The Transit Light Curve Project. XIII. Sixteen Transits of the Super-Earth GJ 1214b

1.2 MB·English

by  Joshua A. Carter

Draftversion January28,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THE TRANSIT LIGHT CURVE PROJECT.XIII. SIXTEEN TRANSITS OF THE SUPER-EARTH GJ 1214b
Joshua A. Carter1,2,4, Joshua N. Winn1, Matthew J. Holman2, Daniel Fabrycky2,3,4,
Zachory K. Berta2, Christopher J. Burke2, Philip Nutzman2,3
Draft version January 28, 2011
ABSTRACT
We present optical photometry of 16 transits of the super-Earth GJ 1214b, allowing us to refine
1 the system parameters and search for additional planets via transit timing. Starspot-crossing events
1 are detected in two light curves, and the star is found to be variable by a few percent. Hence, in
0 our analysis, special attention is given to systematic errors that result from star spots. The planet-
2 to-star radius ratio is 0.11610 0.00048, subject to a possible upward bias by a few percent due to
the unknown spot coverage. E±ven assuming this bias to be negligible, the mean density of planet
n
can be either 3.03 0.50 g cm−3 or 1.89 0.33 g cm−3, depending on whether the stellar radius is
a
± ±
J estimated from evolutionary models, or from an empirical mass-luminosity relation combined with
the light curve parameters. One possible resolution is that the orbit is eccentric (e 0.14), which
7 ≈
would favor the higher density, and hence a much thinner atmosphere for the planet. The transit
2
times were found to be periodic within about 15 s, ruling out the existence of any other super-Earths

] with periods within a factor-of-two of the known planet.
P Subject headings: stars: planetary systems — planets and satellites: individual (GJ 1214b) — stars:
E individual (GJ 1214) — techniques: photometric
.
h
p
1. INTRODUCTION in compact arrangements (Lo Curto et al. 2010), and it
-
o The recently discovered planet GJ 1214b (Charbon- wouldbeinterestingtoknowifGJ1214bisanothersuch
r neau et al. 2009) is the smallest known exoplanet for example.
st which the mass, radius, and atmospheric properties are This paper is organizedas follows. Section 2 describes
a the observations and data reduction. Section 3 presents
allpossibleto studywithcurrenttechnology. Itis there-
[ the light curve model, taking into account the effects of
fore a keystone object in the theory of planetary inte-
starspots. Section4discussesthemethodbywhichwees-
2 riors and atmospheres, and has been welcomed as the
timatedthemodelparametersandtheirconfidenceinter-
v harbinger of the “era of super-Earths” (Rogers & Sea-
vals. Section5discussestheresultsfortheplanet-to-star
6 ger 2010).
radiusratio. Section6presentstwodifferentmethodsfor
7 Theplanet’sdiscoverersestimatedthemassandradius
3 ofGJ1214btobe6.55 0.98M⊕and2.68 0.13R⊕,giv- determining the stellar radius (and hence the planetary
0 ingameandensityof1±.87 0.40gcm−3(C±harbonneauet radius), which give discrepant results. Some possible
2. al. 2009). This is such a l±ow density that it would seem resolutions of this discrepancy are discussed. Section 7
presents our analysis of the measured transit times, and
1 impossible for the planet to be solid with only a thin,
constraints on the properties of a hypothetical second
0 terrestrial-like atmosphere. Rather, it seems necessary
planet. Finally, in Section 8, we discuss the implications
1 to have a thick gaseous atmosphere, probably composed
of our analysis on the understanding of GJ 1214b and
: ofhydrogenandheliumbutpossiblyalsoofcarbondiox-
v more broadly on M dwarf transit hosts.
ide or water (Charbonneau et al. 2009, Rogers & Seager
i
X 2010, Miller-Ricci & Fortney 2010). 2. OBSERVATIONSANDDATAREDUCTION
In this paper, we report on observations of additional
r Our data were gathered during the 2009 and 2010 ob-
a transits of GJ 1214b. As in other papers in the Transit
serving seasons. Thirteen transits were observed with
Light Curve (TLC) series (Holman et al. 2006, Winn et
the 1.2m telescope at the Fred L. Whipple Observatory
al. 2007), one of our goals was to refine the basic sys-
(FLWO) on Mount Hopkins, Arizona, using Keplercam
tem parameters, and thereby allow for more powerful
and a Sloan z′ filter. The blue end of the bandpass was
discriminationamongmodelsoftheplanet’sinteriorand
set by the filter (approx. 0.85 µm) and the red end by
atmosphere. Another goal was to check for any non-
the quantumefficiency ofthe CCD (ranging fromnearly
periodicity in the transit times, as a means of discover-
100% at 0.75 µm to 10% at 1.05 µm). The first two of
ing other planets in the system, through the method of
the FLWO transits were already presented by Charbon-
Holman&Murray(2005)andAgoletal.(2005). Super-
neau et al. (2009); those data have been reanalyzed for
earthshavefrequentlybeenfoundinpairsoreventriples
thiswork. Anotherthreetransitswereobservedwiththe
1DepartmentofPhysics,andKavliInstituteforAstrophysics 6.5m Magellan (Baade) telescope at the Las Campanas
and Space Research, Massachusetts Institute of Technology, ObservatoryinChile, withthe MagIC andIMACS cam-
Cambridge,MA02139 eras and a Sloan r′ filter (approx. 0.55-0.70 µm)
2Harvard-Smithsonian Center for Astrophysics, 60 Garden We used standard procedures for bias subtraction and
St.,Cambridge,MA02138
3Department of Astronomy and Astrophysics, University of flat-fielddivision. Circularaperturephotometrywasper-
California,SantaCruz,CA95064 formedforGJ1214andseveralcomparisonstars(5-6for
4Hubblefellow the FLWO images, and 2 for the Magellanimages). Cir-
2 Carter et al. 2010
cular annuli centered on each star were used to estimate in the usual way. In this work we use the formulas of
the background level. The flux of GJ 1214 was divided Mandel & Agol (2002) for a quadratic limb-darkening
by the sum of the fluxes of comparisonstars, to produce law. Thus, our model adds one new parameter ǫ specific
a relative flux time series for GJ 1214. The radii of the to each transit. A similar model was used for Corot-2b
apertures and annuli were varied in order to seek those by Czesla et al. (2009).
