To Orbit…and BeyondTo Orbit…and Beyond(Intro to Orbital Mechanics)(Intro to Orbital Mechanics)

Scott SchonemanScott Schoneman

6 November 036 November 03

AgendaAgenda

Some brief history - a clockwork universe?Some brief history - a clockwork universe? The BasicsThe Basics

What is really going on in orbit: Is it really zero-G?What is really going on in orbit: Is it really zero-G? Motion around a single bodyMotion around a single body Orbital elementsOrbital elements Ground tracksGround tracks

PerturbationsPerturbations J2 and gravity modelsJ2 and gravity models DragDrag ““Third bodies”Third bodies”

Why is this important?Why is this important? The physics of orbit mechanics makes launching spacecraft difficult and complex: It’s The physics of orbit mechanics makes launching spacecraft difficult and complex: It’s

difficult to get there! (with current technology)difficult to get there! (with current technology)

Orbit mechanics touches the design of essentially all spacecraft systems:Orbit mechanics touches the design of essentially all spacecraft systems: Power (shadows? Distance from Sun?)Power (shadows? Distance from Sun?) Thermal ( “ )Thermal ( “ ) Attitude Control (disturbance environment)Attitude Control (disturbance environment) Propulsion systems (launch, orbit maneuvers - indirectly affects structures)Propulsion systems (launch, orbit maneuvers - indirectly affects structures) Radiation environment (electronic design)Radiation environment (electronic design) All of the above can affect softwareAll of the above can affect software

Practical problems: Practical problems: Where will the satellite be & when can I talk to it?Where will the satellite be & when can I talk to it? When will it see/not see it’s mission target? 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 How do I get it to see its mission target or ground stations (attitude, propulsion

maneuvers)?maneuvers)?

Earth-Centered & Sun-CenteredEarth-Centered & Sun-CenteredThe Universe must be perfect! All motion must be The Universe must be perfect! All motion must be

based on spheres and circles (Aristotle)based on spheres and circles (Aristotle)Ptolemy (c. 150 AD)Ptolemy (c. 150 AD) worked out a system of worked out a system of

“ “epicycles”, “eccentrics” and “equants” based on epicycles”, “eccentrics” and “equants” based on

circlescirclesFit observations for many centuriesFit observations for many centuries

Copernicus (1543)Copernicus (1543) published his sun-centered universe published his sun-centered universe““Mathematical description only”Mathematical description only”Described retrograde motion well, but still used circles and Described retrograde motion well, but still used circles and

epicycles to fit observational detailsepicycles to fit observational details

Observations & EllipsesObservations & EllipsesTycho Brahe (1546 - 1601):Tycho Brahe (1546 - 1601): Foremost observer of his Foremost observer of his

dayday

Most accurate and detailed observations performed Most accurate and detailed observations performed up to that timeup to that time

Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)

Used Tycho’s observations in attempt to fit his sun-centered Used Tycho’s observations in attempt to fit his sun-centered system of spheres separated by regular polyhedrasystem of spheres separated by regular polyhedra

Could not fit the observations to systems of circles and Could not fit the observations to systems of circles and spheresspheres

Resorted to other shapes, eventually settling on the ellipseResorted to other shapes, eventually settling on the ellipse

Kepler’s LawsKepler’s LawsKepler made the leap to generalize 3 laws for planetary motion: Kepler made the leap to generalize 3 laws for planetary motion:

1) Planets move in an ellipse, with the sun at a focus1) Planets move in an ellipse, with the sun at a focus

2) The motion of a planet “sweeps out” area at a constant rate2) The motion of a planet “sweeps out” area at a constant rate

(thus the speed is not constant)(thus the speed is not constant)

3) Period3) Period22 is proportional to (average distance) is proportional to (average distance)33

“ “The harmony of the worlds”The harmony of the worlds”

““My aim in this is to show that the celestial machine is ...... a clockwork”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. Note that these were purely EMPIRICAL laws - there’s no “physics” behind them.

Halley and NewtonHalley and NewtonEdmond Halley (1656 - 1742)Edmond Halley (1656 - 1742) sought to predict the motion of sought to predict the motion of

comets, but couldn’t fit modern observations with older comet comets, but couldn’t fit modern observations with older comet theoriestheories

Suspected inverse-square law for force, but sought Newton’s Suspected inverse-square law for force, but sought Newton’s helphelp

Helped Newton (technically & financially) publish “Principia”Helped Newton (technically & financially) publish “Principia”

Isaac Newton (1643 - 1727)Isaac Newton (1643 - 1727) proved inverse-square law proved inverse-square law yields elliptical motionyields elliptical motion

Published “Principia” in 1687, bringing together gravity Published “Principia” in 1687, bringing together gravity on Earth and in space (between the Sun, planets, and on Earth and in space (between the Sun, planets, and comets) into a single mathematical understanding comets) into a single mathematical understanding

Also developed differential and integral calculus, derived Also developed differential and integral calculus, derived Kepler’s three laws, founded discipline of fluid Kepler’s three laws, founded discipline of fluid mechanics, etc. mechanics, etc.

