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# Hohmann transfer orbit

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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