values that minimize the scatter in the residuals, while With infinite photometric precision, ǫ could be deter-
also producing a minimal level of time-correlated noise, minedfromasingletransitlightcurvebasedontheslight
as determined by the wavelet-basedalgorithm described differences in the ingress or egress data as compared to
by Carter & Winn (2009). Because the amplitude of the a spot-free model. However,the differences are typically
correlated component was always . 1% of that of the nogreaterthana few partsper million, andwillbe diffi-
white component, in what follows we assume the noise cult to detect in practice (see the right panel of Fig. 3).
tobe uncorrelatedintime. Eachmeasurementwasasso- Consequently,thereisastrongdegeneracybetweenǫand
ciatedwitha time stampgivenby the midexposuretime the planet-to-star radius ratio R /R , in the sense that
p ⋆
expressed in the BJD system (Eastman et al. 2010). increasing either parameter results in a deeper transit
TDB
Fig. 1 shows the light curves, after correcting for dif- with no other observable changes. If the spots produce
ferential airmass extinction as described in 4. Fig. 2 an overall darkening (ǫ 0) then the measured transit
§ ≥
gives a clearer view of two of the Magellan light curves depth δ (corrected to remove the effects of limb darken-
(epochs 5 and 260), which show evidence for anomalous ing) can only be used to place an upper bound on the
brightening events during the transit. The data points radius ratio:
correspondingtotheseeventsareshownwithunfilledcir-
R
cles, rather than gray circles. Correlations were sought p = δ(1 ǫ) √δ. (3)
between the flux observed during these events, and var- R⋆ − ≤
ious image statistics such as the median X and Y pixel p
If the spot coverage is time-variable, then the transit
position, the shape parameters of the point spreadfunc-
depth will vary fromevent to event. One may hope that
tion, and the background level, but none were found.
the shallowest observed transit corresponds to a time
We interpret each of those events as the passage of the
when the visible disk was nearly free of spots, thereby
planet in front of a dark starspot, as has been observed
allowing the planet-to-star radius ratio to be measured
by Rabus etal.(2009)andothersfor differenttransiting
with minimal bias. However, there is no guarantee that
systems.
the spot coverage ever falls to zero.
3. TRANSITLIGHT CURVEMODEL
3.2. Spot crossings during transit
The evidence for starspots on GJ 1214 caused us to
make a few changes to the usual models for transit pho- When the planet transits a cool spot, the fractional
tometry. Coolstarspotson the visible hemisphere of the loss of light temporarily decreases, and a brightening is
star can produce two different effects: when they are on observed in the light curve. The duration and ampli-
the transit chord, they cause brightening events in the tude of the brightening event depends on the size of the
light curve such as those we observed; and when they spot and the intensity contrast with the unspotted stel-
are away from the transit chord, they reduce the over- lar photosphere. If the spot is approximately the same
all brightness of the star and increase the fractional loss size as the planet, and is completely occulted, then the
of light due to the planet. The latter effect causes the brightening event will have a duration of approximately
planettoappearlargerthanitisinreality,byproducing twice the transit ingress duration. For a circular spot,
a deeper transit. the amplitude of the brightening event is
3.1. Loss of light due to starspots Rs 2 s
δ = 1 I (4)
spot
During a transit, the flux received from a spotted star (cid:18)R⋆(cid:19) (cid:18) − I⋆(cid:19)
can be written
whereR and aretheradiusandmeanintensityofthe
s s
I
F(t)=F [1 ǫ(t)] ∆F(t), (1) spot, and is the mean intensity that the photosphere
0 − − would havIe⋆at that location in the absence of spots.
where F is the flux that would be received from an
0 Rather than modeling the spots (e.g., Wolter et al.
unspottedanduntransitedstar,ǫ(t)isthefractionalloss
2009)andfittingforthebrighteningevents,wechosethe
of light due to starspots, and ∆F(t) is the flux that is
simple approachofidentifying the brighteningeventsvi-
blocked by the planet. The factor ǫ(t) changes on the
sually,andassigningthosedatazeroweightinthefitting
timescaleofthestellarrotationperiod( months),while
process. For the Magellan data, the identification could
∼
∆F(t)changesonthemuchshortertimescaleofthetran-
beperformedwithoutmuchambiguity,andthe“excised”
sit ( hours).
dataareshowninFigs.1and2. FortheFLWOdata,no
∼
Itisconvenienttonormalizethelightcurvebytheflux
brighteningeventswereobvioustotheeye. Thisisprob-
measured just outside of the transit,
ably due to a combination of the lower precision of the
1 ∆F(t) FLWO data, and the weaker contrast between the spots
f(t) 1 , (2) and the photosphere at the longer wavelengths of the
≡ − 1−ǫ F0 FLWO observations (z′ band as compared to r′ band).
