A Brief Introduction to
Astrodynamics
Shaun Gorman
Iowa State University
Ames, Iowa
Topics Discussed
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Coordinate Systems
Orbital Geometry
Classical Orbital Elements
Classes of Orbits
Two-line Element Sets
Coordinate Systems
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Heliocentric-Ecliptic Coordinate System
Geocentric-Equatorial Coordinate System
Right Ascension-Declination System
Perifocal Coordinate System
Heliocentric-Ecliptic Coordinate
System
• Origin at the center of the sun.
• X-Y plane coincides with the earth’s plane
of revolution
• X axis points in the direction of the vernal
equinox
• Z axis points in the direction of the suns
north pole
Geocentric-Equatorial Coordinate
System
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•
•
•
•
•
Also called Earth Centered Inertial or ECI
Origin at the center of the earth
X-Y plane coincides with the earth’s equator
X axis points in the direction of the vernal equinox
Z axis points in the direction of the north pole
I, J and K unit vectors lie along the X, Y and Z
axes
Right Ascension-Declination System
Perifocal Coordinate System
• Origin at the center of the earth
• P-Q plane coincides with the satellites orbit
plane
• P axis points in the direction of the vernal
equinox
• Q axis is 90o from the P axis in the direction
of satellite motion
• W axis is normal to the satellite orbit
ECI Coordinate Systems
• Several different types of ECI coordinate
systems.
–
–
–
–
–
Fixed
J2000
B1950
TEME of Epoch
TEME of Date
ECI Coordinate Types
Classical
Orbital
Elements
(COE)
Uses the traditional osculating
Keplerian orbital elements to
specify the shape and size of an
orbit.
Cartesian
Uses the initial X, Y and Z
position and velocity components
of the satellite.
Fixed
• X is fixed at 0 deg longitude, Y is fixed at
90 deg longitude, and Z is directed toward
the north pole.
• Only Cartesian type of coordinates can be
used.
J2000
• X points toward the mean vernal
equinox and Z points along the mean
rotation axis of the Earth on 1 Jan 2000
at 12:00:00.00 TDB, which
corresponds to JD 2451545.0 TDB.
• Can use either Cartesian or COE.
B1950
• X points toward the mean vernal equinox
and Z points along the mean rotation axis of
the Earth at the beginning of the Besselian
year 1950 (when the longitude of the mean
Sun is 280.0 deg measured from the mean
equinox) and corresponds to 31 December
1949 22:09:07.2 or JD 2433282.423.
• Can use either Cartesian or COE.
TEME of Epoch
• X points toward the mean vernal
equinox and Z points along the true
rotation axis of the Coordinate Epoch.
• Can use either Cartesian or COE.
TEME of Date
• X points toward the mean vernal
equinox and Z points along the true
rotation axis of the Orbit Epoch.
• Can use either Cartesian or COE.
Orbital Geometry
• Apoapsis- farthest point in an orbit
• Periapsis- nearest point in an orbit
• Line of Nodes - The point where the vehicle
crosses the equator
• Radius - distance from the center of the
Earth to the orbit
Orbital Geometry
Classical Orbital Elements
a - Semi-major Axis-a constant defining the size of the orbit
e – Eccentricity-a constant defining the shape of the orbit (0=circular, Less
than 1=elliptical)
i – Inclination-the angle between the equator and the orbit plane
W - Right Ascension of the Ascending Node-the angle between vernal
equinox and the point where the orbit crosses the equatorial plane
w - Argument of Perigee-the angle between the ascending node and the
orbit's point of closest approach to the earth (perigee)
v - True Anomaly-the angle between perigee and the vehicle (in the orbit
plane)
C.O.E. (continued)
Vector Re-fresher
• Before we start lets go over some basic
vector math
a  a I I a J J a K K
I  1I  0 J  0 K
a  a  aI  aJ  aK
2
2
2
I
a b  a I
J
aJ
K
a K  a J b K  b J a K  I  a I b K  b I a K  J  a I b J  b I a J  K
bI
bJ
bK
Determining Orbital Elements
• Let’s say that a ground
station on the earth is
able to provide the
position and velocity
of a satellite by
providing us with
vectors r and v.
Conversion from Cartesian to COE
• Given the position and velocity vectors:
r and v
• Determine the six classical orbital elements:
e, a, i, W, w and v
Setting up a coordinate system
• We will use the geocentric
equatorial coordinate
system.
• The I axis points towards
the vernal equinox.
• The J axis is 90o to the
east in the equatorial
plane.
• The K axis points directly
through the north pole.
Determining Orbital Elements
• The expression, h  r v which is called
specific angular momentum, must be held
constant due the law of conservation of
angular momentum.
• Thus:
I
J
K
h  rI
vI
rJ
vJ
rK  h I I  h J J  h K K
vK
Determining Orbital Elements
• An important thing to remember is that h is
a vector perpendicular to the plane of the
orbit. The node vector is defined as. n  K h
• Thus:
I
J
K
n 0
hI
0
hJ
1  n I I  n J J  n K K  h J I  h I J
hK
Determining Eccentricity
• The eccentricity vector is just a function of
the gravitational parameter m and the r and
v vectors
3
16 ft
• For the Earth m  1.407646882 10
2
sec

1  2 μ 
e   v  r  r v v
μ 
r

e e
Determining Semi-major Axis
• The equation for the semi-major is a
function of the velocity and radius vectors
along with the gravitational parameter m
2 v
a   
r μ
2
• If e=1, a=inf.



