Radiation Pressure and Dust Particle Dynamics

ICARUS 49, 347-366 (1982)

Radiation Pressure and Dust Particle Dynamics

FRANCOIS MIGNARD 1

Cornell University, Space Sciences Building, Ithaca, New York 14853

Received September 4, 1981; revised January 7, 1982

The dynamics of small dust grains orbiting a planet are investigated when solar radiation pressure forces are added to the planet's gravitational central field. In the first part a set of differential equations is derived in a reference frame linked to the solar motion. The complete solution of these equations is given for particles lying in the planet's orbital plane, and we show that the orbital eccentricity may undergo considerable variation. At the same time the pericenter longitude librates or circulates according to initial conditions. With this result we establish a criterion for any orbiting particle (because of its highly eccentric orbit) to collide with its planet 's atmosphere. The case of inclined orbit is studied through a numerical integration and allows us to draw conclusions related to the stability of the orbital plane. All solutions are periodic, with the period being independent of the initial conditions. This last point permits us to investigate the different time scales involved in that problem. Finally, the Poynting-Robertson drag is included, along with the radial radiation pressure forces, and the secular trend is considered. A coupling effect between the two components is ascertained, yielding a systematic behavior in the eccentricity and thus in the pericenter distance. Our solutions generalize the results of S. J. Peale (1966, J. Geophys. Res. 71, 911-933) and J. A. Burns, P. Lamy, and S. Soter ( 1979, Icarus 40, 1-48) by allowing eccentricities to be large (of order 1) and inclinations to be nonzero and by considering Poynting-Robertson drag.

I. INTRODUCTION

This article discusses the dynamical consequences of the forces due to solar radiation on a small particle orbiting a planet. By dynamical consequences we mean the orbital evolution over very long periods of time, or the secular evolution, but likewise over times of intermediate duration, say, some ten orbital periods.

The advent of artificial satellites strongly encouraged astronomers to undertake in- vestigations dealing with the dynamical effects of solar radiation on spacecraft motion (Shapiro, 1963). However, the motion of an artificial satellite is, with the exception of a big balloon like Echo, usually weakly disturbed by radiation pressure compared to the magnitude of the oblateness perturbation or, to a lesser extent, to the luni-solar perturbations. Only the nearness of the typ-

1 Permanent address: CERGA, Av. Copernic, 06130 Grasse, France.

ical spacecraft, combined with accurate observations, has allowed radiation pressure effects to be detected in its motion and af- terwards to be included in any model of forces. More recently specific satellites have been designed to investigate radiation pressure from the slight inequalities observed in their paths (Bernard et al., 1979); presently, a project called BIRAMIS is under study to determine the Earth's radiation balance from those tiny accelerations expe- rienced by a satellite that come from the Earth's rediffused radiation.

So in the satellite case, the action of radiation pressure is characterized by its weak- ness, the true orbit not departing noticeably from that about an oblate planet. A very different situation can arise once particles are micron-sized or smaller: this occurs because, at least for large objects, the radiation pressure forces are proportional to the particle's cross section, while the gravitational force involves its mass, so that the ratio of the two increases as particles be-

347

0019-1035/82/030347-20502.00/0 Copyright © 1982 by Academic Press. Inc.

All rights of reproduction in any form reserved.

348 FRANCOIS MIGNARD

come smaller. Therefore the orbit of micron-sized particles could be significantly different from a Keplerian path.

At this level it might be useful to pay a little attention to the terminology to be used throughout this paper. So far we have mentioned radiation pressure forces without talking about what kind of forces they are. The interaction between radiation and matter is an age-old problem and, as far as forces are concerned, has been fully described and updated by Burns et al. (1979), who deri~,e a new and more accurate expression for the involved forces. The authors compare the interaction, including both the absorbed and scattered light by the particle. They find these forces by using a classical formalism, as well as the very powerful relativistic four-vector formulation, even though the radiation pressure is basically a classical force. According to this calculation the radiation pressure force can be split into a radial component along the radiative source-particle line and a drag component opposed to the velocity. Hence- forth, for the sake of brevity, but abusively, we will always call the radial force the radiation pressure force and the transverse force the Poynting-Robertson drag or P.R. drag after the discoverers who first pointed out the existence of a tangential component (Poynting, 1903) and found the proper expression (Robertson, 1937).

Each of these components gives rise to particular orbital perturbations involving two very different time scales. It is partly because of this fact, as well as due to the difficulty of carrying out a complete analytical treatment, that the previous studies of the orbital history of a dust particle have dealt solely with the radial forces or with the P.R. drag alone.

In fact the Poynt ing-Rober tson drag is generally invoked to derive time scales for the stability of orbiting particles either around the Sun (Wyatt and Whipple, 1950) or around a planet (Burns et al., 1979). The latter case has recently received more attention in connect ion with planetary rings.

For example, Goldrcich and Tremaine (1979) have pointed out that the P.R. drag must spread a planetary ring, comprising millimeter-sized particles, in a time scale far shorter than the age of the solar system. (For larger ring members collisions between particles cause the ring to expand too.) It is on the basis of this instability that several models (Goldreich and Tremaine, 1979; Dermott et al., 1980) have originated for confining the narrow Uranian ring.

The continuous orbital evolution of a particle moving about the Sun is, as far as the radiation forces are concerned, relatively easy to understand. From the particle point of view the Sun appears less massive than it really is inasmuch as the radial radiation force is directed outward; accordingly, the orbit keeps on being a conic section which can even be a hyperbola leading to escape. When escape is not possible the elliptical orbit is affected by P.R. drag and its semimajor axis decreases as the orbit becomes more circular (Burns et al., 1979). In that case the radial and transverse effects are completely uncoupled.

On the other hand we have to be careful in extending this simple result to a planeto- centric orbit. The main motivation of this paper is to investigate the dynamical effect of the radial radiation pressure force in a more accurate and more complete way than in previous studies of this question. Then we wish to establish that the coupling between the two components of the radial forces causes the eccentricity to vary in a way more complex than a mere periodic inequality about a mean value.

This paper is organized in three sections. In the first section we develop a Hamilto- nian formalism to obtain a set of four differential equations which rule the orbital evolution of a circumplanetary dust particle. The second section is devoted to the solution of this set of equations. We give a complete solution for both prograde and retrograde orbits lying in the planet 's orbital plane. Some partial results are given, as well as when the inclination is taken into

CIRCUMPLANETARY DUST DYNAMICS 349

account. In the course of Section III, the P.R. drag is introduced and its dynamical consequences are studied by means of the adiabatic invariant technique.

