Model Mercury Orbit w/Relativity: Force of Gravity Calc Explained

[edguy99]

This question relates to building a computer model of gravity under relativity in a frame by frame type animation where the force on an orbiting object is calculated between each frame and applied to the animation.

Its pretty easy to model normal planet orbits using Newtons inverse square law f = g1 * (m2)/r² where m2 is the mass of the other object, r is the distance and g1 is the gravitational constant. The more accurate the numbers, the better an orbit you get and they all form nice ellipses that follow Kepler's laws.

Due to relativity, the orbit of Mercury precesses 43 arc seconds per century. In order to model precession, its a common programming trick to introduce a r³ term, ie. calculate the force f = g1 * (m2)/r² + g2 * (m2)/r³. This works great and you can adjust the value of g2 to get any amount of precession you want, specifically you can make the orbit precess at 43 arc seconds per century and model Mercury's orbit with a great deal of accuracy.

The question is why is this method so accurate in modelling the force of gravity under relativity? I have assumed that whatever the correct equation to use to calculate the force of gravity on a planet from the sun that includes relativity can be expressed as some kind of an infinite Taylor series along the lines of f = g1 * (m2)/r² + g2 * (m2)/r³ + g3 * (m2)/r⁴ + g4 * (m2)/r⁵ ... hence the r³ term is simply reflecting the better accuracy.

Is there an expert here that know this for sure? Ie. What is the correct equation that should be used to calculate the force between a planet and the sun under relativity in a frame by frame type calculation? Can it be expressed in this form?

 

[Orodruin]

What is the correct equation that should be used to calculate the force between a planet and the sun under relativity in a frame by frame type calculation?

This statement is not very meaningful. In relativity, the orbits of planets are not due to a force but to the curvature of space-time.

The fact that you can change a single parameter to fit a single parameter should not be very surprising. Since orbits are elliptical in a 1/r potential but not in a general potential, any small disturbance is going to lead to a perihelion precession. Introducing this perturbation and fitting the perturbation parameter can of course get you whatever value you want for the precession (as long as it is small enough to be considered a perturbation). The correct thing to do in GR would be to start from the equations of motion in the Schwarzschild metric.

 

[edguy99]

The correct thing to do in GR would be to start from the equations of motion in the Schwarzschild metric.

Thanks, I agree. The inverse r³ works pretty good, but would like to get the exact equation. Looking for this in a format that could be used in a step by step animation (imagine you are in a spacecraft high above the orbit of Mercury, watching it from above over a lot of ortbits) or maybe someone here has already done a precessing orbit?

 

[Orodruin]

Thanks, I agree. The inverse r³ works pretty good, but would like to get the exact equation. Looking for this in a format that could be used in a step by step animation (imagine you are in a spacecraft high above the orbit of Mercury, watching it from above over a lot of ortbits) or maybe someone here has already done a precessing orbit?

The equations of motion according to GR is, as I already said, just the geodesic equations for the Schwarzschild metric. You can easily find this online.

 

[Mentz114]

Thanks, I agree. The inverse r³ works pretty good, but would like to get the exact equation. Looking for this in a format that could be used in a step by step animation (imagine you are in a spacecraft high above the orbit of Mercury, watching it from above over a lot of ortbits) or maybe someone here has already done a precessing orbit?

The exact solution is here

G. V. Kraniotis, S. B. Whitehouse,
Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi's Inversion
problem.
http://128.84.158.119/abs/astro-ph/0305181v3

Also if you search this forum you will find online applications that plot orbits.

 

[Nugatory]

The inverse r³ works pretty good, but would like to get the exact equation.

Take a look at http://www.fourmilab.ch/gravitation/orbits/ - it has the equations you are looking for, an animation that looks a lot like what you're trying for, and downloadable source code.

 

[bcrowell]

This may be of interest: https://www.physicsforums.com/threa...-schwarzschild-spacetime.808318/#post-5075017