F2023
Astrodynamics Homework 1
Solution Guide
/
Dr. Alicia Petersen
.
EAS 4510 Astrodynamics Unit 1 Homework
Assigned: November 01, 2023
Due: November 8, 2023 by 11:59 PM
INSTRUCTIONS
Show all of your work. Partial credit is possible.
Define your variables and justify your assumptions.
Give your final answers accurate to at least 4 significant figures.
Begin each problem on a new page and label the problem number at the top.
Submit a single PDF to Canvas.
1. Given the position and velocity vectors of an orbit in the Earth-centered inertial
frame, calculate the following orbit properties and sketch a diagram of the orbit.
~r = 2500 î + 16000 ĵ + 4000 k̂ (km)
~v =
3 î
ĵ + 5 k̂ (km/s)
a) Orbital Elements: a, e, i , ⌦, !, ⌫
-
-
↑
b) Orbital Period, ⌧
c) Eccentric Anomaly, E
d) Is the satellite currently closer to periapsis or apoapsis? Explain how you determined this answer in 1-2 sentences.
e) Is the satellite moving away or towards periapsis? Explain how you determined
this answer in 1-2 sentences.
f) Sketch a diagram of the appropriate orbit. (You may break this into multiple
diagrams of 2D orbits, or one singular 3D orbit, or a combination thereof.) Label
and identify the apoapsis, periapsis, true anomaly and current position vector.
1)
12500
=
i
[- 3
=
1)Ill
11 ill
5
.
=
D27km
.
41608
Angular
n
15)
16688
=
=
-
2000]T
16000
Momentum
rxi
(80
=
,
000
-
25
,
50053500]
1) ill-48 623kmG/s
,
Inclination
i
Line
=
cr" (i)
Noder
of
N
In Il
=
art)) =c) =
Vector
kxn
=
=
=
I
N
(
K
8
J
I
2008
-
24508
I
-5530
[205500 0 000
=
by
T km
%
07500km %
RAAN
2
)
E360-cor"(i)
:
Nx
Ny
=
=
N.
Nog
Upo
Nyc
Ny
1
=
=
N j
.
=
0000
()") )
=
-
Ny
3
>
0
F2023 EAS 4510 Astrodynamics Unit 1 Homework
2. Say that you are studying the motions of a planet on an elliptical orbit in a distant
star system. You are able to observe that the line of sight between you and the planet
must be in the orbital plane of the planet. You are also able to measure vaway , the
component of the planet’s velocity directed along the line of sight away from you.
0
However, vtot
is the planet’s velocity in the star-centered inertial frame. (You can
assume the star system is sufficiently far away that your line of sight with the star is
equivalent to your line of sight with the planet.)
Answer each of the following:
a) What is it that you must be observing about the planet in order determine that
your line of sight with the planet is in its orbital plane? Draw a diagram to
demonstrate this.
0
b) What would need to be true about the orbit for the planet’s vtot
to be equal
to vaway ? Where in the orbit would this occur? Explain your answer in 2-4
sentences.
c) If you were to observe an entire revolution of the planet and determine the
orbital period, which orbital element(s) would you be able to find a unique
solution for? Justify your answer for each element.
2
2b)
Taway
is
which
a)
is
direction (as selected
+y
wherever
orbital
y
the
orbit of the planet
viewed
from
figure
tangent
the
above
to the
orbit
&
of the
only
one
answer
T
If
orbital
=
you
may
with
a
elements
unique
part
as
,
Not
when
shown
in
when
y
is
positive.
is
Depending onyouryes
2)
clockwise
Taway=
,
in
I Tawayl=lvol
,
y-plane
X,
partd then
in
is
velocity
tangential
is
axis
path of the planet
If the
the
the planet's
component of
the
in
Then
the
to
the
labels for
a, e ,
solution
partta e
a
ie wer (0
-
,
,
only
meaning
axis
6)
,
the
one
true
a
, is the
semi-major
=
if
know the mass of the star
GM
,
you
chose to assume
,
knew that constant or not
is
,
2π
change
you
your
Justifications for the
determination of the
other elements
answer
.
may vary.
