Homework 2
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School
University of California, Los Angeles *
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Course
161A
Subject
Aerospace Engineering
Date
Jan 9, 2024
Type
Pages
3
Uploaded by PrivateHyenaMaster979
MAE 161A
Due Thursday, Oct. 19 (at the beginning of class)
D. R. Boone, UCLA
1
Homework # 2
Use this page as your coversheet
Instructions:
Show all steps for credit.
All homework submissions must be
neat and legible. Please staple multiple sheets of paper and write your name
on the first page. Show all your work and circle the final answer. Points will
be deducted if the final answer has missing or incorrect units.
Name:
SID:
Problem 1.
Orbit Determination.
Consider an inertial reference frame with the center
of the earth as its origin and the
i-j
plane in the equatorial plane.
Directly above the
equator, a satellite is at position
𝒓 = 6500 ? km
and has an inertial velocity
𝑣 = 5.4 ? + 1.0 ? − 5.4 ? km/s
.
On a separate piece of paper:
a.
Draw the orbit
b.
Calculate the eccentricity vector and draw it
c.
Calculate and draw the specific angular
momentum vector
d.
Calculate the vector position of perigee and
draw it.
e.
Calculate the vector position of apogee and
draw it
f.
Calculate the angle between the specific angular
momentum vector and the equatorial plane.
Draw it.
Problem 2.
Space Shuttle.
The Space Shuttle orbiter (R.I.P.) will deploy a satellite into
Earth orbit in 40 minutes. The perigee altitude of the orbit is 300 km, the apogee altitude
is 600 km, and the current shuttle true anomaly
=330
. What is the true anomaly at the
moment the satellite is deployed?
Problem 3.
Planetary Orbit.
The orbit of Mars has a semi-major axis of 227.9x10
6
km
and an eccentricity of
e
=0.0935. What is the period of Mars
’s orbit? What is the
minimum distance between Mars and the Sun? What is Mars
’s maximum
orbital speed?
MAE 161A
Due Thursday, Oct. 19 (at the beginning of class)
D. R. Boone, UCLA
2
Problem 4.
K
epler’s E
quation.
Neglecting the eccentricity of Neptune’s orbit, how
many Earth years in each Pluto orbit is Pluto closer to the Sun than Neptune? Use the
following constants a
N
=4.495x10
9
km, a
P
=5.87x10
9
km, e
P
=0.2444.
Problem 5.
Numerical Integration.
A satellite is in an elliptical orbit around the earth.
At perigee, the satellite is at 500 km altitude and moving 9.0 km/s.
a.
Complete the following table
b.
Using Kepler’s Equation, determine
the time, T
50
, for the satellite to move from
perigee to a true anomaly of θ =
50°.
c.
Implement a time step method in Matlab (or program of your choice) and the
restricted
two-body equation of motion
to estimate the true anomaly of the
satellite at T
50
i.
Write down (derive) the system of first order ODEs that describes the
restricted two-body problem. (Hint: One should be position and one
should be velocity)
ii.
Calculate a time step that is 1/25000 of an orbit period
iii.
Apply Euler’s method to the problem us
ing Matlab. Run your simulation
up to T
50
. Turn in your code and report the estimated value of the true
anomaly at this time.
iv.
Is there a difference between (iii) and (b)?
If so, what is the error and
how can it be improved?
Problem 6
.
Hyperbolic Meteoroid.
(Curtis 2.38) A meteoroid is first observed
approaching the Earth with a true anomaly
of θ =
150°. If the speed of the meteoroid at
that time is 2.23 km/s, calculate:
a) the eccentricity of the orbit
b) the altitude at closest approach
c) the speed at closest approach
Semi-major axis
Magnitude of Eccentricity
Distance to apogee
Orbital period
MAE 161A
Due Thursday, Oct. 19 (at the beginning of class)
D. R. Boone, UCLA
3
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