The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
doi:
10.1088/0004-637X/794/2/155
C
2014. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
HUBBLE SPACE TELESCOPE
NEAR-IR TRANSMISSION SPECTROSCOPY OF
THE SUPER-EARTH HD 97658B
Heather A. Knutson
1
, Diana Dragomir
2
,
3
, Laura Kreidberg
4
, Eliza M.-R. Kempton
5
, P. R. McCullough
6
,
Jonathan J. Fortney
7
, Jacob L. Bean
4
, Michael Gillon
8
, Derek Homeier
9
, and Andrew W. Howard
10
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA;
hknutson@caltech.edu
2
Las Cumbres Observatory Global Telescope Network, Goleta, CA 93117, USA
3
Department of Physics, Broida Hall, UC Santa Barbara, CA 93106, USA
4
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA
5
Department of Physics, Grinnell College, Grinnell, IA 50112, USA
6
Space Telescope Science Institute, Baltimore, MD 21218, USA
7
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
8
Institut d’Astrophysique et de G
́
eophysique, Universite
́
edeLi
́
ege, Li
́
ege 1, Belgium
9
Centre de Recherche Astrophysique de Lyon, F-69364 Lyon, France
10
Institute for Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA
Received 2014 March 17; accepted 2014 August 27; published 2014 October 2
ABSTRACT
Recent results from the
Kepler
mission indicate that super-Earths (planets with masses between 1–10 times that of
the Earth) are the most common kind of planet around nearby Sun-like stars. These planets have no direct solar
system analogue, and are currently one of the least well-understood classes of extrasolar planets. Many super-Earths
have average densities that are consistent with a broad range of bulk compositions, including both water-dominated
worlds and rocky planets covered by a thick hydrogen and helium atmosphere. Measurements of the transmission
spectra of these planets offer the opportunity to resolve this degeneracy by directly constraining the scale heights
and corresponding mean molecular weights of their atmospheres. We present
Hubble Space Telescope
near-infrared
spectroscopy of two transits of the newly discovered transiting super-Earth HD 97658b. We use the Wide Field
Camera 3’s (WFC3) scanning mode to measure the wavelength-dependent transit depth in 30 individual bandpasses.
Our averaged differential transmission spectrum has a median 1
σ
uncertainty of 23 ppm in individual bins, making
this the most precise observation of an exoplanetary transmission spectrum obtained with WFC3 to date. Our data
are inconsistent with a cloud-free solar metallicity atmosphere at the 10
σ
level. They are consistent at the 0
.
4
σ
level with a flat line model, as well as effectively flat models corresponding to a metal-rich atmosphere or a solar
metallicity atmosphere with a cloud or haze layer located at pressures of 10 mbar or higher.
Key words:
binaries: eclipsing – planetary systems – techniques: spectroscopic
Online-only material:
color figures
1. INTRODUCTION
The
Kepler
mission has resulted in the discovery of more
than 3000 transiting planets and planet candidates (Batalha
et al.
2013
; Burke et al.
2014
) to date, with a majority of the
sample consisting of sub-Neptune-sized planets. Analyses of
this set of Kepler planet candidates indicate that planets with
radii intermediate between that of Neptune and the Earth appear
to be the most common kind of extrasolar planet orbiting nearby
FGK stars, with a peak at radii between 2–3
×
that of the Earth
(Howard et al.
2012
; Fressin et al.
2013
; Petigura et al.
2013
).
Planets in this size range are typically referred to as “super-
Earths,” although they could potentially form with a broad
range of compositions including primarily rocky with a thin
atmosphere (true “super-Earths”), a rocky or icy core surrounded
by a thick hydrogen atmosphere (“mini-Neptunes”), or water-
dominated with a thick steam atmosphere (“water worlds”).
Many of these super-Earths are found in close-in, tightly packed
multiple planet systems (e.g., Lissauer et al.
2011
; Fabrycky
et al.
2012
; Steffen et al.
2013
), and there is an ongoing debate
as to whether these systems formed in place or migrated in from
more distant orbits (Hansen & Murray
2012
; Chiang & Laughlin
2013
; Raymond & Cossou
2014
).
Detailed studies of super-Earth compositions offer important
clues on their origins: presumably water-rich planets must have
formed beyond the ice line, while in-situ formation models
predict primarily rocky compositions with relatively water-
poor hydrogen-dominated atmospheres (e.g., Raymond et al.
2008
). By combining radius measurements from
Kepler
with
mass estimates obtained using either the radial velocity or
transit timing techniques, it is possible to constrain the average
densities and corresponding bulk compositions of the super-
Earths in the Kepler sample (e.g., Lithwick et al.
2012
; Hadden
& Lithwick
2014
; Weiss & Marcy
2014
; Marcy et al.
2014
).
These observations indicate that the super-Earths in the Kepler
sample display a broad range of average densities, with a
transition toward denser, primarily rocky compositions below
1
.
5–2 Earth radii (Weiss & Marcy
2014
; Marcy et al.
2014
).
For the larger, lower-density super-Earths in the Kepler sample
it is possible to match their measured densities with a broad
range of compositions, including both water-rich and water-
poor scenarios, simply by varying the amount of hydrogen in
their atmospheres (Seager et al.
2007
; Valencia et al.
2007
,
2013
;
Rogers & Seager
2010a
; Zeng & Sasselov
2013
). Although it
can be argued that some compositions are unlikely based on
models of planet formation and atmospheric mass loss for close-
in planets, this still leaves a wide range of plausible models (e.g.,
Rogers & Seager
2010b
; Rogers et al.
2011
; Nettelmann et al.
2011
; Heng & Kopparla
2012
; Lopez et al.
2012
; Lopez &
Fortney
2013
; Fortney et al.
2013
).
1
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
Measurements of the transmission spectra of super-Earths
allow us to directly estimate the mean molecular weight
of the planet’s atmosphere (e.g., Miller-Ricci et al.
2008
;
Fortney et al.
2013
), which in turn provides improved con-
straints on its interior composition. Planets with cloud-free, hy-
drogen dominated atmospheres will have relatively large-scale
heights and correspondingly strong absorption features during
transit, while planets with hydrogen-poor atmospheres will have
relatively small-scale heights and weak absorption. Unfortu-
nately, the majority of the super-Earths detected by Kepler orbit
faint stars (
m
K
>
10), making it difficult to accurately measure
their transmission spectra using existing facilities. There are cur-
rently three super-Earths known to transit relatively bright stars,
including GJ 1214b (
m
K
=
8
.
8; Charbonneau et al.
2009
), 55
Cancri e (
m
K
=
4
.
0; Winn et al.
2011
; Demory et al.
2011
), and
HD 97658b (
m
K
=
5
.
7; Dragomir et al.
2013
).
GJ 1214b is currently the only one of these three systems
with a well-characterized transmission spectrum, which is flat
and featureless across a broad range of wavelengths (e.g., Bean
et al.
2010
,
2011
;D
́
esert et al.
2011
; Berta et al.
2012
; Kreidberg
et al.
