Perihelion advance expressed differently

[TrickyDicky]

I was wondering if there is an easy formula to express the perihelion advance of Mercury in terms of angular velocity, instead of degrees per century, Perhaps this is already been done.

My interest in this comes from thinking of the motion of the planet not from the point of view the sun as a center but looking at the planet's trajectory from far above, in this way you can observe globally the sun's path with Mercury following the sun, sort of meandering by its sides, faster than the sun at one side and slower at the other. In this view the precession is more like a tiny acceleration wrt an external non-inertial observer.

Not sure if this makes much sense so any comment is welcome.

 

[Mentz114]

'Degrees per century' is angular velocity ( of the major elliptical axis in this case). I'm not sure what you mean. Maybe you meant 'angular acceleration' ? The effects of various factors have been taken out of Mercury's orbital precession before GR accounts for the residual. So some of the precession is caused by accelerations from asymmetric factors.

 

[D H]

My interest in this comes from thinking of the motion of the planet not from the point of view the sun as a center but looking at the planet's trajectory from far above, ...

?

in this way you can observe globally the sun's path with Mercury following the sun, sort of meandering by its sides, faster than the sun at one side and slower at the other.

You are describing basic Keplerian motion. That is not what perihelion advance is about. In Newtonian mechanics, bound orbits as viewed by an inertial observer are in the shape of an ellipse. Perihelion will always occur at the same location. Add other bodies such as other planets to the mix and the shape is no longer elliptical. Because the Sun is so much more massive than any of the other planets, the shape of a planet's orbit will look very much like an ellipse. One key difference is that perihelion will no longer occur at the same location from one orbit to the next. It will move over time, but rather slowly.

Do note that 43 arc seconds per century is a very, very slow rotation rate. Another way to write this is 1 revolution per 3 million years.

 

[atyy]

http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html

See the figure below "flower pattern". The orbit is very nearly a closed ellipse in a plane. However the ellipse, moves a little each period, while remaining in the same plane.

 

[TrickyDicky]

Thanks for the answers, specially the formula atyy links is what I asked for in the first part of my post.

The rest of my post was probably poorly worded.
What i meant is that the precession due to spacetime curvature is basically calculated from the geodesic deviation equation applied in a Schwartzschild spacetime (therefore with absent Ricci curvature) for a timelike test mass (not going into the details of perturbative Newtonian approximation methods used also).
Now this geodesic deviation that is the basic expression of curvature in GR, can be thought of as an acceleration as in the usual GR textbook treatment of geodesic deviation as the relative acceleration two particles subject to the gravitational influenced of a source, each of them feels from its particular local inertial frame (tidal force).

So I was wondering if the precession we observe of Mercury due to curvature can be expressed in terms of the acceleration of Mercury's (local inertial frame) timelike geodesic wrt our own (local inertial frame) timelike geodesic. It's probably already calculated and if not perhaps it's not so hard with the relative acceleration for the geodesic deviation equation

[PLAIN]http://www.kevinaylward.co.uk/gr/reimann/Image45.gif

 

[atyy]

Geodesic deviation is the deviation of two geodesics that intersect at a point in spacetime. It is related to the spacetime curvature at that point in spacetime. So there isn't a geodesic deviation for mercury's geodesic and ours.

In curved spacetime, there is no global inertial frame. However, in this case, since they want to compare against the Newtonian prediction, they probably made an effort to put the results in as close to a Newtonian global inertial frame as possible. Usually D H knows this sort of thing.

 

[TrickyDicky]

Geodesic deviation is the deviation of two geodesics that intersect at a point in spacetime. It is related to the spacetime curvature at that point in spacetime. So there isn't a geodesic deviation for mercury's geodesic and ours.

In curved spacetime, there is no global inertial frame. However, in this case, since they want to compare against the Newtonian prediction, they probably made an effort to put the results in as close to a Newtonian global inertial frame as possible. Usually D H knows this sort of thing.

I see what you mean, I guess Mercury and the Earth geodesics should intersect at some point to apply them the geodesic deviation equation and get a relative acceleration.
But I'm quite sure the precession due to curvature is obtained with the geodesic deviation equation in static empty spacetime, so I guess what it does is calculating the geodesic deviation from an infinitessimally near geodesic that is approximated to the Newtonian-Keplerian global inertial frame, does this make more sense?

 

[D H]

One way to derive the relativistic precession of Mercury is to start with the Schwarzschild metric and derive equations of motion for radial distance and angular deviation. This results in an ugly mess, but a lot of that ugly mess is very small terms. Dropping all but the very largest results in Kepler's equations. Not going quite so far eventually yields an expression for the relativistic precession (ofttimes, with further simplifications).

An alternative approach is obtained via a post-Newtonian expansion, again keeping only the largest terms. The very largest is of course Newton's law. The next largest can be expressed as a 1/r⁴ perturbative force. After way too much math to portray here, and after discarding periodic terms and smallish terms, one gets a secular effect that once again yields the relativistic precession.

There are lots of derivations in various texts and papers on the 'net. None of them, as far as I can see, worry about Mercury versus Earth geodetics.

