# Experimental proof that gravity ~ 1/R^2 ?

by NSEFF
Tags: 1 or r2, experimental, gravity, proof
 P: 9 What is the experimental proof of Newton's law of universal gravitation? Specifically, how has it been established that the gravitational force between two masses varies inversely as square of the distance between them ( as opposed to, say, as 1/R^(2+x) where x is small |x|<<1 )?
 P: 731 Short range tests are done using the torsion balance: http://www.npl.washington.edu/eotwash/sr
 P: 208 Wouldn't the applications for which Gauss' Law gives valid results also be a sort of support because that whole mathematical principle require that the force fall off exactly as the inverse of the square and not some other exponent?
Mentor
P: 15,401

## Experimental proof that gravity ~ 1/R^2 ?

Planetary orbits also provide a stringent test. For a 1/r^(2+x) force, the semimajor axis of the ellipse does not precess only if x=0. Non-observation of precession allows limits to be set on x. (And indeed, for Mercury this was used to show GR made more accurate predictions than Newton)
P: 3,413
 Quote by NSEFF |x|<<1
You can always add some x << 1 into a formula, and claim the accuracy of measurement is not sufficient to pick it up.
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P: 2,725
 Quote by NSEFF What is the experimental proof of Newton's law of universal gravitation? Specifically, how has it been established that the gravitational force between two masses varies inversely as square of the distance between them ( as opposed to, say, as 1/R^(2+x) where x is small |x|<<1 )?
It hasn't been established. All that observation ever does is set an upper bound on x; but by now we know that if x is non-zero, its value is very small indeed.

Textbooks tend to gloss over this point, not surprising if you think about it. If you were trying to teach someone about gravity, would you start with $\frac{1}{r^2}$ or start talking about mysterious tiny maybe zero magic x values? But you'll notice that published scientific papers are generally much more careful; instead of saying that they've "proved" something they'll say that they've established a new smaller limit on how wrong the the theory could be.

And it turns out that Newton's law isn't exactly right anyways. There's a very small deviation predicted by General Relativity. It's only (just barely) noticeable in the orbit of Mercury, which is close enough to the sun for the deviation to be strong enough to see. That doesn't stop us from using $\frac{1}{r^2}$ to navigate spaceships and predict planetary motion - and it shouldn't.
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PF Gold
P: 38,706
 Quote by NSEFF What is the experimental proof of Newton's law of universal gravitation? Specifically, how has it been established that the gravitational force between two masses varies inversely as square of the distance between them ( as opposed to, say, as 1/R^(2+x) where x is small |x|<<1 )?
If |x| is very small, it might well be. No physical theory is certain, there is always the possiblity (or perhaps certainty) that, as we learn more, we might have to change the theory. But both astronomical and laboratory measurements indicate that, if you are going to use that formula, the exponent must be very, very close to 2.

I said "if you are going to use that formula" because back in the beginning of the 19th century, observations of the orbit of Mercury indicated that formula was not perfectly correct. But that resulted in a whole new theory, not just a non-zero value for "x".
 P: 9 TURTLEMEISTER: My understanding, from a cursory reading of the cited paper, is that it is directed primarily at detecting deviations from an inverse-square law at sub-millimeter distances. However, I was unable to access the arXiv.org papers detailing the experiment. So it is unclear to me whether this experiment actually confirmed the 1/R^2 dependence or just showed that the torsion balance behaved consistently (within experimental error) as the separation distance was varied -- as might be the case if the actual dependence was, say, 1/R^(2.000001)? SCHAEFERA: True, Gauss' Law suggests a 1/R^2 behavior -- and this Law is adequately confirmed at laboratory separations. But at larger distances it has to be taken on faith. NUGATORY: Actually, while 1/R^2 is used to navigate spacecraft, deviations are continually observed and the orbits must be continually corrected. From these remarks it appears that there is no empirical confirmation of the 1/R^2 dependence (except possibly at sub-millimeter distances); that this dependence is merely a postulate; and that it is primarily supported by comparing observed planetary orbits to calculated orbits. Deviations from Newton's Law due to GR effects are not pertinent to my question. These effects only become apparent for masses moving in strong gravitational fields. I am more curious about 1/R^2 behavior of the gravitational field of an isolated mass at interstellar distances. Of course I don't expect any experimental confirmation of 1/R^2 at these separations; but I have found no reference to experimental verification of gravitational 1/R^2 behavior at any separation -- except perhaps the paper cited by TurtleMeister. Nor have I found any mention of an upper bound on my suggested |x| factor. However, in the past I have run across suggestions that a small deviation from Newton's Law might provide an explanation for certain astronomical observations.
P: 1,549
 Quote by NSEFF However, in the past I have run across suggestions that a small deviation from Newton's Law might provide an explanation for certain astronomical observations.
So you want to discuss MOND (modified Newtonian dynamics). It is a shot at getting the rotation curves for spiral galaxies correct. Stars far from the center are way faster than expected. However, that typically does not investigate the 1/r^2 part. You have the standard gravitational acceleration acting on a star: ma=Gm/r^2.

In MOND, one typically assumes that Newton's law breaks down at some scale, so that for constant masses, you do not get the standard F=ma, but F=ma/mu, where mu is a function of a/a0. a0 is assumed to be very small - on the order of 10^-10 m/s^2, so that in everyday life and also in almost any experiment, this modification does not give any difference. On large scales, however, it may be. One could of course try to shift the modification to the inverse square law, bu that seems rather pointless. In a nutshell, MOND is an alternative to dark matter.

