A Science Metric for Direct Characterization of Known Radial-Velocity Exoplanets
Robert A. Brown
Space Telescope Science Institute
rbrown@stsci.edu
June 17, 2013
Abstract
Known RV exoplanets are viewed as prime targets for a future high-contrast imaging
mission, which may be able to detect and characterize these enigmatic objects in reflected
starlight. To help define and differentiate the candidates for such a mission on the basis of
scientific performance, and to help set realistic expectations, we develop a science metric,
NRV, defined as the estimated number of planets that would be detected and characterized
by such a mission. In this report, we estimate upper limits to NRV for missions with
apertures in the range 1–2.4 m. We treat both star-shade and coronagraphic missions.
1. Introduction
We define the science metric NRV to be the number of known RV exoplanets that satisfy
four criteria:
#1
#2
#3
#4
permitted pointing: during observations, the angle between the host star and the
sun ( g ) must be greater than the solar avoidance angle, g > g 1 and, for a starshade mission, it must also be less than the angle at which the star shade appears
illuminated, g < g 2
systematic limit: the flux ratio of the planet to the host star, expressed in delta
magnitudes ( Dmag ), must be better than the systematic limit in delta magnitudes
( Dmag0 ), i.e. Dmag < Dmag0
wavelength: criteria #1–2 must be satisfied at all wavelengths deemed critical for
planet detection and characterization, and particularly for the coronagraph, at the
longest critical wavelength (lc)
time: criteria #1–3 must be satisfied at some time during the mission for at least as
long as the exposure time required for detection or characterization of a planet as
faint as the systematic limit.
The input catalog of RV exoplanets—plus working assumptions about the photometric
properties of exoplanets—provide the natural parameters for evaluating criteria #1–4.
The mission design and science operations provide the technical parameters. In this
report, we develop the methods for computing NRV from these parameters. The algorithms
and methods were developed earlier by Brown (2004a, 2004b, 2005) and Brown &
Soummer (2010) for estimating direct-imaging search completeness and for studying
design reference missions (DRMs).
2. Natural Parameters: The Input Catalog
On May 10, 2013, we drew 423 known RV exoplanets from www.exoplanets.org,
comprising all planets satisfying the search term “PLANETDISCMETH == 'RV'.” We
reduced the list to 419 planets by dropping three host stars with unknown distances (HD
13189, BD+48 738, and HD 240237) and one planet with missing orbital information
(Kepler-68). The final catalog includes 406 planets with unknown orbital inclination
angle (i) and 13 planets with known i. The input catalog includes 347 unique host stars.
The input catalog includes 13 planets with known i, only one of which—HD 192310 c—
achieves sufficient separation to be observed by the optical systems considered in this
report. For current purposes, we have simply carried HD 192310 c along as if we did not
know its inclination.
The input catalog contains the following parameters for each RV exoplanet:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
stellar right ascension ( a in hours) and declination ( d in degrees)
stellar distance (d) in pc
mass of the star (ms) in solar masses
minimum planetary mass (mp sini) in Jupiter masses
semimajor axis (a) in AU
orbital eccentricity ()
argument of periapsis of the star ( w s ) in degrees. (p = s + 180°)
orbital period (T) in days
epoch of periapsis (T0) in JD
In addition, we introduce the following natural photometric parameters for the planet:
(x)
(xi)
(xii)
geometric albedo of the planet (p)
planetary phase function ( F(b ) )
planetary radius (Rp)
In this report, we choose to investigate NRV using a Jupiter-like exoplanet as the test body:
p = 0.5, F = F L , the Lambert phase function, and Rp = RJupiter . Such an object is a
maximal planet in terms of flux of reflected light relative to the host star, which we
express in delta stellar magnitudes (mag).
In their Figure 9, Cahoy, Marley, & Fortney (2010) suggest that p of a Neptune-like
exoplanet orbiting at 5 AU would be 40% smaller in the visible to near-infrared
wavelength range than a Jupiter-like exoplanet. Also taking relative size into account, a
Neptune analog would be 2.9 magnitudes—a factor of 14—fainter than a Jupiter analog
at the same phase. Because the maximum brightness of a Jupiter observable by the
current program is typically mag = 20–21, a Neptune would exhibit mag = 24–25 at
brightest. In §2, part of the reason for setting the instrumental requirement for the
systematic limit of detectability at Dmag0 = 25 is a perceived scientific imperative to
2
reach—find and study—exoplanets at least as small as Neptune. An associated
requirement is the ability to accommodate the long exposure times necessary to reach
limiting objects—objects at the systematic limit of detectability—with a robust signal-tonoise ratio (S/N).
