ASEN 5050 Space Flight Dynamics – Fall 2013
Homework #9 – ASEN 5050
Due: Monday, 12/1/2014
Name: ______________________________________
Note: Use Appendix D of the book for all constants not given in the problem.
1.
(40 pts) We wish to plan an interplanetary mission from Earth to Mars using the
Patched Conics technique described in class and in the book. We initiate the mission
from a low Earth parking orbit that is circular at an altitude of 185 km. We wish to place
the spacecraft into a 300 km altitude circular orbit about Mars. For the purpose of this
homework, assume that Earth and Mars are in circular coplanar orbits about the Sun
using the following radii:


Earth’s fictitious circular orbit about the Sun: r = 149,598,023 km
Mars’ fictitious circular orbit about the Sun: r = 227,939,186 km
a) What are the radii of the Spheres of Influence for Earth and Mars relative to the Sun?
Build the Laplace SOIs as we described in class.
SOI of Earth:
SOI of Mars:
The Moon is treated as a separate body when we’re near the Earth, but when we’re far
away it’s fine to treat the Earth-Moon system as a single mass at the Earth-Moon
barycenter. If we include the Earth’s and Moon’s masses, what is the radius of the EarthMoon SOI?
SOI of Earth-Moon:
b) Construct a Hohmann transfer between Earth and Mars (again, this is fictitious!).
Recall how the Patched Conics method ignores the gravity of Earth and Mars, and just
connects their heliocentric positions in space. What is the heliocentric velocity that the
spacecraft will require to depart the Earth’s heliocentric position onto this transfer? What
is the heliocentric velocity that the spacecraft will have when it arrives at Mars? Ignore
all effects of the Earth’s and Mars’ gravity when computing these values.
Heliocentric departure velocity at Earth’s orbit:
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
Heliocentric arrival velocity at Mars’ orbit:
c) Given the Hohmann transfer constructed in (b), what is the spacecraft’s Earthdeparture V¥ magnitude? What is the spacecraft’s Mars-arrival V¥ magnitude?
Earth-departure V¥ magnitude:
Mars-arrival V¥ magnitude:
d) Assuming the spacecraft uses the circular LEO parking orbit described above (h=185
km), what tangential 𝛥𝑉 must the spacecraft perform to escape the Earth and achieve this
required hyperbolic excess velocity?
e) Assuming the spacecraft arrives at Mars and performs a single tangential 𝛥𝑉 to insert
into its targeted 300-km circular orbit, what 𝛥𝑉 must the spacecraft perform?
f) How long does this orbit transfer take? Please provide the answer in Earth solar days
(86400 sec per day).
2.
(20 pts) Consider problem (1), but let’s assume the spacecraft does not insert into
orbit but instead performs a Mars flyby. What are the minimum and maximum
heliocentric velocities that the spacecraft could practically achieve via a gravity assist?
The spacecraft cannot get closer than 200 km to the surface of Mars (assume a spherical
Mars).
ASEN 5050 Space Flight Dynamics – Fall 2013
ASEN 5050 Space Flight Dynamics – Fall 2013
3.
(20 pts) Is it possible for a spacecraft to perform a Hohmann transfer from Earth
to Jupiter, execute a gravity assist by Jupiter, and escape the Solar System? What is the
maximum specific energy (Joules/kg!) that a spacecraft can have after performing this
flyby? Assume that Jupiter’s orbit is circular and coplanar with Earth, at a heliocentric
radius of 778,298,361 km. Assume a minimum flyby altitude of 1 Jupiter radius
(minimum flyby radius of 2x Jupiter’s radius).
4.
(20 pts) Compute the Sphere of Influence (SOI) for Jupiter, relative to the Sun.
The natural satellite furthest from Jupiter (S/2003 J 2) that is known to date is in an orbit
with an average distance to Jupiter of approximately 29,540,000 km. It is commonly
thought that satellites’ orbits are stable within about 67% of the planet’s SOI. Is this
moon’s orbit likely to be stable? How does this relate with Saturn and its furthest natural
satellite (Fornjot), which has a semi-major axis of 24,504,879 km relative to Saturn?
Could additional satellites be discovered based on these numbers?
ASEN 5050 Space Flight Dynamics – Fall 2013