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. In fact sinks that govern the total mass of in- for very small particles (~< 10 -2/zm) the cor- terplanetary matter and in interpreting the responding drag due to solar corpuscular observational evidence. Finally, it is impor- radiation is actually more important than tant in assessing the danger that spacecraft that due to photons. The orbital conse- might incur in hitherto unexplored regions quences of the Poynting-Robertson drag of the solar system (Millman, 1971). and radiation pressure, including possible In general the motion of small particles in loss by orbital collapse or ejection, are also space over short times is dominated by the considered for heliocentric particles. The preponderant gravitational presence of the influence of these same forces on particles Sun. However, the long-term evolution of circling planets is discussed. In addition we such bodies is commonly produced by non- outline the coupling between the dynamics gravitational forces (radiation forces or col- of particles and their sizes (which shrink lisions) because these change a body's total with time owing to sublimation and sputter- mechanical energy and angular momentum ing). A short qualitative description is given 4 BURNS, LAMY, AND SOTER

for the Yarkovsky effect, which is impor- which has been the subject of considerable tant for particles in the meter-to-kilometer controversy and misunderstanding since range, particularly for those made of stony the beginning of the century, receiving its materials. Also briefly mentioned is a new fundamentally correct--although incom- effect, the differential Doppler effect, dis- plete and abstruse--treatment by Robert- covered by McDonough (1975), which may son (1937). be appreciable in other planetary systems. Unfortunately, Robertson's derivation Although certainly important for particles was unnecessarily difficult because it relied of special size and composition, the direct on the metric of special relativity even interaction with the interplanetary elec- though, as shown below, the result can be tromagnetic field of the solar wind will not derived and understood in classical terms be discussed because to date an accurate without invoking space-time dilatation. treatment of the electric potential of in- Furthermore, Robertson's expression ex- terplanetary objects is not available (Be- plicitly contained an important assump- lton, 1966), although recent progress has tion which is usually overlooked: he con- been made (see papers in Grard, 1973); also sidered only perfectly absorbing particles the dynamics for a particle with a given whereas, in general, small particles scatter electric charge are not well developed (cf. or transmit most of the energy incident on Parker, 1964; papers in Weinberg, 1967; them. This deficiency in Robertson's ex- Standeford, 1968; Lamy, 1975; Levy and pression was noted by Mukai et al. (1974) Jokipii, 1976) but advances (Consolmagno and Lamy (1976a), who (incorrectly) con- and Jokipii, 1977; Consolmagno, 1978) are sidered the drag to be caused by the ab- occurring. It seems possible that such sorbed energy only; we shall see below that forces are responsible for the north-south the scattered energy is just as important. To asymmetry seen in the zodiacal light cloud our knowledge, a general expression for this of the inner solar system by Helios (C. term has not previously been included cor- Leinert, personal communication, 1976). rectly, although Lyttleton (1976) derived We review here only those forces that in- the forces for a specific, but artificial, scat- volve the momentum carded by radiation, tering law. H~meen-Anttila (1962) instead and so neglect such effects as, for example, erred by merely multiplying the radiation those due to the stochastic collisions be- force by a constant. tween orbiting particles (see Whipple, 1955; Perhaps it is in part due to Robertson's Kresfik, 1976) or the nonuniform evapora- abstruse derivation and in part to the sub- tion of rotating comet nuclei (Marsden, sequent neglect of scattering that most 1974). We also do not account for the influ- textbook explanations of the Poynting- ence of any gravitational force other than Robertson effect have been confusing that due to a point mass primary (i.e., quad- and/or inaccurate. Even though the effect rupole terms and planetary perturbations is essential to an understanding of the or- are ignored; see Misconi and Weinberg, bital evolution of small interplanetary (and 1978). circumplanetary) particles, we are not aware of any simple but accurate and com- II. RADIATION PRESSURE AND plete derivation except for our prior devel- POYNTING-ROBERTSON DRAG opments (Soter et al., 1977; Burns and So- Bodies in interplanetary space are not ter, 1979), from which the complete absorp- only attracted to the Sun by gravity but are tion case of the derivation below closely fol- also repelled from it by radiation pressure lows, although the approaches of Best and due to the momentum carded in solar Patterson (1962) and Harwit (1973, p. 176) photons. The orbits of such particles are are somewhat similar to it. also modified by the velocity-dependent In order to clarify the problem, we pre- Poynting-Robertson effect, the nature of sent two derivations, one heuristic and the RADIATION FORCES IN THE SOLAR SYSTEM 5

other formal. The heuristic derivation is ap- proached in two stages, first considering ,-%n only perfectly absorbing particles in order © r ~ to obtain Robertson's expression (but on a much simpler physical basis), and then ap- Orbit plying an even more elementary model of colliding particles to find the correct general FIG. 1. A schematic diagram illustrating an orbiting expression for a particle that scatters, particle being struck by solar radiation of flux density S ergs cm-2 sec -1. The particle is of mass m and cross transmits, and absorbs light. These results section A ; it is located at r and is moving with velocity will be supported by rederiving them in a v. The unit vector parallel to the radiation beam is S. more rigorous and methodical manner using transformation laws from special relativity. force, as can be understood by recalling After developing and discussing the correct Newton's second law: a time rate of change expressions, we provide numerical values of linear momentum requires an impressed of the solar radiation forces felt by small force. particles of cosmochemically important The absorbed energy flux S'A is continu- compositions. ously reradiated from the particle. Since the reradiation is nearly isotropic (small parti- III. HEURISTIC DERIVATION OF RADIATION cles being effectively isothermal), there is FORCES no net force exerted thereby on the particle Perfectly Absorbing Particles in its own frame. However, the reradiation is equivalent to a mass loss rate of S'A/c 2 The force on a perfectly absorbing parti- from the moving particle--which has veloc- cle due to solar (photon) radiation can be ity v as seen from the inertial frame of the viewed as composed of two parts: (a) a Sun--and, as measured in the solar frame, radiation pressure term, which is due to the this gives rise to a momentum flux from the initial interception by the particle of the in- particle of -(S'A/c2)v, schematically cident momentum in the beam, and (b) a shown in Fig. 2. Since the particle is losing mass-loading drag, which is due to the ef- fective rate of mass loss from the moving Incident particle as it continuously reradiates the in- "~ Radiot ion cident energy. The total amount of energy intercepted per second from a radiation beam of inte- @ Iv grated flux density S(ergs cm -2 sec -1) by a stationary, perfectly absorbing particle of IN PARTICLE'S FRAME geometrical cross section A is SA. If the particle is moving relative to the Sun with velocity v, we must replace S by Incident S' = S(I -- r/c), (1) Radiation -v/c)Po where I; = v • S is the radial velocity (Fig. IN REST FRAME OF SUN 1), S is a unit vector in the direction of the FIG. 2. A schematic illustration that reradiation pre- incident beam, and c is the speed of light; ferentially emits S'Av/c ~ more momentum in the for- the factor in parentheses is due to the Dop- ward direction, as seen from the solar frame of refer- pler effect, which alters the incident energy ence, because the frequencies (and momenta p) of the flux by shifting the received wavelengths. quanta reradiated (or scattered) in the forward direc- tion are increased over those in the backward direc- The momentum removed per second from tion. The length of the vectors in the reemitted radia- the incident beam as seen by the particle is tion pattern show the momenta in the two frames of then (S'A /c) S. This is the radiation pressure reference. 6 BURNS, LAMY, AND SOTER

momentum while its mass is in fact always over the actual solar spectrum. These coef- conserved, the particle is decelerated: it ficients will be calculated later from Mie suffers a dynamical drag. The momentum theory. loss per unit time represents, according to The scattering diagram resulting from the impulse-momentum theorem, a force unpolarized light incident on a spherical on the particle of the same amount; this is particle has rotational symmetry about the sometimes called the Poynting-Robertson radial direction S. The intensity of the scat- drag. tered light thus depends only on the angle ot The net force on the particle is then the it makes with the incident beam direction sum of the forces due to the impulse (where a---0 corresponds to forward- exerted by the incident beam and the scattering). The energy scattered into the momentum density loss from the moving incremental solid angle annulus d× = 2,r particle or, for a particle of mass m, sinot dot in the direction ot is proportional to f(ot)dx, wheref(ot) is the phase function of rn~ = (S'A/c)S - (S'A/cZ)v the scattering particle, normalized so that ~- (SA/c)[(I - i'/c)S - v/c], , (2) f(ot)dx integrated over all directions is to terms of order v/c. This last expression is unity. Then the net energy flux anisotropi- equivalent to Robertson's (1937) result. cally scattered per second into the forward direction is S'AQsea(cos a)S, where the Scattering Particles anisotropy parameter, In general, as Fig. 3 illustrates, a small (spherical) particle of geometrical cross sec- (cos ot) -= f4f(a) cos ot d X, (3) tion A will scatter an amount of light equiv- alent to that incident on an area AQs~a, and can be calculated, again from Mie theory. absorb that incident on AQ~b~, where Qs~a When the scattering on the average is in the and Q,b~ are defined as the scattering and direction of the beam, (cos ot) is a positive absorption coefficients, respectively; that number whereas it is negative for reflected is, they correspond to the fractional radiation. amounts of energy scattered and absorbed. The radiation pressure coefficient, For a given particle, Qsea, Qabs, and Qp~ (to Qpr-~ Qabs + Qsca(l - (cos ot)), (4) be defined) depend on wavelength; in the derivations that follow, we take them to be will appear in the expressions for the forces average values, computed by integrating felt by particles having general optical properties. For a stationary particle (at least) this factor multiplies the incident momentum flux to transform it into the radiation pressure force. In perfect forward-scattering, Qpr = Qabs; with iso- tropic scattering, Qpr = Qabs + Qsca; while Q~~ Q)roa$+ Osea (COS tl~ for perfect back-scattering, Qpr = Q~bs + 2Q~e~. With more general scattering FIG. 3. An illustration of the efficiency factors of optics. One unit of energy is seen entering the upper laws one can consider a portion of Qsc~ to be left-hand edge: Qam is absorbed by the particle, Qsea is scattered isotropically while the remainder scattered in a pattern that is symmetric about the radi- has (cos ot) = +-1. ation beam while Qtram is transmitted. In the scattered In order to compute the radiation forces beam a momentum of Qsca (cos a) is sent in the for- felt by particles which both scatter and ab- ward direction, where (cos a) is the asymmetry fac- tor, i.e., the weighted average of the angular depen- sorb radiation, we consider a simple model dence of the scattered radiation. Qabs + Qsc, + which involves colliding particles and mass Qtraas = 1, ignoring diffraction. exchange; these processes each cause RADIATION FORCES IN THE SOLAR SYSTEM 7

momentum transfer. This model contains scattered, gM' passes directly through m the two features that were found to be im- while, if backscattered, it undergoes perfect portant in the perfect absorption case: a reflection (see Fig. 4c). Based on our ex- pressure along the incident beam's direction perience with totally absorbing particles, and a drag, due to mass loading, along the we permit the particle's velocity to be mod- particle's velocity. At the top of Fig. 4 a ified following the interaction with the beam particle of mass m, moving with velocity v, in two ways: AV per second in the direction is about to be struck by a beam of projec- of the beam's original momentum is the ac- tiles, M per second, which travel relative to celeration due to the radiation pressure the stationary frame with velocity c, not whereas Av per second along particle's necessarily ,the speed of light. In the middle velocity vector is the deceleration due to panel the same scene is viewed one second mass loading. Although these two compo- after the first projectile in the stream strikes nents are not orthogonal, obviously they m. In the meantime M' = M(I - v cos ~/c) give a complete description of the particle's mass has collided with m; a fractionfM' of response since this is a two-dimensional this has been absorbed but is continuously problem. reemitted (or, equivalently, isotropically Conservation of linear momentum in the scattered) so as to keep the particle's mass x and y directions will provide Av and AV. rn constant. The remaining mass gM' which For this we consider that fM' still resides in strikes the particle leaves it with the same m, as shown in Fig. 4b, since it has no relative vector velocity with which it ar- momentum relative to the particle while its rived; to mimic the two cases of interest for momentum density seen in the stationary radiation, this means that, if forward- frame is identical to the particle's. Equating the momentum in the y direction before and (a) Before after the interaction, we have M 00000 ,V my sin ~: = (m + fM')(v + Av) sin + gM'[(-W-)v sin sr + v sin ~:], where here, and throughout this section, (b) After the top sign corresponds to forward- O'~OOc~M' V + AV scattering and the bottom sign to back-scat- (...')o ° (~'-~ Av tering. The first part of the bracketed term "~m+ fM' is then the vertical velocity of the scattered beam as measured relative to the particle; in magnitude, it is the y component of the

(C) Velocities Relotive to Particle received velocity according to the particle. The last part of this term accounts for the v sin~ ident projectiles fact that the previous measurement is ac- c-v c~;( "qr.. back-scattered complished relative to a moving frame scattered ""(~word whose velocity is v sin st; we ignore any change in v during the interaction. Realizing FIG. 4. A nonrelativistic model to demonstrate radia- tion pressure and Poynting-Robertson drag. (a) The that f + g = 1 and assuming that Av ~ v, c top diagram shows a beam of projectiles on its way to and that M ~ m, we find that strike a particle. (b) A fraction fM' of the projectiles has been absorbed and then isotropically reemitted (or Av = -(1 ~ g)M'v/m; isotrooically scattered). The remainder gM' is either this slowing of the particle along its path we forward-scattered or back-scattered; here it is shown back-scattered. (e) The interaction as viewed from the interpret as being caused by a mass-loading particle where only the incident and the scattered radi- drag. ation are seen. We can now determine AV in the same 8 BURNS, LAMY, AND SOTER

