Detection of visible light from the darkest world

Mon. Not. R. Astron. Soc. 000, 1–6 (2011) Printed 12 August 2011 (MN L TEX style file v2.2)
A

Detection of visible light from the darkest world

David M. Kipping1 & David S. Spiegel2
1 Center for Astrophysics, 60, Garden Street, Cambridge, MA 02138 [E-mail: dkipping@cfa.harvard.edu]
2 Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544
arXiv:1108.2297v1 [astro-ph.EP] 10 Aug 2011

Accepted 2011 August 1. Received 2011 July 25; in original form 2011 June 29

ABSTRACT
We present the detection of visible light from the planet TrES-2b, the darkest exoplanet
currently known. By analysis of the orbital photometry from publicly available Kepler
data (0.4-0.9 µm), we determine a day-night contrast amplitude of 6.5 ± 1.9 ppm,
constituting the lowest amplitude orbital phase variation discovered. The signal is
detected to 3.7σ confidence and persists in six different methods of modelling the data
and thus appears robust. In contrast, we are unable to detect ellipsoidal variations or
beaming effects, but we do provide confidence intervals for these terms. If the day-
night contrast is interpreted as being due to scattering, it corresponds to a geometric
albedo of Ag = 0.0253±0.0072. However, our models indicate that there is a significant
emission component to day-side brightness, and the true albedo is even lower (< 1%).
By combining our measurement with Spitzer and ground-based data, we show that
a model with moderate redistribution (Pn ≃ 0.3) and moderate extra optical opacity
(κ′ ≃ 0.3 − 0.4) provide a compatible explanation to the data.
Key words: techniques: photometric — stars: individual (TrES-2)

1 INTRODUCTION In this letter, we investigate the hot-Jupiter orbiting the
G0V star TrES-2 (O’Donovan et al. 2006), where we detect
Orbital photometric phase variations have long been used in
a reflected/emitted light amplitude of (6.5 ± 1.9) ppm to a
the study and characterisation of eclipsing binaries (Wilson
confidence of 3.7σ, or 99.98%. We also measure the ellip-
1994), where the large masses and small orbital radii re-
soidal variation and relativistic beaming amplitudes to be
sult in phase variations at the magnitude to millimagnitude
(1.5 ± 0.9) ppm and (0.2 ± 0.9) ppm respectively, which are
level. The three dominant components of these variations
broadly consistent with theoretical expectation.
are i) ellipsoidal variations, due to the non-spherical nature
If our detected signal is interpreted as being purely
of a star caused by gravitational distortion (e.g. Welsh et al.
due to scattering, then the corresponding geometric albedo
2010) ii) relativistic beaming, due to the radial motion of the
would be Ag = 0.0253 ± 0.0072 (using system parameters
star shifting the stellar spectrum (e.g. Maxted et al. 2000)
from Table 2, column 2 of Kipping & Bakos 2011 (KB11),
iii) reflected/emitted light, which varies depending on what
as will be done throughout this work), meaning that just
phase of a body is visible (e.g. For et al. 2010).
four months of Kepler ’s exquisite photometry has detected
The visible bandpass orbital phase variations of a star
light from the darkest exoplanet yet found. Extrapolating to
due to a hot-Jupiter companion are much smaller - around
a 6 year baseline, one can expect to detect albedos 0.1 (to
the part-per-million (ppm) level - and thus have eluded de-
3σ confidence) at similar orbital radii down to RP ≃ 3.0 R⊕ .
tection until relatively recently. The high precision space-
This clearly highlights the extraordinary potential which
based photometry of CoRoT (0.56-0.71 µm) (Baglin et al.
would be granted by an extended mission for Kepler.
2009) and Kepler (Basri et al. 2005) have opened up this
exciting new way of studying exoplanets for first time, with
2 OBSERVATIONS & ANALYSIS
several detections recently reported in the literature:
2.1 Data Acquisition
CoRoT-1b (Snellen et al. 2009); reflected/emitted light
amplitude 126 ± 36 ppm We make use of “Data Release 3” (DR3) from the Kepler
HAT-P-7b (Welsh et al. 2010); ellipsoidal amplitude Mission, which consists of quarters 0, 1 and 2 (Q0, Q1 &
37 ppm, reflected/emitted light amplitude 63.7 ppm Q2). Full details on the data processing pipeline can be
CoRoT-3b (Mazeh & Faigler 2010); ellipsoidal ampli- found in the DR3 handbook. The data includes the use of
tude (59 ± 9) ppm, beaming amplitude (27 ± 9) ppm BJD (Barycentric Julian Date) time stamps for each flux
Kepler-7b (Demory et al. 2011); reflected/emitted light measurement, which is crucial for time sensitive measure-
amplitude (42 ± 4) ppm ments. All data used are publicly available via MAST.