II. H A M I L T O N I A N F O R M U L A T I O N

1. B a c k g r o u n d

Several methods can be used to obtain the evolution rate for the Keplerian elements of a particle in orbit about a planet (Peale, 1966; Burns et al., 1979; Chamber- lain, 1979). In this problem the Sun-planet line constitutes a natural axis of symmetry which we can take advantage of by using a rotating frame of reference so that the Sun's position remains fixed. As long as the eccentricity of the planet's orbit is neglected, the speed of rotation will be uniform and equal to the planet's mean motion around the Sun. A complication associated with the selection of a rotating reference system is that, in such a frame, inertial forces have to be included. Thus a Hamiltonian formulation can be easier to employ than using the planetary perturbation equations in their Gauss form.

The basic way to study the motion of a particle resulting from the action of a force, in addition to Newtonian gravity, is the so- called method of the variation of parame- ters (or constants), originally developed in the 18th century by the geometers, L. Euler and J. L. Lagrange. This method involves a set of osculating elliptical elements, which are completely defined by the particle's position and velocity in some chosen reference frame. Thus, when the method is applied in a rotating coordinate system, a set of elliptical elements that is not directly connected to the elliptic orbit in the inertial frame is generated. For example, consider a simple two-body problem. Since the particle's mechanical energy (which in an inertial system defines the orbital semimajor axis) is not the same in the rotating frame, the osculating semimajor axis in the rotating frame is not that of the actual Keplerian ellipse described by the particle, even

though the true trajectory of the particle is a rotating ellipse.

On the other hand this drawback can be avoided by working with a Hamiltonian formulation. In such a case the conjugate coordinates are still the components of the abso- lute velocity but are referred to the moving frame. Accordingly, the solution of the equations of motion for a particle attracted by a point mass leads to an elliptical orbit of the same semimajor axis as seen in the fixed frame, but now the orbit has a retrograde motion of its pericenter.

In short, there are at least two separate ways to investigate the effect of the radiation pressure. First, one can treat the problem in a fixed frame of reference by using a time-dependent force to allow for the Sun's motion, in which case the Gauss equations are more appropriate (Burns et al., 1979; Peale, 1966). On the other hand, with a conservative force, and in a rotating frame, a Hamiltonian formulation is better suited and in this paper the latter approach has been selected; nevertheless it should be noted that many of our results have been confirmed by employing the more usual treatment in a fixed frame.

2. Hami l ton ian Funct ion

There are only two forces to be taken into consideration, namely, the gravitational force due to the planet's central field plus the radiation pressure force. The latter force---or, correspondingly, the acceleration as used here---can be expressed as

F = ( E / c ) ( Q / m ) n (1)

for a particle with mass m, where E denotes the light energy flux going through the particle's cross section, c stands for the velocity of light, and Q is a global coefficient which expresses the particle surface properties related to the incoming light ( cf. Burns et al. , 1979). The unit vector n is directed outward from the Sun along the Sun-particle line. Equation (1) can be reformulated in a more convenient way by introducing the coefficient/3, the ratio of the radiation force

350 F R A N C O I S M I G N A R D

to the solar gravitational force at a distance r:

Here G stands for the gravitational constant and M for the Sun's mass. Since E decreases according to the inverse square law, /3 is independent of the solar distance and synthesizes in a certain way the rele- vant surface and bulk properties of the particle (Burns et al., 1979). Let us assume the particle's orbit about the planet to be small compared to that of the planet around the Sun, and so we disregard any change in radiation forces as the distance to the Sun varies slightly. Thus the distance from the particle to the Sun will always be taken

equal to the planet's semimajor axis; hence (1) can be written as

F = X/3Von, (3)

where V e is the orbital velocity of the planet and k its mean motion around the Sun. The value of/3 has been computed by Burns e t a / . (1979), among others, for various materials as a function of the diameter of a particle; their results are reproduced in Fig. 1. For numerical illustration we will consider values of/3 in the range of 0.01 to 1, corresponding to particle sizes ranging from 0.01 to 10/zm.

In the above-mentioned rotating frame, the force given in Eq. (3) is constant in magnitude and direction. It corresponds to

t0

~ . I ~ . . ~ . I ron . . . . . ~:.- x.. Basalt ...........

. . t " ~ \ \ Ideal Mater ia l ' . . t " ~ .. (Q : t , p : 5 ) . . . . .

,,=,,B ~ ° ~ • . >_.-.--" _ ~ \ Geometrical Optics , / , " - - - " ~ ". (p = 3 ) - -

s" X~ \ 4 -- / x \ -

t . . . .

, , ~ , , ~ . , . / " / . . . . . . . . ..... ~ ~. ' - \

o," ~ %.o . -

,I ..." ",, - • " \ " ° . / ....--" ,, \',,. o., .i / , , ....%,,,.. - :" \ ", , , -=

,/..." \ \ , , , , "'-. X \ . ' " \ %. .

j . / .-" ~ "'..

• • o. / / "- 0.0t ' " , , , . . . . . I , , . . . . , , I . . . . . . .

0.0t 0.t t t0 Par t i c le Radius s(#~rn) ,

Flo. 1. A log-log plot of the/3 parameter (relative radiation pressure) as a function of the particle sizes for three cosmically abundant materials and two comparison standards. Reproduced with permission from Burns et al. (1979).


a uniform field of force and so can be derived from the force function

U = Fn "r , (4)

where r is the radius vec tor of the particle and F is the radiation pressure force ex- erted on a particle of unit mass; it is related to the physical properties by

F = h/3V o. (5)

In Eq. (4) the sign of the force function is chosen so that the components o f the force are precisely the partial derivatives of the force function. This potential is superposed on the central field of the planet and it is its existence which allows us to calculate a Hamiltonian function for the problem.

In Eq. (4) no allowance was made for the planetary shadow; nevertheless its effect is assumed to be small for particles orbiting a planet at several planetary radii, even though we do not have any real quantitative evaluation of its influence. In any case, a particle orbiting the planet in the equatorial plane at four planetary radii spent only 8% of its time in the shadow, and it seems reasonable to neglect this effect in a first approach. Ultimately our confidence is based on the study by Radzievskii and Ar tem'ev (1962), who compared the influence o f solar radiation pressure on the motion of an artificial Ear th satellite with and without shadow. They concluded that the perturbations are alike and only the magnitude is changed.

The Hamiltonian function for a particle moving in the central field of a planet and disturbed by radiation pressure is obtained by applying the standard procedures described in any text book on analytical dynamics (e.g., Goldstein, 1980). The reference frame used is rotating at a constant angular velocity h so that the Sun remains at a fixed coordinate on the X axis. I f we locate the particle 's position in terms o f spherical coordinates (Fig. 2), the Hamilto- nian for a unit mass is

P~ ix h P . + F r sin 0 cos ~, (6) K - 2 r -

P

I"

:~SUN FIG. 2. The moving coordinate system. The origin is

the planet center of mass.

where, in terms of its components, the momentum squared is

p 2 = p 2 + _ _ + r 2 sin2________ ~

and ~ equals GMp (G is the Newtonian constant of gravity and Mp the planet 's mass). The third term on the right-hand side of (6) comes from the rotation of the coordinate system, and contains P . , the z component o f the angular momentum.

