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Consider a US Air Force launch of its latest GPS satellite from Cape Canaveral. Radar tracking shows the following J2000 state vector (X,Y,Z and Vx,Vy,Vz) at burnout.

26 Feb 2022 17:10:00.000

(X,Y,Z):

5210.345121 -549.481941 4300.883291

(Vx,Vy,Vz):

-1.451280 7.391098 2.690198

Calculate the minimum total delta V required to maneuver this spacecraft into a circular orbit with a 12-hour period and an inclination of 61 degrees, assuming impulsive maneuvers and two-body dynamics.

Please solve using equations from Howard Curtis's Orbital Mechanics for Engineering Students, 4th Edition. Or if not possible name the equations you use. Also please use matlab syntax for solving or write the equations out clearly. Thanks!

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Can you solve this without the transformation matrix and if thtas not possible explain what they do. Also can you explain why you need to change from the J2000 state to geocentric equatorial frame. also would this code work to solve the problem?:

% Step 1: Define the initial state vector at burnout

r0 = [5210.345121, -549.481941, 4300.883291]; % Position vector [km]

v0 = [-1.451280 7.391098 2.690198]; % Velocity vector [km/s]

% Step 2: Calculate the initial velocity magnitude and specific angular momentum

v0_mag = norm(v0); % Initial velocity magnitude [km/s]

h0 = cross(r0, v0); % Specific angular momentum vector [km^2/s]

h0_mag = norm(h0); % Magnitude of specific angular momentum [km^2/s]

% Step 3: Calculate the semi-major axis of the target circular orbit

T_target = 12 * 3600; % Period of the target orbit [s]

mu = 398600.4418; % Earth's gravitational parameter [km^3/s^2]

a_target = (mu * (T_target / (2 * pi))^2)^(1/3); % Semi-major axis of target orbit [km]

% Step 4: Calculate the velocity magnitude required for the target orbit

v_target = 2 * pi * a_target / T_target; % Velocity magnitude for target orbit [km/s]

% Step 5: Calculate the required change in velocity (delta V) for the maneuver

delta_v = v_target - v0_mag; % Change in velocity (delta V) [km/s]

% Step 6: Calculate the inclination change delta V

i0 = asind(h0(3) / h0_mag); % Initial inclination [degrees]

i_target = 61; % Target inclination [degrees]

delta_i = abs(i_target - i0); % Inclination change [degrees]

delta_v_inclination = 2 * v0_mag * sind(delta_i / 2); % Delta V for inclination change [km/s]

% Step 7: Calculate the total delta V required

total_delta_v = sqrt(delta_v^2 + delta_v_inclination^2); % Total delta V required [km/s]

% Display the result

disp(['Total Delta V Required: ' num2str(total_delta_v) ' km/s']);

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