Interplanetary dust dynamics

Interplanetary dust dynamics

ICARUS 66, 280--287 (1986) Interplanetary Dust Dynamics I. Long-Term Gravitational Effects of the Inner Planets on Zodiacal Dust B O A . S. G U S T A...

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ICARUS 66, 280--287 (1986)

Interplanetary Dust Dynamics I. Long-Term Gravitational Effects of the Inner Planets on Zodiacal Dust B O A . S. G U S T A F S O N AND N E B I L Y. M1SCONI Space Astronomy Laboratory, University of Florida, 1810 N W 6th Street, Gainesville, Florida 32609

Received March 22, 1985; revised April 23, 1985 Whereas the inner planets' perturbations on meteoroids' and larger interplanetary bodies' orbits have been studied extensively, they are usually neglected in studies of the dynamics of smaller particles producing the zodiacal light through scattering of sunlight. Forces acting on these dust particles are fairly well known and include radiation forces and interaction with the solar wind. This article is the first in a series aimed at improving our knowledge of the dynamical evolution of dust in interplanetary space by studying the combined effects of these perturbations including gravitational perturbations by the planets Venus, Earth, Mars, and Jupiter. The necessity of including effects of the inner planets in dust dynamics investigations is established. Sample trajectories are presented to illustrate commonly occurring phenomenae, such as nonmonotonic changes in semimajor axis, eccentricity, inclination, and in the line of nodes. These perturbations are shown to be due to the inner planets as opposted to Jupiter or nongravitational forces. ,o 1986AcademicPress,Inc.

Studies of the zodiacal light in the last decade have supplied the first detailed information on the large-scale properties of the interplanetary dust complex, and are suggestive of planetary influences. We adopt c o m m o n practice and refer to particles with radii less than 100 /zm as dust. This is convenient b e c a u s e the large-scale distribution of this size particles can be studied from observations of the zodiacal light, which is produced through scattering of sunlight by primarily I0- to 100-ttm-size dust (Giese and G ~ n , 1976; R6ser and Staude, 1978). The spatial density distribution and other properties of the dust cloud were determined f r o m m e a s u r e m e n t s by optical, impact, and other dust detection instruments on b o a r d the Pioneer 10 and 11, and the Helios A and B space probes (Weinberg and Sparrow, 1978). The plane of m a x i m u m dust density ( s y m m e t r y plane) of the zodiacal cloud was found to vary in inclination with respect to the ecliptic plane as a function of heliocentric distance (Misconi and Weinberg, 1978; Misconi, 1979; Leinert et al., 1980). This suggests that we are dealing with a multiplicity of s y m m e t r y

" p l a n e s " which tend toward the orbital planes of the planets (Fig. 1). Consequently, the suggestion was made that longterm gravitational perturbations by Jupiter and the inner planets are influencing the spatial density distribution of interplanetary dust (Misconi, 1977; Misconi and Weinberg, 1978). Morrill and GrOn (1979) have suggested that the L o r e n t z force resulting f r o m the interaction of the dust with the interplanetary magnetic field is influencing the dust density distribution. This influence would give a s y m m e t r y toward the solar equatorial plane. One difficulty with Morrill and GrOn's suggestion is that the L o r e n t z force b e c o m e s negligible for large particles ( > l0/~m) f r o m which m o s t of the zodiacal light is now thought to arise. Additional difficulties arise from uncertainties in the value for the charge of these particles and in the nature and b e h a v i o r of the interplanetary magnetic field through lifetimes of the particles of the order of 104-105 years. Finally, if the L o r e n t z force is responsible for the o b s e r v e d s y m m e t r y plane, then one would expect the plane to have a smooth decrease in inclination a w a y f r o m

28o 0019-1035/86 $3.00 Copyright © 1986by Academic Press, Inc. All rights of reproduction in any form reserved.

