Regular n-gon as a model of discrete gravitational system

Rosaev A.E.OAO NPC NEDRA, Jaroslavl Russia,

E-mail: hegem@mail.ru

IntroductionIntroduction

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 point-mass modeling. But specific of potential expansion gives an ability to study more complex (non-homogenous) systems on base of considered model.

DefinitionsDefinitions

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.

Definition [1]: A configuration of the n particles is a central configuration at time t if there exist some scalar such that irir for i=1,...,n.

iii r

Urm

0

ii

mrrU

ji

ji

rrmm

U

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 (librationpoints) appear in a system (Fig.1).

Fig. 1. Libration points positions for 3-body (left) and N-body (right) relative equilibrium configurations.

N

j j

j

RxRx

xRGm

RGMR

12322

2

22

12sin2

2sin2

1

12322

2

222

12sin2

2sin2N

j j

j

RxRx

xRGm

xGm

RGMR

j j N 2 0

The libration point coordinates can be determined from equations [2]:

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]):

2/1

1

1

10 2sin

41

N

jjM

m 30 R

GM

The equations for the collinear libration points are the most interesting. They can be reduced to a fifth degree polynomial:

(2+A)x5+ (32R + 2 AR + B)x4 + (32R2 + R2A+ 2BR)x3+ (BR2Gm)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:

1

133 2sin8

3

2sin2

1N

j jjR

GmA

1

12 2sin

14

N

j jRGmB

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.

SummationSummation

N 100 2.408618 2.404013500 2.404358 2.404113

1000 2.404182 2.4041131500 2.404146 2.4041132000 2.404132 2.4041142500 2.404126 2.4041143000 2.404123 2.404114

3))/sin(2/(1 Ni 3/2 i

Coefficients A of expansion depending on number of particles. For collinear case:

333

1

133

)2/(...6541137.1sin8

3sin2

1RGmN

NjNjRGmA

N

j

33

1 1

3333 )2/(...404101.2/2)2/())/sin(2/(1 NiNNiN

i

N

i

For non-collinear case::

21

1

12

/1)/(sin

14 R

GmiNNjR

GmBN

i

N

j

By analogy with three body problem, solution of equations for collinear libration points can be calculated as a series:

...239

1233

123

3/23/1

1

BAMm

BAMm

BAMmx

...239

1233

123

3/23/1

2

BAMm

BAMm

BAMmx

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 1014 x5 0.766177581022708 1014 x4 38309057078071 x3 := 0.2928612775186 107 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

Fig. To a non-collinear pointscoordinates determination

:= J , ,0 3 2 2 A 3 4 4 2 A2 ( )2 A

3 2 2 A 3 4 4 2 A2 ( )2 A

Equation for non-collinear libration pointshas trivial solution x=0 and non-trivial solution of cubic equation:

which have at least one real root. For case B=0 we have three real roots if A sufficiently large (small central mass):

0)23()23()(

2222

2232

BRxBRARRxBARRxA

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 Nwe have chaotic motion due to close encounters. If N sufficiently large, effect of initial phase not significant.

outermnPpRr 1:/)/( 2/3

innermnPpRr 1:/)/( 2/3

Conjunction. At fixed total mass of N-gon, m=miN , interaction test particle with regular N-gon strongly depends on number of N. At conditions:

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 Ndependence.

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 bykeplerian angular velocity (L - angular momentum):

))cos(1(02 emr

Ldtdconst

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

ji

jijiij yyxxU2/122log

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

Thompson HeptagonThompson Heptagon

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.

Results and conclusionsResults and conclusions

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, 45,163-168.6.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.

Few ReferencesFew References

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