

(iv) minimum planetary mass (mp sini) in Jupiter masses (v) semimajor axis (a) in AU (vi) orbital eccentricity (ε)

(vii) argument of periapsis of the star (ω s ) in degrees. (ωp = ωs + 180°) (viii) orbital period (T) in days

(ix) epoch of periapsis (T0) in JD

In addition, we introduce the following natural photometric parameters for the planet:

(x) geometric albedo of the planet (p) (xi) planetary phase function ( Φ(β))

(xii) planetary radius (Rp)

In this report, we choose to investigate NRV using a Jupiter-like exoplanet as the test body: p = 0.5, Φ = ΦL , the Lambert phase function, and Rp = RJupiter . Such an object is a maximal planet in terms of flux of reflected light relative to the host star, which we express in delta stellar magnitudes (Δmag).

In their Figure 9, Cahoy, Marley, & Fortney (2010) suggest that p of a Neptune-like exoplanet orbiting at 5 AU would be 40% smaller in the visible to near-infrared wavelength range than a Jupiter-like exoplanet. Also taking relative size into account, a Neptune analog would be 2.9 magnitudes—a factor of 14—fainter than a Jupiter analog at the same phase. Because the maximum brightness of a Jupiter observable by the

2 current program is typically Δmag = 20–21, a Neptune would exhibit Δmag = 24–25 at brightest. In §2, part of the reason for setting the instrumental requirement for the systematic limit of detectability at Δmag0 = 25 is a perceived scientific imperative to reach—find and study—exoplanets at least as small as Neptune. An associated requirement is the ability to accommodate the long exposure times necessary to reach limiting objects—objects at the systematic limit of detectability—with a robust signal-to- noise ratio (S/N).

2. Technical Parameters of the Mission Design

We adopt the following values for five technical parameters that are relatively noncontroversial:

(a) γ 1 = 45° (coronagraph), γ 1 = 45° and γ 2 = 80° (star shade). These values of γ are being used in current NASA-sponsored studies.

(b) mission timing: January 1, 2020, to December 31, 2022. A duration of three years seems about right for a 1–2.4 m class mission. The chosen absolute timing seems reasonable for implementing a mission of this scale.

(c) Δmag0 = 25. In recent years, NASA-sponsored studies, laboratory tests, and experience with space instruments have built confidence in the community that

optical performance at the level of Δmag0 = 25 can now be achieved for usefully small angular separations (s) at visible and near-infrared wavelengths. In practical

terms, the achievable Δmag0 is ultimately determined by the brightest unstable speckles of starlight at the field position of the planet (Brown 2005).

(d) λlc = 785 nm, which is the red end of the reddest filter, centered at 766 nm.

(e) spectral resolving power R = 5 at 550 nm for search exposures, and R =20 for

spectroscopy near 730 nm, which approximately matches the width of the CH4 spectral feature (see Figure 10 in Cahoy, Marley, & Fortney 2010).

The parameters of limiting search observations (LSOs for detection) and limiting characterization observations (LCOs for characterization) are defined in Table 1.

Type Δmag0 λ (nm) R S/Ngoal q min # LSO 25 550 5 5 2 1 LCO 25 693, 730, 766 20 10 2 3

Table 1. Parameters for LSOs and LCOs. q is the required number of exposures in one limiting observation. The observations are always taken in pairs of consecutive exposures at slightly different telescope rolls, and the data are subtracted, shifted, and added to estimate the signal. The science program sets min #, which is the minimum number of limiting observations: one LSO for a search, and three LCOs for spectral characterization

3 of a massive extrasolar planet. Of the latter, one LCO is centered on the absorption feature of CH4 at 730 nm and two LCOs sample the adjacent continuum, centered at 693 and 766 nm. The red edge of the reddest filter determines λlc = 785 nm. The single LSO and the three LCOs are assumed to constitute a minimum spectral characterization of an RV exoplanet.

NRV depends on the minimum angle at which the suppressive performance Δmag0 is achieved. This angle is called the inner working angle (IWA). For the coronagraph, typical apparent separations of RV exoplanets from their host stars are smaller than a few

Airy rings at λlc, and for this reason, IWA for the coronagraph is controversial.

In this report, we study various values of the coronagraph’s IWA in the range

(f) IWA = 0.134–0.645 arcsec .

