## Transcribed Text

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

clear;clc;

%Earth's orbital elements in inertial heliocentric frame

a_Earth=149599802; %in km

e_Earth=0.0167;

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

e_Mars=0.093;

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];

v0_Earth=[xdot_Earth,ydot_Earth,zdot_Earth];

i=1;

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);...