ICARUS 72, 568581 (1987)
Interplanetary Dust Dynamics II. PoyntingRobertson Drag and Planetary Perturbations on Cometary Dust B . / k . S. G U S T A F S O N , N. Y. M I S C O N I , AND E. T. R U S K Space Astronomy Laboratory, University of Florida, Gainesville, Florida 32609 R e c e i v e d O c t o b e r 2, 1986; r e v i s e d M a y 28, 1987
A statistical analysis of a portion of the calculated orbital evolution of some 200 cometary dust particles as they spiral from just inside the orbit of Jupiter to within 0.3 AU from the Sun reveals an increase in the descent rate as calculated for drag forces alone (i.e., PoyntingRobertson drag and corpuscular drag). This increase appears when planetary perturbations are accounted for in the orbital integrations and are, at least in part, due to planetinduced orbital eccentricities leading to enlarged drag. Trajectories originating from Comet P/Encke are thought to be representative of debris from shortperiod comets due in part to a redistribution in orbital elements while the dust crossed Jupiter's orbit. Statistical methods indicate that any dependence of the increased descent rate on orbital elements other than semimajor axis and eccentricity were insignificant. There are indications that the planetinduced acceleration of the mean descent rate decreases with decreasing eccentricity. The increase in mean descent rate is only a few percent inside 1 AU; it increases to 610% in the asteroid belt and reaches 50% in the 3/1 resonance with Jupiter. By contrast, the mean rate is slower near the 4/1 Jovian commensurability than in nearby regions of phase space. An empirical relation is presented for the descent rate of cometary dust particles subjected to planetary perturbations in addition to PoyntingRobertson and corpuscular drags. The analytic expression is suitable to improve estimates of the contribution from shortperiod comets' to the zodiacal dust, and the resulting particle number density distribution. ©1987AcademicP..... Inc. Before ultimately being destroyed or thrown out of the solar system, dust particles in interplanetary space generally spiral toward the Sun. The angular m o m e n t u m is primarily lost to solar electromagnetic radiation through the P o y n t i n g  R o b e r t s o n drag (a relativistic effect resulting from absorption and scattering of sunlight; R o b e r t s o n 1937). While it was shown in the first p a p e r of this series (Gustafson and Misconi 1986, hereafter p a p e r I) that transfer of angular m o m e n t u m to dust particles through gravitational perturbations by the planets m a y at least temporarily cancel this effect or reverse the spiraling motion of individual dust particles, the overall trend is for the dust particles to migrate toward the Sun. In
p a p e r I we also showed that all orbital elements are affected as the dust particles spiral past a planet. In this p a p e r we discuss aspects of the evolution of the interplanetary dust cloud as the constituent particles spiral from just inside Jupiter's orbit to 0.3 A U from the Sun. The d u s t ' s orbital evolution is compared to that predicted based on the Poynti n g  R o b e r t s o n effect and corpuscular drag (resulting f r o m collisions with solar wind ions) but excluding planetary effects. Inconsistencies b e t w e e n observations of the zodiacal light on one hand and estimates of the budget of the interplanetary dust cloud and the theoretical particle n u m b e r density evolving f r o m likely sources on the other 568
00191035/87 $3.00 Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
POYNTINGROBERTSON AND PLANETARY EFFECTS hand (see paper I) triggered investigations of the effects of combinations of the PoyntingRobertson effect with other perturbations. Combinations with the Lorentz force (Morrill and GrOn 1979, Mukai and Giese 1984) or with dustdust collisions (Trulsen and Wikan 1980, Gr(in et al. 1985) lead to representations of the number density of dust particles in the ecliptic plane by a simple power of heliocentric distance. Such a dependence is supported in the inner solar system by Helios results (Leinert et al. 1981) but the distribution appeared more complex from Pioneer 10 as it probed the dust beyond 1 AU (Schuerman 1980). The aim of this second paper in a series is to deduce an empirical expression, in analytic form, for the average rate at which 30/zmsized cometary dust particles spiral toward the Sun under the influence of the PoyntingRobertson effect combined with corpuscular drag and planetary perturbations. The 10 7_g dust particles are as defined in paper I except for the cross section for radiation pressure which has been increased from Cpr = 2.12 x 105 cm 2 to the same value as the cross section for ion impact Cio, = 2.83 x 10 5 cm 2. These particles represent the size bin 10100/zm, producing most of the zodiacal light by scattering of sunlight (Giese and GriJn 1976, R6ser and Staude 1978). The sought relation may be used to improve estimates of shortperiod comets' contribution to the interplanetary dust complex. With the addition of an expression for the accompanying change in inclination, the method of Briggs (1962) may be used to estimate planetary effects on the particle number density distribution. This study is based on numerical calculations of trajectories corresponding to over 200 particles released from Comet P/Encke. The release times represent a 20,000year period, which spans the PoyntingRobertson lifetimes of these particles, during which the longitude of P/Encke's perihelion is confined to a 200deg interval and the geometry corresponds preferentially to