We define

Ko = t n / 2 - I~/r, (7)

to which the perturbing potential

KI = - h P ~ + F r sin 0 cos • (8)

is added. Equation (7) corresponds to the classical two-body problem and leads to six constant Keplerian elements (often taken as the semimajor axis a, eccentricity e, inclination i, longitude of node fl, argument of pericenter to, and mean anomaly at a given epoch M0). Now the disturbing function (8) is expressed in terms of the instantaneously fixed values o f these elements and is ex- actly the opposite o f K1,

R = h[/xa(1 - e2)] 1/z cos i

- Fr[cos(to + v) cos

- cos i sin(to + v) sin l]], (9)


where the true anomaly v and the radius vec tor r are implicitly expressed in te rms of the Keplerian elements .

For our purposes we do not need to pro- ceed further in this expression. Here we are only concerned with the secular change of the orbital e lements . To a first approximation, the perturbat ion equations as averaged over an orbit can be directly obtained f rom the average disturbing function (see, for example , the von Zeipel method or the L i e - H o r i t ransform). This procedure will no longer be valid if the characteris t ic t ime for significant orbital changes is of the same

order as the orbital period; this criterion will be derived later [Eq. (37)]. The mean disturbing function is

lye:. ( R ) = ~ R a M

= h[/xa(1 - e2)] 112 cos i

+ (3 /2 )F ea[cos to cos l-I

- cos i sin to sin II]. (10)

Using this expression, we can evaluate the six partial derivat ives required for obtain- ing the set of governing differential equations.

O< R ) = o, O M

O(R..___~) = ~- na(1 - e2) an cos i Oa 2

O ( R > _

O e

oi

O ( R ) _ Ol-I

O ( R ) _

Oto

+

h n a 2 e (1 - e2) x t2c°s i +

- - = - h n a 2 ( l - e2) 112 sin i +

3 Fe[cos to cos l-I - cos i sin I I sin to],

Fa[cos to cos I I - cos i sin I I sin to],

3 e a F sin i sin I I sin to,

3 e a F [ c o s to sin ~ + cos i sin to cos 1)], 2

3 e a F [ s i n to cos f~ + cos i cos to sin ll]. 2

(11)

In (11) we have used n for the particle orbital mean motion. Insert ing (11) in the p lanetary equat ion (Brouwer and Cle- mence, 1961, p. 289) we obtain

da dt - 0, (12)

dth _ Man ~ [ c o s 1) sin to d t

+ cos i sin 12 cos to],

dto [ cos I I cos to dt - M a n ~ [ - t ~

cos i sin to sin fl] s i n r c o s r ] '

dD, dt

- h + Man ~ tan ~b sin to sin 1~,

(13)

di - Man ~ tan ~b sin i sin l'l cos to. (16)

dt

In Eqs. (12)-(16) we put

e '-- sin

and

(17)

3 Vo. (18) tan ~ = ~ 13 n---a

3. C o m m e n t s on These E q u a t i o n s

Since, by (12), the semimajor axis is unchanged by radiation pressure, ~ is con-

(14) stant during the motion and is a characteristic of the particle, as well as its distance f rom the Sun and the planet. For example , the case of highly per turbed hydrogen

(15) a toms investigated by Chamberlain (1980)


corresponds to an acceleration F = 0.75 cm/sec -2, whence by Eq. (3)/3 = 1.25, so that at a distance of five Earth radii tan xlt = 15 from Eq. (18). We can see in Fig. 1 that/3 varies substantially according to the particle's diameter and thus plays a major role in determining the value of xp. So for dust particles orbiting the Earth at a few terrestrial radii, xlt varies from 0 to about 10. When xI t = 0 (no perturbation), we obviously find a regressing ellipse as the solution to ( 12)-(16), the angular rate of regression being equal to the angular speed of the Sun around the Earth's center. Note that Chamberlain (1979, 1980) does not include the solar motion, so that our Eq. (15) differs from his version [Chamberlain, 1979, Eq. (20)].

On the right-hand side of (15) the term - h follows directly from the motion of the Sun. The other h's preceding tan • are nothing but a convenient way to represent the planetary semimajor axis by the means of Kepler's third law. Thus, when neglecting the Sun's motion in the set of differential equations (12) to (16), only the term - h in Eq. (15) must be dropped.

Let us take the inclination to be small and write the evolution equation for the longitude of pericenter ~ -- ~0 + l~ by adding Eqs. (14) and (15),

d& _ tan W dt h + X ~ cos ~. (19)

Had the Sun's motion been disregarded, this equation would have taken the simpler form

d~b tan dt - X ~ cos &. (20)

This last equation is Chamberlain's (1979) Eq. (28), notwithstanding the fact that he considers only small eccentricity. It is clear by comparing Eqs. (20) and (19) that the tendency of pericenter to drift toward a stable position at ¢b = 90 °, as noted by Cham- berlain, will not materialize when inertial forces are incorporated. In an inertial

frame, the same conclusion is reached once allowance is made for the variable direction of the radiation pressure force. Thus, as the pericenter approaches 1r/2, the importance of the Coriolis term represented by -X in the right-hand side of Eq. (15) increases in a relative sense; this prevents pericenter from being locked at oJ + f/ = 90 °.

1II. SOLUTION OF THE EQUATIONS OF MOTION

I. Cop/anar Orbits

Let us focus now on a particular solution of the system (12)-(16) for the case of small inclinations, namely, cos i = 1. Before going further with this approximation it might be valuable to examine the possibility of keeping a particle in the planet's orbital plane, that is, to find out whether the trajectories i = 0 or I = 7r are solutions of (12)- (16) or not. Inserting either of these two values of the inclination in Eq. (16) clearly shows that a particle initially moving in the planet's orbital plane keeps orbiting the planet in this plane; this behavior is readily understood in terms of the forces that act. Consequently we are allowed to restrict the study of the trajectory's solution of (12)- (16) to i = 0, the longitude of the node becoming meaningless and the two remain- ing variables, the eccentricity and the longitude of the pericenter, being determined by Eq. (13) plus that obtained by adding Eqs. (14) and (15).

To extend the solutions we are about to find to the case of nonzero, but small, inclinations implies that the solution i = 0 must be stable; this is by far a more difficult problem than the existence of that solution. Some indications about the possibility of such an extension will be given in the next section. At this point it can be said that the solution i = 0 is stable as long as the particle's orbital eccentricity remains less than about 0.95. Therefore the planar solution is likely to be useful in numerous cases of practical interest in planetology for describ- ing the motion of small particles which start close to their parent planet's orbital plane.


A. Trajectories in phase space. The tra- jector ies of the solution to Eqs. (13) and (19) can be drawn in (~b, &) space; their equat ion is given by the first integral, Eq. (10), by putting cos i = 1:

c o s ~ c o s 4~

+ s i n W s i n ~ b c o s & = C. (21)

For a given value of C, Eq. (21) described a curve in the (~b, &) plane and, by varying C, we obtain an array of trajectories. I f we consider a retrograde orbit, with cos i = - 1 , the set o f trajectories is similar to that found for a direct orbit by changing & into zr - &, as can be easily checked in Eqs. (13) and (19).

