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

1. ### NSEFF

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 )?

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3. ### schaefera

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?

18,482
Staff Emeritus
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)

5. ### A.T.

6,462
You can always add some x << 1 into a formula, and claim the accuracy of measurement is not sufficient to pick it up.

### Staff: Mentor

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.

7. ### HallsofIvy

41,265
Staff Emeritus
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".

8. ### NSEFF

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.

9. ### Cthugha

1,717
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.

10. ### stevendaryl

3,522
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.

11. ### stevendaryl

3,522
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.

12. ### A.T.

6,462
Also see:
http://en.wikipedia.org/wiki/Bertrand's_theorem

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

The designers of this 1 pound note think otherwise.

### Staff: Mentor

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.

14. ### TurtleMeister

821
If you have a pdf reader you should be able to access the arXiv papers. From the paper dated February 2, 2008:
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.

15. ### stevendaryl

3,522
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.

### Staff: Mentor

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.

17. ### TurtleMeister

821
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.

18,482
Staff Emeritus
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.

19. ### TurtleMeister

821
It is my understanding that GR is no better at explaining the flat rotation curve of spiral galaxies than Newtonian gravity is, without using the unconfirmed existence of dark matter. So, as NSEFF has already stated, it seems that the law of 1/R^2 for gravity is merely a postulate at very long ranges.

20. ### NSEFF

9
CTHUGHA: Yes, MOND was what I had in mind, though it has been a long time since I ran across this concept and I am not familiar with its current details.

TURTLEMEISTER: While the arXiv article appeared to download, I was unable to open the pdf file. This has happened to me before. I think it is a configuration problem with the Adobe Reader which I'll have to look into.

NUGATORY: Newton postulated his Law, which had previously been suggested by others, and found the calculated orbits of planets closely matched the observations. I too pondered whether this could be deemed an empirical confirmation, but, logically, "If F ~ 1/R^2 => Calculated orbits closely match observations"; does not imply that "Observed orbits matching calculations => F ~ 1/R^2". That is, unless STEVENDARYL's reference to Bertrand's theorem is completely applicable: "that noncircular orbits are not stable except for a perfect inverse square law".

STEVENDARYL: What's nice about this forum is that often one encounters new ideas. Bertrand's theorem is new to me. I found no reference to it in any of my classical mechanics books. It must be something one encountered by astrophysicists.

Bertrand's theorem is predicated on stable, exactly closed orbits. Questions: How stable & exactly closed is Earth's orbit? How do you quantify a deviation from stability and being exactly closed (this would impact on estimating a bound on any deviation from inverse square)? Since the Sun is not stationary, does a finite propagation time for gravity need to be incorporated? Given that Pluto's orbital period is 246 years, has its orbit been sufficiently observed to be deemed stable and exactly closed? If not, and if Bertrand's theorem holds, it can only confirm inverse square to solar system dimensions?

DALESPAM: Good synopsis of theory testing. So, how do you test for 1/R^(2.000001) as opposed to 1/R^2? I presume, given experimental errors, it can't be done??

My original query was about whether direct empirical tests of the inverse square law exist. Other the short sub-millimeter torsion balance tests mentioned there don't seem to be any!