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Spitzer Transit and Secondary Eclipse Photometry of GJ 436b

We report the results of infrared (8 micron) transit and secondary eclipse photometry of the hot Neptune exoplanet, GJ436b using Spitzer. The nearly photon-limited precision of these data allow us to measure an improved radius for the planet, and to detect the secondary eclipse. The transit (centered at HJD = 2454280.78149 +/-0.00016) shows the flat-bottomed shape typical of infrared transits, and it precisely defines the planet-to-star radius ratio (0.0839 +/-0.0005), independent of the stellar properties. However, we obtain the planetary radius, as well as the stellar mass and radius, by fitting to the transit curve simultaneously with an empirical mass-radius relation for M-dwarfs (M=R). We find Rs=Ms=0.47 +/-0.02 in solar units, and Rp=27,600 +/-1170 km (4.33 +/-0.18 Earth radii). This radius significantly exceeds the radius of a naked ocean planet, and requires a gasesous hydrogen-helium envelope. The secondary eclipse occurs at phase 0.587 +/-0.005, proving a significant orbital eccentricity (e=0.15 +/-0.012). The amplitude of the eclipse (5.7 +/-0.8e-4) indicates a brightness temperature for the planet of T=712 +/-36K. If this is indicative of the planet's physical temperature, it suggests the occurrence of tidal heating in the planet. An uncharacterized second planet likely provides ongoing gravitational perturbations, to maintain GJ436b's orbit eccentricity over long time scales.


Introduction
The transit and secondary eclipse of an extrasolar planet allow us to deduce its physical properties to a degree that is not possible in other observing geometries ). GJ 436b (Butler et al. 2004) was recently discovered to be the first transiting Neptune-sized planet (Gillon et al. 2007a), opening new parameter space for exoplanet studies. It orbits an M-dwarf star lying 10 pc from our solar system (Maness et al. 2007).
In order to constrain the bulk composition and internal structure of transiting planets Fortney et al. 2007), precise radii and temperature measurements are needed. The relatively small size of GJ 436A (∼ 0.4 solar radii) enhances the planet-to-star contrast during transit and eclipse. Nevertheless, the shallow depth of the GJ 436b transit (0.006) is a challenge for ground-based photometry. Although ground-based observers are achieving impressive levels of precision (Winn et al. 2007), photometry from space-borne platforms remains the preferred observational technique for the highest-precision transit measurements. This is especially true for secondary eclipse, where Spitzer measurements have been dominant (Charbonneau et al. 2005;Deming et al. 2005Deming et al. , 2006Harrington et al. 2007; Knutson et al. 2007).
In this Letter, we report Spitzer 8 µm transit and eclipse observations of GJ 436b, and we use these data to refine estimates of the planet's radius, temperature, and internal structure.

Observations
The announcement of GJ 436b transits (Gillon et al. 2007a) was fortuitously concurrent with a window of observability using Spitzer. Accordingly, we immediately scheduled observations of one transit, and one secondary eclipse, under our GO-3 Target of Opportunity (ToO) Program (J. Harrington, P.I.). Since the precision required for measurements of this type is daunting, observations must be carefully designed to limit instrumental systematics (Harrington et al. 2007). Moreover, the reported eccentricity of the GJ 436b orbit (Maness et al. 2007) adds significant uncertainty to the timing of the secondary eclipse observations. Our community ToO program thus works with cooperating teams to design observations and analyze data in line with the best practices gleaned from experience. We solict collaborations from cooperating teams that discover suitable targets.
Both observational sequences for GJ 436 used the IRAC instrument (Fazio et al. 2004) in subarray mode, at 8 µm only. The transit sequence consisted of 0.4-second exposures in blocks of 64, obtaining 445 blocks (204 minutes). The secondary eclipse sequence was the same, but used 780 blocks (356 minutes). We planned the eclipse observations based on 10 4 bootstrap trial fits to the Doppler data (Maness et al. 2007), to define the probability distribution of eclipse time. Because of the well known ramp-up in the sensitivity of the IRAC 8 µm detector during long observing sequences (Knutson et al. 2007;Harrington et al. 2007), we offset the transit observations to begin ∼ 2 hours before transit center.

