The obliquity of a terrestrial planet is an important clue about its formation and critical to its climate. Previous studies using simulated photometry of Earth show that continuous observations over most of a planet's orbit can be inverted to infer obliquity. However, few studies of more general planets with arbitrary albedo markings have been made and, in particular, a simple theoretical understanding of why it is possible to extract obliquity from light curves is missing. Reflected light seen by a distant observer is the product of a planet's albedo map, its host star's illumination, and the visibility of different regions. It is useful to treat the product of illumination and visibility as the kernel of a convolution. Time-resolved photometry constrains both the albedo map and the kernel, the latter of which sweeps over the planet due to rotational and orbital motion. The kernel's movement distinguishes prograde from retrograde rotation for planets with non-zero obliquity on inclined orbits. We demonstrate that the kernel's longitudinal width and mean latitude are distinct functions of obliquity and axial orientation. Notably, we find that a planet's spin axis affects the kernel-and hence timeresolved photometry-even if this planet is East-West uniform or spinning rapidly, or if it is North-South uniform. We find that perfect knowledge of the kernel at 2-4 orbital phases is usually sufficient to uniquely determine a planet's spin axis. Surprisingly, we predict that East-West albedo contrast is more useful for constraining obliquity than North-South contrast.

We observed the hot Jupiter HAT-P-32b (also known as HAT-P-32Ab) to determine its optical transmission spectrum by measuring the wavelength-dependent planet-to-star radius ratios in the region between 518 -918 nm. We used the OSIRIS instrument at the GTC in long slit spectroscopy mode, placing HAT-P-32 and a reference star in the same slit and obtaining a time series of spectra covering two transit events. Using the best quality data set, we were able to yield 20 narrow-band transit light curves, with each passband spanning a 20 nm wide interval. After removal of all systematic noise signals and light curve modeling the uncertainties for the resulting radius ratios lie between 337 and 972 ppm. The radius ratios show little variation with wavelength suggesting a high altitude cloud layer masking any atmospheric features. Alternatively, a strong depletion in alkali metals or a much smaller than expected planetary atmospheric scale height could be responsible for the lack of atmospheric features. Our result of a flat transmission spectrum is consistent with a previous ground-based study of the optical spectrum of this planet. This agreement between independent results demonstrates that ground-based measurements of exoplanet atmospheres can give reliable and reproducible results despite the fact that the data often is heavily affected by systematic noise, as long as the noise source is well understood and properly corrected. We also extract an optical spectrum of the M-dwarf companion HAT-P-32B. Using PHOENIX stellar atmosphere models we determine an effective temperature of T eff = 3187 +60 −71 K, slightly colder than previous studies relying only on broadband infra-red data.

Aims. As a sub-Uranus-mass low-density planet, GJ 3470b has been found to show a flat featureless transmission spectrum in the infrared and a tentative Rayleigh scattering slope in the optical. We conducted an optical transmission spectroscopy project to assess the impacts of stellar activity and to determine whether or not GJ 3470b hosts a hydrogen-rich gas envelop. Methods. We observed three transits with the low-resolution Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS) at the 10.4 m Gran Telescopio Canarias, and one transit with the high-resolution Ultraviolet and Visual Echelle Spectrograph (UVES) at the 8.2 m Very Large Telescope. Results. From the high-resolution data, we find that the difference of the Ca ii H+K lines in-and out-of-transit is only 0.67 ± 0.22%, and determine a magnetic filling factor of about 10-15%. From the low-resolution data, we present the first optical transmission spectrum in the 435-755 nm band, which shows a slope consistent with Rayleigh scattering. Conclusions. After exploring the potential impacts of stellar activity in our observations, we confirm that Rayleigh scattering in an extended hydrogen-helium atmosphere is currently the best explanation. Further high-precision observations that simultaneously cover optical and infrared bands are required to answer whether or not clouds and hazes exist at high-altitude.