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Sun-Synchronous Circular Orbit, Inclination vs Altitude (LEO,J2 perturbed)
This code is a MATLAB script that can be used to design and analyze Sun-synchronous orbits. A Sun-synchronous orbit is a geocentric orbit which combines altitude and inclination in such a way that an object in this orbit has an a nodal regression rate which is equals to Earth’s orbital rotation speed around the Sun. The object in this orbit constantly illuminated by the Sun.
Output: Inclination vs Altitude Plot
clc; clear all; mu = 398600.440; % Earth’s gravitational parameter [km^3/s^2] Re = 6378; % Earth radius [km] J2 = 0.0010826269; % Second zonal gravity harmonic of the Earth we = 1.99106e-7; % Mean motion of the Earth in its orbit around the Sun [rad/s] % Input Alt = 250:5:1000; % Altitude,Low Earth orbit (LEO) a = Alt + Re; % Mean semimajor axis [km] e = 0.0; % Eccentricity
h = a*(1 - e^2); % [km] n = (mu./a.^3).^0.5; % Mean motion [s-1] tol = 1e-10; % Error tolerance % Initial guess for the orbital inclination i0 = 180/pi*acos(-2/3*(h/Re).^2*we./(n*J2)); err = 1e1; while(err >= tol ) % J2 perturbed mean motion np = n.*(1 + 1.5*J2*(Re./h).^2.*(1 - e^2)^0.5.*(1 - 3/2*sind(i0).^2)); i = 180/pi*acos(-2/3*(h/Re).^2*we./(np*J2)); err = abs(i - i0); i0 = i; end
plot(Alt,i,'.b'); grid on;hold on; xlabel('Altitude,Low Earth orbit (LEO)'); ylabel('Mean orbital inclination'); title('Sun-Synchronous Circular Orbit,Inclination vs Altitude(LEO,J2 perturbed)'); hold off;
Hohmann vs Bi-elliptic transfer
Contents
% In this example we compare be-elliptic and Hohmann transfer. % The total speed change that required for spacecraft transfer from % geocentric circular orbit with radius Ri to a higher altitude Rf clc; clear all; Rf = 125000; % [km] Final circular orbit Ri = 7200; % [km] Initial circular orbit Rb = 190000; % [km] Apogee of the transfer ellipse mu = 398600; % [km^3/s^2] Earth’s gravitational parameter % For initial circular orbit Vc = (mu/Ri)^0.5; a = Rf/Ri; b = Rb/Ri;
Hohmann transfer
For Hohmann transfer total speed change
dV_H =Vc*(1/(a)^0.5 -(2/(a*(a+1)))^0.5*(1-a) - 1); % Semimajor axes of the Hohmann transfer ellipse a_h = (Rf + Ri)/2; % Time required for Hohmann transfer t_H = pi/(mu)^0.5*(a_h^1.5); %[s] fprintf('Total speed change = %4.4f [km/s]\n',dV_H); fprintf('Time required for transfer = %4.2f [hours]\n\n',t_H/3600);
Total speed change = 3.9878 [km/s] Time required for transfer = 23.49 [hours]
Bi-elliptic transfer
For Bi-elliptic transfer total speed change
dV_BE = Vc*((2*(a+b)/(a*b))^0.5 - (1+1/a^0.5) - ((2/(b +b^2))^0.5*(1-b))); % Semimajor axes of the first transfer ellipse a1 = (Ri + Rb)/2; % Semimajor axes of the second transfer ellipse a2 = (Rf + Rb)/2; t_BE = pi/(mu)^0.5*(a1^1.5+a2^1.5); %[s] fprintf('Total speed change = %4.4f [km/s]\n',dV_BE); fprintf('Time required for transfer = %4.2f [hours]\n',t_BE/3600);
Total speed change = 3.9626 [km/s] Time required for transfer = 129.19 [hours]
Circular orbital speed and period as a function of altitude for LEO
mu = 398600; % Earth’s gravitational parameter [km^3/s^2]
R_earth = 6378; % Earth radius [km] % Plot the speed and period of a satellite in circular LEO as a function % of altitude % Low Earth orbit(LEO) h = 160:1:2000; %[km] v = (mu./(R_earth+h)).^0.5; %[km/s] T = 2*pi*(R_earth+h).^1.5/mu^0.5; %[s] T = T/60; %[min] % Plots figure(1); hold on;grid on; plot(h,v); xlabel('Altitude [km]'); ylabel('Speed [km/s]'); title('Circular orbital speed as a function of altitude,LEO'); figure(2); hold on;grid on; plot(h,T); xlabel('Altitude [km]'); ylabel('Period [min]'); title('Circular orbital period as a function of altitude,LEO');
Small Satellites 