ON THE ORBIT OF EXOPLANET WASP-12b

Preprint typeset using LATEX style emulateapj v. 11/10/09 WASP-12b ON THE ORBIT OF EXOPLANET WASP-12B C HRISTOPHER J. C AMPO , J OSEPH H ARRINGTON 1 , RYAN A. H ARDY 1 , K EVIN B. S TEVENSON 1 , S ARAH N YMEYER 1 , DARIN R AGOZZINE 2 , NATE B. L UST 1 , DAVID R. A NDERSON 3 , A NDREW C OLLIER -C AMERON 4, JASMINA B LECIC 1 , C HRISTOPHER B. T. B RITT 1 , W ILLIAM C. B OWMAN 1 , P ETER J. W HEATLEY 5 , T HOMAS J. L OREDO 6 , D RAKE D EMING 7 , L ESLIE H EBB 8 , C OEL H ELLIER 3 , P IERRE F. L. M AXTED 3 , D ON P OLLACO 9 , AND R ICHARD G. W EST 10 arXiv:1003.2763v2 [astro-ph.EP] 10 Dec 2010 1 1Planetary Sciences Group, Department of Physics, University of Central Florida, Orlando, FL 32816-2385, USA 2Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA 3Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK 4School of Physics and Astronomy, University of St. Andrews, North Haugh, Fife KY16 9SS, UK 5Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 6Department of Astronomy, Cornell University, Ithaca, NY 14853-6801, USA 7NASA’s Goddard Space Flight Center, Greenbelt, MD 20771-0001, USA 8Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA 9Astrophysics Research Centre, School of Mathematics & Physics, Queen’s University, University Road, Belfast, BT7 1NN, UK and 10Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK (Received 2010 Mar 13; Accepted 2010 Oct 06) ApJ, in press. ABSTRACT We observed two secondary eclipses of the exoplanet WASP-12b using the Infrared Array Camera on the Spitzer Space Telescope. The close proximity of WASP-12b to its G-type star results in extreme tidal forces capable of inducing apsidal precession with a period as short as a few decades. This precession would be measurable if the orbit had a significant eccentricity, leading to an estimate of the tidal Love number and an assessment of the degree of central concentration in the planetary interior. An initial ground-based secondary eclipse phase reported by López-Morales et al. (0.510 ± 0.002) implied eccentricity at the 4.5σ level. The spectroscopic orbit of Hebb et al. has eccentricity 0.049 ± 0.015, a 3σ result, implying an eclipse phase of 0.509 ± 0.007. However, there is a well documented tendency of spectroscopic data to overestimate small eccentricities. Our eclipse phases are 0.5010 ± 0.0006 (3.6 and 5.8 µm) and 0.5006 ± 0.0007 (4.5 and 8.0 µm). An unlikely orbital precession scenario invoking an alignment of the orbit during the Spitzer observations could have explained this apparent discrepancy, but the final eclipse phase of López-Morales et al. (0.510 ± +0.007 -0.006 ) is consistent with a circular orbit at better than 2σ. An orbit fit to all the available transit, eclipse, and radial-velocity data indicates precession at < 1σ; a non-precessing solution fits better. We also comment on analysis and reporting for Spitzer exoplanet data in light of recent re-analyses. Subject headings: planetary systems — stars: individual: WASP-12 — techniques: photometric 1. INTRODUCTION When exoplanets transit (pass in front of) their parent stars as viewed from Earth, one can constrain their sizes, masses, and orbits (Charbonneau et al. 2007; Winn 2009). Most transiting planets also pass behind their stars (secondary eclipse). This allows atmospheric characterization by measurement of planetary flux and constrains orbital eccentricity, e, through timing and duration of the eclipse (Kallrath & Milone 1999). WASP-12b is one of the hottest transiting exoplanets discovered to date, with an equilibrium temperature of 2516 K for zero albedo and uniform redistribution of incident flux (Hebb et al. 2009). It also has a 1.09-day period, making it one of the shortest-period transiting planets. The close proximity to its host star (0.0229 ± 0.0008 AU, Hebb et al. 2009) should induce large tidal bulges on the planet’s surface. Tidal evolution should quickly circularize such close-in orbits (Mardling 2007). Hebb et al. (2009) calculate a circularization time for WASP-12b as short as 3 Myr, much shorter than the estimated 2 Gyr age of WASP-12 or even the circularization times estimated for other hot Jupiters, given similar planetary tidal dissipation, though this calculation was based on a formalism [email protected] (Goldreich & Soter 1966) that ignores the influence of stellar tides and the coupling of eccentricity and semi-major axis in the evolution of the system. The influence of stellar tides could prolong the dissipation timescale to well over the age of the system (Jackson et al. 2008). The non-Keplerian gravitational potential may cause apsidal precession, measurable as secondary eclipse and transit timing variations over short time scales. WASP-12b also has an abnormally large radius (Rp = 1.79 ± 0.09 Jupiter radii, RJ , Hebb et al. 2009) compared to those predicted by theoretical models (Bodenheimer et al. 2003; Fortney et al. 2007) and to other short-period planets. Tidal heating models assume non-zero e, and the heating rate can differ substantially for different values of e. WASP12b’s inflated radius may result from tidal heating, but this is difficult to justify if the orbit is circular (Li et al. 2010). Ground-based observations by López-Morales et al. (2009) detected a secondary-eclipse phase for WASP-12b of 0.510 ± 0.002, implying an eccentric orbit at the 4.5σ level (LópezMorales et al. 2010 revised the uncertainty to +0.007 -0.006 ). Radial velocity data (Hebb et al. 2009) find e = 0.049 ± 0.015, a 3σ eccentricity, and predict an eclipse phase of 0.509 ± 0.007. Given an eccentric orbit and the fast predicted precession time scale, WASP-12b makes an excellent candidate for the first 2 Campo et al. direct detection of exoplanetary apsidal precession. Such precession has been detected many times for eclipsing binary stars (Kreiner et al. 2001). Against an orbit established by transit timings, precession would be apparent in just two eclipses, if sufficiently separated in time. For eccentric orbits, the eclipse-transit interval can differ from the transit-eclipse interval, and for precessing orbits this difference varies sinusoidally over one precession period. If the difference is insignificant, it places an upper limit on e cos ω, where ω is the argument of periapsis. In the case of WASP-12b, which is expected to precess at a rate of 0.05◦ d-1 (Ragozzine & Wolf 2009), if the orbit is observed when ω ∼ ±90◦ and the effect on the eclipse timing is maxmized, and assuming a timing precision of 0.0007 days, then secondary eclipse observations situated five months apart could detect precession at the 3σ level (see Equation 8). We note that the method of Batygin et al. 2009, based on the work of Mardling 2007 and extended to the three-dimensional case by Mardling 2010, is an indirect assessment of apsidal precession, since no orbital motion is actually observed. The technique, which only applies to multi-planet systems with a tidally affected inner planet and a nearby, eccentric, outer planet, cannot currently be applied to WASP-12b. Paired with the López-Morales et al. data, our Spitzer Space Telescope (Werner et al. 2004) eclipse observations provide a one-year baseline. Spitzer’s high photometric precision also allows an accurate assessment of e cos ω. One can solve for e and ω separately given e sin ω from precise radial velocity data. The following sections present our observations; photometric analysis; a dynamical model that considers parameters from this work, the original and revised parameters of LópezMorales et al., Hebb et al., new transit times from the WideAngle Search for Planets (WASP), and transit times from a network of amateur astronomers; and our conclusions. 