where ǫ is taken here to be a constant for the duration In particular, the anomalies we observed at epochs
of the transit. The quantity ∆F(t)/F is the fractional 5 and 260 have a duration of approximately twice the
0
loss of light due to the transit, which can be calculated ingressduration,andthereforethetypicalspotsarelikely
Sixteen Transits of GJ 1214b 3
x
u 1.00
Fl
e
ativ 0.99
el
R
0.98 Epoch 0 Epoch 2 Epoch 5 Epoch 14
−Cpt] 05
O[p −5

x
u 1.00
Fl
e
ativ 0.99
el
R
0.98 Epoch 183 Epoch 195 Epoch 200 Epoch 212
−Cpt] 05
O[p −5

x
u 1.00
Fl
e
ativ 0.99
el
R
0.98 Epoch 212 Epoch 214 Epoch 236 Epoch 238
−Cpt] 05
O[p −5

x
u 1.00
Fl
e
ativ 0.99
el
R
0.98 Epoch 243 Epoch 250 Epoch 255 Epoch 260
−Cpt] 05
O[p −5
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
Time since midtransit [hr]
Fig. 1.—GJ1214transitlightcurvephotometry. Eachpanel showstransitlightdata(after correctingforairmassvariation), thebest-
fitting model,andthe residuals(observed −calculated, inparts perthousand). Unfilledcirclesrepresentthe suspected “spot anomalies”
that were assigned zero weight in the fitting process. The data from epochs 5, 212, and 260 are based on r′-band observations with the
MagellanClay6.5mtelescope. Theotherdataarebasedonz′-bandobservationswiththeFLWO1.2mtelescope.
tobecomparableinsizetotheplanet. Theobservedam- 3.3. Using transit durations to estimate the radius ratio
plitudesof2–3ppt(partsperthousand)giveaconstraint
Asdiscussed,theeffectofdarkspotsonthevisibledisk
on the spot size and the intensity contrast. The lowest
ofthestaristoincreasethefractionallossoflightduring
possibleintensitycontrastisobtainedforthelargestpos-
thetransit,causingtheplanettoappearlargerthanitis
sible spots, with R R . In that case, assuming the
s p in reality. This may be counteracted to some degree by
≈
unspotted photosphere and the spots to be described
the “filling in” of the transit due to spot occultations, if
by blackbody spectra, the spot temperature would be
those events cannot be recognized and excised from the
T 2900 K as compared to the photospheric temper-
s data. Thiscausedustowonderwhatonecouldstilllearn
≈
ature of 3026 K (Charbonneau et al. 2009). Had we
about the planet if the transit depth cannot be trusted.
observedthose sameevents in the z′ band, the brighten-
The transit light curve provides three primary observ-
ing amplitude wouldhave been 1.5 ppt, just below the
ables: thetotaltransitdurationT ,theingressoregress
1σ level of the noise in each da≈ta point. Smaller spots tot
durationτ,andthedepthδ. Ifonealsoknowstheorbital
would need to have a greater intensity contrast, leading
period P, eccentricity e, and argument of pericenter ω,
to a greater temperature difference and an even smaller
then one can translate the transit observables into three
predicted amplitude in the z′ band.
parameters more directly related to the star-planet sys-
In this light it is not surprising that we did not detect
tem. One possible choice of three “physical” parameters
similar starspot events in our z′-band data. In what fol-
is the planet-to-star radius ratio R /R , the orbital in-
lows we assume that the z′-band data are unaffected by p ⋆
clination i, and the mean stellar density ρ (see, e.g.,
spot-crossingevents. Ofcourseitispossiblethatsmaller ⋆
Carter et al. 2008 or Winn 2010). Usually, the value of
or less luminous starspots were transited during our ob-
ρ derived in this way is used to improve the character-
servations, and that the transit depths are “filled in” to ⋆
ization of the host star (see, e.g., Sozzetti et al. 2007,
somedegreebybrighteningeventsthatcannotbeidenti-
Holman et al. 2007). However, if one is willing to adopt
fied visually in the light curves. However, the reasoning
avalueforρ basedonexternalinformation(suchasthe
in the preceding paragraph suggests that this effect is ⋆
star’s observed luminosity and spectrum), then the two
minor. Furthermore, we did not find any evidence for
observables T and τ are sufficient to specify the other
time correlated noise in our model residuals. tot
4 Carter et al. 2010
x
u 1.00
Fl
e
v
ati 0.99
el
R
0.98 Epoch 5 Epoch 260

2
−Cpt] 0
Op
[
−2
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
Time since midtransit [hr] Time since midtransit [hr]
Fig. 2.—Close-upsofthetwolightcurveswithsuspectedspotanomalies. Bothlightcurvesarebasedonr′-bandobservationswiththe
MagellanClay6.5mtelescope. Theplottingconventions arethesameasinFigure1.
two physical parameters i and R /R . riodtobe1.5803925days(Charbonneauetal.2009). We
p ⋆
Hence one may derive R /R based on T and τ also allowed the out-of-transit magnitude on each night
p ⋆ tot
alone,withoutanyreferenceto δ. The resultofstraight- to be a linear function of airmass, in order to allow for
forward algebra is color-dependent differential extinction between GJ 1214
and the bluer comparison stars.
4[(3TPto/t(a8lπ−2Gτ))]τ2/3 =(cid:18)RRp⋆(cid:19)ρ−⋆ 23(1+1e−sien2ω)2 afi/cAiReln⋆l,ttso)ig,,euuthreara,nntddhevvrre w((ttehhreee7rz′0′--bbmaaonndddellliipmmabbraddmaaerrtkkeeernns:iinnRggpcc/ooRee⋆ff--,
z z
ρ R 4 ficients), ǫ (for j = 1 to 15), t (for j = 1 to 16),
+O p p , (5) j c,j
ρ R and two constants specifying the linear airmass correc-
" ⋆ (cid:18) ⋆(cid:19) # tion for each of the 16 transits. We determined the pos-
where G is the gravitational constant. For cases where teriorprobabilitydistributionsfortheseparameterswith
star spots are likely to be present, this “duration-based” aMarkovChainMonteCarlo(MCMC)algorithm,using
method of deriving R /R is a useful alternative to the Gibbs sampling and Metropolis-Hastings stepping (see,
p ⋆
usual “depth-based” method, and it may be more ac- e.g.,Tegmarket al.2004,Holman etal.2006). The like-
curate in situations when ρ is well constrained. The lihood was taken to be exp( χ2/2), with
⋆ −
duration-based method is immune to the bias that is
causedbystarspotsoutsideofthetransitchord,although χ2= [Fk−F(tk)]2 (6)
one must still be wary of starspot anomalies that oc- σ2
cur during ingress or egress. As we will describe, for Xk k
GJ 1214 we considered both depth-based and duration- where F is the measured flux, is the calculated flux,
based methods to estimate Rp/R⋆. andσkisroot-mean-squareofthFeresidualsfromthebest-
fitting model specific to each night. The data from the
4. TRANSITLIGHT CURVEANALYSIS suspected spot-induced anomalies during epochs 5 and
We modeled the data that are plotted in Fig. 1 using 260 were excluded from this sum.
the approach represented by Equation (1). Each transit Figures 1 and 2 show the minimum-χ2 solution (black
was assigned a parameter ǫ, representing the fractional curve) and the residuals for each night. Figure 4 shows
loss of light due to spots on that night. Since a degener- the time-binned composite z′ and r′ light curves, with
acypreventsthedeterminationofalloftheǫvaluesalong 2 minute sampling. The data were combined after cor-
withR /R , we fixedǫ=0 forthe transitwith the shal- recting the depths for spot variability [such that F
p ⋆ 7→
lowest depth (epoch 236, UT 2010 June 5) and allowed 1 (1 F)(1 ǫj)]. Table 1 gives the results for each
− − −
all the others to be free parameters. This is equivalent parameter, based on the 15.8%, 50%, and 84.2% values
to assuming that the visible stellar disk had no spots on of the cumulative posterior distribution for each param-
that particular night. The results can be scaled appro- eter,aftermarginalizingovertheotherparameters. This
priately for any assumed value of the spot coverage on table also gives some other parameters based on subse-
that night. quent steps in the analysis, described in the sections to
We used the Mandel & Agol (2002) equations to cal- follow.
culate∆F/F ,assumingaquadraticlimb-darkeninglaw. In order to assess any possible transit duration varia-
0
The orbit was assumed to be circular. The time of con- tions, we also performed a second analysis, in which the
junction t for each transit was allowed to be a free pa- orbital inclination i was allowed to be a free parameter
c
rameter, but for the limited purpose of computing the specific to each night. Table 2 gives the transit depth
scaledorbitaldistance a/R , we assumedthe orbitalpe- [(R /R )2/(1 ǫ )],timeofconjunction,andtotaltran-
⋆ p ⋆ j
−
Sixteen Transits of GJ 1214b 5
m] 6
1.000 p
p
Planet odel [ 4 ε = 10%
M 5%
0.995 e 3%
e 2
−fr 1%
ot
F(t) 0.990 −fit Sp 0
st
e −2
B
Star Spot −
0.985 el
od −4
M
ot
p −6
S
−0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
Time since midtransit [hr] Time since midtransit [hr]
Fig.3.—Theeffect of untransitedstarspots. Left.—Thelightcurves of aplanet transitingastarwith(dashed line)andwithout(solid
line) a starspot on the visible disk [ǫ = 5% in Eqn. (1)]. In both cases the out-of-transit flux was set equal to unity. Right.—Residuals
betweenthelightcurvesandthebest-fittingspot-freelightcurve.
1.000
t
e 0.995
s
f
f
O

+ 0.990

x
u
Fl
0.985
e
v
ti
a
el 0.980
R
0.975
−1.0 −0.5 0.0 0.5 1.0
Time since midtransit [hr]
Fig.4.—GJ1214compositetransitlightcurves,inthez′-band(top) andr′-band(bottom), after mergingallthe dataandresampling
intotwo-minutebins. Theblackcurvesshow thebest-fitting modellightcurves. Thermsresidualis270ppmforther′-bandlightcurve,
and560ppmforthez′-bandlightcurve.
sit duration for each epoch. The depths are plotted as transitdepth,whichisbeneathourtypicalmeasurement
a function of epoch in Figure 5. Figure 6 shows the du- precision of 5%, and hence it is not surprising that no
rations, and the residuals from a linear ephemeris, as a such trend was observed.
function of epoch. Based upon this finding, one way to estimate the
planet-to-star radius ratio is to assume that spots have
5. THEPLANET-TO-STARRADIUSRATIO notsignificantlyaffectedthetransitdepth. Wemaythen
calculatetheradiusratiobasedonthevariance-weighted
Despite our concerns about the effects of starspots on
average of the 16 transit depths δ (R /R )2(1
the light curves, we did not find any significant vari- j ≡ p ⋆ j −
ǫ )−1:
ation of the transit depth or and duration with time, j
and we did not find any significant departure from a
linear ephemeris. In addition, the out-of-transit flux of R 1 16 δ
p j
GJ1214measureddirectlyfromtheimageswasconstant = (7)
to within a few percent over the course of our observa- R⋆ vuu σδ−j2 Xj=1 σδ2j
tions (see Figure 7). There was apparently a decline by =0t.1P1610 0.00048.
1–2%between2009and2010,followedbyagradual1–2% ±
risethroughoutour2010observations. This shouldhave where the averageis taken over all epochs j. This result
been accompanied by 1–2% variations in the measured forthe radiusratiois slightlysmallerthanthat reported
6 Carter et al. 2010
TABLE 1
System Parametersof GJ 1214
Parameter Value Comment
TransitParameters
Orbitalperiod(day) 1.58040482±0.00000024 a
Transitephemeris(BJDTDB) 2,454,980.7487955 ±0.000045 a
Planet-to-starradiusratio,Rp/R⋆ 0.1161±0.0048 b
Orbitalinclination,i(deg) 90+0.0 c
−1.5
Scaledsemimajoraxis,a/R⋆ 14.71+−00..3373 c
Totaltransitduration,TTotal (min) 52.73+−00..4395
Transitingressduration,τ (min) 5.80+0.41
−0.45
Starspot-unaffected quantity, (ρ⋆/ρ⊙)−2/3(Rp/R⋆) 0.017±0.001 d
r′ quadraticlimb-darkeningcoefficients,(ur,vr) (0.49±0.07,0.48±0.22)
z′ quadraticlimb-darkeningcoefficients, (uz,vz) (0.17±0.12,0.64±0.30)
OrbitalParameters
Radialvelocitysemiamplitude,K⋆ (m/s) 12.2±1.6 e
Orbitaleccentricity, e 0 fixed
Stellarprojectedrotational velocity,vsini(km/s) <2.0 e
StellarPhotometricParameters&Parallax
J 9.750±0.024 e
H 9.094±0.024 e
K 8.872±0.020 e
Parallax,π (mas) 77.2±5.4 e
Stellar&PlanetaryParameters
MethodA† Comment MethodB‡ Comment
Stellarmass,M⋆ (M⊙) 0.157±0.012 0.156±0.006
Stellarradius,R⋆ (R⊙) 0.210±0.007 0.179±0.006
Stellarmeandensity,ρ⋆ (gcm−3) 24.1±1.7 f 38.4±2.1
Stellareffectivetemperature, Teff (K) – 3170±23
Stellarsurfacegravity,logg⋆ (cgs) 4.99±0.04 5.12±0.01
Stellarmetallicity([Fe/H]) – 0 fixed
Planetarymass,Mp (M⊕) 6.45±0.91 6.43±0.86
Planetaryradius,Rp (R⊕) 2.65±0.09 2.27±0.08
Planetarymeandensity,ρp (gcm−3) 1.89±0.33 3.03±0.50
Planetarysurfacegravity,gp (ms−2) 9.0±1.5 g 12.2±1.9
Note. —†Parameterscalculatedassuminganempiricalluminosity-massrelationship(see§6.1fordetails). ‡Parametersconstrainedto
theBaraffeetal.(1998)stellarevolutionisochrones(see§6.2fordetails). a=Determinedfromalinearfittothemidtransittimeslisted
inTable2. b=DeterminedbytakingthesquarerootofthevarianceweightedaverageofthedepthslistedinTable2(see§5fordetails).
c = Assuming epoch 236 coincides with the spot-free stellar surface. d = As calculated with Eqn. 5. e = Charbonneau et al. (2009). f
=Estimatedfromtransitparameters asρ⋆=(3π)/(GP2)(a/R⋆)3. g=Estimatedfromtransitparametersas(2π/P)K⋆(a/Rp)2(sini)−1
(Southworth etal.2008).
TABLE 2
Epoch specific transitlightcurve results
Epoch Depth,(Rp/R⋆)2(1−ǫ)−1 Midtransittime(BJDTDB) Duration(min)
0 0.01332±0.00057 2,454,980.74857±0.00015 52.30±0.56
2 0.01388±0.00059 2,454,983.90982±0.00016 52.70±0.48
5 0.01355±0.00059 2,454,988.650808±0.000049 52.84±0.29
14 0.01333±0.00063 2,455,002.87467±0.00019 52.59±0.42
183 0.01384±0.00059 2,455,269.96299±0.00016 52.37±0.46
195 0.01281±0.00081 2,455,288.9282±0.0011 52.64±6.23
200 0.01345±0.00067 2,455,296.83013±0.00023 52.40±0.73
212a 0.01416±0.00066 2,455,315.79485±0.00023 52.76±0.65
212b 0.01380±0.00061 2,455,315.794693±0.000080 52.82±0.31
214 0.01303±0.00062 2,455,318.95523±0.00017 51.61±0.53
236 0.01271±0.00055 2,455,353.72387±0.00018 52.44±0.66
238 0.01377±0.00058 2,455,356.88495±0.00015 52.59±0.42
243 0.01313±0.00059 2,455,364.78700±0.00015 52.69±0.33
250 0.01299±0.00053 2,455,375.84997±0.00013 52.76±0.31
255 0.01376±0.00060 2,455,383.75205±0.00013 52.56±0.31
260 0.01414±0.00061 2,455,391.654105±0.000059 52.72±0.36
Meanc 0.01348±0.00011
a Sloanz′
b Sloanr′
c Varianceweightedmean.
Sixteen Transits of GJ 1214b 7
1.50%
Sloan z’ Sloan r’
1.46%
1.41%
h
t
p
e 1.37%
D

t
si 1.32%
n
a
r
T
1.28%
1.23%
1.19%
0 2 14 183 195 200 212 214 236 238 243 250 255 5 212 260
Epoch Epoch
Fig.5.—Transitlightcurvedepthasafunctionofobservationepoch(usingvaluestabulatedinTable2).
