# N-Body Problem - Initial Values

Hello everyone,
I'm new here so let me introduce myself first:
I'm a games engineering student and my main field is programming

Right now I'm struggeling with the N-Body Problem, well I think actually solved parts of it:
I've implemented the basic equations with a leapfrog algorithm and they seem to be working correctly

Now I have 2 questions:

1.)
When I place a star at a certain position, I want to calculate the ideal velocity in order to produce an orbit. How is this achieved?

2.)
How can I calculate the Angular Velocity for an NBody-Problem?

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My current setup (as reference):

sun (position=0,0,0; mass=333, velocity=0,0,0)
earth(position=-50,0,0, mass=1, velocity=0,0,2)

This is producing a stable circular orbit
It is also possible to add more earths/planets with stable orbits
I know that the mass ratio is not quite right, but i want to achieve a "compressed universe"
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I would appreciate your help on this.

Simon Bridge
Homework Helper
Right now I'm struggeling with the N-Body Problem,
everybody does - you realize there is no analytical solution?
well I think actually solved parts of it
...depending on what you mean by "solved" of course :)
I mean to show you that physicists think of the n-body problem a bit differently to comp-scientists ... I'm guessing you are trying some sort of parallel processing?
Now I have 2 questions:

1.)
When I place a star at a certain position, I want to calculate the ideal velocity in order to produce an orbit. How is this achieved?
For a particular stable orbit, the velocity (and speed) will be different for different parts of the orbit. For a circular orbit, you just need to balance centripetal acceleration for both bodies with the force of gravity on them.

2.)
How can I calculate the Angular Velocity for an NBody-Problem?
That would be the angular velocity of each body in the problem?

$$\vec{\omega}=\frac{\vec{r}\times\vec{v}}{|\vec{r}|^2}$$

...which requires that you know the velocities of course.
The exact approach depends on how you are setting up your simulation.

gneill
Mentor
Hi physxfreak; Welcome to Physics Forums.

If you've got a particular set of conditions that produces a circular orbit you can calculate the gravitational parameter for your simulated solar system.

The circular orbit velocity for a planet of negligible mass at distance r from a Sun of mass M is given by:

$$v = \sqrt{\frac{GM}{r}}$$

If you solve for μ = GM, then you can use this parameter to find the required speed for circular orbits of other radii. Note that if you have a known radius and velocity you don't even need the mass of the star to find the parameter μ. Even in "real life", μ = GM for our solar system is known with more accuracy than G or M alone.

everybody does - you realize there is no analytical solution?...depending on what you mean by "solved" of course :)
Yes I know that the N-Body Problem is unsolved for n>=3, but I'm using a numerical solution (with a leapfrog algorithm)

I'm guessing you are trying some sort of parallel processing?
No not at the moment, as i have a quite small problem size (n<100) at the moment

That would be the angular velocity of each body in the problem?

$$\vec{\omega}=\frac{\vec{r}\times\vec{v}}{|\vec{r}|^2}$$

...which requires that you know the velocities of course.
The exact approach depends on how you are setting up your simulation.

Well i'm searching for a way to calculate the "spin" of each planet, like the earth rotating around it's own axis 365 days a year. But I assume that this spin was given by collisions rather than the gravitational force?

@Gneil:
Thanks, that is exactly what i looked for :)

Simon Bridge