Download Lecture 4 - Concord University

PHYS420:Astrophysics
&Cosmology
Dr RichardH.Cyburt
AssistantProfessorofPhysics
Myoffice:402cintheScienceBuilding
Myphone:(304)384-6006
Myemail:[email protected]
Mywebpage:www.concord.edu/rcyburt
Inpersonoremailisthebestwaytogetaholdofme.
MyOfficeHours
TR5:30-17:00am
W5:00-5:00pm
Meetingsmayalsobearrangedatothertimes,byappointment
DouglasAdams
Hitchhiker’sGuidetotheGalaxy
KudostoKepler
Severalpointsworthnoting...
oAnamazingdiscovery–mathematicsunderliestheworkingsofthecosmos!
oOrbitshaveasimplegeometry...butnotsimplest:ellipsenotcircle
oKepler’slawsremain(almost)perfectlyaccuratetothisday– indeed,inslightly
generalizedformwillshowupinmany(most!)situationswheremotionsare
controlledbygravity
oYetnotewhatwestilldon’thave:anunderstandingofwhy Kepler’slawshold
→thatis,whatisthemechanism thatmakesplanetsmovethisway...forthat,
needtowaitforKepler’ssuccessors...
GalileoGalilei:Astronomer
FirsttousetelescopeinAstronomy
Contributions
omountainsonthemoon
omoonsofJupiter
oSunspots
ThesearebadforPtolemy(butmaybe
notdeadly)Q:how?
Galileoreportingtothefundingagency
GalileoGalilei:Astronomer
Crucial,decisiveexperiment:PhasesofVenus
PtolemaicPredictions
ObservedPhases
CapernicanPredictions
observationscontradictedAristotle,supportedCopernicus
“paradigmshift”(Kuhn)
radicalchangeinoutlook/conceptualframework
GalileoGalilei:Astronomer
Note:Galileoputontrial,forcedtorecantheliocentrism
ohiswork,Copernicus,Keplerbanneduntil1832
oofficialsemi-apology(“mistakesweremade”)1992
complex:crackdownasmuchpoliticalastheologicalshowsviewoftheworld
peoplehad
1. reallynotatallobviousthatsunatcenter
2. theparadigmshiftdifficult,challengedoutlook
Whatareyourthoughts?
DescriptionofMotion
Wantpreciselanguagenotjustforplanetsbutallobjects
Speed:rateofmotion𝑠𝑝𝑒𝑒𝑑 = '()*+,-*.-/0)*',
'()*+,-*0-1,
mathematically:𝑣 = 𝑑/𝑡(moretechnically𝑣 = 𝑑𝑥/𝑑𝑡)
𝑑 = 𝑣𝑡 ,distancetraveled=speedxtraveltime
Velocity:bothspeedanddirectionoftravel
◦ Example:10miEastin½hour,velocity=10/(½)=20mphEast
Q:cantwoobjectshavesamespeed,differentvelocities?
Q:doescarspeedometerreallymeasurespeedorvelocity?
Q:turncornerincar,speedometerpeggedat20mph-whassup?
DescriptionofMotion
Acceleration:Changeinvelocity
◦ Changeinspeed
◦ Changeindirectionofmotion
◦ Speeduprate
◦ Slowdownrate
Q:Whatkind(s)ofmotion(s)havezeroacceleration?
Intuitively:accelerationisrateofspeedinguporslowingdown
Sometimesit’susefultodistinguish:
◦ Acceleration=speedingup
◦ Deceleration=slowingdown
GalileoGalilei:Physicist
Studiedmotionofobjectsonearthtwoimportantcases:
SpecialMotionI:“FreeBody”
◦ movingwithnoexternalinfluences(includingfriction,gravity)
◦ →movesinstraightline,constantspeed→constantvelocity
Galileofindsthisisthe“naturalmotion”ofanobject
◦ keepsconstantspeed&directionunlesssomethinghappenstochangethis
ContrarytoAristotle:naturalmotionistocometorest
Q:WhydidAristotlethinkthis?
GalileoGalilei:Physicist
GalileanRelativity
o Motionofobjectsisalwaysdescribedasrelative tosomethingelse.
Forexample:
oYouwalkontheroad
relativetoEarth,but
Earthismovingrelative
totheSun.
oSoyourmotionrelative
totheSunisdifferentfrom
yourmotionrelativetoEarth.
GalileoGalilei:Physicist
GalileanRelativity
oVelocityisonlymeaningfulwrt aframeofreference
oi.e.abodyismovingrelativetoanotherbody
oThereexistsanabsolutespace(i.e.theUniverse)
oAllreferenceframesaremovingrelativetothisabsoluteframe
oThereisauniversal time
oAllclockstickatthesamerateinallreferenceframes
Thefundamentallawsofphysicsarethesameinallframesofreference
movingwithconstantvelocitywithrespecttooneanother.
