Encyclopedia of the Solar System 2nd ed

798 Encyclopedia of the Solar System

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Eccentricity

Time (Jupiter periods)

FIGURE 9 The eccentricity as a function of time for an

object moving in a chaotic orbit started just outside the

7:4 resonance with Jupiter. The plot was obtained by

solving the circular restricted three-body problem

numerically using initial values of 0.6984 and 0.1967 for

the semimajor axis and eccentricity, respectively. The

corresponding position and velocity in the rotating frame

werex 0 =0.56,y 0 =0,x ̇ 0 =0, andy ̇=0.8998.

The rate of divergence of nearby trajectories in such
numerical experiments can be quantified by monitoring
the evolution of two orbits that are started close together.
In a dynamical system such as the three-body problem,
there are a number of quantities called theLyapunov
characteristic exponents. A measurement of the local di-
vergence of nearby trajectories leads to an estimate of the
largest of these exponents, and this can be used to deter-
mine whether or not the system is chaotic. If two orbits are
separated in phase space by a distanced 0 at timet 0 , andd
is their separation at timet, then the orbit is chaotic if

d=d 0 expγ(t−t 0 ), (42)

whereγis a positive quantity equal to the maximum Lya-
punov characteristic exponent. However, in practice the
Lyapunov characteristic exponents can only be derived
analytically for a few idealized systems. For practical prob-
lems in the solar system,γcan be estimated from the results

of a numerical integration by writing

γ=lim

t→∞

ln(d/d 0 )

t−t 0

(43)

and monitoring the behavior ofγ with time. A plot ofγ

as a function of time on a log–log scale reveals a striking

difference between regular and chaotic trajectories. For

regular orbits,d≈d 0 and a log–log plot has a slope of –1.

However, if the orbit is chaotic, thenγtends to a constant

non-zero value. This method may not always work because

γ is defined only in the limit ast→∞and sometimes

chaotic orbits may give the appearance of being regular

orbits for long periods of time by sticking close to the edges

of the islands.

If the nearby trajectory drifts too far from the original

one, thenγis no longer a measure of the local divergence

of the orbits. To overcome this problem, it helps to rescale

the separation of the nearby trajectory at fixed intervals. Fig-

ure 13 shows logγas a function of logtcalculated using this

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Semimajor axis

Time (Jupiter periods)

FIGURE 10 The semimajor axis as a function of

time for an object using the same starting

conditions as in Fig. 9. The units of the semimajor

axis are such that Jupiter’s semimajor axis (5.202

AU) is taken to be unity.