A proposed science metric for RV exoplanet characterization
by a probe-class, direct-detection mission
Robert A. Brown
DRAFT May 11, 2013
Known RV exoplanets are prime targets for a probe-class, direct-detection mission. To
help discriminate between mission concepts, we propose a science metric for RV
exoplanets: the estimated number of such planets characterized during the mission. In this
report, we estimate of metric as (XXX + YYY) ± UUU. This estimate uses simplifying
assumptions, which could be relaxed to improve accuracy.
A sample of known RV exoplanets was drawn from www.exoplanets.org on May 10,
2013. It includes all 423 objects satisfying the search term “PLANETDISCMETH ==
'RV'.” The orbital inclination angle (i) is unknown for 410 of these objects, in which case
we can only estimate the probability of detection. For the 13 objects with known i, we
can compute detectability with certainty, yes or no. Therefore, the estimate of the metric
is the sum of 410 Bernoulli random variables with known probabilities (YYY ± UUU)
plus the number of detectable planets with known i (XXX).
The “detectability” of an RV exoplanet with unknown i is defined to be the maximum
probability of satisfying three criteria, simultaneously, sometime during the mission
duration. The three criteria are (1) object resolved: apparent separation s > IWA, the inner
working angle; (2) adequate contrast: delta magnitude less than the systematic limit,
Dmag < Dmag0 ; (3) permitted pointing: angle between the host star and the sun greater
than the solar avoidance angle, q > q 0 . In this report, we adopt these mission parameters:
(a) mission duration is five years, running from January 1, 2020, to December 31, 2025;
(b) IWA = 0.3 arcsec; (c) Dmag0 = 25, (d) q 0 = 110° . (Rémi, if you like other values, it’s
no problem.)
For each RV exoplanet in our sample, we are provided values of the following
parameters: (i) semimajor axis (a) in AU; (ii) orbital eccentricity ( e ); (iii) orbital period
(T) in days; (iv) time of periapsis (T0) in JD; (v) argument of periapsis of the star ( w s ) 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 coordinates, right
ascension ( a ) and declination ( d ) in decimal hours and degrees, respectively.
With one caveat— that we must know or assume a value of i—we can compute the threedimensional position, including the separation s, the radial distance (r) from star to the
planet, and the phase angle ( b ), which is the planetocentric between the star and the
observer at earth. We can perform this computation for any RV exoplanet at any time (t),
in JD, according to the procedures in §3 of Brown (2004).
To compute Dmag, we need to make additional assumptions about the photometric
parameters: (x) the geometric albedo of the planet (p); (xi) the planetary phase function
( F(b ) ); (xii) the planetary radius (Rp). In this report, we choose p = 0.5, F = F L , the
Lambert phase function, and Rp = RJupiter . (Rémi, again, if you like other values, it’s no
problem. However, I did use these value is Brown (2004).) We compute Dmag from the
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 the sun, in ecliptic
coordinates, 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 were satisfied on at least once.
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, and q for a large number of random values of i drawn from a uniform
distribution of orbital poles on the celestial sphere. This 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 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 probability distribution of number successful detections for the whole
mission is the convolution of the 410 Bernoulli distribution functions with probabilities
Pj. This distribution function is shown in Figure 1. The expectation value and standard
deviation of the distribution is YYY ± UUU.
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
Simplifying assumptions.
This list is not complete.
These assumptions could be relaxed in future studies, which would increase the accuracy
of the metric.
No attempt has been made to prioritize this list.
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 planet radius for smaller masses, which would be computed from the
planet mass by some acceptable mass-radius relation. The planet mass in m sin i is well
determined for any value of i.