way from conservation of the x momentum For a totally absorbing particle this reduces to (2) since then Opr = 1. Equation (5) is Mc + mv cos ~: identical to Robertson's expression except = (m + fM')[(v + av) cos ~: + AV] for being more general by the inclusion of + gM'[±(c- vcos s~) + vcos s~] the important factor Qpr. We note, however, + (M - M')c. from the above derivation that radiation The last term represents the momentum in forces--including the Poynting-Robertson the beam which has not collided with the drag--are fundamentally classical forces and particle because of the latter's motion. The are not produced by relativistic effects as penultimate term assumes once again that commonly believed. the projectiles leave with the identical rela- For heliocentric particles with v = /-i- + tive velocity they had on approach. Using r0~, where i- = S is the orbital radius unit the same approximations as for the drag vector and 0 is normal to i" in the orbit case and substituting for Av, we arrive at plane, we can also write the radiation force AV = (1 • g)M'c/m; as my -~ (SA/c)apr[(1 - 2i'/c)i~ this is produced by the momentum in the - (rO/c)O]. (6) beam. It is the radiation pressure in the case where the impacting beam is composed of Usually we call the velocity dependent por- photons. tion of (6) the Poynting-Robertson drag and The total effect of the projectile beam can the constant radial term the radiation be found by vectorially adding the two ve- pressure; others say that the radial factor in locity changes this expression is the radiation pressure AV + Av = (1 T- g)(M'/m)(cS - v); while the transverse term is the Poynting- Robertson drag; still others call the coeffi- in actuality this is an acceleration since it cient of the S term in (5) the radiation occurs every second. pressure and the term proportional to v, the The transference of this result from col- Poynting-Robertson drag. liding material objects to the case when To gain familiarity with the radiation radiation instead strikes a particle is direct, forces, we consider the right-hand side of since we can associate ~ g with - Qsea (cos ix) the equation of motion (6) in some limiting and f + g = 1 with Qabs + Qsca; we note cases. These "tests" amount to under- however that, owing to diffraction, Q~b~ standing the meaning of Qpr as given by (4). + Qsea need not be unity for microscopic Perfect transmission--that is, no particle-- particles. Any directly transmitted radiation occurs when Qpr = 0. Complete absorption, is the same as scattered radiation with the case considered by Robertson (1937), (cos a) = 1. Therefore from (4) we have has Qpr = Qab~ = 1, even including diffrac- that tion since the diffracted light has zero Era d = m~¢ = QprM'(c - v), contribution (van de Hulst, 1957, p. 225), and so our results agree with the classi- where M' is the "mass" of the photons cal ones. Any forward-scattered radiation which strike the particle or (S'A/c2). In the ((cos o~)= 1) produces neither radiation above form, the radiation force can be pressure, nor a drag, because the beam just viewed as a drag force caused by the rela- passes through the body, continuing ra- tive velocity, c - v, of the particle through dially outwards from the Sun as though a beam of photons. Finally, to terms of there had been no particle. An isotropically order v/c, we can write the net force on the scattering particle, such as a perfect reflec- particle as tor, has Qpr = Qabs + Qsca. A particle mi, ~-- (SA/c)Qpr[(1 - i'/c)S - v/c]. (5) which perfectly backscatters undergoes RADIATION FORCES IN THE SOLAR SYSTEM 9 twice the drag of a totally absorbing or a beam itself but rather from the momentum perfectly reflecting particle. This happens exchange process. because electromagnetic momentum is car- IV. FORMAL CALCULATION OF RADIATION ded by the back-scattered beam in the di- FORCES USING SPECIAL RELATIVITY rection of the particle's motion since its TRANSFORMATIONS component in that direction is due to the In order to lend credence to the above angle (2v/c) of reflection; this momentum is expressions for the radiation pressure force withdrawn from the particle's motion. A and Poynting-Robertson drag, we derive stationary particle feels a radiation pressure the same results here in a more rigorous without a Doppler shift and, of course, has manner. We do this in order to end once and no Poynting-Robertson drag. for all the controversy over the correct coef- Mass Loss and the Principle of Relativity ficients for these expressions. However, a reader who feels comfortable with the pre- The forces we have derived presume the vious derivations and who is not interested particle's mass to be constant; it remains in the fine points of a formal demonstration the same even though the body is continu- is advised to skip over this section. ally radiating energy (and therefore mass) To find the forces experienced by a parti- because the body absorbs just as much cle of constant mass moving through a energy. (It is, by definition, in thermal beam of radiation, we use the principle of equilibrium.) If we consider a particle that conservation of total linear momentum, in- is not in thermal equilibrium and which is cluding electromagnetic in addition to me- radiating into free space, its mass de- chanical momentum. We take into account creases. The loss of mass occurs at pre- the various interactions the particle has cisely the correct rate so that, considering with the radiation as seen in both the solar the momentum loss, the particle velocity and particle frames of reference. In this way stays constant, permitting the principle of we can calculate the electromagnetic relativity to be satisfied. Whether relativity momentum added per unit time to the would hold was a matter of considerable beam; this must equal the mechanical dispute in the early part of the century when momentum lost per unit time by the parti- Larmor and Poynting each had written in- cle. By the impulse-momentum theorem, correct coefficients on the drag term, mean- the latter quantity is the force acting on the ing that they produced results which vio- particle. This very direct, although cumber- lated relativity (cf. Robertson, 1937). some, approach does not appear to have Note that, for any particle in thermal been previously used to derive the equilibrium, the energy radiated away car- Poynting-Robertson force, with one nota- ries with it mass and linear momentum in ble exception: Einstein (1905) computed the exactly the ratio needed to conserve the radiation pressure on a moving perfect re- particle's linear momentum density (i.e., flector in a way very similar to ours and velocity). On the other hand, the absorbed stated, "All problems in the optics of mov- radiation brings in only mass and not a con- ing bodies can be solved by the method here comitant increase in transverse momentum. employed." We humbly agree. This added mass burdens the particle so The following transformation law that it slows down. We emphasize that the (Jackson, 1962, p. 392) for the four-vector Poynting-Robertson drag takes place be- of momentum-energy is used throughout: cause the momentum in the incoming and px = px,, outgoing radiation is not the same: any such body in thermal equilibrium will suffer a Py =py', decelerating drag. The drag does not result P z = Y[Pz, + (v/c)(E'/c)], (7) from the directed character of the solar P4 =- iE/c = iT(E' + Vpz,)/c, 10 BURNS, LAMY, AND SOTER

where y = (1 - v2/c2) -112 and i -= X/Z-i; the p; = (px,,py,,pz,,iE' /c) primed quantities are measured in a frame = (-Ei/c,O,-TE, v/cZ,iTEJc) (9) of reference which moves with a velocity v along the positive z axis (to be specified). by a direct application of the transformation The same transformation can be applied to law (7). We note that the beam's trajectory find quantities in the primed system in makes an angle with respect to the x axis of terms of unprimed values by merely inter- tk = tan-l(Pz/px) = 3w/c as seen by the par- changing primed and unprimed variables, ticle; this agrees with the transformation and changing the sign of v. law for angles (Jackson, 1962, p. 362). This Since the particle velocities relative to then is the radiation which strikes the parti- the Sun are small in comparison to the cle: some is scattered, some is absorbed to speed of light c, we will consider two sepa- be immediately reemitted isotropically, and rate cases--the particle moving with a the rest passes directly through. All these purely transverse velocity v relative to the interactions can be described by the same beam and then having purely radial motion optical cross sections defined earlier (and to u--and merely add the results from them be calculated in Section VI), since the par- together. ticle is stationary in this reference frame. After inserting these cross sections, one Transverse Motion finds that the outgoing radiation leaving the The case of purely transverse motion cor- vicinity of the particle has a four-vector responds to a particle on a perfectly circular momentum flux of orbit, ignoring the centrifugal acceleration. The situation is shown in Fig. 5, where the po = [-(Edc)(1 - Qpr),0, solar frame is centered on the Sun and the -(yEar~c2)(1 - Qpr),iTE,/c], (10) other frame is attached to the particle which moves with uniform velocity v in the z as seen by an observer in the particle's (tangential) direction. frame of reference. Equation (7) permits As seen by a solar observer when a radia- this to be transformed at once to the outgo- tion beam of energy flux E~ strikes the ing momentum flux as seen in the solar particle, the incoming four-vector of fram e, momentum-energy flux contained in it is po = [-(Edc)(1 - apr), p~ = (px,py ,pz,iE/ c) O,(~E~v/c2)Qpr, = (-Ei/c,O,O,iEdc), (8) i(1 + y¢QprvZ/c")EJc]. (11) where the first three components of the row vector are the x, y, z momentum fluxes Equations (8) and (11) express the in- while the fourth is the energy flux in teraction of the radiation beam with the par- momentum units. According to an observer ticle as seen from a specific frame of refer- on the particle, this vector is ence. Since the system is isolated, any change in the electromagnetic momentum flux must be directly associated with a z z" change of opposite sign in the particle's x ::~ x" mechanical momentum flux. But the change un Porficle /~s Ei y~, tv per unit time in the linear mechanical Y momentum of the particle is, by Newton's FIG. 5. The frame of reference attached to the parti- second law, equal to a force. The force felt cle moves with transverse velocity v in the z(z') direc- by a transversely moving particle, then, is tion relative to the frame of reference attached to the Sun. E~ is the energy per second striking the particle as minus the four-vector momentum loss per seenby an observer on the Sun. unit time from the beam, or RADIATION FORCES IN THE SOLAR SYSTEM 11

Fv = -Ap = p~- Po po = {0, 0, (E, Ic)O - O,,r) = [-QprEi/c,O , -(QprEt/c)('y2v/c), [(1 - U/C)l/Zl(1 + /4/C)1/2], -iQprEW2v2/c a] (12) (iE,/c)[(1 - u/c)'lz/(1 + u/c)'12]}, in which E~ = SA. Po = {0, 0, (E,/c) The x term, since it is radial, is the radia- [1 - Qpr/(1 + u/c)], tion pressure force while the z term is the (iE~/c)[1 - Qpru/C(1 + u/c)]}. (14) tangential Poynting-Robertson drag. Equa- tion (12) agrees with our previous result (5) The force experienced by a particle with the substitution E~ = SA and/" = 0. Al- moving in the direction of the radiation though it might appear that energy is not beam is minus the four-vector momentum- conserved because of the beam's energy energy flux loss from the beam gain expressed by minus the fourth compo- nent of (12), this added energy is lost from F u : -- ~p the particle's kinetic energy: /~kEbeam : F • v = [O,O,(Edc)aor/(1 + u/c), = Fzv. Note that the usual "explanation" of (iEJc)Qr, r(U/c)[l/(l + u/c)]. (15) the Poynting-Robertson force as being the transverse component of the radiation The z term now includes a change in the pressure force, caused by the aberration of radiation pressure force because of the sunlight through an angle (v/c) as seen by motion. Once again the fourth term the particle, is correct in the limit v ~ c, as --AEbeam = F ' u = Fzu and so total energy is apparent by a comparison of the first and is conserved. For u <~ c, the denominator third components of (12). of (12) can be accurately expanded to give Parallel Motion Fu = [0, 0, (SA/c)Qor(1 - 2u/c), We now consider the particle to be mov- (iSA/c)Qpr(u/c)(1 - 2u/c)] (16) ing with velocity u in the direction of the in accord with result (6), substituting i" = u radiation beam, i.e., radially as shown in and 0= 0 along with Et= SA(1- u/c). Fig. 6. An identical approach to that used The z component of (16) is the radiation above for transverse motion is applied and pressure force as modified by a double similar expressions written for the incoming Doppler shift; as seen from the heuristic de- four-vectors of momentum-energy flux, rivation, one (u/c) comes from the rate at p~ = [0, 0, E,/c, iE,/c], which energy is received, the other results p: = {O,O,(EJc)[(1 - u/c)/(1 + u/c)] 'lz, from both the reradiated and scattered (iE~/c)[(1 - u/c)/(1 + u/c)]1'2}, (13) energy containing momentum. As long as both u and v < c (which is and for the outgoing four-vectors of mo- easily satisfied in solar system cases), we mentum-energy flux can linearly add the two force expressions (12) and (16) to find the total effect of the radiation on an arbitrarily moving particle. x x' Using ~ = u, and r0 = v, the final equation of motion is Z Z' r?icle y' = (SA/mc)Qpr[(1 - 2i'/c)~ -(rO/c)O], (17) FIG. 6. The frame of reference attached to the parti- cle moves with radial velocity u in the z(z') direction relative to the frame of reference attached to the Sun. which is, of course, identical to the previ- E, is the energy per second striking the particle as seen ous result (6) and to Robertson's (1937) re- by an observer on the Sun. sult when Qpr = 1. 12 BURNS, LAMY, AND SOTER

V. SOLAR WIND CORPUSCULAR FORCES VI. EVALUATIONOF RADIATION PRESSURE EFFICIENCY FACTORS The conservation of momentum and energy approach that we have applied above is also useful in considering the Size and Shape of Interplanetary Dust forces produced by the solar wind, the par- ticulate radiation from the Sun (cf. Whipple, All that remains in order to find the radia- 1955, 1967; Donnison and Williams, 1977b). tion pressure and Poynting-Robertson drag Since the energy-momentum relation for forces actually experienced by small parti- nonrelativistic particles, p = 2E/v~w (where cles in interplanetary space is to evaluate Vsw is the solar wind velocity), differs from the coefficient Qpr. For this calculation, the that for electromagnetic radiation, we must particles are assumed to be spheres. While appropriately change the energy terms that this is obviously done for reasons of compu- appear in the momentum portion of (12). tational simplicity, it has some observa- The momentum and energy flux densities tional justification. The shapes of lunar mi- carded by the solar wind are on the average crocraters, and those found on returned only 2 × 10-4 and 2 × 10-r, respectively, of spacecraft detectors, are often nearly circu- the amounts carried in the electromagnetic lar, which, according to calibration experi- radiation; hence, since the corpuscular ments (Vedder and Mandeville, 1974; force is proportional to the momentum flux Fechtiget al., 1978), means that the impact- density, the total radiation pressure is little ing particles are approximately spherical affected by solar wind particles striking the (Brownlee et al., 1973; HOrzet al., 1975). To dust grain, even including its increased be more precise, if the particles are Coulomb cross section. Nevertheless, the spheroids, typically their axes are in ratios counterpart of the Poynting-Robertson less than 2:1. Furthermore U-2 aircraft, drag caused by solar wind particles can be balloon, and rocket flights through the significant because the aberration angle of stratosphere and above it have retrieved the solar wind, tan-l(v/V~w), is much larger numerous micrometeorites which are than for the classical radiation case. The roughly equidimensional and are rarely of ratio of the two effects is elongated or plate-like form. More often these interplanetary particles are irregular agglomerates built out of approximately (corpuscular drag/radiation drag) spheroidal submicron elements (Brownlee = (Psw/Prad)(C/Vsw)(Co/Qpr), (17") et al., 1975, 1976, 1977; Brownlee, 1978a,b); the closest terrestrial analogs, perhaps, would be clusters of grapes or where Co is the drag coefficient. This ratio sacks of fish eggs. The structures are quite will be considered in more detail in the sec- compact, with empty space comprising tion on the dynamical consequences of much less than half the volume, and typi- Poynting-Robertson drag. In the next sec- cally have densities of 2 to 2.5 g cm-a; this is tion we will find that Qpr is small for metallic in fair agreement with the derived density of particles less than about 0.003/.tm in radius 2 to 4 g cm -3 for the impacting bodies that and for particles of other materials less than cause the lunar microcraters but is in direct 0.03 /.~m. This means that, in the present contradiction to the traditional view of solar wind, such small particles will be "fluffy" meteroroids (cf. Giese et al., 1978). eliminated by the pseudo-Poynting- Space does contain some of the latter Robertson drag of corpuscular radiation. At nonetheless, as evidenced by a fraction of other times, or about other stars, corpuscu- the stratospheric micrometeorites being lar radiation pressure and drag may be even quite porous: p- 1 g cm -3 (Brownlee, more significant. 1978a,b). Most of the particles collected by RADIATION FORCES IN THE SOLAR SYSTEM 13 the U-2 aircraft are opaque, black aggre- few microns and because particles larger gates (4-25 p,m) built out of -1000-A size than tens of microns are severely heated grains, which themselves are built up out of during entry (Brownlee, 1978b; Brownlee et yet smaller grains less than -100 A in size. al., 1977). Size distributions within this The cumulative elemental composition is range have not been published but the total chondritic (Brownlee, 1978a). Extraterrest- number density in the upper atmosphere is rial vuggy spherules tens of microns in size consistent with the interplanetary flux. have been found in deep-sea sediments; Zodiacal light studies suggest to some (e.g., Parkin et al. (1977) propose that their shape Giese and GriJn, 1976; Leinert et al., 1976) may be intrinsic and not result from heating that the scattering particles are "large" on passage through the terrestrial atmo- (-25 p,m) but fluffy or irregular (Zerull sphere. Theoretical support for the choice et al., 1977b; Giese, 1977; Giese et al., of spherical shapes also exists: recent 1978), whereas others (e.g., Dumont, 1976; analog studies, along with numerical model- Lillie, 1976) believe that substantial num- ing, show that a randomly oriented sample bers of submicron particles are required to of irregular particles has, in the mean, much explain the polarization and ultraviolet re- the same optical response as a sphere. The sults in addition to the north-south asym- irregular particles are somewhat more effi- metry of the zodiacal cloud. Micrometeroid cient scatterers per unit mass due to their detectors aboard spacecraft are sensitive to large surface area and they are more iso- particles in the 0.1- to 10-/zm range tropic, with less forward-scattering and less (Fechtig, 1976; Fechtig et al., 1978); they back-scattering (Zerull, 1976; Zerull et al., have been most successful in determining 1977a; cf. Cuzzi and Pollack, 1978). How- the interplanetary and circumterrestrial ever, as can be seen from any scattering number densities as functions of heliocen- diagram, the radiation pressure felt by the tric distances and in identifying the orbital two classes of particles differs by signifi- characteristics of the dust. Particles mea- cantly less than a factor of two in almost all sured by these techniques are those for cases. which radiation pressure and Poynting- Some comment is due here on the charac- Robertson drag are most significant. Meteor teristic sizes of particles found in in- studies (Soberman, 1971; Millman, 1976; terplanetary space. Lunar microcrater stud- Hughes, 1978) give information on objects ies show a monotonically increasing flux of of centimeter size in the vicinity of the particles throughout the size range from 1 Earth. Information on yet larger objects, mm to 10-2/~m; the crater counts for these those for which the Yarkovsky effect will be sizes imply cratering fluxes of 10-5 to 10a most important, comes by extrapolating the cm -2 year-lsr -1, respectively (Horz et al., meteor data up, or the asteroid data down, 1975; Morrison and Zinner, 1977; Le in size (Dohnanyi, 1972; 1976b; Ashworth, Sergeant and Lamy, 1979). A dip in the 1978); a useful review on the possible mass curve is seen somewhere around a few mi- distribution of interplanetary boulders is by crons which, as discussed later, may be due Kres~k (1978). to selective removal of particles about this Contrary to some early impressions size by either radiation pressure or (HOrz et al., 1975), measurements now indi- Poynting-Robertson drag (see, however, cate that the interplanetary dust flux has Chernyak, 1978). Several hundred extrater- remained constant over ~the past 106 years restrial particles have been captured in the or so (Morrison and Zinner, 1977). Earth's upper atmosphere. The technique is successful for a limited range of sizes be- cause contamination by terrestrial materials Definition of [3 is a serious problem for particles less than a The gravitational attraction of the Sun 14 BURNS, LAMY, AND SOTER

