Orbital Mechanics 101

1. Orbital Mechanics 101 Jessie McCartney

2. Agenda • What is an orbit? • Types of Earth orbits • Launching • Space Environment • Orbit Perturbations • Attitude Determination • Maneuvering • Orbit Determination • Satellite End of Life

3. What is an orbit? • Orbits are the result of a perfect balance between the forward motion of a body in space, such as a planet or moon, and the pull of gravity on it from another body in space, such as a large planet or star. –Northwestern University.

4. What is an orbit? • An orbit is a regular, repeating path that one object in space takes around another one. An object in an orbit is called a satellite. A satellite can be natural, like the Earth or the Moon. It can also be man-made, like the Space Shuttle or the ISS.—NASA

5. What is an orbit? • In physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star--Wikipedia

6. Basic Orbit Equations • Circular Orbit Velocity: • Circular Orbit Period: • Escape Velocity (free-fall): • μ=G*M=gravitational parameter Escape velocity interactive

7. Kepler’s Three Laws • Planets move around the Sun in ellipses, with the Sun at one focus • The line connecting the Sun to a planet sweeps equal areas in equal times. • The square of the orbital period of a planet is proportional to the cube of the mean distance from the Sun. Kepler’s 1st and 2nd Law Kepler’s 3rd Law

8. Types of Earth Orbits • LEO (Low Earth Orbit) • ~520-1,500 km altitude • Orbital period ~ninety minutes • MEO (Medium Earth Orbit) • ~20,000 km altitude • Between LEO and GEO • Orbital period ~5-6 hours

9. Types of Earth Orbits • HEO (High Earth Orbit or Highly Elliptical Orbit) • ~40,000 km altitude (at perigee) • Large dwell time over one hemisphere • Orbital period ~12-24 hours • GEO (Geosynchronous or Geostationary Earth Orbit) • ~36,000 km altitude • Orbital period ~24 hours (matches Earth’s rotation)

10. LEO Low Earth Orbit (LEO): Orbiting at an altitude of 600-1,000 km. Path of Satellite 10

11. Example Ground Trace GN/MAE155B

12. Launch vehicles • The most common launch vehicles are rockets. They are referred to as Expendable Launch Vehicles (ELVs). • Other launch methods include air-launching like Pegasus.

13. Launch vehicle limitations • Latitude—Launch site • Mass / Power • Shape/Configuration • Space available • $$$$

14. Post-Launch: Space Environment Impacts to vehicle • Vacuumoutgassing, exposure • Debris • Magnetic fields • Solar radiation • Cosmic Rays • High Energy Particles • South Atlantic Anomaly • Single Event Upsets

15. Orbit Perturbations: Reality is More Complicated Than Two Body Motion

16. Sources of Orbital Perturbations • Several external forces cause perturbation to spacecraft orbit • 3rd body effects, e.g., sun, moon, other planets • Unsymmetrical central bodies (‘oblateness’ caused by rotation rate of body): • Earth: Requator = 6378 km, Rpolar = 6357 km • Space Environment: Solar Pressure, drag from rarefied atmosphere Reference: C. Brown, ‘Elements of SC Design’ GN/MAE155B

17. Relative Importance of Orbit Perturbations Reference: SpacecraftSystems Engineering, Fortescue & Stark • J2 term accounts for effect from oblate earth • Principal effect above 100 km altitude • Other terms may also be important depending on application, mission, etc... GN/MAE155B

18. Orbital Perturbation Effects: Regression of Nodes Regression of Nodes: Equatorial bulge causes component of gravity vector acting on SC to be slightly out of orbit plane This out of orbit plane component causes a slight precession of the orbit plane. The resulting orbital rotation is called regression of nodes and is approximated using the dominant gravity harmonics term, J2 GN/MAE155B

19. Orbital Perturbation: Rotation of Apsides  Rotation of apsides caused by earth oblateness is similar to regression of nodes. The phenomenon is caused by a higher acceleration near the equator and a resulting overshoot at periapsis. This only occurs in elliptical orbits. The rate of rotation is given by: GN/MAE155B

20. Atmospheric Drag • Along with J2, dominant perturbation for LEO satellites • Can usually be completely neglected for anything higher than LEO • Primary effects: • Lowering semi-major axis • Decreasing eccentricity, if orbit is elliptical • In other words, apogee is decreased much more than perigee, though both are affected to some extent • For circular orbits, it’s an evenly-distributed spiral

21. Attitude Determination and Drag Profile • Attitude determines which end is pointing forward and hence the drag profile • Deltas on yaw, pitch, and roll axes • Spin Stabilized (stable rotation about one axis) • Gyro Stabilized/Three axis stabilized using reaction wheels • Unstabilized (tumbling space rock)

22. Maneuvering Impacts to Orbital Position • Fuel calculation (mass of slosh-y tank) • Thruster operation • Orbital position impact and propagation of errors

23. Orbit Determination • Coordinate Systems • Keplerian • Cartesian • Ephemeris • Orbit Tracking

24. Kepler Elements • a is the semi-major axis of the orbit • e is the orbit’s eccentricity • i is the orbit’s inclination with respect to the central body’s plane • ω is the argument of perigee • Ω is the right ascension of the ascending node • ν is the spacecraft’s true anomaly.

25. Earth-Centered Cartesian Coordinates

26. Cartesian Elements • Earth-Centered Inertial (ECI)-- Inertial, in this context, simply means that the coordinate system is not accelerating (rotating). • Earth-Centered Fixed (ECF) or Earth-Centered Earth-Fixed (ECEF)– ECEF is a non-inertial system that rotates with the Earth. Its origin is fixed at the center of the Earth.