569
seven SunJupiterdust configurations owing to a near 7 : 2 orbital resonance between P/Encke and Jupiter (Whipple 1940). Some of these test particles leave the solar system on hyperbolic orbits following near encounters with Jupiter or are otherwise drastically affected by Jupiter in their first few orbits after separation from the comet. These trajectories are not discussed in this paper, instead we concentrate on the orbital evolution of a 186particle subset after the semimajor axes (initially 2.89 AU) have decreased to 2.5 AU so that the particles no longer cross Jupiter's orbit. By this time, orbital elements have been spread over the full range in longitudes of perihelion and the line of nodes. Out of the 186 trajectories 170 were followed until their orbits shrank below 0.5 AU and most were pursued to 0.3 AU. The integration of the remaining 16 dust trajectories were stopped at a perihelion distance of 10 solar radii, below which our modeling breaks down. Little error is expected to result from this truncation because of accelerated sublimation rates near the Sun. COMPUTATIONS AND ACCURACY Misconi (1976) showed that corpuscular drag amounts to 26% of the PoyntingRobertson drag, assuming a radiation pressure efficiency of unity (which is reasonable for this size particle, see Burns et al. 1979). The ratio is approximately independent of heliocentric distance since densities of both the solar wind and the electromagnetic radiation field fluctuate about average values that are proportional to the inverse square of the solar distance. Neglecting other perturbations, a particle of radius s = 30/zm which is subjected to both PoyntingRobertson drag and corpuscular drag would consequently have an orbital history similar to an s/1.26 = 23.8/~m dust grain subjected to PoyntingRobertson drag alone. The time it takes for the orbit of such particles to shrink from one semimajor axis to another was checked analytically following Wyatt and Whipple (1950).
570
GUSTAFSON, MISCONI, AND RUSK
In addition to the P o y n t i n g  R o b e r t s o n effect and corpuscular drag, our computations (described in paper I) included gravitational perturbations due to the planets Jupiter, Mars, Earth, and Venus. The trajectories were evaluated using Encke's method, integrating in 200 steps over one orbital period. The perturbations on an osculating orbit were obtained in this way except near a planet where the orbit was rectified and computations resumed using Cowell's method of directly applying the equation of motion to calculate the dust particle's position and velocity for time intervals of 1/40,000 of an orbital period. For particles spiraling inward due to P o y n t i n g  R o b e r t s o n drag, Wyatt and Whipple (1950) found the expression C = ae
4/5(I

e 2)
(1)
to be constant, where a and e ¢ 0 represent the semimajor axis and eccentricity of the orbit, respectively. The analogous drag from solar wind ions does not change this relationship between a and e, although it causes them to decrease more rapidly. By assigning null masses to the planets the effect of round off errors from the stepwise integration and in the interpolation between stored data could be estimated from a consistency check between the two solutions. Such a test was run for 20,000 orbits and C remained constant to at least four significant figures throughout. STATISTICAL SIGNIFICANCE OF PLANETARY EFFECTS IN THE DATA SAMPLE Any effect of the planets on the rate at which interplanetary dust particles spiral through the solar system has in general been neglected in the literature. At first it appears from Figs. l a  l d , showing the time required to shrink the orbit's semimajor axis by 0.5 AU as a function of the eccentricity for four different initial semimajor axes, that this simplification is justified for the 1 to 0.5 A U range while the planets produce a wider scattering in the outer solar system. The dimensionless ratio
t =
T(e, al,
a2)  TpR(e, al, a2)
(2)
T e a ( e , a l , a2)
is a measure of the planetary perturbations on the elapsed time T ( e , a l , a2) between two semimajor axes al and a2. H e r e TpR(e, al, a~) is the P o y n t i n g  R o b e r t s o n time including corpuscular drag. Plotting the arithmetic mean (t) versus the interval al to a2 (Fig. 2) reveals a tendency for increased rates with semimajor axis. First, we argue based on standard methods that this tendency is statistically significant, then we show that the increase in descent rate depends on both an orbit's semimajor axis and its eccentricity. The coefficient of correlation
~/ r = 
explained variation based on a hypothetical relation ~ ~
(3)
is a dimensionless measure of linear correlation; it assumes the value 1 if the hypothetical relation is a perfect fit and zero if unrelated. F r o m our sample we estimate the probability that the coefficient of correlation r ¢ 0 in the parent distribution of all cometary particles, of which our sample is a subset. The coefficient of correlation rta, a measure of a possible linear relation between t and a ( a = a l , a2 = al  0.1 AU), is  0 . 3 0 in the interval from a = 0.4 to 1.5 A U and decreases in magnitude to 0.21 as the range is extended to 2.0 AU and  0 . 1 2 at 2.8 AU. This signifies that a linear relation between t and a is more likely when the orbital semimajor axis is less than that of Mars (~1.5 AU) than it is in a range encompassing the asteroid belt. The sign indicates a negative correlation, i.e., the perturbation in descent time decreases with a. For the purpose of estimating the probability that t is correlated to a in the parent distribution as well, we consider our sample to be a random subset of dust released from shortperiod comets. The statistic u 
rX/N
2
l x / i  =  Pp '
(4)
P O Y N T I N G  R O B E R T S O N A N D P L A N E T A R Y EFFECTS
571
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FIG. 1. (ad) Time taken for individual dust orbits to shrink in semimajor axis by 0.5 AU from 2.5, 2.0, 1.5, and 1.0 AU, respectively. The elapsed time is plotted versus starting eccentricity; also shown is the PoyntingRobertson time with corpuscular drag represented by the solid line. Dotted lines correspond to +50% deviation and dashed lines to + 10% deviations from the PoyntingRobertson/ corpuscular drag rate. The 30/xm radii particles are representing zodiacal dust.
w h e r e N is the n u m b e r o f data points, m a y then be u s e d to c o m p u t e this probability ( s e e T r u m p l e r and W e a v e r 1953, or any other standard t e x t b o o k ) . F o r the probability to be 95% or higher m u s t e x c e e d 1.65,
lul
r e s p e c t i v e l y , 2.58 for a 99.5% l e v e l o f c o n fidence. W h i l e the s a m p l e is not strictly r a n d o m , w e j u d g e that the correlation bet w e e n t and a is statistically significant b a s e d on the v a l u e s o f u o f  1 4 . 3 ,  1 1 . 7 ,
572
GUSTAFSON, MISCONI, AND RUSK 11 \
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0.8 Orbital
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FIG. 1Continued.
and  8 . 3 in the intervals from 0.4 to 1.5 AU, 0.42.0 A U , and 0.42.8 AU, respectively! In this model, Fisher's transformation can be used to predict, at a 99% confidence level, that the parent distribution's correlation coefficient of t on a is in the range  0 . 2 5 to  0 . 3 6 . We may not yet conclude that t depends
on a; the correlation between t and a could result entirely from a dependence on the eccentricity or any other parameter that is in turn correlated to a. The orbital eccentricity and inclination i are prime candidates for such a dependence. The Poynt i n g  R o b e r t s o n effect is known to reduce e and a simultaneously, and the correlation
POYNTINGROBERTSON AND PLANETARY EFFECTS
573
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a (AU) FIG. 2. Mean rates of infall over 0.1 A U intervals, in terms of the PoyntingRobertson rate with corpuscular drag. The right and left boundaries of each bar correspond to the starting and ending semimajor axes, respectively.
coefficient ria of linear change in i with a was found to be as high as 0.38 over the interval 0.4 to 2.8 AU. While there is a mechanism for dependence of t on i, from the increased probability of planetary encounters with decreasing inclinations, the main reason for increased descent rates appears to arise from perturbations on the orbital eccentricities. Suggestively 22 out of the 27 particles whose descent rate from 2.5 to 2.0 AU decreased when perturbed by the planets have lower eccentricities when they reach 2.0 AU than would result solely from the PoyntingRobertson effect or its combination with corpuscular drag. At lower eccentricities, the drag is less severe and the rate of reduction in semimajor axis decreases. A corresponding increase in orbital eccentricity predominates among the particles that spiral in faster. Gravitational stirring due to planetary perturbations are more likely to reduce the rate of decrease in orbital eccentricities than to increase this rate. Although most orbital eccentricities are reduced, Fig.
