1CARUS 40, 1-48 (1979) Radiation Forces on Small Particles in the Solar System t JOSEPH A. BURNS Cornell University, 2 Ithaca, New York 14853 and NASA-Ames Research Center PHILIPPE L. LAMY CNRS-LAS, Marseille, France 13012 AND STEVEN SOTER Cornell University, Ithaca, New York 14853 Received May 12, 1978; revised May 2, 1979 We present a new and more accurate expression for the radiation pressure and Poynting- Robertson drag forces; it is more complete than previous ones, which considered only perfectly absorbing particles or artificial scattering laws. Using a simple heuristic derivation, the equation of motion for a particle of mass m and geometrical cross section A, moving with velocity v through a radiation field of energy flux density S, is found to be (to terms of order v/c) mi, = (SA/c)Qpr[(1 - i'/c)S - v/c], where S is a unit vector in the direction of the incident radiation,/" is the particle's radial velocity, and c is the speed of light; the radiation pressure efficiency factor Qpr ~ Qabs + Q~a(l - (cos a)), where Qabs and Q~c~ are the efficiency factors for absorption and scattering, and (cos a) accounts for the asymmetry of the scattered radiation. This result is confirmed by a new formal derivation applying special relativistic transformations for the incoming and outgoing energy and momentum as seen in the particle and solar frames of reference. Qpr is evaluated from Mie theory for small spherical particles with measured optical properties, irradiated by the actual solar spectrum. Of the eight materials studied, only for iron, magnetite, and graphite grains does the radiation pressure force exceed gravity and then just for sizes around 0. l/zm; very small particles are not easily blown out of the solar system nor are they rapidly dragged into the Sun by the Poynting-Robertson effect. The solar wind counterpart of the Poynting-Robertson drag may be effective, however, for these particles. The orbital consequences of these radiation forces--including ejection from the solar system by relatively small radiation pressures-- and of the Poynting-Robertson drag are consid- ered both for heliocentric and planetocentric orbiting particles. We discuss the coupling between the dynamics of particles and their sizes (which diminish due to sputtering and sublimation). A qualitative derivation is given for the differential Doppler effect, which occurs because the light received by an orbiting particle is slightly red-shifted by the solar rotation velocity when coming from the eastern hemisphere of the Sun but blue-shifted when from the western hemisphere; the ratio of this force to the Poynting-Robertson force is (Ro/r)2[(w®/n) - I], where R® and w® are the solar radius and spin rate, and n is the particle's mean motion. The Yarkovsky effect, caused by the asymmetry in the reradiated thermal emission of a rotating body, is also developed relying o n new physical arguments. Throughout the paper, representative calculations use the physical and orbital properties of interplanetary dust, as known from various recent measurements. t An invited review paper to Icarus. Current address. 0019-1035/79/100001-48502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. 2 BURNS, LAMY, AND SOTER I. INTRODUCTION zodiacal light (see Leinert, 1975; Weinberg and Sparrow, 1978), a faint glow in the night This article discusses the forces due to sky principally visible along the ecliptic, is solar radiation incident on small particles: due to the scattering of solar radiation by radiation pressure, Poynting-Robertson dust particles, which may derive from com- drag, the Yarkovsky effect, and the differ- etary or asteroidal detritus; as observed ential Doppler effect. The paper also con- from Earth (Dumont, 1976) and by the siders the resulting orbital evolution for par- photopolarimeters on Pioneers 10 and 11 ticles moving about the Sun and about (Weinberg, 1976), the zodiacal light inten- planets, and to a lesser extent attempts to sity peaks near the Sun, decaying rapidly correlate theoretical results with dust ob- with ecliptic latitude and solar elongation servations, principally made by in situ mea- except for a slight brightening in the antiso- surements in space. While the subject has lar direction, the gegenschein. The Pioneer been studied for more than 70 years, the and Helios spacecraft, as well as several details of the relevant processes are not eas- earlier in situ observatories, have directly ily accessible from the literature. We have, measured dust with impact counters of var- therefore, rederived most results so as to ious types (Fechtig, 1976; Fechtig et al., clarify and generalize them. We have also 1978; McDonnell, 1978). Nature itself has carded out more complete numerical calcu- provided integral counters of a sort in the lations of the coefficients that appear in the impacts recorded as surface microcraters expressions. A recent brief review of some found on otherwise smooth glass spheres on of these processes was done by Dohnanyi the lunar surface (Hrrz et al., 1971, 1975; (1978). Hartung 1976; Ashworth, 