27. Ephemerides • Reference epoch (day and time) • Plus six constants of integration (initial conditions) • Keplerian (a, e, i, ω, Ω, v) elements for classical coordinate system • Cartesian coordinates (x, y, z, xdot, ydot, zdot) for ECI or ECF systems • Kepler elements are easier for humans to work with (spot orbital variations more quickly) • Cartesian elements are easier for computers to work with • Most modern systems will easily convert between the two

28. Ephemeris example

29. Orbit Tracking • Accuracy dependant on frequency of position updates • Depends on download speed, how long satellite is in view of ground station(s), readability of transmitted information • If equipped with GPSR (GPS receiver), the GPS position can be equated to “truth” and the change in satellite position determined from there

30. Orbit tracking for the ISS

31. Satellite End-of-Life • Controlled re-entry • Like launch, but much more stressful and in reverse • Don’t hit the humans! • Uncontrolled re-entry • Primarily used for vehicles that will burn up in the atmosphere, or that are no longer operable

32. Backup Slides • In no particular order…

33. Atmospheric Drag • Effects are calculated using the same equation used for aircraft: • To find acceleration, divide by m • m / CDA : “Ballistic Coefficient” • For circular orbits, rate of decay can be expressed simply as: • As with aircraft, determining CD to high accuracy can be tricky • Unlike aircraft, determining r is even trickier

34. Principal Orbital Perturbations • Earth ‘oblateness’ results in an unsymmetric gravity potential given by:where ae = equatorial radius, Pn ~ Legendre Polynomial Jn ~ zonal harmonics, w ~ sin (SC declination) • J2 term causes measurable perturbation which must be accounted for. Main effects: • Regression of nodes • Rotation of apsides Note: J2~1E-3, J3~1E-6 GN/MAE155B

35. Part II: The CubeSat Standard • The CubeSat is a 10x10x10cm, 1kg public picosatellite design specification proposed by Stanford and Cal Poly San Luis Obispo universities in the USA. • To date, low-earth orbit (LEO) CubeSat missions have had typical lifespans of 3-9 months. • Cost to complete a CubeSat mission (inception to launch to operation to end-of-life) ranges from <$100,000 to $1,500,000, depending on a variety of factors. • Working from a standard promotes rapid development and idea sharing • Picosatellites are already a hot topic in aerospace. Worldwide interest is focused on CubeSats in particular, partly because they are becoming a de facto standard.

36. Prograde vs. Retrograde • Prograde – • Any orbit in which the spacecraft moves from west to east • Usual direction of rotation in our Solar System. • Only a handful of objects orbit or rotate in the opposite direction • Retrograde – • Any orbit in which the spacecraft moves from east to west • This is the less usual direction in the Solar System; however, it is not impossible. • For example, Venus has retrograde spin and some comets – notably comet Halley, which was encountered by ESA’s Giotto spacecraft in 1986 – also has a retrograde orbit.

37. Spacecraft Horizon • Spacecraft horizon forms a circle on the spherical surface of the central body, within circle: • Spacecraft can be seen from central body • Line of sight communication can be established • Spacecraft can observe the central body GN/MAE155B

38. GTO orbit GEO orbit Hohmann Transfer • Hohmann transfer is the most efficient transfer (requires the least DV) between 2 orbit assuming: • Only 2 burns allowed • Circular initial and final orbits • Perform first burn to transfer to an elliptical orbit which just touches both circular orbits • Perform second burn to transfer to final circular GEO orbit Initial Circular Parking Orbit

39. “Systems” Engineering • Looking at the “Big” Picture • Requirements: What Does the Satellite Need to Do? When? Where? How? • Juggling All The Pieces • Mission Design: Orbits, etc. • Instruments and Payloads • Electronics and Power • Communications • Mass • Attitude Control • Propulsion • Cost and Schedule

40. Spacecraft Design Considerations • Instruments and Payloads • Optical Instruments • RF Transponders (Comm. Sats) • Experiments • Electronics and Power • Solar Panels and Batteries • Nuclear Power • Communications • Uplink/Downlink • Ground Station Locations • Frequencies and Transmitter Power

41. Spacecraft Design Considerations(Cont’d) • Mass Properties • Total Mass • Distribution of Mass (Moments of Inertia) • Attitude Control • Thrusters: Cold Gas and/or Chemical Propulsion • Gravity Gradient (Non-Spherical Earth Effect) • Spin Stablized • Magnetic Torquers • Propulsion • Orbit Maneuvering and/or Station Keeping • Chemical or ‘Exotic’ • Propellant Supply

42. Spacecraft Design Considerations(Cont’d) • Cost and Schedule • Development • Launch • Mission Lifetime • 1 Month, 1 Year, 1 Decade?

43. NASA Earth-Observing Satellites Low Earth Orbit: Orbiting at an altitude of 600-1,000 km. Ascending Orbit: The satellite is moving South to North when that portion of the orbit track crosses the equator. Sun-Synchronous: The satellite is always in the same relative position between the Earth and Sun. Descending Orbit: The satellite is moving North to South when that portion of the orbit track crosses the equator. 43

44. Satellite Inclination Low Inclination Orbit (often near 57º-- Space Shuttle) no polar coverage High Inclination or Polar Orbit (near 90º) virtually complete global coverage Equator Inclination: The position of the orbital plane relative to the equator. For near-polar orbits, typically about 97º. 44