This is precisely the negative of the centrifugal force so as r goes to infinity the net force is always zero, the orbital linear velocity of the orbit is a constant. So the attractive force for a hyperbolic spiral orbit is
F = – L squared / (m r cubed)
where mu is the reduced mass and where the angular momentum is
L = mu r squared omega
where omega is the angular velocity (the magnitude of the spin connection). So :
F = – mu r omega squared
Here mu is the reduced mass:
mu = mM / (m + M)
so M of the central part of the galaxy enters into the force in this way. This is of course completely different from Newton, not just a small correction to Newton as in the solar system. This type of force law and orbit were first investigated by Roger Coates (1682 – 1716). The received wisdom of the present age of darkness and dark matter in cosmology is that there is a “black hole” at the centre of a galaxy, if so mu would be mathematically indeterminate (infinity divided by infinity). In fact the mass M is finite but very large, so for all practical purposes:
mu = m
so the star of mass m orbits with constant velocity as r goes to infinity without being affected by M. This finding completely contradicts the idea of attraction between m and M. The real reason for the orbit is geometry, in fact the orbit is governed simply by the properties of plane polar coordinates. Note carefully that this is a theory of general relativity because there is a spin connection vector present – the angular velocity vector. Newton is restricted to motion in a straight line.