54.0 60
53.5
40
c]
n] 53.0 se
mi e [ 20
n [ m
nsit duratio 5522..05 midtransit ti −200
Tra 51.5 −C
O
−40
51.0
50.5 −60
0 50 100 150 200 250 0 50 100 150 200 250
Epoch Epoch
Fig.6.— Transit durations and timing residuals. Left.—Total transit duration versus observation epoch. Right.—Differences between
the measured times of conjunction, and the best-fitting model assuming a constant period. Only the complete transits presented in this
work(solidcircles)wereusedtoderivethemodelparameters. Partialtransits(opencircles)wereexcludedfromthefit. Theotherplotted
data(asterisks)arefromCharbonneauetal.(2009).
8 Carter et al. 2010
5%
Comp. Star 1
4% Comp. Star 2
x Comp. Star 3
u
l 3% Comp. Star 4
F

e
v 2%
i
t
a
l
e 1%
R

T
0%
O
O
−1%
−2%
0 2 14 183 195 200 212 214 236 238 243 250 255
Epoch
Fig. 7.— Out-of-transit variability of GJ 1214. Plotted is the flux (measured out-of-transit) of GJ 1214, relative to four different
comparisonstars.
in the discoverypaper (0.11620 0.00067;Charbonneau differentpathwaystotheplanetaryradius,relyingondif-
±
et al. 2009), but the two results are consistent to within ferent assumptions, and found them to give inconsistent
the uncertainties. results. Theresultsofbothofthesemethods,includinga
Due to time variability of starspots at the 1–2% level, number of derived stellar and planetary parameters, are
this estimate is subject to a bias of a few percent, in the given in Table 2. This inconsistency had already been
sensethattheplanet-to-starradiusratiomayactuallybe noted by Charbonneau et al. (2009), but here we delve
a few percent smaller than 0.11610. An even more con- further into the details and discuss possible resolutions.
servative stance would recognize that we cannot exclude
even larger effects due to the time-independent compo- 6.1. Method A: Empirical mass-luminosity relation, and
nent of the starspot coverage (if, for example, the poles transit-derived stellar mean density
of the star were persistently darker than the rest of the
Inthefirstmethod,thestellarmassisestimatedbased
photosphere). In that sense we can only place an upper
on its observedluminosity (parallax,and apparentmag-
bound on the radius ratio: R /R 0.1161 at 95% con-
p ⋆ ≤ nitude). Then, the stellar radius is found by combining
fidence(basedonthemarginalizedposteriordistribution
the stellarmass with the meanstellardensity ρ derived
produced by the MCMC algorithm). ⋆
It should also be noted that the r′-band transit depth from the transit light curve,
(1.383% 0.035%) was found to be slightly larger than 3 −1
the z′-ba±nd transit depth (1.340% 0.017%), although ρ = 3π a 1+ Mp (8)
the difference is only at the 1–2σ±level. A deeper r′- ⋆ GP2 (cid:18)R⋆(cid:19) (cid:18) M⋆(cid:19)
band transit is expected if cool starspots are affecting
which, for M M , is only a function of the
the results. In contrast, models of the atmosphere of p ≪ ⋆
photometrically-determined parameters a/R and P.
GJ1214bgenerallypredictthatthetransitdepthshould ⋆
We begin with the K-band mass-luminosity function
have the opposite wavelengthdependence, with a deeper
of Delfosse et al. (2000). For GJ 1214, with a par-
z′-band transit (see, e.g., Fig. 3 of Miller-Ricci & Fort-
allax π = 77.2 5.4 mas and apparent K magnitude
ney 2010). Those who would attempt to attribute any ±
8.782 0.020, we used the polynomial function given by
slightwavelength-dependenceofthe transitdepthto the ±
planetary atmosphere should beware of the confounding Delfosseetal.(2000)toestimateM⋆ =0.157±0.012M⊙,
where the uncertainty is based only on the propagation
effects of starspots.
of errors in π and K. There should be additional uncer-
taintydue tothe intrinsicscatterinthe mass-luminosity
6. THERADIUSOFTHEPLANET relation, but this intrinsic uncertainty was not quanti-
In order to understand the structure and atmosphere fied. Charbonneau used the same method and reported
of GJ 1214b, we want to know its radius, rather than M⋆ =0.157 0.019 M⊙.
±
just the planet-to-star radius ratio. This requires some Next, using a/R = 14.71 0.35, as derived from our
⋆
±
externallyderivedestimateofthestellarradius,ormass, ensemble of light curves,we found the mean stellar den-
or both. As we will describe, we have investigated two sity to be ρ =24.1 1.7 g/cm3. By combining M and
⋆ ⋆
±
Sixteen Transits of GJ 1214b 9
ρ⋆ we found the stellar radius to be al.(2009). TheresultswereM⋆ =0.131 0.044M⊙ and
±
R⋆ =0.191 0.019 R⊙.
R⋆ =0.210 0.007R⊙ (method A). (9) This meth±od gives a significantly lower mass than the
±
first method. This difference is almost entirely due to
Finally, assuming that our transit depths were not af-
the additional degree of freedom of the stellar age: the
fected by starspots (i.e., ǫ 0) we used the radius ra-
j ≡ theoreticalevolutionarytracksallowforGJ1214tohave
tio derived from our observations, R /R = 0.11610
p ⋆ ± any age, while the Delfosse et al. (2000) relation is a
0.00048,to compute the planetary radius:
consensus result based on a fit to stars of varying age,
Rp =2.650 0.090 R⊕ (method A). (10) most of which are probably older than a few Gyr.