Albert Einstein Albert Einstein

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

The Basics andThe Basics andTwo-Body MotionTwo-Body Motion

Newton’s MountainNewton’s Mountain

Illustration from “Principia”Illustration from “Principia”

““The knack to flying lies in knowing The knack to flying lies in knowing how to throw yourself at the how to throw yourself at the ground and miss.” (paraphrased) ground and miss.” (paraphrased) - Douglas Adams- Douglas Adams

Orbit is not “Zero-G” - There IS Orbit is not “Zero-G” - There IS gravity in space - Lots of it gravity in space - Lots of it

What’s really going on:What’s really going on:

You are in FREE-FALLYou are in FREE-FALLYou are always being pulled You are always being pulled

towards the Earth (or other towards the Earth (or other central body)central body)

If you have enough “sideways” If you have enough “sideways” speed, you will miss the Earth as it speed, you will miss the Earth as it curves away from beneath you.curves away from beneath you.

Gravitational ForceGravitational Force

Newton’s 2nd Law:Newton’s 2nd Law:

Newton’s Law Of Universal Gravitation Newton’s Law Of Universal Gravitation (assuming point masses or spheres):(assuming point masses or spheres):

Putting these together:Putting these together:

amF

12221

2body ur

mGmF

1221

2body ur

Gma

Gravitational Force - SimplifiedGravitational Force - Simplified(Two Bodies, No Vectors)(Two Bodies, No Vectors)

Newton’s 2nd Law:Newton’s 2nd Law:

Newton’s law of universal gravitation Newton’s law of universal gravitation (assuming point masses or spheres):(assuming point masses or spheres):

Putting these together:Putting these together:

maF

221

2body r

mGmF

21

2body r

Gma

The Gravitational ConstantThe Gravitational Constant““G” is one of the less-precisely known numbers in G” is one of the less-precisely known numbers in

physicsphysicsIt’s very smallIt’s very smallYou need to first know the mass and measure the You need to first know the mass and measure the

force in order to solve for itforce in order to solve for it

You will almost always see the combination of “GM” You will almost always see the combination of “GM” togethertogetherUsually called Usually called Can be easily measured for astronomical bodies Can be easily measured for astronomical bodies

(watching orbital periods)(watching orbital periods)

Sun 132712439935.5Mercury 22032.1Venus 324858.8Earth 398600.4Mars 42828.3Jupiter 126711995.4Saturn 37939519.7Moon 4902.8

µ (km3/sec2)

Conic SectionsConic SectionsNewton actually proved that the inverse-square law meant motion on a Newton actually proved that the inverse-square law meant motion on a

“conic section”“conic section”

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

Conic Sections - CharacteristicsConic Sections - Characteristics

Ellipse GeometryEllipse Geometry

a = “semi-major axis”a = “semi-major axis” e = eccentricity = e / c = ( re = eccentricity = e / c = ( raa - r - rpp )/ ( r )/ ( raa + r + rpp ) )

Periapsis = rPeriapsis = rpp , closest point to central body (perigee, perihelion) , closest point to central body (perigee, perihelion)

Apoapsis = rApoapsis = ra a , farthest point from central body (apogee, aphelion), farthest point from central body (apogee, aphelion)

Most Common Orbits are Defined by the Ellipse:Most Common Orbits are Defined by the Ellipse:

The Classical Orbital ElementsThe Classical Orbital Elements(aka Keplerian Elements)(aka Keplerian Elements)

Symbol Name What is it?a Semi-major axis Sizee Eccentricity Shapei Inclination Orientation of Orbit Plane (in space)

or RAAN Right Ascension of Ascending Node Orientation of Orbit Plane (in space) Argument of Perigee Orientation of Perigee (in orbit plane)

or True Anomaly Object's Position on Orbit

Also need a timestamp (time datum)Also need a timestamp (time datum)

State VectorsState VectorsA state vector is a complete description of the spacecraft’s position and velocity, with a timestampA state vector is a complete description of the spacecraft’s position and velocity, with a timestamp

ExamplesExamplesPosition (x, y, z) and Velocity (x, y, z)Position (x, y, z) and Velocity (x, y, z)Classical Elements are also a kind of state vectorClassical Elements are also a kind of state vectorOther kinds of elementsOther kinds of elements

