Orbit smulation and speed of gravitation

[serge]

Just a question from a begginer :

In the simulation of a solar or (galactic) system, when you calculate the position of a planet P at time t+dt, you only know the position of the other bodies in the system at time t, so you calculate the distance d between P and any other body Q at time t, and then the acceleration P receives from Q

If gravitation speed were infinite then that would be enough for accurate simulation

But if gravitation has a finite speed c, then P receives actually an acceleration from the position of Q when it was at time t-d/c, so you should keep all past positions in the memory of the computer ?

How in practice is this problem solved for precise simulation ? is it possible to calculate the speed of graviation this way, and is it equal to the speed of light ? (it could be a priori different)

 

[pervect]

The quick answer is that in a Newtonian simulation, gravity always points at the instantaneous position of the object. Not only is there no need to keep the past history of the particles, doing so in the manner you describe (having the force point towards the retarded position) will give seriously incorrect results, even for a simple simulation of the solar system. So the quick answer is that the way to do correct _Newtonian_ simulations is to have the force point towards the instantaneous position of the particle.

It is useful to note that something very similar happens for the electrostatic columb force. If you simulate the force as pointing towards the "past position" of the particle, you will get errors. The correct procedure in the electrostatic case is to use the Lienard-Wiechert retarded potentials, not retarded forces. One will find by using the LW procedure that for two charges moving at a constant velocity, the direction of the force is towards the instantaneous position of the charge.

In fact, the conservation of angular momentum *demands* that the force be towards the instantaenous position of the the charge/mass, except insofar as some small amount of angular momentum is carried off by electromagnetic (or gravitational) waves.

Some references

Does gravity travel at the speed of light?

Lienard-Wiechert potentials

(The last link is quite terse, but you can google to find more about the LW potentials).

 

[serge]

Thank you very much, pervect.
If i understand, the force on A points not towards B's retarded position, but towards B's "linearly extrapolated" retarded position, but as long particles follow geodesics (no acceleration), this is the same as if gravity were propagating at infinite speed.

So General Relativity and Newtonian simulations give the same result with only one tiny difference in the case of binary pulsars (due to emission of GW).

Question ; for simulating a big object like a cluster or a galaxy, spanning thousands of LYrs, is is accurate enough to do a Newtonian simulation ? i think, yes ?

 

[pervect]

Thank you very much, pervect.
If i understand, the force on A points not towards B's retarded position, but towards B's "linearly extrapolated" retarded position, but as long particles follow geodesics (no acceleration), this is the same as if gravity were propagating at infinite speed.

So General Relativity and Newtonian simulations give the same result with only one tiny difference in the case of binary pulsars (due to emission of GW).

Question ; for simulating a big object like a cluster or a galaxy, spanning thousands of LYrs, is is accurate enough to do a Newtonian simulation ? i think, yes ?

Basically, as long as the system you are simulating isn't at all close to becoming a black hole, you should be OK. If it is close to being a black hole, then you've got an extremely hard problem.

There is an approximation for the amount of gravitational wave energy that a rotating gravitationally bound system of mass M and radius R emits. This is from MTW's gravitaiton, pg 980.

In geometric units it's just (M/R)^5

In standard units that's (GM/Rc^2)^5 * (c^5/G)

What this means is, the larger the radius, the less gravity waves are emitted. Other GR corrections also become unimporatant for large R, so a large radius is good.