Dark matter Consider the rotation curves of a galaxy with density profile p(r). Assuming that sufficiently far from the centre of the galaxy the radial speed v depends on the dis- tance from the centre of the galaxy according to the relation v(r) oc where a is a constant, determine a relation between the density and radius, up to a constant of proportionality which you do not need to determine. If there is a supermassive black hole in the centre of the galaxy with mass MSMBII, how would the velocity depend on r close to the centre of the galaxy? Ignore any relativistic effects. Which condition could you use to estimate the radius at which the velocity profile would start to deviate away from the result you derived for v(r) close to the black hole? Consider the hypothesis that dark matter could be made out of black holes which are stellar remnants. Briefly list six pieces of observational evidence for the existence of dark matter and state clearly which items on your list are compatible with this hypothesis, and which are incompatible.

Dark matter Consider the rotation curves of a galaxy with density profile p(r). Assuming that sufficiently far from the centre of the galaxy the radial speed v depends on the dis- tance from the centre of the galaxy according to the relation v(r) oc where a is a constant, determine a relation between the density and radius, up to a constant of proportionality which you do not need to determine. If there is a supermassive black hole in the centre of the galaxy with mass MSMBII, how would the velocity depend on r close to the centre of the galaxy? Ignore any relativistic effects. Which condition could you use to estimate the radius at which the velocity profile would start to deviate away from the result you derived for v(r) close to the black hole? Consider the hypothesis that dark matter could be made out of black holes which are stellar remnants. Briefly list six pieces of observational evidence for the existence of dark matter and state clearly which items on your list are compatible with this hypothesis, and which are incompatible.