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orbit.m
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orbit.m
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function orbit(r_0, v_0, m_2, t_0, t_f, a)
%% plots the orbit of a small mass around a large central mass
%
% Jeremy Penn
% 22/11/17
%
% function orbit(r_0, v_0, m_2, t_0, t_f)
%
% Inputs: o r_0 - The initial position vector [OPTIONAL]
% o v_0 - The initial velocity vector [OPTIONAL]
% o m_2 - The mass of the orbiting body [OPTIONAL]
% o t_0 - The initial time [OPTIONAL]
% o t_f - The final time [OPTIONAL]
%
% Requires: rkf45.m,
%
S = load('topo');
%% constants
G = 6.6742e-20; %[km^3/kg s^2] gravitational constant
%% inputs
m1 = 5.974e24; % mass of the Earth [kg]
R = 6378; % radisu of the Earth[km]
if nargin < 1
m2 = input('Input the mass of the orbiting body (kg):\n');
r0 = input('Input the initial geocentric position vector of the orbiting body [x, y, z](km):\n');
v0 = input('Input the initial geocentric velocity vector of the orbiting body [vx, vy, vz] (km/s):\n');
t0 = input('Input the initial time (s):\n');
tf = input('Input the final time (s): \n');
else
r0 = r_0;
v0 = v_0;
m2 = m_2;
t0 = t_0;
tf = t_f;
end
%% confirm r and v are row vectors
if iscolumn(r_0)
r0 = r0';
end
if iscolumn(v_0)
v0 = v0';
end
%% numerically integrate the orbit
mu = G*(m1 + m2);
y0 = [r0 v0]';
[t,y] = rkf45(@rates, [t0 tf], y0);
%% plot and print the results
output
return
%% -------subfunctions------------------------
function dydt = rates(t,f) %#ok<*INUSL>
% ~~~~~~~~~~~~~~~~~~~~~~~~
%{
This function calculates the acceleration vector using Equation 2.22
t - time
f - column vector containing the position vector and the
velocity vector at time t
x, y, z - components of the position vector r
r - the magnitude of the the position vector
vx, vy, vz - components of the velocity vector v
ax, ay, az - components of the acceleration vector a
dydt - column vector containing the velocity and acceleration
components
%}
% ------------------------
x = f(1);
y = f(2);
z = f(3);
vx = f(4);
vy = f(5);
vz = f(6);
r = norm([x y z]);
ax = -mu*x/r^3;
ay = -mu*y/r^3;
az = -mu*z/r^3;
dydt = [vx vy vz ax ay az]';
end %rates
function output
%{
This function computes the maximum and minimum radii, the times they
occur and the speed at those times. It prints those results to
the command window and plots the orbit.
r - magnitude of the position vector at the times in t
imax - the component of r with the largest value
rmax - the largest value of r
imin - the component of r with the smallest value
rmin - the smallest value of r
v_at_rmax - the speed where r = rmax
v_at_rmin - the speed where r = rmin
%}
% -------------
[h, e, inc, W, w, ta] = coe_from_rv(r0,v0);
for i = 1:length(t)
r(i) = norm([y(i,1) y(i,2) y(i,3)]); %#ok<*AGROW>
end
[rmax, imax] = max(r);
[rmin, imin] = min(r);
v_at_rmax = norm([y(imax,4) y(imax,5) y(imax,6)]);
v_at_rmin = norm([y(imin,4) y(imin,5) y(imin,6)]);
%...Output to the command window:
clc;
fprintf('\n\n--------------------------------------------------------\n')
fprintf('\n Earth Orbit\n')
fprintf(' %s\n', datestr(now))
fprintf('\n The initial position is [%g, %g, %g] (km).',...
r0(1), r0(2), r0(3))
fprintf('\n\t Magnitude = %g km\n', norm(r0))
fprintf('\n The initial velocity is [%g, %g, %g] (km/s).',...
v0(1), v0(2), v0(3))
fprintf('\n\t Magnitude = %g km/s\n', norm(v0))
fprintf('\n Initial time = %g h.\n Final time = %g h.\n',0,tf/3600)
fprintf('\n The minimum altitude is %g km at time = %g h.',...
rmin-R, t(imin)/3600)
fprintf('\n The speed at that point is %g km/s.\n', v_at_rmin)
fprintf('\n The maximum altitude is %g km at time = %g h.',...
rmax-R, t(imax)/3600)
fprintf('\n The speed at that point is %g km/s\n', v_at_rmax)
fprintf('\n The classic orbital elements:\n')
fprintf('\n\t h = %.2f [km^2/s]',h)
fprintf('\n\t a = %.2f [km]',a)
fprintf('\n\t e = %.5f',e)
fprintf('\n\t i = %.2f [deg]',inc)
fprintf('\n\t W = %.2f [deg]',W)
fprintf('\n\t w = %.2f [deg]',w)
fprintf('\n\t ta = %.2f [deg]',ta)
fprintf('\n\t ep = %.2f [MJ/kg]',-mu / (2*a))
fprintf('\n--------------------------------------------------------\n\n')
%...Plot the results:
% Draw the planet
grs80 = referenceEllipsoid('grs80','km');
figure('Renderer','opengl')
ax = axesm('globe','Geoid',grs80,'Grid','off', ...
'GLineWidth',1,'GLineStyle','-',...
'Gcolor',[0.9 0.9 0.1],'Galtitude',100);
ax.Position = [0 0 1 1];
axis equal off
%view(0,23.5)
geoshow(S.topo,S.topolegend,'DisplayType','texturemap')
demcmap(S.topo)
land = shaperead('landareas','UseGeoCoords',true);
plotm([land.Lat],[land.Lon],'Color','black')
rivers = shaperead('worldrivers','UseGeoCoords',true);
plotm([rivers.Lat],[rivers.Lon],'Color','blue')
% Draw and label the X, Y and Z axes
line([0 2*R], [0 0], [0 0]); text(2*R, 0, 0, 'X')
line( [0 0], [0 2*R], [0 0]); text( 0, 2*R, 0, 'Y')
line( [0 0], [0 0], [0 2*R]); text( 0, 0, 2*R, 'Z')
% Plot the orbit, draw a radial to the starting point
% and label the starting point (o) and the final point (f)
hold on
plot3( y(:,1), y(:,2), y(:,3),'r')
line([0 r0(1)], [0 r0(2)], [0 r0(3)])
text( y(1,1), y(1,2), y(1,3), 'o')
text( y(end,1), y(end,2), y(end,3), 'f')
% Select a view direction (a vector directed outward from the origin)
view([1,1,.4])
% Specify some properties of the graph
grid on
axis equal
xlabel('km')
ylabel('km')
zlabel('km')
end %output
end