Restating Newtons universal law of gravity in terms of volume
Usually Newtons universal law of gravitation is stated in the following way:
- F = Mm/r2
Newton was interested in the force F. [In fact there are two equal but opposite forces: the force from M towards m, and the force from m towards M.]
In the context of galaxys, we're more interested in accelerations than in Forces. So we use:
- F = ma
- a = MG/r2
All of this is valid for a point mass M at rest, and - as Newton showed - also for a homegenous shell around that point mass (with the same center of gravity). So this works really well within our solar system, where 99% of all mass is concentrated in a small sphere, called 'the sun'.
But our galaxy is not a sphere. It is a disk with exponentially diminishing density, with about only a fifth of its mass in the centre (the bulge). It is not obvious how to use the above formula for acceleration, when a galaxy consists of lots of different masses, with a lot of different distances between them, in a certain distribution.
To work around this, I will reformulate Newtons law into Volume-change as a function of time (-interval).
Think a spherical volume around a point mass, with radius r0. The surface area of the sphere is:
- area = 4πr02
- Δs = -1/2 a Δt2
- ΔVol = -2πar02Δt2
Since Δt → 0, also Δs → 0 , and r0 → r
- ΔVol = -2πMGΔt2
It says that the decrease in volume is determined only by time (squared) and by Mass.
The amount of change in volume does not depend on the radius of the sphere.
That means that
-it is independent of the size on the volume I put around my point mass M
-it is independent on the shape of the volume I put around my point mass M,
-it is independent on the distribution of the mass within the shape.
In the case of our own Milkyway (total Mass = 16,6 * 1040 kg) this means that, no matter the distribution of its mass, no matter the surrounding shape of volume I choose, as long as all mass is within that shape, in 1 second that volume will decrease by 69,554 * 1030 (m3). This will be the crux of my line of reasoning.
Please take note that I did not change Newtonian gravity in any way by formulating it in terms of volume and time.