1. Colorado Springs Cadet Squadron Lt Col M. T. McNeely ORBITAL MECHANICS !! INTRO TO SPACE COURSE

2. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

3. ORIGINS Nicholas Copernicus F Revived Helio-centric model F Believed planetary orbits were circles

4. ORIGINS Tycho Brahe F Introduced precision into astronomical measurements F Mentor to Johannes Kepler

5. ORIGINS Johannes Kepler F Derived 3 laws based upon his observations of planetary motion

6. PHYSICAL LAWS Kepler’s 1st Law: Law of Ellipses The orbits of the planets are ellipses with the sun at one focus

7. PHYSICAL LAWS Ellipses FOCI Period (T) Semi-Major Axis (a) Semi-Minor Axis (b)

8. PHYSICAL LAWS

9. PHYSICAL LAWS Kepler’s 2nd Law: Law of Equal Areas The line joining the planet to the center of the sun sweeps out equal areas in equal times T6 T5 T4 T3 T2 T1 A2 A3A4 A5 A6 A1

10. PHYSICAL LAWS Kepler’s 2nd Law: Law of Equal Areas

11. t0t0 t3t3 t1t1 t2t2 Area 1 Area 2 t 1 -t 0 = t 3 -t 2 Area 1 = Area 2 Satellite travels at varying speeds

12. PHYSICAL LAWS Kepler’s 3rd Law: Law of Harmonics The squares of the periods of two planets’ orbits are proportional to each other as the cubes of their semi- major axes: T 1 2 /T 2 2 = a 1 3 /a 2 3 In English: Orbits with the same semi- major axis will have the same period

13. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

14. PHYSICAL LAWS Sir Isaac Newton F Derived three laws of motion F Derived the Law of Universal Gravitation F Explained why Kepler’s laws worked

15. PHYSICAL LAWS Newton’s 1st Law: Law of Inertia F Every body continues in a state of uniform motion unless it is compelled to change that state by a force imposed upon it

16. PHYSICAL LAWS Newton’s 2nd Law: Law of Momentum F Change in momentum is proportional to and in the direction of the force applied F Momentum equals mass x velocity F Change in momentum gives: F = ma F F

17. PHYSICAL LAWS Newton’s 3rd Law: Action - Reaction F For every action, there is an equal and opposite reaction F Hints at conservation of momentum

18. PHYSICAL LAWS Newton’s Law of Universal Gravitation Between any two objects there exists a force of attraction that is proportional to the product of their masses and inversely proportional to the square of the distance between them F g = G ( ) M1m2M1m2 D2D2

19. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

20. INJECTION REQUIREMENTS Speed

21. 5 m 8 km

22. INJECTION REQUIREMENTS Speed 100 miles 17,500 mi/hr

23. INJECTION REQUIREMENTS Altitude Are you moving FASTER or SLOWER the higher your altitude?

24. INJECTION REQUIREMENTS Altitude V C = G(m 1 +m 2 ) aV C = 5.59 km/s V C = 4.56 km/s 2E 3E

25. INJECTION REQUIREMENTS Altitude 2E V E = V C 2 = 7.91 km/s V C = 5.59 km/s V < 7.91 km/s V > 7.91 km/s

26. INJECTION REQUIREMENTS Direction

27. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

28. ORBIT CLASSIFICATION F Size/Period F Location F Shape

29. ORBIT CLASSIFICATION Size/Period F Defined by semi-major axis (a) F Low Earth Orbit (LEO) F High Earth Orbit (HEO) F Semi-synchronous Orbit F Geo-synchronous Orbit

30. ORBIT CLASSIFICATION Location F Equatorial F Polar

31. ORBIT CLASSIFICATION Shape (Conic Sections) Circle Ellipse

32. ORBIT CLASSIFICATION Shape (Conic Sections) Trajectories: Parabola Hyperbola

33. ORBIT CLASSIFICATIONS Circular Orbits F Characteristics – Constant speed – Nearly constant altitude F Typical Missions – Reconnaissance/Weather (DMSP) – Manned – Navigational (GPS) – Geo-synchronous (Comm sats)

34. ORBIT CLASSIFICATIONS Elliptical Orbits F Characteristics – Varying speed – Varying altitude – Asymmetric Ground Track F Typical Missions – Deep space surveillance (Pioneer) – Communications (Polar comm.) – Ballistic Missiles

35. ORBIT CLASSIFICATIONS Parabolic/Hyperbolic Trajectories F Characteristics – Escaped Earth’s gravitational influence – Heliocentric F Typical Missions – Interplanetary exploration (Galileo, Phobos, Magellan)

36. ORBIT CLASSIFICATIONS Orbit Geometry Apogee Perigee cc a Eccentricity = c/a

37. ORBIT CLASSIFICATIONS Eccentricity e = 0 0 < e < 1 e = 1 e > 1

38. ORBIT CLASSIFICATIONS Eccentricity e = 0 a c = 0 0 < e < 1 c a Eccentricity = c/a

39. ORBIT CLASSIFICATIONS Eccentricity Eccentricity = c/a e = 0.75 e =.45 e = 0

40. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

41. COORDINATE SYSTEMS F Defines positions and directions in a consistent manner -- allows communication F Facilitates the description of a satellite’s position and subsequent motion F Proper choice of reference determines the utility of a coordinate system

42. COORDINATE SYSTEMS Ordinates F Origin – Where you’re starting from F Fundamental Plane – Plane which you’re measuring in F Principle Direction – Direction which you’re measuring from

