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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 being the only source of force.

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

The force that particle j exerts on particle i is

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

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 and then building the phase-space vector of the entire system as . Thus the first-order ODE system is

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

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

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

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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