0

According to the table in wikipedia, there are multiple contributing factors that add to the precession of Mercury's orbit.

What I am curious about is, how do we know these factors can be evaluated independently and then added together?

I also have two follow-up questions.

  1. If the effects don't combine linearly, how do we combine the contributions? Is there any reference/paper/textbook that talks about how to combine the effects correctly?
  2. Even if the effects don't combine linearly, why is this such a good approximation? Is there a proof that shows this?
Maximal Ideal
  • 8,292
  • Notice that the sum of the predicted effects in the table don't add to the experimental results; it's a bit off. The tug due to the planets is certainly predicted by using $N$-body (numerical) Newtonian simulations while the general relativistic effect is, well, relativistic. So you can't just add these together since they aren't even from the same framework of physics. – najkim Jul 29 '20 at 15:31
  • @najkim I wonder whether that's due to experimental/observation error or due to the non-additivity of the effects. (It seems like you're almost certainly right that they are not additive, but then I would like to show to myself that they are not. Also, I'd like to show to myself why this ends up being such a good approximation anyways.) – Maximal Ideal Jul 29 '20 at 15:58
  • 1
    I think your question on additivity was brought up a few years ago here and in some of the links therein – najkim Jul 29 '20 at 23:52

2 Answers2

1

We know that the effects combine linearly because they are small perturbations to the orbit. They are given in units of arcsec/Julian century. If you convert them to fractions $p_i$ of an orbit, they will be really small numbers. Then the first non-linear term for combining the fractional effects $p_1$ and $p_2$ will be of the order $p_1 p_2$, which is a really really small number, and we know we can ignore it. That is why simply adding the contributions is so accurate.

This may seem a little unsatisfactory as an answer, but in fact it is obvious if one has actually solved the equations to derive the perturbations, or has worked in any form of perturbation theory. In this case to give the actual solutions would be a major exercise, much more than is possible in a Q&A, but the principle underpins all of perturbation theory and is very well understood. Terms of order $p_1^2$ and $p_2^2$ have already been ignored in the calculation of the given perturbations.

Charles Francis
  • 11,546
  • 4
  • 22
  • 37
  • +1 Thank you so much for the answer! (I would love to know why this is downvoted. This seems exactly the kind of answer I was looking for. Unless there is false info here or someone has a more detailed account, nothing seems to be wrong. I am open to other answers/posts if anyone has more to offer.) – Maximal Ideal Jul 29 '20 at 22:43
0

I think the chain of reasoning goes in the other direction:

  1. We estimate the amount of precession caused by each of the four effects acting independently.

  2. We note that the observed precession is close to the sum of the four separate effects (if the Wikipedia table included uncertainties in the estimated amounts of the four effects then we would have a better idea of just how close).

  3. We conclude that the probability of an additional cause of precession beyond the four that we know of is low and that any non-linear terms are small.

Of course, we might be fooling ourselves. There could be an unexpected cause of precession (a mysterious accumulation of invisible pink unicorns, for example) and non-linear terms and these just happen to almost cancel each other out. But that is when we apply Occam's razor.

gandalf61
  • 52,505