The pull of that mass gravitationally, Fg, is given by the classic equation
Here, m is the mass of our circling object or star, and G is the universal gravitational constant. Let’s express M in terms of the radius and the density of dark matter:
M=4/3 πρr^3→F_g=4/3 πρGrm≡krm
Here, I gathered all the constants into a single constant, k.
The star is moving with a speed v (I call it speed because velocity is a vector, a speed in a specified direction. That direction is constantly changing as the star orbits, but, for a circular orbit we’re considering, the speed stays the same). The centrifugal force, Fc, is then given by another classic equation,