796 Encyclopedia of the Solar System
the particle’s motion is always restricted to thex−yplane.
The existence of the Jacobi constant implies that the parti-
cle is not free to wander over the entire 4-D phase space,
but rather that its motion is restricted to the 3-D “surface”
defined by Eq. (41). This has an important consequence for
studying the evolution of orbits in the problem.
The usual method is to solve the equations of motion,
convertx,y,x ̇, and ̇yinto orbital elements such as semi-
major axis, eccentricity, longitude of periapse, and mean
longitude, and then plot the variation of these quantities
as a function of time. However, another method is to pro-
duce asurface of section, also called a Poincar ́e map. This
makes use of the fact that the orbit is always subject to Eq.
(41), whereCis determined by the initial position and ve-
locity. Therefore if any three of the four quantitiesx,y,x ̇,
andy ̇are known, the fourth can always be determined by
solving Eq. (41). One common surface of section that can
be obtained for the planar circular restricted three-body
problem is a plot of values ofxandx ̇whenevery=0 andy ̇
is positive. The actual value ofy ̇can always be determined
uniquely from Eq. (41), and so the two-dimensional (x,x ̇)
plot implicitly contains all the information about the parti-
cle’s location in the four-dimensional phase space. Although
surfaces of section make it more difficult to study the evo-
lution of the orbital elements, they have the advantage of
revealing the characteristic motion of the particle (regular
or chaotic) and a number of orbits can be displayed on the
same diagram.
As an illustration of the different types of orbits that can
arise, the results of integrating a number of orbits using a
massm 2 /(ml+m 2 )= 10 −^3 and Jacobi constantC=3.07 are
described next. In each case, the particle was started with
the initial longitude of periapse 0 =0 and initial mean
longitudeλ 0 =0. This corresponds tox ̇=0 andy=0.
Since the chosen mass ratio is comparable to that of the
Sun-Jupiter system, and Jupiter’s eccentricity is small, this
will be used as a good approximation to the motion of fic-
titious asteroids moving around the Sun under the effect
of gravitational perturbations from Jupiter. The asteroid is
assumed to be moving in the same plane as Jupiter’s orbit.4.2.1 REGULAR ORBITS
The first asteroid has starting valuesx=0.55,y=0,x ̇=0,
withy ̇=0.9290 determined from the solution of Eq. (41).
Here a set of dimensionless coordinates are used in which
n=1,G=1, andm 1 +m 2 =1. In these units, the orbit of
m 2 is a circle at distancea=1 with uniform speedv=1.
The corresponding initial values of the heliocentric semi-
major axis and eccentricity area 0 =0.6944 ande 0 =0.2065.
Since the semimajor axis of Jupiter’s orbit is 5.202 AU, this
value ofa 0 would correspond to an asteroid at 3.612 AU.
Figure 6 shows the evolution ofeas a function of time.
The plot shows a regular behavior with the eccentricity vary-
ing from 0.206 to 0.248 over the course of the integration.
In fact, an asteroid at this location would be close to an
orbit–orbit resonance with Jupiter, where the ratio of the
orbital period of the asteroid,T, to Jupiter’s period,TJ,is
close to a rational number. FromKepler’s third lawof
planetary motion,T^2 ∝a^3. In this case,T/TJ=(a/aJ)^3 /^2 =
0.564≈4/7 and the asteroid orbit is close to a 7:4 resonance
with Jupiter. Figure 7 shows the variation of the semima-
jor axis of the asteroid,a, over the same time interval as
shown in Fig. 6. Although the changes inaare correlated
with those ine, they are smaller in amplitude andaap-
pears to oscillate about the location of the exact resonance at
a=(4/7)^2 /^3 ≈0.689. An asteroid in resonance experiences
enhanced gravitational perturbations from Jupiter, which
can cause regular variations in its orbital elements. The ex-
tent of these variations depends on the asteroid’s location
within the resonance, which is, in turn, is determined by
the starting conditions.50 100 150 200 250 3000.20.210.220.230.240.25EccentricityTime (Jupiter periods)0FIGURE 6 The eccentricity as a function of time for an
object moving in a regular orbit near the 7:4 resonance
with Jupiter. The plot was obtained by solving the circular
restricted three-body problem numerically using initial
values of 0.6944 and 0.2065 for the semimajor axis and
eccentricity, respectively. The corresponding position and
velocity in the rotating frame werex 0 =0.55,y 0 =0,x ̇=
0, and ̇y=0.9290.