6.46 Binned Data 6.44 Flux (Jy) 6.42 6.40 6.38 6.360 5 15 10 20 25 Time From Start of Preflash (mins) 30 F IG . 1.— Preflash light curve. These are channel-4 (8 µm data, analyzed with aperture photometry at the pixel location of the eclipse observations. The preflash source is bright compared to WASP-12, which allows the array sensitivity to “ramp” up before the science observations. Without a preflash, similar observations generally show a steeper and longer ramp in the eclipse observations. 2. OBSERVATIONS We observed two secondary eclipses of WASP-12b with the Spitzer Infrared Array Camera (IRAC, Fazio et al. 2004) in full-array mode. Observations on 2008 October 29 at 4.5 and 8.0 µm (IRAC channels 2 and 4, respectively) lasted 338 minutes (program ID 50759); those on 2008 November 3 at 3.6 and 5.8 µm (channels 1 and 3, respectively) lasted 368 minutes (Program ID 50517). The IRAC beam splitter enabled simultaneous observations in the paired channels; all exposures were 12 seconds, resulting in 1696 frames in each of channels 1 and 3 and 1549 frames in each of channels 2 and 4. To minimize inter-pixel variability in all channels and the known intra-pixel variability in channels 1 and 2 (Reach et al. 2005; Charbonneau et al. 2005; Harrington et al. 2007; Stevenson et al. 2010), each target had fixed pointing. Prior to the science observations in channels 2 and 4, we observed a 57-frame preflash, exposing the array to a relatively bright source to reduce the time-dependent sensitivity (“ramp”) effect in channel 4 (Charbonneau et al. 2005; Harrington et al. 2007; Knutson et al. 2008, see Figure 1). Each observation ended with a 10-frame, post-eclipse observation of blank sky in the same array positon as the science observations to check for warm pixels in the photometric aperture. 3. DATA ANALYSIS Spitzer’s data pipeline (version S18.7.0) applied both standard and IRAC-specific corrections, producing the Basic Calibrated Data (BCD) we analyzed. Our analysis pipeline masks pixels according to Spitzer’s permanent bad pixel masks. It masks additional bad pixels (e.g., from cosmic-ray strikes), by grouping frames into sets of 64 and doing a two-iteration outlier rejection at each pixel location. Within each array position in each set, this routine calculates the standard deviation from the median, masks any pixels with greater than 4σ deviation, and repeats this procedure once. Masked pixels do not participate in the analysis. The channel-4 data show a horizontal streak of pixels with low fluxes located ∼10 pixels above the star. A similar diagonal streak appears ∼10 pixels below and left of the star. This artifact, which we masked, resulted from saturation in a prior observation. A two-dimensional Gaussian fit found the photometry center for each image (Stevenson et al. 2010, see the Supplementary Information for discussion of centering methods on Spitzer data). The pipeline uses interpolated aperture photometry (Harrington et al. 2007), ignoring frames with masked pixels in the photometry aperture and not using masked pixels in sky level averages. Table 1 presents photometry parameters. We evaluated numerous photometry apertures (see Table 5 in the appendix), choosing the one with the best final light-curve fit in each channel (see below). Because channel 4 had a higher background flux level, the best sky annulus was larger and the photometry aperture was smaller than in the other channels. The channel-4 aperture contained 63% of the point-spread function; the others contained 89% or more. The intra-pixel variation only affects channels 1 and 2, and was only substantial in channel 1 (see Table 1 and Figure 2). We model the intra-pixel effect with a second-order, twodimensional polynomial, VIP (x, y) = p1 y2 + p2 x2 + p3 xy + p4 y + p5x + 1, (1) where x and y are the centroid coordinates relative to the pixel center nearest the median position and p1 , p2 , p3 , p4 , and p5 can be free parameters. We model the ramp for channel 1 with the rising exponential R(t) = 1 − exp(−r1 [t − r2 ]) , (2) The Orbit of WASP-12b 1.05 3 1.01 3.6 4.5 5.8 8.0 1.01 1.00 3.6 4.5 5.8 8.0 1.00 1.00 0.99 0.99 0.98 Normalized Flux 0.98 Normalized Flux Normalized Flux 0.95 0.97 0.90 0.97 0.96 0.96 0.95 0.85 0.94 0.80 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Orbital Phase (1.091-day period) 0.93 0.35 3.6 4.5 5.8 8.0 0.40 0.95 0.45 0.50 0.55 0.60 Orbital Phase (1.091-day period) 0.65 0.94 0.35 0.40 0.45 0.50 0.55 0.60 Orbital Phase (1.091-day period) 0.65 F IG . 2.— Raw (left), binned (center), and systematics-corrected (right) secondary eclipse light curves of WASP-12b in the four IRAC channels, normalized to the mean system flux within the fitted data. Colored lines are the best-fit models; black curves omit their eclipse model elements. A few initial points in all channels are not fit, as indicated, to allow the telescope pointing and instrument to stabilize. Channel 2 Channel 1 -3 RMS RMS 10 10 10 0 10 1 10 2 -3 10 0 Bin Size 10 1 Channel 3 -2 10 2 Channel 4 -2 RMS 10 RMS 10 10 Bin Size -3 10 10 0 10 1 10 2 -3 10 0 Bin Size 10 1 10 2 Bin Size F IG . 3.