% orbit - Program to compute the orbit of a comet. clear all; help orbit; % Clear memory and print header %* Set initial position and velocity of the comet. r0 = input('Enter initial radial distance (AU): '); v0 = input('Enter initial tangential velocity (AU/yr): '); r = [r0 0]; v = [0 v0]; state = [ r(1) r(2) v(1) v(2) ]; % Used by R-K routines %* Set physical parameters (mass, G*M) GM = 4*pi^2; % Grav. const. * Mass of Sun (au^3/yr^2) mass = 1.; % Mass of comet adaptErr = 1.e-3; % Error parameter used by adaptive Runge-Kutta time = 0; %* Loop over desired number of steps using specified % numerical method. nStep = input('Enter number of steps: '); tau = input('Enter time step (yr): '); NumericalMethod = menu('Choose a numerical method:', ... 'Euler','Euler-Cromer','Runge-Kutta','Adaptive R-K'); for iStep=1:nStep %* Record position and energy for plotting. rplot(iStep) = norm(r); % Record position for polar plot thplot(iStep) = atan2(r(2),r(1)); tplot(iStep) = time; kinetic(iStep) = .5*mass*norm(v)^2; % Record energies potential(iStep) = - GM*mass/norm(r); %* Calculate new position and velocity using desired method. if( NumericalMethod == 1 ) accel = -GM*r/norm(r)^3; r = r + tau*v; % Euler step v = v + tau*accel; time = time + tau; elseif( NumericalMethod == 2 ) accel = -GM*r/norm(r)^3; v = v + tau*accel; r = r + tau*v; % Euler-Cromer step time = time + tau; elseif( NumericalMethod == 3 ) state = rk4(state,time,tau,'gravrk',GM); r = [state(1) state(2)]; % 4th order Runge-Kutta v = [state(3) state(4)]; time = time + tau; else [state time tau] = rka(state,time,tau,adaptErr,'gravrk',GM); r = [state(1) state(2)]; % Adaptive Runge-Kutta v = [state(3) state(4)]; end end %* Graph the trajectory of the comet. figure(1); clf; % Clear figure 1 window and bring forward polar(thplot,rplot,'+'); % Use polar plot for graphing orbit xlabel('Distance (AU)'); grid; pause(1) % Pause for 1 second before drawing next plot %* Graph the energy of the comet versus time. figure(2); clf; % Clear figure 2 window and bring forward totalE = kinetic + potential; % Total energy plot(tplot,kinetic,'-.',tplot,potential,'--',tplot,totalE,'-') legend('Kinetic','Potential','Total'); xlabel('Time (yr)'); ylabel('Energy (M AU^2/yr^2)');