Regular n-gon as a model of discrete gravitational system

Regular n-gon as a model of discrete
gravitational system
Rosaev A.E.
OAO NPC NEDRA, Jaroslavl Russia,
E-mail: [email protected]
Introduction
A system of N points, each having mass m, forming a planar regular polygon
(N-gon), and a central mass M, are considered. The motion equations for a testing
particle are given in different coordinates. To provide stability, N-gon must always
rotate, including case zero central mass. The velocity of rotation is calculated.
3N stationary points (libration points) appear in a system. Studying of
libration point behavior is a main result of this paper. It is shown, that the stationary
solution (libration points) in considered system may be determined from algebraic
equation of 5-th degree. In case small m / M coordinates of these points may be
calculated as a generalization of classical gravitation 3-body problem. In present
research, the dependence of libration points coordinates on mass and number of
particles is studied. Obtained results are discussed and compared with another authors.
Area, where potential allows a linearization is described in small vicinity of
N-gon. However, oscillations even in this small area become strongly nonlinear due to
differential rotation of system.
In fact, approximation ring potential by N-gon is an example of a pointmass modeling. But specific of potential expansion gives an ability to study more
complex (non-homogenous) systems on base of considered model.
Definitions
Definition [1]: A configuration of the n particles is a central configuration at time t
r   r for i=1,...,n.
if there exist some scalar  such that
i
mi ri 
i
U
 mri  0
ri
U
ri
In particular every body can move on a circle around the common center of
masses. In these solutions the motion behaves as if the bodies form a rigid body.
These solutions are also denoted as relative equilibrium solutions, being a fixed
point of (1) if we use a rotating frame.
U 
mi m j
ri  r j
Main target – to describe motion infinitesimal
particle perturbed by n-gon
• Determine stationary points in the system
• Show difference between three body and continue
case
• Consider some generalizations
Libration points
A system of N points, each having mass m, placed in vertex
of regular polygon, and a central mass M, is considered. Such
system, forming a planar central configuration, called a relative
equilibrium system [1]. There are 3N-1 stationary points (libration
points) appear in a system (Fig.1).
Fig. 1. Libration points positions for 3-body (left) and N-body (right)
relative equilibrium configurations.
The libration point coordinates can be determined from equations
[2]:
N
GM
 R  
  Gm
R2
j 1
x
2
 R 2  

2 R sin 2  j 2  x
2

 2 R sin  j 2  1  x R 
2
32
2 R sin 2  j 2   x
GM Gm N 1

  Gm
32
2
R2
x2
j 1
x 2  2 R sin  j 2  1  x R 


 j  2 j N   0
Here G – gravity constant, R is the central configuration’s radius,
 j is the angle between particles, and x is a distance between the test
particle and the central configuration. In the first case 0 = /N, x=0
and we have non-collinear libration point. In the second case 0 = 0, x
 0 and the x-coordinate of the collinear points must be determined.
Regular n-gon configuration begets a periodic solution in which the
bodies rotate uniformly about the central mass with rotation speed (Roberts[6],
Rosaev[2]):

m
   0 1 
 4M
1/ 2

sin 1  j 2 

j 1

N 1
0 
GM
R3
The equations for the collinear libration points are the most interesting.
They can be reduced to a fifth degree polynomial:
(2+A)x5+ (32R + 2 AR + B)x4 + (32R2 + R2A+ 2BR)x3+ (BR2Gm)x2  GmR2=0
The sign “+” is for libration point inside central configuration, and “ – “
for outside ones. Equations for non-collinear libration points have a similar
form.
Coefficients are:
A
Gm
R3
B 


1
3




3

j 1 2 sin 
8 sin  j 2  
j 2

N 1
Gm
4R 2
N 1
1
 sin 
j 1
j
2
When A=0 and B=0 we have well known case 3-body problem
(Szebehely1967). Considered equation for determination libration points
coordinates always have only one real root.
Summation
Coefficients A of expansion depending on number of particles. For
collinear case:
N
N
1 /(2 sin(i / N ))
3
i 1

 N 3 /(2 ) 3  2 / i 3 
2.404101...N 3 /(2 ) 3
N
i 1
1 /(2 sin(i / N ))3
100
500
1000
1500
2000
2500
3000
 2 / i3
2.408618
2.404358
2.404182
2.404146
2.404132
2.404126
2.404123
2.404013
2.404113
2.404113
2.404113
2.404114
2.404114
2.404114
For non-collinear case::
A
Gm N 1 
1
3



3
3
R j 1  2 sin j N 
8 sin j N 

B
Gm
4R 2
N 1

j 1
1

 N /( )
sin j N 

Gm
 1.6541137...N 3 /(2 ) 3

R3

N
Gm
1 / i R
i 1
2
By analogy with three body problem, solution of equations for
collinear libration points can be calculated as a series:
m


x1  

 3M  A  2 B 
1/ 3
m


x2  

 3M  A  2 B 
1/ 3
1
m

 

3  3M  A  2 B 
2/3
1
m

 

