Photo 1 shows the general makeup of a typical spiral galaxy. It has the heavier components collected in a relatively thin plane, and lighter components (probably gas and dust) extending away from the plane (in the z direction normal to the plane) like an atmosphere, with something like exponential decay in density. Thickness is important enough to be considered, although first-order answers could be found from a very thin disk. There is no evidence of any large-diameter halos.
Assuming axial symmetry and symmetry in z, and given the density distribution, using Newton's law to find rotation speeds vs radius (the forward problem) looks fairly easy. It could be further simplified if the densities were constant for a given radius, by changing the variable-density section into an equivalent one of constant density that has the same SMD (surface-mass density = thickness x density) and the same gravity effects.
The Sombrero galaxy looks like it has a halo, and it does, but it's not dark matter. It's gas and dust, pushed out to large radii by light pressure. Also the rim can be seen, so there is a finite maximum radius, true for all galaxies, even though there is gas at the rim. The rim could be defined as the maximum radius for a rotation-speed reading.
For Andromeda, photo3, about 300,000 stars have been found in the halo, apparently in out-of-plane orbits. This is not dark matter. They might be the residue of collisions with smaller galaxies. Compared to the billions of stars in the galaxy plane they do not have a first or second order gravity effect, although they could be included in the equivalent-section concept mentioned above.
Clearly any method for the forward problem must be able to handle arbitrary mass distributions. Look at photo 4. A large fraction of the mass is at the outer ring. In general very few, if any, galaxies can be represented by a smooth analytic function for the mass distribution.
The new method
Thickness is necessary to get density, and with a very thin disk the local SMD and total mass would be too high. So a scheme to account for thicknesses of various types is needed. Constant-density rings are easy to deal with in the computing (Nicholson 2000, 2003) and can handle any thickness distribution like bulbous centers changing to thin disks, or even discontinuous examples. But computing with a density in z such as that in photo 1 is quite difficult. So an equivalent constant-density thickness is used that is an accurate substitute for the actual z distribution.
When the equivalent density is found, the procedure is then reversed to find the detailed density change in z. Actually the equivalent density is a good average density for a given radius, and is the data presented in my plots.
This equivalent constant-density thickness distribution for the galaxy envelope is estimated by looking at the photos (as in photo 1) and comparisons with other galaxies. This thickness accounts for the large bodies and gas/dust near the disk plane, and an exponential "atmosphere" extending away from the plane. It has the same SMD and causes almost the same acceleration effects on the test mass as the actual (assumed) density distribution. For galaxies with no "atmosphere", it is assumed that there are about 1/2 large bodies and 1/2 gas/dust together near the disk plane, and for those the equivalent and basic (estimated) thicknesses are equal. For those with an "atmosphere" it is assumed they have about 1/3 large bodies, 1/3 gas/dust near the disk plane with the large bodies, and 1/3 gas/dust in the exponential "atmosphere," and for these the equivalent is 1.491 times the basic thickness as shown by trials.
It is best to remember that thickness is usually an estimate, since there are few direct measurements yet. One such was in an old Bok article for the Milky Way (1981). Edge-on views like photo 1 are a big help, but few are available..