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Orbital Inclination Change
Transfer from a LEO 350 km circular orbit with 53.4 deg inclination to a Geostationary Equatorial Orbit(GEO)
clear all; clc; close all;
Input
R_LEO = 6378 + 350; % km R_GEO = 42164; % km mu = 398600; % km^3/s^2 incl = 53.4; % deg
Method 1: Hohman transfer from LEO to GEO and after inclination change
Rp = R_LEO; Ra = R_GEO; e = (Ra - Rp)/(Ra + Rp); % transfer orbit eccentricity a = (Ra + Rp)/2; % transfer orbit semimajor axis V_LEO = (mu/R_LEO)^0.5; Vp = (2*mu/R_LEO - mu/a)^0.5; Va = (2*mu/R_GEO - mu/a)^0.5; V_GEO = (mu/R_GEO)^0.5; dV_LEO = abs(Vp - V_LEO); dV_GEO = abs(V_GEO - Va); dV_Hoff = dV_GEO + dV_LEO; % Inclination change at GEO dV_incl = 2*V_GEO*sind(incl/2); dV_total1 = dV_incl + dV_Hoff; fprintf('Method 1: Hohman transfer from LEO to GEO and after inclination change\n'); fprintf('dV_Hoff = %6.2f [km/s] \n',dV_Hoff); fprintf('dV_incl = %6.2f [km/s] \n',dV_incl); fprintf('dV_total = %6.2f [km/s] \n',dV_total1);
Method 1: Hohman transfer from LEO to GEO and after inclination change dV_Hoff = 3.87 [km/s] dV_incl = 2.76 [km/s] dV_total = 6.64 [km/s]
Method 2: Inclination change in LEO and after Hohman transfer to GEO
dV_incl = 2*V_LEO*sind(incl/2); dV_total2 = dV_incl + dV_Hoff; fprintf('\nMethod 2: Inclination change in LEO and after Hohman transfer to GEO\n'); fprintf('dV_incl = %6.2f [km/s] \n',dV_incl); fprintf('dV_Hoff = %6.2f [km/s] \n',dV_Hoff); fprintf('dV_total = %6.2f [km/s] \n\n',dV_total2); fprintf('Method 2/Method 1 = %6.2f \n',dV_total2/dV_total1);
Method 2: Inclination change in LEO and after Hohman transfer to GEO dV_incl = 6.92 [km/s] dV_Hoff = 3.87 [km/s] dV_total = 10.79 [km/s] Method 2/Method 1 = 1.63
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;
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 