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:
 Gravitation of the star by the cluster as a whole—this produces an inward motion.
 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} = 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,
d_{og} = 3.77 * (M_{g } / M_{o})^{½ } (ly) 
(1)

y_{g} = x_{g} /d_{og} = (x + x_{po} + x_{pg} ) / d_{og} 
(2)

v_{og} = (2 * G * M_{g} / d_{og} )^{½} 
(3)

v_{ng} = v_{og * }(1 / y_{g} ^{½}  y_{g}½) (Inward) 
(4)

d_{op} = 3.77*(m_{p} / M_{o})^{½} (ly) 
(5)

y_{p} = x_{p} / d_{op} = (x + x_{po}) / d_{op} 
(6)

v_{op} = (2 * G * m_{p} /d_{op} )^{½} 
(7)

v_{np} = (½) * v_{op} * (y_{p}  1/y_{p} ) (outward) 
(8)

v_{n} = v_{np}  v_{ng} 
(9)

Differentiating the velocity expressions with respect to time gives the accelerations:
a_{g} = G * M_{g} * (1/x_{g}²  1/d_{og} ²) (inward) 
(10)

a_{p} = G * m_{p} * (½) * (x_{p} / d_{op}³  d_{op} / x_{p}³) (outward) 
(11)

a_{n} = a_{p}  a_{g} 
(12)

At equilibrium,
a_{n} = 0 
(13)

Let
h = m_{p} / (2 * d_{op}³) 
(14)

i = M_{g} * (1/d_{og}²  1/x_{g}²) 
(15)

j = (1/2) * m_{p} * d_{op} 
(16)

Then, in terms of x_{po}, at equilibrium,
h * x_{po} ^{4} + i * x_{po} ³  1 = 0 
(17)

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
F = m * G * ((1/2) * m_{p} * ((x_{po} + x)/d_{op}³  d_{op} / (x_{po} + x)³)  
 M_{g} * (1 / (x_{po} + x + x_{pg} )²  1 / d_{og}²)) 
(18)

Differentiating F with respect to x gives
dF / dx = m *G * ((1/2) * m_{p} * (1/d_{op}³ + 3 * d_{op} / (x_{po} + x)^{4})  
+ 2 * M_{g} / (x_{po} + x + x_{pg} )³) 
(19)

If x is positive, dF / dx is positive and hence F increases with x.
If x is negative, dF / dx is still positive. Thus
 dF / dx < 0 
(20)

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