<p>The validation is the same as the validation of any astronomical theory: it fits the data.  Newton's law of gravitation was formed <a href="https://pwg.gsfc.nasa.gov/stargaze/Sgravity.htm" rel="nofollow noreferrer">empirically</a>, by observing the motion of the planets.</p>
<p>In particular, he took Kepler's laws, which were observational, and <span class="math-container">$F=ma$</span> 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 <em>which</em> masses he was using, but I would venture a guess that Jupiter and/or its moons were involved).</p>
<p>Indeed, there is a particularly interesting proof that came later, called <a href="https://en.wikipedia.org/wiki/Bertrand%27s_theorem" rel="nofollow noreferrer">Bertrand's Theorem</a>.  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.</p>
<p><em>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.</em></p>

CC BY-SA 4.0
2020-11-05T16:59:20.850Z
Erm, isn't the force (non-inversely) proportional to distance with Hooke's law?
Hagen von Eitzen
CC BY-SA 4.0
2020-11-06T17:38:50.440Z
@HagenvonEitzen Good catch.  I put too many words there.  Fixed!
Cort Ammon
CC BY-SA 4.0
2020-11-05T14:25:07.243Z
2020-11-06T17:38:24.530Z
2020-11-06T17:38:24.530Z
Cort Ammon
<p><a href="https://en.wikipedia.org/wiki/Strong_interaction#Behavior_of_the_strong_force" rel="nofollow noreferrer">&quot;The strong force acts between quarks. Unlike all other forces (electromagnetic, weak, and gravitational), the strong force does not diminish in strength with increasing distance between pairs of quarks.&quot;</a>.</p>
<p>The behaviors of the various forces are different. They are measured, not intuited.</p>

CC BY-SA 4.0
2020-11-05T20:37:00.540Z
2020-11-05T20:37:00.540Z
Keith McClary
<p>Imagine &quot;gravity&quot; spreading out in a sphere, like light from a bulb.</p>
<p>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.</p>
<p>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.</p>
<p>A well known story, related by de Moivre:</p>
<blockquote>
<p>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.</p>
</blockquote>

CC BY-SA 4.0
2020-11-09T05:50:54.473Z
Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/115981/discussion-on-answer-by-james-k-intuition-about-why-gravity-is-inversely-proport).
called2voyage
CC BY-SA 4.0
2020-11-09T22:59:07.150Z
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.
Jack R. Woods
CC BY-SA 4.0
2020-11-04T21:38:43.603Z
2020-11-06T10:11:17.157Z
2020-11-06T10:11:17.157Z
James K
<p>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 ?</p>

CC BY-SA 4.0
2020-11-05T16:43:35.173Z
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.
Paul_Pedant
CC BY-SA 4.0
2020-11-07T20:29:32.323Z
For reference, Wikipedia has a nice article on the Inverse-square law with a nice illustration that touches on this question as well: https://en.wikipedia.org/wiki/Inverse-square_law
stux
<gravity><newtonian-gravity>
Intuition about why gravity is inversely proportional to exactly square of distance between objects
jrp
# Intuition about why gravity is inversely proportional to exactly square of distance between objects

(Asked by: username_0 on 2020-11-04)

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 ?

**Tags:** gravity, newtonian-gravity

> **Comments:**
> **username_1**: 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.  
> **username_2**: For reference, Wikipedia has a nice article on the Inverse-square law with a nice illustration that touches on this question as well: https://en.wikipedia.org/wiki/Inverse-square_law  

---

## 3 answers

### Answer by username_6 on 2020-11-04. Score: 42 | Accepted answer

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:**
> **username_7**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/115981/discussion-on-answer-by-username_6-intuition-about-why-gravity-is-inversely-proport).  
> **username_8**: 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.  

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### Answer by username_3 on 2020-11-05. Score: 4

The validation is the same as the validation of any astronomical theory: it fits the data. Newton's law of gravitation was formed [empirically](https://pwg.gsfc.nasa.gov/stargaze/Sgravity.htm), 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](https://en.wikipedia.org/wiki/Bertrand%27s_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.*

> **Comments:**
> **username_4**: Erm, isn't the force (non-inversely) proportional to distance with Hooke's law?  
> **username_3**: @username_4 Good catch.  I put too many words there.  Fixed!  

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### Answer by username_5 on 2020-11-05. Score: 2

["The strong force acts between quarks. Unlike all other forces (electromagnetic, weak, and gravitational), the strong force does not diminish in strength with increasing distance between pairs of quarks."](https://en.wikipedia.org/wiki/Strong_interaction#Behavior_of_the_strong_force).

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


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