All these curves can be drawn more easily with the help of a geometrical interpretation for Eq. (21). To aid in this, we first note that the analytical form of (21) implies [CI <- 1 so that we can set C = cos 8. With this definition Eq. (21) is seen to be nothing more than the first relation for a spherical triangle with sides (~ , 8, 4), having an en- closed angle &, as shown in Fig. 3. A given t rajectory can be visualized by moving the point S on the sphere while keeping a constant 8; this is equivalent to the diurnal motion about the pole P in which case ~b would be the zenith distance. As is well known, there are two types of stars visible to a given observer : c i rcumpolar stars, which remain above the horizon throughout

z

FIG. 3. A geometrical interpretation of the solution of the Hamiltonian equations. The point S moves on this sphere while keeping a constant value of 8. Ac- cording to the value of • - 8, the pericenter either librates or circulates.

the day, and others which rise and set daily. Similarly we should expect two classes of trajectories according to whether or not ~b reaches rr/2.

Note first that in Eqs. (13) and (19) W is an imposed external pa ramete r and each t rajectory is entirely defined by initial conditions, that is, by an assigned value of the unique constant of integration 8. Thus every point in (W, 8) space corresponds to a particular t rajectory in (~b, tb) space. Even- tually a quadrature will allow us to ascer- tain the time behavior of a point in phase space.

Based on our geometrical description of movemen t over a sphere, there is no diificulty in determining what the phase space looks like. Let us consider the sphere in Fig. 4 pictured with ~ < ~r/4. Each of the curves drawn on that sphere is equivalent to a t rajectory in phase space because 8 is kept constant on it.

Start f rom 8 = 0, when the initial curve reduces to point P, and progressively increase 8. For a while the angle & remains confined within the two ext reme values ~ t ~ m a x SO that the pericenter librates between those values. The eccentrici ty represented by ~b wanders f rom ~bmin = ~ - 8 to (])max = XI/ + 8. The numerical value of tOma x is obtained by expressing that & at the ex- t reme libration, in which condition the spherical triangle ZPS has a right angle at S. [From an analytical point of view this con-

z

P

FIG. 4. The set of trajectories for xl t < rr/4. The curves are labeled by 8 expressed in degrees. The equatorial line corresponds to e = 1.


dition is similar to d~dch = 0 computed in Eq. (21).]

sin &m=, = sin 8/sin ~ ,

which yields the eccentrici ty

cos ~b = cos ~ / cos 8.

Once 8 > ~ , the angle & takes all values from 0 to 2~-, meaning that pericenter circulates. The largest eccentrici ty remains t~max = ~ + 8 but the smallest becomes t~mln = 8 -- ~ when & = rr. For 8 ->,Ir/2 - xt t, ~bmax reaches ~r/2 as the eccentrici ty goes to 1. Since the particle 's semimajor axis remains constant the ellipse evolves to a straight line as the pericenter distance tends to 0. In such a case the particle hits the planet 's a tmosphere. Finally, the maximum possible value of 8 is 8 = ~ + 7r/2, beyond which there is no motion. The same discussion can be made for • > 7r/4 and the different situa- tions are depicted in Fig. 5. The extreme eccentrici ty ~ = zr/2 is reached before the trajectory goes through Z, thus preventing the existence of circulatory curves. These results are diagrammed in Fig. 6, where the various boundary lines follow from the above-mentioned relationships. Of course the whole preceding discussion could have been made on a purely analytical basis from the properties of Eq. (21).

In the phase space a sample o f typical trajectories is given in Fig. 7 for ~ < 7r/4, and in Fig. 8 for • > zr/4. The libration of

1

FIG. 5. The set of trajectories for ~ > ~r/4. The curves are labeled by 8 expressed in degrees. The equatorial line corresponds to e = I.

0 ~0 60 t00 'NO t80 8 (degrees)

FIG. 6. Classification of the motion in the ~ , 6 plane. The l ibration region corresponds to a restricted motion of the pericenter. The region labeled collision means the eccentricity reaches e = 1.

the pericenter corresponds to closed loops in this representation, whereas continuous open curves stand for the circulatory class. All the curves which attain th = 1r/2, corresponding to a colliding particle, have their upper parts dashed because this region of phase space will undergo some change when the inclination is allowed to vary.

These results should be compared against earlier studies. Chamberlain (1980) numeri- cally integrated the motion of a hydrogen atom at a semimajor axis of five Earth radii with initial conditions & -- 0 and e -- 0.1 (tbmi, -- 5°.7); in our formulation this gives tan ",It equal to 15 (or ~ = 86°), so that with ~bmin, 8 = 8~3. Chamberlain 's integration remains very close to our calculation until the longitude of the pericenter comes to 75 ° as might be expected, following the discussion of Eqs. (19) and (20). In Chamberlain 's integration, pericenter moves slowly toward 90 °, while our study shows that in fact it really goes through a maximum value of 81 ° before returning. At this point the particle 's eccentrici ty would be - 0 . 9 if the particle had not been eliminated earlier because of a collision in the Ear th 's a tmosphere.

This problem of the removal of a particle is rather important and deserves a little attention in connection with the radiation pressure. We have already mentioned the possibility of a particle being lost by trans- fer onto an highly eccentric orbit. How- ever, collisions with the central planet are likely to be an even more efficient process for eliminating circumplanetary particles.


9 0 ~ I s

80

70

60

50

4.0

30

20

t0 0

, / I

---, ,... \ I ~ , ~ +~,,,, I

55

t80 t40 t00 60 20 0 20 60 400 t40 480 OJ

FIG. 7. The phase space (@, &) for ~ = 30 °. All curves are labeled with 6 expressed in degrees. For 6 < 3&, per icenter librates. The dashed arcs on the trajectories correspond to regions modified once the inclination is taken into account .

A particle with a given semimajor axis will remain orbiting a planet only as long as its orbital eccentrici ty is small enough to prevent the particle from entering the planet 's a tmosphere. By neglecting the at- mospheric thickness in comparison to the

planet 's radius, the maximum allowed value of e is thus

ema~ = sin +max = I -- R/a.

Therefore a particle will be lost if

4 > 6max,

70

60 I I

3 0 s 0 ~ i i i \ t /o o\

lot- ',1 ~ -I- ~ / \', 0 I 180 140 100 60 20 0 20 60 1£E) t40 80

FIG. 8. The phase space (@, tb) for xp = 60 o. All cu rves are labeled with 8 expressed in degrees. For 8 < 30 °, per icenter librates. The dashed arcs on the trajectories cor respond to regions modified once the inclination is included.


meaning that pericenter is located below the planet 's surface. This inequality can be plotted quite simply on the (~ , 8) plane as shown in Fig. 9.

