To solve this problem we will use the concepts related to gravitational acceleration and centripetal acceleration. The equality between these two forces that maintains the balance will allow to determine how the rigid body is consistent with a spherically symmetric mass distribution of constant density. Let's start with the gravitational acceleration of the Star, which is
Here
Mass inside the orbit in terms of Volume and Density is
Where,
V = Volume
Density
Now considering the volume of the star as a Sphere we have
Replacing at the previous equation we have,
Now replacing the mass at the gravitational acceleration formula we have that
For a rotating star, the centripetal acceleration is caused by this gravitational acceleration. So centripetal acceleration of the star is
At the same time the general expression for the centripetal acceleration is
Where is the orbital velocity
Using this expression in the left hand side of the equation we have that
Considering the constant values we have that
As the orbital velocity is proportional to the orbital radius, it shows the rigid body rotation of stars near the galactic center.
So the rigid-body rotation near the galactic center is consistent with a spherically symmetric mass distribution of constant density