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

The following link is to 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.

http://astronomy.case.edu/steven/hw/astr306/hw6/potgraph.ps(.gif)


Now we need to add a satellite galaxy to our main one. Modeling it as a
point mass, we will start it a distance r_init from the center of the big
one. Provided r_init is greater than 30 kpc (it will be), we find the
escape velocity from the Keplerian potential. As calculated before, this
is sqrt(2|phi(r)|), in our case, v_esc = sqrt(2*GM/r_init + 2*C).

In adapting the previous version of code to this one, we need to calculate
the acceleration at a point in the Keplerian potential. As usual, this is
the negative gradient of the potential, or a_i = -G M_tot i / r^3. All
particles within the cutoff radius of the main galaxy will see the
logarithmic potential, with a_i = -2 v0^2 i / r^2.

The previous assignment generated an exponential disk which was rotating
circularly, with a scale length of 0.3 (model units) and a circular
velocity of sqrt(2) (model units). Now, we multiply all positions by 10/3
to obtain units of kpc (matching the scale length of the logarithmic
potential), and leave the velocities the same (now in km/s), in accordance
with v_circ = sqrt(v0) = sqrt(1 km/s)

The satellite is started 2 truncation radii above the main galaxy, and
offset from the center of the galaxy by one third of the scale length,
to avoid troublesome interactions with the center of the potential. It
is given a variety of masses, in terms of the main galaxy's mass: 0.1, 1,
2, 3, 4, 5%. All simulations are availible as animated gif files at this
website:

http://astronomy.case.edu/steven/hw/astr306/hw6/movie*.gif


The "best matches" for each simulation are graphed in the following folder

http://astronomy.case.edu/steven/hw/astr306/hw6/best/*.ps(.pdf)

to be compared with the Cartwheel Galaxy. From these, it seems that the
companion galaxy in the Cartwheel was probably around 4% of its mass. The
satellites with mass less than 4% make weak or short-lived rings, and none
form the central concentration. The satellite with 4% mass forms a tight
central concentration surrounded by a well-defined ring, as in the Cartwheel
galaxy. Increasing only one percent to 5% yields a galaxy without and
central concentration but with a strong ring (that does not appear to turn
around for quite a while). At satellite masses of 5% and higher, the main
galaxy's potential begins to be dragged away, and that is a result that
should not be believed from this simulation (there is some lurking error
or oversight).