1. Basic Orbital Mechanics
Jeff Crum
23 Aug 01
9/19/2018

2. Introduction
Orbital mechanics is the study of the motions of artificial satellites and space vehicles moving under the influence of forces such as gravity, atmospheric drag, thrust, etc.
Modern offshoot of celestial mechanics, the study of the motions of the moon, planets, and stars
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3. Training Topics Kepler and Newton Orbital Elements Ground Traces
Orbit Types
Orbit Perturbations
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4. Planetary motion
The problem of accurately describing the motions of planets has challenged observers for centuries
Early theories held that the Earth was the center of the universe and all other heavenly bodies traveled in perfect circles about the Earth
In 1543, Nicolaus Copernicus published his heliocentric theory that postulated that the Sun was the center of the universe, and that all planets—including the Earth, revolved about it in perfect circles
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5. Kepler
Johannes Kepler agreed with Copernicus’ revolutionary and highly controversial theory
Using thousands of astronomical observations from the Danish astronomer Tycho Brahe, Kepler tried in vain to fit the motion of the planet Mars to a circular orbit
Kepler finally discovered the answer: planets travel not in circles, but ellipses about the sun
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6. Ellipse
GEOMETRIC CENTER
Semi-Minor Axis (b)
FOCUS
FOCUS
Semi-Major Axis (a)
Focal Length (c)
MAJOR AXIS
MINOR AXIS
Based on the results of his observations, Kepler published his three laws of planetary motion
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7. Kepler’s First Law
The orbit of each planet is an ellipse, with the Sun at one focus
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8. Kepler’s Second Law
The line joining the planet to the Sun sweeps out equal areas in equal times
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9. Kepler’s Third Law
The square of a planet’s orbital period is in direct proportion to the cube of the semi-major axis
Orbital period (P) = time required to make one complete revolution around the sun
P2  a3
For example, Mercury, the closest planet to the Sun, completes an orbit in 88 days
Pluto, the furthest planet from the Sun, completes an orbit every 248 years
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10. Newton
Building upon the work of Kepler and others, Isaac Newton put forward his laws of motion and formulated his law of universal gravitation
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11. Newton’s Laws of Motion
First Law: A body at rest remains at rest, and a body in motion continues to move at a constant velocity unless acted upon by an external force
Second Law: A force (F) acting on a body gives it an acceleration (a) which is in the direction of the force and has a magnitude inversely proportional to the mass of the body (m)
F = ma
Third Law: Whenever a body exerts a force on another body, the latter exerts a force of equal magnitude and opposite direction on the former
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12. Newton in simpler terms
Objects in motion want to travel in straight lines at constant speed (Newton’s first law)
But…
Force of gravity causes the path to curve
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13. Newton in simpler terms
Amount of curve depends on initial speed and direction
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14. Newton in simpler terms
Satellites must have a balance of…
Speed + Gravity = No orbit Orbit No Orbit
(too slow)
(too fast)
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15. Satellite orbits
The initial speed and direction of an orbiting satellite creates an ellipse with the Earth at one focus
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16. Satellite orbits Apogee = highest altitude, lowest speed
Perigee = lowest altitude, fastest speed
Perigee
Apogee
Note: In generic terms, periapsis is the point in an orbit closest to the primary focus, and apoapsis is the farthest point. These terms are usually modified to apply to the body being orbited (e.g., perihelion/aphelion for the Sun, perigee/apogee for the Earth, perijove/apojove for Jupiter, etc.)
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17. Satellite Orbits
Orbit size determines satellite period, or time to make one orbit
DMSP
510 miles
110 minutes
GPS
10900 miles
11 hrs, 58 min
DSP
22300 miles
23 hrs, 56 min
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18. Satellite Orbits Orbital element set Semi-Major Axis……………………….. Size
Eccentricity………………………………. Shape
Inclination……………………………….. Tilt
Right Ascension of the
Ascending Node……………………….. Direction
Argument of Perigee…………………. Rotation
True Anomaly…………………………… Position
Epoch Time………………………….…… Time Stamp
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19. Semi-Major Axis (Size)
Semi-Major Axis (a)
FOCUS
PERIGEE
APOGEE
Focal Length (c)
Focal Length (c)
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20. Eccentricity (Shape) e = .75 e = .45 e = 0 Eccentricity (e) = c/a a 
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21. Inclined Prograde Orbit
Inclination (Tilt)
Inclination Angle
0 to < 90 
Equatorial Plane
Ascending Node
Inclined Prograde Orbit
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22. Inclined Retrograde Orbit
Inclination (Tilt)
Inclination Angle
>90 to < 180 
Equatorial Plane
Ascending Node
Inclined Retrograde Orbit
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23. Right Ascension of Ascending Node (Direction)
Inclined Prograde Orbit
0 = First Point of Aries 0
180
Ascending Node
Line of Nodes
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24. Right Ascension of Ascending Node (Direction)5
GPS constellation has 6 orbit planes, all inclined at 55° 1 6
Each orbit plane is spaced 60° from the previous plane 2 4 3
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25. Argument of Perigee (Rotation)
Argument of Perigee is the angular distance between the ascending node and the point of perigee
Perigee
Equatorial Plane
Ascending Node
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26. True Anomaly
True Anomaly is the angular distance between perigee and some point in the orbit. Usually this is used to describe where an SV is in the orbit at a given time.
Equatorial Plane
Perigee
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27. Epoch Time
The epoch time is an exact specification of the date and time at which a given Keplerian element set is valid
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28. Ground Trace
A ground trace is the projection of a satellite’s orbit onto the earth’s surface
Note that the highest northern and southern latitudes reached by a satellite’s ground trace is equal to the satellite’s orbital inclination
If inclination > 90°, then highest latitude of the ground trace is 180° minus the orbit inclination
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29. Ground Trace
Consider an orbit of a satellite to lie in a plane that passes through the center of a theoretically spherical Earth
The trace of this plane on the surface of a non-rotating Earth is a great circle
If the Earth did not rotate, the satellite would retrace the same ground over and over
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30. Ground Trace
If we consider a rotating Earth, the orbital plane of a satellite remains fixed in space as the Earth turns under it
Effect of Earth’s rotation is to displace the ground trace on each successive revolution of the satellite
Ground trace displaced by the number of degrees the Earth rotates during on orbital period
This displacement is called nodal regression
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31. Nodal Regression
Earth’s rotation causes the ground trace to regress westward for each successive orbit
Earth rotates approximately 15° per hour
Nodal regression = (orbit period in hours) * 15°
A satellite in a polar orbit has the potential to overfly all the Earth’s surface
If the time required for one complete rotation of the Earth is an exact multiple of the satellite’s period, then eventually the satellite will retrace exactly the same path as it did on some previous revolution
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32. GROUND TRACES Westward Regression
Pictures and animation courtesy of Capt Troy Endicott, Det TRS, Schriever AFB, AETC