2014
). Although initial observations were consistent with
either a cloud-free, hydrogen-poor atmosphere or a hydrogen-
rich atmosphere with a high cloud deck (e.g., Bean et al.
2010
),
the most recent near-IR observations obtained by Kreidberg
et al. (
2014
) using the Wide Field Camera 3 (WFC3) instrument
on the
Hubble Space Telescope
(
HST
) are precise enough to
rule out cloud-free models over a broad range of atmospheric
metallicities (also see Benneke & Seager
2013
). The presence
of a high altitude cloud or haze layer means that for this planet,
at least, transmission spectroscopy provides relatively weak
constraints on the mean molecular weight of its atmosphere
and, by extension, on its interior composition.
In this paper, we present
HST
WFC3 near-infrared trans-
mission spectroscopy of the transiting super-Earth HD 97658b.
This planet was first detected using the radial velocity tech-
nique (Howard et al.
2011
), and later found to transit using
MOST
photometry (Dragomir et al.
2013
). It has a mass of
7
.
9
±
0
.
7
M
⊕
, a radius of 2
.
3
±
0
.
2
R
⊕
, and an average density
of 3
.
4
±
0
.
9gcm
−
1
(Dragomir et al.
2013
), making it modestly
denser and more massive than GJ 1214b. HD 97658b orbits its
K star primary with a period of 9.5 days, and has a predicted
zero-albedo temperature between 700–1000 K depending on the
efficiency of energy transport to the planet’s night side. If this
planet has the same atmospheric composition as GJ 1214b, its
modestly higher atmospheric temperature might prevent the for-
mation of the cloud layer detected in GJ 1214b’s atmosphere
(Morley et al.
2013
). Our observations are obtained at wave-
lengths between 1
.
2–1
.
6
μ
m, and are primarily sensitive to the
presence or absence of the water absorption band located at
1.4
μ
m; this feature has now been robustly detected in the at-
mospheres of several hot Jupiters (e.g., Deming et al.
2013
;
Wakeford et al.
2013
; Mandell et al.
2013
; McCullough et al.
2014
), and is expected to be present in super-Earths as well
over a broad range of atmosphere compositions (e.g., Benneke
& Seager
2012
; Kreidberg et al.
2014
; Hu & Seager
2014
). We
discuss our observations in Section
2
and the implications for
the properties of this planet’s atmosphere in Section
3
.
2. OBSERVATIONS
We observed one transit of HD 97658b on UT 2013
December 19 and another on UT 2014 January 7 (GO 130501, PI
Knutson) using the G141 grism on the
HST
WFC3 instrument,
which provides low-resolution spectroscopy at wavelengths
-0.15
-0.10
-0.05
-0.00
0.05
0.10
Time from Predicted Transit Center (d)
0.996
0.997
0.998
0.999
1.000
1.001
Relative Flux
Figure 1.
Raw white-light photometry for the UT 2013 December 19 visit
(top) and UT 2014 January 7 transit (bottom). Different scan directions for the
December visit are indicated as light (forward scan) and dark (reverse scan)
blue filled circles. Scan directions for the January visit are plotted as yellow
(forward scan) and red (reverse scan) filled circles. The two light curves have
been offset by a relative flux of 0.0015 for clarity. The first spacecraft orbit
has been trimmed from each visit, leaving four orbits per transit. We have also
normalized the light curves from each scan direction to one by dividing each
by the median flux value; we do this in order to remove a small offset in the
measured fluxes from the two scan directions.
(A color version of this figure is available in the online journal.)
between 1
.
1–1
.
7
μ
m. Each observation consists of five
HST
orbits with a total duration of approximately seven hours per
visit; our target was visible for approximately half of each
96 minute
HST
orbit. Scheduling constraints resulted in a
slightly shorter first orbit during the January visit, which con-
sisted of 203 spectra instead of the 206 spectra obtained during
the December visit. During the January visit, we also utilized a
series of short exposures at the end of each orbit in order to force
a buffer dump, which avoided the mid-orbit gaps in coverage
visible in the light curves for the first visit in Figure
1
.Wedo
not include these short exposures in our analysis, as they were
taken in imaging rather than spectroscopic mode.
Our spectra were obtained using the 256
×
256 pixel subarray
and SPARS10 mode with four samples, giving a total integration
time of 14.97 s in each image. Our target is one of the brightest
transiting planet host stars, and we therefore utilized the new
scanning mode (e.g., McCullough & MacKenty
2012
;Deming
et al.
2013
; Kreidberg et al.
2014
; Knutson et al.
2014
)
with a scan rate of 1
.
4s
−
1
in order to achieve a higher
observing efficiency while remaining well below saturation.
This results in a scanned spectral image that fills most of the
subarray image, with a fractional coverage similar to that of the
HD 209458 observations from Deming et al. (
2013
). We also
alternated between forward and reverse scan directions in order
to further reduce overheads; this approach was previously used
2
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
0.19
0.20
0.21
0.22
0.23
0.24
0.25
-0.15
-0.10
-0.05
-0.00
0.05
0.10
Time from Predicted Transit Center (d)
0.22
0.23
0.24
0.25
0.26
0.27
Relative Sky Background (% )
Figure 2.
Estimated sky background as a percentage of the total flux in the
white-light aperture (i.e., summed over all wavelengths) for stacked images
created using the method described in Section
2.1
. Sky backgrounds for the
December visit are shown in the top panel and background for the January visit
are shown in the bottom panel.
by Kreidberg et al. (
2014
) for GJ 1214b. The resulting spectra
have peak counts around 40,000 electrons pixel
−
1
, which is
high enough to cause a modestly steep asymptotic ramp in the
measured fluxes within individual orbits (see Wilkins et al.
2014
,
for a discussion of the relationship between the ramp slope and
total flux).
We reduced the data using two independent methods, which
allow us to test whether or not the transmission spectrum we
obtain from our analysis is sensitive to our choice of analysis
technique. We discuss each approach separately below.
2.1. Spectral Template Fitting Method
In this approach, we utilized the spectral template method
first presented in Deming et al. (
2013
), which we describe in
detail in Knutson et al. (
2014
) and summarize here. We begin
with the raw sample up the ramp images from the ima.fits
files, which are bias and dark subtracted, and apply our own
flat-fielding and wavelength calibrations based on the standard
WFC3 pipeline as discussed in Wilkins et al. (
2014
). We then
subtract successive non-destructive readout pairs in order to
create a series of difference images, trim the region around the
spectral scan in each image, and sum the resulting images to
create a composite containing the full scan. The benefit of this
approach is that it minimizes the contributions of pixels that
are not directly illuminated by the star in a given readout time
step (see Deming et al.
2013
, for more discussion on this point).
We estimate that the sky background in our final composite
images is 0
.
18%–0
.
26% of the total flux measured in each
wavelength element (see Figure
2
), and subtract this estimated
sky background from each image.
