Undergrad Modeling Simple N-body in Hypothetical Relativistic Dark Flow w/ Netlogo

[nearc]

Looking for existing examples of or guidance on building computer simulations of simple N-bodies in relativist situations. Of particular interest is the simulation of a planet orbiting a star whilst that trivial solar system is caught in a dark flow of relativist velocity; maybe .8 or .9 C.

Not that it should matter but I plan to start out with Netlogo since that is what most of my students use and if I need more computational power I will switch over to C++ with openMP.

 

[Paul Colby]

Sounds like the solar system part without the dark flow would be solvable as a two body near-Newtonian problem. Do you have an analytic metric as modified by such a dark mater flow? (no expert here, just someone who's numerically integrated a few ODEs in the day)

 

[pervect]

The relativisitc dark matter flow may break some of the usual approximations, which usually state non-relativistic velocities. While I'd guess that it _might_ not matter as long as the bodies are not relativistic, I can't even say that I'm sure it won't.

 

[nearc]

i assume there are already several basic relativity models, at least one for mercury? i would prefer an N-body model one that would allow me to add an additional uniform velocity in one direction, but any model would be a starting point.

 

[pervect]

Folkner, et a's paper, "The Planetary and Lunar Ephemerides DE430 and DE431" might be a starting place for an N-body simulator. But it doesn't have the relativistic dark mater backround features you want.

Some quotes from the paper that address some of your question:

The translational equations of motion include contributions from: (a) the point mass inter-
actions among the Sun, Moon, planets, and asteroids; (b) the effects of the figure of the Sun
on the Moon and planets; (c) the effects of the figures of the Earth and Moon on each other
and on the Sun and planets from Mercury through Jupiter; (d) the effects upon the Moon’s
motion caused by tides raised upon the Earth by the Moon and Sun; and (e) the effects on
the Moon’s orbit of tides raised on the Moon by the Earth.

The gravitational acceleration of each body due to external point masses is derived from the
isotropic, parametrized post-Newtonian (PPN) n-body metric [24–26]. For each body
A, the acceleration due to interaction with other point masses is given by ...

Tracking down references 24-26 above might be the next step in your search.

The PPN approximation is discussed in a lot of textbooks (including MTW's text "Gravitation"), the questions I can't answer are how the dark matter background impacts the use of this approximation. The IAU 2000 recommendations and the various revisions thereof for the metric of the solar system (see for instance resolution B1.3 at https://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm AND the various updates since that date which I haven't linked to) are also potentially of some interest, basically the IAU extended the PPN metric mentioned by Folkner et al to make it easier to convert from barycentric to geocentric coordinates and back.