A particle will be forever maintained in orbit if its initial conditions lie below the dashed curve when a/R -- 3, or below the solid line for a/R = 20. Rather severe con- straints are placed on the particle 's dimen- sions by the above condition. For example, as Chamberlain (1980) pointed out, hydrogen atoms cannot survive more than a few days in the Ear th 's near environment. More precisely, with the initial conditions (8 = 8(E.3 and • = 86 ° ) at a/R = 5, Eq. (28) shows that the limiting value of eccentrici ty is reached in 3.8 days during the particle 's sixth orbit. For a smaller semimajor axis, the perturbations caused by radiation pressure are relatively weaker, but the maximum permitted eccentrici ty is less, and the particle 's lifetime is unchanged. The same hydrogen atom, when created at three Earth radii with an eccentrici ty of 0.1, would strike the Ear th after 3.7 days in the course of its thirteenth orbit.

B. Time evolution. So far we have described the eccentr ici ty his tory with respect to the pericenter location, but no informa- tion is yet available relating to the time behavior of the solutions with the exception of the hydrogen a tom's lifetime in the Ear th ' s a tmosphere. The time evolution for

! ! ! ! i ! i

"2 8o

ZO 40 60 80 ~00 t20 140 t60 180

8 (degrees)

Fro. 9. Regions of the plane (~, 8) for a particle to be lost by colliding with the planet. For initial conditions below the dashed line (corresponding to a / R = 3) or below the continuous line (corresponding to a / R = 20), the pericenter distance is larger than the planet's radius. The reverse holds in the upper-right part of the diagram.

the eccentrici ty and for the pericenter longitude can be found by expressing ~b as a function of ~b in (21); this is then inserted in (13) with cos I = + 1. Actually we can also perform this computat ion through our geometrical interpretation. In Fig. 3, call H the angle SPZ by analogy to an hour angle on the celestial sphere. From a bit of spherical tr igonometry for the triangle SPZ the following relations hold:

sin & sin ~ = sin H sin 8, (22)

cos & sin ~b = cos 8 sin

- sin S cos @ cos H, (23)

cos H sin 8 = cos 4) sin

- sin ~b cos • cos &, (24)

cos 4~ = cos xt, cos 8 + s i n ~ s i n S c o s H . (25)

By differentiating (22) with respect to time and using the equations of motion (13) and (19), we obtain

dH h d---t-= cos------~ " (26)

This equation allows us to orient the trajectory in the sense of increasing time with

h H - c o s ~ t, (27)

obtained by putting t = 0 at the moment when the particle crosses the meridian in our geometric representation; in other words, time is chosen to start at the peri- cen te r -Sun conjunction.

Then from (25),

cos 4~ = cos xI, cos 8

+ sin ~ sin S cos • (28)

Therefore the motion is periodical and, just as with an idealized harmonic oscillator, whatever the amplitude, the period does not change.

The studies by Burns et al. (1979) and Peale (1966) were restricted to an approximate solution in which both ~" and 4) were


small angles. We put ~b = ~b0 for t = 0 in (28) and, following the previous authors, we take cos ~ = 1, so that (28) reduces to

c o s 4~ = c o s 4,0

+ sin • sin(~b0 - ~)(1 - cos h t)

or, with sin ~b = ~b = e,

e 2 __- e o g

This last equation is similar to Eq. (42a) of Burns et al. (1979) or Eq. (4) in Peale 's paper.

Likewise, the time variation o f the pericenter location can be compared though the computat ion is not so straightforward, because pericenter is referred to afixed frame of reference in the two quoted papers while we use a rotating one. Then we have the correspondence

&v--B = (o + ht, (30)

where ~ - B denotes the pericenter longitude in the fixed frame. From (22) and (23) we get

sin H tan ~ -- ~ /~ _ cos H " (31)

Then with (30),

tan ~ + tan H tan ~ B = I - tan 6~ tan H ' (32)

which reduces to

sin H tan &p_~ = cos H - 8 /~ (31a)

This equation is equivalent to the result given by Peale and Burns eta[. , allowance being made for the expression of 6(= e0 - W) and for the fact that H = h t as cos ~ = l. The nature of the approximation made in the previous studies of the motion of dust particles affected by radiation pressure now appears more clearly. At the same time the agreement between the different solutions is a pleasant check for the present theory.

One of the distinctions between the pre-

vious approximate solutions and our general solution is related to the characteristic time of evolution. In (28) the period of evolution is given by

T = To cos qJ, (33)

where To is the planet 's orbital period, while in the approximate (29) and (31) the period is To alone and does not depend on tO. The difference is not only a quantitative one but there is also a qualitative distinction. In (31a), with H = ht , there is a possibility for the pericenter to librate about the zero value provided that b/qJ > 1. Given the initial conditions that yielded (31a) the libration occurs about the per igee-Sun conjunction line. The period of such a libration is equal to the planet 's orbital period.

With the same condition, 8 /6 > 1, the more complete solution (31) leads to a circulating motion with respect to the Sun, the period of which being given by (33). Le t cos tO be one; the combination of the moving frame with the nonuniform circulation, both having the same period, leads to the librating pericenter in the fixed frame. But this conclusion fails as soon as the two periods differ which is physically the case. In that case the circulation in the moving frame occurs more rapidly than the planet 's mean motion and in the fixed frame the pericenter regresses with the rate h(l - cos tO). Super- imposed to this steady motion are periodic inequalities. So, the quite reasonable simplification of Peale (1966) and Burns et al. (1979) proved to have a limited range of validity.

On the other hand (31) shows that a libration motion exists in the rotating frame for 8 /~ < 1. In that case the pericenter remains close to the solar direction and, when viewed in the fixed frame, exhibits a periodic oscillation around the Sun's longitude. Since • is a measure of the magnitude of the radiation pressure as compared to the gravitational forces, the meaning of (33) is clear: the more significant a perturbation is, the more rapid is the evolution of the orbital elements. Chamberlain (1979) derives a cri-


terion for the validity of his averaging procedure which is based on the duration

¢(years) = To" cos ~/sin ~ . (34)

This he considers an estimate of the time required for a major perturbation to occur. From (34) it is obvious that r can be much larger than a planetary period To when weak perturbations act, that is, for small if'. The pair of time durations (33) and (34) which presumably express the same idea are in discordance when sin • can no longer be approximated as 1.