Once we have created a stacked image from each set of
readouts, we apply a filter to correct for bad pixels and cosmic
ray hits as described in Knutson et al. (
2014
). We then sum the
spectrum along the
y
axis in order to extract a one-dimensional
spectrum, using an aperture that extends fifteen pixels above
1.2
1.4
1.6
Wavelength (microns)
2x10
6
4x10
6
6x10
6
8x10
6
Total Counts (electrons)
Figure 3.
Representative spectrum from the UT 2013 December 19 visit; this
spectrum was created from the two-dimensional image by summing in the
y
(cross-dispersion) direction.
and below the edges of the spectrum (200 pixels in total)
in order to include the extended wings of the point-spread
function (see Figure
3
for a representative example). Because
the WFC3 spectra are undersampled (Deming et al.
2013
), we
convolve each of our one-dimensional spectra with a Gaussian
function with a width (FWHM) of four pixels in order to
mitigate effects related to the shifting position of the spectrum
on the detector. Although this modestly degrades the spectral
resolution of our data, we later bin our transmission spectrum
by an equivalent amount in order to increase the signal-to-noise
ratio. We calculate the MJD mid-exposure time corresponding to
each image using the information from the flt.fits image headers,
and convert these times to the BJD
TDB
time standard following
the methods of Eastman et al. (
2010
).
Our next step is to create a spectral template by averaging
the ten spectra immediately before and after the transit. We
then fit this template spectrum to each individual spectrum in
our time series, allowing the relative position and amplitude of
the template spectrum to vary as free parameters. We find no
difference in our results if we create a separate spectral template
for the forward and reverse scan images, and we therefore use
the same template for all images. The resulting series of best-
fit amplitudes are plotted in Figure
1
, and are identical to the
white-light curves obtained by summing the fluxes across all
wavelengths. We find that there is a small flux offset between
the light curves for the two scan directions, which we remove by
dividing each time series by its median flux value. McCullough
& MacKenty (
2012
) suggest that this offset is a consequence
of the order in which columns are read out by the detector
(the “up-stream
/
down-stream effect”); when the scan moves in
the same direction as the readout then the effective integration
time will be slightly longer than in the case of a reverse scan. The
forward and reverse spectra in our images also occupy slightly
different positions on the array, which might also contribute to
this offset.
We subtract the best-fit spectral template from each individual
spectrum in order to create a differential time series for each
individual wavelength element. The benefit to this approach is
that it effectively removes all common-mode detector effects
from the differential light curve. We find that the scatter in our
light curves for individual wavelength elements is within 5%
3
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
of the photon noise limit in all cases, indicating that there is
minimal color-dependence in the systematic noise sources.
2.1.1. White-light Transit Fits
We take the two raw transit light curves plotted in Figure
1
and fit them each with a transit light curve, a linear function of
time, and an exponential ramp in orbital phase:
F
(
t
)
=
c
1
(1 +
c
2
t
+
c
3
e
−
p/c
4
)
F
transit
(
t
)
,
(1)
where
c
1
−
c
4
are free parameters in the fit,
t
is the time from
the measured transit center in days, and
p
is the time in days
from either the first observation in each spacecraft orbit or the
last observation before a mid-orbit buffer dump. We use this
definition because we find that the ramp after a mid-orbit buffer
dump is usually not as steep as the initial ramp at the start
of the orbit, and using this definition provided a good fit to
the observed behavior by reducing the amplitude of the mid-
orbit ramp model. The latter definition is only used for the
December transit observation, which includes two mid-orbit
buffer dumps that cause the ramp effect to reset. We allow the
planet–star radius ratio
R
p
/R
and center of transit time to
vary as free parameters for each individual transit. We find
that the behavior of the instrumental systematics is slightly
different for the white-light curves calculated from the forward
and reverse scan directions (see Figure
1
). We therefore allow
the linear function of time and exponential ramp functions to
vary independently for each scan direction, while keeping the
same
R
p
/R
and center of transit time.
We calculate the transit light curve
F
transit
following Mandel &
Agol (
2002
) where we initially fixed the orbital inclination
i
and
the ratio of the semi-major axis to the stellar radius
a/R
equal
to their best-fit values from Van Grootel et al. (
2014
). We found
that the uncertainty on our reported transit time for the first visit,
which spanned the transit ingress, was particularly sensitive to
our choice of values for these parameters. Our transit timing
uncertainties for this visit ranged between 30 s up to 5 minutes
for the cases where
i
and
a/R
were either fixed or allowed
to vary freely in our fits. We addressed this by acquiring the
normalized 4.5
μ
m
Spitzer
transit photometry from Van Grootel
et al. (
2014
) and fitting this light curve simultaneously with our
white-light
HST
photometry assuming a single global value for
i
and for
a/R
. This ensured that our final reported transit times
correctly accounted for the added uncertainties contributed by
these two parameters. We allowed the
Spitzer
planet–star radius
ratio and center of transit time to vary independently in our
fits, and found that our results for these two parameters were
consistent with those reported by Van Grootel et al. (
2014
). In
our final fits, the uncertainty on the transit time for the first visit
increased from 30 s to one minute as compared to the fixed
i
and
a/R
case and the best-fit transit time was 3.8 minutes earlier.
The second visit does not include observations during ingress
or egress, and has a correspondingly large uncertainty on the
best-fit transit time. We find that for this visit, allowing
i
and
a/R
to vary has a negligible effect on the reported uncertainties
in the best-fit transit time.
We calculate our initial predicted transit times for the
HST
observations using the ephemeris from Van Grootel et al. (
2014
),
and then derive an updated estimate for the orbital period and
center of transit time based on our new observations. We then
repeat our transit fits using this updated orbital period. We
calculate four-parameter nonlinear limb-darkening parameters
(Claret
2000
) for our transit light curves, with the relative
intensity at each position given as
I
(
μ
)
I
(1)
=
1
−
4
∑
k
=
1
a
k
(1
−
μ
k/
2
)
,
(2)
where
I
(1) is the specific intensity at the center of the stellar
disk,
a
k
is the
k
th limb-darkening coefficient, and
μ
=
cos(
θ
)
where
θ
is the angle between the line of sight and the location
of the emerging flux. We calculate our limb-darkening coeffi-
cients using a
PHOENIX
stellar atmosphere model (Allard et al.
2012
), where we take the flux-weighted average of the theo-
retical stellar intensity profile across the band and then fit for
the limb-darkening coefficients. We use the best-fit stellar pa-
rameters from Van Grootel et al. (
2014
), who find an effective
temperature of 5170
±
50 K, a surface gravity of 4
.
58
±
0
.
05,
and a metallicity of
−
0
.
23
±
0
.
03. These values are also con-
sistent with a recent analysis by Mortier et al. (
2013
), although
this study prefers a lower metallicity of
−
0
.
35
±
0
.
02 and does
not take into account the constraints on the stellar density from
the transit light curve. We also tried models with effective tem-
peratures ranging between 5120–5220 K and a metallicity of
−
0
.
