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Astron. Astrophys. 324, 770-777 (1997)

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4. Results

The dynamical behaviour of comet-derived dust grains is strongly influenced by gravitational perturbations of the planets, in particular Jupiter, by radiation/solar-wind pressures and by Poyinting-Robertson drag forces. The gravitational encounters with the planets scatter the particles randomly in the orbital phase space driving them in a chaotic route while the retarding forces (P-R drag), which act as major perturbative forces on grains with radii between 1 and 100 µ m, cause a monotonic decrease of semimajor axis and eccentricity of the particle orbit with a consequent fall-in toward the Sun. Under these effects the dust particles lose their initial orbital elements on a short timescale and spread around the comet orbit.

We have performed a large number of model computations of the dynamical evolution of dust particles ejected by two SPC with the following goals:

1) to predict the spatial distribution of the dust after a long timespan from emission and the contribution to the zodiacal cloud;

2) to evaluate the relevance of the P-R drag compared with the planetary gravitational perturbations;

3) to state the significance of using a detailed model to compute the ejection velocities of the grains in predicting the spatial distribution of the dust at subsequent times;

4) to estimate the fraction of grains lost from the Solar System on hyperbolic orbits.

The two comets under study (P/SW1 and P/GS) have been selected since they have significantly different orbital parameters. Comet SW1 has a moderate eccentricity (0.0447) and the perihelion at about 5.7 AU between the orbits of Jupiter and Saturn, so its orbital evolution is strongly influenced by the gravitational perturbations of the two planets. The perihelion of comet P/GS is located around 1.1 AU and the orbit has an high eccentricity (0.665), so the comet is perturbed by Jupiter only close to the aphelion. A different balancing between P-R drag and gravitational scattering has to be expected for dust particles released by the two comets. Under this point of view, the choice of these comets allows us to cover a wide range of dynamical behaviours distinctive of cometary dust particles.

In Fig. 1 we visualize the spatial distribution of dust grains of two different sizes (10 µ m - filled triangles, 100 µ m - empty squares) obtained numerically integrating the orbits of [FORMULA] 700 particles for a timespan of 5 103 years. The initial orbital elements of the grains have been computed with the Fulle's model assuming an isotropical emission geometry. The positions of the particles are projected on the x-y plane, i.e. the ecliptic plane of J2000. The spreading of the grains around the comet orbit is due to repeated encounters with both Jupiter and, to less extent, Saturn. The contribution of Saturn can be inferred from Fig. 2a where the orbital elements of the grains are plotted together with the Tisserand invariant of the comet at different evolutionary times (1 103, 5 103 and 2 104 years). Jupiter is the main perturbing planet and in fact initially the particles shift along the invariant line, but also Saturn influences the orbits of the grains scattering the particles around the invariant. The amount of spreading is increased significantly after 2 104 while about 7% of the initial grains have been ejected out of the Solar System. This percentage increases to 17% after 5 104. We conclude that only 1% of dust grains emitted by P/SW1 reaches the Sun distance of 1 AU and contributes to the inner zodiacal cloud. About 80% of the grains are spread in a wide region outside 4 AU and are slowly ejected from the Solar System if they are not destroyed on a shorter timescale by collisions. In Fig. 2b the orbital evolution of grains from P/SW1 is computed including in the P-R drag the non-radial term. In this case the ratio between the corpuscular and radiation drag forces is about 1 and, as a consequence, a faster orbital decay is expected. Comparing Fig. 2a with Fig. 2b (only 300 particles have been integrated in the simulation of Fig. 2b against the 800 in Fig. 2a), we see that when the drag is stronger ( Fig. 2b) few particles detach from the Tisserand invariant line, and so from the Jupiter gravitational influence, and drift toward the Sun under P-R drag. However the percentage of grains which escape from the Jupiter influence and evolve only under P-R drag is very small and we can assume that the P/SW1 cometary grains are not sensitive to the drag model.

[FIGURE] Fig. 1. Spatial distribution of dust grains emitted by comet P/SW1 at the 1989 perihelion passage. The orbits of the particles have been integrated for 5 103 years after the ejection. Filled triangles represent 10 µ m size (diameter) grains, empty squares 100 µ m size grains
[FIGURE] Fig. 2a. Orbital distribution of grains in the a-e plane compared with the Tisserand invariant curve, at different evolutionary times (1 103, 5 103, and 2 104 years), for P/SW1.
[FIGURE] Fig. 2b. The same as in Fig. 2a, but including in the drag term the non-radial component of the wind pressure.