— Root-mean-squared (RMS) residual flux vs. bin size in each channel. This plot tests for correlated noise. The straight line is the prediction for Gaussian white noise. Since the data do not deviate far from the line, the effect of correlated noise is minimal. where t is orbital phase and r1 and r2 are free parameters. The remaining channels used a linear model, R(t) = r3 (t − 0.5) + 1, (3) where r3 is a free parameter. The eclipse, E(t), is a Mandel & Agol (2002) model, assuming no limb darkening. The final light-curve model is F(x, y,t) = FsVIP (x, y)R(t)E(t), (4) where F(x, y,t) is the flux measured from interpolated aperture photometry and Fs is the (constant) system flux outside of eclipse, including the planet. To estimate photometric uncertainties, we propagate the values in the Spitzer BCD uncertainty images through the aperture photometry calculation. Since the Spitzer pipeline generally overestimates uncertainties, we fit an initial model with a χ2 minimizer and then scale all uncertainties to give a reduced χ2 of unity (Harrington et al. 2007). We confirm the fit by redoing it with the new uncertainties. The scaling factor is proportional to the standard deviation of the normalized residuals (SDNR) from the models, as reported in Tables 5 (in the appendix) and 1. The ∼2% SDNR variation does not significantly affect the fits. To select among models, we must compare fits made to the same data, including uncertainties. So, we use just one uncertainty scaling factor for all models in each combination of aperture and channel (see Tables 1 and 5 in the appendix). Sivia & Skilling (2006) provide an accessible tutorial to the Bayesian approach of our subsequent analysis. MacKay (2003, chapter 29, and especially section 4) introduces Markov-Chain Monte Carlo (MCMC) and discusses its practicalities. Briefly, the MCMC algorithm calculates χ2 at random locations near the χ2 minimum in the parameter phase space, accepting only some of these steps for later analysis. The density of these accepted points is proportional to the probability of a model at that location, given the data. The attraction of MCMC is that histograms and scatter plots of subsets of interesting parameters from the accepted points display parameter uncertainties and correlations in a way that fully accounts for the uncertainties in and correlations with 4 Campo et al. TABLE 1 J OINT L IGHT-C URVE F IT PARAMETERS Parameter 3.6 µm 4.5 µm 5.7 µm 8.0 µm Array Position (x̄, pix) Array Position (ȳ, pix) Position Consistencya (δx , pix) Position Consistencya (δy , pix) Aperture Size (pix) Sky Annulus Inner Radius (pix) Sky Annulus Outer Radius (pix) System Flux Fs (µJy) Eclipse Depthc (Fp /F∗ ) Brightness Temperature (K) Eclipse Mid-timeb, c (tmid , phase) Eclipse Mid-timed (tmid , BJD - 2,454,000) Eclipse Durationc (t4−1 , sec) Ingress Time (t2−1 , sec) Egress Time (t4−3 , sec) Ramp Name Ramp, Curvaturec (r1 ) Ramp, Phase Offsetc (r2 ) Ramp, Linear Termc (r3 ) Intra-pixel, Quadratic Term in yc (p1 ) Intra-pixel, Quadratic Term in xc (p2 ) Intra-pixel, Cross Term (p3 ) Intra-pixel, Linear Term in yc (p4 ) Intra-pixel, Linear Term in x (p5 ) Total frames Good framese Rejected framese (%) Free Parameters Number of Data Points in Fit BIC AIC Standard Deviation of Normalized Residuals Uncertainty Scaling Factor 25.20 26.98 0.012 0.012 3.75 7.00 12.00 25922 ± 11 0.00379 ± 0.00013 2740 ± 49 0.5010 ± 0.0006 773.6481 ± 0.0006 10615.66 ± 102.95 1266.43 1266.43 Rising Exponential 29 ± 1 0.17747 0 0 -0.140 ± 0.011 0 0.086 ± 0.004 0 1697 1532 9 10 3075 3155.5 3095.7 0.00228716 0.31248 20.24 27.95 0.013 0.013 4.00 7.00 12.00 16614 ± 3 0.00382 ± 0.00019 2571 ± 73 0.5006 ± 0.0007 769.2819 ± 0.0008 10749.97 ± 142.72 1266.43 1266.43 Linear 0 0.5 -0.0102 ± 0.0015 -0.09 ± 0.04 0 0 0 0 1560 1457 6 9 2924 2996.0 2942.2 0.00324027 0.44500 19.35 27.15 0.030 0.018 2.75 7.00 12.00 11129 ± 4 0.00629 ± 0.00052 3073 ± 176 0.5010 ± 0.0006 773.6481 ± 0.0006 10615.66 ± 102.95 1266.43 1266.43 Linear 0 0.5 -0.016 ± 0.004 0 0 0 0 0 1697 1543 9 10 3075 3155.5 3095.7 0.01058880 0.91832 21.45 25.67 0.13 0.14 2.00 12.00 30.00 6111 ± 3 0.00636 ± 0.00067 2948 ± 233 0.5006 ± 0.0007 769.2819 ± 0.0008 10749.97 ± 142.72 1266.43 1266.43 Linear 0 0.5 0.010 ± 0.005 0 0 0 0 0 1560 1467 5 9 2924 2996.0 2942.2 0.01222100 0.62475 a RMS frame-to-frame position difference. Based on the transit ephemeris time given by Hebb et al. (2009). MCMC jump parameter. d Uncorrected for light-travel time in the exoplanetary system (see Dynamics section). e We reject frames during instrument/telescope settling and with bad pixels in the photometry aperture. b c the uninteresting parameters. These are called marginal distributions. We fit equation 4 with a χ2 minimizer and assess parameter uncertainties with a Metropolis random-walk (MRW) MCMC algorithm. Our MRW used independent Gaussian proposal distributions for each parameter with widths chosen to give an acceptance rate of 20 – 60% of the steps. See Figures 6 and 7 for marginal distributions for the final models. The intent of MCMC is to explore the phase space, not to find one optimal model. Even the best model in an MCMC chain is not a good replacement for the model found by a minimizer, because MCMC is unlikely to land exactly on the minimum that a minimizer easily finds to machine precision. If an MCMC chain finds a lower χ2 value than the minimizer’s, then it has entered the basin of attraction around a better local minimum, and a minimizer will almost certainly find an even better χ2 starting from the MCMC’s best value. We thus refit at such points and then restart our MCMC routine from the new minimizer solution. The χ2 used in the information criteria described below refers to the global minimum of a given dataset and not merely the sampled minimum from MCMC. Although the differences may appear to be small, at the extreme precisions required for high-contrast photometry and models with many parameters, parameter values can differ by a significant fraction of 1σ between the global and MCMC minima, even for converged chains. The MCMC routine ran an initial “burn in” of a least 105 iterations to forget the initial starting conditions, and then used two million iterations to sample the phase space near the fit solution. To test for adequate sampling, we ran four independent MCMC chains, three started away from the initial minimizer location, and calculated the Gelman & Rubin (1992) statistic for each parameter. These were all within 1% of unity, indicating the chains converged. We initially fit each channel separately with all free model parameters as MCMC jump parameters (see Table 5 in the appendix). Then we pair the channels observed together, fitting a common eclipse phase and duration (see Table 1). Due to high correlations, the MCMC sampling becomes very inefficient with all the parameters free in the joint fit. Estimates of the interesting parameters (eclipse depth, time, and duration) are unaffected if we freeze r2 and the ingress and egress times at several different values. We set r2 from the independent light-curve fits and the ingress and egress times as predicted by the (Hebb et al. 2009) orbit. A recent re-analysis of older data by Knutson et al. (2009) demonstrates that the complex models required to fit Spitzer’s systematics can have multiple, comparable χ2 minima in different parts of phase space. These minima may change their relative depths given different systematic models (e.g., exponential vs. log-plus-linear ramps), resulting in different conclusions. To control for this, we fit data from a range of photometry apertures with many combinations of analytic model The Orbit of WASP-12b components (see Table 5 in the appendix) before choosing Eq. 1 – Eq. 