The Yarkovsky and YORP Effects David Vokrouhlicky´ Institute of Astronomy, Charles University, Prague William F. Bottke Department of Space Studies, Southwest Research Institute, Boulder Steven R. Chesley Jet Propulsion Laboratory, California Institute of Technology, Pasadena Daniel J. Scheeres Aerospace Engineering Sciences, University of Colorado, Boulder Thomas S. Statler Astrophysical Institute, Ohio University, Athens and Department of Astronomy, University of Maryland, College Park The Yarkovsky effect describes a small but significant force that affects the orbital motion of meteoroids and asteroids smaller than 30 − 40 kilometers in diameter. It is caused by sunlight; when these bodies heat up in the Sun, they eventually re-radiate the energy away in the thermal waveband, which in turn creates a tiny thrust. This recoil acceleration is much weaker than solar and planetary gravitational forces, but it can produce measurable orbital changes over decades and substantial orbital effects over millions to billions of years. The same physical phenomenon also creates a thermal torque that, complemented by a torque produced by scattered sunlight, can modify the rotation rates and obliquities of small bodies as well. This rotational variant has been coined the Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect. During the past decade or so, the Yarkovsky and YORP effects have been used to explore and potentially resolve a number of unsolved mysteries in planetary science dealing with small bodies. Here we review the main results to date, and preview the goals for future work. 1. INTRODUCTION erature through the works of Vladimir V. Radzievskii and his collaborators (Radzievskii, 1952b). While Radzievskii Interesting problems in science usually have a long and was also the first to consider the effects of systematic pho- complex history. It is rare, though, that they have a prehis- ton thrust on a body’s rotation, his concept was based on a tory or perhaps even mythology. Yet, until recently this was variable albedo coefficient across the surface (Radzievskii, the case of the Yarkovsky effect. Ivan O. Yarkovsky, a Rus- 1952a). However, there is no strong evidence of large sian civil engineer born in a family of Polish descent, noted enough albedo variations over surfaces of asteroids or me- in a privately-published pamphlet (Yarkovsky, 1901; Beek- teoroids. Stephen J. Paddack and John O’Keefe pushed the man, 2006) that heating a prograde-rotating planet should idea forward by realizing that irregular shape, and thermal produce a transverse acceleration in its motion and thus help radiation rather than just the reflected sunlight, will more to counter-balance the assumed drag from the then-popular efficiently change the meteoroid’s spin rate. Thence, the ether hypothesis. While this context of Yarkovsky’s work arXiv:1502.01249v1 [astro-ph.EP] 4 Feb 2015 Yarkovsky-O’Keefe-Radzievskii-Paddack effect (YORP was mistaken and he was only roughly able to estimate the for short) was born as an alter ego of the Yarkovsky effect magnitude of the effect, he succeeded in planting the seed little more than half a century after Yarkovsky’s work (see of an idea that a century later blossomed into a full-fledged Paddack, 1969; Paddack and Rhee, 1975; and Rubincam, theory of how the orbits of small objects revolving about 2000 for summing up the history and coining the termi- the Sun are modified by the absorption and re-emission of nology). Radzievskii’s school also briefly touched upon a solar energy. concept of a radiation-induced acceleration of synchronous It is mainly Ernst J. Opik¨ who is to be credited for keep- planetary satellites (Vinogradova and Radzievskii, 1965), ing Yarkovsky’s work alive and introducing it to western an idea that much later re-appeared in a slightly different literature, long after the original pamphlet had been lost form as a binary YORP (or BYORP for short) effect (Cuk´ (Opik¨ , 1951). Curiously, at about the same time, similar and Burns, 2005). ideas also started to appear in Russian regular scientific lit- The three decades from the 1950’s to the 1970’s saw a 1 slow revival and emergence of the concepts that eventually applications, have reached a level that requires more ac- resulted in today’s Yarkovsky and YORP effects. The works curate models to be used. First steps towards these new that led to a major resurgence in these studies, however, oc- models have been taken recently and these are briefly re- curred in the second half of the 1990’s through the work viewed in Sec. 2.2. of David P. Rubincam and Paolo Farinella. Interestingly, both were studying thermal perturbations of artificial satel- 2.1 Classical models lite motion. With that expertise they realized a direct link between the orbital effects acting on the geodynamics ar- The Yarkovsky effect.