Astronautics

Home , Astronautics, Ground track, Orbital decay, Orbital mechanics, Synchronous orbit

•• Introduction Introduction toto spacespace systemssystems conceptsconcepts •Space•Space environmentenvironment •• Orbital Orbital mechanicsmechanics •• Attitude Attitude dynamicsdynamics andand controlcontrol •• Propulsion Propulsion andand launchlaunch vehiclesvehicles •• Space Space lawlaw andand policypolicy •Space•Space industryindustry Sample Space Applications

CommunicationsCommunications Navigation Navigation Relay Relay WeatherWeather Surveillance Surveillance Science Science MaterialsMaterials processingprocessing Search Search && rescuerescue AstronomyAstronomy Weapons Weapons Tourism Tourism SpaceSpace ExplorationExploration Solar Solar PowerPower TransportationTransportation Colonization Colonization Mapping Mapping Basic Elements of Space Missions

•Subject• Subject- - the the “thing”“thing” which which interactsinteracts withwith oror isis sensedsensed byby thethe payloadpayload •Space• Space segmentsegment -- spacecraft spacecraft comprisedcomprised ofof payloadpayloadand and spacecraftspacecraft busbus •Launch• Launch segmentsegment- - launch launch facilities,facilities, launchlaunch vehicle,vehicle, upperupper stage.stage. ConstrainsConstrains spacecraftspacecraft size.size. •• Orbit/constellation Orbit/constellation -- spacecraft’s spacecraft’s trajectorytrajectory oror pathpath throughthrough spacespace •C•C33 ArchitectureArchitecture- - command, command, controlcontrol andand communicationscommunications •Ground• Ground segmentsegment- - fixed fixed andand mobilemobile groundground stationsstations necessarynecessary forfor TT&CTT&C •• Mission Mission operationsoperations- - the the people,people, policiespolicies andand proceduresprocedures occupyingoccupying thethe groundground (and(and possiblypossibly space)space) segmentssegments Bus

Payload Mission Space segment Operations

Orbit and Command, Control and Constellation Communications Architecture Ground Segment

Reference:Reference: LarsonLarson Launch Segment &&MissionMission Wertz,Wertz, AnalysisAnalysisSpaceSpace andand DesignDesign Subject Payload and Bus Subsystems Spacecraft BasicBasic requirementsrequirements •• Payload Payload mustmust bebe Payload Bus pointedpointed

•• Payload Payload mustmust bebe Sensors operableoperable Cameras Power TT&C Antennas Solar arrays Power switching •• Data Data mustmust bebe Radar Batteries Encoder/decoder transmittedtransmitted toto usersusers PMAD Processors •• Orbit Orbit mustmust bebe ADCS Propulsion maintainedmaintained Sensors Orbit injection Actuators Stationkeeping •• Payload Payload mustmust bebe Processors Attitude control “held“held together”together” • Energy must be Thermal Control Structure • Energy must be Coatings Primary structure providedprovided Insulation Deployment Active control mechanisms Deep Space 1 Deep Space 1 Deployed Kepler Kepler Sunside A-Train Formation StarDust Giotto In the LV Lightband MESSENGER Vibration Test Thoughts on Space Some comments overhead at the Officer’s Club

•• It’s It’s aa reallyreally bigbig placeplace withwith nono air.air.

•• There’s There’s nothingnothing outout there,there, isis there?there?

•• How How manymany g’sg’s isis thatthat satellitesatellite pullingpulling whenwhen thethe groundground tracktrack makesmakes thosethose turns?turns?

•Why•Why can’tcan’t II havehave mymy spyspy satellitesatellite permanentlypermanently positionedpositioned overover Moscow?Moscow? Useful Characteristics of Space

•• Global Global perspectiveperspective oror “There’s“There’s nothin’nothin’ there there toto blockblock youryour view”view” •• Above Above thethe atmosphereatmosphere oror “There’s“There’s nono airair toto messmess upup youryour view”view” •• Gravity-free Gravity-free environmentenvironment oror “In“In free-fall,free-fall, youyou don’tdon’t noticenotice thethe gravity”gravity” •• Abundant Abundant resourcesresources oror “Eventually,“Eventually, wewe willwill minemine thethe asteroids,asteroids, collectcollect moremore solarsolar power,power, colonizecolonize thethe moon,moon, .. .. .”.” Global Perspective

AmountAmount ofof EarthEarth thatthat cancan bebe seenseen byby aa satellitesatellite isis muchmuch greatergreater thanthan cancan bebe seenseen byby anan Earth-boundEarth-bound observer.observer.

