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:
Recall that in the Reciprocal System two forces are acting on the star:
My goal in this paper is to derive the expression for the net force acting on the star, to find the equilibrium position (xpo) of the star, and to determine whether or not this position is stable.
Nehrus recent paper  provides the starting point. Some additional symbols are needed:
dog = gravitational limit of the globular cluster
dop = gravitational limit of nearest neigboring star:
yg = non-dimenslonal distance of the star from the mass center of the globular cluster
yp = non-dimensional distance of the star from the mass center of the nearest neighboring stars
vog = zero-point speed of the star relatlve to the globular cluster
vop = zero-point speed of the star relative to the nearest neighboring stars
vng = net inward gravitational speed of the star
vnp - net outward progression speed of the star
vn = net speed of the star
G = universal gravitational constant
Mo = mass of the sun
ag = acceleration from gravitation of the globular ctuster
ap = acceleration from progression away from the nearest neigbors
an = net acceleration of the star
In this notation,
Differentiating the velocity expressions with respect to time gives the accelerations:
Then, in terms of xpo, 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 Mg = 200*Mo , mp = 2*Mo , xg = 40 ly, dog = 53.32 ly, and dop = 5.33 ly produced xpo = 9.29 ly.. Another sample run with Mg = 30000*Mo , mp = 200*Mo ,xg = 400 ly, dog = 652.98 ly, and dop = 33.32 ly produced xpo = 178.94 ly. Input parameters that are physically impossible produce negative distances.
Now lets turn to the question of the stability of this positlon, xpo 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.
This is the definition of instability. Hence, xpo 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.