3 shows that the majority are reduced less than they would be from PoyntingRobertson and corpuscular drag effects. It is seen in the figure that while there is a spread of the order of + 15% in our sample, the eccentricities are typically a few percent higher than expected from the drag forces alone as the dust spirals in from 1.5 to 1.0 AU. The tendency for the particles to spiral in slightly faster under the influence of planetary perturbations is thus expected qualitatively from gravitational stirring alone. The coefficient of multiple correlation Rt.ea of t on e and a, the analog to rt~, is 0.39 in the 0.4 to 1.5 AU interval, a 30% growth from Irt~l. The increase shows that e is a relevant independent variable; by taking it into account we arrived at a better relationship between the variables. The insignificant (less than 0.1%) gain in correlation when the three orbital elements e, a, and i were simultaneously accounted for indicates that the inclination is nearly irrelevant to t, despite its high correlation to a.
574
GUSTAFSON, MISCONI, AND RUSK 60
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50
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i
6
in
8
10
I
~
.1.2 14
Eccentricity
I
I
16
18
~
20
f
i
I
22
24
26
(7.)
FIG. 3. Percentage deviations of orbital eccentricities from those which would result from P o y n t i n g  R o b e r t s o n effect and corpuscular drag alone, after a reduction in semimajor axes from 1.5 to 1.0 AU. Nearly 60% of the trajectories have their eccentricities increased by 13% over PoyntingRobertson effect through planetary perturbations.
The estimate of t corresponding to Rt.ea is given by the leastsquare fit plane in parameter space (t, e, a): test = bt.ea + b t e . a e
+ bta.ea.
(5)
The constants bte.a and bta.e a r e the partial regression coefficients of t on e keeping a constant and on a keeping e constant, respectively; bt.ea is an independent constant. N u m e r i c a l values are given in Table I along with the partial correlation coefficients, from which we conclude that t has a stronger coupling to e than to a. In reality e is strongly coupled to a through the Poynti n g  R o b e r t s o n effect; re~ is 0.66 in the 0.4 to 1.5 A U interval and increases to 0.71 in the interval of 0.4 to 2.0 AU. It is therefore possible to use the m o r e convenient regression equation of t on a, test = bo + bla,
(6)
to a p p r o x i m a t e the effects of planetary perturbations on the descent rate. The numerical values of b0 and bl depend s o m e w h a t on the interval in a. We p r o p o s e use of the
values 0 and  0 . 0 3 ( A U  ' ) which give a good a p p r o x i m a t i o n inside the asteroid belt for dust shed by P/Encke and p r e s u m a b l y other shortperiod comets. The difference t' = t  test
(7)
is a dimensionless m e a s u r e of the accuracy of the estimate. The coefficient of correlation of t' on a, in the parent distribution, is zero in the ideal case. On the stipulation of r a n d o m sample, there is a 99% probability that Irt,a[ < 0.075 when a < 1.5 AU. Yet the correlation to eccentricity remains important. The distribution in t' is almost Gaussian; t' corresponding to the
TABLE I PARTIAL REGRESSION AND CORRELATION
COEFFICIENTS OF I ON e AND a
al
b, .....
b,.,
1.5 to 0.4 AU 0.033 0.089 1.7 to 0.4 AU 0.038 0.)00
b~,,,, (AU ~) 0.007 0.004
r ......
r ......
0.271 0.057 0.216 0.028
POYNTINGROBERTSON AND PLANETARY EFFECTS
575
0.20 0.15 0.10 •
0.05
, :..
.
.~~_
. ",':.'5
• ~ ...:~ . . . . ..
0.00
.......
 ......
2. ~_
. '.'~.~...
'._.
;.:':.;,?,
"

.
