1. MISSION OPERATIONS DIRECTORATE SPACE FLIGHT TRAINING DIVISION
2.6 Basic Orbit Mechanics
September 24, 2004
Copyright © 2004 by United Space Alliance, LLC.  These materials are sponsored by the National Aeronautics and Space Administration under Contract NAS   The U.S. Government retains a paid-up, nonexclusive, irrevocable worldwide license in such materials to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the U.S. Government.  All other rights are reserved by the copyright owner.

2. Basic Orbital Mechanics: Lesson Objectives
As a result of the basic orbital mechanics presentation, the phase I student will be able to do the following:
Describe the characteristics of an orbit using the cannonball analogy
State that the shuttle flies in elliptical orbits
Identify parabolic & hyperbolic orbits as escape trajectories
Match the definition statement for each of the following terms:
· Perigee · Apogee · Semimajor Axis
· Eccentricity · Ascending Node · Descending Node
· Inclination · Line of Apsides · Beta Angle
· Orbital Period · Flight Path Angle · Nodal Regression

3. Basic Orbital Mechanics: Lesson Objectives
Describe the relationship between an orbital period and orbital altitude
Describe the relationship between orbital altitude and orbital velocity
Describe orbital velocity behavior as an orbiting object approaches apogee & perigee
Identify the characteristics of posigrade, retrograde, radial in, and radial out burns
Describe a Hohmann transfer
Describe the primary constraint on changing orbital plane

4. Basic Orbital Mechanics: Lesson Objectives
Recognize the restrictions that launch site latitude places on achievable orbital inclination
Describe the basic components of a launch window graph
Identify the rationale behind launching east from KSC
State the rationale for the restriction on launch azimuth
Describe why the groundtrack of the shuttle traces a sinusoid across a flat map of the Earth
Describe the cause & effect of two possible orbit perturbations (atmospheric drag & nonsphericity of the Earth)

5. What is an Orbit? The key concept of the CANNONBALL ANALOGY:
The ____________ of the falling ball’s __________ matches the ______________ of the Earth’s ________________.
PROPERTIES of an ORBIT:
____________ is INERTIALLY STABLE
Motion is _______________
Freefall = “ZERO-G”

6. The Three Basic Types of Conic Orbits
“In orbit”
“Escape or fly-by”
“Minimum Escape Trajectory”

7. Orbital Elements: Size, Shape, Orientation, & Location
Semi-major Axis
The semi-major axis determines the ______ of an orbit
________ is the closest point of approach to Earth
________ is the farthest point on the orbit from Earth
Eccentricity determines the ________ of an orbit
Time since Perigee passage can provide the ________ of the spacecraft in a given orbit a
Line of Apsides
e = 0
e = 0.88
e = 0.99

8. Orbital Elements: Continued
X & Y lie in the __________
Z is the _____________ X Y Z
M50 Inertial Reference Frame X Y Z i
Plane of Equator
Inclination is the angle between the _________________ and the orbit ________.
Inclination

9. Orbital Elements: ContinuedX Y Z W
The Right Ascension of the Ascending Node (W) is the angle between the M50 _______ and the orbit’s _________________
Perigee Z w
Line of Nodes
The Argument of Perigee (w) is the angle between the ____________ node and the _________ of the orbit Y X
Ascending Node

10. Orbital Elements: Concluded
The six orbital elements define the _____, ______, and ____________ of the orbit and the __________ of the spacecraft on the orbit
Perigee Z t p w
Plane of Equator a i Y X W e
RECORD SHUTTLE VALUES:
a = 3781 nm (h = 337 nm) (STS-82)
e = 0.03 (313x95 nm, STS-109)
28.35° < i < 62°
(STS-49) (STS-36)
Ascending Node

11. Orbital Period vs. Orbital Altitude3
m = Gravitational Parameter T = 2 p m
h = 160n.mi
T = 90 minutes
“High” Earth Orbit
H = 3444 n.mi
T = 4 hours
Geosynchronous Orbit
h = 19,324 n.mi
T = 23 h 56 m 4 s
As orbital altitude increases,
orbital period ___________.

12. Orbital Velocity vs. Orbital Altitude Circular Orbitsm
V = a
h = 160n.mi
V = 25,300 ft/s
“High” Earth Orbit
H = 3444 n.mi
V = 18,341 ft/s
Geosynchronous Orbit
h = 19,324 n.mi
V = 10,087 ft/s
As orbital altitude increases,
orbital velocity __________.

13. Orbital Velocity vs. Orbital Altitude (Elliptical Orbits)1
V = m 2 r a
r = Radial Vector from Center of Earth
h = 19,324 n.mi
V = 5,273 ft/s
h = 160 n.mi
V = 33,320 ft/s
Geosynchronous Transfer Orbit
a = 13,186 n.mi
e = 0.726
On an elliptical orbit, velocity is greatest at ________
and least at ________.