3. The models included quadratic and logarithmicplus-linear ramps and a variety of polynomial intrapixel models. Additionally, we drop a small number of initial points to allow the pointing and instrument to stabilize, which vastly improved the fits. Choices among photometry apertures and numbers of dropped points are choices between different datasets fit with the same models, so we minimize the SDNR, removing the fewest points consistent with low SDNR. The model lines in Figure 2 show the included points. Once we have selected the dataset (by choice of aperture and dropped points according to SDNR), we may apply any of several information criteria to compare models with different numbers of free parameters (Liddle 2007). These criteria have specific goals and assumptions, so none is perfectly general, but two have broad application. The Akaike Information Criterion, AIC = χ2 + 2k, (5) where k is the number of free parameters, applies when the goal is accurate prediction of future data; its derivation is valid even when the candidate models might not include the theoretically correct one (as is the case, so far, for Spitzer intrapixel and ramp modeling). The Bayesian Information Criterion, BIC = χ2 + k ln N, (6) where N is the number of data points, applies when the goal is identifying the theoretically correct model, which is known to be one of those being considered. The best model minimizes the chosen information criterion. The ratio of probabilities favoring one model over another is exp(∆BIC/2), where ∆BIC is the difference in BIC between models, but the difference in AIC between models has no simple calibration to a probability or significance level. These goals give different answers for finite datasets. If the right model is a candidate, the BIC will do better than AIC as the number of points increases; if not, which is better depends on the sample size and on how close the candidate models are to the (absent) correct model. Other information criteria exist, but are either tailored to specific circumstances or are still being vetted by statisticians. The criteria solve different problems, but the goal of a multi-model analysis is not always easily classified as solely predictive or explanatory, so there is some elasticity regarding the choice of an appropriate criterion. We calculate AIC and BIC for hundreds of models, and reject most of them on this basis (see Table 5 in the appendix). For the final decision, we also consider the level of correlation in the residuals. For this, we plot root-mean-squared (RMS) model residuals vs. bin size (Pont et √ al. 2006, Winn et al. 2008) and compare to the theoretical 1/ N RMS scaling. Figure 3 demonstrates the lack of significant photometric noise correlation in our final models. In some cases, we prefer less-correlated models with insignificantly poorer AIC or BIC (e.g., channel 1). Differences in interesting parameter values (eclipse depth, time, and duration) for such near-optimal alternatives are . 1σ. Given the questions raised by re-analyses of certain Spitzer exoplanet datasets (Knutson et al. 2009; Beaulieu et al. 2010), we consider it critical that investigators disclose the details of their analyses both so that readers can assess the quality of the analysis and so that others may make meaningful compar- 5 isons in subsequent analyses of the same data (e.g., did they find a better χ2 ?). It is important to include a full description of the centering, photometry, uncertainty assessment, model fitting, correlation tests, phase-space exploration, and convergence tests. A listing of alternative model fits and their quality may build confidence that there is not a much better model than those tried. One must identify the particular χ2 minimum explored by reporting even nuisance parameter values, such as those in the intrapixel and ramp curves. Finally, the marginal posterior distibutions (i.e., the parameter histograms) and plots of their pairwise correlations help in assessing whether the phase space minimum is global and in determining parameter uncertainties. We present these plots for the astrophysical parameters in Figures 4, 5, 6, & 7. The electronic supplement to this article includes data files containing the photometry, best-fit models, centering data, etc.. We encourage all investigators to make similar disclosure in future reports of exoplanetary transits and eclipses. 4. DYNAMICS Hebb et al. (2009) detect a non-zero eccentricity for WASP12b that should be observable in the timing of the secondary eclipse. Our two secondary eclipse phases (Table 1) are within 2σ of φ = 0.5 for the Hebb et al. (2009) ephemeris, and taken together imply e cos ω = 0.0016 ± 0.0007. This indicates that if the planet’s orbit is eccentric, then ω is closely aligned with our line of sight. Recognizing the unlikelihood of this configuration (which implicitly questions the López-Morales et al. 2009 eclipse phase), this section nonetheless considers the possibility of significant eccentricity, with precession between the López-Morales et al. (2009) eclipse phase and Spitzer’s. Subsequent to the initial submission of this paper, López-Morales et al. (2010) increased their uncertainty by a factor of three. Since the arXiv postings of both LópezMorales et al. (2009) and the submitted version of this paper (arXiv 1003.2763v1) raised some community discussion, we now treat both cases to explain how this adjustment changes our conclusions. We use an MCMC routine to fit a Keplerian model of the planet’s orbit to our secondary eclipse times, radial velocity data (Hebb et al. 2009), transit timing data provided by the WASP team and amateur observers (Table 2), and the groundbased secondary eclipse measurement of López-Morales et al. (2009, 2010). Because López-Morales et al. folded 1.5 complete eclipses, we represent their point as a single observation taken during an orbit halfway between their eclipses (HJD 2455002.8560 ± 0.0024). We remove three in-transit radial velocity points due to Rossiter-McLaughlin contamination, and correct the times of mid-eclipse given in Table 1 and by López-Morales et al. (2009, 2010) for light travel across the orbit by subtracting 22.8 seconds. We note that eclipse observers should report uncorrected times, as the correction depends on the orbit model and, in the future, measurements may be uncertain at the level of model uncertainty. The amateur observers synchronize their clocks to within one second of UTC by means such as Network Time Protocol (NTP) or radio signals from atomic clocks. In pre-publication discussions with Eastman et al. (2010), we determined that the amateurs’ observing software, MaximDL, did not account for leap seconds, nor did the software of most of our professional contributors. We thus made the adjustment ourselves as needed. Campo et al. a b CH1 System Flux (✁Jy) F IG . 