Overview: the basic idea in prose and pictures

Shortest possible version:
The observed rotation curve of our galaxy can be described and predicted by Newtonian gravity.

Slightly longer version:
Newtonian gravity can be reformulated in the following way:
  • given an amount of mass
  • given an arbitrary virtual volume at rest, containing that mass
  • given a time interval [this needs to approach zero - for details: see the long version]
  • we can calculate the decrease in volume of that volume within the time interval, due to gravity
  • the equitation is: |ΔVol| = 2πMGΔt2
  • this decrease in volume is independent of the shape of the virtual volume
  • this decrease in volume is independent of the distribution of the mass within the volume

The above is applied to our galaxy. Around our galaxy I place a virtual, very flat cilinder with a radius of 50 kpc and a height of 0,2 - 0,5 kpc. (calculations are made for several heights).

The cilinder has an upper- and lower surface, and a rim.

I choose a time interval of 1 second.

Given the total mass of the galaxy, we know how much change in volume takes place within that timeframe. (picture below is the intersection of the galaxy; the green area represents the change in volume)

dent in cilinder

With Newtonian gravity, for every point on the upper- and lower surface of the cilinder, I calculate the acceleration, and consequently the distance travelled. Giver the distribution of mass in our galaxy, gravity causes a 'dent' in the centre of these surfaces. (pink)

From this, the change of volume due to the moving of the upper- and lower surface is being calculated. This is less then the total change in volume as calculated previously for all of the mass. (pink is less then green)

The rest of the volume change has te be accounted for by the displacement of the rim of the cilinder. This (average) displacement is being calculated. (blue)

dent in rim

From this displacement the average inward acceleration can be calculated, again with Newtonian physics only.

From thereon, the average rotationcurve (given the height of the cilinder) can be calculated.

The rotationcurve calculated this way, for several heights of the cilinder looks like this:

roation curve calculated

This result can be compared to the emperical data:

ratation curve emperical

The match between the two is striking, especially given the crude way of modelling:
  • both reach a top around 250 km/sec. Cilinder height of 0,25 or 0,30 seem to match best
  • after the top, both curves are nearly flat, indicating an almost lineair decline in accerleration
  • although nearly flat, both curves have small gradient of the same magnitude. This slope is not being predicted neither by theories involving dark matter, nor by theories involving MOND

The long version next