. . . .
~ ,.
' .. "~k:...~:~,
:.;!~..
0.05 0.10 0.15 0.20
I

02.
I
I
I
0.4
I
O.fl

I
I
0.8
e
FIG. 4. D e v i a t i o n s from the regression equation of t(a) (Eq. (6)) versus orbital e c c e n t r i c i t y for particles in the interval 0.4 < a < 1.0 AU. The d e p e n d e n c e on e indicated by the regression line (solid) is a warning that the d e s c e n t rate d e d u c e d for c o m e t a r y dust (high eccentricities) ma y not be applicable to low e c c e n t r i c i t y orbits. D a s h e d lines are 2 SD from the regression lines.
1.0 to 0.4 AU interval is plotted versus e in Fig. 4. The slope of the regression line (  0 . 0 5 5 + 0.003, at 99% confidence level) suggests that the mean descent rate of dust particles in low eccentricity orbits are less affected by planetary perturbations, but the coverage in eccentricity is too narrow to be conclusive on this point. Similar results are obtained for the region outside 1 AU where the coverage in eccentricity is even less. While we expect that our calculations are representative of dust in high eccentricity orbits, we are compelled to caution against extrapolation of our results to low eccentricities. M E A N R A T E OF D E C R E A S E IN S E M I M A J O R AXIS
Turning our attention back to Fig. 2, it is seen that the sample mean rate (t) at which the semimajor axis of an eccentricity e orbit decreases from al to a2 is generally faster than that predicted from the P o y n t i n g Robertson effect combined with corpuscular drag (TpR(e , a l , a2)). The exception from
2.1 to 2.0 AU, in which the rate is 18.5% slower, contains the 4/1 resonance with Jupiter. The amount by which the rate is accelerated increases with heliocentric distance from about 1% in the 0.40.3 AU region to about 20% in the 2.82.5 AU region. The mean time (T(e, al, a2)) (for the semimajor axis to decrease from al to a2) in units of TpR(e, al, a2) when 0.4 (AU) < al < 2.0 (AU) and a2 = al  0.1 (AU)
(8)
depends nearly linearly on al, as expected from the statistical analysis of t. The increase in mean rate at which the dust particles approach the Sun is quite well approximated by the regression equation (6) or 3% per AU so that we may write for the mean elapsed time (Test(e, al, a2)) = TpR(e, al, a2)(1  0.03 al).
(9)
Equation (9) is the analytic form or empirical relation for the average rate at which
576
GUSTAFSON, MISCONI, AND RUSK
cometary dust particles of 30 /xm radii spiral toward the Sun under the predominance of PoyntingRobertson and corpuscular drags combined with planetary perturbations; the relation that we set out to find. Deviations from this relation extrapolated to 2.8 AU are plotted in Fig. 5. The deviations are small out to the asteroid belt. For semimajor axes up to 1.5 AU, they are less than one standard deviation of the random error and the approximation is within 1.5% or three times the error out to 2 AU. The median of our sample followed (9) even more closely. Beyond 2 AU, the rates at which particles accelerate their descent to the Sun deviate from the smooth increase given by (9). This occurrence may be related to the phenomenon of the Kirkwood gaps in the asteroid belt, as the most pronounced deviations correspond to the strongest resonances with Jupiter in the range covered by the computations. The patchy nature is also reminiscent of the zodiacal light results from Pioneer 10 (Schuerman 1980). The 18.5% increase in the average time that
particles spend with a semimajor axis in the 2.1 to 2.0 AU interval is mainly due to a few particles whose rates are up to 12 times slower than in neighboring intervals. The most probable time, at the mode of the distribution plotted in Fig. 6, is actually 20% lower than the time estimated by extrapolating Eq. (9). The orbital periods of particles in this region are in 4/1 commensurability with Jupiter's orbit; the resonance is shifted inward by radiation pressure from the outer edge of the interval to 2.04 AU. The next major deviation from a smooth heliocentric dependence occurs between 2.5 and 2.4 AU, corresponding to the radiation pressureshifted 3/1 resonance (2.47 AU) with Jupiter. Particles cross that interval at mean rates that are over 45% faster than predicted by extrapolation of Eq. (9) and 35% faster than in neighboring intervals. Contrary to the 2.1 to 2.0 AU interval, this deviation is affecting most trajectories, the standard deviation about the arithmetic mean (14.5% of Tpg(e, ai, a2)) being at a local minimum. No correlation could be found in either interval between
20~.
o ,,a a
o~.
a
107,
0
20X
o
30~
I:1
4O7.