(mass M at distance r) upon a spherical par- Mie Calculations ticle of radius s and density P is The scattering of a plane, monochromatic Fg = ~ 7rsapGM/r 2, (18) electromagnetic wave of wavelength h by a homogeneous isotropic sphere of known where G is the gravitational constant. The optical properties is described by Mie's so- radiation pressure force due to solar radia- lution to Maxwell's equations. Numerous tion, as given in (5), is tabulations of the energy absorbed and the energy scattered with its angular distribu- F, = (SA/c)Qpr, tion have been given as functions of the size using S = L/4~rr 2 for the radiation flux den- parameterX -= 2rrs/h and the particle's op- sity at distance r; L is the solar luminosity. tical properties; these are based on series We can also write S = So(ro/r) 2, where expansions in X of analytical expressions So = 1.36 x l0 s ergs cm -2 sec -1 is the solar (see, e.g., van de Hulst, 1957; Hansen and constant and r0 = 1 AU in centimeters. We Travis, 1974). To find the radiation pressure then compare the radiation force to the felt by a real particle in interplanetary gravitational force, defining their ratio by space, one needs to use the Mie solution to the parameter compute the optical efficiency factors cor- responding to the different optical proper- [3 -~ Fr/Fg = (3L/lfrrrGMc)(Qpr/ps) ties of the various wavelength regions and = 5.7 x lO-SQpr/ps, (19) then to integrate these over the energy dis- tributed in the actual solar spectrum. where Qpr is averaged over the solar spec- A method of computing these integrals trum, and p and s are in cgs units. We for spherical grains using the rigorous Mie have assumed that the Sun is a point source theory has been previously described by of radiation; for coronal particles this is not Lamy (1974b, 1975), who also reviews the a good approximation and instead the solid numerical techniques of other authors. We angle l~ subtended by the Sun should be improve Lamy's earlier calculation in two taken as (f~/2~') = 1 - [1 - (Ro/r)~] in ways: (Over, 1958) and, to be even more correct, the radiation pressure and Poynting- (i) The domain of integration has been Robertson forces can be rederived (Guess, extended to 0.15-15 /~m, the lower limit 1962). Limb darkening should also be in- being taken in order to permit the inclusion cluded (Lamy, in preparation). of the sharp increase in the extinction coef- We see that the force ratio is independent ficients of silicates in the ultraviolet. The of the distance from the Sun and thus the integration accuracy has been bettered by orbits remain conic sections even with the reducing the size of the integration steps addition of radiation pressure. We note that used in computation with the trapezoidal [3/QDr is of order unity for particle radii of rule; they are now 0.01 /~m from 0.15-0.4 order 0.5 /zm, or near where most of the ttm, 0.02 /~m from 0.4-0.7 /zm, 0.05 /~m solar energy is contained. Therefore, the from 0.7-1.0/zm, 0.1/.~m from 1.0-6.0 ~m, important interactions begin to occur when and 0.25/zm from 6.0-15.0/~m. Thus, the the particle is roughly the same size as a integrands are evaluated at 138 values in the characteristic wavelength of the incident wavelength range for each of thirteen grain solar spectrum. Unhappily, geometrical op- sizes: s = 0.005, 0.01, 0.05, 0.07, 0.1, 0.2, tics is therefore not valid in the region 0.3, 0.4, 0.5, 1.0, 3.0, 5.0, and 10.0/~m. where the force becomes significant. The (ii) The mean solar flux S(h) at a distance more general Mie calculation must be sub- of 1 AU is now obtained from the recent stituted for sizes smaller than about a mi- and extensive compilation of Vernazza et al. cron. (1976) in the interval 0.15-0.4/~m. Beyond RADIATION FORCES IN THE SOLAR SYSTEM 15

0.4/xm their values appear too low and we polarization of the zodiacal light. Magnetite rely on the data of Makarova and is a black isometric mineral (FeaO4) often Kharitonov (1972) which represent a good found in meteorites and captured in- average in the interval 0.4-1.0 /zm and terplanetary grains (Brownlee et al., 1976; those of Arvesen et al. (1969) in the 1.0- to 1977), and with other iron compounds as a 2.0-/zm domain. Further in the infrared we residue lining lunar microcraters (Nagel et employed the brightness temperature val- al., 1976); it is also considered a likely con- ues reported by Vernazza et al. (1976, Fig. stituent of interstellar grains (Huffman, 11). A total of 138 numerical values of S(h) 1977). Graphite, a strong absorber of visible were introduced to calculate the integrands. light, is introduced in connection with the interstellar grain problem (see Aannestad Materials and Purcell, 1973; Knacke, 1977; and The materials to be considered are cho- Huffman, 1977; but also Hoyle and Wick- sen for several reasons. Briefly, we wish to ramasinghe, 1977) and because of its pre- study materials which might be typical of sence in carbonaceous chondrites. Lastly, interplanetary dust and for which labora- we include a hypothetical ideal absorbing tory experiments have reasonably de- material as a comparison standard and also lineated the complex index of refraction in plot the radiation pressure as though the the wavelength interval of interest. In addi- conditions of geometrical optics were tion at least a few materials should repre- satisfied for all particle sizes. sent each of the possible different optical The specific complex indices of refraction classifications--some should be dielectrics, for the materials considered here will be others conductors; some transparent, oth- tabulated elsewhere (Lamy, in preparation; ers very absorbent. As archetypes which see Lamy, 1975) and so we limit ourselves have significant cosmic abundances, we to a brief mention of this topic. The ma- choose several silicates, water ice, iron, jority of the optical data for amorphous magnetite, and graphite. The three silicates quartz comes from Steyer et al. (1974), chosen are presumed to be representative of Malitson (1965), Heath and Sachet (1966) dust produced from asteroids and comets as and Drummond (1935). For obsidian and well as that found in interstellar space (Day, basalt the results of Pollack et al. (1973), as 1977; Huffman, 1977): obsidian, a volcanic recently extended into the ultraviolet by glass, is a good example of dirty fused Lamy (1978), are employed. The index of quartz; amorphous quartz is chosen in pre- refraction of ice at 100°K has been mea- ference to crystalline quartz because in- sured by Bertie et al. (1969), and already terplanetary grains appear to become employed by Lamy and Jousselme (1976) in amorphous (at least on their surfaces) by order to evaluate the temperature of in- solar wind ion implantation (Bibring et al., terplanetary ice grains. The classical refer- 1974); and basalt, a rock made essentially ence for the optical properties of graphite is from olivine, pyroxene, and feldspar, may Taft and Phillips (1965). The modern mea- be considered as representative of stony surements of Huffman and Stapp (1973) and meteorites. Water ice is selected because of Steyer (1974) provide the optical properties the expectation that it will be injected into of magnetite over the domain 0.09 to 100 interplanetary space by comets; solid water /zm. For iron, we use the optical properties is chosen rather than liquid because parti- as determined by Moravec et al. (1976) in cles of ice sublimate before reaching the the interval from 0.045 to 0.5/~m, by Gor- melting point (Lamy, 1974a). Iron has a high ban et al. (1973) from 0.25 to 1/~m, and by cosmic abundance; it is found in many Lenham and Treherne (1966a,b) in the meteorites and has been invoked by range 1 to 18/zm. Our "ideal particle" is Wolstencroft and Rose (1967) to explain the one that absorbs all radiation for which

RADIATION FORCES IN THE SOLAR SYSTEM 17

9.8 I IIIII] I I I lili~ I I I IIIII] I i I I Ifll b / ~/ IcelOO*K - - ,( Quartz amorphous...... 9.6 " ~,\ / Obsidian ..... II, , 'l Basalt ...... ~1 \\ Iron .... // ! i Magnetjte .... _ 1.4 - /,~ / ~ Ideal Material __ // I ', P'3Ocm-3

l z /! -

"~ o.o -

O.6

0.4

0.2 " -

0.01 0.t t t0 Particle Radius, microns

FIG. 7b. A semilog plot of the relative radiation pressure force fl = Fr/Fg as a function of particle size for six cosmically significant substances and an ideal material See text for details. greater than a few microns, geometrical op- h/2zr, for the solar spectrum is about 0.1 tics holds, since most of the solar energy is /.i,m. concentrated in wavelengths near 0.5 /xm. The metals suffer a large radiation Thus Qor is essentially constant, indepen- pressure because they backscatter radiation dent of particle size, meaning that the /3 ((cosa) < 0) whereas the dielectrics are curves are linear on the log plot of Fig. 7a strong forward-scatterers. Graphite feels a and hyperbolae, proportional to s -t, in Fig. relatively large radiation force for a variety 7b; the different absolute values for the vari- of sizes because, due to its large imaginary ous materials are due to variations in parti- index of refraction, it absorbs radiation over cle density and albedo. At the other side of a wide wavelength range, very much like a the figure, very small particles are virtually metal, and hence even particles for which X unaffected by the momentum in the solar is substantially less than unity absorb effi- radiation, because the characteristic radia- ciently; because such particles have small tion wavelength ~. is relatively too large to mass,/3 > 1. Even though water ice is rela- sustain much absorption or scattering. Most tively transparent, its low density produces curves peak near 0.1/xm, just as is the case a curve similar to the dielectrics. for the ideal material: the important interac- The numerical integrations are termi- tions are for particle sizes comparable to nated at large particle sizes because so ~,/2zr, decreasing sharply for smallerX, and many terms are needed for the Mie series 18 BURNS, LAMY, AND SOTER

expansions to converge (the number of bly more representative of interplanetary terms goes roughly as X) that the computa- debris; however, carbonaceous chondritic tions become too expensive. For that mat- materials should be added by subsequent ter one should be cautious about trying to workers. Our results do not differ signifi- accurately calculate the radiation pressure cantly in character from those of previous of any big particle by a geometrical optics authors with the exception of the early, and approach: once particles become that large, necessarily crude, models of Opik (1956) the surface elements making up the and Shapiro et al. (1966). Singer and Ban- "spheres" become scatterers themselves dermann (1967), Gindilis et al. (1969), since they can be on the order of a charac- Mukai and Mukai (1973), and Schwehm teristic wavelength. (1976) agree with the conclusions that (i) the One might hope that the small-size end of radiation force is less than gravity for most the/3 plot would be expressed by the Ray- materials, (ii) the maximum value of/3 gen- leigh scattering formula, Q t~ s 4 or/3 a s 3 erally occurs at a few tenths of a micron, (Jackson, 1962, p. 573). Unfortunately, al- (iii) metals differ in their response from though one requirement (X~ 1) of the dielectrics, and (iv) very small particles are Rayleigh scattering approximation holds, virtually unaffected by the momentum car- the other -IdXI ~ 1 with d the refractive ded by the radiation field. index, is less well satisfied; for example, The breadth and height of the result for considering iron particles of 10-z-/zm graphite may be significant in two respects. radius, X = l0 -~ but Id)0 ~ 0.4. This in- First, the majority of captured dust parti- applicability of the Rayleigh result is borne cles are observed to be carbonaceous chon- out in Fig. 7. At very small sizes where dritic in composition (especially like C1 Rayleigh scattering might operate, com- chondrites but having some similarities to plex quantum effects, which are not well C2 objects) and low in reflectivity understood (see Huffman, 1977), may need (Brownlee et al., 1976, 1977). Such particles to be considered. There is a tendency, ac- are known to have carbon abundances cording to the Mie results, for/3 to approach of a few percent. In light of the graphite a constantvalue in this region. Of course, at curve, we wonder--and intend to check in a extremely small sizes (namely, an atom or later publication--whether most small in- two), atomic absorption becomes important terplanetary particles are chondritic and are and/3 can rise above zero; nevertheless this therefore particularly susceptible to radia- only is significant for hydrogen for which, tion pressure. Second, graphite is one of the due to the large amount of energy contained materials proposed as making up a substan- in the solar Lyman-ot line,/3 is about 1.25. tial fraction of interstellar grains (Aanne- Calculations of/3 similar to those shown stad and Purcell, 1973; Huffman, 1977; in Figs. 7a and 7b have also been carded out Knacke, 1977) even though for cosmochemi- by Opik (1956), Shapiro et al. (1977), Singer cal reasons it should be less prominent than and Bandermann (1967), Gindilis et al. other materials. However, Fig. 7a suggests (1969), Mukai and Mukai (1973), Lamy that graphite may be preferentially expelled (1974a, 1974b, 1975), Schwehm (1976), and from the neighborhood of stars. Moreover Rajah and Weidenschilling (1977), but we graphite is known to be formed about car- believe that the results presented here are bon stars, which are not especially massive, more complete. Our numerical program is but are fairly luminous. It seems as though more precise and a more thorough search of ejection should be favored unless the opti- the literature has been made to locate the cal properties of graphite change in going to optical properties of the materials in each the longer wavelengths of these cool stars. particular wavelength region. Moreover we This calls attention to the value of carrying consider a range of materials which is proba- out Mie calculations with accurate optical RADIATION FORCES IN THE SOLAR SYSTEM 19 properties similar to the above but for dif- any star of roughly a solar mass: particles ferent stellar classes. Such calculations are more readily ejected by higher luminos- should be completed not only for graphite ity stars and, furthermore, particles of grains but also for water ice and silicate ma- smaller sizes are more susceptible to ejec- terial; the former might account for in- tion when placed near hotter stars. Lastly, terstellar absorption centered at 3/zm while fluffy particles or very irregular particles, the latter would produce the absorption fea- which have large surface-to-mass ratios, ture near 10/zm. A start has been made on suffer relatively greater values of/3 and thus such calculations by Divari and Reznova are more likely to be affected by the solar (1970), Wickramasinghe (1972) as well as radiation; it is improbable that particles of Lamy (1979, in preparation). such morphology with sizes near 1 /.~m By their very definition Figs. 7a,b are valid would be present in interplanetary space only for particles in the present solar radia- today if our calculations have any validity. tion field. If the solar luminosity has In general, due to the mass-luminosity re- changed in the past, the entire suite of lation L oc M 3 for main sequence stars, we curves should be raised or lowered. Pre- would expect that more massive stars are cisely how this occurs depends on the able to expel a wider composition and size spectral irradiance S(X) and on Qpr(X) for range of dust particles. Calculations by Di- each specific material. For example, if the vari and Reznova (1970) show that this in- Sun passed through a more energetic deed appears to be the case. T-Tauri phase, as suggested by many cos- mogonists, then a wider range of particle vii. DYNAMICAL EFFECTS OF RADIATION sizes may have been blown out of the solar PRESSURE system; because such stars radiate mainly in the infrared, larger particles are more Abrupt Orbital Changes; Escape from the likely to have been ejected. Furthermore an Solar System enhanced stellar wind during the T-Tauri Since the radiation pressure force is a phase may dominate radiation effects. Nev- radially-directed, inverse-square force like ertheless, as a result of the quite narrow gravity, it alone does little to complicate the peak in most /3 curves, only a relatively orbital dynamics of dust particles so long as small class of particles would still be so af- its coefficient remains constant. With the in- fected. In this respect we note that micro- troduction of radiation pressure, a particle's craters are seen on the surfaces of small glass motion is now controlled by the net force, spherules found in the interior of the so that, ignoring the small Poynting- Kapoeta meteorite (Brownlee and Rajan, Robertson drag, 1973) and perhaps in other meteorites. The i- = -(1 -/3)p.S/r ~, (20) character and size range of these micro- craters are similar to those of contemporary where/3 is the ratio of radiation repulsion to lunar microcraters, so we have here an in- gravitational attraction and/z -= GM. Since dication of the existence of very small par- the force remains radial and inverse square, ticles in a meteorite which is thought to be the orbits are still conic sections. If/3 < 1, about 4.0 × 109 years old. This may mean the net force is attractive and so elliptic, that the solar luminosity was not much dif- parabolic, and hyperbolic orbits are per- ferent then. Any shift to shorter wave- mitted, depending on the orbital energy. If lengths in the peak of the solar energy spec- /3 > 1, only hyperbolic orbits occur as in trum, corresponding to higher surface tem- the case of electron-electron collisions. peratures, would accordingly move the We have seen that radiation pressure peaks in the/3 curves to smaller sizes. This forces rarely exceed gravity. However, this should also be the case for particles orbiting does not necessarily mean that particles 20 BURNS, LAMY, AND SOTER