4.— Parameter correlations for 3.6 and 5.8 µm. To decorrelate the Markov chains and unclutter the plot, one point appears for every 1000th MCMC step. Each panel contains all the points. 0.120 0.118 0.116 0.114 0.112 0.110 0.0050 0.0045 0.0040 0.0035 0.0030 0.0025 16625 16620 16615 16610 16605 16600 0.0085 0.0080 0.0075 0.0070 0.0065 0.0060 0.0055 0.0050 0.0045 0.0040 6125 6120 ✂ 6115 6110 6105 6100 With Precession -0.065 ± 0.014 0.0014 ± 0.0007 0.065 ± 0.014 -88.8 ± 0.9 0± 0 1.0914240 ± 3×10−7 1.0914240 ± 3×10−7 508.97686 ± 0.00012 224 ± 4 19087 ± 3 101.0 -0.065 ± 0.014 -0.0058 ± 0.0027 0.065 ± 0.014 -95.1 ± 2.3 0.026 ± 0.009 1.091436 ± 4×10−6 1.091521 ± 3×10−5 508.97686 ± 0.00012 224 ± 4 19088 ± 3 97.6 ✂ MCMC Jump Parameter. MJD = JD - 2,454,000. In our model, X  vrv,o − vrv,m 2 X  ttr,o − ttr,m 2 + χ2 = σrv σtr  X tecl,o − tecl,m 2 + , σecl 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 CH4 Eclipse Flux Ratio No Precession CH3 Eclipse Flux Ratio 0.498 0.499 0.500 0.501 0.502 0.503 0.504 0.110 0.112 0.114 0.116 0.118 0.120 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 16600 16605 16610 16615 16620 16625 0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080 0.0085 a CH1 Eclipse Flux Ratio CH4 System Flux ( Jy) e ω0 (◦ ) ω̇ (◦ d-1)a Ps (days)a Pa (days) T0 (MJD)a,b K (ms-1)a γ (ms-1)a BIC Eclipse Duration CH2 System Flux ( Jy) TABLE 3 O RBITAL F ITS e sin ω0 a e cos ω0 a Eclipse Phase CH2 Eclipse Flux Ratio The Amateur Exoplanet Archive (AXA, http://brucegary.net/AXA/x.htm) and TRansiting ExoplanetS and CAndidates group (TRESCA, http://var2.astro.cz/EN/tresca/index.php) supply their data to the Exoplanet Transit Database (ETD, http://var2.astro.cz/ETD/), which performs the uniform transit analysis described by Poddaný et al. (2010). The ETD web site provided the AXA and TRESCA numbers in this table. |Correlation Coefficients| 0.00141 0.00149 0.00014 0.00203 0.00141 0.00131 |Correlation Coefficients| 2455151.82129 2455164.92317 2455172.5620 2455197.6628 2455198.75595 2455219.48996 WASP Team WASP Team WASP Team Hebb et al. (2009) WASP Team WASP Team Veli-Pekka Hentunen, AXA Alessandro Marchini, AXA Bruce Gary, AXA Frantis̆ek Lomoz, TRESCA Yenal Öğmen, TRESCA Jaroslav Trnka, TRESCA Alessandro Marchini, AXA Ramon Naves, AXA Lubos Brát, TRESCA Leonard Kornos and Peter Veres, TRESCA Stan Shadick, TRESCA Stan Shadick, TRESCA Mikael Ingemyr, TRESCA Brian Tieman, TRESCA Brian Tieman, TRESCA Lubos Brát, TRESCA 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.499 0.500 0.501 0.502 0.503 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.0032 0.0034 0.0036 0.0038 0.0040 0.0042 0.0044 25880 25890 25900 25910 25920 25930 25940 25950 25960 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080 0.0048 0.0070 0.0017 0.0002 0.00016 0.0034 0.0006 0.0013 0.001 0.00213 0.0023 0.00132 0.0056 0.001 0.00098 0.00066 CH1 Eclipse Flux Ratio 2453264.7594 2454120.4290 2454129.1600 2454508.9761 2454515.52464 2454552.6218 2454836.4026 2454837.4955 2454840.7704 2454848.41003 2454860.41473 2454860.4176 2454883.33312 2454908.4372 2454931.35739 2455136.54322 CH1 System Flux ( Jy) Sourcea CH3 Eclipse Flux Ratio Uncertainty CH3 System Flux ( Jy) Mid-Transit Time (HJD) Eclipse Duration TABLE 2 T RANSIT T IMING D ATA Parameter 0.117 0.116 0.115 0.114 0.113 0.112 0.111 0.110 0.109 0.108 0.0044 0.0042 0.0040 0.0038 0.0036 0.0034 0.0032 25960 25950 25940 25930 25920 25910 25900 25890 25880 0.0080 0.0075 0.0070 0.0065 0.0060 0.0055 0.0050 0.0045 11145 11140 11135 11130 11125 11120 11115 11110 Eclipse Duration 6 Eclipse Phase Eclipse Duration CH2 Eclipse Flux Ratio CH2 System Flux (✄Jy) CH4 Eclipse Flux Ratio F IG . 5.— Parameter correlations for 4.5 and 8.0 µm. To decorrelate the Markov chains and unclutter the plot, one point appears for every 1000th MCMC step. Each panel contains all the points. (7) where vrv,o , ttr,o , tecl,o are the observed radial velocities, transit times, and eclipse times, respectively; σrv , σtr , σecl are their respective uncertainties, and vrv,m , ttr,m , tecl,m are the respective model calculations. Table 3 gives our best-fit results using the original LópezMorales et al. (2009); differences from our arXiv posting are due to the time corrections and the use of a minimizer to find the true χ2 minimum. The eccentricity of e = 0.065 ± 0.014 may be high due to poor constraints on e sin ω. Our dynamical fits considered only the transit and eclipse times and did not directly fit the light curves, which could additionally have modeled variable eclipse and transit durations. A significantly positive eccentricity implies either extremely low tidal dissipation (e.g., Qp & 108 ; a tidal evolution model could give a better limit, Mardling 2007, Levrard et al. 2009) or a perturber such as another planet. In the latter case, coupling between the two planets could potentially drain energy and angular momentum from the outer orbit to the point where it is not able to maintain a large eccentricity for WASP-12b (e.g., Mardling 2007). Tidal dissipation of a The Orbit of WASP-12b 3000 2500 2500 2000 2000 1500 1500 1000 1000 500 500 500 0 0 0 2000 1500 0.498 0.499 0.500 0.501 0.502 0.503 0.504 1000 3000 Eclipse Phase 3000 2000 2000 1500 1500 1000 1000 500 500 0 0 CH1 System Flux (☎Jy) CH1 Eclipse Flux Ratio 3500 3000 2500 2000 1500 1000 500 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080 0.0085 2500 25880 25890 25900 25910 25920 25930 25940 25950 25960 2500 Eclipse Duration CH3 Eclipse Flux Ratio 0 11110 11115 11120 11125 11130 11135 11140 11145 11150 2500 0.0032 0.0034 0.0036 0.0038 0.0040 0.0042 0.0044 3000 3000 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 3500 7 CH3 System Flux (☎Jy) 500 500 0 0 CH2 System Flux (✆Jy) 0.0045 0.0050 0.0035 0.0040 CH2 Eclipse Flux Ratio 3000 2500 2000 1500 1000 CH4 Eclipse Flux Ratio 500 0 6125 1000 F IG . 8.— Transit times and orbit models of Table 3. Top: Non-precessing case. Bottom: Precessing case. Both diagrams show the difference (observed-minus-calculated, O-C) from a linear ephemeris determined from Ps and T0 given for their respective cases in Table 3. This highlights deviations from the ephemeris. The dashed lines give eclipse times for the eccentric orbits of the fits. In the non-precessing case, these lines are straight and horizontal, and the transits (which carry the most weight in determining the period) scatter about the line, as expected. However, the López-Morales et al. (2009) point suggests a trend in the eclipses consistent with apsidal precession. In the precessing case, the models are opposing sinusoids with a ∼32-minute amplitude and a 33-year period. The curves cross approximately a cos ω0 where ω = −90◦ . Both curves and data were shifted upward by − eP π (about 2 minutes) to adjust for the modification in Eq. 8, so the curves cross where O-C = 0. 