Referenceframes
Coordinatetransformation
◦ 𝑡’ = 𝑡
◦ 𝑥’ = 𝑥– 𝑉𝑡
◦ 𝑦’ = 𝑦
◦ 𝑧’ = 𝑧
Timeisabsolute
Velocitytransformation
◦ 𝑣𝑥’ = 𝑣𝑥– 𝑉
◦ 𝑣𝑦’ = 𝑣𝑦
◦ 𝑣𝑧’ = 𝑣𝑧
V
GalileoGalilei:Physicist
Acceleration
◦ FormulatedbyGalileobasedonhisexperimentswithinclinedplanes.
Galileoincreasedtheinclinationofinclinedplanes.
◦ Steeperinclinesresultingreateraccelerations.
◦ Whentheinclineisvertical,accelerationisatmaximum,thesameasthatofa
fallingobject.
◦ Whenairresistanceisnegligible,allobjectsfallwiththesameunchanging
acceleration.
SpecialMotionII:“FreeFall”
◦ Motionduetogravityonly
GalileoGalilei:Physicist
Q:infreefall,isvelocityconstant?
Eveniffallinginastraightline,speedchanges
◦ Gravitycausesacceleration
◦ SameaccelerationforALLobjects
◦ Independentofsize,massFeathervsHammer
Einsteincalledthisindependencethe“equivalence
principle”crucialinhisinventionofGeneral
Relativity
Note:Galileodescribesmotion(mathematically)
buttoexplainwithatheoryfellto...
Issac Newton1643-1727
WhyKepler’slawsforplanets?
Aretheyspecial?
Canweunderstandusinggeneralrulesforallmotion?
Newconcepts
Mass:“amountofstuff”
◦ measureinkg→1kgofanythinghasthesamemass
Force:pushorpullonobject
◦ canhavemorethatoneacting,indifferentdirections
Motion&Forceslinked
Newton’sLaws:FirstLaw
NewtonI:LawofInertia
◦ anobjectatreststaysatrestifnonetforceactsonit
◦ anmovingobjectgoesinstraightlinew/const speed
◦ ifnoforcesactoniti.e.,“freebody”asperGalileo
sowesay:objectshave“inertia”or“momentum ”
willkeeptheirstateofmotion(i.e.,velocity)unlessanduntilanet
forceacts
Newton’sLaws:SecondLaw
NewtonII:“F=ma”
anetforceactingonanobjectcausesittoaccelerate
◦ 𝑎⃗ ∝ 𝐹⃗ :astrongerforcewillcauseanobjecttoacceleratefaster
◦ 𝑎 ∝ 1/𝑚:amoremassiveobjectwillbemoredifficulttoaccelerate
∑𝒊 𝑭𝒊 = 𝒎𝒂
2ndLawamathematicalmachinewhichpredictsfuture!
Newton’sLaws:SecondLaw
Input:atinitialtime,needtoknow/specify
◦ objectmassm
◦ allofforcesactingonm
1.
Findnetforce𝑭
2.
Turnthemathcrank:𝑎⃗ = 𝐹⃗ /𝑚
3.
Findacceleration =changeinvelocity
4.
Usethistofindnewposition,newvelocityatatmomentalittlelater
5.
Atnewtimeandposition,findnewnetforce...lather,rinse,repeat
Result:findparticlepathinfuture!
Butalso:canmathematically“runthemoviebackwards”andpredictthepast
historyaswell!
Newton’sLaws:ThirdLaw
NewtonI:LawofInertia
aruleabouthowforcesbehavebetweentwoobjects
if2bodiesinteract:
theforce exertedbyobject1onobject2
isequalandoppositeto
theforce exertedbyobject2onobject1
ExplainingKepler
Kepler’sLaws
I.
LawofEllipses:PlanetorbitsareellipseswiththeSunatafocus
II. LawofEqualAreas:Equalareasoftheellipsearesweptthroughinequaltimes
III. LawofHarmonies:𝑷𝟐𝒚𝒓𝒔 = 𝒂𝟑𝑨𝑼
LawsIandIIsayaplanet’svelocitychanges→planetsaccelerate
Needaforce⇒ gravity
UniversalGravitation
◦ gravityisaforce
◦ theforceofgravityonyouisyourweight
◦ gravityactsbeyondearth
◦ gravitydirectedonlineconnectingcentersofbodies
◦ gravitystrengthdecreaseswithdistance
◦ allobjectswithmassaresourcesofgravity
⇒ everythingattractseverythingelse
UniversalGravitation
Cansummarizemathematicallycompactway:
for2bodies,massesm1,m2 centersseparatedbydistanceR
𝐹𝑔𝑟𝑎𝑣 ∝ 𝑚1
gravitationalforce:𝐹𝑔𝑟𝑎𝑣 ∝ 𝑚2
𝐹𝑔𝑟𝑎𝑣 ∝ 1/𝑅2
Newton’s UniversalLawof Gravitation
“inversesquarelaw”
𝐺𝑚1𝑚2
𝐹𝑔𝑟𝑎𝑣 =
𝑅2
whereG isjustafixed,constantnumber,samealways:
𝑚3
YZZ
𝐺 = 6.7×10
𝑘𝑔𝑠2
GravityandPlanetaryMotion
Newton2nd Law:inputisforce,outputismotion
ForplanetsaroundSun,forceisgravity(freefall!)