cannot attain hyperbolic escape orbits. As it may be moving too fast to be bound by first noted by Harwit (1963), and later con- the new potential field (i.e.,/z'). Hence such sidered by Whipple (1967), Dohnanyi (1972, particles are placed on hyperbolic orbits 1978), Zook (1975), Zook and Berg (1975), even though their/3 < 1. In fact, it is worth Kres~ik (1976), and Rajan and Weidenschil- observing that /3p >- 10-5 is all that is re- ling (1977), all that is necessary for ejection quired for solar system ejection from long- from the solar system is for a particle's period comets, and even for Comet Halley specific orbital energy, it is only 0.016; these values make escape following release near perihelion quite E/m = v2/2 - l~/r =- -1~/2a, (21) plausible. to become positive; v is the particle velocity Any small particle produced close to a at solar distance r and a is the semimajor star can be accelerated to enormous radial axis of the elliptical orbit. For particles un- velocities if it does not burn up first. Using affected by radiation, the orbital semimajor conservation of energy [see (21)], we find axis a is constant becaue E is conserved in a that a particle introduced at distance ro with r -2 field. However, consider a small particle the initial velocity Vo will have acquired, by which is suddenly released from a much the time it reaches a distance >> ro, a termi- larger parent body: radiation pressure im- nal velocity v® given by mediately comes into play and the particle's v~ = Vo~ + 2/~(/3 - 1)/ro. (24a) /3 undergoes an abrupt change from essen- tially zero (that of the parent body) to some Thus a particle released from a long-period finite value. Equation (21) still holds except "Sun-grazing" comet at perihelion distance that/~ is replaced by a new value Ro (measured in solar radii) will have a ter- minal velocity /z'------/z(1 - /3); (22) ~)~o = t~esc(~/Ro) 1/2, (24b) if/3 is sufficiently large, E will be positive. To illustrate the size of particles for where Vest = 617 km/sec is the usual es- which ejection becomes possible, consider cape velocity from the surface of the Sun. a particle to be released at perihelion, For example, a particle with constant/3 = 1 where rp= a(1 - e) and vp2 = (/z/a)(1 + released from such a comet at a perihelion e)/(1 -e) for an orbit of eccentricity e. distance of Ro = 10 would have a velocity Ejection then occurs for E -> 0, so that, of nearly 200 km/sec when crossing the from (21) and (22), particles for which Earth's orbit. Its velocity when leaving the solar system would be the same, but the /3p -> (1 - e)/2 (23) particle would become entrained with the are placed on hyperbolic trajectories; the local interstellar gas after moving only a few corresponding limit for aphelion ejection is tens of light years. Even with Vo = 0, the /3a -- (1 + e)/2 (Kresgtk, 1976; Rajan and terminal velocity can be large if /3 > 1. Weidenschilling, 1977). Even particles on Thus a hypothetical particle formed in a circular orbits (e = 0) need only have/3 = ½ stellar atmosphere at Ro stellar radii from for ejection (cf. Zook and Berg, 1975). This the center will have a terminal velocity, ne- is understandable qualitatively since the glecting drag forces, of particle's velocity at the instant of release is v~ = vest[(/3 - 1)/Ro] '/2, (25) governed by its motion before release, that is, by the motion of the large particle that where Vest is now the escape velocity from experiences essentially no radiation force; the surface of the particular star. A calcula- after release the particle's motion is con- tion of ejection velocities from two model trolled by the forces it then feels. If the par- cool stars is presented by Wickramasinghe ticle is of the proper size, following release (1972), who also discusses some conse- RADIATION FORCES IN THE SOLAR SYSTEM 21

quences of the energetic ejection velocities be directly released from comets during which result from this process. heating near perihelion passage. Whipple Numerous particles on hyperbolic orbits, (1967, 1976) believes that most dust comes called /3 meteoroids by Zook and Berg from short-period comets, particularly (1975), appear to exist in interplanetary Encke (e -- 0.847) and Halley (e -- 0.967); space (Berg and Gr0n, 1973). During seven Sekanina and Schuster (1978) disagree. Del- years of data collection Pioneers 8 and 9 semme (1976) finds long-period and new found many more particles to be streaming comets more productive. From the criterion away from the Sun than moving toward it; derived above we see that material released Helios 1 observations (Gri~n et al., 1977) from Encke at its perihelion will be ejected confirm this. These must be hyperbolic par- from the solar system if its/3 > 0.08, from ticles, for if they were on elliptic orbits they Halley if /3 > 0.016, and from the aver- would eventually return to be recorded dur- age short-period comet (~--0.56) when ing their motion toward the Sun. Doppler fl > 0.22. In all these cases, ejection from shifts of the zodiacal light spectrum have the solar system becomes a possibility for a been taken to show many scattering par- much wider size range than classical ideas ticles have orbital velocities greater than would imply. It is noteworthy that most the local escape velocity (Fried, 1978). Fur- meteor showers are associated with short- ther evidence for hyperbolic particles is the period comets, especially those of high e. In hint in the lunar microcrater data for two this case, the size of the "large" particles species of dust at very small particle sizes (those virtually unaffected by the radiation (H6rz et al., 1975; Brownlee et al., 1975; field) may be no more than a centimeter or Morrison and Zinner, 1977; LeSergeant and so. Rapid sputtering and sublimation may Lamy, 1978); one class might be hyper- produce similar phenomena for particles bolic particles while the other wouldbe the near 1/~m (Sekanina, 1976). A final mecha- usual interplanetary dust. The hyperbolic nism to produce very small particles from trajectories of those/3 meteoroids, stream- larger ones is rotational bursting, in which ing out from the Sun's vicinity, which man- small particles are spun up by radiation age to impact the lunar surface must be forces to such high angular velocities that confined more closely to the ecliptic plane. the resultant centrifugal stresses exceed the Thus the orientations of the lunar rock sur- tensile strength of the object; to accomplish faces on which the two classes of micro- this spin-up Radzievskii (1954) suggests a craters are preferentially observed may be "windmill" torque driven by albedo varia- distinctive. This is currently unresolved tions while Paddack (1969; cf. Paddack and with Morrison and Zinner (1975) and Man- Rhee, 1976) proposes a "paddlewheel" deville (1977) not seeing any such effect, in torque caused by surface irregularities. disagreement with Hutcheon (1975) and Many of the above mechanisms invoke the Hartung et al. (1975). breakup of large particles, drifting sunward Small particles may be instantaneously because of Poynting-Robertson drag, to generated in interplanetary space in several supply the source of the small particles possible ways. The most straightforward is which become/3 meteoroids: they are thus by collisions between larger dust grains subject to considerations of mass flux con- (Whipple, 1976; Zook and Berg, 1975; servation and, as LeSergeant and Lamy Dohnanyi, 1976a; Rhee, 1976); this process (1978) have argued for a particular model of should be enhanced near the Sun since par- the dust, these large particles may fail in ticle densities and orbital velocities are producing sufficient outflow. higher there (see, e.g., Zook and Berg, Lastly it should be noted that, if come- 1975, Fig. 2). Another production mecha- tary particles are indeed fluffy, then/3 > 1 nism, already introduced, is for material to for some size range and direct ejection will 22 BURNS, LAMY, AND SOTER

occur without the clever dynamics de- 1974; Dobrovol'skii et al., 1973; Lamy, scribed above. In addition particles which 1974b), or when the stellar luminosity var- condense in the atmospheres of cool stars ies. The latter may be either apparent, as may be rapidly ejected from the system if, with the clearing of a circumstellar nebula shortly following nucleation,/3 > 1; that is (see Dorschner, 1971), or intrinsic, as with a to say, escape from these stars usually oc- variable or flare star. Differentiation of (26) curs before the particles have attained their and (27) with position and velocity constant maximum permitted condensation sizes, yields which may in part explain why interstellar grains are small. da/a = -a(v/l~')2dlz ' (28) and

Continuous Orbital Changes: Gradual de/e = [(H/iz'e)2/a](1 - a/r)dlx'. (29) Modification of Radiation Pressure From (22) and (28), we see that the orbital We have seen that the orbital energy of a size always increases with increasing radia- small particle can be abruptly changed tion pressure (i.e., for dp,'< 0) and de- when it is detached from the parent body creases as /~' becomes smaller. Similarly, which had shielded it from the effects of the term (1 - a/r) in (29) is negative near radiation pressure. In such cases the parti- pericenter and therefore an increase in radi- cle departs along a simple conic section ation pressure at that position causes a (perhaps a hyperbola), smoothly patched to more eccentric orbit. that of its parent body. But a particle with/3 "To compute the trajectory of a particle changing continuously will follow a more that is losing mass by sublimation requires a elaborate trajectory. This occurs most read- calculation of the sublimation rate (see ily for particles sufficiently near the Sun to Lamy, 1975), which is a strong function of undergo gradual mass loss due to sputtering the particle's temperature and surface and/or sublimation. properties, and involves the complex prob- One can write the orbital semimajor axis lem of sputtering. Here we merely illustrate a and eccentricity e as functions of the orbi- an orbit that exhibits this behavior (see tal velocity v = (/'~ + /ar2r-2)l/~ and specific Lamy, 1974a, 1975, 1976b; Mukai and angular momentum H = r~8 (cf. Bums, Mukai, 1973). We have chosen an obsidian 1976), which are themselves functions of particle, because then/3 is always less than the particle position (r,O): unity (so it will not directly escape), and put it near the Sun to allow the sublimation to a = r/(2 - rv2/iz), (26) proceed rapidly. As shown in Fig. 8 the par- e = (1 - l-P/lza) ~2. (27) ticle, after starting in a circular orbit, can be In the typical orbital problem, /~ = GM is the coefficient of the r -z force but, once radiation pressure is included, it becomes Grain Size A pocenter A Iz' = GM(1 -/3). Increasing the radiation ~ pressure decreases/z' and thus the size of the orbit increases; it is as if the dust were \ 0.5 attracted by a continuously less massive /Or bit Size Pericent s Sun.

