PME (SMAD) 2023/24, Homework
General remarks
This homework includes four activities to be developed, some of them related. To develop the work, any good
Orbital Mechanics book can be used as source of the required equations such as Ref. [1,2]. Ref. [3], in
Portuguese, is available online in the provided link.
All activities require software but the goal is to use pre-existing code, not to develop algorithms from scratch.
Therefore, it requires the use of MATLAB, Python, or other appropriate software application. For MATLAB, all
required algorithms should be available directly in the or in the MATLAB repository, that can be found here. It
is not mandatory to use MATLAB, but alternatives (such as Python) require you to have the code beforehand
or to know where to find it (Algorithms for Python are available through the PyKEP package, but I’m not sure it
will cover all activities, please check before selecting an activity in that case)
In the activities one or more of the following three orbits around the Earth will be considered, A, B, and C, with
eccentricity e and semi-major axis a:
A:
(e,a) = (0.1, 7000 km),
B:
(e,a) = (0.9, 70000 km),
C:
(e,a) = (2, –7000 km).
[1] W. Wiesel, Spaceflight Dynamics, McGraw-Hill, 2nd ed, 1997.
[2] H. D. Curtis, Orbital Mechanics for Engineering Students, 4th Ed., Elsevier (2020)
[3] P. J. S. Gil, Elementos de Mecânica Orbital, IST (2015). Available here.
Homework template
Please use these guidelines for the report.
The homework (one report, several tasks) must include:
1- Title, abstract, identification, introduction and goals (keep it short); use sections and subsections as needed
2- Main part of the work – follow the guidelines provided by the tasks; be objective, be concise. Use the
numbering below to answer
6- Conclusions; include estimated time you did take to complete the assignment
7- References; sources not referenced will null the results; all non-original material should be credited to the
respective authors
Notes:
Penalty for too many, unnecessary pages; questions/issues about the project and the homework
should be talked with the instructor.
The homework requires a considerable amount of time to be done; start early; the reports must be
personal.
Don’t do lengthy theoretical introduction; just state the minimum to understand what are you doing
Activities
1. Lambert’s Problem
1.1 Review what exactly is the Lambert’s problem (not available in [3]).
1.2 Search for a Lambert solver and select one; the selection is your responsibility. Developing one from
scratch is too lengthy and time-consuming, so don’t even try.
1.3 Briefly explain why your selection is a good one (examples of good reasons are: is based on a good
algorithm (provide reference, say shortly why it is good); people (who?) say that it works correctly; always
provide references to back your decision).
1.4 Take orbit A and, using the equations of orbital mechanics (Kepler problem) determine a) the time of flight
(tof) between the true anomalies 𝜃 = 30⁰ and 𝜃 = 60⁰ (so, ∆𝜃 = 30°), b) both the initial and final radii, r1
and r2, and c) both velocities at the those points, v1 and v2 (you need to solve Kepler’s equation for that);
this is the inverse of Lambert’s problem, easy to solve.
1.5 Now solve Lambert’s problem using r1, r2, tof, and ∆𝜃 to determine the original orbit, that is, determine:
v1, v2 (vectors), the initial and final true anomalies 𝜃 , 𝜃 , a, and e.
1.6 Check the precision of the previous result (you have the exact solution from 1.4!).
1.7 Is there any parameter in the Lambert solver to increase the precision? If yes, do experiments and get the
best result you can.
1.8 Can you get a solution for orbit A and ∆𝜃 = 180° starting at perigee? If yes, determine it and explain it; if
not, why not?
1.9 Why do you think there can be the need to test the Lambert solver for orbits B and C?
1.10 Do you have information about the limits of validity or limitations of the solver you are using?
1.11 Using the approximation of circular orbits in the same plane build the pork-chop diagram for a transfer
from a circular LEO at 180 km altitude to the Moon, considering the orbit of the Moon circular (use the
average distance) and neglecting the Moon’s attraction.
1.12 Explain why the result you’ve got in 1.11 is not realistic or at least have some problems (one paragraph).
2. Orbital mechanics solver
2.1 Select two numerical algorithms:
I.
A Runge-Kutta 4th order
II.
A higher order solver (R-K 8(9), or a similar R-K Nystrom);
they are adequate to propagate the orbital motion (i.e. to solve the equations of motion)
2.2 Briefly explain why your selection is a good one i.e. what criteria were used (e.g. source, confidence,
suggestions, quality/speed of the algorithm).
