The Maximum Entropy Distribution of Orbiting
Asteroids Forms a Belt
Geoffrey Schiebinger
November 5, 2010
Introduction
The Asteroid Belt is a collection of planetesimals lying between the orbits of Mars and Jupiter. It is comprised
of rocky material that never formed into a planet–perhaps because of gravitational perturbations by the gas
giant Jupiter. It is believed that the belt was originally populated by larger planetoids, but that the collisions
between them were too violent to lead to accretion. Instead these impacts caused the planetesimals to shatter
into the rocky debris that comprises today’s asteroid belt. Indeed, there are groups of asteroids with similar
orbits that are believed to each be the remanants of the break-up a of single large parent body [1-2].
1
Figure 1: The Asteroid Belt [7]. The Main belt is shown in white, and other groups of asteroids are
shown in green and orange. The Trojans and Greeks share Jupiter’s orbit and cluster around the Lagrangian
points of stability 60 degrees ahead and behind Jupiter. They are evidence of Jupiter’s gravitational effects
on the asteroids.
Here we focus attention on the distribution of asteroid orbital radii and show how, under some simplifying
assumptions, the equilibrium distribution of asteroids is in a belt around Sun. We assume that the total
mass, energy and angular momentum are each conserved. We assume the asteroids orbit in the plane of the
ecliptic so that their angular momenta add, and that each asteroid is in uniform circular motion. None of
these assumptions are perfect: gravitational perturbations by Jupiter and other planets can add energy and
angular momentum to the asteroid population, for example. However, the mean eccentricity of the asteroid
belt is on the order of .1 and the large majority of orbits have an inclination of less than 10 degrees [3], so
these assumptions are a reasonable approximation of the true physical setup. These assumptions lead to a
simple physical model that gives a solution with the desired behavior: the asteroids form a belt around the
Sun.
Construction of Maximum Entropy Problem
Consider an asteroid of mass m in orbit about the sun. The asteroid’s total energy is given by the sum of
its potential energy V (r) and its kinetic energy K(r) = 12 mv 2 . We define the energy of a particle at rest
at r = ∞ to be zero. The gravitational potential V (r) is equal to the amount of work done by gravity in
bringing the particle from radius r out to ∞:
V (r)
= Wgrav =
ˆ∞
r
F · ds =
ˆ∞
r
−
GM m
dr
r2
�∞
GM m ��
GM m
=−
r �
r
=
r
m
where F = − GM
is the force exerted by gravity. In a circular orbit the centripital acceleration must be
r2
balanced by the force of gravity:
�
mv 2
GmM
GM
=
⇐⇒ v(r) =
2
r
r
r
√
The angular momentum at radius r is therefore L(r) = mvr = m GM r and the kinetic energy is K(r) =
GM m
2r , which is half the magnitude of the potential energy. The total energy of a particle in orbit at radius
m
r is therefore given byE(r) = − GM
2r .
Now consider an ensemble of N asteroids of mass m. Let there be n1 particles at radius r1 , n2 particles
at radius r2 , . . . , and nb particles at radius rb , where 0 < r1 < r2 < . . . < rb < ∞. Conservation of mass
tells us that
b
�
N=
ni
(1)
i=1
conservation of energy tells us that
E=
b
�
i=1
n i Ei = −
b
�
GM m
2ri
(2)
�
ni m GM ri
(3)
i=1
ni
and conservation of angular momentum requires that
L=
b
�
i=1
ni Li =
b
�
i=1
2
We seek the ni that maximize
�
N
n1 , n2 , . . . , nb
�
N!
n1 ! . . . nb !
=
(4)
which is the number of ways rearranging the N particles such that n1 are at r1 , n2 are at r2 , ... , and nb are
at rb . Since each such arrangement is equally likely, the assignment
� �ofn n1 , . . . , nb that maximizes [4] will be
the equilibrium configuration. By Sterling’s approximation, n! ≈ ne , so we have
� N �N
�n
b �
�
N!
N i
≈ � b e � n � ni =
= eN H(p1 ,p2 ,...,pb )
i
n1 ! . . . nb !
n
i
i=1
i=1
e
where pi = nNi and H(p1 , . . . , pb ) is the information entropy of the distribution pi . Therefore, for fixed N ,
maximizing (4) is equivalent to maximizing
H
�n
1
N
b
�
nb �
=−
pi ln pi
N
i=1
,...
(5)
as shown in [4]. By Theorem 12.1.1 in [4], the distribution p∗i that maximizes (5) subject to the constraints
(1,2,3) is given by
√
λ
+ν ri
eλ̃Ei +ν̃Li
e ri
∗
pi = �b
= �b
(6)
√
λ
λ̃Ei +ν̃Li
r +ν r
i=1 e
i=1 e
where λ and ν are chosen to satisfy the constraints (1,2,3).
Let’s choose ri so that the spacing is uniformly � = |ri − ri+1 | ∀i, and pass to the limit b → ∞. The
limit of the normalizing sum in (6) exists if and only if ν < 0. To see this, note that if λ ≥ 0, then
b
�
λ
e ri
+ν
√
ri
i=1
⇒ lim
b→∞
b
�
λ
e ri
+ν
√
≤ e r1
b
�
ri
λ
λ
eν
√
∀b
i=1
≤ e r1 lim
b→∞
i=1
ri
b
�
eν
√
ri
i=1
<∞
where the last sum converges by the integral comparison test. If λ < 0, then
lim
b→∞
b
�
λ
e ri
+ν
√
ri
b
�
≤ lim
b→∞
i=1
− rλ +ν
e
i
√
i=1
and the sum also converges.
