Extended-body corrections to the inverse-square law for spherically symmetric sources Mradumay Sadh, Lorenzo Gavassino Nicolaus Copernicus Astronomical Center, Warsaw msadh@camk.edu.pl, lorenzo@camk.edu.pl Introduction We attempt to explore the deviations from the standard inverse-square variation of irradiance when one considers the source as an extended-body. The ideal example to study these differences is a planet very close to its host-star. The recent advances in the study of extra-solar planets have shed light on such planets that are very close to their parent stars (Mayor et al. 1995). The irradiation received by these planets plays an important role in understanding the geological and climate properties of such planets. This underscores the importance of deriving a correct model for irradiation in such planets. The inverse-square law has been well established and has been used ubiquitously in previous studies (Kane et al. 2020). However, it doesn’t sufﬁce for extremely close planets. If one simply imagines such a close conﬁguration, it is immediately clear that inverse-square law doesn’t explain irradiation on the night-side of a close exoplanet since it assumes that the star is point-sized. This demands a new model that better explains the irradiation distribution and also incorporates the stellar limb darkening. In our approach we attempt to make a robust model that accurately predicts the irradiation distribution on such planets. Violation of the inverse square law Figure 1: Boundaries of validity for the inverse-square law Considering a star-planet system, the blue region in Fig.1 depicts the set of all those positions from which the star is not entirely vis- ible. An observer (represented in our case by the eye) sitting on an arbitrary blue point would see a partially eclipsed star. Since in this case, some of the photons directed to the observer are screened by the planet, the ﬂux fails to be radial. This opens the door to pos- sible violations of the inverse-square law on the dark green region of the planet. Quantifying the importance of the violations of the inverse-square law in the dark-green region and their dependence on the geometrical parameters of the star-planet system is the main purpose of this work. Geometrical model In our approach, we intend to integrate the irradiance from each surface element from the apparent surface of the star. For a close-in planet, the limits of integration are deﬁned by the region bound be- tween the tangents to the stellar circle from the point of observation on the planet. Additionally, the stellar surface hidden beneath the local horizon is omitted from the total ﬂux received. Figure 2: Star-planet system geometry in spherical coordinates Fig.2 shows the system geometry with λ and δ deﬁning the latitude and longitude of the planet, respectively. ds gives the distance be- tween the surface element on the star to a given point on the planet, and (r + R + d) is the semi-major axis(a). The star is assumed to have a bolometric luminosity Lb. Critical point of symmetry The position at which the light-green region of the planet (where the inverse-square law is valid) meets the dark-green region (where the inverse-square law ceases to hold) is deﬁned as the critical belt of symmetry and each point of it is a critical point of symmetry. The latitude at which the local horizon will be a tangent to the star is given by: λ = cos−1 R + r a  Without loss of generality we can conclude that all critical points on the planet that make an equivalent angle constitute a critical belt of symmetry. Terminator angle The terminator angle, for a given star planet system, determines how far will the terminator extend beyond the day-side of the planet. Mathematically, the terminator angle can be calculated from the slope of the exterior common tangent of the stellar and planetary circles : At = π/2 + sin−1 R −r a  . Analytical formulation The analytical formulation of the new equation for the