IWA is commonly expressed

IWA = κ λ / D , (1)

where κ is a number of Airy rings at wavelength λ, and D is the diameter of the telescope aperture. Equation (1) is nothing more than a degenerate translation between two operational parameters, IWA and λ, and two descriptive parameters

( f1′ ) κ

( f2′ ) D.

In Table 2, we use Equation (1) to explore combinations of values of f1′–2 that produce the studied values of IWA for the coronagraph.

By recognizing that the range of values IWA = 0.134–0.645 arcsec is compatible with at least some noncontroversial combinations of κ , λ, and D, we implicitly satisfy criterion #3. Nevertheless, it is important to keep in mind that the technical difficulty of achieving a value of IWA for the coronagraph has a negative relationship with the value of κ, and the practicality of achieving κ < 3 is currently controversial.

D (m) κ785 IWA(arcsec) comment 4 0.269 conservative 2.4 3 0.202 moderate 2 0.134 controversial 4 0.430 conservative 1.5 3 0.323 moderate 2 0.215 controversial 4 0.645 conservative 1.0 3 0.484 moderate 2 0.323 controversial

Table 2. The cases of D, κ785, and IWA for the coronagraph. The comment is about the degree of difficulty in realizing full starlight suppression at κ785 Airy rings.

For the star-shade mission, we select IWA = 0.075 arcsec, and D = 1, 1.5, and 2.4 m.

3. Criterion #1: Permitted Pointing

We compute γ from α and δ using the fact that the scalar product of the unit vectors from the telescope to the star and the telescope to the sun is equal to cosγ . We calculate the unit vectors in the rectangular ecliptic coordinates for any given time.

Figure 1 shows the zones of permitted pointing on 29 April 2020.

For this study, we implement criterion #1 by creating 346 “validity” lists—one for each host star—of the days during the mission when the star is observable.

For each star, the longest available exposure time—tmax, in days—is equal to the maximum number of consecutive valid days.

Figure 1. Celestial spheres on 29 April 2020, showing the positions of the ecliptic equator (black line), the vernal equinox, the sun, and the host stars of the 419 RV planets in our input catalog (blue dots). Left: for a star-shade mission (γ 1 = 45° , γ 2 = 80° ); right: for a coronagraph mission (γ 1 = 45° , γ 2 = 180° ). To provide a concrete example, the host star HD 220773 is shown as a yellow dot. If a star lies in the green region, it can be observed, but not if it lies in the red. As time passes, the sun, the coordinate grid, and the red/green zones remain fixed, while the vernal equinox and host stars revolve at constant ecliptic latitude, clockwise as seen from the north ecliptic pole, at the rate of one revolution per year. The yellow line shows the positions of HD 220773 when it is observable for one year following 29 April 2020. For a star-shade mission, HD 220773 is observable for two 36-day intervals during the year. For a coronagraph mission, it is observable for one 275- day interval.

4. Criterion #2: Systematic Limit

The following four steps constitute the master computation of the systematic limit of detectability. The steps evaluate criterion #2 by asking the logical question

s > IWA  Δmag < Δmag0 ? (2)

The computations answer this question, yes or no, for all 419 planets on every day on the host star’s validity list, for two values of i: “face-on” (0.9°) and “edge-on” (89.1°).

In the first step, we compute the three-dimensional position of the planet in space relative to the center of mass, from parameters ii–ix, for both the face-on and edge-on inclinations. Second, knowing the planet’s position, we compute the apparent separation (s), the radial distance (r) from star to the planet, and the phase angle ( β ), which is the planetocentric angle between the star and earth. Third, we compute Δmag from parameters x–xii using Eq. (19) in Brown (2004b). Fourth, for each planet, we select the values of the largest apparent separation (smax) with an associated delta magnitude Δmagmax < Δmag0.

We assume the planet is detectable according to criterion #2 if smax > IWA and

Δmagmax < Δmag0 for any valid time.

We distinguish three possible outcomes for any planet:

Outcome #1: smax > IWA and Δmagmax < Δmag0 at no valid time, neither for the face-on nor the edge-on inclinations. All such planets—the vast majority of the 419

planets in our sample—contribute nothing to NRV.

Outcome #2: smax > IWA and Δmagmax < Δmag0 at some valid times, for either the face- on or the edge-on inclinations, but not both. These planets are obscured or too faint at all valid times for some values of i, but are detectable for some other

values of i at some valid times. This is the outcome for a number of planets N2. It is possible that N2 = 0. Each planet with outcome #2 contributes either zero or a fractional value, between zero and one, to the estimate of NRV, depending on whether tmax is less than or greater than the exposure time for a LSO or LCO.

