Updated: Sep 11, 2021
The concentration of mass in the central region of a galaxy would, according to Newton's gravitational law, mean that the velocities of the planets decrease going further from the center. However, in spiral galaxies, the velocities of the stars far from the center are much faster than expected (see diagram below). Larger centrifugal sources farther from the central region would mean that these fast stars would be ripped out of their orbits and injected into space. Yet, the stars appear to be in stable orbits. This would suggest the presence of unseen, or dark, matter that is keeping these stars in place.
New particles have been invented, and Newton's law has been modified in order to explain why orbital speeds do not fall off there farther one goes from the central region. Are they really necessary?
The spiral nature of the galaxies would implicate an inverse cube law instead of an inverse square law. Cotes' spirals is one such case, and it has been known that
"If the attraction were inversely proportional, not to the square of the distance, but to the cube of the same, one could integrate the [virial] equation."
The potential in the equation would vanish leading to a secular increase in time so that "the solar system would break up".
These conclusions were presented by Jacobi's in his Lectures in Dynamics (1842-3), and have withstood the test of time. Or have they? The curve for which the aberrancy is constant is the logarithmic spiral. According to Jakob Bernoulli (1692) it is a curve 'although changed, still remaining the same!' An orbiting planet would do so at a constant angle less than pi/2, which would apply to a circular orbit.
The orbital equation for a logarithmic spiral is
where k is a constant. Introducing the conservation of angular momentum h=r^2(omega), gives
dr/dt= v =hk/r.
Differentiating in time results in
the inverse cube law. Now introducing the velocity results in
a = dv/dt=-v^3/hk=-Lp/hk=-Lcs/hk, (1)
where we introduced the conservation of angular momentum in v-space, L=v^3/p, and the radius of curvature, p=cs, where s is the arc-length, and c a constant, which we shall set equal to k, the other constant.
Differentiating (1) with respect to time gives
d^2v/dt^2+(Omega)^2 v=0, (2)
where the frequency is (Omega)=(L/h)^1/2, the square-root of the ratio of the conserved angular momenta. It is an easy matter to show that there is zero torque; the condition is
From Eq. (2), we know much more than this. Set v=1/f'^(1/2), then (2) is transformed into
On the rhs is the Schwarzian derivative. The Schwarzian derivative is curvature, and (2') says that v-space is one of positive, constant curvature, (Omega)^2. The Schwarzian will vanish only for geodesic motion, v=const.
So, in v-space the solution is periodic. Velocity and acceleration are related by
a=(Omega) v, (3)
which is strikingly similar to Hubble's law,
where v=dD/dt is the recessional velocity, D, the proper distance, and H_0 is referred to Hubble's parameter, a frequency analogous to (Omega).
The empirical law governing the rotational law of the spiral galaxies is the Tully=Fisher law stating that v varies as the fourth power of the luminosity, instead of the square-root, as it would if Newton's law was in full vigor. MOND introduces a critical acceleration a_0, such that at large scales
in accordance with the Tully-Fisher relation.
Now, if we introduce our "Hubble" law, (3) into (4), we get
v^3=GM (Omega), (5)
but since v=(Omega r), (5) becomes
(Omega)^2 r^2=v^2=GM/r, (6)
which is the classical expression.
To see what has happened, we take the square-root of (6) to get
From the first equality we would conclude that the angular velocity should drop as
1/r^(3/2); however from the second equality in (7) we see that the acceleration
a=(GM/r)^(1/2) (Omega) (8)
only decreases as 1/r^(1/2) instead of as 1/r^2 since (Omega), the ratio of two angular momenta, is constant. And, most important of all, v increases with r instead of decreasing as 1/r^(1/2).
Thus, the excursion to v-space was not superfluous, for without it we would have come to the conclusion that there could be no stable orbits in spiral galaxies.
For most of the stars, the velocities range between 200 km/s and 250 km/s. MOND has determined the critical acceleration a_0=1.2 x 10^(-13) km/s^2, or about 100 billion times smaller than the acceleration due to gravity at the earth's surface. The velocities in question are of the order of 10^2 km/s so that their ratio is of the order of 10^(-15) hz, or a fhz. This is 57 octaves lower than middle-C, or a frequency of over a million billion times deeper than the limits of human hearing. This should be a consequence of the fact that v-space angular momentum is much smaller than c-space angular momentum.
Such sound waves have been "heard" in the heart of the Perseus Cluster. It has been puzzling that so much hot gas in galaxy clusters is found and so little cool gas. The hot gases should be carried away from the core center by energetic X-rays leading to a cooling in the densest part of the cluster which is at the center. [If the same should be true of gravitational waves then there is a real problem for their detection.] With the cooling of the central gas, the lower temperature gas from the outer reaches should be drawn to the center. In the tendency to reach thermal equilibrium, stars ought to be formed in the process of thermal equilibration. Yet there has been no evidence that flows of cooler gas toward the center occur, and of star formation.
A similar phenomenon, although less dramatic, should occur in the spiral arms of the galaxies. The large velocities found in the outer regions would impede fluxes of cooler gas to diffuse to the center. Hence, star formation in the spiral arms should be impeded, analogous to the Perseus Cluster....but that is just speculation.