35, but found that these had a negligible effect (2 ppm or
less) on our resulting transmission spectrum. The inclusion of
limb-darkening in our fits creates a small offset in the average
transit depth as compared to fits without limb-darkening, and
has a negligible effect on the slope of the resulting transmission
spectrum across the bandpass. We find that the uncertainty in
the stellar effective temperature contributes a systematic error
of 1 ppm to our estimate of the differential transit depths.
We estimate the uncertainties in our fitted white-light pa-
rameters using a residual permutation (“prayer bead”) method,
the covariance matrix from our Levenberg–Marquart minimiza-
tion, and a Markov Chain Monte Carlo (MCMC) analysis with
10
6
steps. In our initial fits with fixed
i
and
a/R
,wesetthe
uncertainties on individual points in our
HST
white-light time
series equal to the standard deviation of the residuals from our
best-fit solution. We find that the residual permutation tech-
nique results in uncertainties that are two to three times larger
than the corresponding values from both the covariance ma-
trix and MCMC analysis. This is expected, as the noise in our
white-light curves is dominated by time-correlated instrument
effects, while the MCMC and covariance methods implicitly as-
sume random Gaussian-distributed noise (Carter & Winn
2009
).
We plot the two-dimensional probability distributions from the
MCMC analysis and confirm that there are no significant cor-
relations between any of the fit parameters in this case. For
the case where we fit the
Spitzer
and
HST
transits simultane-
ously while allowing
i
and
a/R
to vary as free parameters,
we increase the per-point uncertainties on the
HST
transit light
curves by a factor of two in order to more accurately reflect the
uncertainties contributed by the time-correlated component of
the noise. We selected this scaling factor by requiring that the
uncertainties on
R
p
/R
derived from the covariance matrix be
equal to the uncertainties from the prayer bead method. This
step is critical for the joint fits, because otherwise the
HST
data
have a disproportionate influence on the preferred values of
i
and
a/R
. We find that in this version of the fit the values of the
inclination and a
/
R* show a high degree of correlation, which
is a well-known property of this particular parameterization of
the transit light curve shape. None of the transit parameters in
either version of the fits are correlated with the parameters used
to describe the time-varying instrumental signal.
4
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
Table 1
Transit Parameters from Joint
HST
and
Spitzer
Fits
Parameter
Value
Global values
i
(
◦
)
89.85
±
0.48
a
a/R
26.24
±
1.21
P
(days)
b
9.489264
±
0.000064
T
0
(BJD
TDB
)
b
2456665.46415
±
0.00078
UT 2013 Aug 10 Spitzer transit
c
R
p
/R
0.02778
±
0.00073
T
c
(BJD
TDB
)
2456523.12537
±
0.00049
UT 2013 Dec 19 HST transit
R
p
/R
0.03012
±
0.00087
T
c
(BJD
TDB
)
2456646.48556
±
0.00071
UT 2014 Jan 7 HST transit
R
p
/R
0.03111
±
0.00080
T
c
(BJD
TDB
)
2456665.4584
±
0.0056
Notes.
a
We limited the value of the inclination to be less than or equal to
90
◦
in our fits. We find a 1
σ
lower limit on the inclination of 89
.
◦
65,
and a 2
σ
limit of 88
.
◦
79.
b
We calculate our updated ephemeris using the published transit
times from Dragomir et al. (
2013
) and Van Grootel et al. (
2014
)
and our two new transit measurements.
T
0
is the zero epoch transit
center time from our best-fit ephemeris, and
T
c
is the measured
transit center time from a given observation.
c
We find that for the
Spitzer
data, the uncertainties from the
covariance matrix are larger than those from the prayer bead analysis,
and we report the covariance errors here.
We give our best-fit transit parameters for the joint
HST
and
Spitzer
fits in Table
1
, and plot the normalized transit light curves
after dividing out our best fit for detector effects in Figure
4
.The
white-light curve residuals from UT 2013 December 19 and UT
2014 January 7 have an rms of [73, 68] ppm and [63, 57] ppm,
respectively, where we calculate this number separately for data
from the forward and reverse scan directions. We show the
residuals from these fits in Figure
5
, and the offset between the
observed and calculated transit times from our updated orbital
ephemeris in Figure
6
.
2.1.2. Differential Transit Fits
We estimate the planet’s wavelength-dependent transit depth
by fitting the differential time series for each individual wave-
length element. In this case, we fit the differential time series
for all five orbits rather than the four used in the white-light
fits, as we find that the differential light curves do not show
any detectible systematic trends during the first orbit. The light
curves for the forward and reverse scans are offset by a con-
stant flux value, so we divide each light curve by its median
value before carrying out our fit. We fit each light curve with a
linear function of time and a transit function calculated as the
difference between the transit light curve in that band and the
white-light transit curve, where we fix the values of
i
and
a/R
to their best-fit values from the white-light analysis. We also
tried fits where we allowed an independent linear function of
time for each scan direction, but found that this gave a transmis-
sion spectrum that was indistinguishable from the case where
we assumed the same linear function for both scan directions.
We use the best-fit transit time from the white-light fits, and
fit for the planet–star radius ratio
R
p
/R
corresponding to each
wavelength element. The best-fit transit depths reported here are
simply the square of these values.
-0.15
-0.10
-0.05
-0.00
0.05
0.10
Time from Predicted Transit Center (d)
0.9970
0.9975
0.9980
0.9985
0.9990
0.9995
1.0000
Relative Flux
Figure 4.
Normalized white-light photometry with best-fit detector effects
removed for the UT 2013 December 19 visit (top) and UT 2014 January 7
transit (bottom). Different scan directions for the December visit are indicated
as light (forward scan) and dark (reverse scan) blue filled circles. Scan directions
for the January visit are plotted as yellow (forward scan) and red (reverse scan)
filled circles. The first spacecraft orbit has been trimmed from each visit, leaving
four orbits per transit, and the two light curves have been offset by a relative
flux of 0.0015 for clarity.
(A color version of this figure is available in the online journal.)
-400
-200
0
200
400
-0.15
-0.10
-0.05
-0.00
0.05
0.10
Time from Predicted Transit Center (d)
-400
-200
0
200
Relative Flux (ppm)
Figure 5.
White-light residuals after the best-fit detector and transit light curves
are removed for the UT 2013 December 19 visit (top panel) and UT 2014
January 7 transit (bottom panel). Different scan directions for the December
visit are indicated as light (forward scan) and dark (reverse scan) blue filled
circles. Scan directions for the January visit are plotted as yellow (forward scan)
and red (reverse scan) filled circles.
(A color version of this figure is available in the online journal.)
We calculate the appropriate four-parameter nonlinear limb-
darkening coefficients for each wavelength element using the
same PHOENIX model used for the white-light fits (see Table
2
),
where we convolve the model spectrum at each position on the
star with the same Gaussian function used on our data before
5
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
300
400
500
600
700
Transit Center [BJD-2456000]
-20
-15
-10
-5
0
5
O-C (min)
Figure 6.
Observed minus calculated transit times using our updated ephemeris.
Previously published
MOST
(Dragomir et al.
2013
)and
Spitzer
transit times (Van
Grootel et al.
