Problem Set 4#

Data#

You may find the following data useful to complete 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.

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#

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.

# Your coded answers here (and create more Code cells if you wish to)

Problem 2#

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.

# Your coded answers here (and create more Code cells if you wish to)

Problem 3#

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.

# Your coded answers here (and create more Code cells if you wish to)

Problem 4#

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.

# Your coded answers here (and create more Code cells if you wish to)