ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Final Exam
Due: Thursday, 12/19/2013 at 10:00 am
(yes, morning!)
CAETE same due date
Read everything on this page before you take the exam!
This is a take-home exam. You may use whatever books and notes you have at your disposal. However, you may not communicate with other people (students or otherwise)
about the exam. You may want to use a computer to help check the calculations you
have made, but you will be graded based on the work you show in your answer. You
may attach any code you have written in support of the exam, but it will in general not be
used to grade your exam.
In general, a correct answer will not give you credit for a question unless you show your
work and/or describe your methods. An incorrect answer may still give you partial credit
if you show a correct process. You can skip steps in the process as long as you make it
clear what you are doing (some steps are rather tedious to show in full detail).
Use Appendix D of the book for all constants not given in the problem.
You may use all the time you have to complete the exam (no 24 hour requirement this
time, like there was for the mid-term, since the CAETE students have the same time requirement).
The exam may be handed in at my office, either in my hand or under my door; or it may
be submitted via D2L anytime before the due date.
Sign the following statement, reflecting the Honor Code (electronic signatures are acceptable):
On my honor, as a University of Colorado at Boulder student, I have neither given nor
received unauthorized assistance on this work.
______________________________
(Signature)
____________________________
(Printed Name)
ASEN 5050 Space Flight Dynamics – Fall 2013
_____
(Date)
ASEN 5050 Space Flight Dynamics – Fall 2013
ANSWER SHEET – for ease of grading. Please show your work in your own space and
fill in more descriptions as needed – this is mostly to identify if an answer is correct or
not. We will check your work, especially if you show an incorrect answer.
1. (10 points) Net acceleration at mean sea level: _______________________________ m/s2
2. (30 points)
a: ____________________________________ km
e: ____________________________________
i: ____________________________________ deg
Ω,: ____________________________________ deg
ω: ____________________________________ deg
ν: ____________________________________ deg
perigee radius: ____________________________________ km
apogee radius: ____________________________________ km
height above release: ______________________________
3. (10 points)
Velocity: ____________________________________ km/s
Direction: ____________________________________________________________
ΔV: ___________________________________________ km/s
4. (30 pts)
a. Sphere of Influence: ________________________________________________________ km
b. Period of orbit: __________________________________________________________ (hours)
c. Radius: ___________________________________________________________ km
d.
ΔV R: ____________________________________________ m/s
ΔV S: ____________________________________________ m/s
ΔV W: ___________________________________________ m/s
ΔV Magnitude: ____________________________________________ m/s
e. Do the operators need to be concerned?
Which orbital elements will be affected in a secular way?
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
5. (20 pts)
a. Heliocentric specific energy: _________________________________________ km2/s2
⃗∞ vector x_________________________________________________ km/s
𝑉
⃗∞ vector y_________________________________________________ km/s
𝑉
⃗∞ vector z_________________________________________________ km/s
𝑉
⃗∞ magnitude _________________________________________________ km/s
𝑉
Venus-centered specific energy: ______________________________________ km2/s2
b. Drawing in your own space (or down below)! Answer some questions
with the drawing.
Answers for clockwise assist:
Radius of periapse:
___________________________________ km
Heliocentric velocity post-flyby: ________________________________________ km/s
Heliocentric specific energy: __________________________________________ km2/s2
Answers for counter-clockwise assist:
Radius of periapse:
___________________________________ km
Heliocentric velocity post-flyby: ________________________________________ km/s
Heliocentric specific energy: __________________________________________ km2/s2
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
1. (10 points) The standard gravitational acceleration at mean sea level, g0, is defined to be 9.80665 m/s2 (see http://physics.nist.gov/Pubs/SP330/sp330.pdf),
originally estimated to be the acceleration an object experiences when located at a
geodetic latitude of 45 deg, though this no longer matches observations perfectly.
Use Newton’s Law of Gravitation, balanced with centripetal acceleration from the
rotation of the Earth to estimate what the actual acceleration on an object is at mean
sea level for an object located at a latitude of 45 degrees.
