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 (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} = nondimenslonal distance of the star from the mass center of the globular cluster y_{p} = nondimensional distance of the star from the mass center of the nearest neighboring stars v_{og} = “zeropoint speed” of the star relatlve to the globular cluster v_{op} = “zeropoint 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.
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
