Modeling of the Yarkovsky effect. I. Thermal dependence
We investigate how the Yarkovsky effect created by one flat surface element on the asteroid depends on the thermal parameter. Then we integrate the Yarkovsky force over the surface of a spherical asteroid. In the next articles of the series, we will investigate the Yarkovsky force for bodies of more complicated shape and compute the change of the asteroid's orbit. Using perturbation theory, we analytically solve the one-dimensional heat conduction equation under the surface element and use the solution to analytically compute the Yarkovsky effect in the limiting cases of big and small thermal parameters. Then we analytically sew the two solutions to obtain an approximate expression for the Yarkovsky effect valid for both large and small thermal parameters. We verify it by comparing it to the results of the numerical solution of the heat conduction equation. Finally, we fit the parameters in an approximate analytical expression to improve the agreement with the numerical results. We obtain a fully analytical expression for the Yarkovsky effect of one surface element, which is composed of 4 terms and is precise for any value of the thermal parameter up to the accuracy of 15%. We also propose a fitted expression of the same form, which is precise up to 0.15%. They both much supersede the traditionally used 3-parameter fit, which produces an error up to 50% and, in particular, is about a factor of 2 wrong for small values of thermal parameters. Finally, we apply the new formalism to evaluate an analytical expression for the Yarkovsky force acting on a spherical asteroid, ending up in a 0.1%-accurate fitted expression for an asteroid with 0 obliquity on a circular orbit and a less accurate expression for the general case.