Outcome #3: smax > IWA and Δmagmax < Δmag0 at valid times, for both the face-on and

edge-on inclinations. This is the outcome for N3 planets. Such planets can always be detected according to criterion #2 on some valid days. Each planet with

outcome #3 contributes either zero or one to the estimate of NRV, depending on whether tmax is less than or greater than the exposure time for a LSO or LCO.

Appendix A demonstrates that full orbital calculations are required to accurately evaluate criterion #2. It also provides salient examples of RV planets with outcomes #1–#3.

5. Criterion #4: Time

For the coronagraph, we find that 26 RV exoplanets satisfy criterion #2 (i.e., outcome #2 or #3) for at least one of the values of IWA in Table 2. These planets have 25 host stars, shown in Table 3a.

For the star shade, we find 79 RV exoplanets satisfy criterion #2 (i.e., outcome #2 or #3) for IWA = 0.075 arcsec. These planets have 72 host stars, shown in Table 3b.

Appendix B explains how we compute exposure times t0 for LSOs and LCOs. The values are the same for both the coronagraph and the star shade.

We make the optimistic assumption that the full tmax is available for a LSO or LCO. Therefore, if t0 < tmax , we assume criterion #4 is satisfied, and not if it is not.

log log log log log log log star tmax t0,LSO,1 m t0,LSO,1.5 m t0,LSO,2.4 m t0,LCO,1 m t0,LCO,1.5 m t0,LCO,2.4 m 1 beta Gem 2.44 -0.432 -0.807 -1.23 0.517 0.144 -0.274 2 gamma Cep 2.56 0.574 0.112 -0.362 1.49 1.04 0.572 3 epsilon Eri 2.47 0.872 0.372 -0.131 1.86 1.36 0.861 4 upsilon And 2.47 1.11 0.580 0.0483 2.32 1.76 1.21 5 47 UMa 2.47 1.74 1.14 0.529 2.90 2.28 1.65 6 mu Ara 2.47 1.81 1.20 0.579 2.91 2.29 1.66 7 HD 10647 2.56 2.10 1.47 0.811 3.32 2.68 2.00 8 HD 39091 2.56 2.19 1.56 0.890 3.38 2.73 2.04 9 HD 192310 2.44 2.25 1.61 0.939 3.23 2.60 1.93 10 HD 190360 2.56 2.25 1.61 0.939 3.32 2.68 1.99 11 HD 30562 2.47 2.28 1.64 0.964 3.44 2.79 2.10 12 55 Cnc 2.44 2.43 1.78 1.08 3.41 2.76 2.07 13 HD 217107 2.44 2.59 1.93 1.22 3.66 3.00 2.28 14 HD 134987 2.44 2.82 2.15 1.42 3.93 3.26 2.52 15 14 Her 2.56 2.93 2.25 1.52 3.91 3.24 2.50 16 HD 154345 2.56 3.04 2.37 1.62 4.13 3.45 2.70 17 HD 33636 2.45 3.23 2.55 1.79 4.43 3.74 2.98 18 HD 220773 2.44 3.28 2.59 1.83 4.41 3.73 2.96 19 HD 87883 2.45 3.68 2.99 2.21 4.60 3.91 3.14 20 HD 181433 2.54 4.32 3.62 2.82 5.21 4.51 3.71 21 GJ 832 2.48 4.54 3.84 3.04 4.49 3.81 3.03 22 GJ 676 A 2.47 5.28 4.58 3.77 5.47 4.78 3.98 23 GJ 649 2.56 5.39 4.68 3.87 5.32 4.63 3.83 24 GJ 849 2.44 5.95 5.24 4.43 5.89 5.19 4.38 25 GJ 179 2.45 7.18 6.48 5.66 6.85 6.15 5.34

Table 3a. Time results for a coronagraphic mission. tmax is the longest possible continuous exposure, in days. t0 is the required exposure time, computed as described in Appendix B. The red cases fail to satisfy the condition t0 < tmax of criterion #4. The values of t0 are independent of IWA, the planetary orbit, and the strategy for starlight suppression, coronagraph or star shade.