NORAD Two-Line-Elements (TLE’s) (Classical Elements with a particular way of interpreting perturbations) NORAD Two-Line-Elements (TLE’s) (Classical Elements with a particular way of interpreting perturbations) Latitude, Longitude, Altitude and VelocityLatitude, Longitude, Altitude and Velocity

Mathematically conversion possible between any of theseMathematically conversion possible between any of these

Orbit TypesOrbit Types LEO (Low Earth Orbit): Any orbit with an altitude less than about 1000 kmLEO (Low Earth Orbit): Any orbit with an altitude less than about 1000 km

Could be any inclination: polar, equatorial, etcCould be any inclination: polar, equatorial, etcVery close to circular (eccentricity = 0), otherwise they’d hit the EarthVery close to circular (eccentricity = 0), otherwise they’d hit the EarthExamples: ORBCOMM, Earth-observing satellites, Space Shuttle, Space StationExamples: ORBCOMM, Earth-observing satellites, Space Shuttle, Space Station

MEO (Medium Earth Orbit): Between LEO and GEOMEO (Medium Earth Orbit): Between LEO and GEOExamples: GPS satellites, Molniya (Russian) communications satellitesExamples: GPS satellites, Molniya (Russian) communications satellites

GEO (Geosynchronous): Orbit with period equal to Earth’s rotation periodGEO (Geosynchronous): Orbit with period equal to Earth’s rotation periodAltitude 35786 km, Usually targeted for eccentricity, inclination = 0Altitude 35786 km, Usually targeted for eccentricity, inclination = 0Examples: Most communications satellite missions - TDRSS, Weather SatellitesExamples: Most communications satellite missions - TDRSS, Weather Satellites

HEO (High Earth Orbit): Higher than GEOHEO (High Earth Orbit): Higher than GEOExample: Chandra X-ray Observatory, Apollo to the MoonExample: Chandra X-ray Observatory, Apollo to the Moon

InterplanetaryInterplanetaryUsed to transfer between planets: the Sun is the central bodyUsed to transfer between planets: the Sun is the central bodyTypically large eccentricities to do the transferTypically large eccentricities to do the transfer

Ground TracksGround TracksGround Tracks project the spacecraft position onto the Earth’s (or other body’s) surfaceGround Tracks project the spacecraft position onto the Earth’s (or other body’s) surface

(altitude information is lost)(altitude information is lost)

Most useful for LEO satellites, though it applies to other types of missionsMost 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 providesGives a quick picture view of where the spacecraft is located, and what geographical coverage it provides

Example Ground TracksExample Ground TracksLEO sun-synchronous ground trackLEO sun-synchronous ground track

Example Ground TracksExample Ground TracksSome general orbit information can be gleaned from ground tracksSome general orbit information can be gleaned from ground tracksInclination is the highest (or lowest) latitude reachedInclination is the highest (or lowest) latitude reachedOrbit period can be estimated from the spacing (in longitude) between orbitsOrbit period can be estimated from the spacing (in longitude) between orbitsBy showing the “visible swath”, you can estimate altitude, and directly see what the spacecraft can see on the groundBy showing the “visible swath”, you can estimate altitude, and directly see what the spacecraft can see on the groundExample: swathExample: swath

Geosynchronous and Molniya Orbit Ground Geosynchronous and Molniya Orbit Ground TracksTracks

GEO ground track is a point (or may trace out a very small, closed path)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 longitudesMolniya ground track “hovers” over Northern latitudes for most of the time, at one of two longitudes

Perturbations: Reality is More Complicated Perturbations: Reality is More Complicated Than Two Body MotionThan Two Body Motion

Orbit PerturbationsOrbit Perturbations

J2 and other non-spherical gravity effectsJ2 and other non-spherical gravity effectsEarth is an “Oblate Spheriod” Not a SphereEarth is an “Oblate Spheriod” Not a Sphere

Atmospheric DragAtmospheric Drag““Third” bodiesThird” bodiesOther effectsOther effects

Solar Radiation pressureSolar Radiation pressureRelativityRelativity

J2 Effects - PlotsJ2 Effects - Plots J2-orbit rotation rates are a function of:J2-orbit rotation rates are a function of:

semi-major axissemi-major axis inclinationinclinationeccentricityeccentricity

(Regresses West)(Regresses East)

Applications of J2 EffectsApplications of J2 Effects

Sun-synchronous OrbitsSun-synchronous OrbitsThe regression of nodes matches the Sun’s longitude The regression of nodes matches the Sun’s longitude

motion (360 deg/365 days = 0.9863 deg/day)motion (360 deg/365 days = 0.9863 deg/day)Keep passing over locations at same time of day, same Keep passing over locations at same time of day, same

lighting conditionslighting conditionsUseful for Earth observationUseful for Earth observation