Nevertheless Chamberlain is only concerned with the effect of Lyman a radiation on hydrogen atoms (~ = 85 °) and thus his z approximately equals our T. In a more general case, however, the difference between the two time scales ~" and T stems from the solar motion, the effect of which becomes predominant as the perturbation decreases in magnitude. At first glance it seems that as the magnitude of the radiation pressure forces tends to zero (for larger particles, for example), the perturbation remains per- fectly defined since the period given by (33) does not present any singularity when approaches zero. In contrast (34) increases to infinity in similar conditions. Indeed the magnitude of all the perturbations [see (13)- (16)] is multiplied by tan ~ and so approach zero with the progressive disappearance of the radiation pressure forces as might be expected. Moreover the residual period T = To is nothing more than the period of the regression of the nodal line for a rotating observer. Therefore a convenient criterion for the nonvalidity of the averaging procedure worked out in (10) may be given by

T -< P, (35)

where P is the particle's orbital period around its planet for a Keplerian orbit. Us- ing then expressions (18) and (33), (35) becomes

Pq 1 +Biff s a -> 1, (36)

where Ai and Be are numerical coefficients listed in Table I, the subscript i stands for a given planet, and Re for that planet's radius. Solving Eq. (36) leads to a critical value of the particle's semimajor axis beyond which the averaged equations cannot be used, since the orbital period is of the same order as the perturbation time scale. Does that mean that the averaging procedure remains acceptable for all orbits lying inside the critical orbit? Probably not, and a refined criterion is desirable.

In a second paper on this subject, Cham- berlain (1980) has compared a numerical integration of the instantaneous rate of change for the eccentricity, the longitude of node, and the longitude of pericenter to the integration of the averaged equations. His integrations were performed for hydrogen atoms (/3 = 1.25) and semimajor axes of 5, 10, and 15 terrestrial radii. From his study it turns out that the averaging procedure is very good for a highly bound orbit such as a i r = 5, fairly acceptable for a/R = 10, but becomes unreliable when aiR = 15. In this case the ratio of the period T to the particle orbital period is slightly less than 4. Conse- quently, in place of (35), we adopt, as a preferred criterion for the validity of the averaging procedure,

T ----- 5P. (37)

The solutions of (35) and (37) are given in Table II for the different planets. The sec-

TABLE I

COEFFICIENTS IN EQ. (36) FOR THE VARIOUS PLANETS

MER VEN EAR MARS JUP SAT URA NEP

A~ 4.3 E-7 7.2 E-8 2.6 E-8 1.0 E-8 8.6 E-10 3.0 E-10 1.4 E-11 2.5 E-12 B I 714 51 32 100 0.2 0.3 0.45 0.23


T A B L E II

THE DISTANCE (IN PLANETARY RADII) AT WHICH THE ORBITAL EQUATIONS CAN BE AVERAGED

Planet

2 1 0.5 0.1 0.01

T = P T = 5P T = P T = 5P T = P T = 5P T = P T = 5P T = P T = 5P

M E R 5.3 VEN 16 E A R 23,4 MARS 22.4 JUP 195 SAT 230 U R A '500 N E P 800

2.4 7.5 3.4 10.7 4,7 23.8 10.8 72 31 7.2 23 10.2 32 14.5 71 32 191 74

10.5 33 15 47 27 104 46 274 106 10 31.6 14 44 20 100 45 294 121

87 275 122

ond number of each group provides the value of a for which orbital evolution occurs rapidly for a specified P.

In Table II only the upper-left portion has any meaning; the other figures are so extreme that it is not worth considering particles so distant from their planets. The giant planets strongly attract particles, whereas radiation pressure is reduced in the outer solar system. The combination of these two facts accounts for the large numbers in the lower section of the table. Nevertheless, in the case of a planet like Mercury it is clear that the averaged equations for the motion of a tiny particle should be applied with caution.

Moreover, the value of (B0 in is simply tan * evaluated at the planetary surface for a particle having/3 = 1. Thus the distinction between the importance of radiation forces for the inner and the giant planets obviously appears in Table I. And, if we look at Fig. 1, we can conclude that for the terrestrial objects, particles with diameters ranging between 0.05 and 1 /~m undergo large variations in eccentricity so as to eventually strike the planet. On the other hand, around Jupiter, similar particles can be kept within 5 to 10 Jovian radii and, even there, will be subjected only to the gradual orbital collapse caused by the Poynting-Robertson drag. This point will be investigated in more detail below.

2. Inclined Orbits

Hitherto the search for solutions of the differential system (12)-(16) was limited to orbits lying in the planet's orbital plane and a complete analytical solution, as well as a tractable solution, was found to be feasible. Unfortunately neither a simple, nor even a complex, analytical solution of the same differential system exists for high inclinations.

Therefore in this section we restrict ourselves to presenting some global properties of the general solution, obtained with the help of a numerical integration. The phase space is now a four-dimensional space based on the variables ~b, to, II, and i and cannot be easily pictured. We have integrated the system (12)-(16) for about 50 different initial conditions and for values of tan W of 0.1,0.5, 1, 2, 5, and 15. A standard fourth-order Runge-Kutta technique was used throughout this numerical computation to adjust the step length to the values of the time derivatives. The routine was im- plemented on a pocket calculator and the four equations were simultaneously integrated, along with a frequent evaluation of the energy integral given by Eq. (21) for checking the quality of the numerical procedure and for changing the step length if need be.

First of all, the numerical integrations


carried out with i = 0 ° and i = 18& have confirmed the analytical solution obtained in the preceding section. More specifically the periodicity of the solutions is a real fact and was checked after one orbit at the level of 10 -3 , which is the ultimate accuracy of our program.

Let us at tempt to draw some order from a pile of figures. First consider an initial orbit moderately inclined, say, less than 45 ° . Pro- vided that the theoretical coplanar maximum eccentrici ty ~bmax = ~ + 8 is smaller than 75 °, the orbital plane experiences only weak variations about an average value, so that the orbital plane is fairly stable. This proper ty was invoked in Section III to stress the importance of the coplanar solution. Moreover the maximum eccentrici ty reached during the evolution is less than it would be in the zero-inclination case, and becomes smaller yet when the inclination is larger. In Figs. 7 and 8 that means the trajectories are somewhat flattened in the vi- cinity of high eccentricity.

For example, with tan ~ = 1, 8 = 20% and an inclination o f 10% the maximum value of ~b is 58 ° instead of the 65 ° found for the planar case. Simultaneously the inclination evolves between 10 and 20 °. In a case with tan • = 0.5 but an inclination of 45 °, the maximum eccentrici ty expected for the set o f initial conditions w a s ~max = 40°,

whereas this angle does not go above 27 ° , while the inclination ranges from 43 to 47 ° .

A quite different situation takes place if the eccentrici ty rises too much. Equation (16) shows that there is a coupling between the inclination behavior and the eccentricity:

di d--t a t a n ~b.

Hence, as the eccentrici ty grows, so does the inclination rate o f variation and eventually the orbital plane becomes unstable, whatever the initial inclination.

To illustrate this general proper ty , consider the initial conditions

tan ~ = 2, ~b = 10%

i = 7 °, c b = 0 .