c 2011 RAS

2 Kipping & Spiegel
We use the “raw” (labelled as “AP RAW FLUX” in
the header) short-cadence (SC) data processed by the DR3
pipeline and a detailed description can be found in the ac- 2.188 107
companying release notes. The “raw” data has been subject
to PA (Photometric Analysis), which includes cleaning of 2.186 107
cosmic ray hits, Argabrightenings, removal of background

Flux
flux, aperture photometry and computation of centroid po-
2.184 107
sitions. It does not include PDC (Pre-search Data Condi-
tioning) algorithm developed by the DAWG (Data Analysis
2.182 107
Working Group). As detailed in DR3, this data is not rec-
ommended for scientific use, owing to, in part, the potential
for under/over-fitting of the systematic effects. 2.180 107
54 960 54 970 54 980 54 990
BJDUTC 2400000
2.2 Cleaning of the Data
The raw data exhibit numerous systematic artifacts, includ- 2.176 107
ing pointing tweaks (jumps in the photometry), safe mode
recoveries (exponential decays) and focus drifts (long-term
2.174 107
trends). The first effect may be corrected by applying an off-
set surrounding the jump, computed using a 30-point inter-
polative function either side. Due to the exponential nature 2.172 107

of the second effect, we chose to exclude the affected data Flux
rather than attempt to correct it. The third effect may be 2.170 107
corrected for using a detrending technique.
For this latter effect, we use the cosine filter
2.168 107
utilised to detect ellipsoidal variations in CoRoT data by
Mazeh & Faigler (2010). The technique acts as a high-pass 55 020 55 040 55 060 55 080
BJDUTC 2400000
filter allowing any frequencies of the orbital period or higher
through and all other long-term trends are removed. Thus,
we protect any physical flux variations on the time scale Figure 1. “Raw” (PA output) flux from DR3 of the Kepler
of interest. We applied the filter independently to Q0+Q1 pipeline for Q0&Q1 (upper panel) and Q2 (lower panel) of the
data and then Q2 data. This is because the Kepler space- star TrES-2. Overlaid is our model for the long-term trend, com-
craft was rotated in the intervening time and so the long- puted using a discrete cosine transform for each data set. Outliers
term trend will not be continuous over this boundary. After and discontinuous systematic effects have been excluded.
removing 3σ outliers with a running 20-point median and
transits using the ephemeris of KB11, we applied the filter,
with the resulting fitted trends shown in Fig 1. Our final dummy term which should be zero and ensures the ellip-
cleaned data consists of 154,832 SC measurements with a soidal variation is detected with the correct phase.
mean SNR≃ 4408. We also tried a third model, M3 , where the a1c term is
replaced by the reflection caused by a Lambertian sphere:
2.3 Three Models
We first define our null hypothesis, M1 , where we employ a sin |φ| + (π − |φ|) cos |φ|
0.5a1c cos φ → a1c . (2)
flat line model across the entire time series, described by a π
constant a0 . For a physical description of the orbital phase
variations, we first tried the same model as that used by
Sirko & Paczynski (2003) and Mazeh & Faigler (2010). This 2.4 Three Data Modes
simple model is sufficient for cases where one is dealing with
low signal-to-noise and reproduces the broad physical fea- In addition to three models, we have three data input modes.
tures. The model, M2 , is given by The first is simply corrected for detrending and nothing else,
denoted D1 . The second mode renormalises each orbital pe-
riod epoch. This renormalisation is done by computing the
M2 (φ) = a0 + 0.5a1c cos(φ) + a1s sin(φ) median of each epoch and dividing each segmented time
+ a2c cos(φ/2) + a2s sin(φ/2) , (1) series by this value and we denote this mode as D2 . Fi-
nally, we tried allowing each orbital period epoch to have its
where φ is the orbital phase (defined as being 0 at the own variable renormalisation parameter, which is simulta-
time of transit minimum) and ai are coefficients related to neously fitted to the data along with the orbital phase curve
the physical model. a0 is simply a constant to remove any model. This parameter is dubbed a0,j for the j th orbital pe-
DC (direct-current) component in the data. a1c corresponds riod epoch. Denoting this data input mode as D3 , the fits
to the reflection/emission effect and is expected to be have a now include an additional 51 free parameters.
negative amplitude. a1s corresponds to the relativistic beam- The models are fitted to the unbinned data using a
ing effect and is expected to be positive. a2c corresponds to Markov Chain Monte Carlo algorithm described in KB11
the ellipsoidal variations and should be negative. a2s is a (method A) with 1.25 × 105 accepted trials burning out the

