# Restating Newtons universal law of gravity

Usually Newtons universal law of gravitation is stated in the following way:

1) | F = GMm/r^{2} |

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/r^{2} |

^{2}.]

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 r

_{0}. The surface area of the sphere is:

4) | area = 4πr_{0}^{2} |

*a*during the interval may be taken to be constant. In that case, starting from rest:

5) | Δs = -1/2 a Δt^{2} |

6) | ΔVol = -2πar_{0}^{2}Δt^{2} |

_{0}à r

7) | ΔVol = -2πMGΔt^{2} |

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 * 10

^{40}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 * 10

^{30}(m

^{3}). 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.