According to the coplanar solution, we get 8 = • - ~b = 53 ° and the expected planar solution looks like the curve 8 = 55 ° in Fig. 8, and so produces a collision orbit. It must be emphasized that the initial inclination is very small and makes this orbit very similar to a coplanar orbit. In fact, with the complete solution the eccentrici ty begins to increase as in Fig. 8, while the inclination retains a small value. But, as soon as the angle ~b passes 75 °, the orbital plane strongly tilts, becoming retrograde for a while. More important is the behavior o f the eccentricity: the largest value of ~b is 86 °. Therefore, owing to a coupling effect between the eccentricity and the inclination, the collision orbit disappears once the inclination is allowed to vary.

Consequently, the upper part of the trajectories in Figs. 7 and 8 are in reality more bent than is shown and therefore remain inside the diagram. From a physical point of view there is a little difference between a parabolic orbit and an osculating ellipse with an eccentrici ty of 0.999, but mathe- matically only two kinds of trajectories are found: the closed curves with a librating pericenter and the open curves leading to a circulation of the pericenter. This relation- ship between inclination and the large eccentricity was already noted by Bertaux and Blamont (1973) in their study of the lifetime of hydrogen atoms in the Earth'~ exosphere.

To conclude the question about the tra- jector ies we can adopt the following rules. For a weakly inclined orbit, compute the initial conditions as for a coplanar orbit. I f the angle ~b constantly remains below 75 °, or e < 0.96, we can trust the coplanar theory and the orbital plane is stable around the initial value. If, with the same hypoth- eses, ~b goes over 75 ° the orbital plane will suffer severe variations, while the eccentricity will remain smaller than 1. No gen-


eral rule has been discovered concerning the angular distance between the pericenter and the Sun's direction which would be valid for every inclination. However, if the pericenter can librate or circulate, these oc- currences are not obviously related to the energy as in the coplanar case.

The time behavior is probably the most surprising result of this numerical study. All the trajectories investigated remain periodic, which means that phase space is filled with closed curves. In addition the period is the same as that previously disclosed in the coplanar calculation, the expression of which is given in Eq. (33). This result was quite unexpected and deserves further study from the theoretical mechanics point of view. Surely the two degrees of freedom Hamiltonian involved with the radiation pressure forces is integrable and the solution has two equal fundamental frequencies. So far this investigation is still in progress but only unproven conjectures have been obtained[

IV. SLOW ORBITAL CHANGES DUE TO POYNTING-ROBERTSON DRAG

So far the radiation pressure force has been represented by a simple radial force directed along the Sun-particle line. It was not until 1937 that Robertson disclosed the complete and more subtle interaction between the solar radiation and small particles. Burns et al. (1979) investigated the dynamical consequences of the P.R. drag on a particle orbiting either the Sun or a planet. In the latter case it turns out that the particle spirals inward toward the planet with a characteristic collapse time given by

T(years) = 530 R~/13,

where R is the planet's semimajor axis in AU while/3 remains the coefficient defined in Section II. For all members of the solar system this collapse time is far greater than the period involved in the orbital evolution caused by the radial radiation pressure and given in Eq. (33). Accordingly, at a specific instant and for an elapsed time on the order

of some planetary periods, the orbital evolution of a dust particle is determined by the radiation pressure forces studied in the previous sections, in which the value of tan is computed by taking the P.R. drag into account, since the mean orbital velocity of the particle increases as its orbit shrinks. Then the set of trajectories is not modified globally but instead slowly changes as decreases. Consequently the orbital evolution of a given particle is just a little bit more complicated than might be expected after reading Sections II and III.

Let us restrict ourselves to the coplanar case and consider two different epochs tl and t2 corresponding for a given particle to ~1 and ~2. During the interval t2 - tl the particle's semimajor axis has shortened because of the P.R. drag and ~2 < xltl if t2 > tl. We can easily figure out what the sets of trajectories are at tl and t2 since their as- pects depend only on the values ~ and ~2. Moreover, given an initial condition at t~ we can place the particle on a particular trajectory, namely, at a fixed value of 81. But what about 82, the value of 8 at tz? If infor- mation is available for the global set of trajectories at t2, one still must determine a precise trajectory at t2 for a given 81 at tl. If we recall the fact that 8 is a constant of the motion stemming from conservation of energy, it is clear that the present difficulty arises due to energy loss when the P.R. drag is operating.

Owing to the existence of two very different time scales in this problem, the technique of adiabatic invariants seems well suited to investigate the long-term evolution and to answer the question concerning 8~. Few references exist on the adiabatic invariant technique and the best are all connected with the old quantum theory after Ehrenfest pointed out that an atom's sta- tionary state is adiabatically invariant with respect to weak external perturbations. The theory of adiabatic invariance as a part of Hamiltonian mechanics may be found in a book by Born (1960) or in the French trea- tise by Boll M. Salomon (1900). To a lesser


extent a short account is given by Landau and Lifshitz (1960) in the first volume of their course of theoretical physics.

We have shown in Section II that the motion of a dust particle could be described by a Hamiltonian function involving a set of canonical variables. The variables (4~, &) used in the previous discussion of the trajectories are not canonical but the following a r e :

G = cos ~b, g = tb.

These variables are nothing but a particular pair of the so-called Delaunay variables widely used in celestial mechanics and well known to be canonical.

We now consider the change of variables

(G, g) ----> (J, H)

defined by means of the following transformation:

J = G" cos + ( 1 - Gn) ln sin ~ cos g, (38)

tan H

(1 - G~) v~ sin g G sin xF -(1 - Gr2) m cos ~ cos g

(39)

The comparison of Eq. (38) to Eq. (21) makes clear that J -- cos 8 while H is the same as that in Section III, as can be checked from (22) and (24). These two variables are canonical if, and only if, the following relation between their partial derivatives is satisfied:

OJ OH OJ OH o ~ " o--~ - O--G 0 g = t . (40)

A direct but somewhat tedious computation shows that (40) holds for the transformation generated by (38) and (39). Since this change of variable is canonical and time independent, the Hamiltonian function is invariant and the new Hamiltonian expressed in terms of J and H is obtained by inserting (38) and (39) into (10):

R = h/cos W • J. (41)

So R is independent of H and the two differential equations of motion reduce to

dJ OR dt OH O, (42)

dH OR h d-)--= 0---] = cos ~ " (43)

The solution of these equations is trivial,

J = Jo, H = H0 + M c o s ~ t .

The two variables J and H are the two action-angle variables of the problem under investigation. J is the action variable and remains constant during the motion while H as an angle variable varies linearly with time. Notice that the fundamental frequency v = h/cos • reappeared and, as was pointed out in the preceding section, the frequency does not depend on J.

So far we merely have a new presentation of the integration already carried out, and no progress has been made in understanding the P.R. drag nor have any previously unknown results been added. Actually the meaning of the two variables 8 and H pri- marily introduced for geometrical purposes turns out to be deeper than expected at first since they are canonically conjugates. But the most important fact is that the action variable has the property of being adiabatically invariant [see Born (1960), or Landau and Lifschitz (1960)] when an external parameter slowly changes. The idea of "s lowly" is defined by reference to the period of the motion, namely, 2~- cos ~/h. Slow change means that variation of an external parameter, say, the semimajor axis, can be considered negligible over any such period.

