AAE251 HW 4
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School
Purdue University *
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Course
251
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
Pages
6
Uploaded by JusticeAtomBear31
1
Name
Team Number
AAE 251: Introduction to Aerospace Design
Homework 4—Orbit Transfer
Due February 15
th
, 11:59 PM ET on Gradescope
Instructions
Show all the work and clearly box your final answer. You will not receive full credit for
a correct final answer if you don’t show your work.
You must use the template provided. Anything that you want the graders to grade must
be written legibly within the boxes provided to you.
There are multiple steps involved to complete your submission on Gradescope. You can
follow the step-by-step guide posted on Brightspace.
Select the relevant pages of your
final answer for each question in your Gradescope submission.
2
Question 1 (10 points)
A geostationary Earth orbit (GEO) is a circular orbit 35,786 km above the Earth's
equator, following the direction of the Earth's rotation. Earth’s gravity is the only
force keeping an object in the circular orbit (in this case, a GEO orbit). So, the
gravitational force equals the force due to the object’s centripetal acceleration.
Centripetal acceleration (
!
!
) is the object’s acceleration while in uniform circular
motion.
(a)
Equating these two forces, derive the formula for the period of a circular orbit
in terms of the radius of the orbit (
"
) and standard gravitational parameter
(
#
).
(4 points)
Hint: Refer to Kepler’s third law to verify your answer.
(b)
Calculate the period of a satellite S1 orbiting in GEO at a height of 35,800 km
above the Earth. Approximately how many days will it take for satellite S1 to
complete one revolution?
(3 points)
(c)
Another satellite S2 is orbiting in the medium Earth orbit (MEO) with a period
of 13.9 hours. MEO’s are circular orbits ranging anywhere from 2,000 km to
35,786 km above Earth’s surface. Calculate the height above the Earth for
satellite S2.
(3 points)
Earth’s radius (
R
E
)= 6371 km
Earth’s gravitational constant or parameter (μ
E
or GM) is
3.986 × 10
"
km
#
/s
$
3
Answer 1:
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4
Question 2 (5 points)
A spacecraft orbiting a planet of mass
5.5 × 10
$%
kg wants to move from a circular
orbit at a height of 6,700 km from the center of mass of the planet to a circular orbit
at a height of 23,000 km from the center of mass of the planet. It does this by firing its
thrusters directly "behind" it to boost itself to an elliptical orbit. This orbit is chosen
so that when the spacecraft reaches the radius of the new circular orbit it is moving
such that its thrusters can again be fired directly behind it to boost itself into the new
circular orbit. Assume that these boosts occur instantaneously and that the mass of
fuel lost is negligible compared to the mass of the rocket and remaining fuel.
What is the total ΔV of this transfer (i.e., what is the sum of the increase in speed
needed for each boost)?
Universal Gravitational Constant (G) is 6.67
× 10
&''
Nm
$
/kg
$
Answer 2:
5
Question 3 (12 points)
You are now ready to launch from Earth to LEO. Your mission is to end up in an orbit
coplanar with the ISS, to assist in maintenance and repairs.
Your launch site is a remote island at a latitude
29.45°
, and you wish to put your
satellite in a circular orbit at a height of 450 km, and an orbital inclination of
52°
.
Assuming the total losses due to steering, atmospheric drag, and gravity losses are a
combined
25%
of
Δ9
()*
, compute the following:
a)
Launch Azimuth values (in degrees).
[2 points]
b)
Δ9
+,-./0
for the launch.
[4 points]
c)
Suppose you are launching for a different mission, one with an orbital
inclination of
180° − 52°
. What is the launch azimuth?
[1 point]
d)
Compute the new
Δ9
+,-./0
. Is it different from your answer in b)? Why, or
why not?
[5 points]
6
Answer 3:
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