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THE LAGRANGE COEFFICIENTS

The task we are trying to solve is with a given initial position and velocity of an orbiting body at a given instant calculate the position and velocity at any later time. Assume an earth satellite moves in the xy plane of an inertial frame with origin at the earth’s center. Initial position and velocity of the satellite at that frame are

clc;
clear all;
R0 = [8000.2 -6900.5];      %[km]
V0 = [0.450 8.9000];        %[km/s]
% Compute the position and velocity vectors when the change in satellite
% true anomaly equals to 100 [deg].
dfi = 100;                  %[deg]
% Magnitude of R0 and V0 vectors
r0 = norm(R0);              %[km]
v0 = norm(V0);              %[km/s]
% Radial component of V0
vr0 = R0*V0'/r0;             %[km/s]
% Magnitude of the constant angular momentum
h = r0*(v0^2-vr0^2)^0.5;    %[km^2/s]
mu      = 398600;           % Earth’s gravitational parameter [km^3/s^2]
r = h^2/mu*1/(1+(h^2/mu/r0 -1)*cosd(dfi)- h*vr0/mu*sind(dfi));
% Lagrange coefficients in terms of the change in true anomaly
f = 1 -  mu*r/h^2*(1-cosd(dfi));
g = r*r0/h*sind(dfi);
fd = mu/h*(1 - cosd(dfi))/sind(dfi)*(mu/h^2*(1 - cosd(dfi)) -1/r0 - 1/r);
gd = 1 - mu*r0/h^2*(1 - cosd(dfi));
% Resultant position and velocity vectors
R = f*R0 + g*V0;
V = fd*R0 + gd*V0;
fprintf('R = %4.2f*i + %4.2f*j [km] \n',R(1),R(2));
fprintf('V = %4.4f*i + %4.4f*j [km/s] \n',V(1),V(2));
R = 3634.07*i + 6101.27*j [km] 
V = -7.6623*i + 7.5831*j [km/s]

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