1. Suzuki's Symplectic Integrator
applied to the Long-term Evolution
of the Solar System
Colin D. D’Elia
Advisor: Vincent Moncrief
Senior Project completed April 24, 1995
At Yale University

2. Suzuki's Symplectic Integrator applied to the
Long-term Evolution of the Solar System
Colin D. D'Elia
Suzuki (1991) has recently shown how to compose symplectic integration
schemes of arbitrarily high order. Here, his fourth order scheme has been
applied to a pared-down version of the solar system consisting of the Sun
and the outer five planets. This system was numerically integrated over a
165 million-year period. An estimate of the Lyapunov time for two test
particle Plutos was in line with those of other researchers. Non-
symplectic integrations of the solar system have shown a linear growth in
Energy while this one does not.
The question of the stability and predictability of the solar
system has intrigued scientists, including such notables as Laplace,
Newton, and Euler for more than 200 years. Today, computers are
allowing researchers to examine this question in a new light. In
1951, Eckert, Brouwer, and Clemence completed a 400-year
numerical integration of the outer five planets. Cohen and
Hubbard published a 120,000-year numerical integration of the
outer planets in 1965 which was soon advanced to 1 million years
by Cohen, Hubbard, and Oesterwinter (1973). In 1984 Kinoshita
and Nakai integrated the outer planets for 5 million years The
Digital Orrery, a special purpose computer designed to
numerically integrate systems with a small number of bodies
which move in nearly circular orbits, allowed Applegate, Douglas,
Gursel, Sussman, and Wisdom (1986, hereafter referred to as
ADGSW) to integrate the outer planets for 200 million years This
was exceeded by an 845 million year integration by Sussman and
Wisdom (1988) using that same computer Each of these
computations was performed with a non-symplectic integrator
and involved only the five outer planets.

3. The most notable long-term integration of the solar system
utilizing a symplectic scheme was performed by Wisdom and
Holman (1991). Theirs was a 1.1-billion-year integration of the
outer planets which utilized a second order symplectic scheme.
Sussman and Wisdom (1992) integrated the entire solar system for
100 million years using that same integrator.
Symplectic Integrators
Symplectic integrators have a number of advantages over
traditional numerical methods. These are enumerated below:
(1) Volume in phase-space is preserved satisfying
Liouville's Theorem.
(2) Both the total angular momentum and the total linear
momentum are invariant.
(3) Symplectic schemes are time reversible.
Each of these is subject to computer round-off error. Round-off
error enters into any numerical calculation since only a finite
number of digits may be kept from one iteration to the next. The
most well-known second-order symplectic scheme is the Trotter
formula;
𝑆2( 𝑡 + 𝜀) = 𝑆 (
𝐴𝜀
2
) ∘ 𝑆( 𝐵𝜖) ∘ 𝑆 (
𝐴𝜀
2
)
where the Hamiltonian is divided into two separably integrable
parts A and B. The most natural division of the n-body
gravitational Hamiltonian, which is the one used in this
integration, is to separate the kinetic and potential energy terms
so that

4. 𝐴 = ∑
𝜌𝑖
2
2𝑚 𝑖
𝑛−1
𝑖=0 and B = - ∑
𝐺𝑚 𝑖 𝑚 𝑗
𝑟𝑖𝑗
𝑖<𝑗
To evolve the dynamical variables forward a time-step ε, a "leap-
frog" method is used. In canonical variables the positions are
evolved according to
𝑞( 𝑡 + 𝜀) = 𝑞 ( 𝑡) + 𝜀̇ 𝑞̇(𝑡)
and the momenta by
𝑝( 𝑡 + 𝜀) = 𝑝 ( 𝑡) + 𝜀̇ 𝑝̇( 𝑡)
𝑞̇ and 𝑝̇ are obtained from the Hamiltonian. Hence;
𝑞̇ =
𝜕𝐻
𝜕𝑝
and 𝑝̇ =
𝜕𝐻
𝜕𝑞
In summary, this means that first the momentum of each body in
the system is evolved with a half time-step while keeping the
positions constant. Then the positions are evolved a whole-step
while the momenta are kept constant and then once again the
momenta are evolved another half time-step keeping the
positions constant. This completes one iteration and has been
shown to agree to second order with an expansion of the full
solution. The final positions and momenta of each iteration are
then used to begin the next iteration. Until recently, there had not
been a general method for obtaining symplectic integration
schemes greater than second order. Suzuki (1991) showed how to
write a symplectic scheme to any desired order by composing the
n order scheme from the (n-2) scheme in the following way;
𝑠 𝑛( 𝑡 + 𝜀) = 𝑠 𝑛−2[ 𝑘 𝑛−1 𝜀] ∘ 𝑠 𝑛−2[(1 − 2𝑘 𝑛−1) 𝜀] ∘ 𝑠 𝑛−2[ 𝐾 𝑛−1 𝜀]

