1. To Orbit…and Beyond (Intro to Orbital Mechanics)
Scott Schoneman
6 November 03

2. Agenda Some brief history - a clockwork universe? The Basics
What is really going on in orbit: Is it really zero-G?
Motion around a single body
Orbital elements
Ground tracks
Perturbations
J2 and gravity models
Drag
“Third bodies”

3. Why is this important?
The physics of orbit mechanics makes launching spacecraft difficult and complex: It’s difficult to get there! (with current technology)
Orbit mechanics touches the design of essentially all spacecraft systems:
Power (shadows? Distance from Sun?)
Thermal ( “ )
Attitude Control (disturbance environment)
Propulsion systems (launch, orbit maneuvers - indirectly affects structures)
Radiation environment (electronic design)
All of the above can affect software
Practical problems:
Where will the satellite be & when can I talk to it?
When will it see/not see it’s mission target?
How do I get it to see its mission target or ground stations (attitude, propulsion maneuvers)?

4. Earth-Centered & Sun-Centered
The Universe must be perfect! All motion must be
based on spheres and circles (Aristotle)
Ptolemy (c. 150 AD) worked out a system of
“epicycles”, “eccentrics” and “equants” based on
circles
Fit observations for many centuries
Copernicus (1543) published his sun-centered universe
“Mathematical description only”
Described retrograde motion well, but still used circles and epicycles to fit observational details

5. Observations & Ellipses
Tycho Brahe ( ): Foremost observer of his day
Most accurate and detailed observations performed up to that time
Johannes Kepler ( )
Used Tycho’s observations in attempt to fit his sun- centered system of spheres separated by regular polyhedra
Could not fit the observations to systems of circles and spheres
Resorted to other shapes, eventually settling on the ellipse

6. Kepler’s Laws
Kepler made the leap to generalize 3 laws for planetary motion:
1) Planets move in an ellipse, with the sun at a focus
2) The motion of a planet “sweeps out” area at a constant rate
(thus the speed is not constant)
3) Period2 is proportional to (average distance)3
“The harmony of the worlds”
“My aim in this is to show that the celestial machine is a clockwork”
Note that these were purely EMPIRICAL laws - there’s no “physics” behind them.

7. Halley and Newton
Edmond Halley ( ) sought to predict the motion of comets, but couldn’t fit modern observations with older comet theories
Suspected inverse-square law for force, but sought Newton’s help
Helped Newton (technically & financially) publish “Principia”
Isaac Newton ( ) proved inverse-square law yields elliptical motion
Published “Principia” in 1687, bringing together gravity on Earth and in space (between the Sun, planets, and comets) into a single mathematical understanding
Also developed differential and integral calculus, derived Kepler’s three laws, founded discipline of fluid mechanics, etc.

8. (Newton is close enough for most engineering purposes)
Albert Einstein
Showed that Newton was all wrong (or at least not quite right), but we won’t talk about that.
(Newton is close enough for most engineering purposes)

9. The Basics and Two-Body Motion

10. Illustration from “Principia”
Newton’s Mountain
“The knack to flying lies in knowing how to throw yourself at the ground and miss.” (paraphrased) - Douglas Adams
Orbit is not “Zero-G” - There IS gravity in space - Lots of it
What’s really going on:
You are in FREE-FALL
You are always being pulled towards the Earth (or other central body)
If you have enough “sideways” speed, you will miss the Earth as it curves away from beneath you.
Illustration from “Principia”

11. Gravitational Force Newton’s 2nd Law:
Newton’s Law Of Universal Gravitation (assuming point masses or spheres):
Putting these together:

12. Gravitational Force - Simplified (Two Bodies, No Vectors)
Newton’s 2nd Law:
Newton’s law of universal gravitation (assuming point masses or spheres):
Putting these together:

13. The Gravitational Constant
“G” is one of the less-precisely known numbers in physics
It’s very small
You need to first know the mass and measure the force in order to solve for it
You will almost always see the combination of “GM” together
Usually called m
Can be easily measured for astronomical bodies (watching orbital periods)
µ (km3/sec2)

14. Conic Sections
Newton actually proved that the inverse-square law meant motion on a “conic section”

15. Conic Sections - Characteristics

16. Ellipse Geometry Most Common Orbits are Defined by the Ellipse:
a = “semi-major axis”
e = eccentricity = e / c = ( ra - rp )/ ( ra + rp )
Periapsis = rp , closest point to central body (perigee, perihelion)
Apoapsis = ra , farthest point from central body (apogee, aphelion)

17. The Classical Orbital Elements (aka Keplerian Elements)
Also need a timestamp (time datum)

18. State Vectors
A state vector is a complete description of the spacecraft’s position and velocity, with a timestamp
Examples
Position (x, y, z) and Velocity (x, y, z)
Classical Elements are also a kind of state vector
Other kinds of elements
NORAD Two-Line-Elements (TLE’s) (Classical Elements with a particular way of interpreting perturbations)
Latitude, Longitude, Altitude and Velocity
Mathematically conversion possible between any of these

