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How can I calculate the rotation curve, [tex]v(R)[/tex], for test particles in circular orbits of radius [tex]R[/tex] around a point mass [tex]M[/tex]?
Ok, I guess this is just the velocity function
[tex]v(R)=\sqrt{G\frac{M}{R}}[/tex]
but how about test particles in circular orbits of radius [tex]R[/tex] inside a rotating spherical cloud with uniform density?
Yeah thanks, then
[tex]v(R) = \sqrt{\frac{4}{3} \rho G \pi R^2}[/tex].
But what if the test particle is rotating inside a spherical halo with density [tex]\rho(r) \propto 1/r^2[/tex]?
Then you proceed as before except that the mass contained within a radius R won't simply be [itex] \rho \frac{4}{3} \pi R^3 [/itex]. You will have to do a (simple) integral to find the mass contained within a radius R, namely
[tex] M(R) = 4 \pi \int_0^R dr r^2 \rho(r) [/tex]
Notice that something special happens to v(R) when the density has the radial dependence you gave....Which has some connection with observations fo rotation curves of galaxies and dark matter.
Patrick