In the solar system the orbital velocities of the planets are inversely proportional to the square root of their orbit radius. This is just a corollary to Kepler's Law. This velocity profile prevails because almost all of the mass of the solar system is contained in the sun and thus each planet experiences nearly the same gravitational attraction. If the mass of the astronomical system is spread out, as in a galaxy, then the tangential velocity profile is quite different.

Consider an astronomical body in which the mass is distributed spherically. Let M(r) be the amount of mass that is contained within a distance r of the center of the body. For circular orbits the gravitational attraction for a mass m has to balance the centrifugal force for that mass in the orbit. This means that

GmM(r)/r² = mv²/r
 

where G is the gravitational constant and v is the orbital velocity. The mass m cancels out and is irrelevant. Thus the orbital velocity is given by

v² = GM(r)/r
 

Where M(r) is a constant as in the solar system this reduces to

v² = GM₀/r
and hence
v = (GM₀)^½/r^½
 

If the mass is uniformly distributed at a density ρ then M(r)=(4/3)πρr³ so

v² = G(4/3)πρr²
and hence
v = [(4/3)Gπρ]^½r
 

and thus tangential velocity increases with distance from the center.

Although there are spherical galaxies the more common shape is disk-like. The analysis of gravitational attraction in a disk requires first analyzing the gravitational attraction due to a ring element.

The attraction of an element of the ring Rdθ on a point at X is a function of the distance D, which is given by

D² R²sin²(θ) + (X − Rcos(θ))²
= X² + R² − 2XRcos(θ)
 

The force per unit mass at X due to the infinitesimal mass element ρRdθ at θ is equal to (GρRdθ)/D², where G is the gravitational constant, ρ is the lineal mass density in the ring. But this force is directed at an angle ψ so the force magnitude must be multiplied by the cosine of ψ, where

cos(ψ) = (X−Rcos(θ))/D
 

Thus the horizontal magnitude is given by

dF = (GρR)[(X−Rcos(θ))/D³]dθ
 

The net force at X is obtained by integration over θ from 0 to 2π; i.e.,

F(X) = (GρR)∫₀^2π[(X−Rcos(θ))/D³]dθ
 

The total force at X, T(X), is obtained by integrating over all values of R from 0 to the maximum radius of the disk, R_m.

T(X) = ∫₀^R_m(GρR)∫₀^2π[(X−Rcos(θ))/D³]dθdR
 

In general the density factor ρ may be a function of R. This dependence may due to variation in the volume density with R and/or a variation in the disk thickness with R.

The equating of gravitational and centrifugal force gives:

T(X) = V²/X
and hence
V² = XT(X)
 

Due to the complexity of T(X) it is not possible to determine analytically the dependence of V on R. Some preliminary numerical analysis will be given later.

In the case of spherical shells there are theorems which allow the effect of all shells within the radius X to be replaced by the mass concentrated at the center. The shells with a radius greater than X have no effect. Such theorems do not exist for cylindrical shells. However if they did exist then for a disk of constant thickness and areal density ρ

M(r) = ρπr²
and thus
V² = Gρπr²/r = Gρπr
and hence
V = (Gρπ)^½r^½
 

If the areal density drops off linearly; i.e., ρ(r) = ρ₀−σr,

then

M(r) = ρ₀πr² − σ(2/3)πr³
and therefore
V² = ρ₀πr − σ(2/3)πr²
 

This indicates a velocity profile which is increasing for a low values of r and then level and then decreasing for higher values of r.

If the thickness and hence areal density is of the form ρ=σ/r then the galaxy would have a form such as

For this case

M(r) = ∫₀^r(σ/s)(2πs)ds = 2πσr
and hence
V² = G2πσr/r = 2Gπσ
and hence
V = (2πσ)^½
 

The above indicates that the velocity profile depends upon the thickness and density profiles of a galactic disk. The numerical computation of the force in a galactic disk of constant thickness and uniform density gives the following profile.

For a balance of gravitational attraction and centrifugal force the tangential velocity squared V² must be directly proportional to the product of distance and force, XF(X). Therefore the velocity profile is as shown.

Thus merely having the mass of a galaxy distributed evenly in a disk would account for an essentially constant tangential velocity profile. For more details on the computation see gravitational forces in a galactic disk

A more realistic shape for a galaxy would involve a spherical globule in the center with an annulus disk around it such as shown below.

From the preceding analysis the tangential velocity would rise linearly within the radius of the sphere and then remain constant in the annulus disk.

This is pretty much what is seen in the empirical velocity profiles such as the one shown below.

Of course, not all galaxies have a velocity profile of this shape.

(To be continued.)