2014
) are shown as open circles, with our new
HST
observations
are shown as filled circles. The second
HST
transit has a larger uncertainty on
the transit time because it does not include any points during ingress or egress,
which provide the strongest constraints on the measured center of transit time.
fitting for the limb-darkening coefficients. We find that varying
the stellar effective temperature by
±
50 K changes the resulting
transit depths by 1 ppm, which we include in our analysis as a
separate systematic error term.
We estimate uncertainties in the best-fit planet–star ra-
dius ratios for each wavelength element using both a resid-
ual permutation method and the covariance matrix from our
Levenberg–Marquart minimization, where we set the uncer-
tainties on individual data points in a given wavelength channel
equal to the standard deviation of the residuals in that channel
after the best-fit model has been subtracted. We find that the
residual permutation errors for both visits are on average higher
than the corresponding covariance matrix errors, suggesting the
presence of time-correlated noise in our light curves. However,
the residual permutation errors also display a greater variation in
size from one wavelength channel to the next, likely as a result
of sampling error due to the limited number of data points in
each light curve. We therefore take a conservative approach and
select the larger of the two error estimates in each wavelength
channel as our final error estimate, which allows us to account
for time-correlated noise in our data while avoiding possible
under-estimation of uncertainties due to the limited number of
measurements available for the residual permutation analysis.
Next, we bin the resulting wavelength-dependent radius
values by a factor of four to create the transmission spectrum
showninFigure
7
, where we average the estimated radii and
corresponding errors in each of the twenty eight bins. Because
we have smoothed our spectra in wavelength space prior to
fitting for the transit depths in each individual channel, the
transit depths in adjacent wavelength channels are correlated
with each other and the errors on the binned transit depths do
not decrease as the square root of the number of wavelength
channels in the bin. We estimate the effect of this smoothing
on the binned errors by creating a simulated data set with the
same time sampling and number of wavelength channels as our
original data. We add noise to these simulated data and fit them
in exactly the same way as our original data in order to derive a
binned transmission spectrum. We then repeat this analysis with
the same simulated data, but including an additional step where
we smooth the simulated “spectra” using the same Gaussian
function that we apply to our actual spectra. When creating our
binned transmission spectrum, we find that we must average the
Table 2
Binned Four Parameter Nonlinear Limb-darkening Coefficients
a
Wavelength
b
c
1
c
2
c
3
c
4
1.133
0.6247
−
0.4071
0.6890
−
0.3099
1.152
0.6489
−
0.4619
0.7545
−
0.3395
1.171
0.6710
−
0.5019
0.7856
−
0.3530
1.190
0.6925
−
0.5644
0.8445
−
0.3743
1.208
0.7275
−
0.6442
0.9250
−
0.4065
1.227
0.7631
−
0.7246
1.0242
−
0.4507
1.246
0.8020
−
0.8191
1.1222
−
0.4894
1.265
0.8576
−
0.9453
1.2552
−
0.5437
1.284
0.4703
0.1247
0.0695
−
0.0947
1.303
0.5182
−
0.01457
0.2336
−
0.1563
1.321
0.5640
−
0.1255
0.3410
−
0.1967
1.340
0.6366
−
0.2849
0.5109
−
0.2661
1.359
0.7125
−
0.4597
0.6892
−
0.3356
1.378
0.8012
−
0.6639
0.8937
−
0.4138
1.397
0.8842
−
0.8546
1.0907
−
0.4911
1.416
0.8901
−
0.8644
1.1018
−
0.4975
1.434
c
1.0237
−
1.1944
1.4386
−
0.6244
1.453
1.1298
−
1.4436
1.6910
−
0.7202
1.472
1.2482
−
1.7152
1.9657
−
0.8248
1.491
c
0.4779
0.4024
−
0.4201
0.1113
1.510
0.5016
0.3492
−
0.3886
0.1055
1.529
0.5557
0.3381
−
0.4372
0.1307
1.547
0.6129
0.2520
−
0.3962
0.1249
1.566
0.6479
0.1523
−
0.3120
0.1000
1.585
0.6325
0.1655
−
0.3350
0.1121
1.604
0.7095
0.0517
−
0.2727
0.0993
1.623
0.7208
0.0239
−
0.2472
0.0905
1.642
0.7129
0.0421
−
0.2933
0.1144
4.5
d
0.8713
−
1.4634
1.4955
−
0.5581
Notes.
a
The spectra extracted from the UT 2013 December 19 and UT 2014
January 7 visits have slightly different wavelength solutions, and we
therefore created custom limb darkening coefficients for each visit.
A full table of both the binned and unbinned coefficients for both
visits is available upon request.
b
These binned coefficients are only used in the white-light residual
fitting analysis. In the spectral template fitting analysis we fit the light
curves for individual wavelength elements and use limb-darkening
coefficients calculated appropriately for this resolution (4
×
higher
than shown here).
c
There is a well-known degeneracy in the values of the four-
parameter nonlinear limb-darkening coefficients; although the values
of the coefficients do not vary smoothly, we find that the correspond-
ing stellar intensity profiles are continuous across all wavelengths
considered here.
d
Limb-darkening coefficients calculated for the 4.5
μ
m
Spitzer
IRAC bandpass, which we use to fit the transit light curve from
Van Grootel et al. (
2014
).
four individual error estimates from the smoothed version of
the data and multiply this average by a factor of 1.22 in order
to obtain the same binned uncertainties as in the unsmoothed
version. We adopt this same scaling when calculating the errors
on the binned transit depth estimates from our real data.
We take the error-weighted average of the two individual
transmission spectra to create the combined spectrum shown in
Figure
8
and in Table
3
. We present two separate estimates for the
measurement uncertainties in Table
3
, including one calculated
using the covariance matrix errors alone and the other taking the
larger of the two error estimates as previously described. As an
additional consistency check, we also carried out a version of our
analysis in which we allowed the light curves derived from the
forward and reverse scans to have different planet–star radius
6
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
1.1
1.2
1.3
1.4
1.5
1.6
Wavelength (micron)
-100
-50
0
50
100
150
Differential Transit Depth (ppm)
Figure 7.
Wavelength-dependent relative transit depths measured for the UT
2013 December 19 (blue filled circles) and UT 2014 January 7 (red filled circles)
visits. The white-light transit depths have been subtracted from each visit.
(A color version of this figure is available in the online journal.)
1.1
1.2
1.3
1.4
1.5
1.6
Wavelength (micron)
850
900
950
1000
Normalized Transit Depth (ppm)
Figure 8.
Wavelength-dependent transit depths averaged over the two visits,
where the depths are defined as the square of the planet–star radius ratio
R
p
/R
in each band. Depths derived using the spectral template fitting technique
(Deming et al.
2013
; Knutson et al.
2014
) are shown as filled circles, and depths
from the white-light residual fitting technique (Kreidberg et al.
2014
)areshown
as open circles. No offset has been applied to either data set, demonstrating that
the average transit depths are also in good agreement.
ratios. We found that we obtained consistent radius estimates
from both scan directions, indicating that the different behaviors
visible in the white-light curves are effectively removed in the
differential time series by our template fitting technique.
