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Astron. Astrophys. 356, 747-756 (2000)

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2. Mechanics of temporary orbital capture

2.1. Orbit mechanics of ejected particles

We assume that a particle is ejected from the surface of a comet at an arbitrary latitude and longitude ([FORMULA] and [FORMULA] respectively), specified in the fixed comet-sun coordinate frame: 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. The semi-major axis of the ejected particle is a free parameter and is assumed to be positive, corresponding to a bound orbit, and can be directly related to the ejection speed of the particle.

The initial position vector of the particle can then be specified as:

[EQUATION]

in terms of the latitude and longitude of the ejection site. The initial velocity vector is determined by two additional angles: the angle [FORMULA] giving the inclination of the velocity respect to the vertical, and the angle [FORMULA] giving the orientation of the velocity vector tangent to the comet surface. Without any additional perturbations these orbits will fall back onto the comet surface as their initial periapsis lies beneath the surface.

We use these initial conditions for ejecta particles with an analytical approach developed by Mignard & Hénon (1984) and Richter & Keller (1995) for averaged particle motion in the presence of solar radiation pressure. This formulation allows us to explicitly compute the orbital elements of a particle at any future date. Due to the SRP force, orbits that initially have periapsis under the surface of the comet can transition into orbits with periapsis radius above the comet surface - allowing the particles to survive for more than one orbit about the comet. Since these solutions are periodic, the particle will eventually re-impact on the comet surface, but the time to re-impact may be many months.

Richter & Keller (1995) develop a very simple and concise form for the general solution of particle evolution under averaged SRP force. This solution was previously investigated in a slightly different form by Mignard & Hénon (1984). These general solutions incorporate the effect of the comet moving on an elliptic orbit about the sun - accommodating both the change in SRP force magnitude and direction. The major drawback is that this solution ignores the perturbations that the particle feels due to the comet gravity field, comet outgassing and solar tide. The first order effect of these perturbations are discussed later. The solution also does not provide any prediction of when the particle lies far enough from the central body to be stripped away by the combination of solar tide and SRP forces. Approximate limits for this limiting semi-major axis are available, however (Hamilton & Burns 1992).

There is only one free parameter that needs to be specified for the solution, combining the comet semi-major axis and eccentricity, the strength of the SRP force, the radius of the ejected particles, the mass of the sun, the mass of the comet and the particle semi-major axis. This parameter is invariant with respect to the comet distance from the sun. Following Mignard & Hénon (1984) we express this parameter as an angle [FORMULA]:

[EQUATION]

where [FORMULA] is the SRP strength parameter, B is the mass to projected area ratio of a sphere of radius [FORMULA], a is the particle semi-major axis about the comet, P is the comet orbit parameter about the sun, µ is the comet mass parameter, and [FORMULA] is the sun's mass parameter. Formula 4 gives this parameter for particles ejected from Tempel 1 with a semi-major axis a (in kilometers) and a particle radius [FORMULA] (in centimeters), see Table 1. The angle [FORMULA] parameterizes the relative strength of the SRP, for a small dust particle the angle approaches [FORMULA] while for a large particle the angle approaches 0.


[TABLE]

Table 1. Assumed parameters for comet Tempel 1


The form of the solution given by Richter & Keller (1995) describes the evolution of the orbit angular momentum vector and apocenter vector as a function of comet true anomaly [FORMULA] about the sun. The averaged solution is periodic in true anomaly with period [FORMULA], thus the averaged solutions we find will repeat themselves after the comet has moved through this angle. Thus when the comet is in an elliptic orbit the period of motion is shorter around periapsis than it is around apoapsis. We note that for stronger perturbations (i.e., for smaller particles) the period of these averaged solutions will decrease. In the limit for a large particle ([FORMULA]) we see that the period of the averaged motion approaches the comet orbit period about the sun.

