3. Corrections to capture and lifetime computations for RPA
The above analysis was carried out making averaging and modeling assumptions which may be violated in the comet or asteroid environment. In this section we consider the possible effect these assumptions have on the basic capture mechanics described above. We first focus on the effect of the averaging assumption, then briefly consider the effects of the solar tide, the non-spherical gravity field, and comet outgassing.
3.1. Comparison between RPA and numerical integration
RPA is an approximate theory based on averaging and thus may have a limited range of validity (Richter & Keller 1995). In cratering events we are dealing with highly eccentric orbits, while the numerical tests performed by Richter and Keller were limited to eccentricities lower than 0.8. For the DI experiment, in particular, the impact on the surface of the comet occurs at the perihelium of the comet, where the true anomaly of the comet changes very rapidly. Before using RPA to study the range of ejection velocities for capture and the lifetimes of temporary orbits, we first tested the reliability of RPA by comparing its predictions with the results of direct numerical integration of dust particle orbits. For our system we used the Tempel 1 definitions given in Table 1.
The numerical integration of the orbits have been performed with the RADAU numerical integrator (Everhart 1985). A fixed inertial reference frame, with x-axis measured positive pointing away from the sun, z-axis positive along the comet's orbit angular momentum about the sun, and y-axis positive along the direction of comet travel have been adopted. In comparing the numerical results with the RPA predictions, we have taken into account that the reference system used in RPA is pulsating with the comet orbital motion. The appropriate trasformations were applied to the initial conditions to account for the difference in the reference systems.
In Fig. 2 we show the comparison between the RPA predictions (dotted line) and the numerical integration (continuous line) for a 1 cm dust particle. The eccentricity evolution predicted by the numerical simulation is well reproduced by RPA and the semimajor axis of the orbit is on average constant (as assumed in RPA) with small variations around the mean. Since the DI experiment occurs at the comet perihelium, the radiation pressure is relatively strong and even large particles in the cm-size range are affected. The dynamical evolution of a 5 cm dust particle is shown in Fig. 3: the difference between the evolution of the eccentricity computed numerically and calculated from RPA is mainly due to the oscillations in the semimajor axis of the orbit and by the solar tide. We can partly obviate this problem by modifing the RPA solution, introducing a mean semimajor axis computed from the numerical simulation. This is equivalent to modifying the modulus of the initial ejection velocity. As an example, in Fig. 3 we include the RPA solution (dashed line) where an averaged semimajor axis is used instead of that derived from the initial conditions of the numerical simulation. A better match between the analytical approximation and the numerical solution is obtained in this case.
If the difference between RPA and numerical simulation is always related to a change in semimajor axis due to the solar tide, then RPA still represents a good approximation on average. The solar tide will systematically modify the RPA predictions by shifting, for a given particle size and ejection site, the range of ejection velocities leading to temporary capture. This is shown in Fig. 4, the dashed line represents the RPA solution and the continuous line the numerical simulation. Above the upper curve, the particles hyperbolically escape while below the bottom curve the particles re-impact on the surface of the comet before completing one single orbit.
There are a few pathological exceptions to this picture. For particular initial conditions, the eccentricity evolves with a small initial bump, as shown in Fig. 5. The numerical integration shows that the eccentricity initially decreases but then rises back to high values on a short timescale. After this initial bump, the eccentricity slowly decreases to low values completing the cycle. RPA instead predicts that the particle will re-impact at the end of the short bump, since the eccentricity gets back to the initial high values and the pericenter falls inside the comet's radius. The difference between RPA and the numerical solution is still due to the solar tide. Fig. 5 shows that the numerical solution follows the evolution in eccentricity predicted by RPA at the beginning, but when the re-impact should occur, the semimajor axis has increased by the effect of the solar tide. The pericenter of the orbit remains larger than the comet radius and the temporary trapping lasts much longer since the grain completes an eccentricity cycle. The predictions of RPA in these cases are still reliable with regard to the interval of ejection velocities leading to temporary capture (Fig. 6). However, the lifetime estimates can be wrong by one order of magnitude. In Fig. 7 we compare the lifetime for a 5 cm particle emitted at different values of . We see that the region where the RPA and numerical predictions do not agree for the pathological behaviour described above, are limited. The effects of the solar tide can also be noted in Fig. 7: the lifetime as a function of from the numerical integration oscillates around the value predicted by RPA due to the change in the orbital energy. Depending on the orientation of the ejected particle (angle ) the solar tide changes its effect.
3.2. Solar tide
As noted above, the solar tide has an effect on the dynamics of ejected particles. We have included the major effect of this force by incorporating Eq. 7 into our analysis, which gives the upper limit on an ejected particle's semi-major axis before it is stripped from the comet.