2.3 Use orbit B, the exact solution of the central-body problem, and integrate the equations of motion of a
satellite around a point-like Earth to study numerical convergence using Alg. I. In this case you can
consider the problem in only two dimensions.
a) Using (a large) time step ∆t = 100 s, integrate for 5 orbits; measure the time it takes to get the result;
Notes: if your you are using a variable step algorithm, try limiting the step in such way that it can be
considered constant; if not possible, try to figure out what minimum step is used for a certain
accuracy goal or use some accuracy goal (but in that case read below to select an adequate one,
taking into ccount the next tasks)
b) Plot the trajectories and the exact elliptic trajectory;
c) Plot ∆𝑟(𝑡) and ∆𝜃(𝑡) i.e. the differences between the analytical trajectory using Kepler’s equation and
the numerical solution for each instant of time.
d) Plot 𝑟⃑
− 𝑟⃑
(𝑡) ;
e) Which error measure do you think is the best, c) or d) above? Why?
2.4 Repeat 2.3 by decreasing ∆t until you cannot improve the result significantly; plot the error measure 2.3 d)
for all considered ∆t together as a function of time. Remember to record the required time for each
calculation.
2.5 Decrease ∆𝑡 until you cannot improve the result; plot the error measure you think is the best (2.3e) for all
considered ∆𝑡 together as a function of time.
2.6 Discuss the results by comparing with the exact orbital ellipse.
2.7 Repeat 2.3-2.6 for Alg. II
2.8 What do you learn when you compare the results of Alg. I and Alg.II?
2.9 Using your Alg. II, integrate for 100 orbits (more, if you can, less, if it is too slow, provide performance
information) with a timestep small enough to get good results and measure the error as function of the
number of orbits; notice in particular the difference of the instant of time of when the perigee or apogee
is reached, as opposed to the exact analytical result.
3. Keplerian orbit perturbed by J2
3.1 Use Alg. II from 2. This should be good enough to reach high numerical accuracy; use a ∆t=1s, or one small
enough to achieve good results. Define orbit B as having an inclination of 30⁰ and perigee at the equator
exactly at the x-axis. Determine the initial conditions for the perigee and numerically integrate orbit B for
10 orbits but including the so-called J2 gravity term (note that the time of perigee passage is the epoch,
inclination is 30⁰, and that both the argument of perigee and right ascension of the ascending node are
zero; you already know a and e. Note: the acceleration due to the “J2 term” in the ECI inertial frame is:
Keep at least time, position, and velocity at each instant of time (see 3.3 before doing the integration).
3.2 Verify that the perigee of successive orbits will not pass at the same point. How much is the difference for
successive orbits? You must use a timestep that results in a numerical error in 2. much smaller than this.
3.3 Compare the result you’ve got in 3.1 with the approximate analytic formula for the classical orbital
elements. For that, use e.g., [3] to transform the results from 3.1 to the classical orbital elements; you can
include that in your routine in 3.1.
3.4 Notice two possible sources of error: the numerical error from the algorithm (Alg. I vs Alg II and/or ∆t size)
and the error from the approximation of the dynamics (point mass worst, include J2, better). How can we
determine each of these errors?
4. Atmospheric flight
4.1 Use a simple atmospheric exponential model (one layer) valid for low Earth orbits; reason about the
length scale and density at reference and select the best you can, explaining your choice. Optional: use
more layers but not too many.
4.2 Check literature for aerodynamic drag at high altitudes and get reference values for the Cd and other
parameters that you think you will need. State and justify them.
4.3 Using the atmospheric model from 4.1 and Alg. II from 2. estimate the time it takes a 10 cm cubesat with 1
kg of mass to fall from a circular orbit with 600 km of altitude. Note: before you do that, be sure that the
calculation does not take too much time. State the time of fall from 400 km, 200 km, and 150 km. If it
takes too much time to do the calculation from 600 km, start at 500 km or 400 km (but state a good
reason or you will be penalized).
4.4 Compare your results with the analytic approximate expression [1, page 87],
where td is the time to fall, h the altitude, H0 the altitude scale factor, µ the gravitational parameter, Rꚛ the
radius of the Earth, B* = Cd A/m is the ballistic coefficient, Cd the drag coefficient, A the reference area of
the satellite towards the motion, and m the mass, and ρ0, is the density at the surface of the Earth.
4.5 (i) How does the lifetime depend on the size and mass of the satellite? (ii) What do you expect to happen
in the case of larger satellites?