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To Orbitand Beyond(Intro to Orbital Mechanics)

- Scott Schoneman
- 6 November 03

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

Why is this important?

- The physics of orbit mechanics makes launching

spacecraft difficult and complex Its 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 its mission target?
- How do I get it to see its mission target or

ground stations (attitude, propulsion maneuvers)?

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

Observations Ellipses

- Tycho Brahe (1546 - 1601) Foremost observer of

his day - Most accurate and detailed observations performed

up to that time

- Johannes Kepler (1571 - 1630)
- Used Tychos 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

Keplers 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 -

theres no physics behind them.

Halley and Newton

- Edmond Halley (1656 - 1742) sought to predict the

motion of comets, but couldnt fit modern

observations with older comet theories - Suspected inverse-square law for force, but

sought Newtons 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 Keplers three laws, founded

discipline of fluid mechanics, etc.

- Albert Einstein
- Showed that Newton was all wrong (or at least not

quite right), but we wont talk about that. - (Newton is close enough for most engineering

purposes)

The Basics andTwo-Body Motion

Newtons 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 - Whats 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

Gravitational Force

- Newtons 2nd Law
- Newtons Law Of Universal Gravitation (assuming

point masses or spheres) - Putting these together

Gravitational Force - Simplified(Two Bodies, No

Vectors)

- Newtons 2nd Law
- Newtons law of universal gravitation (assuming

point masses or spheres) - Putting these together

The Gravitational Constant

- G is one of the less-precisely known numbers in

physics - Its 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)

Conic Sections

- Newton actually proved that the inverse-square

law meant motion on a conic section

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

Conic Sections - Characteristics

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)

The Classical Orbital Elements(aka Keplerian

Elements)

- Also need a timestamp (time datum)

State Vectors

- A state vector is a complete description of the

spacecrafts 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 (TLEs) (Classical

Elements with a particular way of interpreting

perturbations) - Latitude, Longitude, Altitude and Velocity
- Mathematically conversion possible between any of

these

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 theyd 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

Earths rotation period - Altitude 35786 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

Ground Tracks

- Ground Tracks project the spacecraft position

onto the Earths (or other bodys) 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

Example Ground Tracks

- LEO sun-synchronous ground track

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

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

Perturbations Reality is More Complicated Than

Two Body Motion

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

J2 Effects - Plots

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

Applications of J2 Effects

- Sun-synchronous Orbits
- The regression of nodes matches the Suns

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

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, its an evenly-distributed

spiral

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

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)

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)

Lagrange Points Application

- Genesis Mission
- NASA/JPL Mission to collect solar wind samples

from outside Earths magnetosphere - Launched 8 August 2001
- Returning Sept 2004

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