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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

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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;

Sun_sync_vs_Alt

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');

Period_Speed_Altitude_01 Period_Speed_Altitude_02

 

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