Abstract The Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect governs the spin evolution of small asteroids. The axial component of YORP, which alters the rotation rate of the asteroid, is mostly independent of its thermal inertia, while the obliquity component is very sensitive to the thermal model of the asteroid. Here, we develop a semi-analytic theory for the obliquity component of YORP. We integrate an approximate thermal model over the surface of an asteroid, and find an analytic expression for the obliquity component in terms of two YORP coefficients. This approach allows us to investigate the overall evolution of asteroid rotation state, and to generalize the results previously obtained in the case of zero thermal inertia. The proposed theory also explains how a nonzero obliquity component of YORP originates even for a symmetric asteroid, due to its finite thermal inertia. In many cases, this causes equatorial planes of asteroids to align with their orbital planes. The studied nontrivial behavior of YORP as a function of thermal model allows for a new kind of rotational equilibria, which can have important evolutionary consequences for asteroids.

The YORP effect is the thermal torque generated by radiation from the surface of an asteroid. The effect is sensitive to surface topology, including small-scale roughness, boulders, and craters. The aim of this paper is to develop a computationally efficient semi-analytical model for the crater-induced YORP (CYORP) effect that can be used to investigate the functional dependence of this effect. This study linearizes the thermal radiation term as a function of the temperature in the boundary condition of the heat conductivity, and obtains the temperature field in a crater over a rotational period in the form of a Fourier series, accounting for the effects of self-sheltering, self-radiation, and self-scattering. By comparison with a numerical model, we find that this semi-analytical model for the CYORP effect works well for $K>0.1 W/m/K$. This semi-analytical model is computationally three-orders-of-magnitude more efficient than the numerical approach. We obtain the temperature field of a crater, accounting for the thermal inertia, crater shape, and crater location. We then find that the CYORP effect is negligible when the depth-to-diameter ratio is smaller than 0.05. In this case, it is reasonable to assume a convex shape for YORP calculations. Varying the thermal conductivity yields a consistent value of approximately 0.01 for the spin component of the CYORP coefficient, while the obliquity component is inversely related to thermal inertia, declining from 0.004 in basalt to 0.001 in metal. The CYORP spin component peaks at an obliquity of $0^ or $180^ while the obliquity component peaks at an obliquity of around $45^ or $135^ For a z-axis symmetric shape, the CYORP spin component vanishes, while the obliquity component persists. Our model confirms that the total YORP torque is damped by a few tens of percent by uniformly distributed small-scale surface roughness. Furthermore, for the first time, we calculate the change in the YORP torque at each impact on the surface of an asteroid explicitly and compute the resulting stochastic spin evolution more precisely. This study shows that the CYORP effect due to small-scale surface roughness and impact craters is significant during the history of asteroids. The semi-analytical method that we developed, which benefits from fast computation, offers new perspectives for future investigations of the YORP modeling of real asteroids and for the complete rotational and orbital evolution of asteroids accounting for collisions. Future research employing our CYORP model may explore the implications of space-varying roughness distribution, roughness in binary systems, and the development of a comprehensive rotational evolution model for asteroid groups.

caused by surface forces due to mechanical drag of the atmosphere ( in case of low flying satellites), electromagnetism, and solar radiation pressure .