Low-EarthLow-Earth orbitorbit isis closercloser thanthan youyou think.think.

Space isn't remote at all. It's only an hour's drive away if your car could go straight upwards. —Fred Hoyle Instantaneous Access Area

H IAA

IAA= K (1− cosλ) λ A K = 2.55604187×108 km2 Re A R cosλ = e Re + H Example:Example: SpaceSpace shuttleshuttle

Re = 6378 km A friend of mine once sent me a post card λ with a picture of the entire planet Earth H = 300 km taken from space. On the back it said, cos = 0.9551⇒ λ =17.24o “Wish you were here.” — Steven Wright IAA =11,476,628 km2 Above the Atmosphere •• This This characteristiccharacteristic hashas severalseveral applicationsapplications –– Improved Improved astronomicalastronomical observationsobservations –– “Vacuum” “Vacuum” for for manufacturingmanufacturing processesprocesses –– Little Little oror nono dragdrag toto affectaffect vehiclevehicle motionmotion •• However, However, therethere reallyreally isis “air”“air” in in spacespace –– Ionosphere Ionosphere affectsaffects communicationscommunications signalssignals –– “Pressure” “Pressure” can can contaminatecontaminate somesome processesprocesses –– Drag Drag causescauses satellitessatellites toto speedspeed upup (!)(!) andand orbitsorbits toto decay,decay, affectingaffecting lifetimelifetime ofof LEOLEO satellitessatellites Vacuum Effects •• While While spacespace isis notnot aa perfectperfect vacuum,vacuum, itit isis betterbetter thanthan Earth-basedEarth-based facilitiesfacilities –– 200 200 kmkm altitude:altitude: pressurepressure == 1010-7-7torrtorr = = 1010-5-5PaPa –– Goddard Goddard vacuumvacuum chambers:chambers: pressurepressure == 1010-7-7torrtorr •• Outgassing Outgassing –affects–affects structuralstructural characteristicscharacteristics –– possibility possibility ofof vaporvapor condensationcondensation

Wake Shield Facility (shuttle experiment) Atmospheric Drag •• Can Can bebe modeledmodeled samesame asas wiwithth “normal”“normal” atmospheric atmospheric flightflight D = 1 C A V 2 2 D ρ ρ −h / H ≈ ρSLe

•• Key Key parameterparameter isis thethe “ballistic“ballistic coefficient”:coefficient”:

m /(CD A)

•• Larger Larger ballisticballistic coefficientcoefficient (small(small massivemassive satellite)satellite) impliesimplies slowerslower orbitalorbital decaydecay •• Smaller Smaller ballisticballistic coefficientcoefficient impliesimplies fasterfaster orbitalorbital decaydecay •• Energy Energy lossloss perper orbitorbit isis ≈ 2πrD

Better not take a dog on the space shuttle, because if he sticks his head out when you're coming home his face might burn up. — Jack Handey Weightlessness ThisThis illustrationillustration fromfrom JulesJules Verne’sVerne’s RoundRound thethe MoonMoon showsshows thethe effectseffects ofof “weightlessness”“weightlessness” on on thethe passengerspassengers ofof TheThe GunGun Club’sClub’s “bullet”“bullet” capsule capsule thatthat waswas firedfired fromfrom aa largelarge gungun inin Florida.Florida.

TheThe passengerspassengers onlyonly experiencedexperienced thisthis atat thethe half-half- wayway pointpoint betweenbetween thethe EarthEarth andand thethe Moon.Moon. PhysicallyPhysically accurate?accurate? Zero-Gravity?

