Restating Newtons universal law of gravity
Usually Newtons universal law of gravitation is stated in the following way:
|1)||F = GMm/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:
|2)||F = ma|
|3)||a = GM/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:
|4)||area = 4πr02|
|5)||Δs = 1/2 a Δt2|
|6)||ΔVol = 2πar02Δt2|
|7)||ΔVol = 2πMGΔt2|
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 shape will shrink 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.