 Rotation Curve Fitting

The MOND procedure in a nutshell

First, obtain data! One needs for galaxies the rotation curve and a photometry of the stars and gas.
Onse also needs to measure the inclination of the galaxy. This is used to correct both the rotation curve [V(r) = vobs(r)/sin(i)] and the surface photometry.

Traditionally, we fit ellipses to the images to obtain an azimuthally averaged surface brightness profile (the solar luminosity per square parsec as a function of radius along the major axis of the ellipses). [I say traditionally because it is now possible to use the full 2D information with programs like DISKFIT rather than azimuthally averaging around each ellipse.] The surface brightness profile is corrected to face-on for a measured inclination and a guess as to what the galaxy would look like edge-on. (i.e., we usually assume some finite thickness for the galactic disk. However, real disks are observed to be pretty thin, about 8:1 on average, so simply assuming a thin disk [S(r) = sobs(r)/cos(i)] is already pretty good.)

The face-on surface brightness profile is converted to a mass surface density profile assuming a mass-to-light ratio for the stars. This is later treated as a fit parameter - we do not know M*/L a priori. For the HI gas, this conversion is known (there is no free parameter). Most galaxies are treated as two components, a thin disk of stars plus a thin disk of gas. Some additionally have a central bulge component which might be treated as a 3D blob of stars with a separate M*/L from the disk.

From the mass surface density profile, the Poisson equation is numerically solved to obtain the rotation curve predicted by each component: stars and gas. i.e., for the observed distribution of stars, Newton predicts we should observe V*(r), and the same for the gas Vg(r). There are tasks in the GIPSY package to do this. [If you wish to write your own, read up on it first to save yourself from rediscovering some numerical traps.] 