Integrating polarized light over a planetary disk applied to starlight reflected by extrasolar planets

Abstract

We present an efficient numerical method for integrating planetary radiation over a planetary disk, which is especially interesting for simulating signals of extrasolar planets. Our integration method is applicable to calculating the full flux vector of the disk-integrated planetary radiation, i.e. not only its observed flux (irradiance), but also its state of polarization (linear and circular). Including polarization is important for simulations of the light reflected by a planet, in particular, because this will generally be polarized. Our integration method is based on the expansion of the radiation field of a spherical, horizontally homogeneous planet into generalized spherical functions. With the expansion coefficients, the flux vector of the disk-integrated, reflected starlight can be obtained rapidly for arbitrary planetary phase angles. We describe the theory behind the disk-integration algorithm and results of accuracy tests. In addition, we give some illustrative examples of the application of the algorithm to extrasolar planets.Key words. methods: numerical -polarization -radiative transfer -stars: planetary systems IntroductionDuring the past decades, the spatial resolution of the observations of planets in our solar system has increased significantly. This increase stems from planetary missions such as Voyager, Galileo, and Cassini-Huygens, from space-bound telescopes like the Hubble Space Telescope, and from the development of ground-based adaptive optics systems. Together with this increasing spatial resolution, the spatial resolution of numerical simulations for the interpretation of observations of solar system planets has increased, too. Consequently, efficient numerical methods to integrate reflected starlight across a planetary disk have received little attention lately. Recent discoveries of extrasolar planets, however, have renewed interest in such numerical integration methods.Because extrasolar planets are very faint compared to their parent star, and because the angular distance between a star and an orbiting planet is very small, observing the planet itself by detecting the stellar light it reflects or the thermal radiation it emits is extremely difficult. Consequently, almost all of the known extrasolar planets have been found by indirect methods, in which not the planet itself but rather its influence on its parent star is detected. Although very useful for detecting an extrasolar planet, indirect detection methods give, however, little information on the planet itself, apart from its mass and some orbital elements. Information on the physical structure and chemical composition of a planet, for example, can be derived from direct observations of the planetary radiation. To succeed in detecting the very faint planetary radiation, dedicated instruments and space missions are being designed, such as the Planetfinder instrument that has been designed for use on one of ESO's VLTs and ESA's Darwin mission (Fridlund 2004) with space telescopes flying in formation and performing infra...

Looking for the rainbow on exoplanets covered by liquid and icy water clouds

“…The flux vector πF of stellar light that has been reflected by a spherical planet with radius r at a distance d from the observer (d r) is given by (Stam et al 2006) πF(λ, α) = 1 4…”

“…It is based on the same efficient adding-doubling algorithm (de Haan et al 1987) used by Stam et al (2006);Stam (2008). To calculate flux and polarization signals of horizontally inhomogeneous planets, we divide a model planet in pixels with a size such that we can assume that the surface and atmospheric layers are locally plane parallel and horizontally homogeneous.…”

Flux and polarisation spectra of water clouds on exoplanets

“…Q > 0) the light is polarised parallel to the reference plane. The flux vector πF of stellar light that has been reflected by a spherical planet with radius r at a distance d from the observer (d r) is given by (Stam et al 2006) πF(λ, α) = 1 4…”

“…the angle between the star and the observer as seen from the center of the planet. Furthermore, S is the 4×4 planetary scattering matrix (Stam et al 2006) and πF 0 is the flux vector of the incident stellar light. For a solar type star, the stellar flux can be considered to be unpolarised when integrated over the stellar disk (Kemp et al 1987).…”

Pre-conditioned backward Monte Carlo solutions to radiative transport in planetary atmospheres

“…In such cases, more flexible treatments including Mie and other non-Rayleigh forms of scattering are needed. Thus, Stam et al (2006) have devised an efficient technique for disk integration based on the plane-parallel approximation that can deal with arbitrary scattering particles for planets with horizontally uniform atmospheres.…”