The above expressions are convenient for QO , ~ , ~ ~ ~; ',~, , ;o ~ considering cases in which /z' is not con- Number of Revolutions stant, such as can occur when the particle's FIG. 8. The orbital evolution of an obsidian particle size changes because of sputtering and/or (initial size = I p,m) near the Sun. The separating solid sublimation/condensation (see Mukai et al., lines show aphelion and perihelion distances. RADIATION FORCES IN THE SOLAR SYSTEM 23

seen drifting slowly toward the Sun owing The orbital elements specifying the osculat- to the Poynting-Robertson drag; this drift ing orbit are the osculating elements; gov- will be computed later. The particle size is erned by the perturbation equations of ce- relatively constant until the temperature lestial mechanics, these change slowly reaches a critical value at which the particle owing to dF and represent a gradually starts to sublimate quickly; the decrease in evolving orbit. This problem has been con- size accelerates as the temperature in- sidered by celestial mechanicians for cen- creases due to the growing proximity of the turies. In place of more complicated tech- Sun. As the size approaches that at which/3 niques, Burns (1976) has simply derived the begins to be significant, the orbital charac- perturbation equations in terms of the dis- ter (a,e) changes. The orbit grows as the turbing force components by differentiating effect of (28) overbalances the Poynting- expressions for the orbital elements written Robertson decay. At the same time, as seen in terms of the orbital energy and angular by (29), the orbit also becomes eccentric. momentum per unit mass and then sub- The pericenter continues to be dragged in stituting for the manner in which the energy but the rapid growth in apocenter, which and angular momentum change with time would become infinite if/3 -> 0.5, is halted under the action of the perturbation. He once the particle shrinks to such a size that finds (after we correct two typographical the maximum value of/3 is passed. The ec- errors) that the changes in the orbital ele- centricity then decreases because d/x' is ments of interest are now positive. Collapse continues under the effects of both Poynting-Robertson drag da/dt = -aE/E = 2(a2/mH) and decreasing a by virtue of (28). Before [FRe sin f+ Fr(1 + e cosY)I, (30a) total orbital collapse can take place, the de/dt = (e 2 - 1)(2/-//H +/~/E)/2e particle disappears: its lifetime is ended. = (H/mlz)[Fa sin f Continuous Orbital Changes: Effects of + FT(cosf + COS ~)], (30b) Perturbing Forces di/dt We know that an inverse-square radial = (ISI/H- IYtz/Hz)[(H/Hz) z - 1] -'/2 force field, like that of the gravity field of a = (r/mH)FN cos/9, (30c) spherical mass, produces orbits that are and conic sections. Any small deviation from this ideal field, however, means that the or- dto/dt + cos i(dll/dt) = (H/ml~e) bits approximate, but differ slightly from, [--FR COS f+ FT sin f(2 + e cosy)/ conic sections. To describe the manner in (1 + e cosy)], (30d) which the orbit is modified, we use the per- turbation equations of celestial mechanics, where E = -izm/2a is the orbital energy, which show the variation of the orbital ele- (FR,FT,FN) are the (radial, transverse, ments with time under the action of a small normal) components of the perturbing force, perturbing force dF above and beyond the H = [a/z(1 - e2)] 1/2 gravity force. If a particle in an elliptical orbit is suddenly subjected to such a per- is the angular momentum per unit mass and turbation, the particle's orbit is no longer Hz is its component perpendicular to the elliptical; however an auxiliary orbit, the reference plane (the plane above which i is osculating orbit, may now be defined. This measured),f is the particle's true anomaly is the (elliptical) path that the particle would and ~ its eccentric anomaly [cos ~ = (e + follow if dF suddenly vanished, say at cosf)/(l + e cosy)], O = to +fis the or- t = tl, and the particle continued along its bital longitude of the particle, and fl is the way with r(h) and v(tl) as initial conditions. longitude of node (see Fig. 9). To arrive at 24 BURNS, LAMY, AND SOTER

^ may orbit on its own or as part of a plane- tary ring system. Soter (1971) has shown that some ejecta from impacts on the tiny Mariian moons, which lie deep within their planet's gravitational potential well, can easily escape the satellites but find it much more difficult to leave the vicinity of Mars. As first pointed out by McDonough and Brice (1973), constituents of satellite atmo- FIG. 9. Unit circle, centered on the planet P, show- ing the particle's orbit plane with inclination i relative spheres leak away from the satellites but to the orbital plane of the planet; to is the argument of remain in toroids centered around their or- pericenter; fl is the longitude of the node (measured bits. Furthermore, sputtering by energetic relative to vernal equinox ~, and & = co + [~ is that of magnetospheric particles kicks material off the particle's pericenter; fi is a unit vector from the satellite surfaces into orbits about their planet to the orbit's pericenter; 6 also lies in the orbit plane, perpendicular to fi and therefore points along planets; presumably this process accounts the semiminor axis; c = a × b and is parallel to ~. (eR, for the clouds of sodium and other elements ~r, 6N) are the (radial, transverse, normal) unit vectors observed near Io's orbit (Fang, 1976; Mor- located at the particle whose true anomaly is f. The rison and Burns, 1976; Brown and Yung, solar motion is given by h = no t. 1976). And now there is the possibility of volcanic gases being ejected by Io. the secular change of the orbital elements, It is of some interest therefore to com- we take a and e to be constant over an pute how the trajectories of planetocentric orbit, assuming the perturbations are small, particles may be modified by radiation and time-average the perturbation equa- pressure, Poynting-Robertson drag and tions. For example, other forces. The redistribution of such ma- terial may provide an important loss mecha- (/~) = (I/P0)_[0 P° adt nism, and may account in diverse ways for = (1/27r) ~ tidd~, (31a) the unusual albedo patterns seen on the satellite systems of Jupiter and Saturn (cf. where Po = 21r/n is the particle's orbit pe- Pollack et al., 1978; Cruikshank, 1979). riod and d/is the mean anomaly, defined by This problem has been addressed previ- dd~ = ndt. Alternatively, we can integrate ously for Earth-orbiting dust (Shapiro et over the true anomaly f, using conservation al., 1966; Peale, 1966; Lidov, 1969), for de- of angular momentum, H = r~j~, assuming it bris from Phobos and Deimos (Soter, 1971), to hold over one orbit; thus, and qualitatively for particles orbiting (i~) = (n/27rH) ~ ar2df. (31b) planets of the outer solar system by Morri- .I son and Burns (1976) and by Mendis and Axford (1974), who emphasized the signifi- Orbits about Planets cance of electromagnetic forces. Small particles are known to orbit the Poynting-Robertson decay times have been Earth. These probably come from several taken by Goldreich and Tremaine (1979) to sources: (i) interplanetary meteoroid debris place a lower bound on particle sizes in the that was captured by the Earth during pass- Uranian ring system. Further discussion of age through its upper atmosphere, (ii) scat- the character of the orbital evolution of par- tered lunar ejecta, and (iii) terrestrial mate- ticles circling planets is given by Shapiro rials that are randomly--although rarely-- (1963) and Burns (1977); a simple review of perturbed upward. Similar dust undoubt- the literature on this problem as pertaining edly circles other planets, particularly those to the motion of artificial satellites is pro- further out in the solar system, where it vided by Sehnal (1969). Radiation forces on RADIATION FORCES IN THE SOLAR SYSTEM 25

particles in Saturn's rings are discussed by b- g = cos fl' sin to Lumme (1972) and Vaaraniemi (1973). + sin Fl' cos to cos i, (33b) We investigate the motion of cir- and cumplanetary particles using the perturba- tion equations (30) of celestial mechanics /: • g = -sin fl' sin i, (33c) described above. Compared to heliocentric orbits, planetocentric orbits present the where i is the inclination of the particle's added complication that the radiation unit orbit plane with respect to that of the vector S no longer lies exlusively in the par- planet. (For an equatorially orbiting parti- ticle's orbit plane. We must then average cle, as in Saturn's rings, i is therefore the the perturbations over the parent planet's obliquity of the planet.) orbital period (i.e., the period of the motion In order to simplify the problem, we of S relative to the particle's orbit plane) as make the following assumptions: (i) Sun- well as over that of the particle. The vecto- light reflected from the planet's surface is rial celestial mechanics introduced by Her- ignored; numerical integrations by Shapiro rick (1948) is well suited to this task; it has (1963) show its effect to be small. (ii) The been profitably employed in the treatment solar flux is taken to be constant, i.e., the of this problem by Musen (1960) and Allan planet's eccentricity is ignored; the mean (1961, 1962, 1967). The orthogonal coordi- motion n G then approximates the true angu- nate system (fi, I~, 6) to be used is shown in lar motion of the Sun. Because the plane- Fig. 9, where the unit vectors fi and 6 are tary orbit is assumed to be circular and be- directed along the major and minor axes, cause satellite orbits are small compared to respectively, and where 6 = fi × I~ is nor- those of planets, changes in the radiation mal to the orbit plane in the direction of the forces as a particle approaches and then re- orbital angular momentum. Note that I~ is cedes from the Sun are negligible. (iii) Pass- parallel to the particle's velocity vector at age through the planetary shadow (see Fig. pericenter but antiparallel to that at apo- 10) is usually ignored; numerical integra- center. Also from Fig. 9, we see that tions by Allen (1962) demonstrate that its inclusion does not modify the character of FR = Fa cosf + Fb sin f, the perturbed motion but only decreases its FT = -Fa sinf + Fb cos f, (32) magnitude, and then by at most tens of per- and cent (see also Radzievskii and Artem'ev, 1962). Particularly for the more distant or- F N =Ft., biting particles, it is unimportant. (iv) In- whereFa = F.fi, Fb = F.l~,andFc = F- are the components of the perturbing force F along the axes of our coordinate system. The position of the orbital plane relative to inertial space is given by fl, the longitude of the particle's orbital node measured with respect to vernal equinox; thus ~'l' = fl - h is the longitude with respect to the Sun- planet direction S, where h = not is the mean longitude of the Sun along its orbit. Then, if to is the argument of pericenter, we find from the figure and spherical trigonometry that Sun FIG. 10. Passage of particle through planetary ~i • g = -cos t~' cos to shadow. The particle generally enters into, and + sin~' sin to cosi, (33a) emerges from, the shadow at different solar distances. 26 BURNS, LAMY, AND SOTER

teractions with the planetary magnetic field Equivalent expressions have been previ- are not considered (see Shapiro and Jones, ously derived by Musen (1960), Bryant 1961). (v) The planet is taken to be spheri- (1961), Allan (1962) and Chamberlain cal, which means that resonance effects be- (1979). tween, for example, orbital precession rates We see first that the semimajor axis of a and the motion of the planetary shadow are circumplanetary orbit is unchanged by not included (see Shapiro, 1963, and Allan, radiation pressure. This happens because 1967); in addition the gravitational perturba- (30a) shows that work must be done on the tions of planetary satellites are ignored. orbit in order to affect a. In any conserva- Accepting these simplifications, we re- tive force field, work is only accomplished write Eqs. (30) in terms of (32), so that by absolute displacements through the field. A constant force field, such as the assumed da/dt = 2(a2/mH)[-Fa sin f solar radiation field, is a conservative force + Fb(e + cos J)], field and thus no work is done by it as long de/dt = (H/mlz)[-Fa sin f cos to as, following one complete orbit, the parti-

+ Fb(1 - cosfcos to)], cle returns to its initial position. But, ac- di/dt = (r/mH)Fc(cos to cos f cording to the perturbation assumption, the particle does return to approximately where - sin to sin/), it had been and so the orbit size is un- dto/dt + cos idt~/dt changed. This ignores the likelihood that = H[mlze(1 + cos])] -~ the orbit passes through the planetary [-Fa(1 + sin2f+ e cos j0 shadow, entering into and emerging from it + Fb sin f cos f], at different projected distances f • S (see Fig. and we time average these rates over an 10). Thus, generally, work can in fact be done over a single orbit. Nevertheless, orbit, as indicated in (31). Note that for radiation pressure, since the orbit precesses in its orbit plane under the action of radiation pressure (see Fa = F~" S, Fb =FI~'S, Fc = F~ " S, below), all possible shadow orientations are sampled after the longitude of pericenter where F = (SA/c)Qpr; by the basic assump- tion of the perturbation technique, these are has rotated through 2~-. Thus, even includ- constant over an orbit. The time-averaged ing shadowing (as long as resonance effects integrations of the first three are then most are ignored), the integrated result of the simply performed in the eccentric anomaly radiation force on the orbit size adds to zero when taken over enough orbits. tO, making use of the standard relations (cf. Next we note from (36) that changes in Danby, 1962, pp. 127 and 152) the eccentricity vary with the period of d~t = (r/a)dtO = (1 - e cos tO)dtO, (34a) I) • S, that is, of the sine of the angle be- tween the solar direction and the line of sinfd~t = (1 - e2) in sin todtO, (34b) apses. It is therefore periodic both with the cosfdJ/t = (cos tO - e)dtO. (34c) parent planet's orbital motion and with the motion of pericenter relative to inertial Thus we have space; it is the motion of pericenter, under (da/dt) = 0, (35) the combined action of radiation pressure and planetary oblateness, that allows reso- (de /dt) = }(HF/ml~)b " S, (36) nance effects to come into play. However, if other perturbations are ignored, the mo- (di/dt) = -~(aF/mH)(e cos to)~:- S, (37) tion of pericenter is a function only of the (dto/dt + cos id~/dt) parent planet's motion, as we shall see, and = -~(HF/ml~e)a. S. (38) thus (de/dt) has the period of the planetary RADIATION FORCES IN THE SOLAR SYSTEM 27

orbit alone. The maximum value of (de/dt) We can obtain approximate solutions to occurs when 6 • S = 1, i.e., when the parti- (39) and (41) by assuming Z to be constant, cle's velocity at pericenter is parallel to that i.e., by neglecting terms of order e 2. We of the incident sunlight. In contrast, when then introduce the new variables the particle is heading directly into the inci- dent beam at pericenter, (de/dt) is most h = (e/Z) sin tb, k = (e/Z) cos &, negative. All of these statements can be eas- so that ily understood from the left-hand portion of e" = T'(h" + k") (30b) which, since the orbital energy is con- served under radiation pressure forces and [compare (30a) and (35)], reads ~ = -H/eH = Z(h sin 6J + k cos ~). for small e. The change in the magnitude of the angular momentum is the torque Comparing with this (39) and (41), we have rFr; when time averaged over an orbit, the equivalent equations this in general is nonzero because the elongation of the orbit (see Fig. 10) gives a k = no cos h, /~ = -nG sin h, smaller moment arm at pericenter and the which, integrated from (ho, k0) at h = 0 time spent there is less too. The time- (i.e., t = 0), yield averaged torque will be most positive (i.e., (~) most negative) when the line of apses is h= h0 + sinh, k= k0- 1 + cosh. normal to the solar radiation direction, with These solutions are substituted above to a leading S. The periodicity of (~) occurs give simply because H is periodic since the planet progresses along its orbit and the e 2 -- eo' dust particle's orbit reorients. = 2Z"[(1 - ko)(1 - cos h) + h0 sin hi. Finally, we see from (33c) and (37) that i changes most rapidly when it is large. This, If we specify that t = 0 when &o = 0 (i.e., in conjunction with the expectation that time starts at pericenter passage through most sources of circumplanetary debris lie the subsolar direction; as seen directly be- at low inclinations, implies that inclinations low, this is not restrictive), then h0 = 0, are small during most of the particle's ko = eo/Z and so lifetime (cf. Peale, 1966). Hence, we choose e 2 _ eo2 cosi ~ 1 and therefore, from (33b) and = 2Z2(1 - eo/Z)(l - cos not). (42a) (36), (de/dt) -~ noZ(sin ~ cos k In addition, - cos btb sin h), (39) tan ~b = h/k = sin not/ (cos not- l + eo/Z); (42b) where n o = dh/dt is the mean motion of the parent planet about the Sun, tb -- to + fl is it was this variation in pericenter location the longitude of pericenter, and that was invoked earlier to claim that the planetary shadow samples all pericenter Z = ~-[a(1 - e~)/l~]U2F/mno. (40) orientations. From this expression (plotted Furthermore, making the same approxima- in Peale, 1966, Fig. 3), we see that the lon- tion of low inclination, we have from (33a) gitude of pericenter moves at a nonuniform and (38) rate; nevertheless, since the orbit spends the same amount of time at an orientation (dto/dt + cos id~/dt) & - not as it does at -(& - nd), the effect d&/dt -~ +(noZ/e)(cos tb cos h on (ti) of passing through the planetary + sin tb sin h). (41) shadow integrates to zero (see Fig. i0). 28 BURNS, LAMY, AND SOTER