3500 0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080 0.0085 1000 16625 1500 16615 1500 16620 2000 16610 2000 16605 2500 16600 2500 Eclipse Duration 6120 3000 0.118 Eclipse Phase 6115 3000 0.116 0 0.114 0 0.110 500 0.497 0.498 0.499 0.500 0.501 0.502 0.503 0.504 500 0.112 1000 1000 6110 1500 1500 6105 2000 2000 0.120 2500 4000 3500 3000 2500 2000 1500 1000 500 0 0.0025 2500 6100 3000 0.0030 F IG . 6.— Parameter histograms for 3.6 and 5.8 µm. To decorrelate the Markov chains, the histograms come from every 100th MCMC step. CH4 System Flux (✆Jy) F IG . 7.— Parameter histograms for 4.5 and 8.0 µm. To decorrelate the Markov chains, the histograms come from every 100th MCMC step. non-zero eccentricity could account for the inflated radius of WASP-12b. If the orbit is actually circular, the bloated size (Hebb et al. 2009) requires either an energy source or new interior models. As noted above, the planet’s proximity to its star must raise huge tidal bulges (Ragozzine & Wolf 2009) that significantly contribute to an aspherical planetary gravitational potential. This would induce apsidal precession measurable over short timescales through transit and eclipse timing variations. The rate of precession is proportional to the tidal Love number, k2p , which describes the concentration of the planet’s interior mass (Ragozzine & Wolf 2009). A lower k2p implies more central condensation, but k2p alone does not define a unique density profile (Batygin et al. 2009). A nominal value of k2p = 0.3 yields precession of ∼0.05◦d-1 for the orbit of WASP12b. A precise measurement of the precession rate will therefore constrain the planet’s internal structure, as long as the eccentricity is significantly non-zero (Ragozzine & Wolf 2009). Conversely, the absence of observable precession limits the eccentricity. We added a constant precession term, ω̇, to our model, and took the inclination to be ∼90◦, as the timing effects due to inclination should be negligible and the available timing data cannot directly constrain this quantity. With these assumptions we modified Eq. 15 of Giménez & Bastero (1995) such that Ttr = T0 + PsE − ePa (cos ωtr − cosω0 )], π (8) where Ttr is the time of mid-transit, T0 is the transit time at orbit zero, Ps is the sidereal period, and Pa is the anomalistic period, or time between successive periastron passages. The right bracket indicates truncation of a series. Furthermore, Pa 8 Campo et al. is related to Ps by Pa = Ps . ω̇ 1 − Ps 2π (9) E is the number of elapsed sidereal periods since T0 and ωtr = ω̇(Ttr − T0 ) + ω0 , where ω0 is ω at T0 . We expand the equation to fifth order in e and solve iteratively for Ttr . We compute the eclipse time as a function of e, ωtr , Pa , and Ttr ; radial velocity is computed as a function of ω(t). Fitting this model to the data with the López-Morales et al. (2009) point, we found that ω̇ = 0.026 ± 0.009◦d-1, a 3σ result. This corresponds to a precession period of 33 ± 13 years and implies that k2p = 0.15 ± 0.08 (see Table 3). This result depended on an unlikely alignment of the orbit with our line of sight during the Spitzer observations. The revised López-Morales et al. (2010) uncertainty dashed hopes for detecting precession, however, as the model fit with that point (Table 4) yields a marginal precession, and BIC prefers the non-precessing case. Even if the 4σ eccentricity stands, measurement of precession awaits a longer observational baseline. of the eclipse depth in channel 1 is over 29, second only to that for HD 189733b), WASP-12b has emerged as a highly observable exoplanet. Madhusudhan et al. (2010) report our analysis of the planet’s atmospheric composition. Its phase curves, already in Spitzer’s queue, will enable the first observational discussion of atmospheric dynamics on a prolate planet. We thank the observers listed in Table 2 for allowing us to use their results, and the organizers of the Exoplanet Transit Database for coordinating the collection and uniform analysis of these data. The IRAC data are based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. The point TABLE 4 R EVISED O RBITAL F ITS 5. CONCLUSIONS The timing of the Spitzer eclipses is consistent with a circular orbit, and our best fit, including RV data and transit and eclipse times, does not detect precession. Although the López-Morales et al. (2010) eclipse phase is now marginally consistent with zero eccentricity, we note that this 0.9-µm observation could be affected by a wavelengthdependent asymmetry in the planet’s surface-brightness distribution that manifests itself as a timing offset (Knutson et al. 2007). This offset has a maximum possible value of Rp /vp ≈ 9 minutes, where vp is the planet’s orbital velocity. This is somewhat smaller than the observed variation in eclipse timing between López-Morales et al. (2010) and Spitzer. While we have not yet measured precession, the possible prolateness should be measurable in high-accuracy, infrared transits and eclipses (Ragozzine & Wolf 2009), such as we expect will be available from the James Webb Space Telescope. This would provide another constraint on interior structure, one that does not depend on an elliptical orbit. As this paper was in late stages of revision, Croll et al. (2010) published three ground-based secondary eclipses and Husnoo et al. (2010) produced additional radial velocity data. These datasets are consistent with a circular orbit for WASP12b. As the quality of these data attests (the signal-to-noise ratio a b Parameter No Precession With Precession e sin ω0 a e cos ω0 a e ω0 (◦ ) -1 ◦ a ω̇ ( d ) Ps (days) a Pa (days) T0 (MJD)a,b K (ms-1a ) γ (ms-1a ) BIC -0.063 ± 0.014 0.0011 ± 0.00072 0.063 ± 0.014 -89.0 ± 0.8 0± 0 1.0914240 ± 3×10−7 1.0914240 ± 3×10−7 508.97683 ± 0.00012 225 ± 4 19087 ± 3 90.1 -0.065 ± 0.015 -0.0036 ± 0.0045 0.065 ± 0.015 -93 ± 5 0.017 ± 0.019 1.0914315 ± 7×10−6 1.0914872 ± 7×10−5 508.97685 ± 0.00012 224 ± 4 19088 ± 3 92.8 MCMC Jump Parameter. MJD = JD - 2,454,000. of M. Ingemyr is based on observations made with the Nordic Optical Telescope (NOT), operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos Instituto Astrofisica de Canarias, and ALFOSC, which is owned by the Instituto Astrofisica de Andalucia (IAA) and operated at the NOT under agreement between IAA and NBlfAFG of the Astronomical Observatory of Copenhagen. We thank contributors to SciPy, Matplotlib, and the Python Programming Language, W. Landsman and other contributors to the Interactive Data Language Astronomy Library, the free and opensource community, the NASA Astrophysics Data System, and the JPL Solar System Dynamics group for free software and services. APPENDIX TABLE 5 C ANDIDATE M ODELS Model Apa NFPb BICc AICc SDNRc Channel 1, All Intrapixel Parameters Free: 1553 points, uncertainties multiplied by 0.30946 Linear 3.00 11 1622.8 1564.0 Quadratic 3.00 11 1612.1 1553.3 Log+Linear 