7 log log log log log log log star tmax t0,LSO,1 m t0,LSO,1.5 m t0,LSO,2.4 m t0,LCO,1 m t0,LCO,1.5 m t0,LCO,2.4 m 1 beta Gem 1.85 -0.432 -0.807 -1.23 0.517 0.144 -0.274 2 gamma Cep 2.13 0.574 0.112 -0.362 1.49 1.04 0.572 3 iota Dra 2.07 0.619 0.152 -0.326 1.44 0.991 0.530 4 omicron UMa 2.05 0.660 0.188 -0.294 1.67 1.20 0.718 5 epsilon Eri 1.93 0.872 0.372 -0.131 1.86 1.36 0.861 6 7 CMa 2.08 1.01 0.496 -0.0236 1.89 1.39 0.889 7 upsilon And 1.94 1.11 0.580 0.0483 2.32 1.76 1.21 8 kappa CrB 2.17 1.57 0.989 0.399 2.48 1.91 1.33 9 47 UMa 1.95 1.74 1.14 0.529 2.90 2.28 1.65 10 mu Ara 1.93 1.81 1.20 0.579 2.91 2.29 1.66 11 HD 147513 1.88 1.99 1.37 0.722 3.14 2.51 1.85 12 HD 10647 2.16 2.10 1.47 0.811 3.32 2.68 2.00 13 HD 39091 1.95 2.19 1.56 0.890 3.38 2.73 2.04 14 HD 192310 1.87 2.25 1.61 0.939 3.23 2.60 1.93 15 HD 190360 2.18 2.25 1.61 0.939 3.32 2.68 1.99 16 HD 30562 1.93 2.28 1.64 0.964 3.44 2.79 2.10 17 HD 169830 1.86 2.38 1.73 1.05 3.64 2.98 2.26 18 HD 4732 1.92 2.38 1.73 1.05 3.32 2.68 2.00 19 HD 38529 1.89 2.42 1.77 1.08 3.47 2.82 2.12 20 55 Cnc 1.86 2.43 1.78 1.08 3.41 2.76 2.07 21 tau Gru 2.03 2.48 1.83 1.13 3.65 2.99 2.27 22 HD 216437 2.17 2.49 1.83 1.13 3.62 2.96 2.25 23 HD 217107 1.85 2.59 1.93 1.22 3.66 3.00 2.28 24 16 Cyg B 2.09 2.65 1.99 1.27 3.78 3.12 2.39 25 HD 196885 1.94 2.76 2.09 1.37 4.02 3.35 2.60 26 HD 11964 1.89 2.78 2.11 1.39 3.80 3.13 2.40 27 HD 134987 1.85 2.82 2.15 1.42 3.93 3.26 2.52 28 HD 210277 1.85 2.87 2.20 1.47 3.92 3.25 2.51 29 14 Her 2.14 2.93 2.25 1.52 3.91 3.24 2.50 30 HD 154345 2.09 3.04 2.37 1.62 4.13 3.45 2.70 31 HD 50554 1.86 3.11 2.43 1.68 4.31 3.63 2.86 32 HD 197037 2.16 3.13 2.45 1.70 4.45 3.76 2.99 33 HD 168443 1.86 3.17 2.49 1.73 4.25 3.57 2.81 34 HD 33636 1.90 3.23 2.55 1.79 4.43 3.74 2.98 35 HD 164922 2.18 3.24 2.56 1.80 4.27 3.59 2.83 36 HD 89307 1.86 3.25 2.56 1.81 4.44 3.75 2.99 37 HD 220773 1.86 3.28 2.59 1.83 4.41 3.73 2.96 38 HD 222155 2.19 3.32 2.64 1.88 4.48 3.79 3.02 39 HD 70642 2.16 3.36 2.68 1.91 4.47 3.79 3.02 40 HD 50499 2.16 3.39 2.71 1.94 4.57 3.88 3.11 41 HD 117207 1.90 3.43 2.75 1.98 4.52 3.83 3.06 42 HD 10180 2.15 3.49 2.80 2.03 4.65 3.96 3.19 43 HD 106252 1.86 3.55 2.86 2.09 4.71 4.02 3.24 Table 3b. Time results 44 HD 12661 1.87 3.57 2.88 2.10 4.67 3.98 3.20 for a star-shade 45 HD 72659 1.89 3.59 2.90 2.13 4.77 4.08 3.30 46 HD 128311 1.90 3.61 2.92 2.14 4.52 3.83 3.06 mission. The unit of 47 HD 7449 1.87 3.62 2.93 2.16 4.84 4.14 3.36 48 HD 24040 1.86 3.62 2.93 2.16 4.77 4.08 3.29 time is log days. For 49 HD 87883 1.90 3.68 2.99 2.21 4.60 3.91 3.14 the same star, the 50 HD 111232 2.17 3.69 3.00 2.22 4.80 4.11 3.32 51 HD 47186 2.18 3.70 3.01 2.23 4.80 4.11 3.32 values of t0 are the 52 HD 74156 1.87 3.71 3.02 2.24 4.91 4.22 3.43 same as for a 53 HD 13931 1.93 3.71 3.02 2.24 4.87 4.17 3.39 54 HD 37124 1.86 3.77 3.07 2.29 4.90 4.20 3.42 coronagraphic mission, 55 HD 142022 2.15 3.78 3.09 2.31 4.82 4.13 3.34 but the values of t 56 HD 81040 1.86 3.80 3.10 2.32 4.92 4.22 3.44 max 57 HD 187123 2.17 3.88 3.19 2.40 5.02 4.33 3.54 are shorter for the star 58 HD 183263 1.95 3.91 3.21 2.43 5.03 4.34 3.55 59 HD 204313 1.85 4.01 3.32 2.53 5.12 4.42 3.63 shade. 60 HD 79498 1.86 4.03 3.34 2.55 5.14 4.44 3.65 61 HD 131664 2.17 4.12 3.43 2.63 5.25 4.56 3.76 62 HD 66428 1.89 4.22 3.52 2.72 5.31 4.61 3.82 63 HD 181433 2.12 4.32 3.62 2.82 5.21 4.51 3.71 64 HD 30177 1.88 4.34 3.65 2.85 5.39 4.70 3.90 65 HD 204941 1.85 4.38 3.68 2.88 5.35 4.66 3.86 66 GJ 832 1.97 4.54 3.84 3.04 4.49 3.81 3.03 67 HD 37605 1.88 4.55 3.85 3.05 5.56 4.86 4.06 68 GJ 676 A 1.93 5.28 4.58 3.77 5.47 4.78 3.98 69 GJ 649 2.18 5.39 4.68 3.87 5.32 4.63 3.83 70 GJ 849 1.86 5.95 5.24 4.43 5.89 5.19 4.38 71 GJ 179 1.89 7.18 6.48 5.66 6.85 6.15 5.34 72 GJ 317 2.05 7.21 6.51 5.69 7.15 6.45 5.63