From (42) we see that the eccentricity os- particle must have collided with, or escaped cillates with the planetary orbital period from, the planet. A lower limit on /3 for and has a characteristic amplitude -Z. As elimination of a particle (having a mean cir- first pointed out by Peale (1966), this means cumplanetary velocity v) in this way can be that a particle in the size range that suffers found by assuming that an ideal alignment significant perturbations can develop an ec- (1~ parallel to S) is maintained throughout centricity during its planet's orbit period time Po/2. Then, for e to reach 1, sufficient to cause its elimination by colli- sion with the planet. The maximum eccen- /3c = ~(v/vo). (45) tricity that can be attained before a collision occurs is emax = 1 - RJa, where Rp is the For a more normal motion of pericenter,/3 planet's radius. would have to exceed this limit, probably The solution (42) describes the general by a factor of several. Thus/3 ---/3c is a nec- periodic nature of the perturbation on e for essary condition for a particle to have the small and moderate values of e, but it can- possibility to be lost from circumplanetary not formally be applied in the region of orbit by radiation pressure-induced pertur- greatest interest (where e approaches unity) bations of its eccentricity. We emphasize, because, from (40), Z then is not constant however, that the possibility of elimination and in fact can vanish. We therefore derive does not mean the certainty of loss: it all another approximate solution but one that depends on the coupling between e and 6) applies in the region of large eccentricity. variations. This is a subject meriting further From (19) and (36), we can write the per- study. Preliminary results (Halamek and turbation equation Bums, unpublished, 1979), based on numeri- cally integrating the complete equations of (de/dt) = ~-[a(1 - e2)/iz]~lZ(F/m)f~ • ¢3 motion, suggest that ejection is assured, re- = ~(1 - e~)ll~/3no(vQ/v)b • ~, (43) gardless of initial conditions, once where vo and v are the mean orbital ve- /3 > (2 or 3)Bc. Consider now the case of a particle locities, respectively, of the planet around ejected from a planetary satellite (by a crat- the Sun and the particle around the planet. ering impact, say). Since most impact Although emm may be quite small (e.g., in ejecta that escape the satellite will have the case of a particle ejected at low velocity velocities not much larger than the mini- from a satellite with small eccentricity), in mum escape velocity, we assume that the most cases emin :~ 0. However, in order to particle's initial circumplanetary orbit has obtain an upper limit on emax, let us assume low eccentricity and a velocity v nearly the that e = 0 at t = 0 and that e has its maxi- same as that of its parent satellite. We see mum value emax at t = ½P0- Then integrating from (45) that, for a given planet, such a (43) between these limits, particle ejected from an inner satellite (with sin -1 emax = ~/3(v®/v)n®'r, higher orbital v) requires a larger/3 in order to be in a position to be eliminated than where does a particle from a more distant satellite. Thus, for example, a particle ejected from ~. = f0P0j~ b . ~3 dt. one of the outermost satellites of Jupiter Since 0 < I). S < 1 when e is increasing, (with a/Rp >~ 300) stands a much better we have 1- < ½P0, so that chance of developing a large eccentricity and thereby colliding with Jupiter or escap- emax < sin [~'r/3(vdv)]. (44) ing the system than does an identical parti- The largest admissible eccentricity is of cle ejected from the innermost satellite course emax = 1, before which point the (JV, aiR, = 2.55). This is because it has less RADIATION FORCES IN THE SOLAR SYSTEM 29 orbital energy and so is more readily per- its orbital semimajor axis which in turn les- turbed by radiation pressure than is a parti- sens the amplitude of the radiation cle orbiting deeper in the planet's potential pressure-induced eccentricity excursions well. It is worth pointing out that the proba- (44) because v is increasing. Since the ble loss flows for debris from the inner and Poynting-Robertson drag induces no ec- the outer satellites can be in quite different centricity variation on its own, this means directions if the criterion for ejection is only that such a particle will slowly spiral in to- barely satisfied. For the former case, as e ward the planet with ever diminishing maxi- builds up but well before it reaches 1 (when mum eccentricity until it collides with an escape occurs), the particle will strike the inner satellite or at last with the planet it- planet since then e need only be 1 - Rp/a; self. However, if solar wind (or magneto- that is to say, essentially all the debris from spheric plasma) sputtering is sufficiently Amalthea will strike Jupiter. The same is rapid, a large particle, originally little af- not true for debris from an outer satellite fected by radiation pressure, may be de- since then Rp/~ 1; in this circumstance, creased in size enough to enter the regime in half the particles will strike the planet (i.e., which radiation pressure will pump up its those that reach e ~ 1 when on their way eccentricity and thereby limit its lifetime. into the planet) while the other half escape On the other hand, sputtering may directly into solar orbit. cause the demise of very small particles be- For particles ejected from solar system fore their orbits collapse onto the planet. satellites in this way, the range of values of Bertaux and Blamont (1973) have shown ~(v/vo) is from 0.011 for the Moon to 0.676 through numerical experiments that the for Amalthea. Referring to Fig. 7, this radiation pressure produced by solar means that lunar ejecta with -0.02 Lyman-a radiation striking circumterrest- /zm 0.1] over the Any particle with radiation pressure too first half of their planet's orbit about the weak to allow its rapid elimination by ec- Sun. Moreover, even particles making sev- centricity perturbations will remain in cir- eral orbits of their primary could escape if, cumplanetary orbit but will be subjected to due to a particular orientation of the starting the long-term Poynting-Robertson drag as orbit, the shadow effect displayed in Fig. 10 described in the next section. This reduces happened fortuitously to continuously add 30 BURNS, LAMY, AND SOTER

energy during the first half of the planet's VIII. DYNAMICAL CONSEQUENCES OF orbit about the Sun. In each of these cases POYNTING-ROBERTSON DRAG the orbital energy gained would have been Heliocentric Orbits ultimately lost from the orbit had the entire cycle been completed but, before the pro- We note in (5) that the Poynting- cesses reversed to finish the cycle, the par- Robertson force is in the opposite direction ticle escaped and so was not there to benefit to the particle velocity. This takes both from the second half of the cycle. energy and angular momentum from the Chamberlain (1979), in an analytical in- particle orbit, decreasing both the orbital vestigation of the possibility that radiation semimajor axis and the eccentricity. To forces might redistribute circumterrestrial compute the consequences of these changes hydrogen atoms, disputes some conclusions we return to the perturbation equations of of Bertaux and Blamont (1973). He does not celestial mechanics; the perturbation forces address the issue of direct escape of atoms, per unit mass for our problem are given by because his interest lies with planetary (6). The radial and transverse parts of the exospheres whose particles, being nearby Poynting-Robertson acceleration are Fa = the planet, are first lost by collisions into its - 2(SAQpr/mC2)~ and FT = -- (SAQpr/mC")rO, atmosphere. His results suggest three main respectively, where S = So(ro/r) 2 is the effects: (a) high-inclination, eccentric orbits solar flux at distance r from the Sun. There move swiftly to the ecliptic plane; (b) is no normal force component, demonstrat- pericenters of direct orbits drift rapidly to- ing immediately by (30c) that there is no ward stable locations roughly westward change in the orbital inclination for a (but eastward for retrograde orbits) of the heliocentric particle under the action of planet; and (c) satellite orbits near such sta- radiation pressure and Poynting-Robertson ble points are quickly eliminated by colli- drag. The secular changes in a and e can be sion with the planet's atmosphere through a found by substituting FR and F+ into (30) lowering of their pericenters. The analysis and time averaging the resulting expres- however does not include the solar motion. sions for/~ and b, as in (31a), using The higher average eccentricity caused by radiation forces also means that colli- i" = [l~/a(1 - e2)]l/2e sinf (46a) sions with satellites become more probable. and The movement of dust particles between the Martian satellites due to radiation forces r0 = [/~/a(1 - e2)]1/2(1 + e cosy') (46b) has been considered by Soter (1971). Soter from Burns (1976). This procedure then (1974) also showed that, under the action of yields these forces, debris eroded from Phoebe, the retrograde outermost satellite of Saturn, could be streaming into the leading hemi- = -(~/a)Qpr(2 + 3e2)/(1 - e2)al2 (47) sphere of Iapetus, thus accounting for the latter's brightness asymmetry (cf. Cruik- and shank, 1979; Pollack et al., 1978). The im-

the orbit plane, the only other secular varia- drag [see Section V, Eq. (17")] will consid- tion of an orbital element is the advance of erably shorten these times, since the drag perihelion whose change, even accounting due to the particulate radiation dominates for relativistic effects, is much slower than that produced by electromagnetic radiation the variations of a ore (Wyatt and Whipple, at particle sizes less than about a tenth of a 1950). The orbital parameters of cosmic micron. This is shown in Fig. 11, which dust particles detected by Pioneers 8 and 9 gives the ratio of these drags as a function compare favorably with those expected if of size for an obsidian grain located at 1 the particles had evolved due to AU. The actual ratio varies somewhat Poynting-Robertson drag (Weidenschilling, owing to changes in the solar wind charac- 1978). teristics with heliocentric distance and with Except for the coefficient Qp~, these ex- time; of particular importance in the latter pressions are identical to those derived by variations are sporadic solar storms whose Wyatt and Whipple (1950), who considered effects are averaged in this plot. The drag Robertson's totally absorbing particle. ratio is small (of order 10-1 ) and constant They are also, with a redefinition of some for particles larger than a micron or so since coefficients, the same as the dynamical equa- the collision cross sections of the dust grain tions investigated by Donnison and Wil- to radiation and to the solar wind particles liams (1977a,b) who investigated the orbits are the same and essentially invariant with of perfectly absorbing and perfectly scatter- respect to particle size (since geometrical ing particles moving through the solar wind. optics holds for both). Notice that in the limit of small eccentricity, The minimum at a size somewhat less the decay rate for e is ~ that for a. The than a micron reflects the growth in /3(s) characteristic orbital decay time with e = 0 relative to the geometric optics result in the is easily found by integrating (47): same region (see Fig. 7a). The drag ratio increases at sizes smaller than this since Qpr tp-R = a2/4rlQpr = 7.0 x 106spR2/Qpr years is unity for particle collisions, whereas it is

= 400 R2/~ years, (50)

where R is a measured in AU. It is readily NG- ROBERTSON DRAGS shown that a=/'O is the time it takes for a t0 E particle at a to be struck by its own equiva- Q lent mass in solar radiation; that the col- ._8 lapse time is of the same order as, but smaller than this, is understandable in light of our view that the Poynting-Robertson drag results from the absorption of mass but o not momentum. The integration of da and de for finite e is ~ ~' not so simple but can be accomplished in Obsidian Grain at 1AU closed form as shown by Wyatt and Whip- ple (1950); their results, when divided by f(~2 I I I Qor, produce characteristic Poynting- t()t I t0 Robertson lifetimes for very small particles Particle Radius s(/=m ) that become long, contradicating classical FIG. l I. Ratio of drag caused by the solar wind to notions. that due to radiation on an obsidian grain at I AU. However, the inclusion of solar wind Based on a similar plot given by Lamy (1975). 32 BURNS, LAMY, AND SOTER

much less than one for the photons, as seen to compute the rate at which sources must in Figs. 7a and b. The slope of the line in supply the particles producing the zodiacal this area differs for the various materials, light. The most fundamental way of doing owing to the specific optical properties of this calculation, originally devised by each substance. Of course when dust grains Whipple (1955), is to use conservation of comprised of only a few hundred atoms or mass and find out how rapidly particles are less are considered, then our model of a being lost from the system. Besides steady small drag is over-idealized; in such Poynting-Robertson collapse, the destruc- a case the momentum transfer to the parti- tion of particles by collisions or sublima- cle is infrequent, abrupt, and relatively tion, and the ejection of particles by radia- large in comparison to the grain's tion pressure must also be taken into ac- momentum. count. To estimate the Poynting-Robertson We note in (50) that tp-R o:/3 -1 and there- loss, Whipple (1955) calculated the energy fore from Fig. 7 that there should be a pre- contained in the scattered sunlight and di- ferential removal of particles in the size rectly related this to the energy lost by the range 0.02-1/~m due to the hastier collapse decaying orbits of the particles. This ap- of the orbits of such particles. The lunar proach bypasses any computation of the microcrater data do indeed show a defi- dynamics and shows that Poynting- ciency: for example, lunar rocks 60095 Robertson drag eliminates about 108 g (Brownlee et al., 1975), 12054 and 76215 sec -t. However, not too much weight ought (Morrison and Zinner, 1977), as well as five to be given to the precise loss rate since the other rocks (H6rz et al., 1975), lack in a integrated flux of zodiacal light was merely relative sense microcraters of pit diameter estimated and since Whipple's model im- -5-10 /zm, corresponding to a projectile plicitly assumes the only particle orbits are radius in the range 2-10 /~m (Mandeville slowly decaying ellipses. For comparison a and Vedder, 1971), if one accepts that the recent dynamical calculation (LeSergeant laboratory simulation of collision by polys- and Lamy, 1978) indicates a mass efflux of tyrene into soda-lime glass at 5-14 km sec -1 3 x 104 g sec -1 for particles larger than 2 is an appropriate analog (cf. Vedder and /zm assuming a particle size distribution Mandeville, 1974). Unfortunately the dif- based on lunar microcrater data; this could ference in the observed and the modeled be produced by about l0 short period com- size ranges appears too large to be ex- ets like Encke or d'Arrest (Sekanina and plained by Poynting-Robertson loss. In- Schuster, 1978a,b) each year. Whipple stead it could be that the deficiency of parti- (1955) proposes that other effects, in par- cles smaller than 10/~m is the immediate ticular Jupiter's perturbations and colli- consequence of the ejection from the solar sional destruction, are likely to be even system by radiation pressure, as previously more important than the Poynting- mentioned. Chernyak (1978) has suggested Robertson effect in removing particles. rather that such particles could have been vaporized during their passage through an Planetocentric Orbits earlier, now-extinct, lunar atmosphere. Or Any calculation of the effects of the the relative absence of 1- to 10-~m particles Poynting-Robertson force on the orbit of a could simply be a fact of nature, if circumplanetary particle has to account for LeSergeant and Lamy (1978) are correct in the fact that S does not in general lie in the their belief that two independent dust popu- orbit plane. We again use the vectorial ce- lations, both of which have low number lestial mechanics approach, applied above densities in the 1- to 10-tzm region, exist. to the orbital evolution of circumplanetary The rate of loss by Poynting-Robertson particles acted on by radiation pressure, spiralling has been used by several authors and invoke the same simplifications. In ad- RADIATION FORCES IN THE SOLAR SYSTEM 33

dition we employ the eccentric anomaly the particle's mass, so we have which, from r = a(l -e cos ~) and nt = -e sin qs, has a time derivative ~b = (aa) = -~(So/R"c") na/r, where n is the particle's mean mo- (Q,r/pS)(5 + cos'-' i), (54) tion. In terms of the vectors a = a~i and where So = 1.36 × 106 ergs cm -2 sec-' is b = a(1 - eZ)'l~6 (see Fig. 9), it is readily the solar constant and R is the heliocentric shown (Allan, 1962) that r = a(cos ~- distance in AU. Similar results are available e) + b sin tk and in Shapiro (1963), Peale (1966), and Allan f = ~b(-a sin ~ + b cos IV). (51) (1967). We note that ~i does not contain e, unlike the heliocentric case, and that (54) is The Poynting-Robertson force, being the independent of orbital size. Integrating (54), velocity dependent part of (5), may be writ- we can write the exponential decay time for ten circumplanetary particles with i = 0 as

