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Science Metric for RV Exoplanet Characterization
Robert A. Brown
DRAFT May 16, 2013
Introduction
Known RV exoplanets are prime targets for a probe-class, direct-detection mission,
which could spectroscopically characterize their atmospheres or surfaces in reflected
starlight. To help differentiate mission concepts, and to help develop reasonable
scientific expectations, we propose a science metric for RV exoplanets. This metric is the
estimated number of such planets detected and characterized during the mission, and its
standard deviation. In this report, we derive the metric and estimate its value as (XXX +
YYY) ± UUU. We use simplifying assumptions, which could be relaxed in the future to
improve the systematic accuracy of the metric.
We drew our sample of known RV exoplanets from www.exoplanets.org on May 10,
2013. It includes all 423 objects satisfying the search term “PLANETDISCMETH ==
'RV'.” Because it is not constrained by the RV technique, the orbital inclination angle (i)
is unknown for the 410 of these objects with no additional information. For those
exoplanets, 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. The three criteria are
#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
In this report, we adopt these mission parameters relevant to the metric:
(a)
(b)
(c)
(d)
mission duration = three years: January 1, 2020, to December 31, 2025
IWA = 0.165 arcsec
Dmag0 = 26
g 1 = 45° (coronagraph), g 1 = 45° and g 2 = 80° (star shade)
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 coordinates, right ascension ( a ) and declination ( d ) in decimal hours and
degrees, respectively
From these parameters, with the caveat that we must know or assume a value of i, we can
compute the three-dimensional position of the exoplanet, in space, relative to the star, and
from 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. We can
compute the position for any RV exoplanet at any time (t) in JD, following the procedure
in §3 of Brown (2004).
To compute Dmag , we must make additional assumptions about the 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 (2004).
To compute q , we compute the unit vectors from earth to the star and to the sun, 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 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.
Criterion #1: Permitted Pointing
Figure 1 shows the pointing restrictions on a day chosen at random, June 20, 2009.
Figure 1. The celestial sphere on June 20, 2009, showing the positions of the ecliptic
equator (black ring), vernal equinox, sun, and host stars of the 423 RV planets in our
sample (blue dots). As an example, the host star HD 2952 is shown as a larger dot. Left:
star shade ( g 1 = 45° , g 2 = 80° ); 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. In this figure, as time
passes, the sun, coordinate grid, and red/green zones remain fixed, while the vernal
equinox and stars revolve at constant ecliptic latitude—counterclockwise in this view—at
the rate of one revolution per year.
In computing our metric, we implement the pointing restrictions of Criterion #1 by
creating 423 “validity” lists of days, one list for each host star in our sample. Each
validity list contains all—and only—the days during the mission when it is permitted to
point at that star.
During the mission, there are 1097 observing times separated by one day. If there were
no pointing restrictions ( g 1 = 0° , g 2 = 180° ), there would be a total of
1097 ´ 423 = 464,031 valid observing times. With the actual pointing restrictions, the
sum of the cardinalities of the validity lists for the star-shade mission is 122,027, and for
the coronagraphic mission, the number of valid observing times is 393,689.
We compute Criteria #2 and #3 only at times on the validity lists.
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
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.