INNER PLANETS AND ZODIACAL DUST

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FIG. 1. A sketch of the relative inclinations from the ecliptic plane as a function of elongation: for the solar equatorial plane, the orbital plane of Venus and the invariable plane. Also shown is the position of the symmetry plane found by Leinert et al. (1980, dashed line) and Misconi's (1979) combined results over this range of elongation. the Sun, whereas the observations show a rapid decrease in inclination outside the orbit of Venus (Fig. l). The gravitational perturbations of the inner planets and Jupiter on asteroids, comets, and meteoroids in close encounters and in planet-crossing orbits are well established in the literature. To name a few, 0 p i k (1951, 1963, 1966) has developed the statistical theory of perturbations due to close encounters with the planets, which includes probability of survival or elimination of these objects. Other pioneering work was carried out by Wetherill and Williams on the long-term orbital evolution of minor planets and Earth-crossing Apollo objects (see, for example, Williams and Wetherill, 1973; Wetherill, 1976). Integrations of perturbations on the orbits of planet-crossing bodies were carried out by Janiczek e t al. (1972) for minor planets and the possibility of resonance with, and close approaches to the Earth and Venus were scanned. Numerous other research papers on the interaction between asteroids, comets, and meteoroids with the inner planets are also included in the above references. This rather impressive work applies to larger bodies than interplanetary dust. On the other hand, studies on dust dynamics generally took into account radiation forces and solar wind effects while they ignored gravitational effects of the planets [with few exceptions such as our preliminary results

(Gustafson and Misconi, 1983)]. This, along with the intriguing observational results on the s y m m e t r y plane of the zodiacal cloud, triggered a need to study the c o m b i n e d perturbations due to Jupiter, the inner planets, radiation forces, and corpuscular forces. The aim of this first paper in a series of papers on interplanetary dust dynamics is to establish the necessity of including effects of the inner planets in investigations o f dust dynamics. In subsequent papers overall effects on the shape and other observable features of an interplanetary dust cloud is discussed. THE METHOD Interplanetary dust dynamics does not involve a static central force field such as in traditional celestial mechanics, but a complex noncentral field that changes rapidly in both time and space. Aside from the gravitational forces of the Sun and planets, dust particles are primarily perturbed by radiation pressure, ion drag, Poynting-Robertson drag, and the L o r e n t z force resulting from the interplanetary magnetic field (see, for example, Dohnanyi, 1978). These perturbations are strongly dependent on heliocentric distance. Planetary perturbations and the Lorentz force depend on the detailed location. In addition, the latter and the drag terms also depend on the velocity in three-dimensional space. E v e n if we were to restrict the problem to

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GUSTAFSON AND MISCONI

a static central force field, there are only l0 known integrals of motion in the n-body problem. These were already known to Euler but no further integrals have ever been discovered. Indeed, Bruns and Poincare proved that apart from the integrals of energy, area, and the center of mass, no other integrals of the many-body problem exist that give equations involving only algebraic or integral functions of the coordinates and velocities of the bodies that are valid for all values of the masses and which satisfy the equations of motion. Nevertheless, analytical expressions that are valid for larger bodies in a given time interval may be obtained by general perturbation techniques of the two-body solution. This is, in fact, a powerful method used in astrodynamics; however, the problem discussed here involves longer time periods (a few times l04 years). It is also important to note that the orbital nodes of the planets Venus through Jupiter precess with periods of the same magnitude [104-105 years (Cohen et al., 1973)]. This and other variations in the perturbing planets' orbital elements must be taken into account for the study to be relevant. The problem is further complicated by dissipative forces (Poynting-Robertson and ion drag). Analytical approximations, valid for larger bodies and for electrically neutral smaller particles for shorter time periods may be derivable. However, we are faced with particles whose orbits develop from the vicinity of Jupiter's orbit or beyond to inside the orbit of Venus; thus, either the perturbations must be strong or the time period must be very long. Analytical approximations appear therefore to be inadequate. Instead, this paper presents a subset of some 200 particle trajectories obtained from numerical integration using special perturbation techniques. The particle trajectories are approximated with a series of conics, which is suitable when a central force dominates a particle's motion. The orbit is rectified after each revolution. The changes in orbital elements are