2. Technical Parameters of the Mission Design
We adopt the following values for five technical parameters that are relatively
noncontroversial:
(a)
g 1 = 45° (coronagraph), g 1 = 45° and g 2 = 80° (star shade). These values of  are
being used in current NASA-sponsored studies.
(b)
mission timing: January 1, 2020, to December 31, 2022. A duration of three years
seems about right for a 1–2.4 m class mission. The chosen absolute timing seems
reasonable for implementing a mission of this scale.
(c)
mag0 = 25. In recent years, NASA-sponsored studies, laboratory tests, and
experience with space instruments have built confidence in the community that
optical performance at the level of mag0 = 25 can now be achieved for usefully
small angular separations (s) at visible and near-infrared wavelengths. In practical
terms, the achievable mag0 is ultimately determined by the brightest unstable
speckles of starlight at the field position of the planet (Brown 2005).
(d)
lc = 785 nm, which is the red end of the reddest filter, centered at 766 nm.
(e)
spectral resolving power R = 5 at 550 nm for search exposures, and R =20 for
spectroscopy near 730 nm, which approximately matches the width of the CH4
spectral feature (see Figure 10 in Cahoy, Marley, & Fortney 2010).
The parameters of limiting search observations (LSOs for detection) and limiting
characterization observations (LCOs for characterization) are defined in Table 1.
Type 
LSO
25
LCO
25

R S/Ngoal q min #
 (nm)
550
5
5
2
1
693, 730, 766 20
10
2
3
Table 1. Parameters for LSOs and LCOs. q is the required number of exposures in one
limiting observation. The observations are always taken in pairs of consecutive exposures
at slightly different telescope rolls, and the data are subtracted, shifted, and added to
estimate the signal. The science program sets min #, which is the minimum number of
limiting observations: one LSO for a search, and three LCOs for spectral characterization
of a massive extrasolar planet. Of the latter, one LCO is centered on the absorption
feature of CH4 at 730 nm and two LCOs sample the adjacent continuum, centered at 693
and 766 nm. The red edge of the reddest filter determines lc = 785 nm. The single LSO
3
and the three LCOs are assumed to constitute a minimum spectral characterization of an
RV exoplanet.
NRV depends on the minimum angle at which the suppressive performance mag0 is
achieved. This angle is called the inner working angle (IWA). For the coronagraph,
typical apparent separations of RV exoplanets from their host stars are smaller than a few
Airy rings at lc, and for this reason, IWA for the coronagraph is controversial.
In this report, we study various values of the coronagraph’s IWA in the range
(f)
IWA = 0.134–0.645 arcsec .
IWA is commonly expressed
IWA = k l / D,
(1)
where  is a number of Airy rings at wavelength and D is the diameter of the telescope
aperture. Equation (1) is nothing more than a degenerate translation between two
operational parameters, IWA and , and two descriptive parameters
( f1¢ )
( f2¢ )

D.
In Table 2, we use Equation (1) to explore combinations of values of f1–2
¢ that produce the
studied values of IWA for the coronagraph.
By recognizing that the range of values IWA = 0.134–0.645 arcsec is compatible with at
least some noncontroversial combinations of  , and D, we implicitly satisfy criterion
#3. Nevertheless, it is important to keep in mind that the technical difficulty of achieving
a value of IWA for the coronagraph has a negative relationship with the value of , and
the practicality of achieving  < 3 is currently controversial.
D (m)
2.4
1.5
1.0
785
4
3
2
4
3
2
4
3
2
IWA(arcsec)
0.269
0.202
0.134
0.430
0.323
0.215
0.645
0.484
0.323
comment
conservative
moderate
controversial
conservative
moderate
controversial
conservative
moderate
controversial
Table 2. The cases of D, 785, and IWA for the coronagraph. The comment is about the
degree of difficulty in realizing full starlight suppression at 785 Airy rings.
4
For the star-shade mission, we select IWA = 0.075 arcsec, and D = 1, 1.5, and 2.4 m.
