# Talk:Newton's theorem of revolving orbits

Does "twice as small" (in Precession of the Moon's orbit) mean half the size or twice the size? R.e.b. (talk) 04:57, 19 August 2008 (UTC)

Sorry, it means "half the size", roughly 1.5° predicted versus 3.0° observed.  Please don't quote me on these numbers; I'll try to find the exact values and add them to the article.  Thank you very much for catching that! :) Willow (talk) 23:25, 19 August 2008 (UTC)

This review is transcluded from Talk:Newton's theorem of revolving orbits/GA1. The edit link for this section can be used to add comments to the review.


### Protonk comments
As I noted with Problem of Apollonius, I am not a mathematician, so this will be a non-specialist review.  Errors in understanding are likely mine. :)

Thank you very much, Protonk!  I'll try to incorporate your suggestions, which are great as always. :) Willow (talk) 18:09, 20 August 2008 (UTC)
Images Image tags check out.  Images are largely clear and helpful.  Is it the intent of the editors here to use a black background for the videos?  I think white might look better but that may be a personal choice.
I might add an image of an elliptical orbit (especially one that is more clear to the reader than the otherwise very helpful ogg videos.  I'll look for one once I'm done with the review.
I chose black because I wanted to show the shadowing of the Sun's light on the back side of the planet, so that the viewer could better see how the planet was rotating.  Ummm, not to push my own work, but if you were looking for an illustration of a purely elliptical orbit, there's a top-down view and this funky tilted animation, which I like a lot. :) Willow (talk) 18:09, 20 August 2008 (UTC)
Style/MOS
The lead moves too quickly into exposition and illustration of the content with respect to the special case of inverse square forces.  Try to treat that paragraph in the lead as a general overview of the sub-topic rather than delving right into it.
the lead does not mention the illustrative examples or the alternate derivations (aside from Clairaut's).
I need to brood over this a little longer before I can see a solution. Willow (talk) 19:35, 20 August 2008 (UTC)
"...k-fold faster..." Is "k-fold" used in the literature?  Would "k-times faster" be improper?
I changed all but one instance to "k times faster"; the one left over was a "k-fold change". Willow (talk) 19:35, 20 August 2008 (UTC)
"
  
    
      
        
          θ
          
            1
          
        
        ≡
        ∫
        d
        t
         
        
          ω
          
            1
          
        
        (
        t
        )
      
    
    {\displaystyle \theta _{1}\equiv \int dt\ \omega _{1}(t)}
  
", should this read: "
  
    
      
        θ
        ≡
        ∫
        ω
        (
        t
        )
        d
        t
         
      
    
    {\displaystyle \theta \equiv \int \omega (t)dt\ }
  
"?  Partially because your previous sentence is defining the angle variable generally and partially because I thought the integrand went before the infintesimal.  at least removing the subscripts would make the sentence following the equation redundant.
I put the infinitesimal dt last for clarity, as you suggested.  Thanks! :) Willow (talk) 19:35, 20 August 2008 (UTC)
Is it still your intent to leave the expression particularized for 
  
    
      
        
          θ
          
            1
          
        
      
    
    {\displaystyle \theta _{1}}
  
? Protonk (talk) 19:40, 20 August 2008 (UTC)
That was actually my intention.  I was worried about introducing a general variable, because in the article, we generally specify whether the angle variable pertains to particle 1 or 2.  I wouldn't want readers to get confused or have to wonder about it.  A little redundancy isn't too bad, is it?  We could use a general subscript, such as 
  
    
      
        
          θ
          
            s
          
        
      