2.2. White- light Residual Fitting Method
We also obtained an independent estimate of the transmission
spectrum following the approach of Kreidberg et al. (
2014
)
and Stevenson et al. (
2014
). Beginning with the raw images,
this analysis used an entirely different set of codes than those
described in the previous sections. In this version of the analysis,
we treated each up-the-ramp sample in the ima.fits images
as an independent subexposure. For each subexposure, we
interpolated all rows to a common wavelength scale to account
for the changing dispersion solution with spatial position on the
detector. We estimated the background by making conservative
masks around the stellar spectra and taking the median of the
unmasked pixels. We subtracted the background and optimally
extracted the spectra. To combine the data from individual
Table 3
Wavelength Dependent Transit Depths from Template Fitting Method
Wavelength
Depth
White Noise Error
a
Total Error
a
(
μ
m)
(ppm)
(ppm)
(ppm)
1.132
912
20
27
1.151
896
20
34
1.170
916
20
22
1.188
917
19
19
1.207
913
20
20
1.226
897
19
20
1.245
898
18
19
1.264
934
18
18
1.283
948
18
19
1.301
952
19
20
1.320
952
19
21
1.339
936
18
25
1.358
933
18
25
1.377
925
19
29
1.396
914
19
20
1.415
922
19
32
1.433
981
20
27
1.452
946
19
25
1.471
956
18
22
1.490
945
18
23
1.509
964
20
27
1.528
965
20
21
1.546
947
19
20
1.565
946
21
28
1.584
963
21
23
1.603
973
20
22
1.622
968
20
23
1.641
953
24
30
Notes.
a
White noise measurement errors are estimated using
the covariance matrix, which implicitly assumes that individual
measurement errors are Gaussian distributed and uncorrelated. The
total measurement errors are calculated by comparing the covariance
errors to a residual permutation method that better accounts for time-
correlated noise and taking the larger of the two in each wavelength
channel. Uncertainty in the limb-darkening models from the stellar
effective temperature contributes an additional error of 1 ppm in each
bin, which we do not include here.
subexposures, we summed the spectra by column. The final
step in the reduction process is to correct for drift of the
spectra in the dispersion direction over the course of a visit.
We used the first exposure from the first visit as a template
and shifted all subsequent spectra to the template wavelength
scale. The spectra shifted by a total of 0.3 pixels over the five
orbits contained in our observations, which is larger than the
approximately 0.01 pixel drift observed in previous scanning
mode observations of GJ 1214b (Kreidberg et al.
2014
). This
increased drift may be related to the longer scan length and
faster scan rate utilized for these observations as compared to
GJ 1214b.
We binned the spectra in four-pixel-wide channels, yielding
26 spectrophotometric light curves between 1.15 and 1.61
μ
m.
The light curves show orbit-long ramp-like systematics that are
characteristic of WFC3 data. We correct for these systematics
using the divide-white technique, which assumes that the
observed effects have the same shape across all wavelengths.
We fit each spectroscopic light curve with a transit model
multiplied by a scaled vector of systematics from the best-fit
white-light curve and a linear function of time. We fix
i
and
a/R
to the values reported in Table
1
. The fit to each channel
7
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
Table 4
Wavelength Dependent Transit Depths from White-light Residual Method
Wavelength
Depth
White Error
a
Total Error
a
Reduced
χ
2
(
μ
m)
(ppm)
(ppm)
1.145
909
20
23
0.81
1.163
940
19
24
0.77
1.182
896
19
23
0.93
1.200
928
19
22
0.77
1.218
910
19
26
0.82
1.237
895
18
24
0.82
1.255
922
18
20
0.76
1.274
936
19
23
0.88
1.292
944
18
30
0.83
1.311
970
18
29
1.04
1.329
933
18
21
0.87
1.348
922
18
26
0.80
1.366
929
18
26
0.81
1.384
922
18
28
0.83
1.403
915
18
18
0.89
1.421
983
18
18
0.84
1.440
978
18
19
0.85
1.458
960
18
21
0.75
1.477
936
19
26
0.68
1.495
924
19
26
0.89
1.513
962
19
25
0.91
1.532
941
19
29
0.89
1.550
942
19
32
0.97
1.569
960
20
42
0.95
1.587
970
20
26
0.87
1.606
1001
20
29
0.87
Notes.
a
White noise measurement errors are estimated using a Markov
Chain Monte Carlo analysis, which implicitly assumes that individual mea-
surement errors are Gaussian distributed and uncorrelated. Total measure-
ment errors are calculated by comparing the MCMC errors to a residual
permutation method that better accounts for time-correlated noise and tak-
ing the larger of the two in each wavelength bin.
has six free parameters: one transit depth, four scaling factors
for the systematics (one for each visit and scan direction), and
the linear slope. As before, we calculate the four-parameter
nonlinear limb-darkening coefficients using a
PHOENIX
stellar
atmosphere model where we take the flux-weighted average of
the theoretical stellar intensity profile within each photometric
bandpass. We set the uncertainties on individual points equal to
the sum of the photon noise and read noise in quadrature. We
list the best-fit values and corresponding errors in Table
4
.
We report uncertainties on the transit depths corresponding
to 1
σ
confidence intervals from either an MCMC fit, which
implicitly assumes white (Gaussian and uncorrelated) noise,
or a residual permutation analysis that better accounts for any
time-correlated noise present in the data. We take the larger of
the two errors in each wavelength bin as our final uncertainties
and provide the MCMC only errors separately in Table
4
for
comparison. Our residual permutation errors are on average
30% larger than those obtained with MCMC. This suggests that
there is some time-correlated noise in the light curves, which
is most likely the result of imperfect corrections for visit- and
orbit-long systematic trends in the data.
3. DISCUSSION AND CONCLUSIONS
We used the WFC3 instrument on the
HST
to observe
transits of the super-Earth HD 97658b at wavelengths between
1
.
1–1
.
7
μ
m. Our white-light transit fits produce consistent
1.1
1.2
1.3
1.4
1.5
1.6
Wavelength (micron)
15
20
25
30
35
Transit Depth Uncertainty (ppm)
Figure 9.
Comparison of uncertainties on the reported transit depths under
the assumption of either white, Gaussian noise (open circles) or allowing for
time-correlated noise using the residual permutation method for estimating
uncertainties (filled circles). Errors derived using the spectral template fitting
technique (Deming et al.
2013
; Knutson et al.
2014
) are shown as blue circles,
and depths from the white-light residual fitting technique (Kreidberg et al.
2014
)
are shown as red circles.
(A color version of this figure is available in the online journal.)
estimates of the transit times and planet–star radius ratios
between the two visits. Our errors for these parameters are
dominated by systematic noise, as illustrated in Figures
1
–
3
.We
combine our measured transit times with previously published
values from
MOST
and
Spitzer
and show that they are consistent
with a linear ephemeris. We also derive an improved estimate
for the planet’s orbital period with an uncertainty that is a factor
of 30 smaller than in Van Grootel et al. (
2014
). Our average
radius ratio is 2
.