c 2011 RAS, MNRAS 000, 1–6

The Darkest World 3
first 25,000. In total, there are nine ways of combining the Between the two models, there is negligible difference in
three models with the three data modes. All nine models are the goodness-of-fit, as seen in Table 1, for all three data
fitted and results are given in Table 1, with our preferred modes. Including the Lambertian model takes some power
model description being M2 , D3 (since thermal emission is away from the ellipsoidal variations though and thus the cur-
likely dominant over scattering, see §4). rent data does not yield a preference between a Lambertian
sphere model or stronger ellipsoidal variations.
3 RESULTS
3.2 Occultation Measurement
3.1 Orbital Photometry
The duration of the transit, and thus occultation since
Table 1 presents the results of fitting the detrended Kepler TrES-2b maintains negligible eccentricity, is equal to 4624 ±
photometry. Our models make no prior assumption on the 42 seconds (defined as the time between when the planet’s
sign or magnitude of the ai coefficients. The orbital period centre crosses the stellar limb to exiting under the same
and transit epoch are treated as Gaussian priors from the condition). In contrast, the orbital period of TrES-2b is
circular orbit results of KB11. 2.470619 days. We therefore obtain ∼46 times more integra-
When considering statistical significance, what one is tion time of the orbit than the occultation event. This indi-
really interested in is the confidence of detecting each phys- cates that we should expect to be able to reach a sensitivity
√
ical effect i.e. reflection/emission, ellipsoidal and beam- of 46 times greater, purely from photon statistics. The un-
ing. For this reason, model comparison tools, such as the certainty on our phase curve measurement is ±1.9 ppm. We
Bayesian Information Criterion (BIC) or an F-test are inap- therefore estimate that one should have an uncertainty on
propriate. This is because these methods evaluate the pref- the occultation depth of ∼ 13 ppm. If we assume the night-
erence of one hypothesis over another, where the two mod- side has a negligible flux, then the depth of the occultation
els would be a null-hypothesis and a hypothesis including is expected to be 6.5 ppm (i.e. equal to the day-night con-
reflection/emission, ellipsoidal variations, beaming and the trast), and this already suggests that the present publicly
dummy term. Thus, any inference drawn from this would be available Kepler photometry will be insufficient to detect
for the entire model and not for each individual effect. In the the occultation. To test this hypothesis, we will here fit the
analysis presented here, simple inspection of the posteriors occultation event including the Q0, Q1 and Q2 data.
from Fig 2 shows that only one effect is actually detected (re- To perform our fit, we use the same Gaussian priors on
flection/emission), but a model comparison method would P and τ as earlier. We also adopt priors for other important
evaluate the significance of all four physical effects (includ- system parameters from KB11, such as b = 0.8408 ± 0.0050,
ing the dummy term) versus no effect. p2 = 1.643 ± 0.067% and T1 = 4624 ± 42 seconds. We stress
A more useful statistical test would consider the signif- that these are all priors and not simply fixed parameters.
icance of each physical effect individually from a joint fit. We also make use of the priors on the a0,j coefficients from
An excellent tool to this end is the odds ratio test discussed the M3 , D3 fit. Data are trimmed to be within ±0.06 days
