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A Symplectic Integration Scheme that Allows Close
Encounters between Massive Bodies
J. E. Chambers, Armagh Observatory
Mixed­variable symplectic integrators provide a fast, moderately accu­
rate way to study the long­term evolution of a wide variety of N­body sys­
tems (Wisdom & Holman 1991). They are especially suited to planetary and
satellite systems, in which a central body contains most of the mass. How­
ever, in their original form, they become inaccurate whenever two bodies
approach one another closely. Here, I will show how to overcome this diffi­
culty using a hybrid integrator that combines symplectic and conventional
algorithms.
A symplectic integrator works by splitting the Hamiltonian, H , for an
N­body system, into two or more parts H = H 0 + H 1 + \Delta \Delta \Delta, where ffl i =
H i =H 0 Ü 1 for i = 1; 2 : : :. An integration step consists of several substeps,
each of which advances the system due to the effect of one part of the
Hamiltonian only. The error incurred over the whole step is ¸ ffl Ü n+1 , where
Ü is the timestep, n is the order of the integrator, and ffl is the largest of ffl i .
A symplectic algorithm is efficient provided that the ffl factors are small.
In the planetary system, this is usually achieved by making H 0 the un­
perturbed Keplerian motion of the planets about the Sun, and H 1 ; H 2 etc.
the perturbations between planets. For example, using mixed coordinates
(heliocentric positions and barycentric velocities), H is split up as
H 0 =
N
X
i=1
/
p 2
i
2m i
\Gamma Gm fi m i
r ifi
!
H 1 = \GammaG
N
X
i=1
N
X
j=i+1
m i m j
r ij
H 2 = 1
2m fi
/ N
X
i=1
p i
!2
(1)
where N is the number of planets, m denotes mass, and r, p are position
and momentum respectively. Note that each part of the Hamiltonian can
be integrated analytically in the absence of the others, and H 0 contains all
of the large terms, provided that the planets are widely separated.
Now consider a close encounter between bodies a and b. During the
encounter, the distance r ab is small, and the corresponding term in H 1 is
1

large. This means that ffl 1 is no longer small and the integrator becomes
inaccurate. An approximate solution to this difficulty is to transfer the
offending term from H 1 to H 0 for the duration of the encounter. This
ensures that ffl 1 is always small. However, each time H 0 and H 1 are changed
in this way, the system undergoes a shift in energy, and the integrator's
symplectic property is lost.
A better solution is to split each of the interaction terms between H 0
and H 1 as follows
H 0 =
N
X
i=1
/
p 2
i
2m i
\Gamma Gm fi m i
r ifi
!
\Gamma G
N
X
i=1
N
X
j=i+1
m i m j
r ij
[1 \Gamma K(r ij )]
H 1 = \GammaG
N
X
i=1
N
X
j=i+1
m i m j
r ij
K(r ij ) (2)
where the function K is chosen so that K ! 0 when r ij is small, and K ! 1
when r ij is large. This ensures that ffl 1 is always small, without requiring
that terms move from one part of the Hamiltonian to another.
When all of the separations r ij are large, H 0 can be advanced analytically
as before (since 1 \Gamma K = 0). If two bodies undergo a close encounter, the
terms in H 0 due to these objects must be integrated numerically, but all the
remaining terms can still be advanced analytically. By trial and error, I find
that a good expression for K is
K =
8 ? !
? :
0 for y ! 0
y 2 =(2y 2 \Gamma 2y + 1) for 0 ! y ! 1
1 for y ? 1
where
y =
` r ij \Gamma 0:1 r crit
0:9 r crit
'
and r crit is the larger of 3 Hill radii and 0:5Üv max , where v max is the maxi­
mum likely orbital velocity of any of the objects.
Reference
ffl Wisdom J., Holman M., 1991, AJ, 102, 1528.
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