5
σ
larger than the
Spitzer
4.5
μ
m measurement
from Van Grootel et al. (
2014
), and is in good agreement with
the
MOST
visible-light value from Dragomir et al. (
2013
). It is
unlikely that this offset is due to stellar activity as this star has a
log(
R
HK
)value between
−
4
.
95 and
−
5
.
00 (Howard et al.
2011
)
and varied by less than 0
.
2% in brightness over a single 1.5 day
visible-light observation by
MOST
(Dragomir et al.
2013
).
We obtain nearly identical versions of the transmission spec-
trum using the spectral template fitting technique (Deming et al.
2013
; Knutson et al.
2014
) and the white-light residual method
(Kreidberg et al.
2014
), as shown in Figure
8
. Our median
uncertainties on the differential wavelength-dependent transit
depth are 23 ppm from the spectral template fitting and 26 ppm
from the white-light residual method (see Figure
9
), making
these observations the most precise measurement of a planetary
transmission spectrum that we are aware of with WFC3 to date.
Both versions of our spectrum display a discontinuity around
1.42
μ
m, which may be related to a slight upward slope visible
toward redder wavelengths. These features could be instrumen-
tal in nature, as the columns that make up the wavelength bins
closest to the discontinuity are located on a boundary between
quadrants on the WFC3 array. The statistical significance of
these features is marginal, and we therefore do not consider
them in our subsequent comparison to atmosphere models for
this planet.
In Figure
10
, we compare our measured transmission spec-
trum from the spectral template fitting technique to several
representative atmosphere models for HD 97658b, which are
calculated following Kempton et al. (
2012
). The effective tem-
peratures (720–730 K, depending on the model) and corre-
sponding pressure-temperature profiles of these models are cal-
culated assuming full redistribution of energy to the planet’s
night side and an albedo which varies according to composition.
8
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
1.1
1.2
1.3
1.4
1.5
1.6
Wavelength (micron)
850
900
950
1000
Normalized Transit Depth (ppm)
Figure 10.
Wavelength-dependent transit depths averaged over the two visits
(black filled circles). Four different atmosphere models are shown for compari-
son: a solar metallicity model (red solid line), a 50
×
solar metallicity model (red
dashed line), a pure water model (solid blue line), and a solar metallicity model
with an opaque cloud deck at 1 mbar (solid green line). The average depth for
each model has been normalized to match the average measured transit depth
in this plot.
(A color version of this figure is available in the online journal.)
We consider both cloud-free models with solar and 50
×
solar
metallicities, as well as a pure water model and a solar metal-
licity model with a high cloud deck located at one mbar. We
calculate the significance with which our data can rule out a
given model using the equation from Gregory (
2005
):
Significance
=
χ
2
−
ν
√
2
ν
,
(3)
where
χ
2
is calculated by comparing our averaged transmission
spectrum to each model and
ν
is the number of degrees of
freedom in the fit (27 in our case, as there are 28 points and
we normalize the models to match the average measured transit
depth of our data). This metric assumes that our measurement
errors are Gaussian and uncorrelated from one wavelength bin
to the next; although this is almost certainly untrue at some
level, it represents a reasonable starting point for comparing
different models. Following this approach we find that our
measured transmission spectrum is inconsistent with the solar
and 50
×
solar cloud-free models at the 10
σ
and 9
σ
levels,
respectively. It is equally well described by the water-dominated
(0
.
6
σ
) model and the solar metallicity model with optically
thick clouds at a pressure of one mbar (0
.
6
σ
), as well as a flat
line at the average transit depth across the band (0
.
4
σ
). We
find that a solar metallicity model with clouds at 1 mbar is
also consistent with the data (1
.
8
σ
), indicating that the clouds
could be located slightly deeper in the atmosphere. We note that
there are any number of high metallicity atmosphere models
that could provide a fit comparable to that of the pure water
model; all our data appear to require is either a relatively metal-
rich atmosphere with a correspondingly small-scale height, or
the presence of a high cloud deck that obscures the expected
water absorption feature in a hydrogen-dominated atmosphere.
We constrain the maximum hydrogen content of the atmosphere
in the first scenario by considering a series of cloud-free models
with varying number fractions of molecular hydrogen and water,
and find that in this scenario the atmosphere has to be at least
20% water by number in order to be consistent with our data at
the 3
σ
level. We list the
χ
2
values and level of disagreement for
all models in Table
5
.
Table 5
Atmosphere Model Comparison
Model
χ
2
Level of Disagreement
a
Flat line
29.9
0.4
σ
Solar metallicity
102.9
10.3
σ
50
×
solar
92.2
8.9
σ
Solar with cloud deck at 1 mbar
31.2
0.6
σ
Solar with cloud deck at 10 mbar
40.4
1.8
σ
10% H
2
O
53.6
3.6
σ
20% H
2
O
42.2
2.1
σ
30% H
2
O
37.5
1.4
σ
40% H
2
O
35.1
1.1
σ
100% H
2
O
31.1
0.6
σ
Notes.
a
This is the significance with which we can rule out a given model,
calculated according to Equation (
3
). Significance levels less than 3
σ
mean that
the data are consistent with that model within the reported errors.
The conclusion that HD 97658b’s transmission spectrum ap-
pears to be flat at the precision of our data places it in the same
category as both the super-Earth GJ 1214b (Kreidberg et al.
2014
) and the Neptune-mass GJ 436b (Knutson et al.
2014
).
As with these two planets, a more precise measurement of
HD 97658b’s transmission spectrum will eventually allow us
to distinguish between high clouds and a cloud-free, metal-rich
atmosphere. Our constraints on the atmospheric scale height in
the cloud-free scenario are relatively weak compared to those
obtained for GJ 1214b and GJ 436b, despite the fact that we
achieve smaller errors (20 ppm versus 30 ppm) in our measure-
ment of the differential transmission spectrum. This is primarily
because HD 97658b has a smaller planet–star radius ratio than
either of these systems, and the predicted amplitude of the trans-
mission spectrum is correspondingly small. Fortunately, it also
orbits a brighter star than either GJ 1214b or GJ 436b, making
it possible to achieve high precision transit measurements with
relatively few observations. Unfortunately, this makes ground-
based observations particularly challenging as the nearest com-
parison star with a comparable brightness is located more than
40
away. For space telescopes such as
Hubble
and
Spitzer
,
achieving the precision required to study this planet in detail
will mean pushing the systematic noise floor to unprecedent-
edly low levels. There is every reason to believe that this level
of performance should be achievable, and given the unique na-
ture of this planet it is likely that this will be put to the test in
the near future.
We are grateful to STScI Director Matt Mountain and the
Director’s Discretionary Time program for making it possible
to obtain these observations prior to the next regular
HST
pro-
posal cycle. We also thank Beth Perriello and the rest of the
HST
scheduling team, as they worked overtime in the days leading
up to the Christmas holiday in order to make sure that we ob-
tained the best possible data from this program. We also wish
to acknowledge Valerie Van Grootel, who was kind enough to
share an advance copy of her paper prior to publication. H.K.
acknowledges support from NASA through grant GO-13501.
