Formula for Rotation curves of Galaxies To ask a more specific one for the rotation curves of elliptical galaxies, and hope from there to later understand the dynamics of spiral galaxies. 


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*Treating the galaxy as an isothermal gravitational gas sphere, what is the equation of Density for an elliptical galaxy? 

*Assuming the above density and modelling by the Virial Equation, what are the Velocities?
Note: I’ve drawn on a paper by Fritz Zwicky 1937 “On The Masses of Nebulae and of Clusters of Nebulae” 
 A: The Keplerian formula assumes that only a central (point) mass exerts gravity on the orbiting mass. This is an excellent approximation for each individual planet in our Solar System, for instance, but it is totally wrong when applied to star orbits in a galaxy.
In that latter case, you consider basically all the material inside a given object's orbit. Therefore, the effective mass contributing to the orbits of test masses at various radii increases with radius, so the formula
$$ v =  \sqrt{\frac{G M}{R}}$$
becomes
$$ v =  \sqrt{\frac{G M_{(R)}}{R}}$$
where $M_{(R)}$ is the mass enclosed as a function of radius.
For a spiral galaxy approximated as a cylinder, $$  M_{(R)} = \int_0^R \rho_{(r)} 2 \pi r h * dr $$ $\rho$ is the density of stars, $h$ is the height of the cylinder (disk thickness), and $r$ is radius. $\rho$ is roughly proportional to $1/r$, leaving $M$ roughly proportional to $R$, leaving $v$ approximately independent of $R$.
PS: Curve A in the link is what you get for calculating rho according to visible matter only, Curve B for visible + dark matter. Or if you replace traditional gravitational calculations with MOND, of which I am personally very fond, but which is waaaaaaay beyond the scope of this answer.
A: It looks pretty similar to me, here is the wolfram alpha plot of $$v \propto 1/\sqrt{R}$$  wolf alpha graph