To compute the solution we derive the initial angular momentum and apocenter vectors from the particle position and velocity. Then we compute the values of the angular momentum and apocenter vector as functions of time using the theory. We note that the semi-major axis of each particle orbit is assumed to be constant on average (Mignard & Hénon 1984). We then have an explicit prediction for the evolution of the particle orbit as a function of the SRP parameter [FORMULA] and its ejection conditions.

2.2. Capture and escape conditions

Using the analytical solution we generate conditions for a given particle to not re-impact after one orbit period, and hence be "lofted" into a longer term orbit about the comet. First, we choose an impact site (angles [FORMULA] and [FORMULA]), an initial semi-major axis (a) for the orbit, and an initial pair of angles [FORMULA] and [FORMULA] for the ejection direction. Choosing the semi-major axis is equivalent to choosing the ejection speed [FORMULA] at the surface of the comet:

[EQUATION]

where [FORMULA] is the mean radius of the comet.

Given the orbit semi-major axis (or the ejection speed on the surface) we have defined the orbit period for the particle, and can then derive a constraint on the orbit eccentricity after one period so that the particle will not re-impact on the comet surface. If we specify the orbit period as [FORMULA], and the eccentricity after one orbit period as [FORMULA], then the constraint to not impact on the comet surface after the initial orbit is:

[EQUATION]

which is derived by making the orbit periapsis lie above the surface of the comet.

If the radius of the ejected particle, [FORMULA], is specified we can compute the SRP parameter [FORMULA] and explicitly compute [FORMULA]. Note that to perform this computation the change in comet true anomaly over a time period T must be computed, for which we must solve Kepler's equation for the comet's motion about the sun. Once a particle is captured into orbit it can complete a number of orbits before re-impacting on the nucleus. Since eccentricity varies significantly over one particle orbit period, at any periapsis passage it will have different values. If the eccentricity satisfies Eq. 6 at periapsis the particle will perform another full orbit, and so on. The period of the eccentricity evolution depends on the size of the particle, larger grains will have longer eccentricity cycles. Since re-impact on the surface will occur when the eccentricity returns to its starting value at the end of the cycle, larger particles may have longer capture periods. This is mitigated, however, by having a slower change in eccentricity, meaning that there is a longer "window" of opportunity for impact to occur whenever eccentricity returns to its initial value.

With this analytical approach, we can easily compute, for a given particle radius and ejection geometry, the interval of ejection velocities for temporary orbital trapping and their relative lifetimes. There is an upper bound to the ejection velocities against escape, which is generally less than escape speed proper. This is evaluated accounting for the effect of solar tide and solar radiation pressure with the formula:

[EQUATION]

(Scheeres & Marzari submitted) where a is the semimajor axis (km), and d is the comet-sun distance.

2.3. Averaged solution for a special case

To provide a simple, yet concrete, example of the solutions we use for the evolution of the particles under the SRP force we present the closed-form formula for the angular momentum and apocenter vector of a particle initially ejected from the comet on a rectilinear orbit (i.e., with [FORMULA] and [FORMULA]). The form of the solution given by Richter & Keller (1995) describes the evolution of the orbit angular momentum [FORMULA] and apocenter vector [FORMULA] (where [FORMULA]) as a function of comet true anomaly [FORMULA] about the sun. Instead of the comet true anomaly, we use a scaled parameter [FORMULA]. The solution is then periodic in H with period [FORMULA], thus the averaged solutions we find will repeat themselves after the comet has moved through a true anomaly of [FORMULA]. Evaluating the constants for our special initial conditions:

[EQUATION]

results in the closed form solution:

[EQUATION]

We then have an explicit prediction for the evolution of the particle orbit as a function of the SRP parameter [FORMULA] and the ejection latitude and longitude. A particular solution for an ejected particle at Tempel 1 is shown in Fig. 1.

[FIGURE] Fig. 1. Single particle orbit trajectory, trapped around the comet for 3.8 years; 5 cm particle ejected with a speed of 2.23 m/s.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000
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