± This naturally raises the question: what is known
ThisisinagreementwiththatfoundbyCharbonneauet about the age of GJ 1214? As noted by Charbonneau
al.(2009),whofoundRp =2.678 0.130R⊕. Takingthe et al. (2009), and confirmed in this work, the optical
±
moreconservativestancethatspotsmayhavebiasedthe lightvariationsofGJ 1214areonlya few percentinam-
radius ratio measurement, we may set an upper bound plitude,suggestiveofamorematurestar. Moreover,age
on the radius of the planet: Rp 2.81 R⊕ with 95% estimates based on the star’s space velocity and the es-
≤
confidence. timated stellar rotation period also favor an older star,
leading Charbonneau et al. (2009) to estimate an age
6.2. Method B: Stellar evolutionary models of 3–10 Gyr. To account for this evidence, we repeated
The mass-luminosity relationship of Delfosse et ourisochroneanalysisbutwith apriorconstraintonthe
al. (2000) has the virtue of being highly empirical, as it stellar age. Specifically, we used a Gaussian prior on
is based on observations of astrometrically resolved, de- log10(age) with a median value of 10 and a standardde-
tachedeclipsingbinarieswithmeasuredparallaxes. How- viation of 0.4. With this extra constraint, the stellar
ever, with only 16 systems as inputs, the empirical rela- mass was found to be M⋆ =0.156 0.006 M⊙, in excel-
±
tion cannot be expected to account for all the relevant lent agreement with the results of the first method.
variables such as age and metallicity. An alternate ap- This substantiates our claim that the main difference
proachis to trust theoretical models of stellar evolution, between the two methods of estimating the stellar mass
which predict the radius of a star as a function of its isthe treatmentofstellarage. However,there remainsa
mass, age, and metallicity, and are calibrated by obser- significant discrepancy in the stellar radius. The evolu-
vations when possible. tionary models with the “old age” prior give
To try this approach, we used the stellar evolution-
ary models of Baraffe et al. (1998), which are presented R⋆ =0.179±0.006R⊙ (method B). (13)
as a series of stellar parameters as a function of age which is about 15% smaller than the result of the first
(isochrones)for agivenmetallicity. Since the metallicity method. Again taking the radius ratio to be R /R =
p ⋆
of GJ 1214 is unknown we assumed a solar metallicity. 0.11610 0.00048,the planetary radius is
We assigned a likelihood (j) exp[ χ2(j)/2] to every ±
L ∝ −
isochrone entry j such that Rp =2.270 0.080 R⊕ (method B). (14)
±
∆p2
χ2(j)= k (11) 6.3. Possible resolutions of the two methods
σ2
k pk To summarize the preceding results, we have inves-
X
tigated two methods of estimating the stellar mass and
where p is the set of stellar parameters that is sub-
{ k} radiusbasedontheavailableinformation. Thetwometh-
ject to observational constraints (e.g. absolute magni-
ods can be made to agree on the stellar mass, but they
tudes in certain bandpasses), σ are the corresponding
pk refuse to agree on the stellar radius. The first method is
1σuncertainties,and∆p arethedifferencesbetweenthe
k based on an empirical mass-luminosity relation, and the
isochrone values and the “observed” values. Then, the
stellarmeandensityderivedfromthetransitlightcurve.
mostlikely value fora givenstellarparameterp (suchas
Thesecondmethodisbasedonstellarevolutionarymod-
stellar radius) was found by taking a weighted average
els constrained by the infrared absolute magnitudes and
over all the isochrone points,
anassumedage&1Gyr. Thefirstmethodgivesastellar
p(j) (j) radius that is larger by 15%, which is approximately 4.5
p = j L . (12) times larger than the internal uncertainty in either esti-
h i (j)
P jL mate. The discrepancy is even starker if one compares
the mean stellar density derived from the transit light
The uncertainty in thisPparameter is taken to be
curve (24.1 1.7 g cm−3) with that derived from the
the square root of the weighted variance, i.e., σp = stellar evolu±tionary models (38.4 2.1 g cm−3), as they
p2 p 2whereaveragesaretakenasdefinedabove.5 ±
h i−h i disagree by 7σ.
We used this technique with constraints based on the
It has long been appreciated that stellar evolutionary
pobserved apparent magnitudes in the J, H, and K
models have trouble predicting the masses and radii of
bandpasses and the parallax, all from Charbonneau et
Mdwarfsindetachedeclipsing binarysystems (see,e.g.,
Hoxie1973,Lacy1977),inthesensethatthemodelstend
5Thisapproachiscloselyrelatedtothetechniqueadvocatedby to underpredict the observed radius for a given mass by
Torresetal.(2008). TherelativelycoarsesamplingoftheBaraffe
about 10% (Ribas 2006). It was for this reason that
et al. (1998) isochrones precluded the use of more sophisticated
numericalintegrationtechniquessuchasthosedescribedbyCarter Charbonneau et al. (2009) discounted the results based
etal.(2009). ontheevolutionarymodels,andwhysubsequentauthors
10 Carter et al. 2010
havedonethesameintheirdiscussionsofGJ1214. How- It is conceivable that unidentified starspot anomalies
ever, it has recently been argued that the failings of the have significantly biased our estimate of a/R from the
⋆
evolutionarymodels are confined to the mass range 0.3– transitlightcurves(see 3.1). Couldthis be responsible
§
0.7 M⊙, and even more specifically to stars that have forthe7σ discrepancyinρ⋆ betweenthe twomethods of
been tidally spun up (and made more active) by a close characterizing the star? If this were the case, then the
stellar companion (see, e.g., Torres et al. 2006, Lo´pez- moretrustworthyestimateofρ wouldbethevaluefrom
⋆
Morales2007,Moralesetal.2008). Starsbelowathresh- the isochrone analysis.
