4. Application to the Deep Impact experiment
4.1. Comet model
To simplify the discussion of this problem we use the Deep Impact nominal impact event as a test bench for our computations. This represents an extreme case, since the orbit of comet Tempel 1 is highly eccentric and the event occurs at comet perihelion. The basic measured and assumed parameters for comet Tempel 1 are given in Table 1.
4.2. Ejection conditions from the surface
To model the impact scenario on the comet surface we constrain a number of the initial conditions. First, we specify that the impact occurs when the comet is at perihelion. The nominal impact site is at a latitude of -22 degrees (measured from the comet orbital plane) and a longitude of 116 degrees (measured from the anti-sun direction). The ejected particles are assumed to have the same density as the comet, and to have a spherical shape.
We model the initial conditions of the ejected particles as elliptic orbits with sub-escape speeds. In general, all particles that initially have greater than escape speed will not return or orbit the comet (some exceptions to this could occur due to the interaction of the solar tide with the particle). If no additional perturbations act on these sub-escape orbits they will, of course, re-impact on the surface after one orbit period. We assume that the ejection occurs at an angle of with respect to vertical, which is a standard result derived from impact experiments. Note that this ejection geometry assumes that the shape of the comet is spherical. A small comet like Tempel 1 may have a very irregular shape that may invalidate the spherical approximation, however, the range of choices for the ejection geometry is in this case so wide that it would be impossible to consider all the alternatives. Our simulations, on the other hand, are intended to give a general view of the problem. The specific computations can only be done when the shape of the body and its rotation state is known in detail.
4.3. Global computations for the Deep Impact cratering event
The advantage of having a working analytical theory for computing the dust particle evolution resides in the capability of fast computations and a better understanding of how the physical parameters of the problem influence the motion. We can easily calculate the time evolution of a large number of orbits and derive the global behaviour of the dust cloud lofted by an impact. This is particularly useful if we intend to explore in detail the 3-dimensional parameter space (ejection velocity, and ejection angle ) that characterizes the orbits of dust particles ejected at a given impact site.
Fig. 10 is a 3-dimensional histogram where we plot the lifetimes of 1 cm particles as a function of their ejection speeds and ejection angle . The sharp cut in the lifetimes for velocities larger than 2.21 m/s is related to the hyperbolic escape of faster particles. In Fig. 11 we plot the lifetimes as a function of size and ejection velocity. The 3-dimensional surface of Fig. 12 envelopes the capture regions in the x x space. From the three figures we see that larger size grains have to be ejected at larger velocities to get captured in temporary orbits. As a consequence, after an impact large particles will, on average, have higher semimajor axes and, for most of the time, will orbit farther from the nucleus than smaller particles. The density of the dust cloud around the comet will have a gradient in the size of the grains, with smaller particles filling the zones close to the comet surface. From Fig. 11 we also notice that smaller particles have short lifetimes as compared to the larger grains. The outer layers of the dust envelope that surrounds the nucleus after the impact will then survive longer and will be made of larger particles. From Fig. 12 we can also notice that all the ejection directions lead to trapping for larger grain sizes while smaller particles are trapped only if is restricted around and .
These useful statistical results are obtained running a simple numerical code based on RPA that gives the results in less than 5 min. of CPU time.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000