ThisThis plotplot showsshows howhow gravitygravity dropsdrops offoff asas altitudealtitude increases.increases. NoteNote thatthat atat LEO,LEO, thethe gravitationalgravitational accelerationacceleration isis aboutabout 90%90% ofof thatthat atat Earth’sEarth’s surface.surface. Microgravity

•• Weightlessness, Weightlessness, freefree fall,fall, oror zero-gzero-g •• Particles Particles don’tdon’t settlesettle outout ofof solution,solution, bubblesbubbles don’tdon’t rise,rise, convectionconvection doesn’tdoesn’t occuroccur •• Microgravity Microgravity effectseffects inin LEOLEO cancan bebe reducedreduced toto 1010-1-1 gg (1(1 µµg)g)

On Earth, gravity-driven buoyant convection causes a candle flame to be teardrop-shaped (a) and carries soot to the flame's tip, making it yellow. In microgravity, where convective flows are absent, the flame is spherical, soot- free, and blue (b). A Brief History of Orbital Mechanics

AristotleAristotle (384-322 (384-322 BC)BC) PtolemyPtolemy (87-150(87-150 AD)AD) NicolausNicolaus Copernicus Copernicus (1473-1543) (1473-1543) TychoTycho Brahe Brahe (1546-1601) (1546-1601) JohannesJohannes KeplerKepler (1571-1630) (1571-1630) GalileoGalileo GalileiGalilei (1564-1642) (1564-1642) SirSir IsaacIsaac NewtonNewton (1643-1727) (1643-1727) Kepler’s Laws

I.I. TheThe orbitorbit ofof eacheach planetplanet isis anan ellipseellipse withwith thethe SunSun atat oneone focus.focus. II.II. TheThe lineline joiningjoining thethe planetplanet toto thethe SunSun sweepssweeps outout equalequal areasareas inin equalequal times.times. III.III. TheThe squaresquare ofof thethe periodperiod ofof aa planet’splanet’s orbitorbit isis proportionalproportional toto thethe cubecube ofof itsits meanmean distancedistance toto thethe sun.sun. Kepler’s First Two Laws I.I. TheThe orbitorbit ofof eacheach planetplanet isis anan ellipseellipse withwith thethe SunSun atat oneone focus.focus.

II.II. TheThe lineline joiningjoining thethe planetplanet toto thethe SunSun sweepssweeps outout equalequal areasareas inin equalequal times.times. Kepler’s Third Law

III.III. TheThe squaresquare ofof thethe periperiodod ofof aa planet’splanet’s orbitorbit isis proportionalproportional toto thethe cubecube ofof itsits meanmean distancedistance toto thethe sun.sun. π a3 T = 2 µ HereHere TT isis thethe period,period, aa isis thethe semimajorsemimajor axisaxis ofof thethe ellipse,ellipse, andand µµ isis thethe gravitationalgravitational parameterparameter (depends(depends onon massmass ofof centralcentral body)body) µ 5 3 −2 ⊕ = GM ⊕ = 3.98601×10 km s 11 3 −2 µsun = GM sun =1.32715×10 km s Earth Satellite Orbit Periods

OrbitOrbit Altitude Altitude (km)(km) Period Period (min)(min) LEOLEO 300 300 90.5290.52 LEOLEO 400 400 92.5692.56 MEOMEO 3000 3000 150.64 150.64 GPSGPS 20232 20232 720 720 GEOGEO 35786 35786 1436.07 1436.07

You should be able to do these calculations! Don’t forget to add the radius of the Earth to the altitude to get the orbit radius. Canonical Units •• For For circularcircular orbits,orbits, wewe commonlycommonly letlet thethe orbitorbit radiusradius bebe thethe distancedistance unitunit (or(or DUDU)) andand selectselect thethe timetime unitunit soso thatthat thethe periodperiod ofof thethe orbitorbit isis 2π TUs 2π TUs π a3 T = 2 µ •• Clearly Clearly thesethese choiceschoices leadlead toto aa valuevalue ofof thethe gravitationalgravitational parameterparameter ofof µµ== 11 DUDU33/TU/TU22 •• For For thethe Earth’sEarth’s orbitorbit aroundaround thethe Sun,Sun, 11 DUDU == 11 AUAU ≈≈ 90,000,00090,000,000 milesmiles •• What What isis thethe durationduration ofof 11 TU?TU? Examples •• The The planetplanet SaturnSaturn hashas anan orbitorbit radiusradius ofof approximatelyapproximately 9.59.5 AU.AU. WhatWhat isis itsits periodperiod inin years?years?

