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While the gravitational force between two bodies is directly proportional to their masses, and inversely proportional to the distance between them is understandable / seems logical, how did Newton decide that the gravitational force between two bodies is "inversely" proportional to "square" of the distance between them, and not "cube" or "any other power" of the distance?.

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  • $\begingroup$ This question (v2) must have been asked before. $\endgroup$
    – Qmechanic
    Commented May 30 at 14:57
  • $\begingroup$ experiments determined the result $\endgroup$
    – Gauransh
    Commented May 31 at 2:00

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As it turns out, only an inverse square law of gravitation leads to Kepler's laws of planetary motion, which were established prior to Newton's time. Newton had worked out the details privately and later published them in his Principia, partially as a result of his encouragement to do so by the astronomer Edmond Halley.

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Newton arrived at the $\frac{1}{r^2}$ law empirically quite a while before analyzing the planetary motion in the solar system.

He did this by comparing the acceleration $a$ experienced by bodies at different distances $r$ from the center of the earth. (See also "The Physics Classroom - The Apple, the Moon, and the Inverse Square Law".) The acceleration of free falling bodies near the surface of the earth was already measured by Galilei some 50 years before Newton. And the acceleration of the moon in its orbit can be calculated as the centripetal acceleration $a=\frac{v^2}{r}=r\omega^2=r\left(\frac{2\pi}{T}\right)^2$ (with $T=27.32$ days being the sidereal month).

$$\begin{array}{c|c|c} & r & a \\ \hline \text{falling body near surface of earth} & 6370\text{ km} & 9.81\text{ m/s}^2 \\ \hline \text{moon orbiting the earth} & 384000\text{ km} & 0.00272\text{ m/s}^2 \\ \hline \text{Ratio} & 1:60.3 & 3607:1\\ \hline \end{array}$$

By looking at the ratios of $r$ and $a$ the obvious guess is $a\propto\frac{1}{r^2}$. Of course concluding this law from only 2 data points is kind of a bold guess.
But later this law was confirmed, when Newton applied it to the planets moving around the sun and showed that Kepler's laws of planetary motion can be derived from it.

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Newton thought of light as particles ("corpuscles") radiating out from the source. Any radiating source naturally follows an inverse square law because the area of the expanding radiation sphere is the square of the radius. It would be natural for Newton to think of gravity in similar terms, as an influence radiating out from a central gravitational source so it would be reasonable to conclude that it also follows a inverse square law. As others have already mentioned this would be mathematically consistent with the already known Kepler laws and the measurements of Galilei.

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From an experimental point of view, imagine that you do measurements of the gravitational forces at various distances. You would quickly see that half the distance does not just double the force, it quadruples it. And a third of the distance does not just tripple the force, it increases it by a factor of $9$. And so on. Do many such measurements, maybe plot them on a graph, and you quickly realise that there is not a simple inverse-proportionality relationship.

After trying to fit other types of curves to the data points, you might realise that an inverse square relationship seems to fit quite well. This is now your hypothesis. You do a lot more testing in many more experiments. Since we have never seen any results from any such experiments* that do not agree with the inverse square law, then we have chosen to believe it to be the truth. We call it a theory - or a law because it is so impactful and so established - of nature.

From an intuitive point of view, you might be able to realise that an inverse square law rather than just an inverse proportionality law is in fact a more sensible hypothesis to test. Think of a light source (like the Sun) that sends out light as photons, and you are holding up a solar collector. If you halve the distance to the light source, is your solar panel then hit by twice as many or four times as many photons?

Geometrically, it has to be four times as many. Because, the amount of the photons you will be catching is "doubled both vertically and horizontally", so to say. Or you could geometrically consider it more mathematically: Your solar panel is located within a sphere that all photons pass through as they are sent out from the light source. Your solar panel is taking up a piece of this sphere area. When you halve the distance, so you are now considering a sphere with half the radius, does your solar panel now cover double as large a portion of the sphere area? No, it is covering a $4$ times as large portion. This is easy to see from the sphere-area formula: $$A_\text{sphere}=4\pi r^2.$$ The area scaled with the square of the radius, not proportional to the radius.**

By considering a gravitational field in the same way as such "solar/photon field", we arrive at this same conclusion: that it makes sense to expect an inverse square law of some sort as a hypothesis to test for.


* Actually, we do see other behaviour happening when relativistic effects play a role. So the inverse square law has in fact been adjusted for the extreme cases of very high speeds or very large masses.

** By the way, within mathematics such a vector field with the form $\mathbf V=\frac{k}{r^2}\hat r$ (for a $k\geq0$, where $r$ is the distance to the point of evaluation, and where $\hat r$ is a unit direction vector from that point), so a field that has a magnitude that lives up to the inverse square law, $||\mathbf V||=\frac{k}{r^2}$, is called a Coulomb field. It turns out that physical fields that live up to conservation laws typically will result in this type of field. This is the case for gravitational fields, the "solar" field I described, but also electric fields, general electromagnetic fields, and more.

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