– Absorbed and directly reflected tificial satellites such as LAGEOS and the orbital effects sunlight does not tend to produce long-term dynamical ef- on small meteoroids (e.g., Afonso et al., 1995; Rubincam, fects as far as orbital motion is concerned (e.g., Vokrouh- 1995, 1998; Farinella et al., 1998). licky´ and Milani, 2000; Ziˇ zkaˇ and Vokrouhlicky´, 2011). From there, a momentum was gained and a wealth of The Yarkovsky effect thus fundamentally depends on emit- new results appeared, with applications extending to a dy- ted thermal radiation and requires a body to have a non- namics of small asteroids and their populations (e.g., Bottke zero thermal inertia. Any meaningful evaluation of the et al., 2002a, 2006). Studies of the Yarkovsky effect were Yarkovsky effect, therefore, requires a thermophysical soon followed by those of the YORP effect (Rubincam, model of that body. Fortunately, an evaluation of the 2000). Today, both effects belong to a core culture in plane- Yarkovsky effect imposes a minimum of requirements on tary sciences, as well as beyond (e.g., http://www.youtube. the shape of the body; even a simple spherical model pro- com/watch?v=kzlgxqXtxYs), and have become an impor- vides us with a fair approximation of how the body will tant part in the agenda of space missions (e.g., Lauretta orbitally evolve. et al., 2015). Especially after the spectacular discovery of While the Yarkovsky effect results in variations to all of the “once lost” Yarkovsky pamphlet in Russian archives by the orbital elements, what is distinct from most other pertur- Dutch amateur astronomer George Beekman (see Beekman, bations is the secular effect in the semimajor axis a. There- 2006), it seems timely to review the current knowledge of fore, we only discuss this contribution. Assuming (i) a lin- the Yarkovsky and YORP effects. This effort could start earization of the surface boundary condition, (ii) a rotation with a translation, and perhaps a commented edition, of the about a spin axis fixed in the inertial space (at least on a Yarkovsky work (presently available in the original form timescale comparable with the revolution about the Sun), as an Appendix of Miroslav Broz’sˇ Thesis, http://sirrah. and (iii) a circular orbit about the Sun, one easily finds that troja.mff.cuni.cz/˜mira/mp/phdth/). We look forward to fu- the total, orbit-averaged change in a is composed of two ture historians editing the more than a century long story of contributions (e.g., Rubincam, 1995, 1998; Farinella et al., the Yarkovsky and YORP effects, with all the known and 1998; Vokrouhlicky´, 1998a, 1999): possibly hidden roots, into a consolidated picture. Leaving historical issues to their own life, we now turn to da 8 αΦ = − W (R!; Θ!) cos γ ; (1) current scientific issues related to the Yarkovsky and YORP dt diurnal 9 n effects. There are several good technical reviews already existing in the literature (e.g., Bottke et al., 2002a, 2006). the diurnal effect, and While not always possible, we try to avoid discussing the da 4 αΦ = W (R ; Θ ) sin2 γ ; (2) same topics as presented in these previous texts. For in- dt 9 n n n stance, we do not review the elementary concepts of the seasonal Yarkovsky and YORP effects, assuming the reader is famil- the seasonal effect. Here, Φ = πR2F=(mc), where R is the iar with them. Rather, we try to focus on new results and radius of the body, F the solar radiation flux at the orbital ideas that emerged during the past decade and that will lead distance a from the Sun, m is the mass of the body, c is the to research efforts in the next several years. light velocity, n is the orbital mean motion and α = 1 − A, with A denoting the Bond albedo (e.g., Vokrouhlicky´ and Bottke, 2001). The Φ factor is characteristic to any phys- 2. THEORY OF THE YARKOVSKY ical effect related to sunlight absorbed or scattered by the AND YORP EFFECTS surface of the body. Since m / R3, one obtains a typical scaling Φ / 1=R. We start with the simplest analytical models of the More importantly, the diurnal and seasonal components Yarkovsky and YORP effects (Sec. 2.1). This is because of the Yarkovsky effect have a different dependence on the they provide useful insights, such as scalings with several spin axis obliquity γ: (i) the diurnal part is / cos γ, and key parameters, and their results are correct to leading or- consequently can make a positive or negative change of the semimajor axis, being maximum at 0◦ and 180◦ obliquity der. They also allow us to understand why modeling of 2 the YORP effect is inevitably more complicated than mod- values, and (ii) the seasonal part is / sin γ, and conse- quently always results in a decrease in semimajor axis, be- eling of the Yarkovsky effect.