•• The The GPSGPS satellitessatellites orbitorbit thethe EarthEarth atat anan altitudealtitude ofof 20,20020,200 km.km. WhatWhat isis thethe periodperiod ofof thethe orbitorbit ofof aa GPSGPS satellitesatellite inin minutes?minutes? …hours?…hours?

•• The The HubbleHubble SpaceSpace TelescopeTelescope orbitsorbits thethe EarthEarth atat anan altitudealtitude ofof 612612 km.km. WhatWhat isis thethe periodperiod ofof thethe orbitorbit inin minutes?minutes? …hours?…hours? Newton’s Laws

•• Kepler’s Kepler’s Laws Laws werewere basedbased onon observationobservation data:data:“curve “curve fits”fits” •Newton• Newton establishedestablished thethe theorytheory –– Universal Universal GravitationalGravitational LawLaw m GMm r Fg = − 2 r M –– Second Second LawLaw r F = m&rr& AOEAOE 41344134 AstromechanicsAstromechanics coverscovers thethe formulationformulation andand solutionsolution ofof thisthis problemproblem Elliptical Orbits

•• Planets, Planets, comets,comets, andand asteroidsasteroids orbitorbit thethe SunSun inin ellipsesellipses •• Moons Moons orbitorbit thethe planetsplanets inin ellipsesellipses •• Artificial Artificial satellitessatellites orbitorbit thethe EarthEarth inin ellipsesellipses •• To To understandunderstand orbits,orbits, youyou needneed toto understandunderstand ellipsesellipses (and(and otherother conicconic sections)sections) •• But But first,first, let’slet’s studystudy circularcircular orbitsorbits –– a a circlecircle isis aa specialspecial casecase ofof anan ellipseellipse Circular Orbits •• The The speedspeed ofof aa satellitesatellite v inin aa circularcircular orbitorbit dependsdepends a=r onon thethe radiusradius µ v = c r •• If If anan orbitingorbiting objectobject atat aa particularparticular

radiusradius hashas aa speedspeed << vvcc,, thenthen itit isis inin anan ellipticalelliptical orbitorbit withwith lowerlower energyenergy •• If If anan orbitingorbiting objectobject atat radiusradius rr hashas aa speedspeed >>

vvcc,, thenthen itit isis inin aa higher-energyhigher-energy orbitorbit whichwhich maymay bebe elliptical,elliptical, parabolic,parabolic, oror hyperbolichyperbolic Examples

•• The The HubbleHubble SpaceSpace TelescopeTelescope orbitsorbits thethe EarthEarth atat anan altitudealtitude ofof 612612 km.km. WhatWhat isis itsits speed?speed?

•• The The SpaceSpace StationStation orbitsorbits thethe EarthEarth atat anan altitudealtitude ofof 380380 km.km. WhatWhat isis itsits speed?speed?

•• The The MoonMoon orbitsorbits thethe EarthEarth withwith aa periodperiod ofof approximatelyapproximately 2828 days.days. WhatWhat isis itsits orbitorbit radius?radius? WhatWhat isis itsits speed?speed? The Energy of an Orbit

•• Orbital Orbital energyenergy isis thethe sumsum ofof thethe kinetickinetic energyenergy,, mvmv22/2/2,, andand thethe potentialpotential energyenergy,, --µµmm//rr •• Customarily, Customarily, wewe useuseµ thethe specificspecific mechanicalmechanical energyenergy,, EE (i.e.,(i.e., thethe energyenergy perper unitunit massmass ofof satellite)satellite) v2 − µ E = − ⇔ E = 2 r 2a •• From From thisthis definitiondefinition ofof energy,energy, wewe cancan developdevelop thethe followingfollowing factsfacts EE << 00 ⇔⇔ orbitorbit isis ellipticalelliptical oror circularcircular EE == 00 ⇔⇔ orbitorbit isis parabolicparabolic EE >> 00 ⇔⇔ orbitorbit isis hyperbolichyperbolic Examples

•• A A satellitesatellite isis observedobserved toto havehave altitudealtitude 500500 kmkm andand velocityvelocity 77 km/s.km/s. WhatWhat typetype ofof conicconic sectionsection isis itsits trajectory?trajectory?

