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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
On figure of a sublimating cometary nucleus
Yu. D. Medvedev
Institute of Applied Astronomy, St. Petersburg, Russia
The problem of the change of the figure of a cometary nucleus when its matter
is sublimating from the nucleus surface is considered. The sublimation intensity
is taken to be proportional to the solar ray energy falling onto the nucleus surface
facets in the unit of time. For the case when the axis of rotation of the nucleus is
normal to the plane of the cometary orbit and the initial figure of the nucleus has
a rotational symmetry the averaged equations in partial derivatives describing
the changes of the form of the nucleus have been derived both for the cartesian
and the polar coordinates. In the cartesian coordinates with x--y plane laying in
the plane of the cometary orbit this equation is
@x(z; t)
@t
= \GammaA: (1)
For the polar coordinates it looks as
@R(ff; t)
@t
= \GammaA
`
cos ff + sin ff
1
R
@R
@t
'
: (2)
In (1) and (2) A is the constant which depends on the properties of the
sublimated material and the cometary orbital elements, x(z,t) is the distance of
the point of the nucleus surface from the axis of the nucleus rotation (z­axis),
R and ff are the cometocentric distance and latitude of the point of the nucleus
surface, correspondingly. The equation (1) has the unique solution satisfying the
boundary conditions,
x(z; t) = x 0
(z) \Gamma A(t \Gamma t 0
); (3)
where x 0
(z) is the function representing the figure of the initial meridian of the
nucleus. Any initial figures of the cometary nucleus with rotational symmetry are
admissible in (3) if for any permissible z exists only one (there must not be the
selfshadowing effects).
Equation (2) is convenient to use when the initial nucleus has the spherical
figure.
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Its solution for this case is
R =
q
R 2
0 \Gamma A 2 (t \Gamma t 0 ) sin 2
ff \Gamma A(t \Gamma t 0 ) cos ff: (4)
Using (3), (4) the estimates have been obtained for the time of life of the
nuclei of comets moving in different orbits about the Sun.
The detailed numerical model describing the sublimation of matter from the
cometary nucleus having general figure and rotation is developed and realized as
well [1]. The numerical simulation of evolution of the dynamic state and some
physical parameters has been done for the sublimating cometary nucleus. The
evolution of the nucleus figure and rotational parameters is considered taking
into account the dust component in the cometary matter.
The cometary nucleus is approximated by cones with the top in the centre
of inertia of nucleus and bases on the cometary surface. Area, S, vector of ori­
entation, n, and thickness of dust layer, fl, are determined for each base. Two
frames of reference have been used for orientation of these bases in space: the
nonrotating Kenig's frame, OXYZ, with the origin in the center of inertia of the
nucleus and rotating one, oxyz, rigidly connected with the nucleus and with the
axes directed along the main axes of inertia. To connect these frames the matrix
of rotation P is used.
Due to sublimation the nucleus loses matter mainly in the equatorial areas.
When the polar areas of the nucleus contain a dense dust crust the equatorial
ravine arises as a consequence of sublimation. The wideness of this ravine depends
on the dust component distribution in the nucleus. The more dust is maintained
by the nucleus, the narrower ravine appears. The nucleus gets the dumb--bell
shape [2]. During the evolution this ravine becomes more and more deep and at
last the nucleus disintegrates being broken along the ravine.
This research has been supported by the RFFI grant 01­02­17078 of Russian
of Academy of Sciences.
References
1. Medvedev Yu. D. On formation of elongated cometary nuclei. In: Proceedings
of the second international workshop on positional astronomy and celestial
machanics (eds. A. L. Garsia at all), Univesity of Valencia, Spain, 1993,
63--80.
2. Medvedev Yu. D. Evolution of form sublimating cometary nucleus. IAA
Trans., 2001, 6, 318--325 (in Russian).
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