 

[TrickyDicky]

There are lots of derivations in various texts and papers on the 'net. None of them, as far as I can see, worry about Mercury versus Earth geodetics.

Sure, I've got that cleared up from atyy's last post as I said in my last post, it would seem impossible with both planets not having coincident geodesics.

One way to derive the relativistic precession of Mercury is to start with the Schwarzschild metric and derive equations of motion for radial distance and angular deviation. This results in an ugly mess, but a lot of that ugly mess is very small terms. Dropping all but the very largest results in Kepler's equations. Not going quite so far eventually yields an expression for the relativistic precession (ofttimes, with further simplifications).

This is the way I'm referring to. Edit: Novice error I mistook the geodesic equation for the geodesic deviation equation

 

[atyy]

There are lots of derivations in various texts and papers on the 'net. None of them, as far as I can see, worry about Mercury versus Earth geodetics.

Only tangentially related, but I've always wondered whether GR still predicts mercury's perihelion precession after the gravitational effects of all the other planets are taken into account, instead of just the sun's. (After all, the discrepancy between observation and Newton came only after considering the effects of all the planets.)

 

[TrickyDicky]

Only tangentially related, but I've always wondered whether GR still predicts mercury's perihelion precession after the gravitational effects of all the other planets are taken into account, instead of just the sun's. (After all, the discrepancy between observation and Newton came only after considering the effects of all the planets.)

GR predicts the Mercury's precession only after all the other planets gravitatonal effects are taken into account, because it gives you the observed discrepancy not explained by Newtonian theory..

 

[atyy]

GR predicts the Mercury's precession only after all the other planets gravitatonal effects are taken into account, because it gives you the observed discrepancy not explained by Newtonian theory..

Yes, but there should be a calculation that shows how all the other planets are taken into account in GR.

 

[D H]

In Newtonian mechanics, the only way the Sun would cause something other than conic section motion would be if the Sun did not have a spherical mass distribution. It doesn't, of course (it is spinning after all), but the effect is extremely small. The calculations done in the 19th century -- without the aid of any modern computation devices, mind you -- addressed the effects due to all of the planets known to them. It also included the Earth's lunisolar precession. They were observing from a non-inertial frame, after all.

 

[PAllen]

Yes, but there should be a calculation that shows how all the other planets are taken into account in GR.

My understanding is the other planets are taken into accout with Newtonian formulas. Mercury is the only orbit with a detectable difference from Newtonian. In effect, you compute the difference from what would be expected by Newton for mercury and the sun alone, and assume that as deviation from the calculation taking other planets into account using Newton.

This is not so fishy, because the GR weak field expansions justify absence of detectible deviation for other planetary effects.

 

[D H]

Yes, but there should be a calculation that shows how all the other planets are taken into account in GR.

Why? The effects of the planets are already second order effects. GR is just going to add a tiny correction to that.

 

[D H]

Mercury is the only orbit with a detectable difference from Newtonian.

A measurable deviation between the observed precession and that predicted by Newtonian gravity exists for the Earth and Venus as well, and maybe other planets by now. The relativistic precession fits the bill quite nicely for both Earth and Venus. (Venus is a bit tougher because its orbit is so nearly circular.)

 

[atyy]

Why? The effects of the planets are already second order effects. GR is just going to add a tiny correction to that.

The full perihelion precession of mercury is much more than 43", but most of it is accounted for by the other planets in Newton, so that 43" is just the left over. How do we know that the amount accounted for by the other planets in GR is the same as that in Newton? If that number is quite different, then 43" is not the unaccounted amount in GR.

 

[D H]

The full perihelion precession of mercury is much more than 43", but most of it is accounted for by the other planets in Newton, so that 43" is just the left over. How do we know that the amount accounted for by the other planets in GR is the same as that in Newton? If that number is quite different, then 43" is not the unaccounted amount in GR.

That 43 arcseconds per century is an incredibly small number, atyy. Compare that to the Newtonian effect of gravitation on Mercury: 1 rotation per 88 days, or 538 million arc seconds per century.

Since the relativistic effect is, to first order, equivalent to a 1/r⁴ force, the relativistic correction for the effects of the other planets on Mercury's orbit will be immeasurably small. Remember that the planets' Newtonian effect on Mercury's orbit is a precession of 530 arcseconds per century. So talking about the relativistic effect of the other planets is an incredibly small part of an already rather small effect.

 

[atyy]

That 43 arcseconds per century is an incredibly small number, atyy. Compare that to the Newtonian effect of gravitation on Mercury: 1 rotation per 88 days, or 538 million arc seconds per century.

Since the relativistic effect is, to first order, equivalent to a 1/r⁴ force, the relativistic correction for the effects of the other planets on Mercury's orbit will be immeasurably small. Remember that the planets' Newtonian effect on Mercury's orbit is a precession of 530 arcseconds per century. So talking about the relativistic effect of the other planets is an incredibly small part of an already rather small effect.

Yes, I understand the argument. I'm just wondering since this is such a famous piece of evidence, whether it's been checked properly (what is the best upper bound on the relativistic effect of the other planets)?