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Satellite Orbits An Introduction to Orbital Mechanics

Satellite Orbits An Introduction to Orbital Mechanics. KySat Summer Workshop Morehead, KY June 11, 2007. Primary Reference Stanford Spacecraft Design Course (AA236A) Bob Twiggs. Additional References. Chapters 6&7 Other Texts Introduction to Space Dynamics William Tyrerrell Thomson

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Satellite Orbits An Introduction to Orbital Mechanics

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  1. Satellite OrbitsAn Introduction to Orbital Mechanics 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 http://www.braeunig.us/space/orbmech.htm 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

  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? 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 Parameterand 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

  18. 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 At this velocity, how long does it take for one revolution?

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

  20. Orbital Mechanics and Parameters To investigate orbital mechanics and orbital parameters, and ultimately determine TLEs, it is necessary to understand Kepler’s Laws

  21. 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

  22. Kepler’s First Law

  23. 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

  24. Kepler’s Second Law at Work

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

  26. Kepler’s Third Law(Harmonic Law)

  27. 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.

  28. 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

  29. 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.

  30. Geostationary Orbits

  31. 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

  32. 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

  33. Satellite Locations

  34. Satellite Locations

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

  36. Keplerian Elements 1 27651U 03004A 07083.49636287 .00000119 00000-0 30706-4 0 2692 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

  37. Keplerian Elements 1 27651U 03004A 07083.49636287 .00000119 00000-0 30706-4 0 2692 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249 Line 1 Catalog No. 8 1 Security Classification 10 8 International Identification 19 14 YRDOY.FODddddd 34 1 Sign of first time derivative 35 9 1st Time Derivative 45 1 Sign of 2nd Time Derivative 46 5 2nd 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 53 11 Revolutions per Day (Mean Motion) 64 5 Revolution Number at Epoch 69 1 Check Sum Modulo 10

  38. 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

  39. Transfer Orbits

  40. Getting into Orbit: Hohman Transfer

  41. Orbital Trajectories

  42. Orbital Trajectories

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

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