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

Re: Experimental proof that gravity ~ 1/R^2 ?
Short range tests are done using the torsion balance: http://www.npl.washington.edu/eotwash/sr

Re: Experimental proof that gravity ~ 1/R^2 ?
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?

Re: 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. Nonobservation 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)

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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 [itex]\frac{1}{r^2}[/itex] 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 [itex]\frac{1}{r^2}[/itex] to navigate spaceships and predict planetary motion  and it shouldn't. 
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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 nonzero value for "x". 
Re: Experimental proof that gravity ~ 1/R^2 ?
TURTLEMEISTER:
My understanding, from a cursory reading of the cited paper, is that it is directed primarily at detecting deviations from an inversesquare law at submillimeter 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 submillimeter 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. 
Re: Experimental proof that gravity ~ 1/R^2 ?
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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 MaxPlanckinstitute 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. 
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http://en.wikipedia.org/wiki/Bertrand's_theorem Here a simulation of different orbit types: http://megaswf.com/serve/1161536 Quote:
http://bluelyon.files.wordpress.com/...poundnote.jpg 
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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. 
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Re: Experimental proof that gravity ~ 1/R^2 ?
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 arcseconds 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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