given by the standard Gaussian perturbation equations (see, for example, Moulton, 1914) and are integrated along the orbit using Simpson's rule with 200 integration intervals per orbit. Close to the planets or in other regions of space and time where the perturbations are large, the program automatically uses Cowell's method (Moulton, 1914) instead of the Encke-type method described above. Orbital elements of the planets are computed using the method of Brouwer and Van Woerkom (1950). In this first paper evolutions of individual particle trajectories are discussed, as opp o s e d to overall effects on the 200-particle e n s e m b l e to be p r e s e n t e d in p a p e r III. In a

study by Gustafson (1985), and in Paper II in this series, the dynamical evolution of clouds of particles is investigated using algorithms which make use of the spatial distribution of the dust. With this method, time-consuming computations of individual trajectories are avoided. The loss in information content on the shape of the cloud is reduced through use of simple statistical methods. THE DUST Gravitational perturbations on dust orbits are independent of any dust parameters. However, other perturbations interfere, making planetary perturbations indirectly dependent on the dust's mass, cross section for radiation pressure (Cp0, and ion impact (Cion). The acceleration (b) due to interaction with the solar electromagnetic radiation FR = m{) ~

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cle's velocity in the solar reference frame. The velocity-independent term of the radiation force is usually referred to as the radiation pressure, and the remainder is known as the Poynting-Robertson drag. We assume that any nonradial component of the radiation pressure due to asymmetry of the dust particles about the radius vector are negligible, or average out over time periods that are small compared to the orbital pe-

riod. The validity of this frequently made assumption is not well established, however, there appear to be no preferred orientations of the dust in the plane perpendicular to the radius vector as witnessed by the lack of polarization in the gegenschein (Weinberg, 1964; Dumont, 1965). The ratio of ion pressure to radiation pressure on equal surfaces is independent of heliocentric distance and is of the order of 3 × 10 -4

284

GUSTAFSON AND MISCONI

(Misconi, 1975); it is small compared to the uncertainty in Cpr and is therefore neglected. The ratio of ion to Poynting-Robertson drag (Misconi, 1976), which is also independent of heliocentric distance, is of the order of 0.26 (Cion/Cpr). We focus on the particles that are responsible for the bulk of the zodiacal light, because of their size, the charge to mass ratio is assumed to be sufficiently small so that effects of the Lorentz force are negligible. Although the particles are not expected to be uniquely described by their scattering behavior, we note that to date only two models have reasonably reproduced the observed zodiacal light brightness and degree of polarization: the fluffy model of Giese et al. (1978) and the " b i r d ' s n e s t " model of Greenberg and Gustafson (1981). Both lead to densities of the order of 1 g/cm 3. Given this density, the now well-established typical size of 10-100/xm for the zodiacal particles (Giese and Grtin, 1976; R r s e r and Staude, 1978) leads to the 10 -9 to 10 5 g mass range. In these test computations, 10 -7 g particles with Cpr = 2.12 × 10 -5 cm 2 and Cion = 2.83 × 10 -5 cm 2 represent the zodiacal particles. RESULTS In the classical approximation, in which the trajectories for particles moving through interplanetary space are solely perturbed by radiation pressure and the Poynting-Robertson drag, the orbital elements

a and e alone are affected. The P o y n t i n g Robertson drag makes both decreases monotonically with time. Figures 2 a - c and 3 each show an example of the evolution of a single dust particle's orbital elements. All orbital elements are affected; a and e occasionally increase with time. These figures are, in that respect, representative of the 200 trajectories, demonstrating that there are significant perturbations by the planets. The sample test trajectory of a particle depicted in Fig. 2 was integrated from a circular orbit, with ~ = 107 °, i = 1.7 ° (close to the invariable plane) at 1.1 AU from the Sun, starting at the Gregorian calendar year -2000. If unperturbed by the planets, P o y n t i n g - R o b e r t s o n drag would bring the particle to 1.0 AU, 4000 years later, the present time. However, after the fist 6600 years covered in Fig. 2a, the dust particle suffers no loss of orbital momentum. During the first 1600 years, the particle slowly approaches the Earth, and an eccentricity develops. After some 2000 years, the orbit becomes Earth crossing. The dust particle lags behind the Earth at perihelion which occurs near conjunction with the Earth, and orbital angular momentum is transferred from the planet to the grain. The semimajor axis and eccentricity both increase over the next millenium as a result of repeated encounters. The inclination decreases to near 0 °, which triggers rapid changes in the ascending node. At the end of the 6600-year period, the particle's or-

