The applet physics here are pretty simple. We assume two models for mass distribution of the disk and dark halo, calculate the predicted rotation curves for each component, add them together, and plot them on top of the data.
Disk model:Halo model:
We presume the disk is characterized by an exponential distribution of stars:
: disk mass-to-light ratio
: central luminosity density
light: radial luminosity profile = exp(-r/h)
h: disk scale length
and h come from observations of the galaxy, while is a free parameter.
This mass distribution gives rise to the following rotation curve:
where y = r/2h , I and K are Bessel functions.Adding velocities:
The model we use for the halo is that of the isothermal sphere:
: halo central density
rc : halo core radius
this model has a constant density core and a density at large radius which drops as
r-2, giving a flat rotation curve.
This mass distribution gives rise to a rotation curve which looks like:
The last thing to point out is that velocities add in quadrature:
because we are really adding two mass distributions, and mass is proportional to velocity squared.