For fixed �, our problem is (���)
ri
<∞
∞
�
pi ln pi
i=1
∞
�
pi
maximize
−
subject to
α=
β=
i=1
∞
�
ri
√
p i ri
i=1
1=
∞
�
pi
i=1
2E
where we have defined α = − N GM
m and β =
L
√
.
N m GM
λ
p∗i
e ri
= �∞
i=1
3
And the solution is
+ν
e
√
ri
√
λ
r +ν r
Let’s now pass to the limit � → 0 of infinitesimal spacing between√the radial buckets. We determined above
λ
that we must have ν < 0. Note that the function f (r) = Ae r +ν r is integrable over [0, ∞) if and only if
λ < 0 and ν < 0, since
ˆδ
λ
e r dr < ∞ ⇐⇒ λ < 0
∀δ > 0
0
So f is a probability density on [0, ∞) for λ < 0, ν < 0 and A chosen appropriately. If X � ∼ p�i , and X ∼ f ,
then X � is a quantized version of X. Since f (r) is Reimann integrable, then H(X � )+ln � → h(f ) = H(X),
as � → 0 by Theorem 8.3.1 in [4]. For each �, maximizing the objective in (���) is equivalent to maximizing
H(X � ) + ln �. Therefore, in the limit that � → 0, our problem is equivalent to maximizing the differential
entropy subject to the energy, mass and angular momentum constraints:
maximize −
subject to
ˆ∞
p(r) ln p(r)dr
(7)
0
α=
ˆ∞
β=
ˆ∞
1=
ˆ∞
p(r)
dr
r
0
√
p(r) rdr
0
p(r)dr
0
which by Theorem 12.1.1 in [4] has the solution
λ
e r +ν
p (r) = f (r) = ´ ∞
∗
0
e
√
r
√
λ
r +ν r
dr
(8)
where λ < 0 and ν < 0 are chosen to satisfy the constraints which depend on the total angular momentum
L and total energy E of the asteroid belt. This does indeed give a peaked radial density: the asteroids will
most likely cluster in a band around the Sun.
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Discussion
Figure 2: Maximum entropy radial density for various angular momenta at fixed energy.
In Figure 2 you can see the maximum entropy radial density p∗ (r) plotted for various values of L. The
value of E was chosen to correspond to an orbital radius of 2.7 astronomical units (AU). In particular, I
chose R0 = 2.7AU and
GM N m
1
⇒α=
2R0
R0
�
�
L = −N m GM R0 (1 + η) ⇒ β = (1 + η) R0
E
= −
for various values of η > 0. Note that as η increases, the distribution spreads out, and as η → 0, the
distribution of orbital radii√becomes sharply peaked near r = R0 . This is because the angular momentum
constraint approaches β = R0 which corresponds to the angular momentum of a single particle in a perfectly
circular orbit for α = R10 . This is the minimum value of L possible for energy E. Any lower, and there is
√
no solution: the asteroids would spiral in towards the sun. And for the critical values α = R10 , β = R0 ,
√
there cannot be any spread in the distribution p(r): it is a sharp peak at r = R0 . Otherwise, since r is
1
concave and r is convex, the angular momentum would be lower and the energy higher and there would be
no solution.
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Figure 3: Empirical Distribution (adapted from [5])
The shaded region in Figure 3 shows the observed distribution of orbit radii for a subpopulation of the
asteroid belt (those with diameter >50km). The solid line shows the distribution obtained obtained from
a 4Gyr dynamical simulation. Note that the empirical distribution has the same shape as p∗ (r) : there
is a quick initial rise followed by a relatively long tail. There is some obvious additional structure here,
though, that our simple model could not predict. The low-density gaps near 2.5, 2.8 and 3.3 AU are called
the Kirkwood Gaps and are likely caused by gravitational perturbations by Jupiter. They correspond to
orbital periods that are an integral ratio of Jupiter’s orbital period–which is related to the semimajor axis
by Kepler’s law. For example, an asteroid in the 2:1 gap will be directly aligned with Jupiter every other
time it orbits the Sun, and this resonance will disrupt its orbit.
Lastly, note that the width of the empirical distribution is on the order of 1-2 AU. By our earlier
observation that the width of the peak in p∗ (r) depends on the value of η, it might be possible to characterize
the eccentricity of the orbits of the original planetesimals that broke up to form the asteroid belt.
Summary
The maximum entropy principle is a type of statistical inference that gives the least biased estimate consistent
with the known information. In the words of E. T. Jaynes, it is “maximally noncommital with regard to
missing information” [6]. Here we use this technique to calculate the distribution of orbital radii of an
ensemble of asteroids in orbit around the Sun. We show how the maximal entropy problem (7) is constructed
by maximizing the number of configurations the asteroids can take over a discrete set of radii and then passing
to the appropriate limit. We solve (7) to get the maximum entropy density
λ
e r +ν
p (r) = f (r) = ´ ∞
∗
0
√
r
√ dr
λ
e r +ν r
and find that it is one with a sharp peak at a particular value of r. The asteroids form a well-defined belt
around the Sun.
Bibliography
1. Michel, P., Benz, W., Richardson, D. C., Disruption of fragmented parent bodies as the origin of
asteroid families, Nature. 2003.
2. Nesvorny, D., Bottke, W. F., Dones, L., & Levison, H. F., The recent breakup of an asteroid in the
main-belt region, Nature. 2002.
3. Robbins, S., McDonald, D., “Asteroids” http://burro.astr.cwru.edu/stu/advanced/asteroid.html
4. Cover, T., Thomas, J., Elements of Information Theory, Wiley. 2006.
5. Minton, D., Malhotra R., A record of planet migration in the main asteroid belt, Nature. 2009.
6. Jaynes, E. T., Information Theory and Statistical Mechanics, Phys. Rev., 1957
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