Relativistic rotation of eccentric planetary orbits

[Jadzia]

I read last week how the relativistic effect of star gravity can warp space, and how planets with elliptic orbits (such as our own Mercury) dipping in and our of warped space, causes the planet's elliptic axis to rotate. We do witness this effect with Mercury.

We are told that the gravitiational effects of venus and the other planets are taken into consideration, and these are responsible for 50% of Mercury's axis rotation, with relativity making up the other 50%.

A little Newtonian simulation yesterday revealed that yes, venus does have by far the biggest effect on Mercury's orbit, and it is as stated, about 50% of the measured phenomenon. Venus counters the sun's gravity a little as it makes its near pass, so Mercury is carried forward a little and the elliptic axis is rotated forwards.


Well playing devil's advocate, I was thinking in bed this morning of other explanations (other than relativity) for the other 50% of this axis rotation.

I came up with an idea that seemed worth investigating, and a simulation later seems to do away with the relativity explanation.

My theory takes into consideration the fact that neither the sun or mercury are point source masses. The day-side of mercury is closer to the sun than the night-side, so the sun's gravity is slightly stronger on the day-side than the night side.

Because gravity is inverse-square, the gravitational differences from these two sides don't cancel perfectly out, and the error term is inverse-cube. The overall effect is that the centre-of-gravity of Mercury (in the sun's gravity) is shifted slightly closer to the sun than it's centre-of-mass.

The centre-of-gravity is the point which defines the distance where gravitational forces are calculated from. Only in a uniform/linear gravitational field are these two centres concurrent but tend to be used interchangeably.

You do the same thing for the sun too.

In other words, the sun's gravity is felt slightly stronger than inverse-square, and in my model, it alone causes the orbit to rotate forwards - apparently the 50% of the measured effect that we are told is due to relativity.

For the earth, I calculated this difference to be about 30 metres or so. ie, when calculating the sun's gravitational force on earth, subtract 30 metres from the distance of 150 million kms. For the earth this extra gravity is negligible. For mercury it is not.

Now I might have messed up somwehere and am oblivious to it. But I think we should investigate this further.


Jadzia

 

[Haytil]

Do you have code that we can look at to see what you did/check for errors?

 

[Shaw]

Sounds like an interesting project, but what methods in Classical Mechanics are you using? Strict Newtonian mechanics does simple orbits okay, but for complex multi-body systems it is best to embark on such an endeavor using the right tools for the job. In this case Lagrangian mechanics would be best and Lagrange's methods were developed specifically for this type of problem.

And any study of this should also start with the work of the mathematician who discovered the issue with Mercury's orbit (and discovered Neptune using similar methods), Le Verrier. Since much of Lagrangian mechanics (and calculus of variations) was well established by the time that the issue with Mercury was discovered, I would venture a guess that most of the mathematics you would need to consider first would be found in the attempts by Le Verrier to explain the issue that he continued to make for the rest of his life.

What is interesting to note is that the mathematics that would eventually solve the Mercury issue (Riemannian Geometry) was being developed at about the same time that Le Verrier was making his attempts. The actual mathematical structural elements wouldn't show up until after Einstein and Hilbert's original work was put forward, and Levi-Civita is generally credited with some of thoughs break throughs.

There is (of course) no shortage of text on the subject of Classical Mechanics. Just a cursory review of the table of content of this text or this text shows that they cover most of what you would need in this type of study. And there is a nice 8 lecture series by Stanford's Prof. Susskind available for free here if you wanted to watch some nice lectures on the subject.


While General Relativity may seem mystical and out of reach to most people, by the time you have a solid foundation in Classical Mechanics enough to tackle the issue you're attempting here, an understanding of General Relativity becomes a small hurdle.


This is an admirable project, and I would be interested in not only seeing what you find out , but also to see it compared and contrasted with what Le Verrier had put forward. It would be mainly a comparing of 21st century methods against those of the 19th century. I look forward to your progress with great interest.