As noted above there is also a systematic effect of solar tide. Based on results in the existing spacecraft dynamics literature (Yamakawa et al. 1993) we can show that the particles ejected in the first and third quadrants of longitude (between angles 0 and 90° or 180° and 270°) will have a decrease in orbit energy and angular momentum, while particles ejected in the second and fourth quadrant of longitude (between angles 90 and 180° or 270° and 360°) will suffer a slight increase in orbit energy and angular momentum due to the solar tide interaction. Thus, for a given ejection site this will systematically shift the capture conditions of a set of particles. This does not play too large a role for statistical studies as we have here, as this is just a uniform shift in the ejection velocity that leads to capture. Future studies that look at capture conditions over the entire surface should take this effect into account, however.
3.3. Gravity field perturbations
Whenever an ejected particle has a close passage to the comet (generally within 5 - 10 mean radii) there is the potential for the comet gravity field to alter the particle orbit - resulting in either an increase or decrease in the orbit energy and angular momentum, depending on the precise conditions at closest approach. The effect of these interactions have been studied previously (Scheeres et al. 1996, 1998; Scheeres 1999) - the results of those analyses being applied here. Tempel 1 may have a fairly elongated shape, which corresponds to a large gravity term and hence a greater efficiency in altering a particle orbit. It also has an assumed slow rotation period of 24 hours. This combination actually increases the effect of the comet nucleus gravity field on nearby orbits - meaning that the interaction may be important out to distances of 20 km. While the net effect of these interactions will average out over long time spans (i.e., over many close approaches), over finite numbers of interactions we often see strong transient changes in orbit energy and angular momentum. The short term effect of these interactions can be to either cause the particle to escape from the comet or cause the orbit to become more closely bound to the nucleus. The latter effect would also serve to effectively decrease the SRP strength parameter and hence diminish the effect of SRP on the orbit evolution. This would, in general, cause the orbit lifetime to increase significantly - barring any premature escape due to interactions with the gravity field.
An explicit example of this type of behaviour is shown in Figs. 8 and 9 which show a trajectory about Tempel 1 over 200 days, with all changes in the orbit being due only to interactions with a rotating non-spherical Tempel 1 shape. In this computation we modeled Tempel 1 as an ellipsoid with half-axes of - a 2:1:1 shape ratio with a mean radius of 3 km. We gave the ellipsoid a constant density of 1 g/cm and a 24 hour period uniform rotation about its maximum moment of inertia. The initial orbit has apoapsis at 500 km and periapsis at 15 km - mimicing a possible initial condition of a particle after SRP has raised its periapsis above the comet surface. We see that even at this distance - 5 mean radii away - the comet gravity field can have a significant effect on the particle orbit and will be an important effect to include in future, detailed analysis of crater ejecta dynamics.
3.4. Comet outgassing
The net effect of comet outgassing on a particle orbit is an issue currently under study. To properly model this effect requires the specification of a comet outgassing field - an endeavor that requires significant modeling assumptions and has few significant data points or measurements - especially for the field close to a comet. Nonetheless, plausible models for the outgassing at a comet exist and have been used to study particle and spacecraft dynamics (Fulle 1997; Scheeres et al 1998; Fulle 1999).
One particularly simple model specification assumes that the outgassing pressure varies as from the comet, acts in the radial direction, and varies continuously with solar phase angle, leading to an effective drag of the form:
where is equivalent to the mass parameter and combines the particle's mass to area ratio and the outgassing pressure, and is a function of the comet-sun distance (dropping to zero at some distance from the sun and rising to its maximum at perihelion), and is the phase angle of the particle as measured from the sub-solar point. This particular force field can be averaged over one particle orbit and the resulting equations can be analyzed (Scheeres et al. 1998). One interesting note, discussed in Scheeres & Marzari (1998), is that the resulting equations can be integrated in terms of the mean anomaly of the particle orbit - yielding a solution for the eccentricity of the particle orbit under outgassing pressure. If we assume that the initial orbit is rectilinear then we find the solution:
where in general for a larger particle. We see that the initial eccentricity is equal to 1, and that eventually, for mean anomaly M large enough, the eccentricity will become greater than 1 - indicating that such an outgassing field will eventually cause all particles to escape. An important consideration to note is that the eccentricity may initially decrease from unity as a function of its ejection site - allowing the comet outgassing to help capture the particle.
Competing models of comet outgassing have the gas emanating from isolated locations on the comet surface. For these models the outgassing pressure will be relatively large inside the jet, causing a particle that crosses its path to receive a strong, impulsive change to its orbit (Scheeres et al. 1998). However, the probability for such a crossing to occur will be fairly low in general, unless there exists some resonance between the particle motion and the inertial attitude of the comet.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000