L.K. received funding for this work from the National Science
Foundation in the form of a Graduate Research Fellowship.
J.L.B. acknowledges support from the Alfred P. Sloan Founda-
tion and NASA through grants NNX13AJ16G, GO-13021, and
GO-13467. D.H. has been supported through the European Re-
search Council advanced grant PEPS awarded to Gilles Chabrier
9
The Astrophysical Journal
, 794:155 (10pp), 2014 October 20
Knutson et al.
under the European Community’s Seventh Framework Program
(FP7
/
2007-2013 grant agreement No. 247060).
REFERENCES
Allard, F., Homeier, D., & Freytag, B. 2012,
RSPTA
,
370, 2765
Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2013,
ApJS
,
204, 24
Bean, J. L., D
́
esert, J.-M., Kabath, P., et al. 2011,
ApJ
,
743, 92
Bean, J. L., Miller-Ricci Kempton, E., & Homeier, D. 2010,
Natur
,
468, 669
Benneke, B., & Seager, S. 2012,
ApJ
,
753, 100
Benneke, B., & Seager, S. 2013,
ApJ
,
778, 153
Berta, Z. K., Charbonneau, D., D
́
esert, J.-M., et al. 2012,
ApJ
,
747, 35
Burke, C. J., Bryson, S. T., Mullally, F., et al. 2014,
ApJS
,
210, 19
Carter, J. A., & Winn, J. N. 2009,
ApJ
,
704, 51
Charbonneau, D., Berta, Z. K., Irwin, J., et al. 2009,
Natur
,
462, 891
Chiang, E., & Laughlin, G. 2013,
MNRAS
,
431, 3444
Claret, A. 2000, A&A,
363, 1081
Deming, D., Wilkins, A., McCullough, P., et al. 2013,
ApJ
,
774, 95
Demory, B.-O., Gillon, M., Deming, D., et al. 2011,
A&A
,
533, A114
D
́
esert, J.-M., Bean, J., Miller-Ricci Kempton, E., et al. 2011,
ApJL
,
731, L40
Dragomir, D., Matthews, J. M., Eastman, J. D., et al. 2013,
ApJL
,
772, L2
Eastman, J., Siverd, R., & Gaudi, B. S. 2010,
PASP
,
122, 935
Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2012, ApJ, in press
(arXiv:
1202.6328
)
Fortney, J. J., Mordasini, C., Nettelmann, N., et al. 2013,
ApJ
,
775, 80
Fressin, F., Torres, G., Charbonneau, D., et al. 2013,
ApJ
,
766, 81
Gregory, P. (ed.) 2005, in Bayesian Logical Data Analysis for the Physical
Sciences (Cambridge: Cambridge Univ. Press),
167
Hadden, S., & Lithwick, Y. 2014,
ApJ
,
787, 80
Hansen, B. M. S., & Murray, N. 2012,
ApJ
,
751, 158
Heng, K., & Kopparla, P. 2012,
ApJ
,
754, 60
Howard, A. W., Johnson, J. A., Marcy, G. W., et al. 2011,
ApJ
,
730, 10
Howard, A. W., Marcy, G. W., Bryson, S. T., et al. 2012,
ApJS
,
201, 15
Hu, R., & Seager, S. 2014,
ApJ
,
784, 63
Kempton, E. M.-R., Zahnle, K., & Fortney, J. J. 2012,
ApJ
,
745, 3
Knutson, H. A., Benneke, B., Deming, D., & Homeier, D. 2014,
Natur
,
505, 66
Kreidberg,L.,Bean,J.L.,D
́
esert, J.-M., et al. 2014,
Natur
,
505, 69
Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011,
ApJS
,
197, 8
Lithwick, Y., Xie, J., & Wu, Y. 2012,
ApJ
,
761, 122
Lopez, E. D., & Fortney, J. J. 2013,
ApJ
,
776, 2
Lopez, E. D., Fortney, J. J., & Miller, N. 2012,
ApJ
,
761, 59
Mandel, K., & Agol, E. 2002,
ApJL
,
580, L171
Mandell, A. M., Haynes, K., Sinukoff, E., et al. 2013,
ApJ
,
779, 128
Marcy, G. W., Isaacson, H., Howard, A. W., et al. 2014,
ApJS
,
210, 20
McCullough, P. R., & MacKenty, J. 2012, Instrument Science Report WFC3
2012-08 (Baltimore, MD: Space Telescope Science Institute)
McCullough, P. R., et al. 2014, ApJ, in press (arXiv:
1407.2462
)
Miller-Ricci, E., Seager, S., & Sasselov, D. 2008,
ApJ
,
690, 1056
Morley, C. V., Fortney, J. J., Kempton, E. M.-R., et al. 2013,
ApJ
,
775, 33
Mortier, A., Santos, N. C., Sousa, S. G., et al. 2013,
A&A
,
558, A106
Nettelmann, N., Fortney, J. J., Kramm, U., & Redmer, R. 2011,
ApJ
,
733, 2
Petigura, E. A., Marcy, G. W., & Howard, A. W. 2013,
PNAS
,
110, 19273
Raymond, S. N., Barnes, R., & Mandell, A. M. 2008,
MNRAS
,
384, 663
Raymond, S. N., & Cossou, C. 2014,
MNRAS
,
440, L11
Rogers, L. A., Bodenheimer, P., Lissauer, J. J., & Seager, S. 2011,
ApJ
,
738, 59
Rogers, L. A., & Seager, S. 2010a,
ApJ
,
712, 974
Rogers, L. A., & Seager, S. 2010b,
ApJ
,
716, 1208
Seager, S., Kuchner, M., Hier-Majumder, C. A., & Militzer, B. 2007,
ApJ
,
669, 1279
Steffen, J. H., Fabrycky, D. C., Agol, E., et al. 2013,
MNRAS
,
428, 1077
Stevenson, K. B., Bean, J. L., Seifahrt, A., et al. 2014,
AJ
,
147, 161
Valencia, D., Guillot, T., Parmentier, V., & Freedman, R. S. 2013,
ApJ
,
775, 10
Valencia, D., Sasselov, D. D., & O’Connell, R. J. 2007,
ApJ
,
665, 1413
Van Grootel, V., Gillon, M., Valencia, D., et al. 2014,
ApJ
,
786, 2
Wakeford, H. R., Sing, D. K., Deming, D., et al. 2013,
MNRAS
,
435, 3481
Weiss, L. M., & Marcy, G. W. 2014,
ApJL
,
783, L6
Wilkins, A. N., Deming, D., Madhusudhan, N., et al. 2014,
ApJ
,
783, 113
Winn, J. N., Matthews, J. M., Dawson, R. I., et al. 2011,
ApJL
,
737, L18
Zeng, L., & Sasselov, D. 2013,
PASP
,
125, 227
10