The Motion of an Asteroid and the Kirkwood Gaps
A plot of the distribution of the distances of asteroids from the Sun reveals the existence of the Kirkwood gaps. These are regions of the asteroid belt where the density of asteroids is significantly lower than average. The explanation for these gaps is that an asteroid in a gap finds itself in resonance with the motion of Jupiter about the Sun. This means that in the time that Jupiter has completed a certain integer number of orbits around the Sun the asteroid has also completed an integer number of orbits around the Sun.
We can investigate the effects of resonances by modeling the orbit of a point mass (the asteroid) under the combined gravitational pull of the Sun and Jupiter. To make the problem simpler, we will assume that the Sun and Jupiter have circular orbits about the center of mass of the Sun-Jupiter system, and that the orbit of the asteroid lies in the same plane as the orbits of Jupiter and the Sun.
We will
investigate the motion in an inertial frame with origin at the center of mass
of the system. Let r be the position of the asteroid, r1
and r2 the positions of the Sun and Jupiter. Let
be the angle that the Sun-Jupiter line makes with the x-axis of our
inertial frame.
Also let M1 and M2 be the masses of the Sun and Jupiter respectively.
From the definition of center of mass
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The equation of motion of the asteroid is
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In terms of co-ordinates, we have
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where
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We will choose our units for mass, length and time in such a way as to reduce the number of parameters in the problem. For the unit of length we choose S, the distance between the Sun and Jupiter, and for the unit of mass we will choose M, the sum of the masses of the Sun and Jupiter.
From the definition of the center of mass, we find
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The angular velocity of the orbital motion of Jupiter and the Sun about the center of mass is
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Choosing
as the unit of time, we obtain
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where
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Also with this time unit, the angular position of the Sun and Jupiter is
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The mass of Jupiter is much less than the mass of the Sun. It is therefore convenient to introduce the parameter
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so that
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The
solution of our equations for the motion of the asteroid depend only on
and the initial position and initial velocity of the asteroid. For Jupiter,
we take
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We need to follow many orbits of the asteroid to see its long-term behavior. To get accurate results, we use Verlet’s method.
Let’s
start by looking for the 2:1 resonance i.e. the asteroid orbits the Sun twice
during the time Jupiter orbits once. The angular velocity of the asteroid needs
to be twice that of Jupiter. In our units, the angular velocity needs to be
2. So we place the asteroid at a distance
(= 0.6300) in our units from the Sun and give it a transverse velocity
(= 1.260). The final consideration is where to place the asteroid relative
to the Sun-Jupiter line. To minimize the gravitational pull of Jupiter at the
initial position, lets put the asteroid on the other side of the Sun. The initial
conditions are
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Here are the results for this simulation. The plot below shows how the distance of the asteroid from the Sun varies with time.
What happens is that the orbit starts out nearly circular but, because of the relatively close passage to Jupiter every second orbit, the orbit gets elongated (i.e. it becomes elliptical). Since the Sun is very near the focus of the ellipse, its closest approach to the Sun (perihelion) is closer than in the original nearly circular orbit and its furthest distance from the Sun (aphelion) is correspondingly further away. If we plot a large number of orbits in the x-y plane, we find that they fill a ring shaped region
So instead of remaining at a fixed distance from the Sun in a simple circular orbit, the asteroid gets pulled around this annulus by Jupiter. Hence the probability of finding it in the gap is low.
For comparison
here are plots away from a resonance (the orbital angular velocity of the asteroid
is
):
Resonances also exist outside the orbit of Jupiter. Here is the 1:2 resonance:
Some scientists believe that such resonances played a role in planet formation in the early solar system. Once a large planet like Jupiter forms it can sweep material into regions between resonances. In these denser regions, other planets form. This is a possible explanation of the Titus-Bode relation.