    
    {\displaystyle \theta _{s}}
  
 where s equals 1 or 2, but that's also complicated for most readers.  Willow (talk) 22:48, 20 August 2008 (UTC)
That sounds great.  I was mostly asking to see if it was intentional. Protonk (talk) 00:17, 21 August 2008 (UTC)
"Newton's theorem holds for all types of the original central force F1(r)." this sentence is unclear to me.  Works for all functional types (cubic, quadratic, etc.)?  All real numbers (as a function output)?  The sentence immediately following it makes me think the latter.
I re-wrote the whole beginning of that section; I hope it's better now? Willow (talk) 19:35, 20 August 2008 (UTC)
"Over time, the long axis of most orbiting bodies rotates gradually, generally no more than a few degrees per complete revolution, because of perturbations from other planets, general relativistic effects, and so on." Perhaps a wikilink to a specific section in Tests of general relativity?
I added the link, and also another source of precession (oblateness with its own wikilink. Willow (talk) 19:35, 20 August 2008 (UTC)
"Newton's method uses this apsidal precession as a sensitive probe of the type of force being applied to the planets." Placed where it is currently, this sentence seems to be a non-sequitur.  Do we mean to say that newton looked at near-circular orbits and used the apsidal precession of the orbits to calculate the central forces applied?  that seems to be the case and if so it should probably not follow a sentence that asserts (correctly) that some of the causes of precession were non-newtonian in nature.
Yes, this is lame. :P  I'm thinking of starting a new section, just after the lead, called "Historical context" where we could go into the whole story of planetary orbits and motivate the theorem from the beginning. Willow (talk) 19:35, 20 August 2008 (UTC)
The historical context section is wonderfully written prose.  Can we source a few of the claims made there, namely:
"Any orbit can be described with a sufficient number of judiciously chosen epicycles, since this approach corresponds to a modern Fourier transform."
"However, this conclusion holds only when two bodies are present (the two-body problem); the motion of three bodies or more acting under their mutual gravitation (the n-body problem) remained unsolved for centuries after Newton, although solutions to a few special cases were discovered. "
Also, Image:Kepler laws diagram.svg would work wonderfuly next to the brief recitation the historical context section gives of Kepler's laws. Protonk (talk) 18:18, 21 August 2008 (UTC)
Thank you, Protonk; your praise means a lot. :)  I'll try to dig up references for all of those.  The image, though, seems a little too complicated for casual readers; I'll try to remake it with just the bare essentials.  I think I'm going to add a glowing line of apsides to the precession animation (Figure 3 now).  Glowing myself, Willow (talk) 19:08, 21 August 2008 (UTC)
"...to be a smooth, continuous function of the orbital eccentricity..." A wikilink to Continuous function and Smooth function or perhaps (to avoid linking to subjects that might be unhelpful) Differentiable#Continuity_and_differentiability, unless the eccentricity may be smooth, continuous but not differentiable.  I don't know.
I linked to the smooth and continuous functions, since I think that suffices for their arguments; that's also what they say in their article. Willow (talk) 19:35, 20 August 2008 (UTC)
"Newton's approach is to expand C(r) in a Taylor expansion in the distance r" I would recommend treating this as Guicciardini does in Reading the Principia].  We should say something akin to "Newton expanded this in a series now referred to as a Taylor series" (p. 80, 92).
I followed your advice.  I also found out something neat; this seems to be the first appearance of the Taylor series, according to the reference. :) Willow (talk) 19:35, 20 August 2008 (UTC)
"...angular scaling factor k for nearly circular orbits equals..." for the "such that r=R" in the following equation, should that read "such that r is approximately R" (don't know how to do the double ~)?  Or am I reading that wrong?
No, it should be evaluated at exactly r=R, if I'm understanding Chandrasekhar's retelling of Newton correctly. Probably it's not significant whether it's evaluated at the mean, min or max radius, if the orbits are nearly circular. Willow (talk) 19:35, 20 August 2008 (UTC)
"A successful model of the Moon's motion would be useful in determining longitude..." the tense in this sentence makes it seem as though we want for such a model today.
I'll try to fix this. Willow (talk) 19:35, 20 August 2008 (UTC)
"(to first approximation)" Any relationship to Orders of approximation?
Yes, thank you!  I made the wikilink. Willow (talk) 19:35, 20 August 2008 (UTC)
In the derivation section, "
  
    
      
        
          θ
          
            2
          
        
        (
        t
        )
        =
        k
        
          θ
          
            1
          
        
        (
        t
        )
      
    
    {\displaystyle \theta _{2}(t)=k\theta _{1}(t)}
  
" is abnormally small and I can't figure out why.
I haven't the faintest idea, either; maybe it's a formatting preference thing? Willow (talk) 19:35, 20 August 2008 (UTC)
It's being turned into HTML by Wikipedia's TeX rendering engine, texvc. The default font for that is unfortunately a little small. I added a thinspace and a negative thinspace at the end of the equation to force it to render as a PNG like the rest of the text.
Oh, by the way Willow, I'm back from vacation so I've read your comments on my peer review. I like the changes you've made, and I don't have any more concerns about the article. FA, here you come! :-) Ozob (talk) 02:37, 21 August 2008 (UTC)
Fixed.  Thanks. Protonk (talk) 02:40, 21 August 2008 (UTC)

