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A spececraft is in a 300 km circular eart orbit. Calculate (a)The orbital delta v required for a Hohmann to a 3000 km coplanar circular earth orbit and (b)The transfer orbit time
clear all; clc; close all; Alt_a = 300; Alt_b = 3000; Re = 6378; R_a = Alt_a + Re; R_b = Alt_b + Re; mu = 398600; V_a = (mu/R_a)^0.5; %Velocity of Circular Orbit at A fprintf('Alt_a = %8.2f Alt_b = %8.2f \n',Alt_a,Alt_b); fprintf('Velocity of Circular Orbit at A = %8.4f km/s \n',V_a); a = (R_a+R_b)/2; %Eliptical Orbit Semimajor Axis fprintf('Eliptical Orbit Semimajor Axis = %8.4f km/s \n',a); Vtp = (2*mu*(1/R_a - 1/(R_a + R_b)))^0.5; %Transfer orbit at Perigee fprintf('Transfer orbit at Perigee = %8.4f km/s \n',Vtp); dVa = Vtp - V_a; fprintf('dV burn at Perigee = %8.4f km/s \n',dVa); Vtb = (2*mu*(1/R_b - 1/(R_a + R_b)))^0.5; fprintf('Transfer orbit at Apogee = %8.4f km/s \n',Vtp); V_b = (mu/R_b)^0.5; %Velocity of Circular Orbit at B fprintf('Velocity of Circular Orbit at B = %8.4f km/s \n',V_b); dVb = V_b - Vtb; fprintf('dV burn at Apogee = %8.4f km/s \n\n',dVb); dVt = dVa + dVb; fprintf('Total delta Velocity required = %8.4f km/s \n',dVt); T = 2*pi*(a^3/mu)^0.5; %Orbital period fprintf('Transfer orbit time = %d m %2.1f s \n\n\n',floor(T/120),rem(T/2,60));
Alt_a = 300.00 Alt_b = 3000.00 Velocity of Circular Orbit at A = 7.7258 km/s Eliptical Orbit Semimajor Axis = 8028.0000 km/s Transfer orbit at Perigee = 8.3502 km/s dV burn at Perigee = 0.6244 km/s Transfer orbit at Apogee = 8.3502 km/s Velocity of Circular Orbit at B = 6.5195 km/s dV burn at Apogee = 0.5734 km/s Total delta Velocity required = 1.1977 km/s Transfer orbit time = 59 m 39.3 s
Assuming the orbits of earth and Mars are circular and coplanar, calculate (a) the time required for a Hohmann transfer from earth orbit to Mars orbit (b) the initial position of Mars ( ? ) in its orbit relative to earth for interception to occur.Radius of earth orbit 1.496 10^8 km. Radius of Mars orbit 2.279*10^8 km.Sun 1.327 10 11 km 3 /s 2 .
R_earth = 1.496E8; R_mars = 2.279E8; mu = 1.327E11; V_earth = (mu/R_earth)^0.5; %Velocity of Circular Orbit of erath fprintf('Velocity of Circular Orbit of Earth = %8.4f km/s \n',V_earth); a = (R_earth + R_mars)/2; %Eliptical Orbit Semimajor Axis fprintf('Eliptical Orbit Semimajor Axis = %8.4f km/s \n',a); Vtp =(2*mu*(1/R_earth - 1/(R_earth + R_mars)))^0.5; %Trans.orbit at Perigee fprintf('Transfer orbit at Periapsis = %8.4f km/s \n',Vtp); dVp = Vtp - V_earth; fprintf('dV burn at Periapsis = %8.4f km/s \n',dVp); Vta = (2*mu*(1/R_mars - 1/(R_mars + R_earth)))^0.5; V_mars = (mu/R_mars)^0.5; %Velocity of Circular Mars Orbit fprintf('Velocity of Circular Mars Orbit = %8.4f km/s \n',V_mars); dVa = V_mars - Vta; fprintf('dV burn at Apoapsis = %8.4f km/s \n',dVa); dVt = dVa + dVp; fprintf('Total delta Velocity required = %8.4f km/s \n',dVt); T = 2*pi*(a^3/mu)^0.5; %Orbital period % The initial position of Mars ( ? ) in its orbit relative to earth for % interception to occur T_mars = 2*pi*(R_mars^3/mu)^0.5; angle_mars = pi/T_mars*T; angle_mars_t =( pi - angle_mars); fprintf('Angle Mars Traveled = %4.3f deg \n\n',angle_mars*180/pi); fprintf('Transfer time = %4.3f days \n',T/(120*60*24)); fprintf('The initial position of Mars = %4.3f deg \n',angle_mars_t*180/pi);
Velocity of Circular Orbit of Earth = 29.7831 km/s Eliptical Orbit Semimajor Axis = 188750000.0000 km/s Transfer orbit at Periapsis = 32.7264 km/s dV burn at Periapsis = 2.9433 km/s Velocity of Circular Mars Orbit = 24.1303 km/s dV burn at Apoapsis = 2.6478 km/s Total delta Velocity required = 5.5911 km/s Angle Mars Traveled = 135.671 deg Transfer time = 258.840 days The initial position of Mars = 44.329 deg
Two geocentric elliptical orbits have common apse lines and their perigees are on the same side of the earth. The fi rst orbit has a perigee radius of r p 7000 km and e 0.3, whereas for the second orbit r p 32,000 km and e 0.5. (a) Find the minimum total delta-v and the time of flight for a transfer from the perigee of the inner orbit to the apogee of the outer orbit. (b) Do part (a) for a transfer from the apogee of the inner orbit to the perigee of the outer orbit.