If I remember correctly, the WMAP results rule out many of the possible ways mu could look like, so MOND needs some tweaking. I am not sure about published experimental checks of MOND or the inverse square law, but for example the Max-Planck-institute for gravitational physics in Hannover is working on such stuff. You might find some experimental tests (or at least a hint where too look for them) on their web page.
P: 1,618
 Quote by NSEFF What is the experimental proof of Newton's law of universal gravitation? Specifically, how has it been established that the gravitational force between two masses varies inversely as square of the distance between them ( as opposed to, say, as 1/R^(2+x) where x is small |x|<<1 )?
There's actually a remarkable fact about the inverse square law, which is that noncircular orbits are not stable except for a perfect inverse square law. So the stability of orbits such as the Earth's around the sun gives a bound on how far the law can be from inverse square.
P: 1,618
 Quote by stevendaryl There's actually a remarkable fact about the inverse square law, which is that noncircular orbits are not stable except for a perfect inverse square law. So the stability of orbits such as the Earth's around the sun gives a bound on how far the law can be from inverse square.
I misspoke: there are two radial force laws compatible with stable noncircular orbits: inverse square, and linear (such as an ideal spring). The latter isn't a real possibility for holding planets in orbit.
P: 3,413
 Quote by stevendaryl I misspoke: there are two radial force laws compatible with stable noncircular orbits
Also see:
http://en.wikipedia.org/wiki/Bertrand's_theorem

Here a simulation of different orbit types:
http://megaswf.com/serve/1161536

 Quote by stevendaryl and linear (such as an ideal spring). The latter isn't a real possibility for holding planets in orbit.
The designers of this 1 pound note think otherwise.

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P: 2,725
 Quote by NSEFF From these remarks it appears that there is no empirical confirmation of the 1/R^2 dependence (except possibly at sub-millimeter distances); that this dependence is merely a postulate; and that it is primarily supported by comparing observed planetary orbits to calculated orbits.
How is "primarily supported by comparing observed planetary orbits to calculated orbits" not a form of "empirical confirmation"? We've observed that reality matches the theory to the limits of experimental accuracy, and that's as good as empirical confirmation ever gets.
P: 731
 Quote by NSEFF TURTLEMEISTER: My understanding, from a cursory reading of the cited paper, is that it is directed primarily at detecting deviations from an inverse-square law at sub-millimeter distances. However, I was unable to access the arXiv.org papers detailing the experiment. So it is unclear to me whether this experiment actually confirmed the 1/R^2 dependence or just showed that the torsion balance behaved consistently (within experimental error) as the separation distance was varied -- as might be the case if the actual dependence was, say, 1/R^(2.000001)?
If you have a pdf reader you should be able to access the arXiv papers. From the paper dated February 2, 2008:
 We conducted three torsion-balance experiments to test the gravitational inverse-square law at separations between 9.53mm and 55um... We find with 95% confidence that the inverse-square law holds $(\left|\alpha\right| \leq 1)$ down to a length scale $\lambda$ = 56um and that an extra dimension must have a size R $\leq$ 44um.
These papers seem to involve concepts in particle physics which go beyond my understanding. So I hesitate to comment on if/how they help in answering your question.
P: 1,618
 Quote by Nugatory How is "primarily supported by comparing observed planetary orbits to calculated orbits" not a form of "empirical confirmation"? We've observed that reality matches the theory to the limits of experimental accuracy, and that's as good as empirical confirmation ever gets.
I think what he might mean is that planetary observations can't really distinguish exponent 2 from exponent 2+/- $\epsilon$ for a small value of $\epsilon$. I'm not sure if anything can do that.
Mentor
P: 16,298
 Quote by NSEFF What is the experimental proof of Newton's law of universal gravitation? Specifically, how has it been established that the gravitational force between two masses varies inversely as square of the distance between them ( as opposed to, say, as 1/R^(2+x) where x is small |x|<<1 )?
Hi NSEFF, I think that you may have a misunderstanding about how theories are tested.

In order to test a theory, you cannot use the theory itself to make predictions. Instead, what you need to do is to make a generalization of the theory which has some free parameter where, for a certain value of the free parameter, your generalized theory reduces to the specific theory. Such a generalized theory is called a test theory.

In your example here, your 1/R^(2+x) theory would be a test theory for Newtonian gravity with Newtonian gravity predicting a value of x=0. Due to experimental errors, you can never prove that x=0 (nor that x equals any other number), but you can prove that x must lie within some range. If that range contains 0 then you say that the experiment confirms Newtonian gravity to within the given precision.

I don't know what test theories of Newtonian gravity are actually used.
P: 731
 Quote by DaleSpam Due to experimental errors, you can never prove that x=0 (nor that x equals any other number), but you can prove that x must lie within some range
Yes, that is the way I understand the Eot-Wash experiments, except that I do not fully understand how that range is defined in the cited paper. It seems to have something to do with $(\left|\alpha\right|\leq 1)$, which seems to have something to do with the term "Yukawa interaction". I agree with the other posters that for any experimental verification of the 1/R^2 law at long ranges, we are pretty much confined to astronomical observation.
 Mentor P: 15,401 Astronomically, there's no point in a test theory of Newtonian gravity, because we know that GR does a better job. There is something called the "PPN Formalism", which has a dozen or so inputs to calculate observable quantities; GR is consistent with observations. Certainly 2 and 2 + ϵ can be distinguished. A change in ϵ by 1 unit will cause a perihelion advance of ~0.5 radian per orbit, which deviates from Newton by 43 million arc-seconds per century. So we have confirmed that |ϵ| < ~10-6 or so astronomically. In the lab, 2 + ϵ is one popular choice. Another is to multiply by an exp(-r/a) term and set limits on a.

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