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bital plane is close to the ecliptic (i ~ 0.2°). The original circular orbit has evolved to a 0.2 eccentricity Earth-crossing trajectory. The particle spends most of the time at heliocentric distances larger than the 1.1 AU from where it was released. Figure 2b shows the next 6600-year period, during which the semimajor axis decreases from 1.1 to 1.0 AU. Perihelion approaches within 0.1 AU of Venus's orbit, and the inclination increases enough to stabilize f~ around 90 °. On its way out from a perihelion passage

near Venus, the grain has a close encounter with Earth while still inside Earth's orbit. The encounter results in a Venus-crossing rather than an Earth-crossing orbit (see Fig. 2c). The inclination starts to rise past 5.5 ° and subsequently levels off. Thereafter the orbital elements change very little before the end of the computation where the particle reaches a mean heliocentric distance of <0.3 AU. Figure 3 shows the trajectory for a similar test particle that was also released at 1.1

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GUSTAFSON AND MISCONI

AU but never had such a close encounter with Earth. The particle seems to interrupt its descent toward the Sun over a 3000-year period as its trajectory takes the dust grain close to Venus. During this period strong perturbations in both i and fl are evident and cannot have been caused by anything but gravitational influence from the planets. These sample trajectories show several of the most common effects of planetary perturbations. To show that this effect is specifically due to the inner planets, and not to Jupiter, Figs. 4a-c show results of computations similar to those in Fig. 2 and 3 except that perturbations due to the planets Mars, Earth, and Venus were neglected. CONCLUSIONS

This set of test computations clearly demonstrates the decisive influence of perturbations by the inner planets. The changes in i and f~ are of special importance since they affect the inclination of the cloud's plane of maximum dust density. Of the other known major perturbing forces, only the Lorentz force can affect i and fl. Our test computations show that gravitational perturbations m u s t be considered in treatments of interplanetary dust dynamics related to the Zodiacal cloud. The error introduced by neglecting them (Fig. 4) has been seriously underestimated. ACKNOWLEDGMENTS We thank Dr. J. A. Burns for stimulating discussions and Dr. J. L. Weinberg for reviewing the manuscript. Thanks also go to Mr. T. Horak for assistance in drafting the computer results. This research is being supported by NSF Grant AST-82f16152. REFERENCES BROUWER, D., AND A. J. J. VAN WOERKOM (1950). The secular variations of the orbital elements of the principal planets. Astron. Pap. 13, part 2, 83-107. COHEN, C. J., E. C. HUBBARD, AND C. OESTERWINTER (1973). Planetary elements for 10,000,000 years. Celestial Mech. 7, 438-448. DOHNANYI, J. S. (1978). Particle dynamics. In Cosmic" Dust (J. A. M. McDonnell, Ed.), pp. 527-605. Wiley, New York.

