Astron. Astrophys. 344, 362-366 (1999)

1. Introduction

The so-called Yarkovsky effect, a recoil force due to thermal radiation from anisotropically heated orbiting bodies, has recently attracted a considerable attention in the frame of the studies on the delivery of meteorites and the dynamics of small bodies in the Solar System. Specific issues for which the Yarkovsky effect is probably relevant are: the cosmic-ray exposure ages of stony and iron meteorites, which are much longer than the dynamical lifetimes of particles delivered from the asteroid belt (Farinella et al. 1998; Hartmann et al. 1998; Morbidelli & Gladman 1998); the overabundance of decameter-sized near-Earth objects (Rubincam 1995, 1998; Vokrouhlický & Farinella 1998a); the dynamical evolution of large ( km) main-belt asteroid fragments and their delivery to Mars- and Earth-crossing orbits (Farinella & Vokrouhlický 1999). In all these cases, the Yarkovsky effect plays the role of a dissipative mechanism, resulting into a significant long-term mobility of the orbital semimajor axis and a complex interaction with resonances.

To assess the relevance of the Yarkovsky effect in Solar System dynamics one needs, as a first step, to develop a reliable physical model of the thermal processes occurring within solid, spinning and orbiting bodies. A significant amount of work has been performed on this problem in recent years, after Rubincam (1995) resurrected the interest in the dynamical consequences of these thermal effects. Most importantly, Rubincam (1995, 1998) and Farinella et al. (1998) recognized the existence of two distinct variants of the Yarkovsky effect: a "diurnal" variant depending on the rotation frequency of the body around its instantaneous spin axis (), and a "seasonal" variant depending on the mean motion frequency of the body around the Sun ().

Technically speaking, the diurnal variant is obtained when one entirely neglects the orbital motion around the Sun (see e.g. Vokrouhlický 1998a,b), whereas in dealing with the seasonal variant one a priori averages all relevant quantities over the (assumedly) fast rotation of the body (e.g. Rubincam 1995, 1998; Vokrouhlický & Farinella 1998b). This classification is meaningful and useful, since the two variants of the Yarkovsky effect result in qualitatively different long-term changes of the semimajor axis. The diurnal version is maximum at zero obliquity and can lead either to semimajor axis decrease or increase, depending on the sense of rotation; on the contrary, the seasonal version is maximum at obliquity and can only result in orbital decay (e.g. Rubincam 1995, 1998; Farinella et al. 1998; Hartmann et al. 1998). At the essence, however, the two variants of the Yarkovsky effect are just two different limiting cases of a single physical mechanism, i.e., the recoil force associated to thermal radiation from a body having an anisotropic temperature distribution on its surface. As their names imply, the diurnal and seasonal variants correspond to different periodicities and geometries of the external illumination on the body's surface. From this perspective, it seems desirable to develop a unified, self-consistent theory for the Yarkovsky effect, including at the same time both the diurnal and the seasonal periodicities, such that the two classical variants can be derived computing suitable mathematical limits.

Although the classical variants of the Yarkovsky effect are present as particular limiting cases, the unified theory inevitably will contain additional, "mixed" terms, depending on both the relevant frequencies and . This conclusion holds even in the frame of a linear theory for the temperature changes, such as that developed in the following sections. Thus, the major novelty of this paper consists of the derivation of these "mixed" (or "diurnal-seasonal") terms. Specifically, we shall show that the "diurnal" variant does not exist as an effect depending on the rotation frequency alone, but inevitably contains a linear combination of the two frequencies. As expected, in the limit of a very rapid spin rate this doublet tends to merge into a single line, depending just on .

Then, we shall assess the contribution of the new terms to the secular changes in the semimajor axis of the body's orbital motion. As noted above, such changes probably play an important role in several problems of astronomical interest, and the quantitative results obtained so far have always been computed as a simple superposition of the diurnal and seasonal effects (e.g. Farinella & Vokrouhlický 1999), neglecting any possible "mixed" effects.

To make the calculations as simple as possible, we shall make three simplifying assumptions: (i) a circular orbit around the Sun; (ii) a spherical shape of the body; and (iii) a commensurability between the rotation and revolution periods. In particular, we shall introduce a parameter , and we shall assume that m is an integer number. However, we stress that while the first two assumptions correspond to physical simplifications, the third one is just a suitable mathematical step to simplify the derivation of our results, and that this assumption can be easily removed by the technique used by Farinella & Vokrouhlický (1996). Therefore, our final results will be valid for any (real) value of the parameter m.

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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