Science Metric for RV Exoplanet Characterization
Robert A. Brown
DRAFT May 19, 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 ≤ 5,
limited by the inner working angle,
IWA º n LWW / D ³ 0.340 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 by
many combinations of the parameters, for example:
n
4
3
LWW (nm)
800
800
D (m)
1.94
1.46
IWA (arcsec)
0.34
0.34
In the future, we could improve the fidelity of the 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
Based on the calculations 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 = 0.20 arcsec
Dmag0 = 26
g 1 = 45° (coronagraph), g 1 = 45° and g 2 = 80° (star shade)
The website www.exoplanets.org provides values of the following 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 equal to 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 the
exoplanet’s 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).
Detection Criterion #1: Permitted Pointing
Figure 1 shows the pointing restrictions on a random day (June 20, 2010).
Figure 1. The celestial sphere on June 20, 2010, 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 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, the coordinate grid, and the red/green zones remain fixed, while the vernal
equinox and host stars revolve at constant ecliptic latitude, counterclockwise in this view,
one revolution per year.
We implement Detection Criterion #1 by creating 417 “validity” lists, one for each host
star, comprising the days during the mission when that star is observable.
During the three-year mission, we consider 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 reduced to 101,528 for the
star-shade mission and 324, 422 for the coronagraphic mission. The numbers are reduced
drastically when Detection Criterion #2–3 are applied.
Detection Criteria #2–3: s > IWA & mag < mag0
It is tempting to estimate s and mag for RV planets as
s = a(1+ e ) / d
(1)
2
æ
æ p ö æ Rp ö ö
,
Dmag = -2.5log ç p F ç ÷ ç
è 2 ø è a(1+ e ) ÷ø ÷ø
è
(2)
and
however this approach is naïf and erroneous. For RV planets, we know all the orbital
elements except i and the longitude of the ascending node, which is irrelevant because it
constrains only the position angle of the exoplanet and does not affect s. From the period
and time of periapsis, we know the mean anomaly, and generally its value is not  as
assumed by Equations (1–2). Furthermore, we know from w p ¹ 0 that the semimajor axis
does not lie in the plane of the sky, and that the planet is not being viewed at dichotomy.
Nevertheless, these simplistic values of s and mag are easily computed, and Table 1
provides them for the six RV planets with greatest values of a(1+ e ) / d . By inspection,
the simplistic estimate of the science metric is N = 5.
a(1+ e ) mag
1
2
3
4
5
6
RV exoplanet
epsilon Eri b
GJ 832 b
55 Cnc d
HD 217107 c
mu Ara c
HD 190360 b
(arcsec)
1.31
0.77
0.45
0.41
0.38
0.33
(=90°)
21.73
21.51
22.34
23.15
22.45
22.19
mission duration: 1/1/20–12/31/2
edge-on orbits
face-on orbits
smax mag
smax mag
(arcsec)
(arcsec)
1.29 21.69
1.23 22.23
0.77 21.70
0.58 23.22
0.45 22.32
0.45 22.34
0.28 22.38
0.13 21.05
0.38 22.41
0.32 21.72
0.26 21.67
0.17 20.69
Table 1. Estimates of smax and mag for the six RV exoplanets with greatest values of
a(1+ e ) / d . Columns 3–4: the simplistic estimate that disregards full knowledge of the
RV orbit. Columns 4–5 & 6–7: mission values of smax and mag for all edge-on orbits (i
= 89.1°) and all face-on orbits (i = 0.9°). In red: cases that fail Detection Criterion #2
when the RV orbit and mission duration are taken into account.
Figure 2. Photo-astrometric plots for the RV exoplanets in Table 1 during the mission.
The orbits are curved or linear for the face-on and edge-on cases, respectively. The
upper/lower numbers on the color key give the range of mag for edge-on/face-on. The
dark circle is IWA = 0.34 arcsec.
Table 2. Details on the six RV exoplanets with greatest values of a(1+ e ) / d .
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.