Hi imatfaal,
They may have accidently used an equation where the reduced Compton wavelenghth was obscured in the formulation. I am looking for circumstances where the reduced Compton wavelength is used instead of just the Compton wavelength.
In the second paper on the OP, the rebuttal paper for the first non dark matter paper, they change the rotation speed of non central band rotating objects from the central bands average to make dark matter appear.
Here is a simple time/distance scale analysis of light from rotating objects.
Pi, as well as being a dimensionless constant, has a common definition in all the different methodologies in cosmology regardless of how it is used so it is a good starting point for a simple observational/thought experiment. This experiment is purely about what should be expected to be observed when we view rotating sources on different time/distance scales when they all share the same elementary geometric ratios within their basic observation structures.
All of the constants in the 3 ratios below can be regarded as time or distance (based on the distance travelled by light in the time).
If I started photographing a light (say a sparkler) in a dark room around 6 and a bit feet away, and the light was being spun in a circle 2 feet in diameter and I captured the light from the spinning light source during one complete circle the ratio ( A ) of the distance between the rotating source and the observer over the diameter of rotation would be roughly equal to Pi.
In this case the ratio ( B ) of the distance between source and observer over the distance travelled by light in a year would be very small and the ratio ( C ) of the observation period over the time it takes for the light source to be rotated once will equal one. All observations should have a width of field that covers the complete diameter of rotation of the source being observed.
If I halve the exposure period I get half a circle and capture half as much light and when I double the exposure period I get 2 circles over each other and twice as much light in my image. If the light is rotated twice as fast I would expect something that looked similar to when I doubled the exposure period but I would also expect to capture the same amount of light as in my original one rotation in the same time despite the doubling of the speed of rotation. If I put two lights together I could halve the exposure time and double the speed of rotation to capture a similar amount of light from the original 1 light doing 1 complete rotation. If the light moved at an angle to me I would observe an oval instead of a circle but the amount of light captured would remain the same as in a complete circle.
In this simplest base context A = Pi, B = tiny, C = 1 and the observer will capture one complete cycle. On any scale where ratio C >= 1 the observer will capture at least one complete cycle despite the size of ratio B.
On any scale where A = Pi * x, B >= 1 and C < 1 the observer will capture the light from B * C = x of one rotation during any observation regardless of the speed of rotation of the same object.
On a galactic year scale where A = Pi * x, B = 230 million and C = 1/230 million you would capture the light from B * C = x rotations or roughly one rotation regardless of the speed of rotation.
Either way, you get a similar discrepancy in mass.