# Intuition about why gravity is inversely proportional to exactly square of distance between objects

What is the intuition behind why gravity is inversely proportional to exactly square of distance between two object and not cube or not some multiplier. Basically how Newton came up that its exactly square of distance? How it was validated it is in fact square of distance ?

• The harder part of the problem was that tricky "distance between two objects" as applied to non-point objects. Their mass only appears to be effectively at the geometric centre for bodies composed of concentric shells, each of uniform density. Luckily, gravity in massive bodies generally causes that distribution of mass. Nov 5 '20 at 16:43
• For reference, Wikipedia has a nice article on the Inverse-square law with a nice illustration that touches on this question as well: en.wikipedia.org/wiki/Inverse-square_law
– stux
Nov 7 '20 at 20:29

Imagine "gravity" spreading out in a sphere, like light from a bulb.

For each doubling of the distance, the sphere has four times the area. The surface area of the sphere is proportional to the square of the radius. If the same gravity is stretched over that sphere, the force of gravity would be inversely proportional to the square of the radius.

This gives sufficient intuition for most physicists of the time to believe that gravity was probably inverse square. Newton's real genius was that he was able to prove using mathematics that a planet moving in an inverse square law would obey Kepler's laws, something that his contemporaries had failed to do.

A well known story, related by de Moivre:

In 1684 Dr Halley came to visit [Newton] at Cambridge. After they had been some time together, the Doctor asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it.

• Comments are not for extended discussion; this conversation has been moved to chat. Nov 9 '20 at 5:50
• The inverse square law seems very intuitive to me for Newtonian gravity as mentioned above. It doesn't seem quite so intuitive to me when looking at "Einsteinian Gravity". I asked a question a long time ago in one of the Stack Exchanges about deriving Newton from General Relativity and a brilliant person showed me how this is done (but it's not easy). I don't think you can look at this geometrically with regards to General Relativity since mass determines the geometry. Nov 9 '20 at 22:59

The validation is the same as the validation of any astronomical theory: it fits the data. Newton's law of gravitation was formed empirically, by observing the motion of the planets.

In particular, he took Kepler's laws, which were observational, and $$F=ma$$ from his physics and determined that the elliptical orbits we see must be caused by something with an inverse-square relationship to distance. The association with mass can be determined by looking at relative perturbation of orbits as they pass various masses. (I cannot find a good source for which masses he was using, but I would venture a guess that Jupiter and/or its moons were involved).

Indeed, there is a particularly interesting proof that came later, called Bertrand's Theorem. If you have a system governed by central forces (forces caused by a body which only pull inwards or push outwards from the center of that body), there are only two kinds of laws which can produce bounded closed orbits. One is when the force is proportional to distance, as we see in springs and Hooke's law. The other is when the force is inversely proportional to the square of the distance, as we see in gravity (among other forces). This says that if one observes closed bound orbits, one can be certain that the forces involved are one of those two. One can then determine which is which by observation of the orbits.

And, of course, there is no proof that the forces are central. There could be small invisible angels carefully leading the planets on their path. However, the central-force based approach proves more satisfying to the scientific community, for obvious reasons.

• Erm, isn't the force (non-inversely) proportional to distance with Hooke's law? Nov 5 '20 at 16:59
• @HagenvonEitzen Good catch. I put too many words there. Fixed! Nov 6 '20 at 17:38

The behaviors of the various forces are different. They are measured, not intuited.