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Logarythmic
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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]?
Logarythmic said: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?
Logarythmic said: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]?
nrqed said: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
A rotation curve is a graphical representation of the orbital velocity of objects that orbit around a central mass, such as stars rotating around a galaxy.
Calculating the rotation curve allows us to study the distribution of mass within a galaxy and better understand its structure and dynamics. It also helps us to test different theories of gravity and the laws of motion.
The rotation curve is calculated by measuring the orbital velocities of test particles at different distances from the center of the galaxy and plotting them on a graph. This information can then be used to create a mathematical model of the rotation curve.
The rotation curve can be affected by the distribution of mass within the galaxy, the presence of dark matter, and the laws of gravity. Other factors such as interactions with neighboring galaxies and the presence of a central black hole can also influence the rotation curve.
The rotation curve can provide evidence for the existence of dark matter, as it often shows that the observed orbital velocities of stars are higher than what can be explained by the visible mass in the galaxy. This suggests that there must be an additional source of mass (dark matter) that is contributing to the gravitational force.