3.00 12 1624.5 1560.3 Rising Exp 3.00 10 1611.8 1558.3 0.00232818 0.00231926 0.00232366 0.00232512 1544 points, uncertainties multiplied by 0.31028 Linear 3.25 11 1613.8 1555.0 Quadratic 3.25 11 1602.7 1543.9 0.00230984 0.00230068 The Orbit of WASP-12b 9 TABLE 5 — Continued Model Apa NFPb BICc AICc SDNRc Log+Linear Rising Exp 3.25 3.25 12 10 1615.1 1602.4 1551.0 1549.0 0.00230511 0.00230655 1536 points, uncertainties multiplied by 0.31130 Linear 3.50 11 1605.7 1547.0 Quadratic 3.50 11 1591.8 1533.0 Log+Linear 3.50 12 1605.9 1541.9 Rising Exp 3.50 10 1592.3 1539.0 0.00229716 0.00228574 0.00229146 0.00229226 1532 points, uncertainties multiplied by 0.31286 Linear 3.75 11 1601.7 1543.0 Quadratic 3.75 11 1588.9 1530.2 Log+Linear 3.75 12 1601.5 1537.5 Rising Exp 3.75 10 1588.8 1535.4 0.00229111 0.00228061 0.00228509 0.00228651 1530 points, uncertainties multiplied by 0.31547 Linear 4.00 11 1599.7 1541.0 Quadratic 4.00 11 1586.5 1527.8 Log+Linear 4.00 12 1599.2 1535.2 Rising Exp 4.00 10 1586.4 1533.0 0.00229468 0.00228382 0.00228842 0.00228974 Channel 1, Intrapixel with only x2 and y free: 1553 points, uncertainties multiplied by 0.30916 Linear 3.00 8 1603.8 1561.0 Quadratic 3.00 8 1593.0 1550.2 Log+Linear 3.00 9 1605.0 1556.9 Rising Exp 3.00 7 1592.7 1555.3 0.00232820 0.00231930 0.00232329 0.00232514 1544 points, uncertainties multiplied by 0.31013 Linear 3.25 8 1594.7 1552.0 Quadratic 3.25 8 1583.9 1541.2 Log+Linear 3.25 9 1596.1 1548.1 Rising Exp 3.25 7 1583.4 1546.0 0.00231106 0.00230213 0.00230627 0.00230874 1536 points, uncertainties multiplied by 0.31106 Linear 3.50 8 1586.7 1544.0 Quadratic 3.50 8 1572.6 1529.9 Log+Linear 3.50 9 1587.0 1539.0 Rising Exp 3.50 7 1573.2 1535.8 0.00229865 0.00228619 0.00229298 0.00229367 1532 points, uncertainties multiplied by 0.31261 Linear 3.75 8 1582.7 1540.0 Quadratic 3.75 8 1569.7 1527.0 Log+Linear 3.75 9 1583.3 1535.3 Rising Exp 3.75 7 1569.6 1532.3 0.00229250 0.00228188 0.00228609 0.00228781 1530 points, uncertainties multiplied by 0.31521 Linear 4.00 8 1580.7 1538.0 Quadratic 4.00 8 1567.3 1524.6 Log+Linear 4.00 9 1583.8 1535.8 Rising Exp 4.00 7 1567.2 1529.9 0.00229595 0.00228496 0.00229263 0.00229094 Channel 2, All Intrapixel Parameters Free: 1465 points, uncertainties multiplied by 0.44456 No Ramp 3.50 9 1521.6 1474.0 Linear 3.50 10 1501.7 1448.8 Quadratic 3.50 11 1506.8 1448.6 Falling Exp 3.50 11 1509.9 1451.7 0.00326693 0.00323655 0.00323340 0.00323775 1460 points, uncertainties multiplied by 0.44663 No Ramp 3.75 9 1516.6 1469.0 Linear 3.75 10 1496.9 1444.1 Quadratic 3.75 11 1502.3 1444.1 Falling Exp 3.75 11 1505.1 1446.9 0.00326495 0.00323477 0.00323189 0.00323590 1457 points, uncertainties multiplied by 0.44875 No Ramp 4.00 9 1513.6 1466.0 Linear 4.00 10 1492.6 1439.7 Quadratic 4.00 11 1497.7 1439.6 Falling Exp 4.00 11 1500.8 1442.7 0.00326355 0.00323181 0.00322873 0.00323302 1449 points, uncertainties multiplied by 0.45211 No Ramp 4.25 9 1505.5 1458.0 Linear 4.25 10 1483.7 1430.9 0.00327075 0.00323780 10 Campo et al. TABLE 5 — Continued Model Apa NFPb BICc AICc SDNRc Quadratic Falling Exp 4.25 4.25 11 11 1488.5 1492.0 1430.4 1434.0 0.00323425 0.00323913 1435 points, uncertainties multiplied by 0.45715 No Ramp 4.50 9 1491.4 1444.0 Linear 4.50 10 1470.1 1417.4 Quadratic 4.50 11 1474.8 1416.9 Falling Exp 4.50 11 1478.4 1420.5 0.00329102 0.00325808 0.00325441 0.00325944 Channel 2, Intrapixel With Only y2 Free: 1465 points, uncertainties multiplied by 0.44695 No Ramp 3.50 5 1496.4 1470.0 Linear 3.50 6 1464.4 1432.6 Quadratic 3.50 7 1470.0 1433.0 Falling Exp 3.50 7 1472.6 1435.6 0.00328991 0.00324507 0.00324266 0.00324626 1460 points, uncertainties multiplied by 0.44969 No Ramp 3.75 5 1491.4 1465.0 Linear 3.75 6 1455.3 1423.6 Quadratic 3.75 7 1461.2 1424.2 Falling Exp 3.75 7 1463.5 1426.5 0.00329263 0.00324297 0.00324081 0.00324417 1457 points, uncertainties multiplied by 0.45194 No Ramp 4.00 5 1488.4 1462.0 Linear 4.00 6 1450.5 1418.8 Quadratic 4.00 7 1456.2 1419.2 Falling Exp 4.00 7 1458.7 1421.8 0.00329209 0.00324027 0.00323791 0.00324155 1449 points, uncertainties multiplied by 0.45480 No Ramp 4.25 5 1480.4 1454.0 Linear 4.25 6 1444.5 1412.9 Quadratic 4.25 7 1449.8 1412.9 Falling Exp 4.25 7 1452.9 1415.9 0.00329562 0.00324584 0.00324293 0.00324720 1435 points, uncertainties multiplied by 0.45965 No Ramp 4.50 5 1466.3 1440.0 Linear 4.50 6 1432.7 1401.1 Quadratic 4.50 7 1437.9 1401.0 Falling Exp 4.50 7 1441.0 1404.1 0.00331453 0.00326654 0.00326347 0.00326791 Channel 3: 1544 points, uncertainties multiplied by 0.91447 No Ramp 2.50 2 1556.7 1546.0 Linear 2.50 3 1553.7 1537.6 Falling Exp 2.50 4 1563.2 1541.9 0.01066331 0.01062468 0.01063552 1543 points, uncertainties multiplied by 0.92247 No Ramp 2.75 2 1555.7 1545.0 Linear 2.75 3 1549.9 1533.9 Falling Exp 2.75 4 1558.7 1537.4 0.01063659 0.01058879 0.01059726 1535 points, uncertainties multiplied by 0.93860 No Ramp 3.00 2 1547.7 1537.0 Linear 3.00 3 1543.1 1527.1 Falling Exp 3.00 4 1551.7 1530.4 0.01080977 0.01076529 0.01077319 1529 points, uncertainties multiplied by 0.95222 No Ramp 3.25 2 1541.7 1531.0 Linear 3.25 3 1538.6 1522.6 Falling Exp 3.25 4 1547.6 1526.2 0.01103641 0.01099631 0.01100564 1524 points, uncertainties multiplied by 0.96551 No Ramp 3.50 2 1536.7 1526.0 Linear 3.50 3 1535.2 1519.2 Falling Exp 3.50 4 1544.0 1522.7 0.01132113 0.01128630 0.01129501 1522 points, uncertainties multiplied by 0.97776 No Ramp 3.75 2 1534.7 1524.0 Linear 3.75 3 1533.8 1517.8 Falling Exp 3.75 4 1542.3 1521.0 0.01163603 0.01160264 0.01161056 1521 points, uncertainties multiplied by 0.99239 No Ramp 4.00 2 1533.7 1523.0 Linear 4.00 3 1533.2 1517.2 0.01201476 0.01198222 The Orbit of WASP-12b 11 TABLE 5 — Continued Model Apa NFPb BICc AICc SDNRc Falling Exp 4.00 4 1541.6 1520.3 0.01198978 1467 points, uncertainties multiplied by 0.64001 No Ramp 2.00 2 1479.6 1469.0 Linear 2.00 3 1470.6 1454.8 Rising Exp 2.00 4 1477.1 1455.9 Log+Linear 2.00 5 1484.4 1458.0 Quadratic 2.00 4 1476.2 1455.0 0.01269320 0.01262427 0.01261620 0.01261728 0.01260762 1467 points, uncertainties multiplied by 0.65283 No Ramp 2.25 2 1479.6 1469.0 Linear 2.25 3 1466.9 1451.1 Rising Exp 2.25 4 1473.8 1452.6 Log+Linear 2.25 5 1479.7 1453.2 Quadratic 2.25 4 1473.1 1452.0 0.01291715 0.01283127 0.01282649 0.01281869 0.01281909 1464 points, uncertainties multiplied by 0.67232 No Ramp 2.50 2 1476.6 1466.0 Linear 2.50 3 1467.5 1451.6 Rising Exp 2.50 4 1474.6 1453.4 Log+Linear 2.50 5 1481.6 1455.1 Quadratic 2.50 4 1474.3 1453.1 0.01333255 0.01326022 0.01325764 0.01325392 0.01325268 1455 points, uncertainties multiplied by 0.68250 No Ramp 2.75 2 1467.6 1457.0 Linear 2.75 3 1459.1 1443.2 Rising Exp 2.75 4 1466.4 1445.2 Log+Linear 2.75 5 1474.2 1447.8 Quadratic 2.75 4 1466.3 1445.2 0.01361319 0.01354198 0.01354054 0.01354051 0.01353948 1448 points, uncertainties multiplied by 0.69698 No Ramp 3.00 2 1460.6 1450.0 