%% Orbiting around fixed stars
% Nick Trefethen, November 2011
%%
% (Chebfun example ode-nonlin/Orbits.m)
% [Tags: #nonlinearODE, #astrophysics, #IVP]
%%
% Suppose a "star" of unit mass is fixed at the origin in the $x$-$y$
% plane, and a planet, also of unit mass, moves around it according to
% Newton's laws with gravitational constant $1$. To be specific, let's
% suppose the planet starts at $(-1,1)$ heading east with speed $v=1$. What
% does the trajectory look like?
%%
% Here is a code to solve this problem using Chebfun's `ode113` command, and
% complex arithmetic for simplicity. We track the orbit for $23$ time units
% and see that it is an ellipse.
tmax = 23; d = [0 tmax];
LW = 'linewidth'; MS = 'markersize';
opts = odeset('abstol',1e-10,'reltol',1e-10);
fun = @(t,u) [u(2); -u(1)./abs(u(1)).^3];
u0 = -1+1i;
v = 1;
uv = chebfun.ode113(fun,d,[u0; v],opts); uv = uv(:,1);
hold off, plot(0,0,'.r','markersize',24), hold on
plot(uv,LW,1.6), axis equal, grid on, shg
plot(uv(0:tmax),'.k',MS,16)
%%
% If we want the initial speed $v$ to be a parameter, we can make an anonymous
% function:
u = @(v) chebfun.ode113(fun,d,[u0; v],opts);
%%
% Here are the orbits for $v = 0.5,\ 0.75,\ 1,\ 1.5,\ 2$. This kind of thing
% is familiar from introductory physics: every orbit is an ellipse, a
% parabola, or a hyperbola.
hold off, plot(0,0,'.r','markersize',24), hold on
for v = [.5 .75 1 1.5 2]
uv = u(v); uv = uv(:,1); plot(uv,LW,1.6)
plot(uv(0:tmax),'.k',MS,16)
end
axis([-3 3 -3 3]), axis square, grid on, shg
%%
% More unusual behavior comes about if we imagine two or more fixed "stars".
% Orbits can now be bounded without being periodic. For example, suppose we
% have one star at $(0,0)$ and another at $(1,0)$, with the planet feeling a
% gravitational tug from each. Here is an orbit over $10$ time units starting
% with $v=1$:
fun = @(t,u) [u(2); -u(1)./abs(u(1)).^3-(u(1)-1)./abs(u(1)-1).^3];
tmax = 10; d = [0 tmax];
u = @(v) chebfun.ode113(fun,d,[u0; v],opts);
v = 1; uv = u(v); uv = uv(:,1);
hold off, plot([0 1],[0 0],'.r','markersize',24), hold on
plot(uv,LW,1.6), axis equal, grid on, shg
plot(uv(0:tmax),'.k',MS,16)
%%
% Here is what happens when the initial speed is reduced to $0.9$:
v = 0.9; uv = u(v); uv = uv(:,1);
hold off, plot([0 1],[0 0],'.r','markersize',24), hold on
plot(uv,LW,1.6), axis equal, grid on, shg
plot(uv(0:tmax),'.k',MS,16)
%%
% How long is the trajectory?
orbit_length = norm(diff(uv),1)
%%
% How close does it come to the origin?
closeness = min(abs(uv))
%%
% Variations on these themes are infinite!