oldmassof 0.32M⊙areexpectedtobefullyconvective, Thisseemsunlikely,partlybecausetheresidualstoour
≈
and seem to be well described by the evolutionary mod- transitlightcurvesdonotdisplayanyanomaliesbeyond
els (see, e.g., Lo´pez-Morales 2007, Demory et al. 2009, the two that we have already identified, and partly be-
Vida et al. 2009). To the extent this is true, we would cause in this scenarioit would be difficult to understand
not expect GJ 1214—a single, low-activity star of mass the collection of transit depths. Specifically, we can use
<0.3M⊙—tobeaffectedbytheproblemsthatplaguethe the value of ρ⋆ from the evolutionary models as an in-
theoriesofearlier-typeMdwarfsinclosebinarysystems. put to Equation (5), to derive the planet-to-star radius
We are thereby motivated to seek alternative resolu- ratio. The result is R /R = 0.15 0.01. This conflicts
p ⋆
±
tions to the discrepancy between the two methods of es- with the upper limit R /R 0.1161that we derived in
p ⋆
≤
timatingthemassandradiusofGJ1214. Ofcoursethere 5,underthe assumptionthatthe transitdepths areaf-
§
isalwaysthepossibilitythatakeyinputsuchasthepar- fectedby coolspots onthe stellar disk. Thus, onewould
allax or infrared magnitude is faulty, but in the sections havetosupposethatnearlyallofthetransitdepthswere
to follow we discuss some possible resolutions in which biased to smaller values by numerous starspot crossings
all of the data are taken at face value. throughout the transits. We do not consider a conspir-
acy of starpot anomalies to be a satisfactory solution to
6.3.1. A metal-poor star?
the problem.
We have worked exclusively with solar metallicity
isochrones. Lower-metallicity isochrones generally pre- 6.3.4. Is the orbit eccentric?
dict a larger radius for a given mass, in the relevant re-
IftheorbitofGJ1214bisnotcircular,thentheparam-
gion of parameter space. Schlaufman (2010) have pre-
eter that is being determined by the light curve analysis
senteda simple methodto estimate the metallicity ofan
is not a/R and our application of Equation (8) is er-
M dwarf based on its observed V and K magnitudes. ⋆
roneous. The correct procedure must take into account
For GJ 1214, his method gives [Fe/H]= 0.16, suggest-
− the speed of the planet at inferior conjunction, which is
ing that the star is only moderately metal-poor. This
a function of the eccentricity and argument of pericen-
value for the metallicity would not affect the theoreti-
ter (see, e.g., Eqns. 16 and 27 of Winn 2010 or Kipping
cally predicted radius by enough to resolve the discrep-
2010). The end result is that the mean density of the
ancy. Lo´pez-Morales (2007) found that the Baraffe et
planet would be modified as follows
al. (1998) isochrones predict a variation in stellar ra-
dius of only about 3% for metallicities ranging from 3
0.0 to 0.5. Moreover, for a sample of low-mass stars √1 e2
with −0.5 <[Fe/H]< 0, Demory et al. (2009) showed ρ⋆ =ρ⋆, circ 1+e−sinω , (15)
!
−
thatthereisnosignificantcorrelationbetweenmeasured
metallicityandthefractionaldifferencebetweenthemea- whereρ isthemeandensitythatiscalculatedunder
⋆, circ
suredstellarradiusandthatfoundassumingsolarmetal- the assumption of a circular orbit. Therefore, it might
licity. Therefore, while it is possible that GJ 1214 is a be possible to reconcile the two different methods for
metal-poor star, we do not consider this to offer a likely estimating the stellar density, for suitable choices of e
resolutionof the discrepancy we havenoted between the and ω.
two methods of estimating the stellar radius. The orbital eccentricity has been assumed to be zero,
because of the expectation that tidal dissipation has
6.3.2. A young star?
damped out any initial eccentricity to 10−3 or below.
If the star were relatively young and still contracting However,thepublishedRVdataonlypermitacoarseup-
onto the main sequence, then the evolutionary models per limit of e<0.27 with 95% confidence (Charbonneau
would predict a larger radius, relieving the discrepancy. etal.2009). Toinvestigatethepossibilityofaneccentric
To investigate this possibility we repeated our isochrone orbit, we assumed that the isochrone-based estimate of
analysis, but this time with a flat prior on the stellar the mean stellar density (38.4 2.1 g cm−3) is accurate.
age and a Gaussian prior on the stellar mean density to We then derived the constrai±nts on the orbital eccen-
conformwith the transitlightcurveanalysis. The result tricity that would be required for consistency with the
was that the age of the star must be about 100 Myr. transitlightcurveanalysis. Theminimumeccentricityis
This would conflict with the evidence for an older age, obtained for the case ω =90◦, corresponding to transits
namely,theobservedlackofstrongchromosphericactiv- occurring at pericenter. In that case, e= 0.138 0.034.
ity and the kinematics (see 6.2). There is also the low Solutions with e < 0.27 can be found for valu±es of ω
probabilitythatwewouldha§ppen toobservethis starat between 25◦ and 155◦.
such an early phase of its long life. For these reasons, a Theonlyobjectionisthattidaldissipationshouldhave
young age for GJ 1214 does not seem to be a promising damped out this eccentricity. The characteristic circu-
solution. larization timescale is τ 10 Myr assuming a tidal
c
∼
6.3.3. A starspot-corrupted estimate of a/R ? quality factor Qp = 100 for the planet, a rough order-
⋆ of-magnitude estimate for a solid planet. This is much