'
507.
      T       r      l ~ ' l  '    ' ~   ~ 0 0.4 0 . 8 . 1.2 1.6 2
k
r 2.4
1
r 2.8
a (AU)
FIG. 5. Mean deviations from the approximate descent rates over 0.1 AU intervals (Eq. (9)). The right and left boundaries o f each bar correspond to the starting and ending semimajor axes, respectively.
POYNTINGROBERTSON AND PLANETARY EFFECTS
577
00
50
40 v
o J
30
o' M
\Nl
20
\
l0
\
0
I
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\N
~
40
~
30
l
I
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I
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I
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40
50
Dev. from. Approx.
ix S I
80
. .I . . 70
I
I
I
80
90
100
(Y.)
FIG. 6. Spread in the deviation t' from approximation (9). The m o s t probable d e s c e n t rate through the 4/1 r e s o n a n c e with Jupiter, corresponding to the node of the distribution, is faster than that on either side of the c o m m e n s u r a b i l i t y and faster than estimated by extrapolation of Eq. (9). By contrast, the m e a n rate is 18% lower than in neighboring intervals and slower than estimated from Eq. (9).
the descent rate and any orbital parameter. The deviation is plotted out to 2.8 AU but possible selection effects may make the extreme points uniquely representative for dust released from Comet P/Encke so that the increased descent rate in the 2.82.7 AU interval corresponding to the 5/2 commensurability (2.79 AU) could be statistically insignificant. The significance of the 3/1 and 5/2 Jovian resonances are clear from their correspondence to the first two gaps discovered in the distribution of asteroids by Kirkwood (1867). With over 3000 asteroid orbits known today, gaps have also been discovered at the 7/3 and 2/1 resonances and clustering at the 3/2, 4/3, and 1/1 resonances but all of them correspond to larger semimajor axes that are not covered in this investigation. Greenberg (1978), who considered particles in resonance with Jupiter and in a dissipative medium, found that, for reasonable early solar system parameters, the gap in the distribution of asteroids near
the 2/1 resonance is cleared on time scales of a few thousand years. Greenberg also suggested a number of mechanisms to attract and trap (in resonances) bodies orbiting under the influence of the PoyntingRobertson effect. The trapping time of small bodies near several resonances has been studied by Gonczi et al. (1983) using numerical integrations in the coplanar and threedimensional restricted threebody problem (SunJupitertest particle) including the PoyntingRobertson drag. Gonczi et al. (1982) proposed that the PoyntingRobertson drag might explain the depletion of asteroids in the 2/1 resonance and the concentration in the 3/2 resonance. Later Gonczi et al. (1983) found that the trapping times near these resonances were of the same order, while they were shorter near the 3/1 and 5/2 resonances, corresponding to gaps in the asteroid distribution. Burkhardt (1985) used a code similar to ours to study resonances with Jupiter including the effect of PoyntingRobertson drag.
578
GUSTAFSON, MISCONI, AND RUSK
DIVERGENCE IN THE RATE OF DECREASE IN SEMIMAJOR AXIS
While the arithmetic mean rate of decrease in semimajor axis is a key quantity needed to estimate the particle number density, this article would hardly be complete without a discussion of the accuracy with which the rate of descent of individual dust particles can be predicted using Eq. (9). To avoid making assumptions about the shape of the parent distribution and the nonrandom nature of our sample, we chose to plot envelopes of the rates corresponding to some percentage of the trajectories in the sample. While the largest deviation corresponds to a time span that is more than 18 times longer than predicted when Eq. (9) is extrapolated beyond 2 AU, the deviations never exceeded 50% inside 1.5 AU nor 150% from 1.5 to 2 AU. Envelopes of the deviation of 90% of the elapsed times are plotted in Fig. 7 in which it is seen that their descent rates are concentrated much closer to (9). Inside 1.5 AU, 90% of the trajecto
ries deviate less than 7% from the approximation; i.e., there is a 90% probability that an orbit chosen at random from the 186 trajectories sampled spiraled toward the Sun at a rate that deviated by less than 7% from the rate predicted through
(9). The sample distributions are nearly symmetric about the median (see Fig. 7), except for the 3:1 and 4:1 resonances with Jupiter. The distributions tend to narrow with decreasing semimajor axis. There is a widening of the distributions for the intervals corresponding to the 3:1 and 4:1 resonances, as well as in the 1.7 to 1.8 AU interval which includes the 5 : 1 Jovian resonance (1.76 AU). The shapes of the distributions in the 2.2 to 2.1 AU and 2.0 to 1.9 AU intervals plotted in Figs. 8a and 8b are also representative of the region inside the 1/1 commensurability with Mars (1.5 AU). The distribution gets flatter (platykurtic) and asymmetric (positively skewed) across the 4/1 resonance with Jupiter (Fig. 6).