In Table A.2, we identified the six known RV exoplanets with highest values of a (1 + ε) / d, which is an incomplete (“raw”) estimate of maximum apparent separation for the planets of interest here. For the case considered in the Appendix A, IWA = 0.34 arcsec, and five planets— epsilon Eri b, 55 Cnc d, mu Ara c, GJ 832 b, HD 217107 c—satisfy the inequality a (1 + ε) / d > IWA, which would produce the incomplete estimate NRV = 5. In Appendix A, we show that a complete treatment of the planetary orbit eliminates the contribution of HD 217107 c and finds that only a fractional contribution is possible from mu Ara c.

Tables 3a–b show how considerations of time impact the four planets still in play for this example: epsilon Eri b Considerations of time do not negate the ability to detect and characterize this planet by either coronagraph or star shade for any aperture. Potentially

contributes one to NRV.

55 Cnc d Coronagraph: detection possible for any aperture, but characterization with D = 1 and 1.5 m are ruled out. Star shade: detection with D = 1.5 and 2.4 m are possible, but characterization ruled out for any aperture.

Potentially contributes one to NRV.

mu Ara c Coronagraph: detection possible for any aperture, but characterization with D = 1 and 1.5 m are ruled out. Star shade: detection possible for any aperture, but characterization with D = 1 and 1.5 m are ruled out.

Potentially contributes a fraction to NRV, as explained in §6.

GJ 832 b Ruled out by time considerations for all missions and apertures under consideration. (For the parameters in Appendix B, light from GJ 832 b at 550 nm is detected in a 2.4 m aperture at the rate of 42 photoelectrons per

day—fewer than two per hour—for 1101 days.) Contributes zero to NRV.

Coronagraph Star Shade Raw LSO LCO LSO LCO 1.0 m 2 + B 1 1 0 5 1.5 m 2 + B 1 + B 1 + B 1 5 2.4 m 2 + B 2 + B 2 + B 1 + B 5

Table 4. Estimates of NRV for IWA = 0.34 arcsec, in the example of Appendix A. B is the estimated contribution of mu Ara c, a fraction from zero to one, explained in §6. The “raw” estimate uses the fallacious criterion a (1 + ε) / d > IWA and ignores matters of time.

6. Estimation of B

Assuming all feasible LSOs and LCOs with possibly positive results are made, the estimate of the metric is

NRV = N3 + B , (3)

where B is the expectation value for the sum of N2 Bernoulli random variables, which are the planets with outcome #2, each of which has a unique probability P.

th We estimate the probability Pk of the k random variable by a Monte Carlo experiment, as follows. First, we create a large sample of random values of i by drawing from the appropriate random deviate, which is arccos(1− 2Q) , where Q is a uniform random deviate on the interval zero to one. Second, we compute smax and Δmagmax for each value of i by the four-step master computation described in §4. Third, we count the number of random values of i that result in satisfying the systematic-limit criterion in Equation (2) on at least one valid day, and therefore are detectable according to criterion #2. Fourth, we estimate Pk as the ratio of this number to the total number of values of i in the experiment—in the current study, for some 50,000 trials for each of N2 cases.

As explained and illustrated by Brown & Soummer (2010), we compute the density of B by convolving the N2 Bernoulli densities for P, which are delta functions at 0 and 1 with weights (1– Pk) and Pk, respectively. The density of B is therefore a sum of N2 + 1 delta functions at integer values in the range 0–N2. The expectation value of B is the mean of the density of B, which is ΣPk . If needed, the standard deviation of B can be computed as 2 the square root of the sum, k = 0 to N2, of (k – ΣPk ) Pk .

In the particular example discussed in §5 and Appendix A, N2 = 1 for mu Ara c, in which case B = P1, which we did not compute.

7. Pre-DRM Results for NRV

Here, we comment on the results in Tables 5a–b, which show the estimates of NRV—and the breakdowns—assuming all feasible LSOs and LCOs with possibly positive results are actually performed. In §8, we will introduce the science-operational constraints imposed by issues of timing, total time, and target choice.

The ability of a coronagraph to achieve full starlight suppression at κ = 2 is currently controversial. Nevertheless, assuming κ = 2, Tables 5a indicate an approximately factor- two improvement in pre-DRM values of NRV as compared with κ = 3. For κ ≥ 3, we find NRV < 10 for all coronagraph programs, and for all cases except D = 2.4 m, NRV ≤ 3. A

10 mission with D = 1.0 m could be called “The epsilon Eri Express,” because epsilon Eri b is the only currently known RV exoplanet that it could hope to observe. The estimated number of RV exoplanets that can be both detected and characterized with D = 2.4 m and is currently NRV = 13.3 for the star shade and NRV = 6.8 for the coronagraph with κ = 3.

Table 5a. NRV results for coronagraphic missions. A “1” indicates that the planet is definitely observable (outcome #3 and t0 < tmax.). The sum of the 1’s in a column is N3. A fractional value (P) means the planet is observable with probability P estimated by Monte Carlo trials of random values of i. In a column, fractional values are combined as explained in §6 to produce B, and NRV = N3 + B. Stars are listed by descending productivity across all combinations of D and κ, measured by the sum of numbers in a row.

11 Table 5b. NRV results for star- shade missions.

12 Design Reference Missions

Each of six columns headed “log t0” in Tables 3a–b is a list of the time cost per star of an LSO or LCO for the coronagraph or star shade, for D = 1, 1.5, or 2.4 m. For the star shade, the six numerical column in Table 5b gives the benefit for that time cost, in terms of the number of RV exoplanets either detected or characterized. For the coronagraph, the same applies to groups of three columns in Table 5a—three because of the break out of κ, on which the time costs do not depend. Therefore, we can assign to each star and choice of instrument a numerical indicator of merit, which is the benefit of observing it divided by the cost.

A design reference mission (DRM) is essentially a “financial” analysis of the information in Tables 3 and 5. For a given optical system, such an analysis seeks to optimize the sequence of “purchases”—selected targets—while keeping track of the budget, which is the total mission time devoted to this program. The mission-level estimate of NRV is the sum of the RV planets that have been “purchased.”