Layout
The reasoning behind the article layout is not clear to me.  Did Newton derive first the results of revolving orbits with central inverse cube forces and then use the resulting predictions to show that inverse square forces drove mechanics?  Or did he assume from the start two particles with the same starting point distinguished by 
  
    
      
        
          ω
          
            2
          
        
        =
        
          
            
              d
              
                θ
                
                  2
                
              
            
            
              d
              t
            
          
        
        =
        k
        
          
            
              d
              
                θ
                
                  2
                
              
            
            
              d
              t
            
          
        
        =
        k
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{2}={\frac {d\theta _{2}}{dt}}=k{\frac {d\theta _{2}}{dt}}=k\omega _{1}}
  
 and derive what force would result in the different angular motions?
I think the former.  He mainly wanted to derive the phenomenological theory of planetary and lunar motions from a physical theory of gravitation, and he needed (1) a force law for gravity and (2) a way to deal with precessing orbits.  This theorem helps him get both. Willow (talk) 19:35, 20 August 2008 (UTC)
I think--in the case that he derived a set of equations to fit Kelper's observations--that we might benefit from arranging this article in a matter in line with that direction of thought.  If newton started with elliptical and circular orbits where objects returned to their starting points with the same velocity then explored different possibilities in order to explain precession, we might do well to follow that order.  I don't know (I should check) if that is how it happened, but if it was, the order could be changed to introduce newton's proof of kepler's laws first and then the "generalization" (committing some intellectual violence there) to non-elliptical orbits (or bodies other than those around the sun, whatever works).  I'll try and read up and see if that is out to lunch or not. Protonk (talk) 19:52, 20 August 2008 (UTC)
Qualitative behavior and orbital precession:  I know it is simple, but this section might benefit from a display of 
  
    
      
        
          F
          
            2
          
        
        (
        r
        )
        −
        
          F
          
            1
          
        
        (
        r
        )
        =
        
          
            
              L
              
                1
              
              
                2
              
            
            
              m
              
                r
                
                  3
                
              
            
          
        
        
          (
          
            1
            −
            
              k
              
                2
              
            
          
          )
        
      
    
    {\displaystyle F_{2}(r)-F_{1}(r)={\frac {L_{1}^{2}}{mr^{3}}}\left(1-k^{2}\right)}
  
 at the top, just to make the form more clear to readers like me.  That way we can easily see the relationship between k2 and the difference in central forces.
I added the formula, with a slight re-write of the beginning. Willow (talk) 19:35, 20 August 2008 (UTC)
"As a final illustration, Newton considers the case of..." the resulting equation for the scaling factor is illustrated, but the article doesn't explain why this example is listed.  Are we just exhausting what he shows in the Principia?  Should an explanation of the shape of the resulting orbit be considered?  Should we remove this?
I'll need to think more about this.  I'd like to talk about all three of the examples Newton discusses in the Principia.  Willow (talk) 19:35, 20 August 2008 (UTC)
Precession of the Moon's orbit section.  This would make a nice breakout session showing how Newton came to his conclusion.  See here.  This is an idle suggestion and has no bearing on the article's GA nomination.
I'd like to hear more about the Moon myself, but I don't know enough as yet.  I'm trying to read up on it. Willow (talk) 19:35, 20 August 2008 (UTC)
Newton's derivation section could do with an image showing the idea behind the geometric proof.  As I mentioned on Willow's talk page, Calculus (book) Ch 17 has some good illustrations.  The proof itself could also be a breakout article.
Yes, that's be nice; I'll try to make an SVG version of Newton's image.  Willow (talk) 19:35, 20 August 2008 (UTC)
POV Fine.
Overall.  This is a great article.  It is further from GA than Problem of Apollonius but that is hardly a slur.  The biggest problems in the article stem from the lead, some sections lacking clarity, and some overall concerns with the layout.  I'm placing this article on hold.  Thanks for the opportunity to review such a good work. Protonk (talk) 00:33, 20 August 2008 (UTC)