GivenNewton’sUniversalLawofGravity,whataretheplanets’motion?
Newton’s2nd Law+Gravity:PropertiesofPredictedOrbits
Orbitiselliptical,withSunatonefocus
Equalareasinequaltimes
𝟐
𝟒𝝅
𝑷𝟐𝒚𝒓𝒔 = 𝒂𝟑𝑨𝑼,actuallyNewtondoesbetter:𝑷𝟐 = 𝒂𝟑
𝑮𝑴⨀
GravityandPlanetaryMotion
TheoryagreeswithObservations!!AllKepler’sLawsverified!
Onlyforceactingonaplanetisgravity 𝑚bc)*,0 𝑎bc)*,0 = 𝐹bc)*,0 = 𝑎bc)*,0 = de⨀ 1fghijk
lm
de⨀
lm
FreefallaccelerationonlydependsonmassofSun,𝑀⨀ andSunplanetorbitaldistanceR.
◦ Independentofplanetmassandsize!
◦ AtsameR,allobjectsacceleratesameway
◦ EquivalencePrinciplepopsoutofNewton’sGravity!!!Wooo-hooo!!!
TestingNewton’sGravity
MoonsofJupiter:orbitsobeyKepler’slawsbutinNewton’smore
generalform,using𝑀𝐽𝑢𝑝𝑖𝑡𝑒𝑟
◦ Jupiter’sgravityworksliketheSun’sandEarth’s…..GravityisUniversal
1830’s:UranusobsersedorbitdidNOT followpredictionsof
Newtoniansolarsystemmodel
◦ ThedeathofNewton’sgravity?
Remember:HavetoagreewithALL data,notjustsomeevenone
clearfailureisenoughtokilltheorye.g.,KeplerandMars:justa
smalldiscrepancyfromcircularbutstillhadtothrowoutcircular
orbits
Whatareyourthoughts?
th
19
CenturyDarkMatter
tosaveNewton’sgravity:needunseenmassiveobject
◦ couldhavebeencalled“darkmatter”!
whosegravityisalteringUranus’motion
basedonUranus’knownpath,canuseNewton’sgravitytopredict
wherethe“darkmatter”shouldbe!
⇒ Crucialexperiment:golooktherewithtelescope!
Thiswasdone,in1846….
NewtonTriumphs!
planetNeptunefoundat“darkmatter”positionpredictedbyNewton’sgravity!
Othermorerecenttests:
BinaryStars:twostarsorbitingeachothermoveinellipses,obeyKepler’slaws
BinaryStarsOrbit
→Gravitytheoryworksoutsidesolarsystem
Cosmology:ProgressReportafterNewton
Recall:Cosmologyisamysterystory
Basedonevidencewecanobserve,wanttounderstand:
◦ structureofUniverse:howbig?mapofcontentsacrossspace?
◦ contentsofUniverse:whatingredientsisitmadeof?
◦ origin&evolutionofUniverse:birth?past?future?
◦ rulesofthegame:whatmakestheUniversethisway?
Cosmologicalprogresssofar:
◦ Copernicus’Sun-centered“Universe”
◦ Keplerpreciselydescribedplanetarymotion
◦ Newtonexplainedplanetarymotion,agreesw/observations
LessonsfromNewtonianTriumph
LessonfromNewtoniansuccess:insightintorulesofthegame
◦ powerfullawsexistexplainingallmotionduetoforces
◦ planetarymotionisduetoforceofgravity
⇒ gravitydeterminesbehaviorofthecosmos!
AlsoimportantimplicationsforstructureofUniverse
Whatareyourthoughts?
LessonsfromNewtonianTriumph
Recall:tonakedeye,stellarparallaxisundetectablestarsdon’tseem
to“wiggle”annuallyonsky
soifparallaxexists,wigglemustbesmall
⇒ starsmustbeveryfaraway!
Butifstarshugelyfaraway,andcanstillbeseentheymustemitalot
oflight
→perhapsstarsareliketheSun!
Wheredowegofromhere?