Kepler's third law of nbody system
Citation: Z.S. She, Kepler's third law of nbody system, Sci. ChinaPhys. Mech. Astron.
Kepler's third law of nbody system
ZhenSu She 0
0 Department of Mechanics and Engineering Science, Peking University , Beijing 100871 , China

Newton’s gravitational law and its derived Kapler’s orbit of
planetary motions are the very first triumph of science, which
confirms that a purely human’s invention of mathematics is
able to accurately describe observations in nature. For the
past three centuries, the frontier of science has moved to
almost all areas of human experience, yet some basic problem
in celestial mechanics has remained unsolved, despite the
effort of many great minds like Newton (1687), Euler (1760)
and Lagrange (1776), among others.
The famous threebody problem, originating from the
SunEarthMoon system under the Newtonian gravitation field, is
particularly attractive, for which Oscar II, King of Sweden,
established a prize in 1887 for anyone who could find the
solution. The prize was finally awarded to Henri Poincare´, who
first found that there is no general analytical solution [1]. It
is now recognized that the ensemble of its orbits has an
enormous complexity. Recently, thanks to the advance of
computational techniques, some breakthroughs begin to emerge:
a figureeight orbit was discovered numerically by Moore in
1993 [
2
], and 13 new distinct periodic orbits were found by
Sˇuvakov and Dmitrasˇinovic´ [
3
] in 2013, when three masses
are equal with zero angular momentum. In 2017, Li and Liao
[
4
], Li et al. [
5
] reported a further major advance: 695
periodic orbits with equal masses [
4
], and 1223 periodic orbits
found later with unequal masses [
5
].
These findings have led to a fundamental conjecture that
the 3body system (m1; m2; m3) may obey a kind of law of
“harmonies”: in analogy to the Kepler’s third law of the
twobody problem, Sˇuvakov and Dmitrasˇinovic´ [
6
] proposed
that a generalized Kepler’s third law could be in the form
T E3=2 = constant, where E denotes the sum of kinetic and
potential energy of the 3body system, while T is the period
of periodic orbit, to be referred to as the harmonics of the
system.
The question then remains whether T E3=2 = constant is
universal, and if not, what form it will takes. More
interestingly, does there exist similar relation for an nbody system?
This is a question that can not be answered by computation
alone, especially for n > 3 in the presence of an even greater
complexity than 3body problem. In this case, an analytic
approach, if successful, would be likely valuable.
Prof. Bohua Sun from the Cape Peninsula University of
Technology, South Africa, obtained an epic result from the
perspective of dimensional analysis [
7
], by working with a
reduced gravitation parameter n, then predicting a
dimensionless relation TnEn3=2 = const × n √ n ( n is reduced
mass). The const= √2 is derived by matching with the
2body Kepler’s third law, and then a surprisingly simple
relation for Kepler’s third law of an nbody system is derived
by invoking a symmetry constraint inspired from Newton’s
gravitational law:
TnEn3=2 = √2
∑n
G i=1
∑nj=i+1(mim j)3 1=2
∑kn=1 mk
;
where the point masses, m1; m2; :::; mN , the gravitational
constant, G, the orbit period, Tn, and the system total energy, En.
This formulae is, of course, consistent with the Kepler’s third
law of 2body system, but yields a nontrivial prediction of
(1)
100
50
0
TE3/2=3.074m3−0.617
Sun
(a)
12
the Kepler’s third law of 3body:
[ (m1m2)3 + (m1m3)3 + (m2m3)3 ]1=2
T3E33=2 = √2 G m1 + m2 + m3 : (2)
These analytic relations, eq. (1) and eq. (2), are obtained
for the first time, and so present a significant step towards
quantitative description of 3body and nbody system. Note
that the expression on the right hand side of eq. (1) or eq. (2)
involves the sum of all masses in the denominator, and the
terms proportional to the product of two masses only in the
numerator; this construction is a guess at the present stage,
possibly involving a deeper symmetry content worthy
further exploration. Fortunately, numerical results of Li et al.
[
5
] are available for a verification: for 3body system with
G = 1 and m1 = m2 = 1 and varying m3, the predicted result,
T3E33=2 = √2 [ 1+22+(mm33)3 ]1=2, is in close agreement with
numerical finding of ref. [
5
]: (T3E33=2)NUM ≈ 3:074m3 −0:617,
as shown in Figure 1.
Two specific features are noteworthy. First, in the limit of
m3 ≫ 1, eq. (2) predicts that T3E33=2 ≈ (m3−1), a linear
relation on m3 providing an explanation for the numerical slope
3.074. Furthermore, eq. (2) predicts that T3E33=2 ≈ =2 > 0
in the limit of m3 → 0, improving the numerical formulae at
that limit, which becomes negative and so inapplicable.
Note that the small but finite difference between eq. (2)
and the numerical results are very interesting, since eq. (2)
is an analytic construction extended from the exact 2body
Kepler’s third law under a symmetry argument. Is the small
difference a signature of the breakingdown of the
symmetry (from 2body to 3body), or the numerical solutions still
present some systematic bias (e.g. for increasing ms)? This
is intriguing to investigate in the future.
Back to the classical 3body problem, eq. (2) partially
answers the conjecture proposed by ref. [
6
]. Note that the
periodic orbits form a subset of all solutions, and it is likely that
eq. (2) captures only a subset of all periodic orbits.
Nevertheless, if proved to be correct, one may capture a very important
subset of all solutions. Furthermore, from a practical point of
view, celestial planetary system is typically nbody system,
eq. (1) would provide a helpful guide in the search for new
periodic orbit of a practical nbody system whose numerical
study is generally very resourcedemanding. Whether eq. (1)
is confirmed or disproved, important physics will be learnt.
It is particularly interesting to test eq. (1) in practical
celestial system like our planetary system. Finally, even if eq. (1)
is confirmed numerically, rigorous mathematical proof is
desired, since the heuristic argument behind eq. (1) needs to find
its limit of applicability. We thus conclude that [
7
] opens a
new avenue for further study of the classical nbody problem
of mechanics.
100 1 H. Poincare ´, Acta Math. 13 , 5 ( 1890 ).
2 C. Moore , Phys. Rev. Lett . 70 , 3675 ( 1993 ).
3 M. Sˇ uvakov , and V. Dmitrasˇinovic´, Phys. Rev. Lett . 110 , 114301 ( 2013 ), arXiv: 1303 . 0181 .
4 X. M. Li , and S. J. Liao , Sci. ChinaPhys. Mech. Astron . 60 , 129511 ( 2017 ), arXiv: 1705 . 00527 .
5 X. M. Li , Y. P. Jing , and S. J. Liao , arxiv.org/abs/1709.04775.
6 V. Dmitrasˇinovic´, and M. Sˇ uvakov , Phys. Lett. A 379 , 1939 ( 2015 ), arXiv: 1507 . 08096 .
7 B. H. Sun , Sci. ChinaPhys. Mech. Astron . 61 , 054721 ( 2018 ).