Sparke & Gallagher problem 3.19

Hint: the energy E is the sum of kinetic and potential energies while the virial theorem tells you how the energy is divvied up between potential and kinetic terms.

Sparke & Gallagher problem 4.2

Sparke & Gallagher problem 4.6

Use equation 4.11 instead of 4.7. WHY is this more appropriate?

Sparke & Gallagher problem 4.13

Note: While you will need both equations 4.24 and 4.25, in order to determine the combined mass m+M, they mean to say to use equation 4.25, NOT 4.24 as is printed in the text. Once you have the total mass, you'll need 4.24 to get the time till future collision.

Do not repeat the exercise as suggested in the second paragraph. Instead, tell me how increasing the mass affects the time to collide. I.e., does more mass take longer to collapse, or less time? Why?

Given even a crude age for the universe, this was one of the early arguments in favor of lots of dark matter in the Local Group.

It is frequently argued (as in the text) that the MW-M31 orbit must be very nearly radial (e = 1), but this is largely supposition: we have no direct measurement of the proper motion of M31. To see why, compute the proper motion M31 would have if its transverse velocity were 100 km/s. (You should find microarcseconds/year to be an appropriate unit.)

Now suppose the orbit is not perfectly radial, but still pretty eccentric. The development parameter doesn't change much from the e = 1 case [to see this, it may help to plot f(η) = (dr/dt)/(r/t_o) = e sin(η)(η-e sin(η)/(1-e cos(η))²]. Adopting e = 0.8 and η = 4.3 for an age of 13.2 Gyr, determine the semi-major axis a of the orbit. Will M31 collide with the Milky Way now? Assume their centers require a clearance of 100 kpc for safe passage. (That's twice the distance to the Magellanic Clouds.) [Recall that the pericenter = a(1-e).] Does the implied mass increase or decrease? Do either of these changes (to a and m+M) seem so dramatic that they would be readily obvious?