Fp_a = -(SA/cZ)apr[(r " S)S + i']. (52) zp-a = 9.3 × lO~R~ps/Q~r years (55) Then the rate of change of a circumplane- with p and s in cgs units. For small particles tary particle's orbital energy is it is more convenient to express this as /~ = Fp-a" f = -(SA/cZ)Qor[(r • ~)z + r . r] zp-R = 530 R2/[3 years, = -(SA/c2)Qor(b"[(a • S)" sin z tk + (b" ~)2 cos" ¢ - 2(a' S)(b" S) using (19);/3 remains the ratio of radiation sine cos ¢ + a2(1 - e z cos 2 ¢)]. pressure to solar gravity. This ratio for planetocentric orbit collapse is essentially This must now be time averaged over the the same as (50), that for heliocentric orbit particle's orbital motion (~b), its advance of collapse; fundamentally this results because pericenter (to), and the (assumed circular) the Poynting-Robertson drag is due to the orbital motion of its parent planet (fl'). The reradiation of the absorbed energy, which is averaging integrations of /~ can be per- nearly the same regardless of whether the formed in any order, but to obtain all ex- planet or the Sun is being orbited. The pressions in closed form it is convenient to slight difference in the coefficients is princi- take os and l~' first. The dependence of/~ on pally due to one being a characteristic ca and f~' is through (a • S) and (b • S), and decay time while the other is the integrated from (33) we have collapse time. Even though this collapse (0i. g)~>0 = <(b. s)")0 -- 1(1 + cos" i), time can be short, the loss of highly per- turbed circumplanetary particles more <(a. ,~)(fi ~)>o= o, commonly happens due to the eccentricity where ( >0 indicates a quantity averaged variations induced by radiation pressure over both ca and I'V. Then forces, described earlier. These account for removal of particles before collapse (~')0 = -(SA/4d')apr(a$)" through Poynting-Robertson drag can ac- (5 + cos 2 i)(1 - e" cos'-' ~b). cumulate since, as seen in (36), the eccen- tricity variations have the same period as We finally average this last expression over the planet's orbital motion. the particle's orbital motion $, using (31a) By a procedure similar to the above, we and (34a), to obtain can examine the effect on the eccentricity of a circumplanetary particle due to the (.E'> = -(SAtz/4ac2)Qor(5 + cos 2 i), (53) Poynting-Robertson force. The torque on where ( ) indicates the compound average. the particle's orbit due to Fo-R is mH = r × From (30a), (a> = 2a"

Red Shifted vectors and integrating over tk, to, and f~', ~e Photons as before, we find that H/H = fi/2a. Ex- ¢~~ /tResultantForce a. ~,~:r ticle pressing b in terms of H and ti from (30a,b), we find that (b) = 0. Sun Photons The orbital inclination can be shown to /f-" 24~ exhibit an oscillation with the orbital period 0 of the planet (Shapiro, 1963). Therefore it is "Blue not of particular interest. FIG. 12. A schematic diagram of the differential IX. DIFFERENTIAL DOPPLER EFFECT Doppler effect. (a) Photons arriving at the particle from the retreating eastern hemisphere carry less momen- Another, more subtle radiation drag, tum on the average (because of red-shifts) than do identified by McDonough (1975), operates those from the approaching western hemisphere. The because light emitted from the retreating particle is pushed forward. (b) The Sun is modeled by point sources on each limb in order to use the relative eastern half of the Sun will be red-shifted, velocity between the limb and the particle. decreasing its momentum, while identical photons from the approaching western half force since most solar energy comes from will have increased momenta due to being regions which have smaller Doppler shifts blue-shifted. This asymmetric delivery of than the limb values; nevertheless the cor- radiation momentum produces an additional rection is less than an order of magnitude. transverse force on an interplanetary parti- The force is small and positive for distant cle (Fig. 12). particles but is a drag for particles within We obtain a meaningful upper bound on the synchronous orbit position. the effect by considering half of the solar If this force were larger than the radiant energy to be emitted at the eastern Poynting-Robertson drag in certain re- limb and the remaining half at the western gions, then particles could be driven inward limb; that is, the particle (assumed on a cir- when FDDE Fp-R and large dust concentrations but rather just two point sources, one could accrue. However, from (17) and (58), slightly red-shifted and the other equally blue-shifted. The relative velocity between FDDE/Fp-R = (gJr)Z[(toJn) - l] (59) the particle and the eastern "light bulb" is and thus the differential Doppler force is Vrel = ~eRo - nr sin sr always less than the Poynting-Robertson = (to o-n)rsin~; (56) drag. It is important only if a particle is near the solar surface (cf. Guess, 1962), where it the orbital angular velocity of the particle is adds to the Poynting-Robertson drag; how- n, while the solar spin rate is too. ever, it could be appreciable for particles The transverse momentum resulting from orbiting a contact binary system (S. J. two photons of rest momentum po/2, emit- Weidenschilling, private communication, ted on each side, striking the particle is then 1979). The effect is only significant because it provides a more profound insight into the P = (po/2)(1 + Vrel/C) sin phenomena that produce the Poynting- -- (p0/2)(1 - /)rel/c) sin Robertson effect itself. = po(vre~/c) sin ~. (57) X. THE YARKOVSKY EFFECT Following the radiation pressure derivation, and substituting for sin ~ and Vr~, we find Elementary Considerations We noted earlier that the transverse drag FDDE ----- (QprSA/c)(R~/rc)(to O - n). (58) known as the Poynting-Robertson effect This substantially overestimates the actual has been interpreted as caused by the aber- RADIATION FORCES IN THE SOLAR SYSTEM 35

Morning Hemisphere by Peterson (1966) in his M.S. thesis. Our (T -A T/2 ) discussion will follow the physical argu-

S°'°r --" ments of Opik (1951) but results will be Radiation ~ compared against those of the more detailed Re-emitted isphere treatment by Peterson (1976). Another re- Radiation -- (T +ATIz ) cent analytical solution is presented by FIG. 13. The Yarkovsky effect. The evening hemi- Hasegawa et al. (1977). sphere radiates extra energy and momentum because it is hotter than the morning hemisphere. For prograde "Fast" Rotation vs "Slow" Rotation; rotation as shown, the particle is thrust forward the "Large" Bodies vs "Small" Ones momentum carded away in the reemitted radiation. The computation of the force will require a knowledge of the surface temperature on ration from the radial direction of the re- a rotating atmosphereless sphere of radius s ceived radiation. An angular displacement having thermal diffusivity K = K/pC, from the radial direction is also present in where O is mass density, K is thermal con- the absorbed and reemitted radiation any ductivity, and C is specific heat. At the out- time a particle rotates and is not isothermal. set we find it valuable to distinguish parti- The temperature pattern is skewed by the cles that are rotating rapidly from those particle's rotation in concert with thermal spinning slowly, because the thermal dis- lags; this can be viewed as an evening hemi- tributions, both in the interior and on the sphere that on the average is slightly surface, in the two cases are qualitatively warmer than the morning hemisphere, just different. A schematic diagram of the sur- as on planets. The warmer evening hemi- face temperatures for the fast and slow sphere radiates more energy, and hence cases is given as Fig. 14. The surface tem- momentum, than the cooler morning hemi- perature of an element in sunlight is deter- sphere. The reaction force due to the ab- mined by a balance of the absorbed solar sorbed and reemitted radiation is found by energy with the surface reemission plus that integrating over the entire surface; in gen- conducted to the interior. Starting on the eral the reaction force will have a trans- left of the figure, at noon the temperature is verse component analogous to the non- at its peak and slowly decreases as the solar gravitational forces which perturb cometary elevation drops, since in this region the sur- orbits (but, of course, the latter forces are face layers are in equilibrium with the solar produced by outgassing of volatiles rather flux. Beginning at dusk the surface cools by than by emitting radiation; see Marsden, 1976; Whipple, 1978). Figure 13 depicts the essentials of this effect. One can anticipate that the force may involve the particle's ro- tation rate, thermal properties and dimen- sions, as well as the solar distance. According to Opik (1951), this process

was first described around 1900 by an East- F-- ern European civil engineer, I. O. Yar- kovsky (for a biographical sketch, see • "~e(o. -'~,

Neiman et al., 1965), in a pamphlet whose Noon DuIsk Down NOOn reference has been lost. The effect was twice independently rediscovered: by Rad- FIG. 14. Sketch of the surface temperature on a rotating body. Two cases are shown: The "'fast" rota- zievskii (1952) who, along with Opik (1951), tion (or "large" body) case is the solid line while the hinted at its dominance among perturbation "slow" rotation (or "small" body) case is the dotted effects for meter-sized particles, and again curve. See text. 36 BURNS, LAMY, AND SOTER

radiation so that the interior conducts its S > (2KP)112; (61) warmer heat there. At fast spins the tem- perature decay at night is essentially linear this is in agreement with the definition of a as shown, the achieved 8T being propor- "large" body used by Opik (1951) and tional to P. However, if the object rotates Peterson (1976); see also Kaula (1968, p. slowly enough, interior temperatures drop 272). It should be remarked that the "large" to the point where heat can be effi- body case is just as appropriately denoted ciently conducted through from the front; the "fast" rotation case. Typical values for this causes the curve for slow spin to start the material parameters of most silicates to turn up near down; it would be symmetric that form the surface layers of the Earth are for a nonrotating particle. Thus the maxi- C= 107ergs°K-lg -1,p = 2.5gcm -a,K = mum asymmetry on the night side will 3.5 × 105 ergs °K-t cm -1 sec -1 (Kaula, 1968; occur at spins near the transition between cf. Peterson, 1976). Then for stony bodies the slow and the fast rotation cases. Once the criterion distinguishing surficial from dawn is reached, T grows quickly as total heating is equilibrium with the incoming radiation is sought. The morning surface temperature s ~> 0.2 p1/2 cm, (62a) on rapidly spinning particles is cooler than with P in seconds. For metal-rich objects, with slow ones because in the fast case the we choose C=5 x 106 ergs °K-lg -t, temperature is not yet in equilibrium and p = 8.0 g cm -3, K = 4 × 106 ergs appreciable heat is still being carded to- °K-lcm -1 sec -1, in which case the criterion ward the center. reads When an object falls into the rapid rota- tion class, its temperature fluctuations, as it S ~> 0.5 pl/2 cm. (62b) turns in the sunlight, are restricted to sur- face layers, while in the slow spin case they If we take P to be the rotation period of the penetrate deeply the particle's interior. We smallest observed asteroids (D ~> 1 km), all take the size dividing these cases to be of which have periods in excess of several hours (Bums and Tedesco, 1979), (62) im- s = h, (60) plies that objects larger than several tens of where h is the characteristic thickness centimeters will fall into the "large" body traversed by a thermal pulse during half the case. Since, if anything, we expect in- rotation period (P/2) = ,r/to. To estimate h, terplanetary boulders to spin even faster we make use of the Green's function, which than asteroids owing to the increased im- is the solution to a partial differential equa- portance of collisions for smaller objects tion where a delta function is the source (cf. Opik, 1951; Dohnanyi, 1976), we be- term. For the heat equation, the Green's lieve that all objects larger than 1 cm, or function for the temperature T exhibits the some number considerably smaller, will be characteristic manner in which T decays "fast" rotators by our definition. If the sur- with time t and distance 5e. Since the face layers of the particles are not solid but Green's function contains the term e -~/4Kt instead are rather porous or textured, as for (Greenberg, 1971, p. 67), the interior tem- a regolith, the right-hand sides of (62) are perature is damped to e -1 the surface value significant overestimates of the critical size. at an approximate depth 2e of 2(Kt) tn. Thus In this connection we note that the thermal when considering the temperature fluctua- inertias (KpC) t/2 for the surfaces of the Gali- tions on a rotating particle, we are dealing lean satellites and Phobos, according to ec- only with a surface effect if the particle has lipse radiometry (Morrison, 1977), are 2 or 3 a radius larger than the characteristic pene- orders of magnitude lower than those given tration depth of a thermal wave, i.e., if by our choices of material parameters. RADIATION FORCES IN THE SOLAR SYSTEM 37

Temperature Changes on Rapidly Rotating midnight portion radiates more than the Bodies midnight-to-dawn side while the dawn-to- When the thermal effect is confined to the noon area absorbs more radiation than does surface layers (i.e., h < s), the temperature afternoon side. We idealize the actual tem- change during a rotation may be estimated perature distribution by considering the en- by equating the energy received with that tire evening hemisphere to be at a tempera- stored in the volume contained in a thermal ture T + AT/2 whereas the morning hemi- penetration depth h. During a complete ro- sphere is at T - AT/2, where ATis given by tation the energy striking an equatorial strip (63). For the force consider a surface ele- of unit height is 2sSP. Thus for surface ma- ment dA at temperature T, radiating isotrop- terial with albedo 8 the energy absorbed per ically into one hemisphere With intensity I; rotation period by a unit area surface ele- i.e., the entire surface is taken to radiate as ment on the equator is (1 - 8)SPlit. The a Lambert surface would. The outward internal energy which is added to the ther- energy flux is f I cos v d x = ~rl = trT', mal penetration layer during a temperature where v is the angle with respect to the sur- rise of AT is phCAT-~ 2AT(crKpC/¢o) v2. face normal, X is solid angle, and tr is the Equating these, the typical temperature Stefan-Boltzmann constant; we assume the fluctuation within the thermal penetration surface emissivity ~(h) ~ 1. Then the radia- depth for a rapidly rotating object is tion reaction upon the element, normal to its surface, is AT ~ (1 - 8)yS(P/27r) v2, (63) dF = f (I cos v)(cos v dA)dx/c where the inverse thermal inertia y is = -~ trT'dA/c. (65) (KoC) -in (Kaula, 1968). We will take this computed variation in the mean tempera- For a spherical particle of radius s having ture of the layer to also measure the change the assumed surface temperature with in the surface temperature during a rota- AT/T <~ 1, the transverse reaction force in tion. Our result is within 25% of Peterson's the orbit plane--that is, the Yarkovsky asymptotic value (AT/T <~ 1), which was force--resulting from the excess emission found by solving the partial differential equ- on the evening side is computed to be ation for the temperature variation on a cy- Fy = ~TrsZ(trT4/c)(AT/T) cos ~; (66) linder's surface. Using the values for the parameters given previously, the tempera- is the particle's obliquity, the angle be- ture difference felt over the surface of a rap- tween its rotation axis and orbit pole. Note idly spinning stony body would be that this perturbation force can be either positive or negative, depending on whether AT-~ ~ P1/Z/R2, (64a) the particle rotates prograde (0 < g < ~r/2) which has the form of Opik's (1951) result; or retrograde (~r/2 < g < It), respectively. his coefficient is ~. However, we know If the rotation is retrograde, the particle's neither the basis of his expression (beyond leading hemisphere has a larger force on it "dimensional analysis") nor his choice of pa- than the trailing side and so the Yarkovsky rameters (beyond "proper values for force then adds to the Poynting-Robertson stones"). For iron, drag, causing a faster inward spiral. How- ever, for prograde rotation, as in Fig. 13, AT-~ ~ PV2/R2. (64b) the Yarkovsky force , opposes the Poynting-Robertson force and, if large enough, can actually produce outward Yarkovsky Force for Rapid Rotation Case spiraling. The Yarkovsky force results from the In order to compare the Yarkovsky force fact, shown in Fig. 14, that the dusk-to- to the Poynting-Robertson drag, we write 38 BURNS, LAMY, AND SOTER the latter from (5), assuming a circular or- from Fig. 14, this will also be the largest bit, as possible force for all situations, since at slower rotation rates, conduction of heat Fp-R = -SAOprv/c 2. (67) starts to equalize the surface temperatures The ratio of (66) to (67) is then on the two sides of the noon-midnight line. The maximum Yarkovsky force then is Fy/Fp_R ~ -~(clv)(AT/T) cos ~. (68) from (68) Crudely, this can be viewed as the ratio of F y / F p -R lmax the asymmetry in the Yarkovsky problem = -W(1 - 8)al4s cos ~/2VZKR, (71) (AT cos g/T) to that of the Poynting- Robertson drag (v/c): the input energy has where AT has taken its value at the transi- cancelled out because it is the same in the tion from "slow" to "fast" rotation. For two cases. Since (AT/T) is a function of or- stony objects this is about 20s/R, where s is bital distance, as well as the composition in centimeters and R in astronomical units; and rotation period of a given particle (size for irons it is about an order of magnitude being unimportant for these rapid rotators), less. while v varies with orbital position, it is possible that Fv = --Fp-R in parts of the inner solar system; for this to occur, AT/T Yarkovsky Force for "Slowly" Rotating need be merely of order 10-4 . In such re- Bodies gions, material could accumulate since par- In the slow rotation case the temperature ticles are driven there, but not removed, by on the night side at the start drops linearly drag forces (see Gold, 1975). with time. In this region Eq. (63) will repre- The average surface temperature T is de- sent AT if the rotation period P there is re- termined such that the absorbed energy placed by the thermal penetration time flux, SAQabs, heats the sphere to the point T = sZ/2K from (61); this decreases the ef- that the reemitted energy flux, 4rrs2cr/~, fect over what it would be if the fast spin equals it. For macroscopic particles, condition were still satisfied. The Yar- Qpr ~ Qabs ~ 1- 8, ignoring any noniso- kovsky force is further reduced due to the tropic scattering. Hence the mean equilib- shifted location of the asymmetry in the re- rium temperature of a body in space is radiation and the absorption; as slower and T = [(SolR2)(1 - 8)/4o-]"4. (69) slower rotations are considered, the radia- tion and absorption patterns become more This is approximate but only valid when and more symmetrical with respect to the AT/T is very small; otherwise significantly direction along which the radiation arrives. more energy appears in this formulation to We account for this feature by multiplying be radiated away than is received. Defining the Yarkovsky force by T/P, which gives W = 2c (So 3 o')lt4(r/R)It2/[3(~" GM) 112] = 1.31 the correct result in the limits P ~ r and x l0 T ergs AU sec -1 °K-~cm -2 and using p ~ 0o; Opik (1951) does not include this (69) in (68) with (63), the Yarkovsky force factor. With these modifications, we find relative to the Poynting-Robertson drag is from (63), (68), and (69) Fy / Fp_R Fv/Fp-R = -W(1 - 8)a14sa cos g/ = -W~/(1 - 8)3/4p v2 cos g/R (70) [P(2)anRKK] (72) for rapidly spinning particles. This will be for slowly rotating objects. For this case considered again following the next section. 0pik (1951) found Fv/Fp-R ~ s cos g/R, The maximum value of the Yarkovsky a result like (71) while a linearization of force for the fast case will occur when AT Peterson (1976) for small P has the force becomes as large as possible; as can be seen ratio -cos g/RP vz. Our disagreements are RADIATION FORCES IN THE SOLAR SYSTEM 39

