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Hohmann vs Bi-elliptic transfer
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% 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]
Bi-Elliptic Hohmann Transfer
In this example we calculate the total change in speed required for a bi-elliptic Hohmann transfer from a geocentric circular orbit of 7200 km radius to circular orbit of 125000 km radius. The apogee of the first transfer ellipse is 190000 km.
clc; clear all; R_i = 7200; % [km] R1_a = 190000; % [km] R_f = 125000; % [km] mu = 398600; % [km^3/s^2] Earth’s gravitational parameter % For initial circular orbit V_i = (mu/R_i)^0.5; % Speed at apogee and perigee for the first transfer ellipse V1_a = (2*mu*R_i/(R1_a*(R1_a+R_i)))^0.5; V1_p = (2*mu*R1_a/(R_i*(R1_a+R_i)))^0.5; % Semimajor axes of the first transfer ellipse a1 = (R_i + R1_a)/2; % Speed at apogee and perigee for the second transfer ellipse V2_a = (2*mu*R_f/(R1_a*(R1_a+R_f)))^0.5; V2_p = (2*mu*R1_a/(R_f*(R1_a + R_f)))^0.5; % Semimajor axes of the second transfer ellipse a2 = (R_f + R1_a)/2; % For target circular orbit V_f = (mu/R_f)^0.5; % For bi-elliptic maneuver the total speed change required dV = abs(V_i - V1_p)+ abs(V1_a - V2_a) + abs(V_f - V2_p); %[km/s] % Time required for transfer t_bi = pi/(mu)^0.5*(a1^1.5+a2^1.5); %[s] fprintf('Total speed change = %4.4f [km/s]\n',dV); fprintf('Time required for transfer = %4.2f [hours]\n',t_bi/3600);
Total speed change = 3.9626 [km/s] Time required for transfer = 129.19 [hours]
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