““Frozen Orbits”Frozen Orbits”At the right inclination, the Rotation of Apsides is zeroAt the right inclination, the Rotation of Apsides is zeroUsed for Molniya high-eccentricity communications Used for Molniya high-eccentricity communications

satellitessatellites

Atmospheric DragAtmospheric Drag

Along with J2, dominant perturbation for LEO satellitesAlong with J2, dominant perturbation for LEO satellitesCan usually be completely neglected for anything higher Can usually be completely neglected for anything higher

than LEOthan LEOPrimary effects:Primary effects:

Lowering semi-major axisLowering semi-major axisDecreasing eccentricity, if orbit is ellipticalDecreasing eccentricity, if orbit is elliptical

In other words, apogee is decreased much more than In other words, apogee is decreased much more than perigee, though both are affected to some extentperigee, though both are affected to some extent

For circular orbits, it’s an evenly-distributed spiralFor circular orbits, it’s an evenly-distributed spiral

Atmospheric DragAtmospheric Drag Effects are calculated using the same equation used for aircraft:Effects are calculated using the same equation used for aircraft:

To find acceleration, divide by mTo find acceleration, divide by m m / Cm / CDDA : “Ballistic Coefficient”A : “Ballistic Coefficient”

For circular orbits, rate of decay can be expressed simply as:For circular orbits, rate of decay can be expressed simply as:

As with aircraft, determining CAs with aircraft, determining CDD to high accuracy can be tricky to high accuracy can be tricky Unlike aircraft, determining Unlike aircraft, determining is even trickier is even trickier

ACVF D2

2

1

m

ACaa DREV

22

Applications of DragApplications of DragAerobraking / aerocaptureAerobraking / aerocapture

Instead of using a rocket, dip into Instead of using a rocket, dip into the atmospherethe atmosphereLower existing orbit: aerobrakingLower existing orbit: aerobrakingBrake into orbit: aerocaptureBrake into orbit: aerocapture

Aerobraking to control orbit first Aerobraking to control orbit first demonstrated with Magellan demonstrated with Magellan mission to Venusmission to Venus

Used extensively by Mars Global Used extensively by Mars Global SurveyorSurveyor

Of course, all landing missions to Of course, all landing missions to bodies with an atmosphere use drag to bodies with an atmosphere use drag to slow down from orbital speed (Shuttle, slow down from orbital speed (Shuttle, Apollo return to Earth, Mars/Venus Apollo return to Earth, Mars/Venus landers)landers)

Third-Body EffectsThird-Body EffectsGravity from additional objects complicates matters greatlyGravity from additional objects complicates matters greatly

No explicit solution exists like the ellipse does for the 2-body problemNo explicit solution exists like the ellipse does for the 2-body problemThird body effects for Earth-orbiters are primarily due to the Sun and MoonThird body effects for Earth-orbiters are primarily due to the Sun and Moon

Affects GEOs more than LEOsAffects GEOs more than LEOsPoints where the gravity and orbital motion “cancel” each other are called the Points where the gravity and orbital motion “cancel” each other are called the

Lagrange pointsLagrange pointsSun-Earth L1 has been the destination for several Sun-science missions Sun-Earth L1 has been the destination for several Sun-science missions

(ISEE-3 (1980s), SOHO, Genesis, others planned)(ISEE-3 (1980s), SOHO, Genesis, others planned)

Lagrange Points ApplicationLagrange Points ApplicationGenesis Mission:Genesis Mission:

NASA/JPL Mission to collect solar wind samples from outside NASA/JPL Mission to collect solar wind samples from outside Earth’s magnetosphereEarth’s magnetosphere

Launched: 8 August 2001Launched: 8 August 2001Returning: Sept 2004Returning: Sept 2004

Third-Body Effects: SlingshotThird-Body Effects: SlingshotA way of taking orbital energy from one body ( a planet ) and giving it to A way of taking orbital energy from one body ( a planet ) and giving it to

another ( a spacecraft )another ( a spacecraft )Used extensively for outer planet missions (Pioneer 10/11, Voyager, Used extensively for outer planet missions (Pioneer 10/11, Voyager,

Galileo, Cassini)Galileo, Cassini)Analogous to Hitting a Baseball: Same Speed, Different DirectionAnalogous to Hitting a Baseball: Same Speed, Different Direction

planet’s orbit velocity

spacecraft incomingto planet

hyperbolic flyby(relative to planet)

spacecraft departing planet

departing sun-centric velocity

incoming sun-centric velocity