3  3M  A  2 B 
2/3
1
m

 
  ...
9  3M  A  2 B 
1
m

 
  ...
9  3M  A  2 B 
for inner point L1 and for outer point L2 respectively. These equations are
valid for small m/M ratio and when N is not large than few hundreds. But in
all cases, solutions of fifth degree equation can be obtained for example, with
Maple.
F := 0.383088197088235 10 14 x50.766177581022708 10 14 x438309057078071 x3
0.2928612775186 10 7 x23.952593
m/M=0.1000000, ln N=5 x=-0.0000469292
m/M=0.1000000, ln N=5 x=0.0000468753
Libration point coordinates on mass ratio dependence
Equation for non-collinear libration points
has trivial solution x=0 and non-trivial solution
of cubic equation:
(2  A) x 3  (32 R  2 AR  B ) x 2 
 (32 R 2  R 2 A  2 BR ) x  BR 2  0
which have at least one real root. For case
B=0 we have three real roots if A sufficiently
large (small central mass):
Fig. To a non-collinear points
coordinates determination
3  22 A 3  44  2 A 3  22 A 3  44  2 A
J := 0 ,
,
2
2 (  A )
2 (  2A )
Motion in field of N-gon
In restricted three body problem (RTBP), each k-th of it’s period, test
particle has strong perturbation – close encounter with planet. For case N+2
body problem strong perturbed encounters take place in N-times more
frequently. There are more significant variations in orbital elements expected
in this case. At limit large N there are no any resonant phenomena. At small N
we have chaotic motion due to close encounters. If N sufficiently large, effect
of initial phase not significant.
Conjunction. At fixed total mass of N-gon,
m=miN , interaction test particle with regular Ngon strongly depends on number of N. At
conditions:
(r / R )3 / 2  p / P  n : m  1 outer
(r / R )3 / 2  p / P  n : m  1 inner
where r, R and p, P – central distances and periods of test particle and
particle of ring respectively, resonance effects in motion test particle take
place.
The longitude (solid line) and eccentricity (dashed line) on number of ring particles N
dependence.
Fig.. Difference between perturbed and unperturbed positions for resonance
case N=19 and for case N=100, close to continuous limit. Orbit with a=1.07
Generalization to
Low-Elliptic case
Put ring of N particles m with small eccentricity in gravity field of central mass M. It is naturally to find
stationary distribution of particles by true anomaly and about influence of mutual perturbations of particles on ring’s
eccentricity.
There are two possibility. For the low-massive particles m<<M, their orbital distribution determined by
keplerian angular velocity (L - angular momentum):
  const
d
L

  0 (1  e cos( ))
dt mr 2
Fig.. Orbital distribution of particles along eccentric ring (small mass particle case)
Fig. Declination of the shape of ring (from best
fitted rotated ellipse) as function of true
anomaly.
Fig.2. Eccentric ring shape change due to gravitational
interactios of particles.
Other applications
Thompson Heptagon

2

2 1/ 2
U     j i log xi  x j    yi  y j 
i j
Model of N point-vortex on sphere by
6.Boatto S, Simo C.: Thompson heptagon: a case of bifurcation ad
infinity., Physica D: Nonlinear Phenomena., Volume 237, Issue
14-17, p. 2051-2055. 2008.
Real stable vortex on Saturn pole
Results and conclusions
In result, the dependence of libration points coordinates on mass and number of particles
is studied. Coefficients A and B have a limit at large N, depends on N and m/M ratio.
Accordingly, libration point coordinates close to a fixed value, slowly depends on mass
ratio. On the other hand, at large m coordinates have a maximal value, which respect a
case infinitesimal central mass. In case small m and small N we have a 3-body problem.
Positions of non-collinear libration points are slightly differ from n-gon described circle.
In conclusions, some generalizations of problem and possible way of future work are
considered. Area, where potential allows a linearization exists in small vicinity of N-gon.
However, oscillations even in this small area become strongly nonlinear due to
differential rotation of system. In fact, approximation ring potential by N-gon is an
example of a point-mass modeling. But specific of potential expansion gives an ability to
study more complex (non-homogenous) systems on base of considered model.
Few References
1.Wintner, A. The Analytical Foundation of Celestial Mechanics, Princeton University
Press (1941).
2.Rosaev A.E. The investigation of stationary points in central configuration dynamics.
Proceedings of Libration Point Orbits and Applications Conference, Parador d'Aiguablava,
Girona, Spain 10 - 14 June, 2002 Eds. by G. Gomez, M. W. Lo and .J.J. Masdemont,
World Scientific Pub, New Jersey - London – Singapore - Hon Kong, (2003). p. 623-637.
3.Rosaev A.E. The application of Computer Algebra to Central configuration dynamics.
Abstr. of 6-th IMACS Int Conf on Applications of Computer Algebra (IMACS ACA
2000), June 25-28, 2000 St.Petersburg, Russia, p.51.
4.Szebehely, V. Theory of Orbits. The Restricted Problem of Three Bodies. Academic
Press (1967).
5.Ollongren A. (1981). On a restricted (2n+3) body problem. Celestial Mechanics, 45,163168.
6.Roberts G. Linear Stability of the 1+n-gon Relative Equilibrium Hamiltonian Systems
and Celestial Mechanics (HAMSYS-98), World Scientific Monograph Series in
Mathematics 6, 303-330, 2000.
7.Boatto S, Simo C.: Thompson heptagon: a case of bifurcation ad infinity., Physica D:
Nonlinear Phenomena., Volume 237, Issue 14-17, p. 2051-2055. 2008.
Future works
•
•
•
•
Test particle motion in N-gon potential
Relative equilibrium in case non-equal masses
Approximation of oblate planet potential by system of N-gons
N-gon as a Toda chain with non-zero curvature
Thank You very much