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Basic N-body Equations

The general astrophysical N-body system consists of N particles moving according to Newton's three laws of motion, with Newton's familiar gravitational law tex2html_wrap_inline4287 being the only source of force.

Let tex2html_wrap_inline4289 be the unit vectors of the standard Cartesian (x,y,z) system. Let tex2html_wrap_inline4293 be the position vector of particle i. Let tex2html_wrap_inline4297 be the vector pointing from particle i to particle j, i.e.,  the position of particle j with respect to particle i. Thus

displaymath4307

The force that particle j exerts on particle i is

eqnarray1517

where G is the gravitational constant, tex2html_wrap_inline4315 are the masses of the particles, tex2html_wrap_inline4317 is the vector pointing from particle i to particle j, tex2html_wrap_inline4323 is the magnitude of this vector, and tex2html_wrap_inline4325 is the unit vector pointing in the direction of tex2html_wrap_inline4317 . Let the total force on particle i be tex2html_wrap_inline4331 . It is the sum of all the forces from all the other particles. Thus the total force tex2html_wrap_inline4331 on particle i is

eqnarray1531

For a 3-dimensional space, this is a set of 3N second-order ordinary differential equations (ODEs), which we translate into a set of 6N first-order ODEs by letting the velocity tex2html_wrap_inline4341 and then building the phase-space vector of the entire system as tex2html_wrap_inline4343 . Thus the first-order ODE system is

displaymath4345

where tex2html_wrap_inline4347 is the vector representing the accelerations of all the particles.

At any given time, the position tex2html_wrap_inline4349 and velocity tex2html_wrap_inline4351 of every particle is known. Thus tex2html_wrap_inline4353 is simply tex2html_wrap_inline4351 , and tex2html_wrap_inline4357 is the set of time-derivatives of the velocity tex2html_wrap_inline4359 of each particle, where

  equation1557

If force softening is used, the denominator instead becomes tex2html_wrap_inline4361 where tex2html_wrap_inline4363 is the softening parameter.


next up previous
Next: Jacobian of an N-M Up: Review of the astrophysical Previous: Review of the astrophysical

Wayne Hayes
Sun Dec 29 23:43:59 EST 1996

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