The high school math explanation is that centripetal force is equal to w^2 x R, w (omega) = radial velocity = 2 x pi / T_orbit, R = distance from central attracting body. Gravitational force goes as 1/R^2. Hence, for a body in circular obit,
w^2 x R = mu/R^2 ,
mu is a characteristic of the central body, its gravitational constant and is proportional to its mass.
More high school math, solve for w:
w = (mu)^(1/2) x (R)^(-3/2)
The gravitational attraction of a spherical shell is zero for points inside of it, and acts as if all its mass were concentrated at its center for points outside. If galaxies are embedded in a spherical soup of particles resembling neutrinos, but more massive, and not interacting with “ordinary” matter at all, except gravitationally, the gravitational attraction as a function of distance from the center of the sphere (assuming uniform density) would go as:
F(R) = G x rho x 4/3 x pi x R^3 /R^2
G = Newtonian gravitational constant
rho x 4/3 x pi x R^3 = mass within sphere of radius R
rho = density, mass per unit volume
pi = 22/7
Canceling terms:
F(R) = (G x rho x 4/3 x pi) x R
and
w^2 x R = F(R) = (G x rho x 4/3 x pi) x R
solving for w
w = (G x rho x 4/3 x pi)^(1/2)
which is independent of R, if rho is independent of R
the stuff making up rho does not show in telescopes, hypothetically because it does not interact with photons (is electrically neutral). If it does not interact with the weak or strong subatomic forces, there is no way of ever detecting them.
Stay tuned for further developments.