3. Criterion #1: Permitted Pointing
We compute g from a and d using the fact that the scalar product of the unit vectors
from the telescope to the star and the telescope to the sun is equal to cosg . We calculate
the unit vectors in the rectangular ecliptic coordinates for any given time.
Figure 1 shows the zones of permitted pointing on 29 April 2020.
For this study, we implement criterion #1 by creating 346 “validity” lists—one for each
host star—of the days during the mission when the star is observable.
For each star, the longest available exposure time—tmax, in days—is equal to the
maximum number of consecutive valid days.
Figure 1. Celestial spheres on 29 April 2020, 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
input catalog (blue dots). Left: for a star-shade mission ( g 1 = 45° , g 2 = 80° ); right: for a
coronagraph mission ( g 1 = 45° , g 2 = 180° ). To provide a concrete example, the host star
HD 220773 is shown as a yellow dot. 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, clockwise as seen from the north ecliptic pole, at the rate of one revolution per
year. The yellow line shows the positions of HD 220773 when it is observable for one
year following 29 April 2020. For a star-shade mission, HD 220773 is observable for two
36-day intervals during the year. For a coronagraph mission, it is observable for one 275day interval.
4. Criterion #2: Systematic Limit
5
The following four steps constitute the master computation of the systematic limit of
detectability. The steps evaluate criterion #2 by asking the logical question
s > IWA
Dmag < Dmag0 ?
(2)
The computations answer this question, yes or no, for all 419 planets on every day on the
host star’s validity list, for two values of i: “face-on” (0.9°) and “edge-on” (89.1°).
In the first step, we compute the three-dimensional position of the planet in space relative
to the center of mass, from parameters ii–ix, for both the face-on and edge-on inclinations.
Second, knowing the planet’s position, we compute the apparent separation (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. Third, we compute Dmag from parameters x–xii using
Eq. (19) in Brown (2004b). Fourth, for each planet, we select the values of the largest
apparent separation (smax) with an associated delta magnitude magmax < mag0.
We assume the planet is detectable according to criterion #2 if smax > IWA and
Dmagmax < Dmag0 for any valid time.
We distinguish three possible outcomes for any planet:
Outcome #1: smax > IWA and Dmagmax < Dmag0 at no valid time, neither for the face-on
nor the edge-on inclinations. All such planets—the vast majority of the 419
planets in our sample—contribute nothing to NRV.
Outcome #2: smax > IWA and Dmagmax < Dmag0 at some valid times, for either the faceon or the edge-on inclinations, but not both. These planets are obscured or too
faint at all valid times for some values of i, but are detectable for some other
values of i at some valid times. This is the outcome for a number of planets N2. It
is possible that N2 = 0. Each planet with outcome #2 contributes either zero or a
fractional value, between zero and one, to the estimate of NRV, depending on
whether tmax is less than or greater than the exposure time for a LSO or LCO.
Outcome #3: smax > IWA and Dmagmax < Dmag0 at valid times, for both the face-on and
edge-on inclinations. This is the outcome for N3 planets. Such planets can always
be detected according to criterion #2 on some valid days. Each planet with
outcome #3 contributes either zero or one to the estimate of NRV, depending on
whether tmax is less than or greater than the exposure time for a LSO or LCO.
Appendix A demonstrates that full orbital calculations are required to accurately evaluate
criterion #2. It also provides salient examples of RV planets with outcomes #1–#3.
5. Criterion #4: Time
6
For the coronagraph, we find that 26 RV exoplanets satisfy criterion #2 (i.e., outcome #2
or #3) for at least one of the values of IWA in Table 2. These planets have 25 host stars,
shown in Table 3a.
For the star shade, we find 79 RV exoplanets satisfy criterion #2 (i.e., outcome #2 or #3)
for IWA = 0.075 arcsec. These planets have 72 host stars, shown in Table 3b.
Appendix B explains how we compute exposure times t0 for LSOs and LCOs. The values
are the same for both the coronagraph and the star shade.
We make the optimistic assumption that the full tmax is available for a LSO or LCO.
Therefore, if t0 < tmax , we assume criterion #4 is satisfied, and not if it is not.
Table 3a. Time results for a coronagraphic mission. tmax is the longest possible continuous
exposure, in days. t0 is the required exposure time, computed as described in Appendix B.
The red cases fail to satisfy the condition t0 < tmax of criterion #4. The values of t0 are
independent of IWA, the planetary orbit, and the strategy for starlight suppression,
coronagraph or star shade.
