How do we know the various sources for perihelion shift for Mercury are independent of one another? How do we know they are additive?

Question

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.

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?
Even if the effects don't combine linearly, why is this such a good approximation? Is there a proof that shows this?

Charles Francis · Accepted Answer

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.

gandalf61 · Answer

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

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

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).

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.

How do we know the various sources for perihelion shift for Mercury are independent of one another? How do we know they are additive?

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