in Kipping et al. (2010). If a parameter was equal to zero, of the expected time of occultation to prevent the phase
we would expect 50% of the MCMC runs to give a positive curve polluting our signal, leaving us with 8457 SC mea-
value and 50% to give a negative value. Consider that some surements. Assuming a circular orbit, the data were fitted
asymmetry exists and a fraction f of all MCMC trial were using an MCMC algorithm.
positive and 1 − f were negative. The reverse could also be The marginalised posterior of the occultation depth
true and so we define f such that f > 0.5 i.e. it represents yields δocc = 16+13 ppm, which is clearly not a significant de-
−14
the majority of the MCMC trials. The odds ratio of the tection. The derived uncertainty of 13-14 ppm is very close to
asymmetric model over the 50:50 model is: our estimation of ∼ 13 ppm and thus supports our hypothe-
sis that the current Kepler data are insufficient to detect the
0.5 occultation of TrES-2b. We also note that the inclusion of
O= (3) the Q2 data does improve the constraints on the occultation
1−f
event (KB11 found δocc = 21+23 ppm using Q0 & Q1 only).
−22
For only two possible models, the probability of the
asymmetric model being the correct one is P(asym) =
4 DISCUSSION
1 − [1/(1 + O)]. We perform this test on each of the four
parameters fitted for, a1c , a2c , a1s and a2s . The associated Hot-Jupiters are generally expected to be dark. Significant
results are visible in the top-left corners of each posterior absorption due to the broad wings of the sodium and potas-
shown in Fig 2, for our preferred model and data mode i.e. sium D lines is thought to dominate their visible spectra
M2 , D3 . To summarise, only one parameter presents a sig- (Sudarsky et al. 2000), leading to low albedos of a few per-
nificant detection - the reflection/emission effect. Here, we cent. Indeed, aside from the recent report of Kepler-7b’s
find a1c ’s posterior is sufficiently asymmetric to have a prob- (38 ± 12)% Kepler -band geometric albedo (KB11), searches
ability of occurring by random chance of just 0.02%, which for visible light from hot-Jupiters have generally revealed
equates to 3.67σ. We consider any signal detected above 3σ mere upper limits (Collier Cameron et al. 2002; Leigh et al.
confidence to merit the claim of a “detection” rather than 2003; Rowe et al. 2008; Burrows et al. 2008).
a measurement and thus we find TrES-2b to be the darkest The 6.5±1.9 ppm contrast (determined from our pre-
exoplanet from which visible light has been detected. ferred model M2 , D3 ) between the day-side and night-side
As discussed in §2.3, we tried two different models for photon flux that we measure for TrES-2b represents the
the reflection/emission effect; a simple sinusoid (M2 ) and most sensitive measurement yet of emergent radiation in
the reflected light from a perfectly Lambertian sphere (M3 ). the visible from a hot-Jupiter, and is a factor of ∼20 and

c 2011 RAS, MNRAS 000, 1–6

4 Kipping & Spiegel

Table 1. Results of three models with three data modes, giving nine sets of results. Emboldened row denotes our favoured solution. Results
do not include the orbital period, P and transit epoch τ , which are treated as Gaussian priors via P = 2.47061896 ± 0.00000022 days and
τ = 2454950.822014 ± 0.000027 BJDTDB . Quoted values are medians of marginalised posteriors with errors given by 1σ quantiles. ∗ =
parameter was fixed. We do not show the a0,j fitted terms, which are simply renormalisation constants and are available upon request.

