GLOBULAR CLUSTER MECHANICS IN THE RECIPROCAL SYSTEM

This paper discusses the forces on stars in a globular cluster. Consider Figure 1; the symbols are defined as follows:

M_g = mass of the stars of a globular ciuster Internal to g that of a particular star

m = mass of that particular star

m_p = mass of the nearest neigboring stars

x_g = distance of the star from the mass center of the globular g cluster

x_{p =} distance of the star from the mass center of the nearest neighboring stars

x_pg = distance of the mass center of the nearest neighboring stars from the mass center of the globular cluster

x_po = equilibrium distance of the star from the mass center of the nearest neigboring stars

x = distance of the star from the mass center of the nearest neighboring stars, relative to the equilibrlum distance

Recall that in the Reciprocal System two forces are acting on the star:

1. Gravitation of the star by the cluster as a whole—this produces an inward motion.
2. Progression of the star away from its nearest neighbors—this produces an outward motion.

My goal in this paper is to derive the expression for the net force acting on the star, to find the equilibrium position (x_po) of the star, and to determine whether or not this position is stable.

Nehru’s recent paper [1] provides the starting point. Some additional symbols are needed:

d_og = gravitational limit of the globular cluster

d_op = gravitational limit of nearest neigboring star:

y_g = non-dimenslonal distance of the star from the mass center of the globular cluster

y_p = non-dimensional distance of the star from the mass center of the nearest neighboring stars

v_og = “zero-point speed” of the star relatlve to the globular cluster

v_op = “zero-point speed” of the star relative to the nearest neighboring stars

v_ng = net inward gravitational speed of the star

v_np - net outward progression speed of the star

v_n = net speed of the star

G = “universal” gravitational constant

M_o = mass of the sun

a_g = acceleration from gravitation of the globular ctuster

a_p = acceleration from progression away from the nearest neigbors

a_n = net acceleration of the star

In this notation,

Differentiating the velocity expressions with respect to time gives the accelerations:

At equilibrium,

Let

Then, in terms of x_po, at equilibrium,

a quartic equation.

The appendix gives a simple computer program written in BASICA to solve equation 17 numerically. (An attempt to solve the equation analytically using the MU MATH AI program failed). A sample run with M_g = 200*M_o , m_p = 2*M_o , x_g = 40 ly, d_og = 53.32 ly, and d_op = 5.33 ly produced x_po = 9.29 ly.. Another sample run with M_g = 30000*M_o , m_p = 200*M_o ,x_g = 400 ly, d_og = 652.98 ly, and d_op = 33.32 ly produced x_po = 178.94 ly. Input parameters that are physically impossible produce negative distances.

Now let’s turn to the question of the stability of this positlon, x_po The net force acting on the star in terms of the distance from equilibrium, x, is

Differentiating F with respect to x gives

If x is positive, dF / dx is positive and hence F increases with x.
If x is negative, dF / dx is still positive. Thus

This is the definition of instability. Hence, x_po is a point of unstable equilibrium. But there is one saving grace: the forces near this point are quite small, so sudden changes in position are precluded.

Globular clusters continually grow by accretion until eventually being absorbed into galaxies. The stars in the clusters must keep changing their temporary equilibrium positions.

Reference

1. K. V. K. Nehru, “The Gravitational Limit and the Hubble’s Law,” presented at the 1986 Convention of ISUS.