7
Table 3b. Time results
for a star-shade
mission. The unit of
time is log days. For
the same star, the
values of t0 are the
same as for a
coronagraphic mission,
but the values of tmax
are shorter for the star
shade.
8
In Table A.2, we identified the six known RV exoplanets with highest values of a (1 + )
/ d, which is an incomplete (“raw”) estimate of maximum apparent separation for the
planets of interest here. For the case considered in the Appendix A, IWA = 0.34 arcsec,
and five planets— epsilon Eri b, 55 Cnc d, mu Ara c, GJ 832 b, HD 217107 c—satisfy
the inequality a (1 + ) / d > IWA, which would produce the incomplete estimate NRV = 5.
In Appendix A, we show that a complete treatment of the planetary orbit eliminates the
contribution of HD 217107 c and finds that only a fractional contribution is possible from
mu Ara c.
Tables 3a–b show how considerations of time impact the four planets still in play for this
example:
epsilon Eri b Considerations of time do not negate the ability to detect and characterize
this planet by either coronagraph or star shade for any aperture. Potentially
contributes one to NRV.
55 Cnc d
Coronagraph: detection possible for any aperture, but characterization
with D = 1 and 1.5 m are ruled out. Star shade: detection with D = 1.5 and
2.4 m are possible, but characterization ruled out for any aperture.
Potentially contributes one to NRV.
mu Ara c
Coronagraph: detection possible for any aperture, but characterization
with D = 1 and 1.5 m are ruled out. Star shade: detection possible for any
aperture, but characterization with D = 1 and 1.5 m are ruled out.
Potentially contributes a fraction to NRV, as explained in §6.
GJ 832 b
Ruled out by time considerations for all missions and apertures under
consideration. (For the parameters in Appendix B, light from GJ 832 b at
550 nm is detected in a 2.4 m aperture at the rate of 42 photoelectrons per
day—fewer than two per hour—for 1101 days.) Contributes zero to NRV.
1.0 m
1.5 m
2.4 m
Coronagraph
LSO
LCO
2.45
1
2.45
1.45
2.45
2.45
Star Shade
LSO
LCO
1
0
1.45
1.45
2.45
2.45
Raw
5
5
5
Table 4. Estimates of NRV for IWA = 0.34 arcsec, in the example of Appendix A.
=
0.45 is the estimated contribution of mu Ara c on the basis of 50,000 Monte Carlo
experiments with random values of i. The “raw” estimate uses the fallacious criterion a (1
+ ) / d > IWA and ignores matters of time.
9
6. Estimation of
Assuming all feasible LSOs and LCOs with possibly positive results are made, the
estimate of the metric is
,
(3)
where
is the expectation value for the sum of N2 Bernoulli random variables, which
are the planets with outcome #2, each of which has a unique probability P.
We estimate the probability Pk of the kth random variable by a Monte Carlo experiment,
as follows. First, we create a large sample of random values of i by drawing from the
appropriate random deviate, which is arccos(1- 2Q) , where Q is a uniform random
deviate on the interval zero to one. Second, we compute smax and magmax for each value
of i by the four-step master computation described in §4. Third, we count the number of
random values of i that result in satisfying the systematic-limit criterion in Equation (2)
on at least one valid day, and therefore are detectable according to criterion #2. Fourth,
we estimate Pk as the ratio of this number to the total number of values of i in the
experiment—in the current study, for some 50,000 trials for each of N2 cases.
As explained and illustrated by Brown & Soummer (2010), we compute the density of B
by convolving the N2 Bernoulli densities for P, which are delta functions at 0 and 1 with
weights (1– Pk) and Pk, respectively. The density of B is therefore a sum of N2 + 1 delta
functions at integer values in the range 0–N2. The expectation value of B is the mean of
the density of B, which is SPk . If needed, the standard deviation of B can be computed as
the square root of the sum, k = 0 to N2, of ( k – SPk ) Pk .
2
In the particular example discussed in §5 and Appendix A, N2 = 1 for mu Ara c, in which
case
= P1 = 0.45.
7. Pre-DRM Results for NRV
Here, we comment on the results in Tables 5a–b, which show the estimates of NRV—and
the breakdowns—assuming all feasible LSOs and LCOs with possibly positive results are
actually performed. In §8, we will introduce a DRM formalism to study the scienceoperational constraints imposed by issues of timing, total time, and sequential target
choice.