1
Determining Inclination
• Since the inclination is the angle between K
and h, the inclination can be found using the
formula:
hk
cos(i) 
h
• Inclination is always between zero and pi.
Determining RAAN
• Since the Right Ascension of the Ascending
Node is the angle between I and n, the
inclination can be found using the formula:
nI
cos( W) 
n
• RAAN is always between pi and two pi.
Determining Argument of Perigee
• Since the Argument of Perigee is the angle
between n and e, the inclination can be
found using the formula:
n e
cos(w ) 
ne
• Argument of Perigee is always between
zero and pi.
Determining True Anomaly
• Since the True Anomaly is the angle
between e and r, the inclination can be
found using the formula:
e r
cos(vO ) 
er
Classes Of Orbits
• Types of rotation
– Prograde
– Retrograde
– Polar
• Types Of Orbital Geometry
–
–
–
–
Elliptical
Circular
Parabolic
Hyperbolic
Prograde
• The Prograde or direct
orbit moves in
direction of Earth's
rotation
• 0o<i<90o
Retrograde
• The retrograde or
indirect moves against
the direction of Earth's
rotation
• 90o<i<180o
Polar
• Direct orbit over north
and south pole
• i=90o
Elliptical
• Eccentricity, 0<e<1
• Semi-major Axis, rp<a<ra
• Semiparameter, rp<p<2rp
Circular
• Eccentricity, e=0
• Semi-major Axis, a=r
• Semiparameter, p=r
Parabolic
• Eccentricity, e=1
• Semi-major Axis, a=inf
• Semiparameter, p=2rp
Hyperbolic
• Eccentricity, e>1
• Semi-major Axis, a<0
• Semiparameter, p>2rp
Two-line Element Sets
• One of the most commonly used methods of
communicating orbital parameters is the Two-line element
sets generated by NORAD. It is important to note that
TLEs were developed for use only with the MSGP-4
propagator. Using TLEs with any other propagator may
invalidate some of the built-in assumptions.
• These elements contain most of the same elements as the
classical orbital elements, along with some additional
parameters for identification purposes and for use in
modeling perturbations in the MSGP-4 propagator.
TLEs
• TLEs contain 12 different variables
– Six for the Classical Orbital Elements
• Four actual C.O.E.s: e, i, W and w
• Two variables that can be used in place of C.O.E.:
M, Mean motion and n, mean anomaly
– Three to describe the effects of perturbations on
n
n
satellite motion: Bstar, 2 and 6
– Two for identification purposes
– One for the time when this data was observed
TLE format
• The following is an example of a Two-line
Element set.
1 1 6 6 0 9 U
8 6 0 1 7 A
9 3 3 5 2 . 5 3 5 0 2 9 3 4
. 0 0 0 0 7 8 8 9
2 1 6 6 0 9
5 1 . 6 1 9 0
1 3 . 3 3 4 0
1 0 2 . 5 6 8 0
0 0 0 5 7 7 0
0 0 0 0 0
0
2 5 7 . 5 9 5 0
1 0 5 2 9 - 3
3 4
1 5 . 5 9 1 1 4 0 7 0 4 4 7 8 6
• This Format looks rather intimidating and is
read the following way
Satellite
Number
1 1 6 6 0 9 U
International
Designator
8 6 0 1 7 A
Inclination
2 1 6 6 0 9
5 1 . 6 1 9 0
n
2
Epoch
Y Y DDD . DDDDDDDD
9 3 3 5 2 . 5 3 5 0 2 9 3 4
Right Ascension
of node
1 3 . 3 3 4 0
. 0 0 0 0 7 8 8 9
n
6
0 0 0 0 0
Eccentricity
Argument of
perigee
Mean Anomaly
0 0 0 5 7 7 0
1 0 2 . 5 6 8 0
2 5 7 . 5 9 5 0
Bstar
0
1 0 5 2 9 - 3
Element
Number
3 4
Mean Motion
1 5 . 5 9 1 1 4 0 7 0 4 4 7 8 6
TLE Classical Orbital Elements
• The two-line element sets provide four of
the classical orbital elements : e, i, W and w.
• Instead of true anomaly the TLE gives the
mean anomaly because it can be calculated
at future time easier.
• This is also true for the substitution of
mean motion for semi-major axis which
will be explained on the next slide.
Mean Motion to Semi-major Axis
• n=15.5911407 revolutions/day
– n=5612.81065 degrees/day
• 1 day=107.088278 TU
– n=52.4129 degrees/TU
• 1 radian=57.2957795 degrees
– n=.9147782342 radians/TU
• a=
m 
 2
n 
1
3
– a=1.061180 ER
• 1 ER=6378.1363 km
– a=6768.357 km
TLE Perturbations Effects
• The three perturbation effects in the TLE’s
are mean motion rate, mean motion
acceleration and B* a drag parameter
• The ballistic coefficient, BC, can be found
from B*
TLE Identification Purposes
• The Satellite number
• The International Designation tells us the
year of the satellite launch, launch number
of year and section
– For this satellite is 86017A, that means it was
the 17th launch of 1986 an it was the A section.
TLE Time
• The epoch is what time the values were recorded
– The Time give was 93352.53502934
– This Translates to the 352nd day of 1993 which was
December 18.
– To find the Hours, minutes and seconds just take the
remainder divide by 24 to get the hours, take the
remainder of that divide by 60 to get the minutes and
take the remainder of that divide by 60 to get the
seconds
– This should translate to 12 h. 50 min. and 26.535 sec.