1. Satellite Orbits An Introduction to Orbital Mechanics
IBM 360 (1964) Uses: data encryption, nuclear research, weather modeling, CFD, other science, government comms (internet)
Dad: nuclear research
KySat Summer Workshop
Morehead, KY
June 11, 2007

2. Primary Reference
Stanford Spacecraft Design Course
(AA236A)
Bob Twiggs

3. Additional References
Chapters 6&7
Other Texts
Introduction to Space Dynamics
William Tyrerrell Thomson
Adverntures in Celestial Mechanics
Victor G. Szebehely
Wilkipedia
IBM 360 (1964) Uses: data encryption, nuclear research, weather modeling, CFD, other science, government comms (internet)
Dad: nuclear research
Robert A. Braeunig, 1997, 2005

4. Overview Importance of Orbital Parameters
Bound and Unbound Orbits– Conic Sections
Underlying Physics
Centripetal Force and Gravity
Circular and Escape Velocities
Orbit Equations
Types of Orbits
LEOs, MEOs, HEOs, GEOs, Molniyas, Lagrangian
Kepler’s Laws
TLEs
Orbits Established from Initial Conditions
Projected Paths
Orbit Perturbations
Orbital Decay
J-Track Orbit Simulator

5. Importance of Orbital Parameters to Mission
When should you start analyzing orbits to satisfy mission requirements?
Can the orbit effect any of the following in the mission design?
Revisit time of satellite to a point on earth?
Amount of data that can be transferred between the satellite and ground?
Space radiation environment?
Power generation for the satellite?
Thermal control on the satellite?
Launch costs?

6. Orbiting Bodies

7. What is an Orbit?
In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity
The CG for the system can be inside the parent body (satellite orbit) or in-between the two bodies (co-orbital bodies) orbiting about a barycenter

8. Orbits are Conic Sections

9. Eccentricity e Shape e = 0 Circle e < 1 Elipse e = 1 Parabola
Hyperbola

10. Eccentricity
In mathematics, eccentricity is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.
The eccentricity of a circle is zero.
The eccentricity of an (non-circle) ellipse is greater than zero and less than 1.
The eccentricity of a parabola is 1.
The eccentricity of a hyperbola is greater than 1 and less than infinity.
The eccentricity of a straight line is 1 or ∞, depending on the definition used
Where    is the length of the semimajor axis ,   the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola…
But…there is an easier way to conceptualize eccentricity…

11. Why Do Satellites Orbit?
Earth’s Gravity Provides a Centripetal Force
Momentum is provided by Launch (maintained by Newton’s First Law)

12. Why Do Satellites Orbit?
IBM 360 (1964) Uses: data encryption, nuclear research, weather modeling, CFD, other science, government comms (internet)
Dad: nuclear research
Satellites “fall” constantly around the Earth
At about 5 mi/sec (18,000 mph) the “drop” of a horizontally thrown ball is exactly equal to the curvature of the Earth

13. Gravitational Force Every mass attracts every other mass
The force is proportional to the product of the two masses and inversely proportional to the square of the distance
F is the magnitude of the gravitational force between the two masses (Newtons)
G is the gravitational constant (6.67 × 10−11 N m2 kg−2)
r is the distance between the two masses (d1 + d2)

14. Standard Gravitational Parameter
Product of the gravitational constant G and the mass M:
The units of the standard gravitational parameter are km3s-2
=398,600.5 km3s-2

15. Standard Gravitational Parameter
For Circular Orbits
Simpler Generalization for Elliptical Orbits

16. Standard Gravitational Parameter and Orbital Speed
                                               
General Case
Elliptic orbit
          
                                           
Parabolic Trajectory
         
                        
Hyperbolic Trajectory                                                        
where:
      is the standard gravitational parameter
      is the distance between the orbiting body and the central body
   is the specific orbital energy
      is the semi-major axis

17. Orbital Speed Can Also Be Given by the Orbit Equation
                                               
Derived by Equating Universal Law of Gravity Equation to Newton’s Second Law

19. Sample Problem
Calculate the velocity of the space shuttle if its orbit is 1560 km. The mass of the earth is 6 x 1024 kg and its radius is 6.4 x 106 m
IBM 360 (1964) Uses: data encryption, nuclear research, weather modeling, CFD, other science, government comms (internet)
Dad: nuclear research
At this velocity, how long does it take for one revolution?

