Two-body problem program gone wrong

Hi everyone... I have written a simple program on matlab that will draw an orbit using ode45 to integrate the two body problem which is rddot = -mue.*rv/R^3

where:

rddot: second derivative of position vector (acceleration)

mue: Earth constant (which is known to be = 398600 km^3/s^2)

rv: position vector

R: position scalar

and my problem is the orbit will complete one circle without a problem but it will decay as it complete more than one period around the earth which I'm sure it's not the realistic case..

for example, completing one period it will look normal:

if I wanted to complete more than one period, the orbit will decay:

this picture is taken after completing 20 rotations.

someone please help if found any problem in my code or my mathematical expression,

this is my code:

r0=[-7072.7292 3150.61 123.8]'; %km
v0=[0.4600 -4.2446 -7.312]'; %km/s
y0=[r0;v0];
T=14332.25;
tspan=[0 T];
odefun=@dfdf;
[t,y]= ode45(odefun,tspan,y0);
xv = y(:,1);
yv = y(:,2);
zv = y(:,3);

plot3(xv,yv,zv)
axis equal

function [dydt]=dfdf(t,y)
rv=y(1:3);
R=norm(rv);
mue=398600.44189;
rdot=y(4:6);
rddot = -mue*rv/R^3;
dydt = [rdot;rddot];
end

[Just Manuel]

More a philosophical answer, thus just as a comment:

In my experience, it is very hard to find parameters that produce a stable orbit. The fact that we are orbiting the sun is thus rather a miracle :) In fact, even the distance to the sun is growing (https://www.forbes.com/sites/startswithabang/2019/01/03/earth-is-drifting-away-from-the-sun-and-so-are-all-the-planets).

So I suspect that there is nothing wrong with your simulation. You just have not found the perfect parameter set. Try tweakind them a bit.

Cheers

Manuel

Answer 1

[Jan]

Decrease the tolerance to reduce the truncation errors:

opt = odeset('AbsTol', 1e-8, 'RelTol', 1e-8);
[t,y]= ode45(odefun,tspan,y0, opt);

[Matthew Lee]

Thank you very much!!

that worked for me!

Answer 2

[darova]

rddot = -mue*rv/R^3;

This line tells that you have non-constant angular acceleration

So it's ok if you orbit is changing

Answer 3

[edoardo de angeli]

How did you manage to iterate the process for more than one orbit?