Model M, a1c [ppm] a1s [ppm] a2c [ppm] a2s [ppm] χ2
Data D (reflec./emiss.) (beaming) (ellipsoidal) (dummy)

M1 , D 1 0∗ 0∗ 0∗ 0∗ 162603.5431
M2 , D 1 −7.2+1.8
−1.8 0.78+0.85
−0.85 −1.42+0.91
−0.92 −0.27+0.85
−0.85 162583.4014
M3 , D 1 −7.3+1.8
−1.9 0.79+0.86
−0.86 −0.77+0.92
−0.91 −0.26+0.86
−0.86 162583.3162

M1 , D 2 0∗ 0∗ 0∗ 0∗ 161875.4005
M2 , D 2 −6.4+1.8
−1.8 0.34+0.86
−0.87 −1.52+0.93
−0.94 0.19+0.87
−0.87 161859.6732
M3 , D 2 −6.4+1.8
−1.9 0.34+0.86
−0.86 −0.95+0.92
−0.92 0.19+0.86
−0.87 161859.6095

M1 , D 3 0∗ 0∗ 0∗ 0∗ 161837.6648
M2 , D 3 −6.5+1.9
−1.9 0.22+0.88
−0.87 −1.50+0.92
−0.93 0.31+0.88
−0.87 161821.7228
M3 , D 3 −6.7+1.8
−1.8 0.23+0.89
−0.88 −0.90+0.91
−0.91 0.32+0.88
−0.88 161821.7232

Theory Expectation −20 → 0 ∼ 2.4 ∼ −2.3 0 -

Figure 2. Top Left: Final fit to the phased photometry. Points without errors are the 2000-point phase binned data. Points with
errors are 5000-point phase binned data. Best-fit model M2 with data mode D3 shown in solid. Note that all fits were performed on the
unbinned photometry. Top Right & Lower Panels: Marginalised posterior distributions for the same model of four fitted parameters.
Unity minus the false-alarm-probability values are provided for each parameter, based upon an odds ratio test described in §3.1.

∼6 dimmer than the corresponding differences for HAT-P- 2008). We leave TiO and VO out of our calculations, how-
7b (Welsh et al. 2010) and Kepler-7b. ever, because of the difficulty of maintaining heavy, conden-
sible species high in the atmospheres (Spiegel et al. 2009).
In order to interpret the visible flux, we use the plane-
Instead, we use an ad hoc extra opacity source κ′ , as de-
tary atmosphere modelling code COOLTLUSTY (Hubeny et al.
scribed in Spiegel & Burrows (2010).
2003). For simplicity, we adopt equilibrium chemistry with
nearly Solar abundance of elements, although we leave tita- We calculate a grid of models with κ′ ranging (in
2
nium oxide and vanadium oxide (TiO and VO) out of the cm g−1 ) from 0 to 0.6 in steps of 0.1 and with redistri-
atmosphere model. These compounds could, if present in the bution Pn ranging from 0 to 0.5 in steps of 0.1 (Pn repre-
upper atmosphere of a hot-Jupiter, strongly affect the atmo- sents the fraction of incident irradiation that is transported
sphere structure and the visible and near infrared spectra, to the nightside, which is assumed in our models to occur
by making the atmosphere more opaque in the visible and in a pressure range from 10 to 100 mbars). For each of these
by leading to a thermal inversion if the stellar irradiation ex- 42 parameter combinations, we calculate a day-side model,
ceeds ∼109 erg cm−2 s−1 (Hubeny et al. 2003; Fortney et al. a night-side model, and a model that has the same tempera-

c 2011 RAS, MNRAS 000, 1–6

The Darkest World 5
ture/pressure structure as the dayside but that has the star
turned off, so as to calculate the emitted (and not scattered)
flux (thus also giving the scattered component). 0.6
We draw several inferences from our models and the >4σ
data. First, the nightside contributes negligible flux in 0.5
the Kepler -band (always <12% of the dayside, and for

Goodness of Fit
most models significantly less than that), meaning that the 3−4σ
0.4

κ′ (cm2/g)
6.5 ppm number is essentially entirely due to day-side flux.
Second, by also including the available infrared sec- 0.3 2−3σ
ondary eclipse data on TrES-2b (O’Donovan et al. 2010;
Croll et al. 2010), we find that in our model set there must 0.2
be some redistribution (but not too much) and there must be 1−2σ
some extra absorber (but not too much). For each model, 0.1
we compute a χ2 value, including 6 data points: Kepler -
band, Ks-band, and the four Spitzer IRAC channels (3.6, 0−1σ
0
4.5, 5.8 & 8.0µm). Fig 3 portrays the χ2 values of our
grid of models, with the colour ranges corresponding to
the χ2 values bounding 68.3% of the integrated probabil- 0 0.1 0.2 0.3 0.4 0.5
ity (1σ), 95.5% (2σ), 99.7% (3σ) and 99.99% (4σ). The P
n
models that best explain the available data correspond to
κ′ ∼ 0.3 − 0.4 cm2 g−1 and Pn ∼ 0.3 (∼30% of incident Figure 3. Goodness-of-fit for a grid of atmosphere models. The
flux redistributed to the night). In particular, models with models that are consistent with available Kepler-band, Ks-band,
no extra absorber are completely inconsistent with obser- and Spitzer IRAC data have moderate redistribution to the night
vations, even on the basis of the Kepler data alone. The side (Pn ) and moderate extra optical opacity (κ′ ). Models with
upshot is that some extra opacity source appears to be re- κ′ = 0 can be ruled out on the basis of the Kepler data alone.
quired to explain the emergent radiation from this extremely
dark world. Owing to this optical opacity, our models that
are consistent with the data have thermal inversions in their Basri, G., Borucki, W. J. & Koch, D. 2005, New Astronomy
upper atmosphere, as in Spiegel & Burrows (2010). We note Rev., 49, 478
that Madhusudhan & Seager (2010) find that the IR data of Burrows, A., Ibgui, L., & Hubeny, I. 2008, ApJ, 682, 1277
TrES-2b may be explained by models both with and with- Demory, B.-O. et al. 2011, ApJL, accepted
out thermal inversions; nevertheless, we believe that optical Collier Cameron, A., Horne, K., Penny, A. & Leigh, C.
opacity sufficient to explain the Kepler data is likely to heat 2002, MNRAS, 330, 187
the upper atmosphere, as per Hubeny et al. (2003). Croll, B., Albert, L., Lafreniere, D., Jayawardhana, R., &
Finally, by computing the scattered contribution to the Fortney, J. J. 2010, ApJ, 717, 1084
total flux, we find that for all parameter combinations the For, B.-Q., et al. 2010, ApJ, 708, 253
scattered light contributes 10% of the Kepler -band flux, Fortney, J. J., Lodders, K., Marley, M. S., & Freedman,
and for the best-fit models the scattered light is 1.5% of R. S. 2008, ApJ, 678, 1419
the total. TrES-2b, therefore, appears to have an extremely Hubeny, I., Burrows, A., & Sudarsky, D. 2003, ApJ, 594,
low geometric albedo (for all models, the geometric albedo 1011
is < 1%, and for the best-fit models it is ∼0.04%). Exact Kipping, D. M. et al. 2010, ApJ, 725, 2017
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tion and the albedo cannot be presently provided because Leigh, C., Collier Cameron, A., Horne, K., Penny, A. &
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as the wavelength dependence of the extra opacity source Madhusudhan, N., & Seager, S. 2010, ApJ, 725, 261
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ACKNOWLEDGMENTS O’Donovan, F. T. et al. 2006, ApJ, 651, L61
O’Donovan, F. T., Charbonneau, D., Harrington, J., Mad-
We thank the Kepler Science Team, especially the DAWG,
husudhan, N., Seager, S., Deming, D. & Knutson, H. A.
for making the data used here available. Thanks to A. Bur-
2010, ApJ, 710, 1551
rows, M. Nikku & the anonymous referee for helpful com-
Rowe, J. F. et al. 2008, ApJ, 689, 1345
ments and I. Hubeny & A. Burrows for the development
Sirko, E. & Paczynski, B. 2003, ApJ, 592, 1217
and continued maintenance of COOLTLUSTY and associated
Snellen, I. A. G., de Mooij, E. J. W. & Albrecht, S. 2009,
opacity database. DMK is supported by Smithsonian Instit.
Nature, 459, 543
Restricted Endowment Funds.
Spiegel, D. S., Silverio, K. & Burrows, A. 2009, ApJ, 699,
1487
Spiegel, D. S. & Burrows, A. 2010, ApJ, 722, 871
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Detection of visible light from the darkest world

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