Linear 3.00 3 1451.3 1435.4 Rising Exp 3.00 4 1458.5 1437.4 Log+Linear 3.00 5 1467.0 1440.6 Quadratic 3.00 4 1458.5 1437.3 0.01404468 0.01396745 0.01396580 0.01396978 0.01396422 1443 points, uncertainties multiplied by 0.71158 No Ramp 3.25 2 1455.5 1445.0 Linear 3.25 3 1443.8 1428.0 Rising Exp 3.25 4 1451.0 1429.9 Log+Linear 3.25 5 1456.9 1430.6 Quadratic 3.25 4 1450.9 1429.8 0.01457611 0.01448370 0.01448177 0.01448402 0.01447941 1440 points, uncertainties multiplied by 0.72888 No Ramp 3.50 2 1452.5 1442.0 Linear 3.50 3 1437.9 1422.1 Rising Exp 3.50 4 1445.1 1424.0 Log+Linear 3.50 5 1451.3 1424.9 Quadratic 3.50 4 1444.9 1423.8 0.01528549 0.01517316 0.01517067 0.01517428 0.01516664 1431 points, uncertainties multiplied by 0.75781 No Ramp 4.00 2 1443.5 1433.0 Linear 4.00 3 1424.0 1408.2 Rising Exp 4.00 4 1431.1 1410.1 Log+Linear 4.00 5 1437.9 1411.6 Quadratic 4.00 4 1431.0 1409.9 0.01694407 0.01679042 0.01678704 0.01679340 0.01678180 Channel 4: a Aperture radius in pixels b c Number of free parameters (k in the text) Compare between the aperture sizes, for the same model, by SDNR. Compare within the aperture sizes by BIC and AIC. REFERENCES Batygin, K., Bodenheimer, P., & Laughlin, G. 2009, ApJ, 704, L49 Beaulieu, J., Tinetti, G., Kipping, D. M., Ribas, I., Barber, R. J., Y-K. Cho, J., Polichtchouk, I., Tennyson, J., Yurchenko, S. N., Griffith, C. A., Batista, V., Waldmann, I., Miller, S., Carey, S., Mousis, O., Fossey, S. J., & Aylward, A. 2010, ArXiv, 1007.0324 Bodenheimer, P., Laughlin, G., & Lin, D. N. C. 2003, ApJ, 592, 555 Charbonneau, D., Allen, L. E., Megeath, S. T., Torres, G., Alonso, R., Brown, T. M., Gilliland, R. L., Latham, D. W., Mandushev, G., O’Donovan, F. T., & Sozzetti, A. 2005, ApJ, 626, 523 Charbonneau, D., Brown, T. M., Burrows, A., & Laughlin, G. 2007, Protostars and Planets V, 701 Croll, B., Lafreniere, D., Albert, L., Jayawardhana, R., Fortney, J. J., & Murray, N. 2010, ArXiv, 1009.0071, ApJ, in press Eastman, J., Siverd, R., & Gaudi, B. S. 2010, PASP, 122, 935 12 Campo et al. Fazio, G. G., Hora, J. L., Allen, L. E., Ashby, M. L. N., Barmby, P., Deutsch, L. K., Huang, J., Kleiner, S., Marengo, M., Megeath, S. T., Melnick, G. J., Pahre, M. A., Patten, B. M., Polizotti, J., Smith, H. A., Taylor, R. S., Wang, Z., Willner, S. P., Hoffmann, W. F., Pipher, J. L., Forrest, W. J., McMurty, C. W., McCreight, C. R., McKelvey, M. E., McMurray, R. E., Koch, D. G., Moseley, S. H., Arendt, R. G., Mentzell, J. E., Marx, C. T., Losch, P., Mayman, P., Eichhorn, W., Krebs, D., Jhabvala, M., Gezari, D. Y., Fixsen, D. J., Flores, J., Shakoorzadeh, K., Jungo, R., Hakun, C., Workman, L., Karpati, G., Kichak, R., Whitley, R., Mann, S., Tollestrup, E. V., Eisenhardt, P., Stern, D., Gorjian, V., Bhattacharya, B., Carey, S., Nelson, B. O., Glaccum, W. J., Lacy, M., Lowrance, P. J., Laine, S., Reach, W. T., Stauffer, J. A., Surace, J. A., Wilson, G., Wright, E. L., Hoffman, A., Domingo, G., & Cohen, M. 2004, ApJS, 154, 10 Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659, 1661 Gelman, A. & Rubin, D. B. 1992, Stat. Sci., 7, 457 Giménez, A. & Bastero, M. 1995, Ap&SS, 226, 99 Goldreich, P. & Soter, S. 1966, Icarus, 5, 375 Harrington, J., Luszcz, S., Seager, S., Deming, D., & Richardson, L. J. 2007, Nature, 447, 691 Hebb, L., Collier-Cameron, A., Loeillet, B., Pollacco, D., Hébrard, G., Street, R. A., Bouchy, F., Stempels, H. C., Moutou, C., Simpson, E., Udry, S., Joshi, Y. C., West, R. G., Skillen, I., Wilson, D. M., McDonald, I., Gibson, N. P., Aigrain, S., Anderson, D. R., Benn, C. R., Christian, D. J., Enoch, B., Haswell, C. A., Hellier, C., Horne, K., Irwin, J., Lister, T. A., Maxted, P., Mayor, M., Norton, A. J., Parley, N., Pont, F., Queloz, D., Smalley, B., & Wheatley, P. J. 2009, ApJ, 693, 1920 Husnoo, N., Pont, F., Hebrard, G., Simpson, E., Mazeh, T., Bouchy, F., Moutou, C., Arnold, L., Boisse, I., Diaz, R., Eggenberger, A., & Shporer, A. 2010, ArXiv, 1004.1809 Jackson, B., Greenberg, R., & Barnes, R. 2008, ApJ, 678, 1396 Kallrath, J. & Milone, E. F. 1999, Eclipsing Binary Stars: Modeling and Analysis (New York: Springer) Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., & Megeath, S. T. 2008, ApJ, 673, 526 Knutson, H. A., Charbonneau, D., Allen, L. E., Fortney, J. J., Agol, E., Cowan, N. B., Showman, A. P., Cooper, C. S., & Megeath, S. T. 2007, Nature, 447, 183 Knutson, H. A., Charbonneau, D., Cowan, N. B., Fortney, J. J., Showman, A. P., Agol, E., & Henry, G. W. 2009, ApJ, 703, 769 Kreiner, J. M., Kim, C., & Nha, I. 2001, An Atlas of O-C Diagrams of Eclipsing Binary Stars (Cracow, Poland: Wydawnictwo Naukowe Akademii Pedagogicznej) Levrard, B., Winisdoerffer, C., & Chabrier, G. 2009, ApJ, 692, L9 Li, S., Miller, N., Lin, D. N. C., & Fortney, J. J. 2010, Nature, 463, 1054 Liddle, A. R. 2007, MNRAS, 377, L74 López-Morales, M., Coughlin, J. L., Sing, D. K., Burrows, A., Apai, D., Rogers, J. C., & Spiegel, D. S. 2009, ArXiv, 0912.2359 López-Morales, M., Coughlin, J. L., Sing, D. K., Burrows, A., Apai, D., Rogers, J. C., Spiegel, D. S., & Adams, E. R. 2010, ApJ, 716, L36 MacKay, D. J. C. 2003, Information Theory, Inference and Learning Algorithms (New York: Cambridge University Press) Madhusudhan, N., Harrington, J., Stevenson, K. B., Nymeyer, S., Campo, C., Wheatley, P. J., Deming, D., Blecic, J., Hardy, R., Lust, N. B., Anderson, D. R., Collier Cameron, A., Britt, C. B. T., Bowman, W. C., Hebb, L., Hellier, C., Maxted, P. F. L., Pollacco, D., & West, R. G. 2010, Nature,, in press Mandel, K. & Agol, E. 2002, ApJ, 580, L171 Mardling, R. A. 2007, MNRAS, 382, 1768 —. 2010, ArXiv, 1001.4079 Poddaný, S., Brát, L., & Pejcha, O. 2010, New Astronomy, 15, 297 Pont, F., Zucker, S., & Queloz, D. 2006, MNRAS, 373, 231 Ragozzine, D. & Wolf, A. S. 2009, ApJ, 698, 1778 Reach, W. T., Megeath, S. T., Cohen, M., Hora, J., Carey, S., Surace, J., Willner, S. P., Barmby, P., Wilson, G., Glaccum, W., Lowrance, P., Marengo, M., & Fazio, G. G. 2005, PASP, 117, 978 Sivia, D. & Skilling, J. 2006, Data Analysis: A Bayesian Tutorial, 2nd edn. (Oxford Univ. Press, USA) Stevenson, K. B., Harrington, J., Nymeyer, S., Madhusudhan, N., Seager, S., Bowman, W. C., Hardy, R. A., Deming, D., Rauscher, E., & Lust, N. B. 2010, Nature, 464, 1161 Werner, M. W., Roellig, T. L., Low, F. J., Rieke, G. H., Rieke, M., Hoffmann, W. F., Young, E., Houck, J. R., Brandl, B., Fazio, G. G., Hora, J. L., Gehrz, R. D., Helou, G., Soifer, B. T., Stauffer, J., Keene, J., Eisenhardt, P., Gallagher, D., Gautier, T. N., Irace, W., Lawrence, C. R., Simmons, L., Van Cleve, J. E., Jura, M., Wright, E. L., & Cruikshank, D. P. 2004, ApJS, 154, 1 Winn, J. N. 2009, in IAU Symposium 253: Transiting Planets, 99–109 Winn, J. N., Holman, M. J., Torres, G., McCullough, P., Johns-Krull, C., Latham, D. W., Shporer, A., Mazeh, T., Garcia-Melendo, E., Foote, C., Esquerdo, G., & Everett, M. 2008, ApJ, 683, 1076