2) (10 points) The dark matter halo of a galaxy can be described by...

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2) (10 points) The dark matter halo of a galaxy can be described by the equation: ATGA(+)== r2 + 2 where p(r) is the density of the halo at distance r from the galactic centre, V is the velocity of an object in the halo, and a is the "length scale" for the halo. Consider an object of mass M orbiting through the dark matter halo of the Milky Way which has a density as described by the equation above and a = 30 kpc. a) (4 points) Using equations derived in our notes, prove that the force opposing the motion of object M through the MW's dark matter halo is: 22 GM² A where: A-Tmas b) (4 points) If the dynamic frictional torque acting on object M (i.e. T = r Fill ) is equal to the rate of change of its momentum (i.e. dL/dt = MV dr/dt), show that the time (ts) it would take for the object to spiral in from a maximum galactocentric distance (Imax at t = 0) to a minimum galactocentric distance (Imin at t = ts) is given by: = 2GM V In A - c) (2 points) The LMC has a mass of M = 2x1010 Mo, a velocity of V = 200 km/s, and is currently at a distance of Imax = 50 kpc. A globular cluster in the MW's halo has M = 106 Mo, a velocity of V = 200 km/s, and a distance of Tmax = 30 kpc. Calculate how long (in yrs) it will take the LMC and the globular cluster to spiral in to a distance of Tmin = 4 kpc.

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