2. (30 points) A person is standing at a park near Boulder and tosses a ball directly
upwards, toward their local zenith, giving the ball a zenith-ward velocity of 2 m/s
relative to the person. Compute the instantaneous orbital elements of the ball (twobody motion, ignoring everything else!): a, e, i, Ω, ω, and ν. Also compute the perigee
and apogee radii. How high will the ball get above the release point? Try it out
yourself and see if your answer agrees with an object you toss in the air!
Show at least 6 digits past each decimal, in units of degrees and kilometers.
Here are the facts you’ll need:
 Treat the Earth as a perfect sphere, rotating once every sidereal day. Ignore
the atmosphere and all perturbations.
 The person is located at a geocentric latitude of 40 deg N, a longitude of 105
deg W (note: West), and an altitude of 5430 feet above the mean sea level
(use 6378.1363 km for mean sea level). The person released the ball 6 feet
above ground.
 The person let go of the ball at 12:15 pm Thursday December 12th, 2013
Mountain Time.
This problem tests a large number of topics from our class. I suggest drawing a picture and computing as much as you can, even if you can’t finish the problem. What
answers do you expect to get?
3. (10 points) Take the setting from Problem 2. Again ignoring all atmospheric effects and all other perturbations, what velocity must the ball achieve from its release location near Boulder from Problem 1 in order to just escape the Earth?
Which direction would you launch the ball to minimize the ΔV needed to obtain that
velocity? You can state your answer in whatever coordinate frame is the easiest.
What is the ΔV needed to just escape the Earth from that location?
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
4. (30 pts) Consider a spacecraft in a circular orbit about a spherical asteroid. The
asteroid has a mass of 1.0×1015 kg and is in a circular orbit about the Sun with a
semi-major axis of 3 AU.
a. What is the sphere of influence of the asteroid? Use the Laplace formulation.
b. The spacecraft’s orbital semi-major axis about the asteroid is 40.0 km. What is
the period of this orbit? Ignore all perturbations.
c. The spacecraft performs a 1.0 m/s burn in the anti-velocity direction in order to
reduce its altitude. What will the spacecraft’s radius be after 6 hours?
d. The spacecraft operators program the spacecraft to execute a maneuver to circularize its orbit at this new orbital radius. What are the RSW coordinates of that circularization maneuver? What is the magnitude of that maneuver?
e. The scientists discover that the asteroid’s gravity field really isn’t spherically
symmetric, but has a sizeable J2 term in its spherical harmonic expansion. The
spacecraft is in a circular, polar orbit (inclination of 90 deg wrt the asteroid’s equator). Do the operators need to be concerned about this new information from a navigation perspective? Which orbital elements will be affected in a secular way by this
J2 term? Make an argument for why / why not the operators will need to be concerned.
5. (20 pts) A spacecraft is approaching Venus with the purpose of flying by and acquiring a gravity assist en route to some other destination in the Solar System. We
have artificially removed all z-axis motion for the purpose of this exercise. At the
moment in time of the gravity assist, Venus’ heliocentric position and velocity are
equal to:
𝑅𝑉𝑒𝑛𝑢𝑠 = [48965315.1 96179438.8 0.0] km
𝑉𝑉𝑒𝑛𝑢𝑠 = [−31.322263 15.730492 0.0] km/s
The spacecraft’s heliocentric velocity at the Venus arrival is:
𝑉𝑆𝐶𝐴𝑝𝑝𝑟𝑜𝑎𝑐ℎ = [−28.123456 8.654321 0.0] km/s
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
a. What is the specific energy (energy per unit mass) of the spacecraft relative to the
⃗∞ vector? What is that
Sun prior to the flyby? What is the spacecraft’s incoming 𝑉
vector’s magnitude? What is the specific energy of the spacecraft relative to Venus
prior to the flyby?
b. Consider a gravity assist that rotates the spacecraft’s velocity such that the outgoing hyperbolic asymptote is 42 degrees away from the incoming hyperbolic asymptote, i.e., the turn angle is 42 degrees. Draw two gravity assists that do this: one
that rotates the V-infinity vector clockwise and one that rotates it counterclockwise.
Which one results in an increase in the spacecraft’s heliocentric energy? Which one
results in a decrease? Build both gravity assists. How close do these gravity assists
get to Venus? After each flyby, what heliocentric velocity does the spacecraft have?
What are the resulting specific energy values relative to the Sun?
ASEN 5050 Space Flight Dynamics – Fall 2013