- #1
Einstein2nd
- 25
- 0
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:
Where W is the function for the acceleration:
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.
http://img15.imageshack.us/img15/5015/2bodyorbitzoomedout.jpg
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:
http://img171.imageshack.us/img171/3607/2bodyorbitzoomedin.jpg
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.
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:
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 onfor 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:
Code:
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.
http://img15.imageshack.us/img15/5015/2bodyorbitzoomedout.jpg
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:
http://img171.imageshack.us/img171/3607/2bodyorbitzoomedin.jpg
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.
Last edited by a moderator: