哈工大天文學(xué)概論-銀河

StellarMassGravityandOrbits(2)NewtonianGravity

anexplanationforplanetarymotionsNewton’s3LawsofMotionstarting-pointNecessaryingredienttocalculatemotionsExactdescriptionofforcesinvolvedHowtoquantifygravity?Galileo’sLawofFallingBodiesNewton,theAppleandtheMoon!Galileo’sMechanicsExperimentsPriortoworkwithtelescope,Galileoperformedfundamentalresearchonmotion.Exploredtherateoffallingbodiesbydroppingdifferentweights,orslidingthemdowninclinedplanes.LawofFallingBodies:Intheabsenceofair,heavyobjectsandlightobjectsfallatthesame,constantrateofacceleration.AstronautDavidR.Scott,Apollo15(Falcon)nearHadleyRille,August1971TheIdeaofUniversalMutualGravitationNewton,inhisPrincipia,formulatedtheLawofUniversalMutualGravitation:GravityisanAttractiveforce:Workstobringmassiveobjectsclosertogether.GravityisaUniversalforce:WorkseverywhereintheUniverse.GravityisaMutualforce:Worksbetweenpairsofmassiveobjects.TowardsanexactdescriptionForceofgravitybetweenanytwoobjectsdependsonlyupon:Themassesofthetwoobjects:Moremassiveobjectsfeelastrongerforce.Thedistancebetweenthem:Objectsclosertogetherfeelastrongerforce.Itdoesnotdependatallontheshapes,colors,orcompositionsofthetwoobjects.TowardsanexactdescriptionNewton’sIdea:ForcedependslinearlyonmassesofobjectsTwiceasmuchmasstwicetheforceHalfasmuchmasshalftheforceNewton’sLawsthenguaranteeGalileo’sobservationofFallingBodiesWhatisthedependenceondistance?ComparinganappletothemoonNewton’sfascinatingthoughtprocess:Assumptions:FallofanappleandpathofmoongovernedbythesameprincipleGravitationalattraction(only!)GravityindependentofshapeofEarthCarefulnumericalanalysisleadstodeeperunderstandingofgravityWhatNewtonknewaboutapplesFallingapplesonEarth:ConstantaccelerationnearsurfaceofEarth:a=9.8meters/second2RadiusofEarth:6378km=6378,000meters(Eratosthenes!)WhatNewtonknewabouttheMoonDistancetotheMoon~380,000km=~60xEarthradiusSiderealOrbitalPeriod28.3daysSpeedofMoononitspath:~1000m/secDeflectionACurvedPathBut,ofcoursetheMoonreallymovesalongacurvedpath:Accordingtothefirstlaw,itisdeflectedfromastraight-linepathbytheforceofgravity.ThiscausesthemoontofallalittlebittowardstheEarth,deflectingitspathintoanarc.ThecurvedpathoftheMoonHowmuchdoesthemoonhavetofallin1secondto‘closetheloop’?Simplegeometry:0.00136meters(about1.4mm!)Newton’scleveridea:comparetoapple’sfall!ComparingappleandMoonAppleonEarthfallsin1second:dApple

=4.8mMoonfallsin1second:dMoon

=0.00136metersRatioofthesedistances=ratioofaccelerations=ratioofforcesdApple/dMoon=4.8m/0.00136m~3600ComparingappleandMoonRatioofforces:FApple/FMoon

~3600RatioofdistancesfromcenterofEarth:Dapple-Earth/DMoon-Earth~60Pricewinningquestion:Whatistherelationbetweenthesetworatios?Newton’sexactformulaTheforceofgravitationalattractionbetweenanytwomassivebodiesisproportionaltotheirmassesandinverselyproportionaltothesquareofthedistancebetweentheircenters.GravitationalForceLawF=forceduetogravity.M1=massofthefirstobjectM2=massofthesecondobjectd=distancebetweentheircenters.G=“GravitationalForceConstant”2M1M2dM1M2d2M12M2dM1M2dM1M22dM1M2d/2Whyisthissuchapowerfulconcept?

