Steven Janowiecki
Astr306 HW6
11/30/07

We begin by redefining our logarithmic potential to have a cutoff radius
of 30 times the scale length. The scale height is 1 kpc, and so the cutoff
radius is 30 kpc. As before, a hole is carved at the center, within 10% of
the scale length, to prevent unrealistic interactions with the potential's
singularity. Thus, to find the mass contained, we integrate the density
(rho = v0^2 / 2 pi G r^2) from a radius of 0.1 kpc out to the cutoff of
30 kpc (dtau = 2 pi r^2 dr). The enclosed mass is then
M = (2*v0^2 / G)*(30 - 0.1 kpc). Plugging in our value of v0 = 220 km/s,
we obtain M_enclosed = 5.56(10^11) M_sun. (v0 = 220km/s in reality, but
for our model, we use v0 = 1km/s)

Outside of this cutoff radius, the potential will be Keplerian (treat-able
as a point mass) and go as -GM_tot/r + C, where the constant is determined
in seaming the logarithmic potential to the Keplerian potential. At the
cutoff radius, the two potentials must be equal, so:

      phi_log(r=30kpc) = phi_kep(r=30kpc)
v0^2 ln (30kpc / r0)^2 = -GM_tot / 30kpc   + C

We solve for C and then substitute in our values for M_tot=1.390(10^7)M_sun,
v0=1km/s, r0=1kpc.

C = v0^2 ln( (30kpc/r0)^2 ) + GM_tot / 30kpc
C = 4.78(10^5) km^2/s^2

Attached is a plot of our resulting potential, as well as the original
logarithmic and Keplerian potentials that compose it. In the plot, the
thick black line shows our composite potential, while the red and green
show the full range of the other two potentials. As expected, the Keplerian
potential is getting increasingly flatter past the cutoff radius, whereas
the logarithmic potential continues to rise. Additionally, the Keplerian
potential begins to misbehave around very small radii, as it is the
potential from a point mass.