This is in answer to the original topic of this thread, by Grav and Cougar on 4/24/08 "galaxy rotation speeds and mass distribution"

The key to finding rotation speeds from galaxy mass distribution is to realize that a disk can be represented by a series of rings. For a single ring a test mass at the center of the ring has no acceleration, but as it goes out toward the ring it is attracted toward the ring more strongly , until very near the ring. Outside the ring the attraction is again strong but reversed, and falls off as the radius increases, becoming asymptotic to that caused by the ring as a point mass. For a zero-thickness "wire" ring the acceleration becomes infinite at the ring in each direction, because the ring has mass but no volume. But a simple thought experiment shows this is wrong for galaxies. Imagine a test mass passing through a "cloudy" ring with thickness in the z direction (perpendicular to the disk) and also a little in the r direction. We know the acceleration should look like that of the wire ring until the test mass gets close to the ring, then the acceleration should smoothly change signs as it passes through the ring. Right at the ring the acceleration should be the same as if the mass of the ring was concentrated at the center (this is not obvious). A proper simulation will include this characteristic.

So the gravitational field for a ring is quite different from that for a sphere. In particular a test mass inside a spherical shell is not attracted at all by the shell. The approach that says the rotation speed at a given radius of a galaxy is set by the mass inside that radius is WRONG. The mass outside that radius can have a strong effect on that inside.

Thus the forward problem (acceleration, and thus rotation speeds, from mass distribution) becomes straightforward, given the geometry (including thickness) and mass or density distribution. Just add up the acceleration effects of all the rings on a test mass at selected radii. To do this I first compute the acceleration a(i,j) of a test mass at each desired radius j caused by the ring i with unit density. I say I here because most people do not consider thickness or density. The matrix solution for the total acceleration A(j) at ring j becomes, with p being the ring density,

a(i,j)*p(i) = A(j) , forward problem

Now the reverse problem (mass distribution from speeds) becomes easy because A(j) is known from the measured speeds, a(i,j) does not change and its inverse ainv(i,j) can be used. The solution is now for density and thus the local SMD (surface mass density, mass per unit surface area).

p(i) = ainv(i,j)*A(j), reverse problem

I think the best place to see more details of this development and application to several galaxies is in my paper, arxiv-ph/0309762 v2. Probably the best place to see how it was done in error is the original source of dark matter, van Albada et al, Apj 295 305 (1985). There they assumed that since galaxy light seemed to fall off exponentially to the rim, and mass was thought to be proportional to light, that ALL galaxy disks must have an exponential mass distribution. When they used the known analytic solution for a disk with exponential loading (correct, Freeman ApJ 160 811F, 1970) they found the measured speeds were much larger. Rather than putting the needed extra mass in the disk (perhaps because they didn't know how) they added spherical shells, centered on the galaxy center, with the density adjusted to cause the combined acceleration effects to approximate those needed. Wikipedia echoes those results.

Since there was no evidence of those shells, they called them dark matter, and it has aquired special properties as the time-and-money-consuming search for it has gone on over the years: it cannot be detected by its own or reflected radiation, only by gravitational effects. Well there is no doubt about that! After all it is imaginary. It's not hard to predict it will never be found. Fritz Zwicky coined the term back in the 30's, but he apparently thought the mass he needed was just ordinary non-star matter.

The dark-matter solution using spherical shells was obviously wrong because it allows an infinite number of solutions, as the ratio of shell matter to disk matter is changed. A good match of speeds can be found using spherical shells alone! Freeman and van Albada were good mathematically (not so their peer reviewer) so that the solution might have been a hoax. If so it has been one of the most successful scientific hoaxes of all time.

Ken Nicholson, knchlsn@sbcglobal.net