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Applied Orbital Mechanics Unless stated otherwise, assume that: to = 1.32712440018 X 1011 km³/s² Problems The Earth has orbital elements a 149599802 km, e@ = 0.0167, ** = 0.0%, ?@= 174.87° W@ = -71.935° in the inertial heliocentric frame. Mars has elements a = 227941896 km. e = 0.093, 10 = 1.84°, 20 = 49.558° wa = 286.5°. 1. Consider a set of transfers where MD = 0° and 1/' = 1°.2° 359°. Generate plots of Ca versus and "100,0" versus vo using the minimum energy solution to Lambert's problem. (a) Include both graphs as subplots in the same figure. (b) If we seek the transfer with the minimum C3. what is the desired (c) If we seek the transfer with the minimum 100,0° what is the desired vo? Note: You will need to add a line or two to your minimum energy Lambert solver to get the velocity at the arrival. To do this, you can compute 9 and use the f and g function solution for V2 2. We are going to generate a variation on a pork-chop plot using the minimum-energy Lambert solver. Instead of considering variations in departure and arrival time, we will instead consider variations in departure and arrival true anomaly v. Initialize an array of departure true anomaly values ve = 0°, 19,2° 359° and arrival values 18 359°. Using the minimum energy solution to Lambert's problem, create the following contour plots and include them in your write-up: (a) C3 required at departure. (b) 100,0° at arrival. (c) toj based on the minimum energy transfer You will need to play around with the definition of the contours to make sure you can see the variations in the quantity of interest. On your contour plot, the z axis must be vo and the y axis is 10 Select the direction of the transfer such that the spacecraft departs in the same direction as the Earth's velocity (see discussion in Lecture 27). Additionally, answer the following questions: 1 (d) What is the Va-Vo combination with the minimum C? (e) What is the Va-Vo combination with the minimum "60,0° ? (f) What combination would you select for your Earth-Mars transfer and why? 2

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%Question 1
%Earth's orbital elements in inertial heliocentric frame
a_Earth=149599802; %in km
i_Earth=0.0;%in deg
omega_Earth=174.87;%in deg
w_Earth=-71.935;%in deg
ta_Earth=0;%in deg

%Mars' orbital elements in inertial heliocentric frame
a_Mars=227941896; %in km
i_Mars=1.84;%in deg
omega_Mars=49.558;%in deg
w_Mars=286.5;%in deg

[x_Earth, y_Earth, z_Earth,xdot_Earth,ydot_Earth,zdot_Earth]=kep2cart(a_Earth, e_Earth, i_Earth,omega_Earth, w_Earth,ta_Earth);
r0_Earth=[x_Earth, y_Earth, z_Earth];
for ta_Mars=1:1:359
    [x_Mars, y_Mars, z_Mars,xdot_Mars,ydot_Mars,zdot_Mars]=kep2cart(a_Mars, e_Mars, i_Mars,omega_Mars, w_Mars,ta_Mars);...

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