Kepler's third law of n-body system
Citation: Z.-S. She, Kepler's third law of n-body system, Sci. China-Phys. Mech. Astron.
Kepler's third law of n-body system
Zhen-Su 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 three-body problem, originating from the
SunEarth-Moon 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 . 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 figure-eight orbit was discovered numerically by Moore in
], and 13 new distinct periodic orbits were found by
Sˇuvakov and Dmitrasˇinovic´ [
] in 2013, when three masses
are equal with zero angular momentum. In 2017, Li and Liao
], Li et al. [
] reported a further major advance: 695
periodic orbits with equal masses [
], and 1223 periodic orbits
found later with unequal masses [
These findings have led to a fundamental conjecture that
the 3-body system (m1; m2; m3) may obey a kind of law of
“harmonies”: in analogy to the Kepler’s third law of the
two-body problem, Sˇuvakov and Dmitrasˇinovic´ [
that a generalized Kepler’s third law could be in the form
T |E|3=2 = constant, where |E| denotes the sum of kinetic and
potential energy of the 3-body system, while T is the period
of periodic orbit, to be referred to as the harmonics of the
The question then remains whether T |E|3=2 = constant is
universal, and if not, what form it will takes. More
interestingly, does there exist similar relation for an n-body 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 3-body 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 [
], by working with a
reduced gravitation parameter n, then predicting a
dimensionless relation Tn|En|3=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 n-body system is derived
by invoking a symmetry constraint inspired from Newton’s
Tn|En|3=2 = √2
∑nj=i+1(mim j)3 1=2
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 2-body system, but yields a non-trivial prediction of
the Kepler’s third law of 3-body:
[ (m1m2)3 + (m1m3)3 + (m2m3)3 ]1=2
T3|E3|3=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 3-body and n-body 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.
] are available for a verification: for 3-body system with
G = 1 and m1 = m2 = 1 and varying m3, the predicted result,
T3|E3|3=2 = √2 [ 1+22+(mm33)3 ]1=2, is in close agreement with
numerical finding of ref. [
]: (T3|E3|3=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 T3|E3|3=2 ≈ (m3−1), a linear
relation on m3 providing an explanation for the numerical slope
3.074. Furthermore, eq. (2) predicts that T3|E3|3=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 2-body
Kepler’s third law under a symmetry argument. Is the small
difference a signature of the breaking-down of the
symmetry (from 2-body to 3-body), 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 3-body problem, eq. (2) partially
answers the conjecture proposed by ref. [
]. 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 n-body system,
eq. (1) would provide a helpful guide in the search for new
periodic orbit of a practical n-body system whose numerical
study is generally very resource-demanding. 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 [
] opens a
new avenue for further study of the classical n-body problem
100 1 H. Poincare ´, Acta Math. 13 , 5 ( 1890 ).
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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. China-Phys. Mech. Astron . 61 , 054721 ( 2018 ).