ASTE 280 Astronautics and Space Environment I
Dan Erwin USC Astronautics and Space Technology Division
Spring 2006
Copyright 2006 by Dan Erwin. All rights reserved.
ii
ASTE 280 S PRING 2006
U NIVERSITY
OF
S OUTHERN C ALIFORNIA
Contents
1 We

ASTE 280
Thursday
22 January 2015
2—Thu-1
Proof of cosine law 2
Prove CL. 2 ﬁrst to show general technique
For arbitrary spherical trian-
gle, choose axes:
o z-aXis through a-b vertex
(pt 1)
o x-aXis in plane of side a
0 => Side a is part of prime
m

ASTE 280
TUesday
13 January 2015
l-Tue-
Coordinate Systems and
Transformations
0 Length scales for space travel
0 Types of coordinate systems
0 Spherical trigonometry
o Rotations and rotation matrices
0 Polar and Cartesian coordinates
0 Proofs and app

ASTE 280 Spring 2015
Homework 5 (40 points)
Due in class Thursday 19 February 2015
1. (15 points) Consider the problem of designing a Sun-synchronous orbit about Mars.
(a) What would the rate of nodal regression d =dt be in order to match Mars average
ang

Chapter 1
Coordinate Systems and Transformations
We will begin our study of astronautics by discussing various ways of specifying locations, usually
in space, but sometimes on the surface of a body such as the Earth or another planet. A location is
comple

Chapter 2
Spacecraft Orbits
In this chapter, we will discuss two-body orbits, i.e., the trajectories of pairs of point masses which
attract each other by gravitation. As we will see, practical orbits such as those of spacecraft about
the Earth are approxi

ASTE 280
Thursday
5 February 2015
4-Thu
-1
Velocity on orbit
from which we get the ; - '
orbit:
This holds for all types of orbits if the proper sign of a is
used. J/‘ 9
£5
4—Thu
—2 Time since periapsis for elliptical orbit
This is derived in Hale;

ASTE 280
Thursday
29 January 2015
% Mink/tummy DOS/UHA1’0U5/ {44/759
0/ Aw w
[N SeArCi/x ,LV‘ ﬁmw "(5 Euler Parameters or magma“;
So far, have two ways of specifying object’s orientation:
0 Roll-pitch-yaw (RPY): Non-unique for pitch of :|:90°
o Euler

ASTE 280
Thursday
15 January 2015
~E>
1 -Thu
-1
Some definitions
Fich/IIJM‘ Orbrl’ w~a+ (OI/uni!
o Astronomical unit (AU): Earth-Sun distance
0 Geocentric, heliocentric coordinates: Origin is at center
of Earth or Sun respectively
0 Inertial coordinat

ASTE 280
TUesday
27 January 2015
61% Cm’ar rif/M/wm : Transformation of body vector to inertial
(my? : $3”
in a L
r|l=Rbirb.
Inverse transformation is the transpose
(All pure rotations: Inverse = transpose)
To transform inertial vector to body coord

ASTE 280 Spring 2015
Homework 2 (40 points)
Due in class Thursday 29 January 2015
1. (10 points) Consider a right spherical triangle, where
sin b D
tan a
:
tan
D 90 . Show that
Hint Express tan as sin = cos and transform numerator and denominator using s

Formulas
Aphelion, perihelion refer to farthest distance, and closest to the Sun. Apogee,
perigee refer to farthest distance and closest to the Earth. (peri is the closest)
a
e
p peri aap
2
p peri
aap
aap p peri
(1 e)
p peri
V p peri
(1 e)
aap
Va ap

ASTE 280 Spring 2015
Homework 3 (35 points)
Due in class Thursday 5 February 2015
1. (10 points) Assuming that the Earths heliocentric ecliptic latitude is zero, show that the angle c between
the North Pole and the Earth-to-Sun vector is given by
cos c D

ASTE 280 Spring 2015
Homework 4 (40 points)
Due in class Thursday 12 February 2015
1. (10 points) A spacecraft is in an Earth orbit with a perigee altitude of 300 km and an eccentricity of 0.7. Find:
(a) The perigee velocity;
(b) The apogee radius;
(c) Th

ASTE 280 Spring 2015
Homework 1 (35 points)
Due in class Thursday 22 January 2015
1. (10 points)
(a) How far, in km, is it from New Yorks Kennedy Airport to Johannesburg, South Africa, on a greatcircle route?
(b) On this route, what is the azimuth when le

ASTE 280
Tﬁﬁesday
3 February 2015 Energy of orbiting body
Total energy: kinetic + potential, per unit spacecraft mass
The kinetic energy per unit spacecraft mass is
V2
E = —
K 2
and the potential energy per unit spacecraft mass due to
gravitational intera