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To Orbit…and Beyond (Intro to Orbital Mechanics)

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To Orbit…and Beyond(Intro to Orbital Mechanics)

Scott Schoneman

6 November 03

- 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”

- 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)?

- 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

- Tycho Brahe (1546 - 1601): Foremost observer of his day
- Most accurate and detailed observations performed up to that time

- Johannes Kepler (1571 - 1630)
- 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

- 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.

- Edmond Halley (1656 - 1742) 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 (1643 - 1727) 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.

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)

The Basics andTwo-Body Motion

- “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”

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

- Newton’s 2nd Law:
- Newton’s law of universal gravitation (assuming point masses or spheres):
- Putting these together:

µ (km3/sec2)

- “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)

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

http://ccins.camosun.bc.ca/~jbritton/jbconics.htm

- 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)

- Also need a timestamp (time datum)

- 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

- 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

- Examples: GPS satellites, Molniya (Russian) communications satellites

- Altitude 35786 km, Usually targeted for eccentricity, inclination = 0
- Examples: Most communications satellite missions - TDRSS, Weather Satellites

- Example: Chandra X-ray Observatory, Apollo to the Moon

- Used to transfer between planets: the Sun is the central body
- Typically large eccentricities to do the transfer

- 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

- LEO sun-synchronous ground track

- 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

- 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

Perturbations: Reality is More Complicated Than Two Body Motion

- 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

(Regresses West)

(Regresses East)

- J2-orbit rotation rates are a function of:
- semi-major axis
- inclination
- eccentricity

- Sun-synchronous Orbits
- The regression of nodes matches the Sun’s longitude motion (360 deg/365 days = 0.9863 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

- 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

- 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

- 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

- Instead of using a rocket, dip into the atmosphere
- 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)

- 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)

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

spacecraft

departing planet

departing sun-centric velocity

hyperbolic flyby

(relative to planet)

planet’s orbit velocity

spacecraft incoming

to planet

incoming sun-centric velocity

- 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