1. A Brief Introduction to Astrodynamics
Shaun Gorman
Iowa State University
Ames, Iowa

2. Topics Discussed Coordinate Systems Orbital Geometry
Classical Orbital Elements
Classes of Orbits
Two-line Element Sets

3. Coordinate Systems Heliocentric-Ecliptic Coordinate System
Geocentric-Equatorial Coordinate System
Right Ascension-Declination System
Perifocal Coordinate System

4. 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

5. Geocentric-Equatorial Coordinate System
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

6. Right Ascension-Declination System

7. 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

8. ECI Coordinate Systems
Several different types of ECI coordinate systems.
Fixed
J2000
B1950
TEME of Epoch
TEME of Date

9. 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.

10. 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.

11. 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 TDB.
Can use either Cartesian or COE.

12. 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 deg measured from the mean equinox) and corresponds to 31 December :09:07.2 or JD
Can use either Cartesian or COE.

13. 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.

14. 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.

15. 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

16. Orbital Geometry

17. 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)

18. C.O.E. (continued)

19. Vector Re-fresher
Before we start lets go over some basic vector math

20. 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.

21. 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

22. 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.

23. Determining Orbital Elements
The expression, which is called specific angular momentum, must be held constant due the law of conservation of angular momentum.
Thus:

24. 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.
Thus:

25. Determining Eccentricity
The eccentricity vector is just a function of the gravitational parameter m and the r and v vectors
For the Earth

26. 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
If e=1, a=inf.

27. Determining Inclination
Since the inclination is the angle between K and h, the inclination can be found using the formula:
Inclination is always between zero and pi.

28. 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:
RAAN is always between pi and two pi.

29. Determining Argument of Perigee
Since the Argument of Perigee is the angle between n and e, the inclination can be found using the formula:
Argument of Perigee is always between zero and pi.

30. Determining True Anomaly
Since the True Anomaly is the angle between e and r, the inclination can be found using the formula:

31. Classes Of Orbits Types of rotation Types Of Orbital Geometry Prograde
Retrograde
Polar
Types Of Orbital Geometry
Elliptical
Circular
Parabolic
Hyperbolic

32. Prograde
The Prograde or direct orbit moves in direction of Earth's rotation
0o<i<90o

33. Retrograde
The retrograde or indirect moves against the direction of Earth's rotation
90o<i<180o

34. Polar
Direct orbit over north and south pole
i=90o

35. Elliptical Eccentricity, 0<e<1 Semi-major Axis, rp<a<ra
Semiparameter, rp<p<2rp

36. Circular
Eccentricity, e=0
Semi-major Axis, a=r
Semiparameter, p=r

37. Parabolic Eccentricity, e=1 Semi-major Axis, a=inf
Semiparameter, p=2rp

38. Hyperbolic Eccentricity, e>1 Semi-major Axis, a<0
Semiparameter, p>2rp

39. 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.

40. 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 satellite motion: Bstar, and
Two for identification purposes
One for the time when this data was observed

41. TLE format The following is an example of a Two-line Element set.
This Format looks rather intimidating and is read the following way

42. 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.

43. Mean Motion to Semi-major Axis
n= revolutions/day
n= degrees/day
1 day= TU
n= degrees/TU
1 radian= degrees
n= radians/TU a=
a= ER
1 ER= km
a= km

44. 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*

45. 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.

46. TLE Time The epoch is what time the values were recorded
The Time give was
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 sec.