TheLawofGravityisUniversal:GovernsthefallofapplesontheEarthGovernsthefalloftheMoonaroundtheEarthGovernsthefalloftheEarth/MoonsystemaroundtheSunGovernsthefalloftheSunaroundthecenteroftheMilkyWayGalaxy.GovernsthefalloftheMilkyWayandAndromedaGalaxiesintheirmutualorbit...TheGravitationalForceConstantTheforceconstant,G,isanumberwhichgivesthesizeofthegravitationalcouplingbetweentwomassiveobjects.Gisverysmall,inmetricunits:G=6.710–11Newtonsmeter2/kilogram2TheNewtonisthemetricunitofforce:4.41Newtons=1poundGhastobemeasuredexperimentally.Example1:WeighingtheEarthMeasuretheaccelerationofgravitybydroppingweights(Galileo):a=9.8meters/second2MeasuretheradiusoftheEarthusinggeometry(Eratosthenes):RE=6378kilometers=6,378,000metersEarth’sMassis:Example2:TheConceptofMutualityWhatistheforceoftheEarthontheappleF=GMEMA/RE2Whatistheapple’sacceleration(2ndLaw):a=F/MA=GME/RE2

=9.8meters/sec2Theaccelerationduetogravityisindependentofthemassoftheapple!Example2:TheConceptofMutualityThethirdlawsaysthatallforcescomeinequalyetoppositepairs.WhatistheforceoftheappleontheEarthF=GMEMA/RE2HowmuchdoestheEarthacceleratetowardstheapple?a=F/ME=GMA/RE2a=9.8m/sec2

(MA/ME)=verysmallamountSecondLawofOrbitalMotionOrbitalmotionsconserveangularmomentum.Thisdoesn’tsoundmuchlike“equalareasinequaltimes”,butinfactitisthesamething.AngularMomentum:L=mvr=constantm=mass,v=velocity,r=distancefromthecenterofmass.AngularMomentum&EqualAreasLisaconstant.Ifthedistancechanges,thevelocitymustchangetocompensate:NearPerihelion:Planetisclosertothesun,hencesmallerrSpeedincreasesproportionallytocompensate.NearAphelion:Planetisfartherfromthesun,hencelargerrSpeeddecreasesproportionallytocompensate.ThirdLawofOrbitalMotionNewton’sGeneralizationofKepler’s3rdLaw:P=periodoftheorbita=semi-majoraxisoftheorbitM1=massofthefirstbodyM2=massofthesecondbodyAThirdLawforEveryBodyTheproportionalitynowdependsonthemassesofthetwobodies.ForplanetsorbitingtheSun,Msunissomuchbiggerthananyplanet(evenJupiter,at1/1000thMsun),werecoverKepler’sversion:MeasuringMassesNewton’sformofKepler’s3rdlawisawaytomeasuremassesfromorbitalmotions!MassoftheSunfromtheEarth’sorbit:Pearth=1year=3.156107secondsaearth=1AU=1.4961011metersUniversalMethodforMassesMeasuremassofJupiterfromtheorbitsoftheGalileanmoons,sinceMJupiter>>MmoonsFindMJupiter

300MearthMeasurethemassoftheEarthandMoonbymeasuringtheirorbitalparameters.Earthisonly~81xmoremassivethantheMoon,soyouhavetousethefullformula.Measurethemassesofbinarystarsusingthefullformula.ThePredictivePowerofGravityNewton’sdescriptionofplanetarypositionsisonlyastart.Italsoallowsquantitativenewpredictions.Halley’sComet:UsingNewtonianGravity,EdmundHalleyfoundthattheorbitofthegreatcometof1682wassimilartocometsseenin1607and1537.Predicteditwouldreturnin1758/59.Itdid,dramaticallyconfirmingNewton’slaws.TheDiscrepantOrbitofUranusIn1781,WilliamHerscheldiscoveredtheplanetUranusorbitingbeyondSaturn.By1840,thediscrepanciesbetweenthepredictedandactualpositionsofUranusgrewlargerthan1arcminute.Twotheorists,AdamsintheUKandLeverrierinFrance,predictedthatthedeviationswereduetothegravitationalinfluenceofanother,unknownmassiveplanetbeyondUranus.TheDiscoveryofNeptuneUsingthedeviantmotionsofUranus,theyindependentlycalculatedwherethisunknown8thplanetshouldbe.AdamswasignoredbyEnglishastronomers.LeverrierconvincedGalleinBerlintosearch.OnSept23,1845,GallefoundNeptuneonly52arcminutesfromwhereAdamsandLeverrierpredicteditwouldbe!TheWhyofPlanetaryMotionsKepler’sLawsaredescriptionsofthemotion:Arrivedatbytrialanderror,andsomevaguenotionsaboutcelestialharmoniesOnlydescribethemotions,withoutexplainingwhytheymovethatway.Newtonprovidestheexplanation:Kepler’sLawsareanaturalconsequenceofNewton’s3LawsofMotionandgravitation.Givesthelawspredictivepower.