14. Flight Path Angle
Orbital
Direction
Local Horizontal -g
Velocity Vector
Local Vertical
Local Horizontal
Local Vertical +g
Velocity Vector
Flight Path Angle is the angle between an orbiting spacecraft’s
________ vector & the local __________.

15. Changing Orbits - The Effects of Burns Posigrade & Retrograde
Orbital
Direction æ 2 1 ö V = m ç - ÷ è r a ø Dh DV DV Dh
A __________ burn will RAISE orbital altitude (max. at 180° from the burn point); a ___________ burn will lower orbital altitude (max. at 180° from the burn point).

16. Changing Orbits - The Effect of Burns Radial In & Radial Out( ) 1
V = m 2 r a
EXAMPLE: Radial In Burn at Perigee
Radial In Burn
Resultant Velocity
Initial Velocity
Radial burns shift the _______________ without significantly altering other orbital parameters

17. Orbital Transfers - The Hohmann
Direction
Burn 2
Circular to circular
Coplanar
Two burns
Least expensive energy wise
Orbit 1
Burn 1
Lowering orbits via two
retrograde burns can
also be accomplished.
Orbit 2

18. Orbital Transfers - Changing PlanesV2 DV V1
Burn point must be intersection of two orbits (“nodal crossings”)
Extremely expensive energywise:
For 160 nmi circular orbits, a 1° of plane change requires a DV of over 470 ft/sec.

19. Orbital Transfers - Going Uphill and Changing Planes
Transfer Orbit
Example:
Transfer of
Comsat to
Geosynchronous
orbit.
Dihigh
Orbit 2
i = 0.0
Orbit 1
i = 28.5
Dilow
Line of Nodes
Given h1 = 160 nmi & h2 = 19,300 nmi:
MINIMUM TOTAL DV achieved with
Dilow = 2.2°
Di high = 26.3°

20. Orbital Transfers - Lambert Targeting2 1
Permits 3-dimensional transfers between two arbitrary orbits in a fixed time.
Given : the start position (R1),
the final position (R2),
and
the time of transfer (DT) 2 1 R2 R1 DT
Lamberts Theorem computes the
___________ ORBIT from R1 to R2. 2
Used in Rendezvous. 1

21. Ground Matters - Launch Site Latitude & Inclination
Equator
Launch Site
Latitude
Orbital Plane
(Minimum Inclination)
(Higher Inclination)
Minimum Inclination
Time to Launch!
Desired
Orbit
Minimum achievable INCLINATION = Launch Site __________

22. Launch Windows Driven By:
Lighting (Launch/ Abort / Landing Satellite Deploy/Retrieve)
Rendezvous (Lighting, Planar Geometry, Phasing)
Planetary Geometry (For Deep Space Missions)
Ground Target Conditions (Lighting, Overflight Timing)
Crew Fatigue Limitations

23. Launch Windows Driven By:
Lighting (Launch/ Abort / Landing Satellite Deploy/Retrieve)
Rendezvous (Lighting, Planar Geometry, Phasing)
Planetary Geometry (For Deep Space Missions)
Ground Target Conditions (Lighting, Overflight Timing)
Crew Fatigue Limitations

24. Ground Matters - Launch Azimuth
True North = 0°
Measured +clockwise
Due East = ____
Due South = ____

25. Ground Matters - Launch Azimuth Determination
From KSC (Latitude = 28.5°) N
Orbit 1
Orbit 1 (i = 57°)
Azimuth = 38.3° (or 141.7°)
Azimuth 2
Orbit 2 (i = 35°)
Azimuth = 68.8° (or 111.2°)
Orbit 2

26. Ground Matters - Launch Azimuth Restrictions
East Coast Limit:
_____ (i = 57°)
Launch Azimuth is
restricted due to
_______ concerns.
Operational Southern Limit:
90° (i = _____)
Caribbean Islands Limit:
_____ (i = 39°)

27. Ground Matters - The Rotating EarthV
Launch
Site
KSC’s easterly velocity is ~ 1000 mph
(1347 ft/sec, or 5.3% of orbital velocity) V
Launch
Site
Launching directly east yields
the maximum payload-to-orbit capability. R
Launch
Site V
Orbital

28. Ground Track
Nodal Shift
Maximum
Latitude
Ground track shifts about 22° west on each orbit as the Earth rotates beneath.
Maximum latitude = Inclination
Nodal Shift = function of orbit size (a)
North/South Symmetry = function of eccentricity & perigee location