As an external parameter we have only mentioned the semimajor axis; of course the semimajor axis decreases as a result of the energy loss caused by the P.R. drag and so does ~ . But in fact W is also affected by any sublimation or sputtering of the particle. A typical particle 1 /~m in diameter is continuously eroded through impacts by


energetic particles or micrometeoroids and eventual ly can be wholly disrupted. In a gentle process the particle size steadily decreases , whence the value of/3 changes, as can be checked in Fig. 1. Accordingly xlt follows the same trend.

Therefore sputtering can be taken into account in allowing for the long-time evolution of a dust grain. Fortunately the same paramete r xtt is affected by either P.R. drag or sputtering, and a unique formalism is likely to work in both cases.

We are now set to answer the question asked at the beginning of this section: What about 8~? Since J = cos 8 is adiabatically invariant, its value remains unchanged while the P.R. drag drives the particle closer to the planet, thereby making ~ ' s value smaller. Then, if each t rajectory in the phase space is labeled with its value of 8, it can be said that the particle stays on the same trajectory until it crashes into the planet. Consequent ly if we start at tl with 8 = 81, we end with 8~ = 81 at t = t2. It is this p roper ty that allows us to draw some conclusions about the behavior of the orbital eccentr ici ty of any particle experiencing radiation pressure forces.

Let us initially take qt < rr/4 and consider a particle moving on a t rajectory characterized by 8 such that • + 8 < rr/2. In Fig. 7 all librational trajectories are of this kind. For this initial condition we have a max imum eccentr ici ty given by

(J~max = ~ A- 8 (44)

and a minimum

d~min = XI' - - 8 . ( 4 5 )

Therefore the amplitude of variation is ~bmax - +rain = 28, which does not depend on xI t. As the grain approaches the planet with t ime, the value of xlt gets smaller and therefore so do d~max and ~bmin; never theless the ampli tude is unaffected as long as ~bm~, > 0, namely , as long as the t rajectory is a librational one. As soon as ~ becomes small enough to allow a circulatory orbit to exist, the t ra jectory enters the class of circulating

trajectories w i t h ~max = xlz¢ + 8 a n d ~min = 8 - ~ . In that case the amplitude of variation is +max - ~bmin = 2 xIt. So, for small values of xI t, the trajectories tend to be straight lines ~b = 8 or, in other words, the eccentrici ty remains constant over the circulation of the apocenter , but the orbit does not evolve toward a circle. This result confirms and extends what was pointed out by Burns et al. (1979) in a limiting case that the P.R. drag lessens the amplitude of the eccentricity excurs ion caused by the radial radiation pressure force. Accordingly the probabil i ty of elimination of that particle by eccentricity per turbat ion decreases too.

As a rule this kind of elimination depends strongly on the pericenter distance p,

p = a(l - emax) = a l l - sin(XI t + 8)], (46)

while a is connected to • by

tan • = (3/2)/3 ( V J V ) oc all2.

Thus the closest approach between the particle and the planet can be expressed in terms of W and 8:

p = ~ t a n zW[1 - sin(W + 8)]. (47)

For a given value of 8, that is, for a given trajectory, the pericenter distance varies as

decreases due to the P.R. drag. But because of the s imultaneous evolution of the max imum eccentrici ty, the pericenter distance does not systematical ly decrease with the semimajor axis and even the unexpected situation of an increase in the pericenter distance can occur for awhile. Even- tually the minimal distance always drops as soon as the semimajor axis is smaller than a critical value. In (W, 8) space a critical curve separates the plane into two regions according to the sign of dp/dt, as illustrated in Fig. 10.

When sputtering acts alone, with the P.R. drag being neglected, a pair o f different cases can be treated. First let us consider a particle initially a few microns in diameter . Due to erosion the coefficient/3 increases as shown in Fig. 1. Thus, for a given semima-


( ] ! i ! i i i i i

7(?

613

4 o

102(](3 J dtl -l 0 20 40 60 80

B (degrees)

FIG. 10. The Poynting-Robertson drag causes the pericenter distance to evolve. In the shaded region, this distance increases while the semimajor axis is decreasing.

jo t axis, ~ becomes larger and if the process of erosion is slow enough we can repeat the discussion made in the P.R. drag case and ascribe a constant value to 8. But since ~ grows the maximum eccentricity given in Eq. (44) rises too, leading to a probable elimination of the particle by a collision with a satellite or with the planet itself.

For intermediate-sized particles, /3 decreases with the particle diameter and the orbital history is akin to the description given for the P.R. drag except that the semimajor axis remains constant during this process. When a particle simultaneously undergoes both sputtering and drag, the two processes compete against each other in opposite directions; the subsequent evolution depends on the particle size and on the density of the surrounding microme- teoritic material which eventually erodes the particle. This situation is somewhat too complex to allow general conclusions to be drawn, since each particle and each planetary environment can produce a particular evolution.

To conclude this section it must be stressed that, even if the P.R. drag by itself does not cause a systematic variation in the orbital eccentricity of a dust particle during

its circumplanetary motion, this result is no longer valid when the radial radiation pressure force is included. Then there is a coupling effect between the secular variation of the semimajor axis and the amplitude of the eccentricity excursion which gives rise to a systematic change in the mean eccentricity.

CONCLUSION

In this paper we have investigated the effect of radiation pressure on a circumplanetary particle's orbit and we have given a more complete solution than those previously known.

In the solar system micron-sized particles have been observed around Jupiter and their diameter has been determined from the scattering pattern of the sunlight. The radiation pressure on these particles is important (Morrill e t al. , 1980) but it competes with other processes, such as plasma drag and spallation (Burns e t al . , 1980). Saturn's F-ring is also known to contain many sub- micron-sized particles from the scattering observed by Voyagers ! and 2. As in the Jovian case, any complete solution of the radiation pressure effect needs to include the planet's quadrupole moment which, by driving the pericenter, modifies the eccentricity coupling. Work on this problem is in progress.

Observations of Iapetus made during the Voyager 2 mission have narrowed the number of possible explanations for its albedo variations. Among the most promising mechanics is that the leading hemisphere has been bombarded by matter coming from Phoebe; this material should suffer both the eccentricity excursions explained throughout this paper as well as the long-term evolution due to the Poynting-Robertson drag.

All these examples come from the most recent planetology; they indicate why an improved understanding of the effect of the radiation on the dust particle dynamics will be valuable.

AC KNOWLEDGMENTS

I am indebted to J. A. Burns for various fruitful discussions we had during the preparation of this paper


and for having read the manuscript and greatly im- proving its content. Remarks by S. J. Peale have pre- vented mistakes from being printed. The author was supported by NATO and the French National Center for Scientific Research during a postdoctoral stay at Cornell University.

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Radiation Pressure and Dust Particle Dynamics

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