Coursework 1 (60 points; 4 questions)

Coursework 1 (60 points; 4 questions)#

Relevant constants#

You may find the follow data useful in computations for this assessment.

Radius of the Moon, \(R_M\) = \(1738\) \(km\)
Moon’s gravitational parameter \(\mu_M = 4903 km^3 s ^{-2}\)

These constants have been defined in the variables R_m, and mu_m for you in the code-cell below. Please make use of these variables in your calculations/formulae.

In the same code cell, we have also provided access to the constant for \(\pi\) (pi in computing font), and access to the sine and cosine functions (sin and cos respectively). As an example, we also demonstrate how to use sin and pi.

Lastly, we also demonstrate how to use abs to get the absolute value of a number (or function). You may find you need this in various calculations.

Note 1#

Alongside this solution, it is recommended that you also submit handwritten work. It will be considered in evaluating your answers here. You can do so by uploading a scanned pdf of your handwritten work in the images folder. Ensure to name the file Handwritten_Solutions.pdf. DO NOT USE ANY OTHER NAME FOR THIS FILE!

R_m = 1738
mu_m = 4903
from math import sin, cos, pi
print("You can see the result for sin(pi) is", sin(pi))
print("The absolute value of -1 is:", abs(-1))

You can see the result for sin(pi) is 1.2246467991473532e-16
The absolute value of -1 is: 1

Note 2#

In developing your solutions for various orbits in this problem set, you will need to create more variables to explain your work and thinking. We will be checking the values of some of these symbols to gauge your understanding of the material. The below table describes some examples of how you should choose variable names.

Parameters	Math symbol	Equivalent computer symbol
semi-major axis of Lunar Orbit	\(a_{lo}\)	`a_lunar_orbit`
semi-minor axis of Lunar Orbit	\(b_{lo}\)	`b_lunar_orbit`
Lunar Orbit apoapsis	\(r_{al}\)	`r_a_lunar_orbit`
Lunar Orbit periapsis	\(r_{pl}\)	`r_p_lunar_orbit`
Lunar Orbit eccentricity	\(e_{lo}\)	`e_lunar_orbit`
change in the inclination angle	\(\Delta i\)	`delta_i`
velocity impulse (magnitude only)	\(\Delta v\)	`delta_v`
Orbital period	\(T\)	`T`
escape velocity (magnitude only)	\(v_e\)	`v_e`
Velocity at apolune (magnitude only)	\(v_a\)	`v_a_lunar_orbit`
Velocity at perilune (magnitude only)	\(v_p\)	`v_p_lunar_orbit`
Circular velocity (magnitude only)	\(v_c\)	`v_c_lunar_orbit`

Problem 1 (15 points)#

The first spacecraft to orbit the Moon was Luna \(10\), launched by the former Soviet Union in \(1966\). The orbit of the satellite about the Moon took it from \(350\) \(km\) above the lunar surface out to \(1017\) km.

Compute the semi-major axis and hence orbital period (in hours) of Luna \(10\). These answers are to be stored in variables name a_lunar_orbit and T, respectively. Create other variables, as needed, for your answers.

# Compute a
r_a_lunar_orbit = R_m + 1017
r_p_lunar_orbit = R_m + 350
a_lunar_orbit = (r_a_lunar_orbit + r_p_lunar_orbit)/2.
print("P1a", a_lunar_orbit)
T = 2/3600 * pi * (a_lunar_orbit**3/mu_m)**0.5
print("P1b", T)

P1a 2421.5
P1b 2.970114083127854

Problem 2 (15 points)#

What is the velocity impulse (i.e., \(\Delta v\)) required at pericentre to escape the lunar gravitational influence? You may neglect the Earth’s gravitational influence. Stored your answer under the variable name delta_v. Create other variables, as needed, for your answers.

# Velocity impusle required
v_p_lunar_orbit = (mu_m * (2/r_p_lunar_orbit - 1/ a_lunar_orbit))**0.5
v_e = (2*mu_m / r_p_lunar_orbit)**0.5
delta_v = abs(v_e - v_p_lunar_orbit)
print("P2", delta_v)

P2 0.5326111942727525

Problem 3 (10 points)#

Having analysed the parabolic orbit, you are now asked to study a rotational maneuver to change the inclination of the satellite’s elliptical orbit by \(60^o\). What is the \(\Delta v\) required for this plane change? Store your answer in variable called delta_v_plane_change. Create other variables, as needed, for your answers.

# final answer
delta_i = pi / 3
delta_v_plane_change = 2 * v_p_lunar_orbit * sin(delta_i /2)
print("P3", delta_v_plane_change)

P3 1.6344975139504192

Problem 4 (20 points)#

A lunar module (LM) lifts off the surface of the Moon and flies in a powered trajectory to its burnout point at an altitude of 30 km. At this point the LM is travelling purely transversal and coasts around the Moon for half an orbit, at which point it is to rendezvous with the orbiting command module (CM), which is in a circular orbit of altitude 250 km.

What is the burnout speed of the LM? Store your answer in a variable named v_burnout_of_lunar_module.

What is the magnitude of the final impulse manoeuvre at rendezvous? Store your answer in the variable named delta_v_rendezvous.

Hint: For this problem, you will need to create variables with naming conventions that allow us to distinguish between the orbit parameters of the LM from that of the CM. As one example, I recommend using r_p_lunar_orbit_of_lunar_module for the perilune of the lunar module. As another example, r_c_lunar_orbit_of_command_module for the circular radius of the command module (remember, it is in a circular orbit). This hint is important because there are two satellites involved in the maneuvers described here.

r_p_lunar_orbit_of_lunar_module = R_m + 30
r_a_lunar_orbit_of_lunar_module = R_m + 250

a_lunar_orbit_of_lunar_module = 1/2* (r_p_lunar_orbit_of_lunar_module + r_a_lunar_orbit_of_lunar_module)
E_lunar_orbit_of_lunar_module  = - mu_m / (2*a_lunar_orbit_of_lunar_module)

v_p_of_lunar_module = (mu_m * (2/r_p_lunar_orbit_of_lunar_module - 1/ a_lunar_orbit_of_lunar_module))**0.5

v_burnout_of_lunar_module = v_p_of_lunar_module

print("P4a", v_burnout_of_lunar_module)

v_a_of_lunar_module = (2*mu_m/r_a_lunar_orbit_of_lunar_module + 2 * E_lunar_orbit_of_lunar_module)**0.5

r_c_lunar_orbit_of_command_module = 250

v_c_of_command_module = (mu_m /r_c_lunar_orbit_of_command_module)**0.5

delta_v_rendezvous = abs(v_c_of_command_module - v_a_of_lunar_module)

print("P4b", delta_v_rendezvous)

P4a 1.7133662676036499
P4b 2.9047854481706317