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I have a question that I believe is relatively easy to answer, I am working on an $N$-body simulation of a fictional star system and am having trouble finding the velocity of moons so that they will remain in near-circular orbits (were they not to be influenced by other moons). So far, I can calculate the velocities of planets relative to their parent star in order to have a near circular orbit at a specific semi-major axis. When I say velocity, I'm looking for a 3-dimensional vector and not just a 1D value. I find the velocity of the planets by calculating the orbital speed like$$ v_0 ~\approx~ \sqrt{\frac{GM}{r}} \,,$$where $r$ is the length of the planet's semi-major axis and $M$ is the mass of the star. This value is then multiplied by a 3-dimensional vector that points tangential to the planet's desired orbit which gives me the velocity. This works fine but I'm having trouble doing this for moons of these planets. I made an attempt by repeating the process that I used for the planets but instead $r$ becomes the moon's semi-major axis from its parent planet and $M$ becomes the mass of the planet. I then find the orbital velocity and multiply it by a vector tangential to the desired orbit relative to the planet and add the planets orbital velocity (as a 3D vector) which doesn't work as many moons get left behind, others fall into the planet, and few remain in stable orbits (without a lot of perturbations from other objects).

While the moon's velocity relative to the planet (so that it would have a circular orbit) is being properly calculated, the moon's velocity relative to the star isn't as apparently simply adding the planet's velocity doesn't work - and this is what I'm looking for. Can anyone help me with finding the velocity (relative to the sun) of a moon so that it will remain in a stable orbit around its parent planet?

EDIT: Code that calculates velocity of moons

double localOrbitalSpeed = sqrt((g * parentObjectMass)/(localSMA));
velocity = normalize(cross(position, Vector3(0, -1, 0))) * localOrbitalSpeed + parentObjectVelocity; 

and the code for planets

double orbitalSpeed = sqrt((g * starMass)/(starSMA));
velocity = normalize(cross(position, Vector3(0, -1, 0))) * orbitalSpeed;

and that's pretty much all that really determines initial velocities, not much room for... while writing this I figured out what I was doing wrong, when calculating the velocity vector, the tangent vector is being determined relative to the star instead of the planet :/ so yeah, simple programming error. I just fixed this and will see if I have any more issues.

EDIT 2: Well now that the velocity is being properly calculated, I find that the moons tend to spiral into the planet or spiral away in only a few orbits (still without influencing each other), I know that moons do, do this but due to tidal forces (which I am not simulating), right? I assume that the integration method I used is causing this problem.

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    $\begingroup$ That...sounds like the correct way to go about it. My money would be on a programming error. Perhaps you should post the velocity calculation code? $\endgroup$ – Brionius May 3 '15 at 3:23
  • $\begingroup$ Note that in a multi body problem like this, the effect of the moons on each other can be quite significant - accelerating some, decelerating others. It is not a stable situation. I agree with Brionius that you may have a simple programming error - but don't rule out that you are seeing real physics. How close do the moons get to each other - and what is their mass relative to the planet? $\endgroup$ – Floris May 3 '15 at 3:39
  • $\begingroup$ I'll try stopping moons from influencing each other and see what happens. I'll post the code if the orbits still don't look right. $\endgroup$ – user3684950 May 3 '15 at 4:24
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    $\begingroup$ To check your code, calculate the total energy and angular momentum of the system. Unless you used the right numerical integrator for this problem, your integration method will not conserve energy and angular momentum and that alone will cause significant instability in n-body problems. $\endgroup$ – CuriousOne May 3 '15 at 4:31
  • $\begingroup$ I am using a symplectic second order integrator and the total energy does rise over time, unfortunately, but I don't think it's causing my moon error. After stopping moons from influencing each other I found that they end up on very eccentric orbits and will either get ejected or collide with their parent planet so my velocity calculation isn't right. I will post the code in the OP shortly. $\endgroup$ – user3684950 May 3 '15 at 5:16
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If you plot the path of the moon with respect to the star you should get a cycloid pattern averaging around the path of the planet. That's what our moon does. The planet and the moon are doing a gyrating dance about their center of mass, with the moon doing the most motion (because it has the smallest mass), while the planet/moon duo orbit the star.

I've found that the Verlet-velocity method works very well for this problem. See Gould and Tobochnik Computer Simulation Methods, Volume 1.

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    $\begingroup$ This is more of a comment than an answer. $\endgroup$ – garyp Sep 30 '17 at 12:48

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