not too important for, as we shall see, the ,o~i i l , , i" .~ jl-- fast rotation case is more probable. The Probable Yarkovsky Force ,o~ ~,o~" L / P.1O4 sec It is beyond the scope of this paper to ,( // calculate probable rotation rates. Neverthe- less, as mentioned earlier, most likely the / J" ~0 /" heating is a skin effect (cf. Opik, 1951; 1o // ,,6// Hasegawa et al., 1977) because small in- terplanetary bodies are believed to rotate i / / i / r" i / /.~)" P"~ "~c 1 I ii / ii / very rapidly, owing either to collisions with i I / fl P= tCT2sec other small bodies (McAdoo and Burns, / x/ / / 1973; Burns, 1975; Harris and Burns, 1979; ~o"2 ~d i ~o to2 Burns and Tedesco, 1979; Opik, 1951; Particle Size , cm Dohnanyi, 1971, 1972, 1976b; Harris, 1979) F[o. 15. The ratio of the Yarkovsky force to the Poynting-Robertson drag for various spin periods and or differential radiation forces (Radzievskii, particle sizes for stony particles at ! AU. The top dot- 1954; Paddack, 1969; Paddack and Rhee, ted line depicts the transition from the fast rotation 1976; Icke, 1973). The potential of each of case to the slow spin case and gives the maximum these processes to produce rapid spins is Yarkovsky force; the numbers on the sloping dotted easily understood. In the first case a collid- lines are the ratio of the Yarkovsky perturbation to gravity. ing cloud of bodies will, when in "thermal" equilibrium, distribute its kinetic energy drag for stony objects at 1 AU; for irons the equally between the random velocity com- fast spin curves would be reduced by a fac- ponents (i.e., the noncircular part of the or- tor of about 4 while their slow spin curves bital velocity) and the intrinsic spins of the would be down by nearly 100. The level objects making up the cloud; enormous ro- lines on the right-hand portion are the solu- tational velocities are so produced unless a tion (70) for fast rotations whereas the left- drag is present or equilibrium has not been hand curves, having a slope of + 3 for small reached. Radiation forces may also be very s, are for slow spins (Eq. 72). The transition effective in spinning up small particles: ac- between these two cases comes from (71); cording to the discussion following (45), as shown, it also forms the envelope of the such particles during their lifetimes are family of solution curves for various spin struck by amounts of radiation whose rates; for irons it would be shifted to the equivalent mass may exceed the particle right by log 2.5. We notice that for most mass. This radiation impacts the particle at objects larger than dust grains the Yar- velocity c and, hence, if there is the kovsky force dominates the Poynting- slightest systematic asymmetry in the way Robertson drag; nevertheless, since the lat- it is absorbed or reflected, it is capable of ter has constant direction whereas the delivering huge amounts of angular momen- former's sign depends on the spin direction, tum: as a result the particles will spin very the Poynting-Robertson drag may dominate fast. Larger particles are not as affected by over long evolution times. these processes and therefore rotate more We remind the reader that the strength of slowly, but they probably still have thermal the perturbation due to the Yarkovsky force penetration depths much less than their is found by comparison against the gravita- radii. Thus we expect that (70) is more ap- tional attraction. It is in this sense that (71) propriate to use for estimating the Yar- gives the maximum force. The Yarkovsky kovsky perturbation than is (72). perturbation relative to gravity at the criti- We show in Fig. 15 the ratio of the Yar- cal size is Fy/Fgravlma x = fl(Fe_rt/Fap ) kovsky force to the Poynting-Robertson (Fv/F,-R) or (20s/R)(v/c)fl for stones. 40 BURNS, LAMY, AND SOTER

Using (71) and considering objects for orbit size caused by the Yarkovsky effect which geometrical optics holds, Fv/ permit secular resonances with Jupiter to be F~ravlmax ~ 5 × 10-8R-3j2, a very small per- achieved and then secular responses allow turbation but one that over 10r-108 years rapid modification into Earth-crossing or- will cause complete orbital collapse if the bits. object has a continuous retrograde spin. The relative perturbation for spin rates XI. SUMMARY other than the critical one are found by scal- We have derived in two original ways the ing and are shown dotted in Fig. 15. radiation pressure force and Poynting- The dynamics resulting from Yarkovsky Robertson drag on small particles in the forces are quite complicated (Dohnanyi, solar ~system; our derivations include the 1978). The orbital evolution produced is de- first accurate, general treatment of the role scribed by a random walk process of vary- of scattering in addition to absorption of ing step size and direction. This occurs be- sunlight. In order to investigate interplane- cause collisions, being statistical vector ad- tary dust, we have used Mie calculations, ditions of angular momentum, are as likely employing actual optical properties and the to produce prograde as retrograde rotations solar spectrum, to compute radiation if much angular momentum is transferred in pressure efficiency factors, to which both such a manner (Burns and Safronov, 1973; forces are proportional. We have found the Harris, 1979). Thus perturbation forces forces felt by 0.005- to 10.0-wm particles occur in random directions and orbits may composed of water ice, graphite, iron, spiral erratically inward or outward (the lat- magnetite, basalt, amorphous quartz, and ter provided Fy/Fp_R> 1). Moreover, obsidian. These results show that only for Slabinski (1977) even questions whether the iron, magnetite, and graphite does the radi- Yarkovsky force will have a cumulative ef- ation pressure force exceed gravity and, fect on objects less than a meter in size; he even then, just for a fairly narrow size points out that solar radiation pressure will range. This implies that only particles of produce precession of the spin axis and quite specific sizes and composition can be thereby will cause the Yarkovsky force to easily ejected from the solar system. Ob- average to zero over long times. The ran- jects of larger sizes are lost by collapse of dom orientation of the particle's angular ve- their orbits under Poynting-Robertson locity vector also means that orbital inclina- drag; smaller sizes are eliminated by solar tions will evolve. wind drag. Although not calculated, it ap- It is important to note that the Yarkovsky pears as though destruction by collisions or force depends upon the body's composition sublimation may be more important loss [see (70) and (72)], such that the maximum mechanisms. We have briefly reviewed the perturbation on a stony object is ten times circumstances under which material can be that on an iron. Thus the orbital evolution ejected, and how orbits evolve with these timescales may differ for stony and iron ob- perturbations as well as with changes in jects. In this regard, Peterson (1976) has particle sizes. We have also described how suggested that the markedly younger the orbits of small particles moving about cosmic-ray exposure ages for stony versus planets are modified by these radiation iron meteorites are produced by the more forces. rapid orbital evolution of the stones under We have presented physical arguments to the Yarkovsky effect as seen by substituting approximately derive two other forces re- (64a) and (64b) in (68). He believes that sulting from solar radiation. One, recently meteorites are delivered to the Earth in a discovered by McDonough (1975), is pro- two-stage process following their produc- duced by the different signs of the Doppler tion in asteroidal collisions: changes in the shifts for the radiation from the two solar RADIATION FORCES IN THE SOLAR SYSTEM 41 hemispheres. We have found that this force C Specific heat is always less than the Poynting-Robertson Co Drag coefficient force and that its sign depends on whether d Refractive index or not its orbit lies within the solar syn- e Eccentricity chronous orbit. We have also derived the E Orbital energy, energy flux Yarkovsky force in a manner more corn- Ei Incoming energy flux plete than the original English language f True anomaly; absorption presentation of 0pik (1951) and more acces- fraction sible than the detailed analytical treatment f(ot) Scattering phase function of Peterson (1976). This force arises owing F Force to asymmetric absorption and reradiation F, Force component in i of the received solar energy for a rotating direction body having thermal lags. The Yarkovsky (Fa,Fv,FN) Radial, transverse, normal effect, depending on material properties, components of force dominates other dissipative perturbations vector for bodies that are meters in size. g Scattered fraction The derivations of the four effects pre- G Universal gravitational sented above have clarified and, in some constant cases, generalized previous developments, h (e/Z) sin go, This should permit a better and broader un- A~ thermal penetration depth derstanding of some of the important forces H Orbital angular momentum that modify the orbits of small objects in the per unit mass solar system. Nevertheless the problem of i Inclination; X/S] determining the orbital evolution of in- I Radiation intensity terplanetary particles is far from resolved, k (e/Z) cos rb New approaches are necessary for calculat- K Thermal conductivity ing the long-term effects of planetary per- 1 Path length for differential turbations, stochastic collisions, and elec- Doppler effect tromagnetic forces, each of which may it- L Solar luminosity self be of overriding significance for certain rn Particle mass particles. Until these problems are solved, M Solar mass; projectile mass the life history of a typical member of the Mean anomaly interplanetary complex will remain obscure. n Mean motion of particle P Momentum flux APPENDIX: NOMENCLATURE pr Pressure P-R Poynting-Robertson a Semimajor axis P Rotation period a Vector along line of apses Po Orbital period in direction of pericenter Qpr,Qabs,Qsea Mie scattering coefficients (see Fig. 9) r Radius to particle A Area ro Earth's orbital radius A Angstrom i" Radial unit vector b Semiminor axis R Orbital radius in AU; solar b Vector along semiminor radius axis (see Fig. 9) S Particle radius c Speed of light; projectile S Solar energy flux density speed S' Doppler-shifted solar Unit vector equaling h × b energy flux density (see Fig. 9) So Solar constant 42 BURNS, LAMY, AND SOTER

Unit vector in direction of 1~ Solar solid angle, longitude beam of node t Time 1)' Longitute of node with re- /P--R Collapse time under spect to subsolar direc- Poynting- Robertson tion = tl-nzt Drag ~G Solar spin rate T Temperature to Spin rate; argument of u Velocity parallel to beam pericenter v Velocity perpendicular to beam; velocity along ACKNOWLEDGMENTS particle path We appreciate the advice and encouragement of R. Av Velocity increment along Bergstrom, J. W. Chamberlain, J. N. Cuzzi, W. beam Hartmann, T. McDonough, C. Peterson, S. Teukolsky, (x,y,z) Coordinates O. B. Toon, and R. Wagoner. The careful reading of an W Coefficient of (70) earlier draft of the manuscript by E. J. Opik and two X Mie scattering parame- other referees greatly improved this version. Much of this work was carried out while J. A. Burns was at ter = 27rs/k NASA-Ames and supported under the Inter- Z [a(1 - e2)//~] 1/2 F/mn governmental Personnel Exchange Program. The con- ¢t Scattering angle (see tribution of S. Soter was funded by NSF Grant Fig. 3) ATM74-20458 A01. Later preparation of the manu- # Fa/Fg, force ratio (19) script was supported by the Viking Guest Investigator Program through NASA Grant NAS1-9683 and by T (1 - v2/c2)-1/2; inverse other NASA grants. thermal inertia 8 Albedo REFERENCES Emissivity Obliquity AANNESTAD, P. A., AND PURCELL, E. M. (1973). In- terstellar grains. Ann. Rev. Astron. Astrophys. ll, Coefficient in (47) 309-362. 0 Orbital longitude ALLEN, R. R. (1961). Satellite orbit perturbations in Transverse unit vector vector form. Nature 190, 615. K Thermal diffusivity ALLAN, R. R. (1962). Satellite orbit perturbations due k Wavelength; mean to radiation pressure and luni-solar forces. Quart. J. Mech. Appl. Math. 15, 281-301. longitude of Sun = not ALLAN, R. R. (1967). Resonance effects due to the GM longitude dependence of the gravitational field of a it' GM (1 - /3) rotating primary. Planet. Space Sci. 15, 53-76. ~m Micron ARVESEN, J. C., GRIFFIN, R, N., JR., AND PEARSON, lp Angle relative to normal B. D., JR. (1969). Determination of extraterrestrial solar spectral irradiance from a research aircraft. Half-angle of Sun (see Appl. Opt. 8, 2215-2232. Fig. 10); trajectory angle ASHWORTH, D. G. (1978). Lunar and planetary impact (see Fig. 4) erosion. In Cosmic Dust (J. A. M. McDonnell, Ed.), to + It, longitude of pp. 427-526. Wiley, Chichester. pericenter BEARD, D. B. (1963). Comets and cometary debris in the solar system. Rev. Geophys. l, 211-229. P Particle density BELTON, M. J. S. (1966). Dynamics of interplanetary 0" Stefan-Boltzmann dust. Science 151, 35-44. constant BERG, O. E., AND GrON, E. (1973). Evidence of hyperbolic cosmic dust particles. Space Res. 13, T ~'o°t2 fa • ¢5 dt; thermal pene- 1047-1055. BERTAUX, J. L., AND BLAMONT, J. E. (1973). In- tration time terpretation of OGO 5 Lyman alpha measurements X Solid angle annulus in the upper geocorona. J. Geophys. Res. 78, 80-91. q, Eccentric anomaly BERTAUX, J. L., AND BLAMONT, J. E. (1976). Possible RADIATION FORCES IN THE SOLAR SYSTEM 43

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