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Universal Kepler’s equation


In this example we solve the universal Kepler’s equation to determine universal anomaly after dt time with a given initial radius r0, velocity v0 and semimajor axis of a spacecraft.

clear all;
mu       = 398600;        % Earth’s gravitational parameter [km^3/s^2]
% Initial conditions
r0       = 12000;  %[km] Initial radius
vr0      = 3;      %[km/s] Initial radial speed
dt       = 7200;   %[s] Time interval
a        = -20000; %[km] Semimajor axis
alfa = 1/a;        %[km^-1] Reciprocal of the semimajor axis;

X0 = mu^0.5*abs(alfa)*dt;  %[km^0.5]Initial estimate of X0
Xi = X0;
tol = 1E-10;              % Tolerance
    zi = alfa*Xi^2;
    [ Cz,Sz] = Stumpff( zi );
    fX  = r0*vr0/(mu)^0.5*Xi^2*Cz + (1 - alfa*r0)*Xi^3*Sz + r0*Xi -(mu)^0.5*dt;
    fdX = r0*vr0/(mu)^0.5*Xi*(1 - alfa*Xi^2*Sz) + (1 - alfa*r0)*Xi^2*Cz + r0;
    eps = fX/fdX;
    Xi = Xi - eps;
    if(abs(eps) < tol )
fprintf('Universal anomaly X = %4.3f [km^0.5] \n',Xi)

% Equations from the book
% Orbital Mechanics for Engineering Students,2nd Edition,Aerospace Engineering
Universal anomaly X = 173.324 [km^0.5]

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