Science Metric for RV ExoplanetCharacterization
Robert A. Brown
DRAFT May 19, 2013
Abstract
Known RVexoplanetsareprime targets for a probe-classmission to directly detect and spectroscopically characterize in reflected starlight. To help discriminate between mission candidates, and to inform—expectations—wepropose a science metric for RV exoplanets: the number (N) of RV exoplanets observed and studied by the mission.Our initial estimate is
N ≤ 15,
limited by the inner working angle,
,
wheren is the Airy ring number, LWW is the longest working wavelength, and D is the diameter of the telescope aperture. The value IWA = 0.2 arcsec could be achieved, for example, by any of these combinations of parameters:
nLWW (nm)D (m)IWA (arcsec)
34851.500.200
37602.350.200
27601.570.200
210002.060.200
In the future, we couldimprove the fidelity of the RV science metricby addressing a list of issues at the end of this report.
Sample of Known RV Exoplanets
We drew 423 known RV exoplanets from on May 10, 2013, including all objects satisfying the search term “PLANETDISCMETH == 'RV'.”We reduced the list to 419objects by dropping three host stars of unknown distance (HD 13189, BD+48 738, and HD 240237), and dropping one star (Kepler-68) with missing orbital information. The final sample includes 406 objects with unknownorbital inclination angle (i), 13 objects with knowni, and 346 unique host stars.
Definition and Computation of Detection Criteria
Based on the completeness calculations and design reference missions (DRMs) in Brown (2004a, b), Brown (2005), and Brown & Soummer(2010), we adoptthree criteria for direct detection:
#1permitted pointing: angle between the host star and the sun () greater than the solaravoidance angle, and less than the angle at which the star shade appears illuminated (star-shade glint),
#2object resolved: apparent separation sIWA, the inner working angle
#3adequate contrast: delta magnitude less than the systematic limit,
We explore four mission parameters relevant to the metric:
(a) mission duration =three years: January 1, 2020, to December 31, 2022
(b) IWA = 0.165 arcsec
(c)
(d) (coronagraph), and (star shade)
Meanwhile, providesvalues of these parameters for each RV exoplanet,:
(i) semimajor axis (a) in AU
(ii)orbital eccentricity ()
(iii)orbital period (T) in days
(iv) time of periapsis (T0) in JD
(v) argument of periapsis of the star () in degrees
(vi) mass of the star (ms) in solar masses
(vii) minimum planetary mass (mp sini) in Jupiter masses
(viii) stellar distance (d) in pc
(ix) stellar right ascension ( in hours)and declination ( in degrees).
We compute from and using the fact that the dot product of the unit vectors from the telescope to the star and sunis .
We compute the three-dimensional positionthe exoplanetrelative to the star from parameters i–viii, withthe caveat that we must know or assume a value of i. Knowing this position, we can computes, the radial distance (r) from star to the planet, and the phase angle (), which is the planetocentric angle between the star and earth.
To compute , we mustintroduce additional, photometric parameters:
(x) geometric albedo of the planet (p)
(xi)planetary phase function ()
(xii) planetary radius (Rp)
In this report, we choose p = 0.5, , the Lambert phase function, and .We compute from the parameter vector {r, , p, , } using Eq. (19) in Brown (2004b).
For XXX of the 13 RV exoplanets with knowni, we computes, , and for each mission day, and we find thatthe three detection criteria are satisfied at least once during the mission.
In the case of the 410 RV exoplanets with unknowni, we proceeded probabilistically, as follows. For each exoplanet, for each day of the mission, we compute we computes, for a large number ofrandom values of idrawn from a uniform density of orbital poles on the celestial sphere. This density is achieved by drawing from the random deviate , where R is a uniform random deviate on the interval 0–1. The detection probability for the jth exoplanet on the kthday (Pj,k) is defined as the fraction of the random values of i for which the three detection criteria are satisfied. We define themission 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.
Detection Criterion #1: Permitted Pointing
Figure 1 shows the pointing restrictions on a day chosen at random (June 20, 2010).
Figure 1. The celestial sphere on June 20, 2010, showing the positions of the ecliptic equator (black line), vernal equinox, sun, and host stars of the 420 RV planets in our sample (blue dots). To provide an example, HD 2952 is shown as a larger dot. At left: star shade (, );at right: coronagraph (, ). If a star lies in the green region, it can be observed,but notif it lies in the red. As time passes, the sun, coordinate grid, and red/green zones remain fixed, while the vernal equinox and host stars revolveuniformly,at constant ecliptic latitude, counterclockwise in this view, at one revolution per year.
We implementDetection Criterion #1 by creating 417“validity” lists, onefor each host star, ofthe days during the mission when a star is observable.
During the three-year mission, there are 1097 evenly spaced observing times separated by one day. If there were no pointing restrictions (, ), then the total number of valid observingopportunities would be . After applying the actual pointing restrictions, the total numbersare101,528 for the star-shade mission and 324,422for the coronagraphic mission.
In evaluating the science metric, weimplement Detection Criterion #1 by computingDetection Criteria #2 and #3only at the times in a host star’s validity list.
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, “Single-visit photometric and obscurational completeness,” ApJ, 624, 1010
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 accuracyof 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 ifor 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 : 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.
RV exoplanetMaxs (arcsec)
1epsilon Eri b1.312
2GJ 832 b0.769
355 Cnc d0.452
4HD 217107 c0.408
5mu Ara c0.378
6HD 190360 b0.329
7HD 87883 b0.301
8HD 39091 b0.300
947 UMa c0.279
10GJ 849 b0.269
11HD 134987 c0.249
12GJ 179 b0.238
13upsilon And d0.237
14HD 154345 b0.237
1514 Her b0.229
16HD 192310 c0.175
Table 1. The 16 known RV exoplanets with largest maximum separation s.