33. Orbit Types
Satellites use a wide variety of orbits to fulfill their missions
Factors determining orbit type:
Mission requirements
Booster capability and cost
Satellite and orbital mechanics
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34. Orbit Types
For a spacecraft to achieve orbit, it must be launched to an elevation above the Earth’s atmosphere and accelerated to orbital velocity
The most energy efficient orbit is a direct, low inclination orbit
To achieve this orbit, the spacecraft is launched in an eastward direction from a site near the Earth’s equator
The rotational speed of the Earth contributes to the spacecraft’s final orbital speed
e.g. A due east launch from the Cape (28.5 deg north latitude) results in a “free ride” of 915 mph
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35. Orbit Types
Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an orbit of higher inclination
High inclination orbits are less able to take advantage of the initial speed provided by the Earth’s rotation
Launch vehicle must provide greater energy to attain orbital velocity
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36. Safety constrains possible launch azimuths
The desired orbit inclination determines the azimuth of launch.
Vandenberg AFB
Patrick AFB
Safety constrains possible launch azimuths
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37. Orbit Types Low Earth Orbit (LEO) Medium Earth Orbit (MEO)
Sun Synchronous
Polar
Medium Earth Orbit (MEO)
Geosynchronous/Geostationary Earth Orbit (GEO)
Highly inclined orbits
Molniya
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38. Low Earth Orbits Up to 520 miles Common missions: Manned (shuttle)
Reconnaissance
Communications
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39. LEO
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40. Polar Orbits Inclination of 90 degrees Missions: Mapping Surveillance
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41. Sun Synchronous 460-520 miles Near-polar inclination
Orbital plane precesses with the same period as the Earth’s solar period
Satellite crosses perigee at the same local time every orbit
Common missions:
Earth sensing (LANDSAT)
Weather (DMSP/NOAA)
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42. Sun Synchronous
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43. Medium Earth Orbit Also called semi-synchronous
10,900 miles; high inclination
Missions:
Navigation (GPS, GLONASS)
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44. MEO
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45. Geosynchronous and Geostationary
Altitude = 22,300 miles
Period = 24 hours
Any inclination
Does not have to be circular
Geostationary
Inclination near zero
Eccentricity nearly zero
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46. GEO
GEO spacecraft appear to hang motionless above one position on the Earth’s surface
Missions:
Communications
Weather
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47. GEO
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48. Highly Elliptical Orbits
degree inclination
200-23,800 mile altitude
Missions:
Comm relay (Molniya)
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49. Molniya Orbits
Many Russian cities are at high northern latitudes where it is impractical to use GEO satellites for telecommunications
GEO satellites appear either low on the horizon or are not visible at all
Molniya orbit has a 12 hour period at high eccentricity and inclination (63.4°)
Satellite spends most of its time near apogee, so for approximately 11 hours of each orbit the satellite is above the horizon for high northern latitudes
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50. Molniya Orbit
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51. GROUND TRACES Highly Elliptical Orbit
Pictures and animation courtesy of Capt Troy Endicott, Det TRS, Schriever AFB, AETC

52. Orbit Perturbations
Any disturbance in the regular motion of a satellite resulting from a force other than those causing regular motion
Non-spherical earth
Atmospheric drag
Sun/moon gravity
Space environment
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53. Summary
TBD
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54. Backup Charts
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55. 9/19/2018