•• A A satellitesatellite isis observedobserved toto havehave altitudealtitude 380380 kmkm andand velocityvelocity 1111 km/s.km/s. WhatWhat typetype ofof conicconic sectionsection isis itsits trajectory?trajectory?

•• What What typetype ofof trajectorytrajectory dodo cometscomets follow?follow? Asteroids?Asteroids? Conic Sections

AnyAny conicconic sectionsection cancan bebe visualizedvisualized asas thethe intersectionintersection ofof aa planeplane andand aa cone.cone.

TheThe intersectionintersection cancan bebe aa circlecircle,, anan ellipseellipse,, aa parabolaparabola,, oror aa hyperbolahyperbola.. Properties of Conic Sections

ConicConic sectionssections areare characterizedcharacterized byby eccentricity,eccentricity, ee (shape)(shape) andand semimajorsemimajor axis,axis, aa (size)(size)

ConicConic eaea CircleCircle ee == 00 aa >> 00 EllipseEllipse 00 << ee << 11 aa >> 00 ParabolaParabola ee == 11 aa == ∞∞ HyperbolaHyperbola ee >> 11 aa << 00 Properties of Ellipses

a b r vacant ae focus focus 2a-p p=a(1-e2)

a a(1+e) a(1-e) Facts About Elliptical Orbits

•• Periapsis Periapsis isis thethe closestclosest pointpoint ofof thethe orbitorbit toto thethe centralcentral bodybody r = a(1-e) rpp = a(1-e) •• Apoapsis Apoapsis isis thethe farthestfarthest pointpoint ofof thethe orbitorbit fromfrom thethe centralcentral bodybody r = a(1+e) raa = a(1+e) •• Velocity Velocity atat anyany pointpoint isis vv== (2(2EE+2+2µµ//rr))1/21/2 •• Escape Escape velocityvelocity atat anyany pointpoint isis v = (2µ/r)1/21/2 vescesc = (2µ/r) Examples •• A A satellitesatellite isis observedobserved toto havehave altitudealtitude 500500 kmkm andand velocityvelocity 77 km/s.km/s. WhatWhat isis itsits energy?energy? WhatWhat isis itsits velocityvelocity atat somesome laterlater timetime whenwhen itsits altitudealtitude isis 550550 km?km?

•• A A satellitesatellite isis inin anan EarthEarth orbitorbit withwith semimajorsemimajor axisaxis ofof 10,00010,000 kmkm andand anan eccentricityeccentricity ofof 0.1.0.1. WhatWhat isis thethe speedspeed atat periapsis?periapsis? …at…at apoapsis?apoapsis?

•• A A MarsMars probeprobe isis inin anan EarthEarth parkingparking orbitorbit withwith altitudealtitude 400400 km.km. WhatWhat isis itsits velocity?velocity? WhatWhat isis itsits escapeescape velocity?velocity? WhatWhat changechange inin velocityvelocity isis requiredrequired toto achieveachieve escapeescape velocity?velocity? Orbital Elements

K^ Equatorial plane ν ω

^ I Ω i e ^ l plan ^ J Orbita n

Orbit is defined by 6 orbital elements (oe’s): semimajor axis, a; eccentricity, e; inclination, i; right ascension of ascending node, Ω; argument of periapsis, ω; and true anomaly, ν Orbital Elements(continued) • Semimajor axis a determines the size of the ellipse • Eccentricity e determines the shape of the ellipse • Two-body problem – a, e, i, Ω, and ω are constant –6th orbital element is the angular measure of satellite motion in the orbit – 2 angles are commonly used: • True anomaly, ν •Mean anomaly, M • In reality, these elements are subject to various perturbations

– Earth oblateness (J2) – atmospheric drag – solar radiation pressure – gravitational attraction of other bodies Ground Track This plot is for a satellite in a nearly circular orbit

ISS (ZARYA)

0 latitude

−30

−60

−90 0 60 120 180 240 300 360 longitude Ground Track This plot is for a satellite in a highly elliptical orbit