80% 7O%
60% 0
50%
el 40% '~ 0 ~, t~ II,
m
30% 20% 10%
~10%
0
~'a
30%
4o% 50%
Q
60% 70%
80%
r
0
I
0.4
i
i
0.8
i
i
1. :~
i
i
1.6
i
i
:~
i
t
Z.4
1
I 2.8
a (AU)
FIG. 7. Deviation envelopes of 90% probability in the sample distribution. Transit times between semimajor axes corresponding to the end values of each bar, differed from approximation (9) by the fraction indicated by the bars or less for 168 out of the 186 test trajectories.
POYNTINGROBERTSON AND PLANETARY EFFECTS
579
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30
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\\ \\
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\\ ~.\ N\
N\ N\ ~.\
0
o' @ L
20
10
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I
50
I
I
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I
20
I0
0
tO
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20
from
30
40
Approx.
50
I
!
I
t
I
60
70
60
90
100
(7,)
60 u
50
\\ \\ \\ \\
40 \\ \\ \\
v
I
30
\\
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20
\\ \\ \\
\\,1
o' I
r..
\\ \ \
10
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\\ \\ \\
I
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I
40  3 0
I
20
~
10
0
10
Dev.
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from
:30
I
I
I
I
I
I
I
40
50
60
70
80
90
100
Approx.
(~.)
FIC. 8. (a and b) Spread in the deviation t' from approximation (9). The distribution over the 4/1 resonance with Jupiter (Fig. 6) is much flatter and less symmetric than the distribution just outside (a) or inside (b) the resonance.
CONCLUSIONS
We have shown how the rates of descent of c o m e t a r y dust particles from 2 to 0.3 A U from the Sun, while subjected to P o y n t i n g Robertson drag and corpuscular drag in
combination with planetary perturbations, depend on the orbital eccentricity and semimajor axis. A n y dependence on other orbital parameters was too weak to be statistically significant based on our 186 trajectory sample. We have also shown that the de
580
GUSTAFSON, MISCONI, AND RUSK
scent rate of the cometary dust particles can be expressed as a linear function of the initial semimajor axis times the PoyntingRobertson time. In the asteroid belt, deviations from this function occur at major resonances with Jupiter. We suspect that these deviations may have nontrivial explanations, analogous to the Kirkwood gaps. While estimating the particle number density distribution is not the thrust of this paper, it is apparent that planetary effects distort the distribution of dust by altering the rate at which dust grains migrate toward the Sun. Heliocentric spherical shells or rings of dust, as well as depletion zones, are anticipated outside of the orbit of Mars in conjunction with Jovian resonances. Whether this phenomenon is related to the observed dust bands in the asteroid belt (Low et al. 1984) is unclear. In contrast, planetgenerated deviations from the PoyntingRobertson and corpuscular drag rate are found to be small throughout the inner solar system. Finally, we have shown that, on the average, the gravitational effects of the planets cause cometary particles to be lost to the inner solar system more rapidly than through PoyntingRobertson and corpuscular drags alone. This makes the issue of supply versus sink of the interplanetary dust complex no less difficult to reconcile with shortperiod comets as the primary suppliers. ACKNOWLEDGMENTS We are thankful to Dr. J. D. Mulholland for stimulating discussions, and to anonymous referees for helpful comments. This research was supported by NSF Grant AST8206152. REFERENCES BRIGGS, R. E. 1962. Steadystate space distribution of meteoric particles under the operation of the PoyntingRobertson effect. Astron. J. 67, 710723. BURKHARDT, G. 1985. Dynamics of dust particles in the Solar System. In Proceedings, I.A.U. Colloquium No. 85, Properties and Interactions o f lnterplanetary Dust (R. H. Giese and P. Lamy, Eds.), pp. 389393. Reidel, Dordrecht/Boston/London.