DUMONT, R. (1965). Srparation des composantes atmosphrrique interplanrtaire et stellaire du ciel nocturne a 5000 A: Application a la photometrie de la lumirre zodiacale de du Gegenschein. Ann. Astrophys. 28, 265-320. GIESE, R. H., AND F. GRON (1976). The compatibility of recent micrometeoroid flux curves with observations and models of the zodiacal light. In Lecture Notes in Physics No. 48, Interplanetary Dust and Zodiacal Light (H. Elsasser and H. Fechtig, Eds.), pp. 135-139. Springer-Verlag, Berlin/Heidelberg/ New York. GIESE, R. H., K. WEISS, R. H. ZERULL, AND T. ()NO (1978). Large fluffy particles: A possible explanation of the optical properties of interplanetary dust. Asiron. Astrophys. 65, 265-272. GREENBERG, J. M., AND B..~. S. GUSTAFSON (1981). A comet fragment model for zodiacal light particles. Astron. Astrophys. 93, 35-42. GUSTAFSON, B. ,~. S. (1985). Planetary perturbations: Effects on the shape of a cloud of dust in circular heliocentric orbits. In Proceedings, IAU Colloquium 85, Properties and Interactions o f Interplanetary Dust (R. H. Giese, Ed.), in press. GUSTAFSON, B. A. S., AND N. Y. MISCONI (1983). Can cometary dust perturbed by the inner planets be an explanation for the observed distribution of interplanetary dust? In Cometary Exploration (T. Gombosi, Ed.), Voi. 2, pp. 121-134. Central Research Institute for Physics, Hungarian Academy of Sciences, Budapest. JANICZEK, P. M., P. K. SEIDELMANN, AND R. L. DUNCOMBE (1972). Resonances and encounters in the inner Solar System. Astron. J. 77(9), 764-773. LEINERT, C., M. HANNER, I. RICHTER, E. PITZ (1980). The plane of symmetry interplanetary dust in the inner Solar System. Astron. Astraphys. 82, 328336. MISCONI, N. Y. (1975). Solar Flare Lffects on the Zodiacal Light. Ph.D. thesis, State University of New York at Albany. MIscoNl, N. Y. (1976). Solar flare effects on the zodiacal light. Astron. Astrophys. 51, 357-365. MISCONI, N. Y. (1977). On the photometric axis of the zodiacal light. Astron. Astrophys. 61,497-504. MISCONI, N. Y. (1979). The symmetry plane of the zodiacal cloud near 1 A.U. In Proceedings, I.A.U. Symposium No. 90, Solid Particles in the Solar System (1. Halliday and B. A. Mclntosh, Eds.), pp. 4953. Reidel, Dordrecht/Boston/London. MISCONI, N. Y., AND J. L. WEINBERG (1978). IS Venus concentrating interplanetary dust towards its orbital plane? Science (Washington, D.C.) 200, 14841485. MORFILL, G. E., AND E. GRON (1979). The motion of charged dust particles in interplanetary space, i. The zodiacal dust cloud. Phmet. Space Sci. 27, 1269-1282.

INNER PLANETS AND ZODIACAL DUST MOULTON, F. R. (1914). An Introduction to Celestial Mechanics. Macmillan, New York. OPIK, E. J. (1951). Collision probabilities with the planets and the distribution of interplanetary matter. Proc. R. Irish Acad. Ser. A 54, 165-199. OPIK, E. J. (1963). The stray bodies in the Solar System. 1. Survival of cometary nuclei and asteroids. A d v a n . Astron. Astrophys. 2, 220-262. t3PiK, E. J. (1966). The stray bodies in the Solar System. 11. The cometary origin of meteorites. Advan. Astron. Astrophys. 4, 301-336. ROSER, S., AND H. J. STAUDE (1978). The zodiacal light from 1500 ,~ to 60 micron: Mie scattering and thermal emission. Astron. Astrophys. 67, 381394.

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WEINBERG,J. L. (1964). The zodiacal light at 5300 A. Ann. Astrophys. 27, 718-738.

WEINBERG, J. L., AND J. G. SPARROW(1978). Zodiacal light as an indicator of interplanetary dust. In Cosmic Dust (J. A. M. McDonnell, Ed.), pp. 75122. Wiley, New York. WETHERILL, G. W. (1976). Where do the meteorites come from? A reevaluation of the Earth-crossing Apollo objects as sources of stone meteorites. Geochim. Cosmochim. A cta 40, 1297-1317. WILLIAMS, J. G,, AND G. W. WETHERILL (1973). Minor planets and related objects. Xlll. Long-term orbital evolution of (1685) Toro. Astron. J. 78, 510515.