19. Orbit Types
LEO (Low Earth Orbit): Any orbit with an altitude less than about 1000 km
Could be any inclination: polar, equatorial, etc
Very close to circular (eccentricity = 0), otherwise they’d hit the Earth
Examples: ORBCOMM, Earth-observing satellites, Space Shuttle, Space Station
MEO (Medium Earth Orbit): Between LEO and GEO
Examples: GPS satellites, Molniya (Russian) communications satellites
GEO (Geosynchronous): Orbit with period equal to Earth’s rotation period
Altitude km, Usually targeted for eccentricity, inclination = 0
Examples: Most communications satellite missions - TDRSS, Weather Satellites
HEO (High Earth Orbit): Higher than GEO
Example: Chandra X-ray Observatory, Apollo to the Moon
Interplanetary
Used to transfer between planets: the Sun is the central body
Typically large eccentricities to do the transfer

20. Ground Tracks
Ground Tracks project the spacecraft position onto the Earth’s (or other body’s) surface
(altitude information is lost)
Most useful for LEO satellites, though it applies to other types of missions
Gives a quick picture view of where the spacecraft is located, and what geographical coverage it provides

21. Example Ground Tracks
LEO sun-synchronous ground track

22. Example Ground Tracks
Some general orbit information can be gleaned from ground tracks
Inclination is the highest (or lowest) latitude reached
Orbit period can be estimated from the spacing (in longitude) between orbits
By showing the “visible swath”, you can estimate altitude, and directly see what the spacecraft can see on the ground
Example: swath

23. Geosynchronous and Molniya Orbit Ground Tracks
GEO ground track is a point (or may trace out a very small, closed path)
Molniya ground track “hovers” over Northern latitudes for most of the time, at one of two longitudes

24. Perturbations: Reality is More Complicated Than Two Body Motion
The International Sun/Earth Explorer 3 (ISEE-3) satellite, launched in 1978, was designed to monitor the interaction between the solar wind, a stream of charged gas emanating from the Sun, and the Earth's magnetic field. Because of this mission, the spacecraft was not equipped with a camera system, but with numerous instruments to measure the characteristics and composition of the solar wind. When the decision was made to change the trajectory for a comet intercept, the spacecraft was renamed the International Cometary Explorer.

25. Orbit Perturbations J2 and other non-spherical gravity effects
Earth is an “Oblate Spheriod” Not a Sphere
Atmospheric Drag
“Third” bodies
Other effects
Solar Radiation pressure
Relativity

26. J2 Effects - Plots J2-orbit rotation rates are a function of:
semi-major axis
inclination
eccentricity
(Regresses West)
(Regresses East)

27. Applications of J2 Effects
Sun-synchronous Orbits
The regression of nodes matches the Sun’s longitude motion (360 deg/365 days = deg/day)
Keep passing over locations at same time of day, same lighting conditions
Useful for Earth observation
“Frozen Orbits”
At the right inclination, the Rotation of Apsides is zero
Used for Molniya high-eccentricity communications satellites

28. 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

29. 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
Density is calculated using various atmospheric models
US Standard atmosphere approximate, no variations
Harris-Priester - dates to 1960’s, incorporates variations from Solar Activity, atmospheric swelling on the day-side
Jacchia-Roberts - various models in late 1960’s, 1970’s. Updated models from Harris Priester, incorporates effects due to Earth magnetic activity
MSIS-86, MSISE-90 - Incorporates seasonal & geographic variations, constituent chemistry model
Many others exist
Even the most complex models have a hard time predicting density to < 10% error
Solar activity and geomagnetic activity very uncertain and have drastic effects
For typical LEO altitudes, Solar activity can change density nearly 2 orders of magnitude
A lot of upper atmospheric dynamics not well understood

30. Applications of Drag Aerobraking / aerocapture
Instead of using a rocket, dip into the atmosphere
Lower existing orbit: aerobraking
Brake into orbit: aerocapture
Aerobraking to control orbit first demonstrated with Magellan mission to Venus
Used extensively by Mars Global Surveyor
Of course, all landing missions to bodies with an atmosphere use drag to slow down from orbital speed (Shuttle, Apollo return to Earth, Mars/Venus landers)

31. Third-Body Effects
Gravity from additional objects complicates matters greatly
No explicit solution exists like the ellipse does for the 2-body problem
Third body effects for Earth-orbiters are primarily due to the Sun and Moon
Affects GEOs more than LEOs
Points where the gravity and orbital motion “cancel” each other are called the Lagrange points
Sun-Earth L1 has been the destination for several Sun-science missions (ISEE-3 (1980s), SOHO, Genesis, others planned)

32. Lagrange Points Application
Genesis Mission:
NASA/JPL Mission to collect solar wind samples from outside Earth’s magnetosphere
Launched: 8 August 2001
Returning: Sept 2004

33. Third-Body Effects: Slingshot
A way of taking orbital energy from one body ( a planet ) and giving it to another ( a spacecraft )
Used extensively for outer planet missions (Pioneer 10/11, Voyager, Galileo, Cassini)
Analogous to Hitting a Baseball: Same Speed, Different Direction
planet’s orbit velocity
spacecraft incoming
to planet
hyperbolic flyby
(relative to planet)
spacecraft
departing planet
departing sun-centric velocity
incoming sun-centric velocity
Analogous to hitting a baseball
The incoming and departing speeds, relative to the bat, are the same
Relative to the ground, the speed and Kinetic Energy has greatly increased
Example: Velocity relative to planet was turned
Same speed, different direction (relative to planet)
Exiting velocity now adds directly to the velocity relative to the Sun, instead of being a component 90 degrees from it
Total velocity relative to Sun was increased