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

Primary Reference Stanford Spacecraft Design Course (AA236A) Bob Twiggs

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 http://www.braeunig.us/space/orbmech.htm Robert A. Braeunig, 1997, 2005

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

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?

Orbiting Bodies

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

Orbits are Conic Sections

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

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…

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

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

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)

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

Standard Gravitational Parameter For Circular Orbits Simpler Generalization for Elliptical Orbits

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

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

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?

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

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

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

Kepler’s First Law

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

Kepler’s Second Law at Work

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

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

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.

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

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.

Geostationary Orbits

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

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

Satellite Locations

Satellite Locations

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

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

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

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

Transfer Orbits

Getting into Orbit: Hohman Transfer

Orbital Trajectories

Orbital Trajectories

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

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

Orbital Decay

Ultimate Fate

NASA’s J-Track 3-D http://science.nasa.gov/Realtime/JTrack/3d/JTrack3D.html