Identifying a Perihelion Shift Solution

[shalayka]

Hello, this is my first post. I am a programmer from Regina, SK Canada who likes to learn about Physics when I have spare time.

My question:
Does anyone know the name of the following solution for the perihelion shift? I do not know who to attribute it to. Thank you for your time.

For Mercury, when the orbit is simplied to being circular:

$$t &=& 88\times24\times60\times60$$

$$r &=& \frac{perihelion + aphelion}{2}$$

$$v &=& \frac{2\pi r}{t}$$

$$n &=& 2\pi[1 - \cos(\arcsin(v/c))]$$

$$\delta &=& n\times360\times60\times60\times415 &=& 43.1$$


For Earth:

$$t &=& 365\times24\times60\times60$$

...

$$\delta &=& n\times360\times60\times60\times100 &=& 4$$


I arrived at the preceding solution while trying to verify if the following acceleration equation works for simulating the transverse gravitation of bodies traveling at less than the speed of light:

$$a &=& \frac{GM[2 - \cos(\arcsin(v/c))]}{r^2}$$

- Shawn

P.S. I will be buying my copy of Gravitation by Misner, et al. soon. Does anyone recommend any other books on this subject of General Relativity?

P.P.S. I already have my copy of Relativity: The Special and the General Theory by Albert Einstein, which includes the equation I used to verify the preceding results:

$$\frac{24\pi^3a^2}{T^2c^2(1 - e^2)}$$

 

[shalayka]

Here is a picture from an exaggerated simulation of Mercury, where:

$$x &=& 10e8$$

$$a &=& \frac{GM[2 - \cos(\arcsin(v/c))^x]}{r^2}$$

http://cwiki.org/uploads/3/35/Orbit.jpg

The Sun is in the centre of the view plane, and Mercury is orbiting counterclockwise.

A transparent line is drawn between the Sun and Mercury at each timestep of the simulation, leading to a gradient buildup.

Mercury can start at aphelion 3-position $$\{0, 69817079e3, 0}\}$$, with a 3-velocity of $$\{-38860, 0, 0\}$$.

 

[pervect]

I am not at this point even sure if your solution is even correct, much less if it has been done before.

A standard approach based on MTW can be found online at http://www.fourmilab.ch/gravitation/orbits/

An online source which gives the Newtonian form of the "effective potential" solution is at

http://physics.ucsd.edu/students/courses/winter2007/physics161/Lectures/p161.1feb07.pdf

though it may be a bit terse.

You'll also find some info in Goldstein, "Classical mechanics", which describes the Newtonian effective potential method in much more detail, and also talks about the relativistic case.

Basically, what happens is this. You get the following equation for $dr/d\tau$ from relativity

$$\frac{d^2 r}{d \tau^2} = E^2 - 1 + \frac{2M}{r} - \frac{L^2}{r^2} + \frac{2 M L^2}{r^3}$$

If you replace the Schwarzschild coordinate r with the Newtonian radius (they're really not the same concept), and replace proper time tau (note: this is not the same as the Schwarzschild time coordinate) with the Newtonian time t, you can draw a formal analogy from the relativistic solution to the Newtonian solution with the addition of an "extra" cubic term, the last term above.

 

[shalayka]

Thank you for all of this information. I have a lot to look into. :)

This simulation I did was mostly a toy, since it uses an instantaneous transfer of gravitational energy between the Sun and Mercury. This simplification was made under the assumption that if the Sun does not move, its potential field will not change.

The goal of this toy was to see if I understood the precession behaviour well enough to model it without the benefits of university-level mathematics.

I am working on a vector + scalar field replacement for this so that I don't have to assume that the force vector points directly toward the emitting body. This way I will also be able to model the Shapiro time delay of light when it travels through the inwardly increasing complexity of the Sun's gravitational field.

I think that the information you've given me, and the information in the MTW book will both give me what I need to make sure I'm doing this all correctly.

I wrote up a couple of papers reviewing the things I went by in order to come up with this approximation method:

http://cwiki.org/uploads/f/f3/Examining_the_Effect_of_Velocity_on_Transverse_Acceleration.zip"
"[URL the perihelion approximation equation, with images of test orbits...[/URL]
http://cwiki.org/uploads/f/f9/Mercury_simulation_source.zip"

 

[pervect]

For some past discussion of a rather sophisiticated GR solar system model, you might look at the following PF thread

https://www.physicsforums.com/showthread.php?t=151751

If you want something fast and lose for a simulator, just add a GM/r^2 potential term to the suns potential field (I think it has a minus sign, i.e. a "pit in the potential", but I'm not positive about the sign).

If you dig up Goldstein's "Classical Mechanics", you can give a justification and source for this approximate method - Goldstein works out the precession of mercury.

 

[shalayka]

Awesome, thank you! I will review the thread and continue working on this.

I really appreciate your time a lot.

Up until now I have had very little contact with the Physics community, and it is nice to feel included in the discussions, and to have somewhere to go to ask questions.

It has been suggested to me that I bring this up with the Educational Outreach Director for the Gravity Probe B Classroom. If I get a reply back, I will post the results here.