Photometry
Because GJ 436 is bright at 8 µm, and the zodiacal background is weak in comparison, simple aperture photometery attains nearly photon-limited precision. Our photometry first applies the calibration information contained in the FITS headers, to convert the signal levels to electrons. Within each 64-frame block, we drop the first frame and the 58th frame, due to known instrumental effects (Harrington et al. 2007;Knutson et al. 2007). We examine the time variation of signal level for each pixel in the remaining 62 frames, and correct pixels in frames that are discrepant by > 4σ to the median value for that pixel (this removes energetic particle hits). We sum the intensity in an 8-by 8-pixel square aperture centered on the star in each frame, including fractional pixels, and sum again over the 62 frames in the block. We varied the aperture size to verify that an 8-pixel box produced the lowest noise, but this dependence is not strong. We fit a Gaussian to the peak in a histogram of pixel intensities for each block to determine, and subtract, the average background level. We used the same background value for all 62 frames in a block. We calculated the expected noise level for the photometry, based on the Poisson electron counting noise (dominant), and read noise (small). Comparing the aperture photometry for the 62 frames within each block, we find that these photometry errors are distributed as Gaussian noise, with a dispersion merely 3.5% greater than predicted. The intensities for the 445 transit blocks are illustrated in Figure 1, top panel. We examined the block-to-block variation in intensity for these points after removing the best transit fit, and we find a Gaussian distribution, with a standard deviation of 7 × 10 −4 . Since we detect ∼ 3.1 × 10 6 electrons per block (Figure 1), we expect a photon-limited precision of 5.7 × 10 −4 . We thus attain about 80% of photon-limited S/N, consistent with previous Spitzer photometry at this wavelength (Knutson et al. 2007;Harrington et al. 2007).
IRAC photometry at 8 µm is known to exhibit a gradually increasing ramp-up in sensitivity, due to filling of charge traps in the detectors (Knutson et al. 2007;Harrington et al. 2007). This ramp is visible in the top panel of Figure 1, but is weaker than usual for the transit data (the ramp varies due to prior usage of the detector). We removed it by masking out the data near transit, and fitting a parabola to the out-of-transit points.
We have considerable experience in fitting to this ramp, via our monitoring of GJ 876 (program 30498). Even strong ramps can be fit by the sum of a linear plus logarithmic function, using linear regression. We applied this more elaborate procedure to the GJ 436 transit ramp, but found no significant difference with the simple parabola fit. In the case of the secondary eclipse (Figure 2), the ramp is stronger, but is still well removed by our full linear + logarithmic fit. We conclude that this ramp is properly reproduced in both cases, and does not contribute significantly to our errors.

Transit Parameters
A feature of IR transit measurements is the virtual lack of stellar limb darkening. Not only does this produce a simple box-like shape for the transit, but Richardson et al. (2006) suggest that it can increase the radius precision for a given level of photometric precision.
Our analysis adopts the (small) limb darkening for GJ 436A based on a Kurucz model atmosphere for 3500/5.0/0.0 in Teff/log(g)/[M/H]. We verified that changing the stellar temperature, gravity, or metallicity within the errors (Maness et al. 2007;Bean et al. 2006) has negligible effect, because the limb darkening remains small over the plausible range. We integrated the stellar center-to-limb intensities in the Kurucz model over the bandpass of the IRAC 8 µm filter to obtain the limb darkening appropriate to this IR transit. Since this small IR limb darkening is not included in the Claret (2000) prescriptions, we generate theoretical transit curves numerically.
We compute theoretical transit curves by tiling the star in a latitude-longitude grid with zone spacing of 0.18 degrees, and applying the IRAC 8 µm limb darkening. We pass the planet across the numerical star in steps of 0.01 stellar radii, with the planet radius and impact parameter specified in units of the stellar radius. To increase precision, stellar zones at the edge of the planet are adaptively sub-sampled in a 10 × 10 finer grid. We verified the code's precision (better than 10 −6 ) by comparing to the Mandel and Agol (2002) analytic non-linear limb darkening cases, and by comparing the depth of synthetic transits to the planet-to-star area ratio (R p Fitting to high precision transit photometry requires a determination, or assumption, of the stellar mass (Brown et al. 2001). Gillon et al. (2007a) adopted 0.44 solar masses for GJ 436A, based on the observed luminosity, and they cited the empirical M-dwarf mass-radius relation from Ribas (2006) (R = M in solar units), to justify 0.44 solar radii for the stellar size. Our fit procedure is somewhat different. Given the lack of limb darkening, we can immediately determine the ratio of planet to stellar radius as 0.0839 ± 0.0005 from the depth of the transit (Figure 1). With this value fixed, we generate a grid of transit curves for a range of impact parameters. At each impact parameter, we vary the adopted stellar mass, and compute the transverse velocity of the planet across the star. This computation uses the orbital elements from a fit to the Doppler data (Maness et al. 2007), constrained by the secondary eclipse time (see below). We vary the stellar radius to convert the radius increments on the abscissa of the synthetic transit curve to orbit phase, using the calculated transverse velocity. We include a shift in phase for the synthetic transit curve, to allow for imprecision in the Gillon et al. (2007a) ephemeris. In this manner, we find the best fit stellar radius versus stellar mass, and a revised transit time. We estimate the stellar radius precision from the variation in χ 2 at a given mass.
We intersect the radius versus mass relation (R ∼ M 0.33 ) from the fitting procedure with the empirical mass-radius relation R = M (Ribas 2006) to find the best stellar mass and radius, and planet radius, at each impact parameter. Repeating this over a grid of impact parameters, we adopt the best fit from the global minimum χ 2 . We determine the error range from δχ 2 , and from visually inspecting the quality of the fits, paying particular attention to ingress/egress. Our fitting always uses the unbinned data (Figure 1, top), but we bin the data for the lower panel of Figure 1, to better illustrate the quality of the fit.
The derived time of transit center is HJD = 2454280.78149 ± 0.00016.

