# Orbit smulation and speed of gravitation

• serge
In summary, when simulating a solar or galactic system, the position of a planet at any given time is calculated based on the position of the other bodies in the system at the same time. If the speed of gravity is infinite, this is enough for accurate simulation. However, if gravity has a finite speed, the acceleration of the planet must also take into account the position of the other bodies at a time in the past. This can be solved using the Lienard-Wiechert potentials, which take into account the retarded position of the bodies. However, in a Newtonian simulation, the force always points towards the instantaneous position of the object, not the retarded position. It is accurate enough to do a Newtonian simulation for
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)

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).

Last edited:
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 ?

serge said:
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.

## 1. How is orbit simulation used in scientific research?

Orbit simulation is used to model and predict the behavior of orbiting objects in space, such as planets, satellites, and asteroids. It allows scientists to study the effects of gravitational forces, atmospheric drag, and other factors on the orbits of these objects.

## 2. What is the role of gravity in orbit simulation?

Gravity is the force that governs the motion of objects in orbit. It is responsible for keeping objects in orbit around a larger body, such as a planet or star. In orbit simulation, gravity is used to calculate the trajectories and velocities of orbiting objects.

## 3. How does the speed of gravitation affect orbit simulation?

The speed of gravitation, also known as the speed of gravity, is the speed at which gravitational forces propagate. In orbit simulation, it is used to calculate the time it takes for changes in gravity to affect the orbit of an object. This is important for accurately predicting the behavior of orbiting objects over time.

## 4. What factors can affect the speed of gravitation in orbit simulation?

The speed of gravitation can be affected by the distance between two objects, their masses, and any intervening objects that may influence the gravitational field. In general, the closer two objects are and the more massive they are, the stronger the gravitational force and the faster the speed of gravitation.

## 5. How do scientists validate the accuracy of orbit simulation models?

Scientists use a variety of methods to validate the accuracy of orbit simulation models, including comparing the predicted trajectories and velocities of objects with actual observations, conducting experiments in a controlled environment, and incorporating new data and observations into the simulation. By continually refining and improving these models, scientists can gain a better understanding of the complex dynamics of objects in orbit.

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