F2023 EAS 4510 Astrodynamics Unit 1 Homework
3. Solve for the position and velocity of a satellite in Earth orbit at time, to , that has
the following orbital elements: a = 7016 km, e = 0.05, io = 45 , ⌦o = 0 , !o = 20 ,
⌫o = 20 .
a) Find position at to : ro
b) Find velocity at to : vo
c) What is the eccentric anomaly, Eo at to ?
d) What is the mean anomaly, Me at time to ?
e) Solve the first iteration for E1 or E (1) . Use either the method given in Lectures
13-14 and in "The Orbit as a Function of Time - EAS 4510 - Rao," Section 3.5,
with no periapsis crossings, or the method in Curtis Section 3.4 and Algorithm
3.1, with an error tolerance of 10 6 . (Note that Curtis uses ✓ for true anomaly
while Rao uses ⌫.)
f) EXTRA CREDIT: Solve for the eccentric anomaly at time t, when t to = 1
hour. Use either the method given in Lectures 13-14 and in "The Orbit as a
Function of Time - EAS 4510 - Rao," Section 3.5, when k=0, or the method in
Curtis Section 3.4 and Algorithm 3.1, with an error tolerance of 10 6 . (Note
that Curtis uses ✓ for true anomaly while Rao uses ⌫.)
Helpful Equations:
Kepler’s Equation:
Me = E
e sin E
For Curtis’ Algorithm 3.1:
Ei+1 = Ei
where
f (Ei ) = Ei
Ei
e sin Ei Me
1 e cos Ei
Me and f 0 (Ei ) = 1
e sin Ei
e cos Ei
For Rao’s fixed point iteration:
E (k+1) = e sin E (k) + C
when
C=
r
µ
(t
a3
to )
2⇡k + (Eo
3
e sin Eo )
F2023 EAS 4510 Astrodynamics Unit 1 Homework
4. The altitude of a satellite in an elliptical orbit around the earth is 2000 km at apogee
and 500 km at perigee. Calculate the following:
a) The eccentricity, e.
b) The magnitudes of the maximum and minimum velocities. Explain where along
the orbit these extrema occur.
c) The period of the orbit, ⌧ .
4
F2023 EAS 4510 Astrodynamics Unit 1 Homework
5. Calculate the orbital inclination, i, required for a satellite to be in a 300 km by 600
km sun-synchronous orbit.
5
F2023 EAS 4510 Astrodynamics Unit 1 Homework
6. In 1-4 sentences answer the following questions and explain them in your own words.
Use diagrams if desired.
a) For a circular orbit can the argument of perogee be uniquely determined?
b) What is the eccentric anomaly?
c) What is the true anomaly?
d) In order to define the Right Ascension of Ascending Nodes (RAAN), explain
each of the following: Right Ascension, Nodes, Line of Ascending Nodes, and
altogether, Right Ascension of Ascending Nodes.
e) What happens to the Right Ascension of Ascending Nodes as the orbital inclination reaches i = 0 ?
2)
the
No
is
the location
perogee
distance to the
origin, every
,
its center
, therefore
value
any
b)
O to
The eccentric
and
(not the
2)
is
vector
location
the
angle
ellipse
an
on
of
argument
Lit
.
anomaly
your position
the
on
a
which
is
between the
minimal
a
equidistant
circle is
perogee
is
to
either undefined
eccentricity
with
respect
to the center of the
the
angle
between
or
rector
ellipse
origin)
The true
anomaly to
is
vector with respect
the
your position
origin and the periapsis position
rector
.
4)
Right
ascension
Nodes
=
Line of
=
the
to
angle corresponding
astronomical coordinate
points of intersection
Ascending
Nodes
=
Right
Ascension of
Ascending
in a
spherical
or
system
.
the line of intersection between
equatorial plane and the
orbiting body ascends (has
the
longitude
orbital
&
a
+z
Nodes
=
plane
through
which
an
velocity component).
from
Longitudinal angle
*
to the line of intersection of the orbital plane with the
has a
equatorial plane through which the
northward
orbiting body
velocity.
e)Whenithe ofascending nodescould
e
6
be
anywher
is
F2023 EAS 4510 Astrodynamics Unit 1 Homework
7. A satellite has a specific angular momentum, h, of 70,000 km2 /s and a specific energy,
", of -10 km2 /s2 . Calculate the apogee and perigee altitudes. Assume the Earth’s
radius is 6378 km.
a) za
b) zp
7