In order to verify that the spatial evolution of the grains is not conditioned by a) the particular geometry of ejection from the cometary nucleus b) by the position of the comet in its orbit at the moment of emission, we have performed two additional numerical simulations. In the first simulation we have assumed a conical ejection geometry for the grains: the opening of the cone is [FORMULA] (half opening [FORMULA]) and the symmetry axis always pointing toward the Sun. In the second simulation we integrate the trajectories of grains ejected at different locations along the comet orbit (t=-782.6, -640.1, -499.1, -359.6, -220.9, -82.7 days before perihelion passage). In Fig. 3 we compare the distribution of the perihelia at t=2 104 years for these different sets of particles. No significant differences can be detected among the three histograms.

[FIGURE] Fig. 3. Perihelia distribution at t-2 104 years for three different sets of dust particles emitted by P/SW1 with different ejection geometry and orbital configurations: isotropical ejection (continuous line), conical ejection (dotted line), multiple ejection (dashed line).

This last result confirms that the evolution of dust grains released by P/SW1 is dominated by chaotic phenomena associated with repeated close approaches both to Jupiter and Saturn. The grain orbits are characterized by random impulsive changes of the orbital elements in correspondence to each close encounter and, as a consequence, they lose memory of their initial orbital parameters on a short timescale ([FORMULA] years). Due to the high frequency of close encounters, the P-R drag is not a relevant pertubing force for dust particles since in most of the cases it does not have enough time to accumulate relevant orbital changes between two subsequent encounters. Only few particles, in particular the smaller ones, avoid close encounters with both planets for an interval of time sufficiently long to be driven by P-R drag toward the Sun.

Different is the behaviour of dust grains released by comet P/GS. We used again the Fulle's model to generate starting conditions for dust grain trajectories, assuming a spherical distribution of the ejection velocities. We had also to consider a single location of the comets in its orbit due to the lack of observative data to be used in the Fulle's model. 20µ m size particles are strongly influenced by P-R drag during the perihelion passage located very close to the Sun and they evolve quite rapidly toward the Sun on a timescale of the order of few thousand years. The spatial distributions displayed in Fig. 4 at t=1 103, t=5 103 and t=2 104 years, respectively, show a progressive shrinking of the grain cloud toward the Sun. In particular after t=2 104 years about 47% of the grains have reached a perihelion distance lower than 0.3 AU from the Sun and have to be considered as completely vaporized. The gravitational perturbations by Jupiter are not very effective compared to the P-R drag as visualized in Fig. 5. Only few grains follow the invariant line while most of the grains exhibit a progressive damping of the eccentricity and decrease of the semimajor axis. When we include the non-radial term in the drag force, the drift of grains toward the Sun is faster and after t=1.2 104 years all the particles in our simulation are fallen into the Sun.

[FIGURE] Fig. 4. Spatial distribution at different evolutionary (1 103, 5 103 and 2 104 years) of 20µm dust grains emitted by the P/GS at 1992 perihelion passage.
[FIGURE] Fig. 5. Orbital distributions at different times (1 103, 5 103 and 2 104 years) of 20 µ m grains emitted by P/GS.

The corresponding spatial and orbital distributions for 200µm size particles are shown in Figs. 6-7a,b. A higher correlation is observed in Fig. 7a, b (in the simulation of Fig. 7b the non-radial term in the drag force has been included) with the invariant line due to the decreased strength of the non-gravitational forces. Jupiter is very effective in scattering the particles when they are at the aphelion and also in simulation b, when the drag force is stronger, the orbital evolution of P/GS grains is dominated by the gravity of the planet. The spatial distribution, compared to the one for 20µm size particles, is more diffuse and less concentrated around the Sun. Only 6% of the grains fall into the Sun after 2 104 years. The initial eccentricity damping trend is reversed by Jupiter perturbations which push particles back to higher eccentricities and larger semimajor axes following the invariant line. The different behavior of 20 and 200 µm particles is evidenced in Fig. 8 that shows the distributions of their perihelia. The histogram for 20µm particles has a maximum around 0.5 AU induced by the high P-R decay rate while the 200µm histogram is uniformly distributed with a peak around 4.5 AU produced by the Jupiter perturbations. The results of these simulations suggest that there is a slow change in the behavior of dust particles emitted by P/GS depending on their size. Particles smaller than 20 µm drift toward the Sun on short timescales ([FORMULA] 2 104 years or less, depending on their size) without being affected by Jupiter. As the size of the dust particles increase, a larger percentage of them are scattered along the Tisserand invariant line by close encounters with Jupiter and contribute to the zodiacal dust cloud for a longer time (on highly eccentric orbits). Their final fate is ejection from the Solar System on a timescale of the order of [FORMULA] 2 105 years, having an orbital evolution similar to that of short period comets.

[FIGURE] Fig. 6. Same as Fig. 4 for 200µm size grains
[FIGURE] Fig. 7a. Same as Fig. 5 for 200µm size grains
[FIGURE] Fig. 7b. Same as Fig. 7a but including in the drag force the non-radial term
[FIGURE] Fig. 8. Perihelia distributions at t=2 104 years for 20µm particles (continuous line) and 200µm particles (dashed line)
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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