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If you assume gravitational flux[1] is conserved, then flux density, and thus force at a given point, must decrease proportional to the increase in surface area over which it distributed. Thus, inverse-square. As for why it's a nice even number, it's because we live in a universe with a nice integral number of dimensions.
- newtonian gravity
Now, invoking Bertrand's theorem with regard to the Kepler...
- newtonian gravity
Now, invoking Bertrand's theorem with regard to the Kepler problem of the motion of a body in a central force, one can show that the only cases for which closed orbits are stable are that of the inverse square law and Hooke's law for the force. I find it kind of remarkable that the requirement of stability of closed orbits is sufficient to constrain the form of the central force to just two cases.
The gravitational force between two masses outside a uniform field is defined by Newton’s Law of Gravitation. Although planets are not point masses, their separation is much larger than their radius. Therefore, Newton’s law of gravitation applies to planets orbiting the Sun. The 1/r 2 relation is called the ‘inverse square law’.
Aug 6, 2024 · That leads to a force law where gravity gets weaker in precisely the same way: as though it’s proportional to ~1/r². The reason gravity follows a ~1/r² force law is because we live in a ...
- Ethan Siegel
E = 1 2mv2 − GMm r. For a circular orbit, the radius R determines the speed (as per Equation (10.1.11)), and hence the total energy, which is easily seen to be E = − GMm 2R. It turns out that this formula holds also for elliptical orbits, if one substitutes the semimajor axis a for R: E = − GMm 2a.
In the context of non-Euclidean geometries and general relativity, deviations from the inverse-square law do not arise from the law itself but rather from the assumption that the force between two bodies is instantaneous, which contradicts special relativity. General relativity reinterprets gravity as the curvature of spacetime, leading particles to move along geodesics in this curved spacetime.
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The intensity of the influence at any given radius r is the source strength divided by the area of the sphere. Being strictly geometric in its origin, the inverse square law applies to diverse phenomena. Point sources of gravitational force, electric field, light, sound or radiation obey the inverse square law.
