A novel mathematical model for computing irradiance on spherically symmetric close-in exoplanets

Abstract

Abstract The inverse-square law for calculating the irradiation arises as a direct consequence of the conservation of the energy, when spherical symmetry is imposed. The law implies that any spherically-symmetric source can be replaced, without changing the energy flux, by a point-sized source located in the center of symmetry. On the other hand, anybody who has seen a sunset knows that, when the center of the Sun moves slightly below the horizon, still a considerable portion of the stellar surface can be visible, irradiating much more than the corresponding point-sized source, which would be hidden under the horizon. This apparent contradiction is immediately solved when one realises that the presence of the planet, whose surface is absorbing part of the photon flux coming from the star, breaks the spherical symmetry, producing violations of the inverse-square in the sunset (or, equivalently, sunrise) region. On exoplanets that are extremely close to their star, this breakdown of the inverse-square law can become very important. In the present paper I calculate the total irradiance from the star by explicitly integrating the irradiance from each surface element of the stellar surface visible from a given latitude on the planet. I assume that the stellar limb darkening can be approximated using a linear law in these calculations. It has been found that the difference in the irradiation calculated by this model and the inverse-square law is significant at the polar regions of the exoplanets. It has also been found that, as a consequence of the geometry of the star, the day-night terminator shifts from poles towards the night side of the exoplanet.