Science Metric for RV Exoplanet Characterization
Robert A. Brown
DRAFT May 16, 2013
Abstract
Known RV exoplanets are prime targets for a probe-class mission to directly detect and
spectroscopically characterize in reflected starlight. To help discriminate between
mission candidates, and to inform—expectations—we propose 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,
IWA º n LWW / D ³ 0.2 arcsec,
where n 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:
n
3
3
2
2
LWW (nm)
485
760
760
1000
D (m)
1.50
2.35
1.57
2.06
IWA (arcsec)
0.200
0.200
0.200
0.200
In the future, we could improve the fidelity of the RV science metric by addressing a list
of issues at the end of this report.
Sample of Known RV Exoplanets
We drew 423 known RV exoplanets from www.exoplanets.org on May 10, 2013,
including all objects satisfying the search term “PLANETDISCMETH == 'RV'.” We
reduced the list to 419 objects 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 unknown orbital
inclination angle (i), 13 objects with known i, and 346 unique host stars.
Definition and Computation of Detection Criteria
For those exoplanets with unknown i, 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.
Based on the completeness calculations and design reference missions (DRMs) in Brown
(2004a, b), Brown (2005), and Brown & Soummer (2010), we adopt three criteria for
direct detection:
#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
We explore four mission parameters relevant to the metric:
(a)
(b)
(c)
(d)
mission duration = three years: January 1, 2020, to December 31, 2022
IWA = 0.165 arcsec
Dmag0 = 26
g 1 = 45° (coronagraph), g 1 = 45° and g 2 = 80° (star shade)
Meanwhile, 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 right ascension ( a in hours) and declination ( d in degrees).
We compute g from a and d using the fact that the dot product of the unit vectors from
the telescope to the star and sun is cosg .
We compute the three-dimensional position the exoplanet relative to the star from
parameters i–viii, with the caveat that we must know or assume a value of i. Knowing
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.
To compute Dmag , we must introduce additional, 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 (2004b).
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.
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 ( g 1 = 45° , g 2 = 80° ); at 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. As time passes, the sun,
coordinate grid, and red/green zones remain fixed, while the vernal equinox and host
stars revolve uniformly, at constant ecliptic latitude, counterclockwise in this view, at one
revolution per year.
We implement Detection Criterion #1 by creating 417 “validity” lists, one for each host
star, of the 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 ( g 1 = 0° , g 2 = 180° ), then the total
number of valid observing opportunities would be 1097 ´ 420 = 460,740 . After applying
the actual pointing restrictions, the total numbers are 101,528 for the star-shade mission
and 324,422 for the coronagraphic mission.
In evaluating the science metric, we implement Detection Criterion #1 by computing
Detection Criteria #2 and #3 only 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 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.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
RV exoplanet
epsilon Eri b
GJ 832 b
55 Cnc d
HD 217107 c
mu Ara c
HD 190360 b
HD 87883 b
HD 39091 b
47 UMa c
GJ 849 b
HD 134987 c
GJ 179 b
upsilon And d
HD 154345 b
14 Her b
HD 192310 c
Max s (arcsec)
1.312
0.769
0.452
0.408
0.378
0.329
0.301
0.300
0.279
0.269
0.249
0.238
0.237
0.237
0.229
0.175
Table 1. The 16 known RV exoplanets with largest maximum separation s.