29. Orbit Perturbations - Atmospheric Drag
BOOM!
Atmospheric density is a function of latitude, solar heating, season, land masses, etc.
Drag also depends on spacecraft attitude
Lower apogee of elliptical orbit
Perigee remains relatively constant, like a retrograde burn each orbit
Once Ha = Hp, relatively constant drag causes steady altitude loss

30. Orbit Perturbations - Gravitational Potential[ ] U r J a P C m S n e nm = - é ë ê ù û ú + ì í î ü ý þ å f l 1 2
(sin( ))
cos( )
sin(
where
Spherical Term
m = Universal Gravitational Constant X Mass of Earth
r = Spacecraft Radius Vector from Center of Earth
ae = Earth Equatorial Radius
P() = Legendre Polynomial Functions
f = Spacecraft Latitude
l = Spacecraft Longitude
Jn = Zonal Harmonic Constants
Cn,m,Sn,m = Tesseral & Sectorial Harmonic Coefficients
Zonal Harmonics
Only the first 4X4 (n=4, m=4) elements are used in shuttle GNC software.
J2 has 1/1000th the effect of the spherical term;
all other terms start at 1/1000th of J2’s effect.
Tesseral Harmonics
Sectorial Harmonics
Just know that the higher order terms are present and the software accounts for them.

31. Orbit Perturbations - “The J2 Effect”
The Earth’s oblateness causes the most significant perturbation of any of the nonspherical terms. h
orbit h J2
Right Ascension of the Ascending Node, Argument of Perigee and Time since Perigee Passage are affected.
100
300
200
500
1000 20 30 40 50 60 70 80 90 10
Typical
Shuttle
Orbits
Nodal Regression is the most important operationally.
-6.7 39
Magnitude depends on orbit size (a), shape (e) and inclination (i).
Posigrade orbits’ nodes regress
Westward (0° < i < 90°)
Retrograde orbits’ nodes regress
Eastward (90° < i <180°)
180
170
160
150
140
130
120
110
100
Inclination, Degrees

32. Northern Hemisphere Over Flight Is Always In Darkness
Solar Beta Angle
Plane of Orbit
Sun
The angle between the Orbit’s b
and
the Line of Sight to the
(POSITIVE is TOWARDS the Angular Momentum Vector)
(The Russian Program uses a angle, where a = -b)
Primary concerns with b angle are THERMAL ENVIRONMENT
and Solar Cell Power Generation Capability
Northern Hemisphere Over Flight Is Always In Darkness
Constant Daylight
Combinations of b angle and inclination result in some interesting situations

33. ORBITAL MECHANICS QUESTIONS
What is the “KEY PHRASE” of the cannonball analogy? ___________________________ _________________________________________________
What kind of orbit does the shuttle fly in? ________________________
What is the closest point of approach on an orbit? ____________ The furthest point? ____________
The size of an orbit is defined by _________________________________
The shape of an orbit is defined by _______________________________
The angle between the plane of the equator and the plane of the orbit is ______________________
When the shuttle crosses the equator, it is passing through a ______________________
As orbital altitude increases, orbital period _______________________
As orbital altitude increases, orbital velocity _________________________
In an elliptical orbit, velocity is greatest at_____________and least at __________________
A posigrade/retrograge burn _____________________ your velocity and _____________________ your altitude on the other side of the earth.
The cheapest orbital transfer is the___________________transfer.

34. ORBITAL MECHANICS QUESTIONS
Orbital inclination cannot be lower than launch site __________________
Name three possible constraints to time of launch: _________________________________ __________________________________________________________________
Launch azimuth is restricted due to ____________________concerns.
 Launching east increases ____________________________ performance because of _____________________.
The groundtrack of the shuttle is shaped like a _____________________________
 On a groundtrack, maximum latitude equals _________________________
 The groundtrack of a shuttle shifts westward on each revolution because the _______________________________
 Orbital decay is caused by _________________________.
The angle between the orbit’s plane and the Line of Sight to the Sun is defined as ___________.

35. ORBITAL MECHANICS ANSWERS
The curve of the falling ball’s trajectory matches the curve of the Earth’s surface.
Elliptical
Perigee, Apogee
the semi-major axis (a).
the eccentricity (e).
the inclination (i).
node.
increases.
decreases.
perigee ,apogee.
increases/decreases , increases/decreases
Hohmann
latitude.
lighting, rendezvous (lighting, plane geometry, phasing, etc), planetary geometry, ground target conditions, crew fatigue limitations, etc.
safety  
payload carrying , Earth’s rotation.
sinusoid.
inclination.
Earth rotates beneath the orbit.
atmospheric drag.
Beta angle.

36. Change History Rev. Letter Description Date Basic Baseline issue
05/31/2003
CPN-1
Inserted copyright statement to cover slide
06/22/2004 A
Updated names, trajectory records, various cosmetic fixes
9/24/2004