Mean temperature of a spherical body on an elliptic orbit

¹ Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
² Institute of Astronomy, Charles University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic

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Received: 30 October 2025
Accepted: 23 January 2026

Abstract

Context. The accurate determination of thermal accelerations acting on small bodies orbiting the Sun requires knowing the surface temperature at any moment. In analytical methods, the computation of the temperature is often simplified by assuming that it varies slightly about a constant value. This ansatz allows us to conveniently linearize the problem.

Aims. However, the mean temperature is constant only in the case of a circular orbit. Our aim is to define a time-dependent temperature that would closely represent the mean temperature of a spherical body revolving around the Sun on an eccentric orbit.

Methods. We adopted a model of the mean temperature with a radial profile inside the body and expressed it by superposing the eigenfunctions of the heat diffusion problem. These were represented with Fourier series in a time domain and spherical Bessel functions in the space domain. The coefficients that weight the contribution of each term were determined from the boundary conditions. Special care was taken to properly account for the non-linearity of the surface energy balance.

Results. We developed a robust algorithm to obtain the coefficients of the mean temperature series and tested the results for various choices in their truncation. Degree eight appears to be adequate for orbits up to eccentricity ≃0.4. The thermal parameter may have an arbitrary value, including limits of both zero and infinite thermal inertia of the surface. The size of the body may also be arbitrary. We provide simplified results for the small- and large-body limits, with the penetration depth of the seasonal thermal wave being the length scale.

Conclusions. Our formulation of the mean temperature offers the possibility to develop an analytical description of seasonal and diurnal variants of thermal accelerations, including their coupling for an eccentric orbit. Previous models are thus generalized.