Vectorized N-Body Equation

Fully vectorized n-body equations of motion in second and first order form.


Updated 2 Jun 2010

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NBODYVECT - Fully vectorized n-body equation
nbodyVect(x0,dx0,mus,t) integrates the n-body equations of motion using either a symplectic second order integrator, or any of the standard built-in or user supplied first order integrators.

x0 - 3n x 1 stacked initial position vectors:
dx0 - 3n x 1 stacked initial velocity vectors
mus - gravitational parameters (G*m_i) where G is the gravitational constant and m_i is the mass of the ith body.
t - time span: either a 2 element array of initial and final time, or array of all time steps to output.

All units must be complementary, i.e., if positions are in AUs, and time is in days, dx0 must be in AU/day and mus must be in AU^3/day^2 (these are the units in solarSystemData.mat). This data was downloaded from JPL's System Web Interface (

To switch to first order integrator, comment line 61, and uncomment lines 64-65. Line 61 should be replaced with your symplectic integrator of choice, such as rkn86 (this needs modifications to accept arbitrary time arrays).

%integrate solar system for one year
ssdat = load('solarSystemData.mat');
[t,x,dx] = nbodyVect(ssdat.p0,ssdat.v0,ssdat.mus,0:1:365);
%plot first 3 orbits

Cite As

Dmitry Savransky (2023). Vectorized N-Body Equation (, MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2009b
Compatible with any release
Platform Compatibility
Windows macOS Linux

Inspired by: rkn86

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