The ability of a coronagraph to achieve full starlight suppression at  = 2 is currently
controversial. Nevertheless, assuming  = 2, Tables 5a indicate an approximately factortwo improvement in pre-DRM values of NRV as compared with  = 3. For  ≥ 3, we find
NRV < 10 for all coronagraph programs, and for all cases except D = 2.4 m, NRV ≤ 3. A
10
mission with D = 1.0 m could be called “The epsilon Eri Express,” because epsilon Eri b
is the only currently known RV exoplanet that it could hope to observe. The estimated
number of RV exoplanets that can be both detected and characterized with D = 2.4 m and
is currently NRV = 13.3 for the star shade and NRV = 6.8 for the coronagraph with  = 3.
Table 5a. NRV results for coronagraphic missions. A “1” indicates that the planet is
definitely observable (outcome #3 and t0 < tmax.). The sum of the 1’s in a column is N3. A
fractional value (P) means the planet is observable with probability P estimated by Monte
Carlo trials of random values of i. In a column, fractional values are combined as
explained in §6 to produce
, and NRV = 0. Stars are listed by descending productivity
across all combinations of D and , measured by the sum of numbers in a row.
11
Table 5b. NRV
results for starshade missions.
12
8. Design Reference Missions (DRMs)
Each of six columns headed “log t0” in Tables 3a–b is a list of the cost in time per star of
an LSO or LCO for the coronagraph or star shade, for D = 1, 1.5, or 2.4 m. For the star
shade, the six numerical columns in Table 5b gives the benefit attained for that cost, in
terms of the number of RV exoplanets either detected or characterized. For the
coronagraph, the same applies to groups of three columns in Table 5a—three because of
the break-out of  (on which the time costs do not depend). Therefore, we can assign to
each star and choice of instrumental parameters a numerical indicator of merit, which is
the ratio of benefit to cost.
A design reference mission (DRM) is essentially a “financial” analysis of the information
in Tables 3 and 5. For a given optical system, such an analysis seeks to optimize the
sequence of “purchases”—observed targets—while keeping track of the budget of time,
including the bottom line, the total mission time devoted to this program. The missionlevel estimate of “value” is NRV, the total number of RV planets that have been
“purchased.”
To illustrate with one example, we build six DRMs to compare a coronagraph with  = 3
and the star shade with IWA = 0.075 arcsec, for the three choices D = 1, 1.5, and 2.4 m.
In this example, we demand the “full Monte” of observations, which is one LSO and
three LCOs, one of which is located on the 730 nm CH4 feature, and the other two are on
the adjacent continuum. The total cost in time of the full treatment is t0, LSO + 3 t0, LCO. We
prioritize the individual observations according to the “planet rate,” which is the planet
increment to NRV for that star divided by the total cost.
The results of the example DRMs are shown in Tables 6a,b and summarized in Table 6.
Coronagraph
Star Shade
D (m)
1.0
1.5
2.4
1.0
1.5
2.4
183
1
1
2
2
3
6
Total exposure time (days)
365
730
1
1
1
2
3.15
4.15
3
4.23
5
6.34
8.23
11.3
Table 6. Mission-level estimates of NRV.
13
Table 7a. Estimates of NRV as a function of total exposure time for the star-shade mission.
Table 7b. Estimates of NRV as a function of total exposure time for the coronagraph
mission.
9. Conclusions
14
Acknowledgments
My gratitude to Rémi Soummer, Howard Bond, Wes Traub, Stuart Shaklan, Tom Greene,
Nick Cowan, Charley Noecker, Marc Postman, and Doug Lisman for helpful remarks.
This research has made use of the Exoplanet Orbit Database and the Exoplanet Data
Explorer at exoplanets.org.
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, “0,” ApJ, 624, 1010
R. A. Brown & R. Soummer 2010, “New completeness methods for estimating exoplanet
discoveries by direct detection,” ApJ, 715, 122
C. J. Burrows, 2003, in The Design and Construction of Large Optical Telescopes, e. P.
Y. Bely (New York: Springer), 433.
C. J. Burrows, R. A. Brown, & E. M. Sabatke, 2006, “Aberrational Delta Magnitude,”
manuscript. http://www.stsci.edu/~RAB/papers/Abrr_Del_Mag.pdf
K. L. Cahoy, M. S. Marley, & J. J. Fortney, 2010, "Exoplanet albedo spectra and colors
as a function of planet phase, separation, and metalicity, ApJ 724, 189.