fprintf('Part A\n'); mu = 398600; R1_p = 7000; %km e1 = 0.3; R2_p = 32000; %km e2 = 0.5; R1_a = R1_p*(e1 + 1)/(1 - e1); h1 = (2*mu*R1_a*R1_p/(R1_a + R1_p))^0.5; V1_p = h1/R1_p; %Velocity at Perigee 1st orbit V1_a = h1/R1_a; R2_a = R2_p*(e2 + 1)/(1 - e2); h2 = (2*mu*R2_a*R2_p/(R2_a + R2_p))^0.5; V2_a = h2/R2_a; %Velocity at Apogee 2nd orbit V2_p = h2/R2_p; x_dist = R2_a + R1_p; R3_p = R1_p; R3_a = x_dist - R3_p; e3 = (R3_a - R3_p)/(R3_a + R3_p); h3 = (2*mu*R3_a*R3_p/(R3_a + R3_p))^0.5; V3_p = h3/R3_p; V3_a = h3/R3_a; dV13_p = V3_p - V1_p; fprintf('\ndV burn at Perigee Orbit 1 = %8.4f km/s \n',dV13_p); dV23_a = V2_a - V3_a; fprintf('dV burn at Apogee Orbit 2 = %8.4f km/s \n',dV23_a); dV_total = dV13_p + dV23_a; fprintf('\nTotal Velocity Increase = %8.4f km/s \n',dV_total); a3 = (R3_p + R3_a)/2; T3 = 2*pi*(a3^3/mu)^0.5; fprintf('Transfer time = %4.3f hours \n\n',T3/(120*60)); % Transfer from the apogee of the inner orbit to the perigee of the % outer orbit. fprintf('Part B'); R4_p = R1_a; R4_a = R2_p; e4 = (R4_p + R4_a)/2; h4 =(2*mu*R4_a*R4_p/(R4_a + R4_p))^0.5; V4_p = h4/R4_p; V4_a = h4/R4_a; dV14_ap = abs(V4_p - V1_a); fprintf('\n\ndV burn at Apogee Orbit 1 = %8.4f km/s \n',dV14_ap); dV42_ap = abs(V4_a - V2_p); fprintf('dV burn at Perigee Orbit 2 = %8.4f km/s \n',dV42_ap); dV_total = dV14_ap + dV42_ap; fprintf('\nTotal Velocity Increase = %8.4f km/s \n',dV_total); a4 = (R1_a + R2_p)/2; T4 = 2*pi*(a4^3/mu)^0.5; fprintf('Transfer time = %4.3f hours \n',T4/(120*60));
Part A dV burn at Perigee Orbit 1 = 1.6989 km/s dV burn at Apogee Orbit 2 = 0.6896 km/s Total Velocity Increase = 2.3885 km/s Transfer time = 16.154 hours Part B dV burn at Apogee Orbit 1 = 1.9708 km/s dV burn at Perigee Orbit 2 = 1.6398 km/s Total Velocity Increase = 3.6106 km/s Transfer time = 4.665 hours
Published with MATLAB® 7.10