BURNS, J. A., P. L. LAMY, AND S. SOTER 1979. Radiation forces on small particles in the Solar System. Icarus 40, 148. GIESE, R. H., AND E. GRCN 1976. The compatibility of recent micrometeroid flux curves with observations and models of the zodiacal light. In Lecture Notes in Physics No. 48, Interplanetary Dust and Zodiacal Light (H. Els~isser and H. Fechtig, Eds.), pp. 135139. SpringerVerlag, Berlin/Heidelberg/New York. GoNCZl, R., CH. FROESCHLE, AND CL. FROESCHLE 1982. PoyntingRobertson drag and orbital resonance. Icarus 51, 633654. GONCZI, R., CH. FROESCHLE, AND CL. FROESCHLE 1983. Evolution of three dimensional resonant orbits in presence of PoyntingRobertson drag. In Asteroids', Comets, Meteors (C.I. Lagerkvist and H. Rickman, Eds.), pp. 137143. Uppsala Universitet Reprocentralen, Uppsala. GREENBERG, R. 1978. Orbital resonance in a dissipative medium. Icarus 33, 6273. GRON, E., H. A. ZOOK, N. FECHT1G, AND R. H. GIESE 1985. Collisional balance of the meteoric complex. Icarus 62, 244272. GUSTAFSON, B. ilk. S., AND N. Y. MISCONI 1986. Interplanetary dust dynamics. I. Longterm gravitational effects of the inner planets on zodiacal dust. Icarus 66, 280287. KIRKWOOD, D. 1867. Meteoritic Astronomy: A Treatise on Shootingstars, Fireballs, and Aerolites, Chap. 13. Lippincott, Philadelphia. LEINERT, C., 1. RICHTER, E. PITZ, AND B. PLANCK 1981. The zodiacal light from 1.0 to 0.3 AU. Astron. Astrophys. 103, 177188. Low, F. J., D. A. BEITEMA, T. N. GAUTIER, F. C. GILLETT, C. A. BEICHMAN, G. NEUGEBAUER, E. YOUNG, H. H. AUMANN, N. BOGGESS, J. P. EMERSON, H. J. HABING, M. C. HAUSER, J. R. HOUCK, M. ROWANROalNSON, B. T. SOFIER, R. G. WALKER, AND P. R. WESSEEIUS 1984. Infrared cirrus: New component of the extended infrared emission. Astrophys. J. 278, Ll9L22. MISCONI, N. Y. 1976. Solar flare effects on the zodiacal light. Astron. Astrophys. 51, 357365. MORFILL, G. E., AND E. GRON 1979. The motion of charged dust particles in interplanetary space. 1. The zodiacal dust cloud. Planet. Space Sci. 27, 12691282. MUKAI, T., AND R. H. GIESE 1984. Modification of the spatial distribution of interplanetary dust grains by Lorentz forces. Astron. Astrophys. 131, 355363. ROBERTSON, H. P. 1937. Dynamical effects of radiation in the Solar System. Monthly Notices Roy. Astron. Soc. 97, 423438. ROSER, S., AND H. J. STAUDE 1978. The zodiacal light from 1500 ,~ to 60 micron: Mie scattering and thermal emission. Astron. Astrophys. 67, 381394. SCHUERMAN, D. W. 1980. Evidence that the properties of interplanetary dust beyond l AU are not
POYNTINGROBERTSON
AND PLANETARY
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EFFECTS
581
TRUMPLER, R. J., AND H. F, WEAVER 1953. Statistical Astronomy. Univ. of California Press, Berkeley. WHIPPLE, F. L. 1940. Photographic meteor studies. III. The Taurid Shower. Proc. Amer. Phil. Soc. 85, No. 5, 711745. WYATT, S. P., AND F. L. WHIPPLE 1950. The PoyntingRobertson effect on meteor orbits. Astrophys. J. 111, 134141.