20. Circular and Escape Velocity
Orbital Velocity = 17,500 mph (29,000 klm/h)
Escape Velocity = 25,000 mph (41,000 klm/h)

21. Orbital Mechanics and Parameters
To investigate orbital mechanics and orbital parameters, and ultimately determine TLEs, it is necessary to understand Kepler’s Laws
IBM 360 (1964) Uses: data encryption, nuclear research, weather modeling, CFD, other science, government comms (internet)
Dad: nuclear research

22. Kepler’s Laws
The orbit of every planet is an ellipse with the sun at one of the foci
A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the planet travels along its orbit
The squares of the orbital periods of planets are directly proportional to the cubes of the major axis (the "length" of the ellipse) of the orbits

23. Kepler’s First Law

24. Kepler’s Second Law
A line joining a planet and the sun sweeps out equal areas during equal intervals of time
This is also known as the law of equal areas

25. Kepler’s Second Law at Work

26. Kepler’s Third Law (Harmonic Law)
Kepler’s Version
General Form
Useful Form (Kepler’s Law derived by Newton)

27. Kepler’s Third Law (Harmonic Law)
Orbit
Altitude above the Earth's surface
Speed
Period/time in space
Sub-orbital
100 km
0.0 km/s
just reaching space
ICBM
up to 1200 km
6 to 7 km/s
time in space: 25 min
LEO
200 to 2000 km
circular: 6.9 to 7.8 km/s elliptic: 6.5 to 8.2 km/s
89 to 128 min
Molniya
500 to 39,900 km
1.5 to 10.0 km/s
11 h 58 min
GEO
35,600 km
3.1 km/s
23 h 56 min
Moon
357,000 to 399,000 km
0.97 to 1.08 km/s
27.3 days

28. Keplerian Elements and Orbital Energy (Specific Orbital Energy or Vis-Viva Equation)
where
      is the orbital speed of the orbiting body;
      is the orbital distance of the orbiting body;
                 is the standard gravitational parameter of the primary body
    is the specific relative angular momentum of the orbiting body;
   is the orbit eccentricity
It is expressed in J/kg = m2s-2 or MJ/kg = km2s-2.

29. Types of Orbits LEO (Low Earth Orbit: 80 km – 2,000 km
MEO (Medium Earth Orbit: 2,000 km – 35,786 km)
HEO (Highly Elliptical Orbit)
Polar (inclination near 90 deg, passes over the equator at a different longitude on each orbit
GEO (35,786 km or 22,236 statute miles)
Molniya (Highly Elliptical Orbit with inclination of 63.4° and orbital period of about 12 hours)
Lagrangian

32. Sun-synchronous Orbits
Combines altitude and inclination in such a way that an object on that orbit passes over any given point of the Earth's surface at the same local solar time.
The surface illumination angle will be nearly the same every time
The uniformity of sun angle is achieved by tuning natural precession of the orbit to one full circle per year.
Because the Earth rotates, it is slightly oblate and the extra material near the equator causes spacecraft that are in inclined orbits to precess;
The plane of the orbit is not fixed in space relative to the distant stars, but rotates slowly about the Earth's axis
The speed of the precession depends both on inclination of the orbit and also on the altitude of the satellite;
By balancing these two effects, it is possible to match a range of precession rates.

36. Geostationary Orbits

37. Molniya Orbits Class of a Highly Elliptical Orbit Inclination of 63.4°
Orbital period of about 12 hours.
Spends most of its time over a designated area of the earth, a phenomenon known as apogee dwell.
Named after a series of Soviet/Russian communications satellites since the mid 1960s

38. Lagrange Point or Libration Point Orbits
Five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects
The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to rotate with them

39. Satellite Locations

41. Satellite Locations

42. Keplerian Elements Inclination ( ) Longitude of the ascending node ( )
Argument of periapsis (    )
Eccentricity (  )
Semimajor axis (    )
Mean anomaly at epoch (      )

43. Keplerian Elements
U 03004A

44. Keplerian Elements
U 03004A
Line 1 Catalog No. 8 1 Security Classification 10 8 International Identification YRDOY.FODddddd 34 1 Sign of first time derivative st Time Derivative 45 1 Sign of 2nd Time Derivative nd Time Derivative 51 1 Sign of 2nd Time Derivative Exponent 52 1 Exponent of 2nd Time Derivative 54 1 Sign of Bstar/Drag Term 55 5 Bstar/Drag Term 60 1 Sign of Exponent of Bstar/Drag Term 61 1 Exponent of Bstar/Drag Term 63 1 Ephemeris Type 65 4 Element Number 69 1 Check Sum, Modulo 10
Line 2 Identification 3 5 Catalog No. 9 8 Inclination 18 8 Right Ascension of Ascending Node 27 7 Eccentricity with assumed leading decimal 35 8 Argument of the Perigee 44 8 Mean Anomaly Revolutions per Day (Mean Motion) 64 5 Revolution Number at Epoch 69 1 Check Sum Modulo 10

45. Getting Into Orbit
An orbit may be determined from the position and the velocity at the beginning of its free flight. A vehicle's position and velocity can be described by the variables r, v, and  , where r is the vehicle's distance from the center of the Earth, v is its velocity, and   is the angle between the position vector and the velocity vector

46. Transfer Orbits

47. Getting into Orbit: Hohman Transfer

48. Orbital Trajectories

49. Orbital Trajectories

50. Orbital Perturbations
Third Body Perturbations
Non-shperical Earth
Atmospheric Drag
Solar Radiation
Micrometeorites
Collisions with Space Debris

51. Chinese ASAT Test Debris Cloud
Third Body Perturbations
on-shperical Earth
Atmospheric Drag
Solar Radiation

52. Orbital Decay

53. Ultimate Fate

54. NASA’s J-Track 3-D