Thank you, it's a real pleasure to have such a nice and insightful reviewer! :) Willow (talk) 19:35, 20 August 2008 (UTC)
A note.  I didn't read the peer review comments before writing mine.  sorry if I have duplicate concerns. Protonk (talk) 00:48, 20 August 2008 (UTC)


#### Updates
Let me know when you think you're done updating the article and I'll give it a quick once over. Protonk (talk) 17:52, 22 August 2008 (UTC)

Hi Proton, I think I may have addressed all of your concerns, at least the ones listed here? However, I changed the article over the past few days, so you should probably re-review it with fresh eyes.  I'll add the image for the Newton derivation early next week.  Thanks again! :) Willow (talk) 22:16, 22 August 2008 (UTC)
It wasn't as hard as I thought, so I uploaded the diagram for the Newtonian derivation this morning.  Thank you! :) Willow (talk) 12:49, 23 August 2008 (UTC)
PS. Oh, could you add the reference for Spivak to "Further reading"?  I can't seem to find that book anywhere. :(  That's perhaps not surprising, being so far in the country. :P Willow (talk) 12:51, 23 August 2008 (UTC)
Done, and I passed the article.  Congrats! Protonk (talk) 14:18, 23 August 2008 (UTC)
Thank you again, Protonk!  I appreciate your careful attention and suggestions; the article is so much better now. :) I'll begin working on X-ray crystallography this weekend, where I think there's even more room for improvement... ;)  Willow (talk) 15:02, 23 August 2008 (UTC)


### Illustrative example: Cotes' spirals
I think the formulae for k and lambda need checking.
For example, cosh of an angle starts with value 1 for angle zero, and increases indefinitely.  cosh of an angle cannot go less than 1.  This means that 1/r cannot go to zero, but can go indefinitely high.  r, therefore, cannot go to infinity, but can only reach a certain maximum distance (although it can get ever smaller).Roo60 (talk) 00:30, 8 February 2009 (UTC)

This does not seem to contradict the article. For the moment I've removed your expert attention template, but if you can clarify what you think is wrong, I'll be happy to look into it. Ozob (talk) 00:17, 10 February 2009 (UTC)
I have gathered the various results stated in this section:
For this problem, there are 3 different solutions:

1.)  
  
    
      
        
          
            1
            r
          
        
        =
        
          
            1
            b
          
        
        cos
        ⁡
         
        
          (
          
            
              
                
                  θ
                  
                    2
                  
                
                −
                
                  θ
                  
                    0
                  
                
              
              k
            
          
          )
        
      
    
    {\displaystyle {\frac {1}{r}}={\frac {1}{b}}\cos \ \left({\frac {\theta _{2}-\theta _{0}}{k}}\right)}
  
 , where the constant   
  
    
      
        
          k
          
            2
          
        
        =
        1
        −
        
          
            
              m
              μ
            
            
              L
              
                1
              
              
                2
              
            
          
        
      
    
    {\displaystyle k^{2}=1-{\frac {m\mu }{L_{1}^{2}}}}
  
 .
2.) 
  
    
      
        
          
            1
            r
          
        
        =
        
          
            1
            b
          
        
        cosh
        ⁡
         
        
          (
          
            
              
                
                  θ
                  
                    0
                  
                
                −
                
                  θ
                  
                    2
                  
                
              
              λ
            
          
          )
        
      
    
    {\displaystyle {\frac {1}{r}}={\frac {1}{b}}\cosh \ \left({\frac {\theta _{0}-\theta _{2}}{\lambda }}\right)}
  
 , where the constant 
  
    
      
        
          λ
          
            2
          
        
        =
        −
        
          
            
              m
              μ
            
            
              L
              
                1
              
              
                2
              
            
          
        
        −
        1
      
    
    {\displaystyle \lambda ^{2}=-{\frac {m\mu }{L_{1}^{2}}}-1}
  
 .
3.) 
  
