Kenneth F Nicholson, retired engineer
rewritten 6/20/06
Key words: Newton, galaxy, rotation, dark matter
Photos by Hubble and NASA
Welcome. Here I start from the basic inputs for galaxies in the form of pictures (courtesy of Hubble and NASA), and develop a new method to find rotation speeds from mass distribution (forward problem), and mass distribution from rotation speeds (reverse problem). Surface brightness, dark matter, elliptic integrals or table lookups, are not needed. The computing is easy, using only sines and cosines and the usual tools available on a home computer. I show how to include the effects of thickness that has a mix of large bodies , gas, and dust, and an "atmosphere" that extends away from the disk. Also I introduce new dimensionless data reduction and a plotting format that allows easy comparisons of galaxies. If you have read my papers in the arXiv, you will find much that is familiar.
First a short history and critique of other methods is useful. When rotation speeds began to be measured, the constant-speed pattern was a big surprise. It was expected that speed patterns would be similar to those of our solar system, as if all disk mass seemed to act at the center. Many thought that a single ring acted like that, and in most texts rotation speeds were computed that way, by adding up ring effects as if they were spherical shells. Actually, if they had correctly worked out the rotation speeds for a flat, constant-thickness disk, the community would not have been surprised by the constant-speed pattern, but that result was apparently not well known if known at all.
Most methods assume axial symmetry with symmetry in z (normal to the disk), and no relativistic, magneto/electric, or gas pressure effects. Only gravity is acting, using Newton's law. My assumptions are the same. There are some papers using general relativity, but so far these show no advantages and add many complications. Also methods that change Newton's law, such as MOND, are not considered.
An early method for the forward problem is shown in Binney and Tremaine (B&T here, Galactic Dynamics, 1987, eq 2-146). It adds up the gravity effects of rings on a test mass in a zero-thickness disk, and the mass of each ring can be arbitrary. Unfortunately the results blow up when the test mass is at a particular ring, but this effect can be dodged by assuming the ring is split into two rings above and below the disk mid plane. However this method uses elliptic integrals that complicate the computing and does not account for thickness correctly. I haven't heard of its use lately, but I think it was on the right track. It could have been used to find the correct rotation speeds for a thin disk of constant SMD for example.
B&T next show a method that uses Bessel functions and Hankel transforms to find analytic solutions for the forward problem. So far as I can tell it has only two usable solutions. Both require the galaxies have "zero" thickness (very thin) and infinite disk radii. The first (B&T eqs 2-161 to 165) is called Mestes' Disk and uses a surface-mass distribution of SMD = A / r . Although SMD is infinite at the center, total mass inside a given r is finite. The resulting rotation speed at r is the same as that caused by the mass inside r acting as a point mass, ie the same as if all the mass were made up of spherical shells. However the mass outside any chosen r is infinite, and when the gravity effects of that are removed (they pull outward), the speeds get much larger toward the rim. This solution seems to be no longer used.
The second solution using Hankel transforms in B&T (eqs2-166 to 170) was derived by Freeman (1970) and has an exponential surface-mass distribution of SMD = A exp(-B r). It can have a negligible mass outside a chosen rim and seems usable. However the rotation speeds are only slightly higher than they would be if the mass distribution used was applied as spherical shells. These solutions are correct. Unfortunately the measured data show speeds nearly constant, or even rising toward the rim and there seem to be no galaxies to match these analytic solutions.
By far the most influential method however has been that of van Albeda et al (1985), that uses Freeman's solution. This method seems to be based on the idea that the local SMD of a galaxy is proportional to the local surface brightness, and since that seems to fall off exponentially to the galaxy rim, the Freeman solution must be correct for the disk part of a galaxy. So two brightness measurements, with a guess as to the mass/light ratio, are sufficient to define the mass distribution and rotation speeds for the Freeman disk. However rotation-speed results, although reasonable near the center, fall far short of measured data away from the center. They compensated for this by adding an analytic distribution of spherical shells, centered on the galaxy center, that allowed approximate matches of the data. Since there was no physical evidence of these shells they said they were made of dark matter. In the figure below, the squares of the disk and halo speeds are added to get the rotation profile. Fritz Zwickey named dar matter first back in the 1930's (as something that should be there), but this method was the driver for the use of the dark-matter spheres.
This started a whole new specialty in astrophysics, searching for dark matter. What is dark matter? It is matter that can be detected only by gravitational effects, and not by its own or reflected radiation. For galaxies in particular its presence is said to be implied because (as discussed above) the predicted rotation speeds are so much lower than the measured, and large spherical halos of dark matter can be used in the computing to make up the difference. For most cases the halos then have much more mass than the in-plane galaxy. Planets are not dark matter. Neither are objects orbiting a galaxy in an out-of-plane halo.