We present exact analytic formulae for the eclipse of a star described by quadratic or nonlinear limb darkening. In the limit that the planet radius is less than a tenth of the stellar radius, we show that the exact lightcurve can be well approximated by assuming the region of the star blocked by the planet has constant surface brightness. We apply these results to the HST observations of HD 209458, showing that the ratio of the planetary to stellar radii is 0.1207 ± 0.0003. These formulae give a fast and accurate means of computing lightcurves using limbdarkening coefficients from model atmospheres which should aid in the detection, simulation, and parameter fitting of planetary transits.

One focus of modern astronomy is to detect temperate terrestrial exoplanets well-suited for atmospheric characterisation. A milestone was recently achieved with the detection of three Earth-sized planets transiting (i.e. passing in front of) a star just 8% the mass of the Sun 12 parsecs away1. Indeed, the transiting configuration of these planets combined with the Jupiter-like size of their host star - named TRAPPIST-1 - makes possible in-depth studies of their atmospheric properties with current and future astronomical facilities1,2,3. Here we report the results of an intensive photometric monitoring campaign of that star from the ground and with the Spitzer Space Telescope. Our observations reveal that at least seven planets with sizes and masses similar to the Earth revolve around TRAPPIST-1. The six inner planets form a near-resonant chain such that their orbital periods (1.51, 2.42, 4.04, 6.06, 9.21, 12.35 days) are near ratios of small integers. This architecture suggests that the planets formed farther from the star and migrated inward4,5. The seven planets have equilibrium temperatures low enough to make possible liquid water on their surfaces6,7,8.

No abstract

"Hot Jupiter" extrasolar planets are expected to be tidally locked because they are close (<0.05 astronomical units, where 1 AU is the average Sun-Earth distance) to their parent stars, resulting in permanent daysides and nightsides. By observing systems where the planet and star periodically eclipse each other, several groups have been able to estimate the temperatures of the daysides of these planets 1-3 . A key question is whether the atmosphere is able to transport the energy incident upon the dayside to the nightside, which will determine the temperature at different points on the planet's surface. Here we report observations of HD 189733, the closest of these eclipsing planetary systems 4-6 , over half an orbital period, from which we can construct a 'map' of the distribution of temperatures. We detected the increase in brightness as the dayside of the planet rotated into view. We estimate a minimum brightness temperature of 973±33 K and a maximum brightness temperature of 1212±11 K at a wavelength of 8 µm, indicating that energy from the irradiated dayside is efficiently redistributed throughout the atmosphere, in contrast to a recent claim for another hot Jupiter 7 . Our data indicate that the peak hemisphere-integrated brightness occurs 16±6 degrees before opposition, corresponding to a hot spot shifted east of the substellar point. The secondary eclipse (when the planet moves behind the star) occurs 120±24 s later than predicted, which may indicate a slightly eccentric orbit.We monitored HD 189733 continuously over a 33.1 hour period using the 8 µm channel of the InfraRed Array Camera (IRAC) 8 on the Spitzer Space Telescope 9 , observing in subarray mode 1 with a cadence of 0.4 s. Our observations spanned slightly more than half of the planet's orbit, beginning 2.6 hours before the start of the transit (when the planet moves in front of the star) and ending 1.9 hours after the end of the secondary eclipse. This gave us a total of 278,528 32 × 32 pixel images. We found that there was a gradual detector-induced rise of up to 10% in the signal measured in individual pixels over time. This rise is illumination-dependent; pixels with high levels of illumination (greater than 250 MJy sr −1 ) converge to a constant value within the first two hours of observations and lower-flux pixels increase linearly over time. We characterize this effect by producing a time series of the signal in a series of annuli of increasing radius centered on the star (masking out a 5-pixel-wide box centered on HD 189733's smaller, fainter M dwarf companion 10 ). This set of curves describes the behavior of the ramp for different illumination levels.To correct our images, we estimate the median illumination for each pixel in the array, and interpolate over our base set of curves (scaling as the natural log of the illumination) to calculate a curve describing the behavior of that pixel. We correct for this instrumental effect by dividing the flux in each pixel in a given image by the value of the interpolated curve. Pixels with illumi...

The transits of a planet on a Keplerian orbit occur at time intervals exactly equal to the period of the orbit. If a second planet is introduced the orbit is not Keplerian and the transits are no longer exactly periodic. We compute the magnitude of these variations in the timing of the transits, dt. We investigate analytically several limiting cases: (i) interior perturbing planets with much smaller periods; (ii) exterior perturbing planets on eccentric orbits with much larger periods; (iii) both planets on circular orbits with arbitrary period ratio but not in resonance; and (iv) planets on initially circular orbits locked in resonance. Case (iv) is perhaps the most interesting case since some systems are known to be in resonances and the perturbations are the largest. As long as the perturber is more massive than the transiting planet, the timing variations would be of order of the period regardless of the perturber mass! For lighter perturbers, we show that the timing variations are smaller than the period by the perturber to transiting planet mass ratio. An earth mass planet in 2:1 resonance with a 3-day period transiting planet (e.g. HD 209458b) would cause timing variations of order 3 minutes, which would be accumulated over a year. These are easily detectable with current ground-based measurements. For the case of both planets on eccentric orbits, we compute numerically the transit timing variations for several cases of known multiplanet systems assuming they were edge-on. Transit timing measurements may be used to constrain the masses and radii of the planetary system and, when combined with radial velocity measurements, to break the degeneracy between mass and radius of the host star. (abstract truncated)Comment: 21 pages, 9 figures, submitted to MNRA