Path of a planet orbiting a star in GR

[backward]

We know that the orbit of a planet and its star is a conic section. For a closed orbit, it will be an ellipse described by
x^2/a+y^2/b =1, or its equivalent equation in r and θ
What would be the equation of the path under GR? and how will it approximate to a conic section when r(s)/r tends to be very small? (r is the tadial coordinate and r(s) its Schwarzschild radius). When r(s)/r is exactly 0, the path should be a straight line.
Can someone please enlighten me?

 

[Bill_K]

As you say, for Newtonian gravity the orbit is a conic section, most conveniently written in terms of the reciprocal radius, u = 1/r. The Newtonian orbit equation is

u - u₁ = (u₂ - u₁) sin²(φ/2)

where u₁ and u₂ are constants. These can be related to the more usual constants, the eccentricity and the semimajor axis. For example the eccentricity is e = (u₂ - u₁)/(u₂ + u₁).

[STRIKE]Closed[/STRIKE] Bounded orbits in the Schwarzschild field can be written in a form which is quite similar. Basically all you have to do is replace the trig function sin by a Jacobi elliptic function, sn.

This has an important effect - whereas the period of sin is 2π, the period of sn is somewhat greater. Its value is 2K where K is the complete elliptic integral. Which causes the orbit to precess, the "advance of the perihelion".

 

[D H]

Closed orbits in the Schwarzschild field ...

You meant bounded orbit, not closed orbit. A closed orbit is periodic: After some time T the orbiting bodies returns to exactly the same position and velocity. Without precession, this can only happen with either an inverse square force (e.g., Newtonian gravity) or with a linear force (e.g., an ideal spring) per Bertrand's theorem. With precession, an orbit will be closed if the ratio of the radial period to the angular period is rational.

Orbits in general relativity cannot be "closed" because radial distance is not periodic. There is a (typically very small) energy loss due to gravitational radiation.

 

[pervect]

The equations of motion of a test particle are given in http://www.fourmilab.ch/gravitation/orbits/ (You'll find the same equations in GR textbooks).

Typically, the equations are given in parametric form, parameterized by proper time. So one gets differential equation for t(tau), r(tau), and phi(tau), tau being proper time.

There are conserved quantites for test particles, rather similar to energy and angular momentum in calssical physics, that make determining the orbital equation much simpler.

It's possible (but messy) to rewrite the equations in terms of coordinate time "t". There's no particular utility to it (that I"m aware of), but it's possible. I'm not aware of any source that's done this correctly (written the orbits as function of t rather than tau) even though it is possible - I've seen some incorrect ones on the WWW though :-(.

 

[Mentz114]

This paper has excellent illustrations

Uros Kostic, Analytical time-like geodesics in Schwarzschild space-time.
General Relativity and Gravitation, 2012.
Preprint :http://arxiv.org/pdf/1201.5611v1.pdf

and this one integrates the elliptical orbits.

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

 

[George Jones]

You meant bounded orbit, not closed orbit. A closed orbit is periodic: After some time T the orbiting bodies returns to exactly the same position and velocity. Without precession, this can only happen with either an inverse square force (e.g., Newtonian gravity) or with a linear force (e.g., an ideal spring) per Bertrand's theorem. With precession, an orbit will be closed if the ratio of the radial period to the angular period is rational.

Orbits in general relativity cannot be "closed" because radial distance is not periodic. There is a (typically very small) energy loss due to gravitational radiation.

Since this problem is already highly idealized and conceptual (i.e., Schwarzschild), I don't think that it is useful to stray from the path of test particles. In this case, there are some very special non-circular closed orbits.

What are these orbits?

 

[Bill_K]

Since this problem is already highly idealized and conceptual (i.e., Schwarzschild), I don't think that it is useful to stray from the path of test particles. In this case, there are some very special non-circular closed orbits.
What are these orbits?

I give up! What Schwarzschild orbits are closed??

 

[George Jones]

I only thought of this after I wrote a program that traced out orbital curves, and I saw that the patterns reminded me of spirograph.

In very, very special circumstances, there are closed "spirograph" orbits.

A condition for a closed orbit is that the precession angle divides evenly into an integral multiple of 360 degrees, i.e., n*360/(precession angle) = m, where n and m are integers. If this is true, then the total precession after m aphelia is n complete circles, hence the repetition.

I wrote a Java applet for Scwharzschild orbits that illustrates this and other things.

 

[backward]

Thanks everybody