by Michael McLaughlin M. Eng. Sc.
e: michael@thecurate.com
The Galaxy Rotation Problem
Figure 1.
The rotation curve of a galaxy can be represented by a graph that plots the orbital velocity of the stars or gas in the galaxy on the y-axis against the distance from the centre of the galaxy on the x-axis. Stars revolve around the centre of galaxies at a constant speed over a large range of distances from the centre of the galaxy. Thus they revolve much faster than would be expected if they were in a free Newtonian potential. The galaxy rotation problem is this discrepancy between the observed rotation speeds of matter in the disk portions of spiral galaxies and the predictions of Newtonian dynamics considering the known mass. This discrepancy is currently thought to betray the presence of Dark Matter that permeates the galaxy and extends into the galaxy's halo. A lot of money has been spent looking for dark matter, but it has never been found. e.g. See Dark matter no-show at sensitive underground lab and Dark matter: The hunt.
From Newtons Law: the required rotational velocity decreases with distance
There is no need to invoke a magical new substance "Dark Matter" to explain this. There is a much simpler explanation. The reason rotational velocity is expected to decrease as distance from the centre increases is because Newtons law of gravity says that the gravitational force on an object rotating around the massive centre of the universe is proportional to the square of the distance to the centre. According to Newton, Gravity, like light, spreads itself out in proportion to the surface area of the sphere surrounding the source of the force. To counteract this force the object needs to be rotating at a speed such that the centripetal force cancels this gravitational force. The required speed to cancel the gravitational pull reduces the further you go out.
 v is the velocity of object
mo mass of object
mg mass of amount of galaxy encompassed by objects orbit
r distance to centre of galaxy
G gravitational constant
The above equation assumes that the mg is constant as you move out. This would not be true if the orbit, as it gets bigger, encompasses more and more matter, e.g. dark matter.
Another Solution?
Another fairly simple and fairly obvious way to solve this problem would be for the square of the distance in the equation for gravitational force to change to just the distance. Now the velocity becomes a constant.
 But this gravitational law contradicts all the evidence. We still don't know the mechanism, but we do know that the gravitational field radiates in three dimensions in space and therefore gets weaker with the square of the distance. For it to get weaker as a function of the distance alone, it would need to radiate in only two dimensions. Many precise measurements, e.g. the rotation of the planets about the Sun have verified Newton's inverse square law.
It is argued here, however, that measurements of the rotation of stars about the centre of the galaxy contradict this. How can both contradictory measurements be reconciled?
Gravity on a galactic scale propagates in two dimensions in a planar galaxy
The mechanism for the propagation of gravity is still unknown, but any theories there are assume it is by interaction of matter. The measurements that gave the graph in Figure one were carried out on stars and gases orbiting in disk portions of spiral galaxies. What if gravitional field lines get gradually bent in the plane of these disks by the stars in the disks, so that at the level of the solar system, the field radiates in three dimensions, but beyond a certain scale the propagation is effectively two dimensional. This would mean that the force would be inversely proportional to the distance for galactic distances, the second set of equations would be the appropriate ones to use and there would be no need to invent Dark Matter.
Verification.
Is there any experimental evidence to backup this theory? If this theory is correct then galaxies that have stars which are distributed in three dimensions, rather than the planar distributions of spiral and elliptical galaxies, should behave the way all galaxies were originally assumed to behave. A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. According to http://en.wikipedia.org/wiki/Dark_matter globular clusters show little evidence that they contain dark matter, i.e. the speed of rotation of the stars in these clusters reduces with the distance from the centre as predicted by the inverse square of distance gravitational law. More contradictions to Dark Matter Three other observations that contradict the theory of Dark Matter: |