1978) and in some Before describing the radiation forces meteorites (Brownlee and Rajan, 1973). that act on interplanetary dust, it will be Not only is dust present in the solar system, valuable to outline the observational evi- but its existence in clouds throughout in- dence for such particles and to list various terstellar space is strongly implied by the review articles summarizing this informa- extinction, reddening, and polarization of tion. Meteors, curved comet tails, and the starlight (see Aannestad and Purcell, 1973; zodiacal light are all indicators of in- Huffman, 1977; Ney, 1977; Day, 1977; terplanetary dust visible to the naked eye. Greenberg, 1978). Dust can be produced in Sporadic and shower meteors (see Sober- circumstellar space and can escape into in- man, 1971; Millman, 1976; Hughes, 1978) terstellar space (cf. Dorschner, 1971). It are observable when interplanetary may even lie in the intergalactic void (Mar- meteoroids pass through the upper atmo- golis and Schramm, 1977). Although the sphere; the survivors of this journey ac- number densities, composition, and origin count for the extraterrestrial dust found in of dust particles may be controversial, their cores taken from the ocean floor and from existence is not. polar ice. Such dust has even been captured The orbits of larger particles are also in- by high-altitude aircraft (Brownlee et al., fluenced by radiation forces. The impacts of 1977) and discovered in deep-sea sediments such solar-orbiting particles are registered (Ganapathy et al., 1978). Arcuate comet by lunar seismometers (Duennebier et al., tails are composed of particles which es- 1976); similar bodies impacting the Earth cape the cometary nucleus and move on are found as meteorites (see Wetherill, their own slightly different solar orbits due 1974). Other evidence for interplanetary to nongravitational forces (cf. Beard, 1963; boulders (5-200 m) is indirect (Kres~ik, Finson and Probstein, 1968a, 1968b; 1978). Dohnanyi (1972) summarizes infor- Marsden, 1974; Vanysek, 1976; Whipple mation on the number density of all in- and Huebner, 1976; Whipple, 1978). The terplanetary masses. RADIATION FORCES IN THE SOLAR SYSTEM 3 The precise origin of the interplanetary [see the orbital perturbation equations as particles causing the above phenomena is developed by Burns ( 1976)]. In this review, unclear (see Whipple, 1976). Certainly we discuss, derive, and explain the forces some interplanetary material in sizes less that arise from interactions between small than a centimeter or so comes from comets, particles and solar radiation. The explana- as attested to by Type II tails (Whipple, tion of these radiation forces was a problem 1955, 1978; Vanysek, 1976; cf. Sekanina that attracted the notice of several eminent and Schuster, 1978a,b; Parthasarathy, 1979) scientists around the turn of the century, and the correlation between meteor show- notably Poynting, Larmor, and Plummer. It ers and cometary orbits. Dust must be gen- was even included briefly in Einstein's erated as well in collisions between larger (1905) classic paper on special relativity. meteoroids (Dohnanyi, 1971; 1976b). Other The correct expression for radiation interplanetary particles may be captured or pressure and Poynting-Robertson drag was transient interstellar grains (Best and Pat- the subject of some dispute at the time, as terson, 1962; Radzievskii, 1967; Bertaux summarized by Robertson (1937) and Lyt- and Blamont, 1976; cf. Levy and Jokipii, tleton (1976), who give pertinent historical 1976). Direct condensation in interplanetary references. space, or perhaps even in sunspots We first provide a simple heuristic deriva- (Hemenway, 1976; cf. Mullah, 1977), might tion of the radiation pressure and be a further source. Large particles--but Poynting-Robertson drag felt by a perfectly still objects whose orbits are affected by absorbing particle, as preparation for a new radiation forces--probably derive from ex- derivation of the complete expression tinct cometary nuclei or represent the small which considers scattering as well as ab- end of the asteroidal size distribution. Not- sorption. The results are confirmed by de- withstanding these other possibilities, the riving them again in a more formal manner, current view is that most dust comes from using the transformation laws for energy cometary matter while large rocky particles and momentum from special relativity. are from the asteroids. Numerical values are presented and dis- To select among the proposed origins for cussed for the forces experienced by small any sample of interplanetary dust (as re- spherical particles composed of real mate- trieved from a balloon, spacecraft, or core rials that are irradiated by the actual solar sample), one must understand the dynamics spectrum; these demonstrate that radiation of such particles and the evolution of their pressure and Poynting-Roberston drag in orbits. Such knowledge is valuable also in the solar system are significant only for par- determining the size of the sources and ticles in a rather narrow size range.