1997065B 90

0 latitude

−30

−60

−90 0 60 120 180 240 300 360 longitude Algorithm for SSP, Ground Track

•• Compute Compute positionposition vectorvector inin ECIECI θ θ •• Determine Determine GreenwichGreenwich SiderealSidereal TimeTime θgg atat epoch,epoch, θg0g0 δ = sin-1-1(r /r) •• Latitude Latitude isis δss = sin (r33/r) L = tan-1-1(r /r )- θ •• Longitude Longitude isis Lss = tan (r22/r11)- θg0g0

•• Propagate Propagate positionposition vectorvector inin “the“the usualusual way”way” θ θ +ω (t-t ) •• Propagate Propagate GSTGST usingusing θgg == θg0g0 +ω⊕⊕(t-t00) ω wherewhere ω⊕⊕ isis thethe angularangular velocityvelocity ofof thethe EarthEarth •Notes:•Notes: http://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdfhttp://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdf http://aa.usno.navy.mil/data/docs/WebMICA_2.htmlhttp://aa.usno.navy.mil/data/docs/WebMICA_2.html http://tycho.usno.navy.mil/sidereal.htmlhttp://tycho.usno.navy.mil/sidereal.html Elevation Angle

ElevationElevation angle,angle, εε,, isis measuredmeasured upup fromfrom horizonhorizon toto targettarget R⊕λ 0 ε D λ MinimumMinimum elevationelevation η angleangle isis typicallytypically basedbased ρ onon thethe performanceperformance ofof anan antennaantenna oror sensorsensor TheThe angleangle ηη isis calledcalled thethe nadirnadir angleangle IAA is determined by IAA is determined by TheThe angleangle ρρ isis calledcalled thethe apparentapparent same formula, but the same formula, but the EarthEarth radiusradius EarthEarth centralcentral angle,angle, λλ,, TheThe rangerange fromfrom satellitesatellite toto targettarget isis isis determineddetermined fromfrom thethe denoteddenoted DD geometrygeometry shownshown Geometry of Earth-Viewing

R⊕λ 0 ε D λ η ρ •• Given Given altitudealtitude HH,, wewe cancan statestate sin ρ cos λ R / (R +H) sin ρ == cos λ00 == R⊕⊕ / (R⊕⊕+H) ρ+ρ+ λλ00 == 90°90° •• For For aa targettarget withwith knownknown positionposition vector,vector, λλ isis easilyeasily computedcomputed coscos λλ == coscos δδss coscos δδtt coscos ∆∆LL ++ sinsin δδss sinsin δδtt •Then•Then tantan ηη == sinsin ρρ sinsinλλ // (1-(1- sin sin ρρ coscosλλ)) R •And•And ηη ++ λλ ++ εε == 9090°°andand DD == R⊕⊕ sinsinλλ // sinsinηη Earth Oblateness Perturbations •• Earth Earth isis non-spherical,non-spherical, andand toto firstfirst approximationapproximation isis anan oblateoblate spheroidspheroid •• The The primaryprimary effectseffects areare onon ΩΩ andand ωω:: 2 m 3J 2nRe Ω& = cosi r 2a2 (1− e2 )2 2 M − 3J nR 2 ω& = 2 e (4 − 5sin i) 4a2 (1− e2 )2 The Oblateness Coefficient −3 J 2 = −1.0826×10 Main Applications of J2 Effects

•• Sun-synchronousSun-synchronous orbitsorbits:: TheThe raterate ofof changechange ofof ΩΩ cancan bebe chosenchosen soso thatthat thethe orbitalorbital planeplane maintainsmaintains thethe samesame orientationorientation withwith respectrespect toto thethe sunsun throughoutthroughout thethe yearyear •• CriticalCritical inclinationinclination orbitsorbits:: TheThe raterate ofof changechange ofof ωω cancan bebe mademade zerozero byby selectingselecting ii≈≈63.4°63.4° Sun-Synchronous Orbit

