# Error in N-Body simulation

by Einstein2nd
Tags: error, nbody, simulation
 P: 25 Hello all. I'm currently writing a program in Matlab that works out the positions of N bodies under the influence of gravity. The code is setup such that it only requires a mass and an initial position and velocity for each body. Any number of bodies can be entered but for testing purposes I am just using two. The code uses either Euler or RK4 (fourth-order Runge-Kutta integration). The problem I am getting occurs with either type of integration. Firstly, here is the code: clearvars % UNIVERSAL CONSTANTS G = 6.673*10^-11; % gravitational constant Msun = 1.98892*10^30; Mjup = 1.8987*10^27; Mearth = 5.9742*10^24; Mlunar = 7.349*10^22; AU = 1.49598*10^11; Vearth = 2.9786*10^4; Vlunar = 1.076*10^3; Tearth = 3.15576*10^7; Tmoon = 2.36058*10^6; % USER INPUTS IntegrationMethod = 2; % 1 = Euler, 2 = RK4 N = 2; % The number of bodies m = [1*Msun 1*Mearth]; % Mass vector maxtime = 5*Tearth; % Maximum time to integrate over npts = 500; % Number of integration points % SETUP INTEGRATION H = maxtime/npts; % APPLY INITIAL CONDITIONS r_old(:,:,1) = [0 0 0]; % Sun starts at the origin r_old(:,:,2) = [0 AU 0]; % Earth starts 1 AU away drdt_old(:,:,1) = [0 0 0]; % Sun is initially motionless drdt_old(:,:,2) = [1*Vearth 0 0]; % Earth is moving perpendicular to the line joining it to the Sun % SETUP FIGURE close all figure(1) set(get(gca, 'xlabel'),'FontSize',20) set(get(gca, 'ylabel'),'FontSize',20) set(get(gca, 'title'),'FontSize',20) colourarray = ['r','b','g','g']; % DO INTEGRATION xlabel 'x' ylabel 'y' title 'N-Body Simulation' axis equal hold on for i = 1:npts if IntegrationMethod == 1 drdt_new = drdt_old + H*W(N,G,m,r_old); r_new = r_old + H*drdt_new; elseif IntegrationMethod == 2 k1 = zeros(1,3,N); k2 = zeros(1,3,N); k3 = zeros(1,3,N); k4 = zeros(1,3,N); l1 = zeros(1,3,N); l2 = zeros(1,3,N); l3 = zeros(1,3,N); l4 = zeros(1,3,N); k1 = drdt_old; l1 = W(N,G,m,r_old); k2 = drdt_old + H*l1/2; l2 = W(N,G,m,r_old + H*k1/2); k3 = drdt_old + H*l2/2; l3 = W(N,G,m,r_old + H*k2/2); k4 = drdt_old + H*l3; l4 = W(N,G,m,r_old + H*k3/2); drdt_new = drdt_old + (H/6)*(l1 + 2*l2 + 2*l3 + l4); r_new = r_old + (H/6)*(k1 + 2*k2 + 2*k3 + k4); end for j = 1:N scatter(r_new(1,1,j)/AU,r_new(1,2,j)/AU,colourarray(j),'.') end pause(0.0000001) r_old = r_new; drdt_old = drdt_new; end Where W is the function for the acceleration:  function [accel] = W(N,G,m,r) accel = zeros(1,3,N); for j = 1:N for i = 1:N if i ~= j accel(:,:,j) = accel(:,:,j) + G*m(i)*(r(:,:,i)-r(:,:,j))/norm(r(:,:,i)-r(:,:,j))^3; end end end end Now here are the results. I have used Euler's method for these results but the RK4 results are pretty similar. RK4 is much more accurate than Euler integration but the error here is (I am pretty sure) not due to that. I have the Earth and Sun as the two bodies. The above image looks pretty good yeah? The Earth orbits the Sun in pretty much a circular orbit. Now, let's zoom in on the Sun. Remember that the Sun also orbits the centre of mass of the Sun/Earth system but it will remain very close to the centre of mass because the Sun/Earth mass ratio is so large. I don't know about you though, but this doesn't look right: I cannot explain this. Can someone please help? Just take the Euler integration...the code is so simple that I can't find the error. I have tried making the Sun body 1 and the Earth body 2, just in case I was referring to an index incorrectly or something but that did nothing.
 P: 1,396 edguy99 is right, If you start the earth moving at speed v, and the sun motionless, there is an initial momentum of $m_{earth} v$, wich means you have a net drift of the center of mass by $$v \frac {m_{earth} } {m_{earth} + m_{sun}}$$ The orbit of the earth will drift as well, but the yearly drift is only $$2 \pi r \frac { m_{earth} } {m_{earth} + m_{sun}}$$ about (1/50000) AU The graph produced for the sun is a cycloid, a superposition of a rotation round the center of mass and a linear motion with (nearly) the same speed. You get the same graph if you graph the position of a point on the circumference of a rolling wheel.