5. Every even ordered scheme n agrees with the n-1 scheme. S3 = S4
for example. It is remarkable that whenever a new scheme is
composed, two orders of accuracy are gained instead of just one.
This lends itself very naturally to numerical computations. The
integration being reported here utilized Suzuki's fourth order
scheme. I began with the invariable frame center of mass
coordinates and momenta listed in ADGSW (1986) for
“JD2430000.5”. The masses were also copied from this source. The
masses of the inner planets were added to that of the Sun with
Pluto integrated as a test particle. This 165-million-year run was
performed on the Vax alpha machine at Yale University and took
about two weeks of computation time. A somewhat arbitrary time-
step of 40 days was chosen with the output recorded at
approximately 5 million year intervals.
The following three questions were investigated;
(1) Does the estimate of the Lyapunov time for Pluto agree with
that of other researchers?
(2) How well is the Energy of the system conserved and how
does this compare with that of other long-term solar system
integrations?
(3) How accurate can the initial conditions be recovered after
running the system forward a time t and then backwards
that same time t?
The Lyapunov Time
The Lyapunov time is defined as the time it takes for the paths of
two bodies to become separated in phase-space by a factor of e. A
positive Lyapunov time is considered to be the hallmark of Chaos.
Formally, this definition of the Lyapunov time can only be applied
when the paths are diverging exponentially and in the limit as

6. time -> ∞ Any numerical estimate then, can only give an
approximation of the Lyapunov time. Furthermore, it may be that
the estimate is sensitive to uncertainties in the initial positions and
momenta though this has not yet been investigated.
Phase-space distance is measured by the Euclidean six-
dimensional norm. Since the positions are quoted in Astronomical
Units (A.U.) and the velocities as A.U./day, the velocities are small
compared to the distances in these units. On that basis, I have
chosen to estimate the Lyapunov time by the time it takes for the
test-particle Plutos to become separated in Cartesian 3-Space by a
factor of e.
In this case the Lyapunov time for the trajectory of Pluto is what
was investigated. The Sun, Jupiter, Saturn, Uranus, Neptune, and
two test particle Plutos were integrated by point particle
Newtonian gravitation in rectangular coordinates. The initial and
final positions of the test particle Plutos (in A.U.) are included
below. The number of digits does not reflect the machine accuracy
and are included for completeness only.
time = 0 days
x y z
Pluto 1 -21.3858977531572 A.U. 32.0719104740000 A.U. 2.49245689556095 A.U.
Pluto 2 -21.3858977531572 A.U. 32.0719104739886 A.U. 2.49245689556095 A.U.
time = 6.0225x 108
days
x y z
Pluto 1 -18.9969402226207 A.U. 38.9892747647021 A.U. -7.79468473359073 A.U.
Pluto 2 -21.3858977531572 A.U. 32.0719104739886 A.U. -7.07607867883915 A.U.

7. The initial positions of the two test-particle positions differ by only
1.14x 10-10
A.U. in the y-coordinate. This is about 1.7 meters. The
final positions differed by about 2.66 A.U. or 3.99x109
meters.
Practically, this means that after 165 million years the position of
Pluto cannot be ascertained to within 3.99x109
meters unless the
initial position of Pluto is known to within 1.7 meters.
It is estimated here that the Lyapunov time is greater than 6.3
million years. This is in quite reasonable agreement with the
results of ADGSW (1986) who showed a Lyapunov time of greater
than 6.5 million years for a 110 million year run and Wisdom,
Sussman (1988) who estimated the Lyapunov time as greater than
20 million years in their 845-million-year integration. Wisdom and
Holman (1991) have also estimated Pluto's Lyapunov time as
about 20 million years in their 1.1-billion-year integration. Fig. 1
shows the Lyapunov time plotted vs. time. It should be noted that
the Lyapunov time has not levelled off here and is still growing. A
longer run will have to be performed to see whether this method
is in strict agreement with the results reported earlier in this
paper. That all numerical computations have given a Lyapunov
time of the same order of magnitude is significant and seems to
confirm the chaotic trajectory of Pluto though Wisdom and
Holman (1991) caution that until the dynamical mechanisms are
understood we cannot be certain.

8. Energy Conservation
Traditionally, one method of testing the reliability of Solar system
integrations has been to observe the change in energy with time.
In a "perfect" integration, energy should, of course, be strictly
conserved. Sussman and Wisdom (1988) reported the relative
energy error, defined at time t as the energy at time t minus the
initial energy divided by the initial energy, for a number of non-
symplectic integrations. These included Sussman and Wisdom
(1988), ADGSW (1986), Cohen, Hubbard, and Oesterwinter (1973),
Kinoshita and Nakai (1984), and Project LONGSTOP (A.E. Roy et.al).
Each of these reported a nearly linear relative energy error with
magnitudes ranging from 3.0x10-19
per year to 5x10-16
per year.
It has been expected that long-term symplectic integrations of the
solar system would result in a periodic energy (see Liu, Liao, Zhao,
and Wang (1994)) and this seems to be confirmed here (FIG.2).

