In this example we will show how to compute state vectors R and V in the geo-centric equatorial frame of reference using orbital elements
clear all; clc; % Six orbital elements are: h = 82000; % [km^2/s] Specific angular momentum i = 50; % [deg] Inclination RAAN = 60; % [deg] Right ascension (RA) of the ascending node e = 0.2; % Eccentricity omega= 90; % [deg] Argument of perigee theta= 35; % [deg] True anomaly mu = 398600; % Earth’s gravitational parameter [km^3/s^2] % Components of the state vector of a body relative to its perifocal % reference rx = h^2/mu*(1/(1 + e*cosd(theta)))*[cosd(theta);sind(theta);0]; vx = mu/h*[-sind(theta); (e +cosd(theta));0]; % Direction cosine matrix QXx = [cosd(omega), sind(omega),0;-sind(omega),cosd(omega),0;0,0,1]*... [1,0,0;0,cosd(i),sind(i);0,-sind(i),cosd(i)]*... [cosd(RAAN), sind(RAAN),0;-sind(RAAN),cosd(RAAN),0;0,0,1]; % Transformation Matrix QxX = inv(QXx); % Geocentric equatorial position vector R R = QxX*rx; % Geocentric equatorial velocity vector V V = QxX*vx; fprintf('R = %4.2f*i + %4.2f*j + %4.2f*k [km]\n',R); fprintf('V = %4.4f*i + %4.4f*j + %4.4f*k [km/s]\n',V);
R = -10766.25*i + -3383.89*j + 9095.35*k [km] V = -0.9250*i + -5.1864*j + -2.1358*k [km/s]
Small Satellites 