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When I did my MSc in astrophysics back in '95-'96, I was told that there had been an attempt around the early 20th century to account for the advance of the perihelion of Mercury by altering the Newtonian gravitational equation by changing the exponent of radial distance r to something like 2.0016.

Now obviously this was rapidly debunked by the Einstein theory, but it still strikes me as very odd because in Newtonian theory the exponent has to be exactly 2 and so what could have motivated this complete kludge with no basis in physical or mathematical principle. It is unlike the Nordstrom theory because that, though wrong, is actually mathematically coherent.

My problem is that this was an aside from a lecture 25 years ago and I can't find any details on it. I would like to know more on the reasoning behind this idea and when and by whom this was suggested. If possible I'd like to know if it gained any traction whatsoever.

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  • $\begingroup$ Changing the exponent was proposed by reputed astronomers (Hall and Newcomb) and was seriously considered at the time, along with the planet Vulcan. However, the most popular solution before GR was the influence of a rarefied dust cloud surrounding the Sun proposed by Seeliger, which had the added advantage of explaining the zodiacal light, see Anomalous Precessions chapter from Brown's Reflections on Relativity. $\endgroup$ – Conifold Feb 8 at 2:43
  • $\begingroup$ The discussion of this question clears up something I've been wondering about since the early 1970s (but apparently not since the late 1990s, or I probably would have done an internet search or asked about it in sci.math or something), namely a passage on the middle of p. 119 of Relativity for the Layman by James A. Coleman (I have the 1969 Signet Books revised edition): "If the Newton law of universal gravitation (Equation 8) is modified (continued) $\endgroup$ – Dave L Renfro Feb 8 at 19:06
  • $\begingroup$ so that the results agree with $[\cdots]$ then the corrected formula should be Equation 9: $$F = \frac{G \; mm'}{d^{2.00000016}}$$ which is only slightly different from $[\cdots]$" I read several, probably over 10, such semi-popular books about relativity when I was roughly age 13 to 16, and I never saw this mentioned in any of the other books, and at the time I wondered why complications of curved spacetime were needed when we can just use the exponent $2.00000016$ (even reading this now, I find Coleman's discussion extremely misleading; he definitely wasn't paying attention to his readers). $\endgroup$ – Dave L Renfro Feb 8 at 19:20
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In this page K. Brown writes:

The failure to arrive at a realistic Newtonian explanation for the anomalous precession led some researchers, notably Asaph Hall and Simon Newcomb, to consider the possibility that Newtonian theory itself was at fault, i.e., that perhaps gravity isn't exactly an inverse square law. Hall noted that he could account for Mercury's precession if the law of gravity, instead of falling off as $1/r$ , actually falls of as $1/r^n$ where the exponent $n$ is 2.00000016. However, most people didn't (and still don't) find that idea to be very appealing, since it conflicts with basic conservation laws, e.g., Gauss's Law, unless we also postulate a correspondingly modified metric for space (ironically enough). Several researchers attempted to explain Mercury’s precession by imposing a finite speed of gravity on Newton’s theory, but these efforts too gave unsatisfactory results. (See Section 8.10.)

It seems that the first one to propose this solution was Asaph Hall:
A suggestion in the theory of Mercury, Astronomical Journal, vol. 14, iss. 319, p. 49-51 (1894), which can be found here.

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The idea that the exponent in the law of gravitation is not exactly 2 was around since 18th century when people were trying to work the precise theory of the Moon. At some point, Clairaut thought that deviations in the Moon motion prove that the exponent cannot be 2. A satisfactory theory of the Moon was only developed in the middle of 18th century.

Now, it is not clear why you write that "in Newtonian theory the exponent has to be exactly 2". On what grounds? After all, physical theories have to be checked against experiments and observations, and when a good theory does not account for some small deviations, it must be modified. This is exactly what happened when General relativity was developed. Of course it is a much more elegant modification than just replacing the exponent, but how could one know in advance what kind of modification is required?

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  • $\begingroup$ Err... because Newtonian physics is fundamentally based on Euclidean 3-space + a universal time co-ordinate. From a point source, as you go out, the gravitational flux is spread out over a sphere - 4 pi r^2. $\endgroup$ – NickM Feb 8 at 10:34
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    $\begingroup$ @NickM Sure, but that's an assumption based on empirical evidence. And it clearly runs into problems with the perihelion precession of Mercury. So if allowing for a different exponent makes the model fit with observation, then it is our duty to at least consider adjusting the model. No matter how ugly the adjustment is or how difficult it may be to explain it using a bigger theory or framework, if it fits better with experiments, it is a better model. $\endgroup$ – Arthur Feb 8 at 10:48
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    $\begingroup$ @NickM There is nothing fundamental about gravitational flux being conserved, Euclidean 3-space along with absolute time have nothing to say about that. Without it the inverse square law does not follow. $\endgroup$ – Conifold Feb 8 at 12:04
  • $\begingroup$ Seeing as there is no mechanism suggested as to why grav flux wouldn't be conserved I fail to see how you can think a purely ad-hoc adjustment to fit a specific case in what purports to be a universal theory is at all justifiable. What effect would changing the exponent have on the orbits of other bodies? It seems to me that what my detractors here are suggesting is something akin to the Ptolomaic system with epicycles added as and when. It is continual improvised fire-fighting rather than building the city with proper fire protection and a professional fire brigade. $\endgroup$ – NickM Feb 9 at 22:20
  • $\begingroup$ I have not read the details of these papers, but I suppose the proposed adjustment was sufficiently small not to affect other planets. And I don't understand why you seem to mention Ptolemy system as a negative example. On my opinion, this was a very successful theory in astronomy. $\endgroup$ – Alexandre Eremenko Feb 10 at 1:51

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