43. COORDINATE SYSTEMS Classifications F Inertial – Non-rotating – Time Independent F Non-inertial – Rotating – Time Dependent

44. COORDINATE SYSTEMS Examples F Geographic F Topocentric F Geocentric Inertial F Orbit Inertial

45. COORDINATE SYSTEMS Geographic F Purpose: To locate points on the Earth’s surface

46. COORDINATE SYSTEMS Topocentric F Purpose: To locate a satellite with respect to a specific point on the Earth

47. COORDINATE SYSTEMS Topocentric Elevation

48. COORDINATE SYSTEMS Topocentric Azimuth Range Origin: Antenna FP: Local Horizon PD: True North

49. COORDINATE SYSTEMS Geocentric Inertial F Purpose: To determine the exact orientation of an orbital plane and to locate points in space with respect to the Earth Vernal Equinox Equatorial Plane Ecliptic Plane

50. COORDINATE SYSTEMS Geocentric Inertial Inclination

51. COORDINATE SYSTEMS Geocentric Inertial Vernal Equinox Ascending Node Direction of Satellite motion Right Ascension

52. COORDINATE SYSTEMS Orbit Inertial F Purpose: To fix the satellite orbit in the orbital plane Argument of Perigee Ascending Node Perigee

53. COORDINATE SYSTEMS Review F Geographic – Locates a point on the Earth’s surface – Requires Latitude and Longitude F Topocentric – Locates a satellite with respect to a site – Requires Azimuth, Elevation, Range

54. COORDINATE SYSTEMS Review F Geocentric Inertial – Locates orbital plane with respect to the Earth – Requires Right Ascension and Inclination F Orbit Inertial – Locate orbit within orbital plane – Requires Argument of Perigee

55. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

56. ORBITAL ELEMENTS Definition F A set of mathematical parameters that enables us to accurately describe satellite motion

57. ORBITAL ELEMENTS Purpose F Discriminate one satellite from other satellites F Predict where a satellite will be in the future or has been in the past F Determine amount and direction of maneuver or perturbation

58. ORBITAL ELEMENTS Keplerian Elements F Semi-Major Axis (Size) F Eccentricity (Shape) F Inclination F Right Ascension F Argument of Perigee F Epoch Time (Location within orbit) – True Anomaly (Orientation)

59. ORBITAL ELEMENTS Keplerian Elements: Inclination Orbital Plane Equatorial Plane Inclination ( i )

60. ORBITAL ELEMENTS Keplerian Elements: Right Ascension i Line of Nodes Right Ascension of the Ascending Node (  ) First Point of Aries (  )

61. ORBITAL ELEMENTS Keplerian Elements: Argument of Perigee  i Line of Nodes  Argument of Perigee (  )

62. ORBITAL ELEMENTS Keplerian Elements: True Anomaly True Anomaly ( ) Direction of satellite motion

63. ORBITAL ELEMENTS Keplerian Elements: True Anomaly i Line of Nodes   

64. ORBITAL ELEMENTS Keplerian Elements: Inclination Equatorial: i = 0 or 180 Polar: i = 90 Prograde: 0  i < 90 Retrograde: 90  i ú 180

65. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

66. Ground Tracks Westward Regression - Earth rotates east under a satellite => satellite appears to walk west - Earth rotates 360 degrees in 24 hours (15 degrees per hour)

67. Ground Tracks Westward Regression 030-30-60-90-120 AB C A - time zero B - after one orbit C - after two orbits 60

68. Ground tracks Inclination 60 30 0 60 45N 45S Inclination = 45 degrees Eccentricity ~ 0

69. Ground Tracks Eccentricity Ground Track for Molnyia orbit eccentricity =.7252

70. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

71. PERTURBATIONS F Definition – A disturbance in the regular motion of a satellite F Types – Gravitational – Atmospheric Drag – Third Body Effects – Solar Wind/Radiation Effects – Electro-magnetic

72. PERTURBATIONS Gravitational: Libration F Ellipticity of the Earth causes gravity wells and hills F Stable points: 75E and 105W -- Himalayas and Rocky Mountains F Unstable points: 165E and 5W -- Marshall Islands and Portugal F Drives the requirement for stationkeeping

73. PERTURBATIONS Electro-Magnetic F Interaction between the Earth’s magnetic field and the satellite’s electro-magnetic field results in magnetic drag

74. ORBITAL MECHANICS Lesson 1 F Origins F Physical Laws F Requirements for Injection F Classifications of Orbits F Coordinate Reference Systems F Orbital Elements F Ground Tracks F Perturbations F Launch Considerations

75. LAUNCH CONSIDERATIONS Launch Windows F The period of time during which a satellite can be launched directly into a specific orbital plane from a specific launch site F Window duration driven by safety, fuel requirements, desired injection points, etc. F Window is centered around optimal launch time

76. PLACING SATELLITES IN ORBIT F Booster Types DELTA II

77. PLACING SATELLITES IN ORBIT F Booster Types ATLAS 2AS

78. PLACING SATELLITES IN ORBIT F Booster Types TITAN IV

79. PLACING SATELLITES IN ORBIT F Booster Types TAURUS

80. PLACING SATELLITES IN ORBIT F Booster Types The SHUTTLE BOOSTER

81. PLACING SATELLITES IN ORBIT F Booster Types PEGASUS

82. PLACING SATELLITES IN ORBIT F Launch Constraints