    
      
        
          
            1
            r
          
        
        =
        A
        
          θ
          
            2
          
        
        +
        ε
      
    
    {\displaystyle {\frac {1}{r}}=A\theta _{2}+\varepsilon }
  
 , where A and ε are arbitrary constants.
For case 1, the possible values of the parameter k may range from zero to infinity, which corresponds to values of μ ranging from negative infinity up to the positive upper limit, L12/m. 
For case 2, The possible values of λ range from zero to infinity, which corresponds to values of μ less than the negative number -L12/m. 
Now, there seems to be an overlap of μ values for these two cases, between negative infinity and  -L12/m.  So, for the forces with μ values in this region, there are two possible orbits for the second particle.  If I have learned my physics correctly, this can only happen in QM.  Can someone please explain what's happening here? Thanks!--LaoChen (talk) 06:20, 10 August 2010 (UTC)

Yes, there's a typo. There was an extra minus sign in the expression for λ. I've removed it, so the article is OK again. Incidentally, the two formulas are really instances of the same formula: Since cos ix = cosh x, you can deduce the second formula from the first (or vice versa) if you assume that the formula ought to remain true for all μ.  Ozob (talk) 23:52, 11 August 2010 (UTC)


### Problem with equation in Generalization section
I don't think the following force equation for the particle 2 is correct:

  
    
      
        
          F
          
            2
          
        
        (
        
          r
          
            2
          
        
        )
        =
        
          
            
              a
              
                3
              
            
            
              
                (
                
                  1
                  −
                  b
                  
                    r
                    
                      2
                    
                  
                
                )
              
              
                2
              
            
          
        
        
          F
          
            1
          
        
        
          (
          
            
              
                a
                
                  r
                  
                    2
                  
                
              
              
                1
                −
                b
                
                  r
                  
                    2
                  
                
              
            
          
          )
        
        +
        
          
            
              L
              
                2
              
            
            
              m
              
                r
                
                  2
                
                
                  3
                
              
            
          
        
        
          (
          
            1
            −
            
              k
              
                2
              
            
          
          )
        
        −
        
          
            
              b
              
                L
                
                  2
                
              
            
            
              m
              
                r
                
                  2
                
                
                  2
                
              
            
          
        
        
        
      
    
    {\displaystyle F_{2}(r_{2})={\frac {a^{3}}{\left(1-br_{2}\right)^{2}}}F_{1}\left({\frac {ar_{2}}{1-br_{2}}}\right)+{\frac {L^{2}}{mr_{2}^{3}}}\left(1-k^{2}\right)-{\frac {bL^{2}}{mr_{2}^{2}}}\,\!}
  
 .
Take the case 

  
    
      
        a
        =
        0
        ,
        
        k
        =
        1
        
        
      
    
    {\displaystyle a=0,\qquad k=1\,\!}
  
 .
Then, particle 2's orbit is a circle with radius 
  
    
      
        1
        
          /
        
        b
        
        
      
    
    {\displaystyle 1/b\,\!}
  
 :

  
    
      
        
          r
          
            2
          
        
        (
        t
        )
        =
        1
        
          /
        
        b
        
        
      
    
    {\displaystyle r_{2}(t)=1/b\,\!}
  
 .
Also, the angular velocities for both particle 1 and particle are the same:

  
    
      
        
          θ
          
            1
          
        
        (
        t
        )
        =
        
          θ
          
            2
          
        
        (
        t
        )
        
          /
        
        k
        =
        
          θ
          
            2
          
        
        (
        t
        )
        
        
      
    
    {\displaystyle \theta _{1}(t)=\theta _{2}(t)/k=\theta _{2}(t)\,\!}
  
 .
The angular momentum for particle 2 should be a constant for a central force:

  
    
      
        
          L
          
            2
          
        
        =
        m
        
          r
          
            2
          
          
            2
          
        
        
          
            
              
                θ
                ˙
              
            
          
          
            2
          
        
        
        
      
    
    {\displaystyle L_{2}=mr_{2}^{2}{\dot {\theta }}_{2}\,\!}
  
 .
So, the angular velocities for both particle 1 and particle are same constant

  
    
      
        
          
            
              
                θ
                ˙
              
            
          
          
            1
          
        
        =
        
          
            
              
                θ
                ˙
              
            
          
          
            2
          
        
        =
        
          L
          
            2
          
        
        
          b
          
            2
          
        
        
          /
        
        m
        
        
      
    
    {\displaystyle {\dot {\theta }}_{1}={\dot {\theta }}_{2}=L_{2}b^{2}/m\,\!}
  
 .
Now, the force for particle 1 is arbitrary, let's say of the following inverse square distance form:

  
    
      
        
          F
          
            1
          
        
        (
        
          r
          
            1
          
        
        )
        =
        −
        μ
        
          /
        
        
          r
          
            1
          
          
            2
          
        
        
        
      
    
    {\displaystyle F_{1}(r_{1})=-\mu /r_{1}^{2}\,\!}
  
 .
So, the usual orbit for particle 1 should be an ellipse with radius 
  
    
      
        
          r
          
            1
          
        
        (
        t
        )
        
        
      
    
    {\displaystyle r_{1}(t)\,\!}
  
 not been a constant. Then, the angular momentum for particle 1 is not constant either:

  
    
      
        
          L
          
            1
          
        
        =
        m
        
          r
          
            1
          
          
            2
          
        
        
          
            
              
                θ
                ˙
              
            
          
          
            1
          
        
        =
        
          L
          
            2
          
        
        
          b
          
            2
          
        
        
          r
          
            1
          
          
            2
          
        
        
        
      
    
    {\displaystyle L_{1}=mr_{1}^{2}{\dot {\theta }}_{1}=L_{2}b^{2}r_{1}^{2}\,\!}
  
 .
But, for central force, angular momentum should be a constant.  There must be something wrong.  Please help!--LaoChen (talk) 05:51, 9 September 2010 (UTC)


### Can someone elaborate on the case of an imaginary coefficient?
Where it says ″By contrast, if k2 is less than one, F2−F1 is a positive number; the added inverse-cube force is repulsive″, I wonder what is the physical meaning of k2 being negative, which obviously implies that k is imaginary. Can some expert elaborate on this case, and explain how can k, the ratio between the angular speeds, be imaginary? Thanks a lot.  — Preceding unsigned comment added by 94.209.165.16 (talk) 16:11, 22 September 2014 (UTC) 

It's not "negative" but less than one. The value k (the article should probably not say it can be "any constant", but I'll look into that) is the ratio of the angular speed of one particle to another. As such it can be 0 (exactly the same speed) or any positive number (some multiple of the other particle's speed). Protonk (talk) 17:27, 22 September 2014 (UTC)

I've tried, but can't at all see the relevance of all the stuff about ancient observations of planetary motion, retrogradation, mention of epicycles, to the subject of this article.  
This article, in its essentials, is about a (very significant, to be sure) point within Newton's rational-mechanical analysis of orbits under different conceivable laws of central force – e.g. including exactly inverse-square, an inverse power law with an index a little more than 2, an index a little less than 2, and so on with other power laws.  He shows how different power laws reveal themselves in the characteristics of the orbit, e.g. the rotational motion (or lack of it) of the direction of the long axis of a slightly elliptical orbit.  One of his purposes was to provide a test, very sensitive for his time, by which observations could furnish a check on the reality and accuracy, or otherwise, of the inverse square law in the physical world.
The recent research explores numerous extensions of the applicable conditions in various ways; it shows among other things what happens if approximative assumptions made by Newton are not made; what happens if the restriction 'slightly' on the eccentricity of the ellipses (i.e. restriction to very small values of eccentricity) is removed. 
It's not apparent how there is any real nexus here with the basics of planetary motion and retrogradation still less the epicycles.  I'd suggest that for the basics and that part of the history, a wikilink to the relevant stuff would be clearer, and would also help underline how the topics are distinct.  The present collection of historical stuff seems irrelevant and confusing in its aggregate in the context, rather than enlightening. Terry0051 (talk) 20:17, 1 October 2009 (UTC)

I concur. NOrbeck (talk) 08:33, 13 August 2010 (UTC)

It has been a while since this article has been reviewed, so I took a look and noticed the following:

There are uncited statements throughout the article, including entire paragraphs. While some are not necessary per WP:CALC, other statements should be cited if they are making definitive statements that are not calculations.
There are lots of sources listed in the "Bibliography" that are not used as inline citations and in the "Further reading" section. Does this article cover all major aspects of the topic?
Some sections are too long and make it difficult to navigate (especially on mobile) such as "Historical context", "Precession of the Moon's orbit", "Proposition 43; Problem 30" and "Modern derivation". These should be summarised more effectively or split up using additional headers.
Should this article go to WP:GAR? Z1720 (talk) 22:44, 9 November 2025 (UTC)