To illustrate, we build DRMs to compare a coronagraph with κ = 3 and the star shade with IWA = 0.075 arcsec, for D = 1, 1.5, and 2.4 m.

We demand the full treatment, calling for one LSO and three LCOs, one of which is on the 730 nm CH4 feature and the other two are on the adjacent continuum. The time cost of the full treatment is t0, LSO + 3 t0, LCO. The benefit, in terms of incremental planets contributing to NRV, is one for an outcome #3 planets or a fraction, for an outcome #2 planet.

The results are shown in Figures 6a,b.

Rémi, I stopped here on Friday. I intend to work on this over the weekend.

Table 6a. DRM mission-level estimates of NRV for the star-shade mission.

Table 6b. DRM mission-level estimates of NRV for the coronagraph mission.

14 Acknowledgments

My gratitude to Rémi Soummer, Howard Bond, Wes Traub, Stuart Shaklan, Tom Greene, Nick Cowan, Charley Noecker, Marc Postman, and Doug Lisman for helpful remarks. This research has made use of the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org.

References

R. A. Brown 2004a, “Obscurational completeness,” ApJ, 607, 1003. R. A. Brown 2004b, “New information from radial-velocity data sets,” ApJ, 610, 1079. R. A. Brown 2005, “0,” ApJ, 624, 1010 R. A. Brown & R. Soummer 2010, “New completeness methods for estimating exoplanet discoveries by direct detection,” ApJ, 715, 122 C. J. Burrows, 2003, in The Design and Construction of Large Optical Telescopes, e. P. Y. Bely (New York: Springer), 433. C. J. Burrows, R. A. Brown, & E. M. Sabatke, 2006, “Aberrational Delta Magnitude,” manuscript. http://www.stsci.edu/~RAB/papers/Abrr_Del_Mag.pdf K. L. Cahoy, M. S. Marley, & J. J. Fortney, 2010, "Exoplanet albedo spectra and colors as a function of planet phase, separation, and metalicity, ApJ 724, 189. Ch. Leinert, et al., 1998, “The 1997 reference of diffuse night sky brightness,” Astron. Astrophys. Suppl. Ser. 127, 1.

15 Appendix A: On The Need for Complete Orbital Computations

In the past, computing orbits has sometimes been avoided by comparing IWA with the maximum possible apparent separation:

sx ≡ a(1+ ε) / d (A.1)

and by comparing Δmag0 with the delta magnitude at quadrature

⎛ 2 ⎞ ⎛ π ⎞ ⎛ Rp ⎞ Δmagx = −2.5log⎜ p Φ⎜ ⎟ ⎜ ⎟ ⎟ . (A.2) ⎝ ⎝ 2 ⎠ ⎝ a(1+ ε)⎠ ⎠

This approach is incomplete or “raw”, because overlooks the effect of ωp when ε ≠ 0 and i ≠ 0, which in the current research would result in a significant overestimate of the yield ΝRV .

In the following example, we assume IWA = 0.34 arcsec.

Table A.1 compares values of (sx, Δmagx) and (smax, Δmagmax)—the latter for both the face-on and edge-on inclinations—for the six planets in the input catalog with largest value of sx. Table A.2 provides their basic information. Figures A.1–A.6 show their face- on and edge-on orbits on valid days.

The incomplete computation produces the estimate NRV = 5, referring to epsilon Eri b, GJ 832 b, 55 Cnc d, HD 217107 c, and mu Ara c.

In Figures A.1–A.3, we recognize the outcome #3 planets as epsilon Eri b, GJ 832 b, and 55 Cnc d, because each is detectable on some valid days for both edge-on and face-on orbits. In Figure A.4, we see mu Ara c is an outcome #2 planet, because its edge-on orbit of mu Ara c does not separate to IWA = 0.34 on any valid day. In Figure A.5, we see that HD 217107 c is an outcome #1 planet, never reaching a separation of 0.34 arcsec on any valid day. This is also true for HD 190360 b, in Fig A.6, although in this case we know that this planet never separates to 0.34 arcsec, because sx = 0.33 arcsec.

The first of three problems with the incomplete computation is that the mission duration (three years) is short compared with the orbital periods of planets that can achieve large s. This means that the largest possible separation may not be realized during the mission. By inspection, this true for epsilon Eri b, HD 217107 c, and HD 190360 b—and it may be true of others. The incomplete computation wrongly assumes that sx is achieved during the mission.

