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Debate:What is the exponent of r in Newtonian gravity? (view source)
Revision as of 01:35, February 27, 2010
, 01:35, February 27, 2010Newcomb's theory
:::::: Marnick, it's not necessary or helpful to run a simulation to test a logical truth. One doesn't "look for evidence" that "2+2=4", and your claim that the planets "will crash" into the sun (within a reasonable time) for any value of the exponent slightly different from 2 is mathematically fallacious. For starters, it implies an absurd discontinuity in the limit as the exponent approaches 2. Open your mind, please, and let logic in.--[[User:Aschlafly|Andy Schlafly]] 10:39, 26 February 2010 (EST)
:::::: Marnick, it's not necessary or helpful to run a simulation to test a logical truth. One doesn't "look for evidence" that "2+2=4", and your claim that the planets "will crash" into the sun (within a reasonable time) for any value of the exponent slightly different from 2 is mathematically fallacious. For starters, it implies an absurd discontinuity in the limit as the exponent approaches 2. Open your mind, please, and let logic in.--[[User:Aschlafly|Andy Schlafly]] 10:39, 26 February 2010 (EST)
I'm quite skeptical of Marnik's claimed simulation. Crashing into the Sun in a single orbit is just not credible for an exponent not wildly different from 2. And "moving through the Sun" is incredible. It implies that he entered the Sun's own radius as one of the simulation parameters. In any case, the planet would have to lose nearly all of its angular momentum. (This is what makes Solar exploration spacecraft trajectories so difficult.)
Running simulations of gravity is not easy. I admit I haven't done this, so I suppose I could be accused of closed-mindedness on this issue. But I ''have'' done an analysis of the equations. The results are below. I agree with Andy that the only thing that will happen, for small perturbations in the exponent, is that the orbits will precess slowly.
The paper to which Andy refers near the top of this page was presumably the one by Simon Newcomb, who proposed an exponent other than 2 for the purpose of explaining the anomalous precession of the perihelion of Mercury. The value he gave was 2.0000001574. This was one of a number of proposed explanations for the precession, prior to general relativity. So, in the last couple of hours (when I should have been thinking about multivariable calculus!) I worked out the equations. Let the gravitational force be formulated as
::<math>F = \frac{G M m}{r^{2+k}}</math>
Now if we let δ be the precession, as a fraction of an orbit, per orbit (that is, if δ is 1/360, the orbit will precess by 1 degree for each of the planet's own years) then (to first order):
::<math>\delta = \frac{k}{2}</math>
This is (to first order) independent of the orbital eccentricity, and independent of the radius!
For Mercury, δ is observed to be .077E-6 (43 arcseconds per (Earth) year.) This gives k=.154E-6, in good agreement with Newcomb's figure.
So why was Newcomb's theory discarded? Because it doesn't work in general. It was first found to be in error in analyzing the orbit of the Earth's moon. But it also gives the same precession for all the planets! We now know that the anomalous precession for the other planets (the δ value) is much less than the value for Mercury.
[[User:SamHB|SamHB]] 20:35, 26 February 2010 (EST)