Secondary Eclipse
The secondary eclipse is shown in Figure 2. The top panel plots the bulk of the data (omitting some points at the outset); the eclipse occurs near the end of the observational sequence, at phase 0.587 ± 0.005, with amplitude 5.7 ± 0.8 × 10 −4 in units of the stellar intensity. Like the transit, all of our fits to this event were made on the original, unbinned data. However, for clarity, the lower panel of Figure 2 shows binned data, expands the phase scale, and overplots the best fit eclipse curve. Our fit constrains the duration of eclipse to equal the duration of transit, finding only the amplitude and central phase.

Results and Discussion
Our result for the stellar mass and radius is M = R = 0.47 ± 0.02 in solar units.
We are encouraged that these are close to values (0.44) constrained by independent data (Maness et al. 2007). Our derived planet radius is R p = 27, 600 ± 1170 km (4.33 ± 0.18 Earth radii). We conclude that this planet is larger than originally indicated by ground-based photometry (Gillon et al. 2007a). Since this radius is significantly larger than all planets of exclusively solid composition Fortney et al. 2007), GJ 436b must have a significant, gaseous, hydrogen-helium envelope. After our transit analysis was complete, we became aware of an independent analysis of the transit data (but not the eclipse data) by Gillon et al. (2007b). These authors do not vary the stellar mass in their fit, but they obtain a very similar radius for this planet, and arrive at essentially the same conclusion.
The observed phase of the secondary eclipse, φ = 0.587 ± 0.005 indicates that the orbit of GJ 436b is significantly eccentric. Assuming a longitude of pericenter ̟ = 0, the magnitude of the observed timing offset indicates a minimum orbital eccentricity, e min = 0.137 ± 0.007.
Using the constraints provided by the observed times of central transit, T c = 2454222.616 HJD (Gillon et al. 2007a), our T c = 2454280.78149 HJD, and the observed secondary eclipse at T s = 2454282.33 ± 0.01 HJD, we obtained a set of single-planet Keplerian fits to the radial velocity data published by Maness et al. (2007). A straightforward bootstrap resampling procedure (Press et al. 1992) yields e = 0.150 ± 0.012, ̟ = 343 ± 14 • , and As a consequence of its non-zero orbital eccentricity, GJ 436b is likely experiencing asynchronous rotation. Hut (1981) gives an expression for the spin period of a zero-obliquity spin pseudo-synchronized planet: (1 + 3e 2 + 3 8 e 4 )(1 − e 2 ) 3/2 1 + 15 2 e 2 + 45 8 e 4 + 5 16 e 6 P orbit For GJ 436b, we find P spin = 2.32 d, which yields a 19-day synodic period for the star as viewed from a fixed longitude on the planet. The large orbital eccentricity also indicates that a significant amount of tidal heating must be occurring. To second order in eccentricity, the tidal luminosity of a spin-synchronous planet (Peale & Cassen 1978;Mardling 2007) is where k 2 is the planetary potential Love number of degree 2, n is the orbital mean motion, a is the orbital semimajor axis, and Q is the planet's effective tidal dissipation parameter. The analysis of Levrard et al. (2007) indicates that the tidal luminosity of an asynchronously rotating planet with e ∼ 0.15 will exceed the value implied by the above expression by a small amount.
If we adopt T eff = 3350K for GJ 436A, take a zero albedo for the planet, and assume a uniform re-radiation of heat from the entire planetary surface, we obtain a planetary T eq = 642 K. The somewhat higher temperature (T = 712 ± 36K) implied by the secondary eclipse depth could arise from inefficient transport of heat to the night side of the planet, from a non-blackbody planetary emission spectrum, from tidal luminosity, or from a combination of the three.
We can estimate Q by assuming A = 0, uniform re-radiation, and a blackbody planetary SED to find a fiducial tidal luminosity of 4.7 × 10 26 ergssec −1 , and Q k 2 = 2.1 × 10 4 . Assuming k 2 = 0.34 (the Jovian value), gives Q = 7140. This value is in rough accord with the Q-values measured for Uranus and Neptune. Banfield and Murray (1992) derive 1.2 × 10 4 < Q N < 3.3 × 10 5 for Neptune, whereas Tittemore and Wisdom (1989) employ the Uranian satellites to derive Q U < 3.9 × 10 4 . If the planet has maintained e ∼ 0.15 for billions of years, Q ∼ 7000 indicates that the planet has radiated tidal energy comparable to the orbital energy and in excess of 100 times its own gravitational binding energy.
Furthermore, Q ∼ 7000 implies a circularization timescale e de/dt ∼ 3.0 × 10 7 yr. Indeed for any range of Q -which is uncertain even for solar system bodies -the circularization timescale is < 10 8 years. It is thus highly likely that an as-yet uncharacterized second planet is providing ongoing gravitational perturbations that allow GJ 436b's eccentricity to be maintained over long timescales.
We are grateful to the staff at the Spitzer Science Center for their prompt and efficient scheduling of our observations. Facilities: Spitzer.  comparison to the best fit secondary-eclipse curve, whose amplitude is 5.7 ± 0.8 × 10 −4 .