雙星和恒星的質(zhì)量

(17.9)1.雙星(binarystars)由在彼此引力作用下以橢圓軌道互相繞轉(zhuǎn)的兩顆恒星組成的雙星系統(tǒng)。大部分的恒星位于雙星和聚星系統(tǒng)中。研究雙星的意義→驗證萬有引力定律→測量恒星質(zhì)量→研究恒星結(jié)構(gòu)（形狀、大小、大氣）→研究恒星演化→研究物質(zhì)交流和吸積過程2.目視雙星和恒星質(zhì)量的測定

(1)目視雙星(visualbinaries)在望遠(yuǎn)鏡內(nèi)能夠分辨出兩顆子星的雙星系統(tǒng)。Krueger60ABinaryStarSystem雙星的軌道運動

兩顆子星圍繞公共質(zhì)心作橢圓運動，半長徑分別為a1和a2.公共質(zhì)心位于橢圓的焦點上，子星在運動時與公共質(zhì)心始終位于一條直線上。橢圓軌道的大小與子星的質(zhì)量有關(guān)，

M1a1＝M2a2如果以一顆子星以參照點，另一顆子星的相對運動也是一個橢圓，其半長徑為

a＝a1+a2aa1a2CMRemovingthetiltfromavisualbinary'sorbit

目視雙星質(zhì)量的測定

利用Kepler第三定律和Newton萬有引力定律得到：以太陽－地球系統(tǒng)為參照其中a,P為雙星的軌道半長徑和周期。(2)天體測量雙星(astrometricbinaries)

某些雙星的一顆子星較暗，很難觀測到，但通過較亮子星的自行軌跡的變化推測其伴星的存在。雙星系統(tǒng)的質(zhì)心以直線運動，但每一顆子星的運動軌跡是波浪形的， 如天狼星（Sirius）。TheMotionofSiriusAandB3.分光雙星(spectroscopicbinaries)

通過子星軌道運動引起的譜線的Doppler位移確定其雙星性質(zhì)。 雙線、單線分光雙星。

譜線位移量也與雙星軌道傾角的大小有關(guān)。AABBBAAB視向速度曲線

(radialvelocitycurve)由子星譜線的Doppler位移得到的子星的視向速度隨時間的變化曲線。如子星1的軌道運動速度為V1,0，雙星軌道平面的法線與視線的夾角為i,它的視向速度為由于得到

且

4.食雙星(eclipsingbinaries)子星相互交食造成亮度變化的雙星。光變曲線(lightcurve)：子星間的相互交食造成雙星亮度的變化曲線。由光變曲線可以得到： 兩顆子星的溫度比、軌道傾角（→恒星質(zhì)量）和恒星的大小。

EclipsingBinaryTime31Brightness421324Algol(BetaPersei),thePrototypeEclipsingBinaryDipsfrommagnitude2.1to3.4every2.87days.Eacheclipse,includingthepartialphases,takesnearly10hours.

MeasureMassofaSingleStar

Astronomershavedirectlymeasuredthemassofasinglestar—thefirsttimeforanysolitarystarotherthanourownSun.Themeasurementhasbeendoneonasmallredstarlocatedsome1,800light-yearsfromEarth,byuseoftheeffectofmicrolensing.5.主序星的質(zhì)光關(guān)系和質(zhì)量－半徑關(guān)系

恒星的質(zhì)量和密度分布：~<0.1

~>10010-6

1.4

106褐矮星超巨星太陽