The second problem—a concern for planets with elevated values of ε and with values of i and ωp away from 0° and 180°, the maximum possible apparent separation may actually be closer to a (1 – ε) / d than to a (1 + ε) / d.

The third problem is that Δmagmax will be significantly different from Δmagx—greater or smaller—for eccentric orbits oriented to produce significantly reduced separations at apoapsis.

mission duration: 1/1/20–12/31/22 edge-on orbits face-on orbits

Δmagx smax Δmagmax smax Δmagmax sx RV exoplanet (arcsec) (β = 90°) (arcsec) (arcsec) 1 epsilon Eri b 1.31 21.73 1.29 21.69 1.23 22.23 2 GJ 832 b 0.77 21.51 0.77 21.70 0.58 23.22 3 55 Cnc d 0.45 22.34 0.45 22.32 0.45 22.34 4 HD 217107 c 0.41 23.15 0.28 22.38 0.13 21.05 5 mu Ara c 0.38 22.45 0.38 22.41 0.32 21.72 6 HD 190360 b 0.33 22.19 0.26 21.67 0.17 20.69

Table A.1. Estimates of smax and Δmagmax for the six RV exoplanets with greatest values of a(1+ ε) / d . Columns 3–4: the incomplete estimates, which disregard full knowledge of the RV orbit, particularly ε and ωp, and don’t account for the mission duration being significantly shorter than the orbital periods involved. Columns 4–5 and 6–7: mission values of smax and Δmagmax for edge-on orbits (i = 89.1°) and face-on orbits (i = 0.9°). In red: cases that fail criterion #2 when the full knowledge of the RV orbit and the mission time span are taken into account.

periapsis RV mpsini period JD a 1$Ε d exoplanet d pc m m a au Ε Ω days #2450000 arcsec s ! m! p 1 epsilon Eri b 3.22 0.82 1.05 3.38 0.25 186.00 2500. #1060.00 1.31 2 GJ 832 b 4.95 0.45 0.64 3.40 0.12 124.00 3416. 1211.00! 0.77! "# 3 55 Cnc d 12.34! " 0.91! " 3.54! " 5.47! " 0.02 74.00 4909.! " 3490.00" 0.45! " 4 HD 217107 c 19.86 1.11 2.62 5.33 0.52 18.60 4270. 1106.32 0.41 5 mu Ara c 15.51 1.15 1.89 5.34 0.10 237.60 4206. 2955.20 0.38 6 HD 190360 b 15.86 0.98 1.54 3.97 0.31 192.93 2915. 3541.66 0.33

Table A.2. Details on the six RV exoplanets with largest values of a(1+ ε) / d .

17 2.0 21.69face!on 20.61 1.0 21.52face!on 21.31 27.94edge!on 20.23 1: epsilon Eri b 30.08edge!on 23.23 2: GJ 832 b

1.5

1.0 0.5 " " 0.5 arcsec arcsec ! !

y 0.0 y 0.0

!0.5

!1.0 !0.5

!1.5 !1.5 !1.0 !0.5 0.0 0.5 1.0 1.5 !0.5 0.0 0.5 x arcsec x arcsec

! " ! " 1.0 22.32face!on 22.22 1.0 22.38face!on 20.64 22.61edge!on 21.07 3: 55 Cnc d 30.57edge!on 19.74 4: HD 217107 c

0.5 0.5 " " arcsec arcsec ! !

y 0.0 y 0.0

!0.5 !0.5

!0.5 0.0 0.5 !0.5 0.0 0.5 x arcsec x arcsec

! " ! " 1.0 22.41face!on 22.33 1.0 21.67face!on 20.78 21.72edge!on 21.16 5: mu Ara c 29.84edge!on 20.08 6: HD 190360 b

0.5 0.5 " " arcsec arcsec ! !

y 0.0 y 0.0

!0.5 !0.5

!0.5 0.0 0.5 !0.5 0.0 0.5 x arcsec x arcsec

Figures A.1–6. Photo-astrometric! " plots for the RV exoplanets in! Tables" A.1–2. Only valid days for the coronagraph are shown. The orbits are curved or linear for the face-on or edge-on cases, respectively. The upper or lower numbers on the color key give the range of Δmag for edge-on or face-on. The dark circle is IWA = 0.34 arcsec.