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Astrophys. Suppl. Ser. 127, 1.
15
Appendix A: On The Need for Complete Orbital Computations
In the past, computing orbits has sometimes been avoided by comparing IWA with the
maximum possible apparent separation
sx º a(1+ e ) / d
(A.1)
and by comparing Dmag0 with the delta magnitude at quadrature
2
æ
æ p ö æ Rp ö ö
.
Dmagx = -2.5log ç p F ç ÷ ç
è 2 ø è a(1+ e ) ÷ø ÷ø
è
(A.2)
This approach is incomplete or “raw”, because overlooks the effect of p when 0 and
i ≠ 0, which in the current research would result in a significant overestimate of the
yieldRV .
In the following example, we assume IWA = 0.34 arcsec.
Table A.1 compares values of (sx, magx) and (smax, magmax)—the latter for both the
face-on and edge-on inclinations—for the six planets in the input catalog with largest
value of sx. Table A.2 provides their basic information. Figures A.1–A.6 show their faceon and edge-on orbits on valid days.
The incomplete computation produces the estimate NRV = 5, referring to epsilon Eri b, GJ
832 b, 55 Cnc d, HD 217107 c, and mu Ara c.
In Figures A.1–A.3, we recognize the outcome #3 planets as epsilon Eri b, GJ 832 b, and
55 Cnc d, because each is detectable on some valid days for both edge-on and face-on
orbits. In Figure A.4, we see mu Ara c is an outcome #2 planet, because its edge-on orbit
of mu Ara c does not separate to IWA = 0.34 on any valid day. In Figure A.5, we see that
HD 217107 c is an outcome #1 planet, never reaching a separation of 0.34 arcsec on any
valid day. This is also true for HD 190360 b, in Fig A.6, although in this case we know
that this planet never separates to 0.34 arcsec, because sx = 0.33 arcsec.
The first of three problems with the incomplete computation is that the mission duration
(three years) is short compared with the orbital periods of planets that can achieve large s.
This means that the largest possible separation may not be realized during the mission.
By inspection, this true for epsilon Eri b, HD 217107 c, and HD 190360 b—and it may be
true of others. The incomplete computation wrongly assumes that sx is achieved during
the mission.
The second problem—a concern for planets with elevated values of  and with values of i
and p away from 0° and 180°, the maximum possible apparent separation may actually
be closer to a (1 – ) / d than to a (1 + ) / d.
16
The third problem is that magmax will be significantly different from magx—greater or
smaller—for eccentric orbits oriented to produce significantly reduced separations at
apoapsis.
sx
RV exoplanet (arcsec)
1 epsilon Eri b 1.31
2 GJ 832 b
0.77
3 55 Cnc d
0.45
4 HD 217107 c 0.41
5 mu Ara c
0.38
6 HD 190360 b 0.33
magx
mission duration: 1/1/20–12/31/22
edge-on orbits
face-on orbits
smax magmax
smax magmax
(= 90°) (arcsec)
21.73
1.29 21.69
21.51
0.77 21.70
22.34
0.45 22.32
23.15
0.28 22.38
22.45
0.38 22.41
22.19
0.26 21.67
(arcsec)
1.23 22.23
0.58 23.22
0.45 22.34
0.13 21.05
0.32 21.72
0.17 20.69
Table A.1. Estimates of smax and magmax for the six RV exoplanets with greatest values
of a(1+ e ) / d . Columns 3–4: the incomplete estimates, which disregard full knowledge of
the RV orbit, particularly  and p, and don’t account for the mission duration being
significantly shorter than the orbital periods involved. Columns 4–5 and 6–7: mission
values of smax and magmax for edge-on orbits (i = 89.1°) and face-on orbits (i = 0.9°). In
red: cases that fail criterion #2 when the full knowledge of the RV orbit and the mission
time span are taken into account.
Table A.2. Details on the six RV exoplanets with largest values of a(1+ e ) / d .
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Figures A.1–6. Photo-astrometric plots for the RV exoplanets in Tables A.1–2. Only
valid days for the coronagraph are shown. The orbits are curved or linear for the face-on
or edge-on cases, respectively. The upper or lower numbers on the color key give the
range of mag for edge-on or face-on. The dark circle is IWA = 0.34 arcsec.
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