Three Body Problem: Dark Matter's Impact

[AntonVrba]

http://count.ucsc.edu/~rmont/papers/list.html

Above a link to papers using mathematics to define the orbits and below a link to Java animation of n-body orbits

http://www.soe.ucsc.edu/~charlie/3body

Khavel how does your work and above correlate?

Some very fancy orbits do exist - completely unexpected, like a celestial ballet

 

[khavel]

Very interesting web sites. Nice simulations!

The unrestricted simulations done by my program are interactive. You can edit the values of the initial conditions of the bodies and observe their amended trajectories.

 

[PeteSF]

Hi khavel,
Your simulations work incrementally, right?
As in each step is calculated from the previous step?
And the smaller the time increment, the more accurate the simulation, right??

 

[PeteSF]

Quote from khavel's site (http://www.grevytpress.com/enbody.pdf ):

In an n-body gravitational system, the trajectory of anybody generally depends on the masses, positions, and velocities of all remaining bodies.
To determine the trajectories of all bodies, we set a time interval deltaT for the recalculation. Obviously, the shorter the time interval results in more precise calculation of the trajectories. At the beginning of the time interval we know the masses, positions, and velocities of all bodies. During time interval deltaT we calculate for each body the sum of accelerations imparted by gravitation of all other bodies, from their masses and positions. Then, for each body, we respectively integrate the sums of the accelerations over time interval deltaT, to obtain the increment of its velocity. We add the increment of its velocity to its previous velocity, to obtain its new velocity. Then we integrate its new velocity over time interval deltaT, to obtain the increment of its position. We add the increment of its position to its previous position, to obtain its new position. Since acceleration, velocity, and position are vectors, all calculations are done with their x and y components.
We keep repeating all calculations, over successive time intervals deltaT, using the new positions and velocities of all bodies. This method of calculation is valid for any number of bodies. Needless to say, the complexity of calculation increases with the number of bodies.

'nuff said. Khavel has produced an incremental numerical simulation, not an analytic one. Nothing new here.