9. It is quite impressive that after 200 million years the energy is
virtually where it started. It appears to be cyclic with a period of
about 120 million years. It is not known whether that particular
length of time has a special significance. The difference between
the maximum and minimum energies is only 3.75x10-14
S.M.*(A.U.)2
/(day). FIG. 3 shows the relative energy error vs. time. It
is clear that unlike the results of non-symplectic integrations, the
relative energy error is not linear.
FIG. 3. Relative Energy Error vs. Time

10. There were actually two separate runs with the only difference
being the initial positions of the test-particle Plutos. These were
run in separate batches to speed the integration process. As a
result, one of the runs was actually 200 million years long and it
was from this run that FIG. 2 and FIG.3 were composed.
Forward/Backward Run
A separate run was performed in order to see how well the initial
positions could be recovered after integrating forward and then
backward three million years. This is an excellent indicator of the
degree to which round-off error enters into the calculation since
the scheme is otherwise time-reversible.
At the end of the run the error in recovering the positions of the
planets ranged from 1.7x104
A.U. for Pluto to 8.4x10-3
A.U. for
Jupiter. Sussman and Wisdom (1988) performed a
forward/backward run of three million years and were able to
recover the initial position for each of the planets with an error of
about 10-5
for each planet. Most of this difference is probably due
to the defined machine precision. Sussman and Wisdom used a
special purpose computer with pseudo-quadrupole precision
variables. All the variables in this calculation were defined as
double precision.
Conclusions
This research, as well as that of others, shows symplectic schemes
to be performing well. To this author's knowledge, no long-term
evolution of the solar system has been performed utilizing a sixth
order or higher symplectic scheme.

11. Until higher order symplectic schemes are used to integrate the
solar system, direct comparison between symplectic schemes and
non-symplectic schemes will be difficult. All but one of the non-
symplectic integrations reported in this paper employed a 12th-
order integrator.
Additionally, Wisdom and Holman (1991) suggest that the division
of the Hamiltonian which was used in this paper is not the optimal
one. Instead they recommend H=HKEPLER + HINTERACTION where the
first term is the Keplerian motion of each planet with respect to
the sun and the second includes the perturbations which the
planets have upon each other. This division, they claim, will allow
larger time-steps to be used thus saving valuable computation
time. Wisdom and Holman were able to perform their 2nd-order
symplectic integration with a time-step of one year.
Since the dynamical variables at each time t are calculated from
the time (t-ε) positions and momenta, this integration method is
well suited for parallel processing where a different processor
could be assigned to each planet. I estimate that this could speed
the computation time by a factor of four to six times. Each two
orders of magnitude one wishes to gain using Suzuki's method
increases the computation time by a factor of three.
Given that the 165 million year 4th-order integration reported
here took roughly 14 days, it is conceivable that with the above
mentioned improvements one could repeat Wisdom and
Holman's 1.1 billion year, second order integration of the outer
planets at 8th-order on the Yale Vax in a little over two weeks’
time. This is the same time scale in which they performed their
2nd-order calculation

12. As judged by the 3 million year forward and backward runs the
integration method used here still falls short of the best solar
system integrations. Since this scheme is time reversible, the only
difference must be in the precision with which the variables are
defined. Keeping more digits from one iteration to the next,
though, costs computation time. Combining the methods of
reducing computation time mentioned above should effectively
allow greater machine precision to be defined. The energy seems
to be conserved to a greater degree of accuracy here than with
integrations using non-symplectic methods. It is encouraging that
the energy seems to be bounded with this integrator. With
traditional methods the relative energy error will continue to
increase or decrease roughly linearly with time. All integrations
indicate a chaotic trajectory for Pluto. The effect which this has on
the rest of the solar system must still be fully investigated. It is
extremely unlikely that integrating Pluto as a full particle would
have much of an effect upon the conclusions here though this
should still be investigated. We are just beginning to see long-term
integrations of the entire solar system which use the full equations
of motion. These are allowing researchers to investigate the
stability of the solar system as a whole. In short, there are still
many interesting questions which can be investigated and as top-
speed computers become increasingly accessible it can only be
expected that they will.

13. REFERENCES
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C.J. Cohen, E.C. Hubbard, and C. Oesterwinter, 1973, Astron.
Papers Amer. Ephemeris 22,1
Eckert, Brouwer, and Clemence, 1951, Astron. Papers Amer.
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