哈工大天文学概论――银河

StellarMassGravityandOrbits(2)NewtonianGravityanexplanationforplanetarymotions•Newton’s3LawsofMotionstarting-point•Necessaryingredienttocalculatemotions–Exactdescriptionofforcesinvolved–Howtoquantifygravity?–Galileo’sLawofFallingBodies–Newton,theAppleandtheMoon!Galileo’sMechanicsExperiments•Priortoworkwithtelescope,Galileoperformedfundamentalresearchonmotion.–Exploredtherateoffallingbodiesbydroppingdifferentweights,orslidingthemdowninclinedplanes.•LawofFallingBodies:–Intheabsenceofair,heavyobjectsandlightobjectsfallatthesame,constantrateofacceleration.AstronautDavidR.Scott,Apollo15(Falcon)nearHadleyRille,August1971TheIdeaofUniversalMutualGravitation•Newton,inhisPrincipia,formulatedtheLawofUniversalMutualGravitation:–GravityisanAttractiveforce:•Workstobringmassiveobjectsclosertogether.–GravityisaUniversalforce:•WorkseverywhereintheUniverse.–GravityisaMutualforce:•Worksbetweenpairsofmassiveobjects.Towardsanexactdescription•Forceofgravitybetweenanytwoobjectsdependsonlyupon:–Themassesofthetwoobjects:•Moremassiveobjectsfeelastrongerforce.–Thedistancebetweenthem:•Objectsclosertogetherfeelastrongerforce.–Itdoesnotdependatallontheshapes,colors,orcompositionsofthetwoobjects.Towardsanexactdescription•Newton’sIdea:–Forcedependslinearlyonmassesofobjects•Twiceasmuchmasstwicetheforce•Halfasmuchmasshalftheforce–Newton’sLawsthenguaranteeGalileo’sobservationofFallingBodies–Whatisthedependenceondistance?Comparinganappletothemoon•Newton’sfascinatingthoughtprocess:–Assumptions:•FallofanappleandpathofmoongovernedbythesameprincipleGravitationalattraction(only!)•GravityindependentofshapeofEarth–CarefulnumericalanalysisleadstodeeperunderstandingofgravityWhatNewtonknewaboutapples•FallingapplesonEarth:–ConstantaccelerationnearsurfaceofEarth:a=9.8meters/second2–RadiusofEarth:6378km=6378,000meters(Eratosthenes!)WhatNewtonknewabouttheMoon•DistancetotheMoon–~380,000km=~60xEarthradius•SiderealOrbitalPeriod–28.3days•SpeedofMoononitspath:~1000m/secDeflectionACurvedPath•But,ofcoursetheMoonreallymovesalongacurvedpath:–Accordingtothefirstlaw,itisdeflectedfromastraight-linepathbytheforceofgravity.–ThiscausesthemoontofallalittlebittowardstheEarth,deflectingitspathintoanarc.ThecurvedpathoftheMoon•Howmuchdoesthemoonhavetofallin1secondto‘closetheloop’?•Simplegeometry:–0.00136meters(about1.4mm!)•Newton’scleveridea:comparetoapple’sfall!ComparingappleandMoon•AppleonEarthfallsin1second:–dApple=4.8m•Moonfallsin1second:–dMoon=0.00136meters•Ratioofthesedistances=ratioofaccelerations=ratioofforces–dApple/dMoon=4.8m/0.00136m~3600ComparingappleandMoon•Ratioofforces:–FApple/FMoon~3600•RatioofdistancesfromcenterofEarth:–Dapple-Earth/DMoon-Earth~60•Pricewinningquestion:–Whatistherelationbetweenthesetworatios?Newton’sexactformula•Theforceofgravitationalattractionbetweenanytwomassivebodiesisproportionaltotheirmassesandinverselyproportionaltothesquareofthedistancebetweentheircenters.GravitationalForceLaw•F=forceduetogravity.•M1=massofthefirstobject•M2=massofthesecondobject•d=distancebetweentheircenters.•G=“GravitationalForceConstant”122GMMFd2M1M2d121222(2)2GMMGMMFddM1M2d122GMMFd2M12M2d121222(2)(2)4GMMGMMFddM1M2dM1M22d1212221(2)4GMMGMMFdd122GMMFdM1M2d/21212224(/2)GMMGMMFddWhyisthissuchapowerfulconcept?•TheLawofGravityisUniversal:–GovernsthefallofapplesontheEarth–GovernsthefalloftheMoonaroundtheEarth–GovernsthefalloftheEarth/MoonsystemaroundtheSun–GovernsthefalloftheSunaroundthecenteroftheMilkyWayGalaxy.–GovernsthefalloftheMilkyWayandAndromedaGalaxiesintheirmutualorbit...TheGravitationalForceConstant•Theforceconstant,G,isanumberwhichgivesthesizeofthegravitationalcouplingbetweentwomassiveobjects.•Gisverysmall,inmetricunits:•G=6.710–11Newtonsmeter2/kilogram2•TheNewtonisthemetricunitofforce:•4.41Newtons=1pound•Ghastobemeasuredexperimentally.Example1:WeighingtheEarth•Measuretheaccelerationofgravitybydroppingweights(Galileo):•a=9.8meters/second2•MeasuretheradiusoftheEarthusinggeometry(Eratosthenes):•RE=6378kilometers=6,378,000meters•Earth’sMassis:222411(9.8)(6,378,000)5.9810kg(6.6710)EEaRMGExample2:TheConceptofMutuality•WhatistheforceoftheEarthontheapple–F=GMEMA/RE2•Whatistheapple’sacceleration(2ndLaw):–a=F/MA=GME/RE2=9.8meters/sec2•Theaccelerationduetogravityisindependentofthemassoftheapple!Example2:TheConceptofMutuality•Thethirdlawsaysthatallforcescomeinequalyetoppositepairs.•WhatistheforceoftheappleontheEarth–F=GMEMA/RE2•HowmuchdoestheEarthacceleratetowardstheapple?–a=F/ME=GMA/RE2–a=9.8m/sec2(MA/ME)=verysmallamountSecondLawofOrbitalMotion•Orbitalmotionsconserveangularmomentum.–Thisdoesn’tsoundmuchlike“equalareasinequaltimes”,butinfactitisthesamething.–AngularMomentum:•L=mvr=constant•m=mass,v=velocity,r=distancefromthecenterofmass.AngularMomentum&EqualAreas•Lisaconstant.Ifthedistancechanges,thevelocitymustchangetocompensate:–NearPerihelion:•Planetisclosertothesun,hencesmallerr•Speedincreasesproportionallytocompensate.–NearAphelion:•Planetisfartherfromthesun,hencelargerr•Speeddecreasesproportionallytocompensate.ThirdLawofOrbitalMotion•Newton’sGen