Orbit rotates to maintain same a angle with sun Equinox a Winter a Summer a a

Sun

Equinox

a a Basic Space Propulsion Concepts •• Three Three functionsfunctions ofof spacespace propulsionpropulsion –– place place payloadpayload inin orbitorbit ((launchlaunch vehiclesvehicles)) –– transfer transfer payloadpayload fromfrom oneone orbitorbit toto anotheranother ((upperupper stagesstages)) –– control control spacecraftspacecraft positionposition andand pointingpointing directiondirection ((thrusters)thrusters) •• Rockets Rockets provideprovide aa changechange inin momentummomentum accordingaccording toto thethe impulse-momentumimpulse-momentum formform ofof Newton’sNewton’s 2nd2nd law:law: r r r r r F(t2 − t1) = p2 − p1 ⇔ F∆t = ∆p –– usually usually wewe talktalk aboutabout thethe changechange inin r velocityvelocityprovided provided byby aa rocketrocket ⎯⎯ itsits “Delta“Delta Vee”Vee” ∆V –– generally generally involvesinvolves changechange inin magnitudemagnitude andand directiondirection Thrust, Mass Flow Rate, & Exhaust Velocity •• The The simplestsimplest modelmodel ofof thrustthrust generatedgenerated byby aa rocketrocket isis F = −m& Ve m& isis thethe raterate atat whichwhich propepropellantllant isis ejectedejected (negative)(negative) Ve isis thethe exhaustexhaust velocityvelocity relativerelative toto thethe rocketrocket •• An An importantimportant performanceperformance measuremeasure isis thethe SpecificSpecific ImpulseImpulse I sp = Ve / g

I sp hashas unitsunits ofof secondsseconds –– is is thethe numbernumber ofof poundspounds ofof thrustthrust forfor everyevery poundpound ofof propellantpropellant burnedburned inin oneone secondsecond

•• Using Using I sp thethe thrustthrust generatedgenerated byby aa rocketrocket isis

F = −m& gI sp •The•The ggthatthat isis usedused isis ALWAYSALWAYS thethe standardstandard meanmean accelerationacceleration duedue toto gravitygravity atat thethe Earth’sEarth’s surfacesurface Typical Functional Requirements

•• Launch Launch fromfrom EarthEarth toto LEO:LEO: ∆V ≈ 9.5 km/s –– initial initial velocityvelocity isis ≈≈00 v =√(µ/r) ≈ 7.7 km/s –– LEO LEO circularcircular orbitorbit velocityvelocity isis vcc=√(µ/r) ≈ 7.7 km/s –– add add aa bitbit forfor losseslosses duedue toto dragdrag

•• LEO-to-GEO LEO-to-GEO orbitorbit transfer:transfer: ∆V ≈ 4 km/s –use– use HohmannHohmann transfer transfer ∆v ≈ 2.4257 km/s –– perigee perigee burn:burn: ∆v11 ≈ 2.4257 km/s ∆v ≈ 1.4667 km/s –apogee–apogee burn:burn: ∆v22 ≈ 1.4667 km/s Hohmann Transfer vc2 •Initial LEO orbit has radius •Initial LEO orbit has radius ∆v v2 r , velocity v 2 r11, velocity vcc11 •Desired•Desired GEOGEO orbitorbit hashas radius r , velocity v radius r22, velocity vcc22 •Impulsive•Impulsive ∆∆vvisis appliedapplied toto getget onon geostationarygeostationary LEO GTO r2 transfertransfer orbitorbit (GTO)(GTO) atat perigee µ perigee µ r1 ∆v = 2 − 2 − µ 1 r1 r1 +r2 r1 v ∆v •Coast to apogee and apply c1 1 •Coast to apogee and apply v1 another impulsiveµ ∆v another impulsiveµ ∆v 2 2µ ∆v = − − GEO 2 r2 r2 r1 +r2 Other Typical Requirements

•• Orbit Orbit maintenancemaintenance forfor GEOGEO satellitessatellites –– East-West East-West stationkeepingstationkeeping 3 3 toto 66 m/sm/s perper yearyear –– North-South North-South stationkeepingstationkeeping 45 45 toto 5555 m/sm/s perper yearyear

•• Reentry Reentry forfor LEOLEO satellitessatellites 120 120 toto 150150 m/sm/s

•• Attitude Attitude controlcontrol typicallytypically requiresrequires 3-10%3-10% ofof thethe propellantpropellant massmass