AN INTRODUCTION TO ASTROPHYSICS AND COSMOLOGY BY ANDREW NORTON

239 Pages·2012·4.36 MB·English

Contents
AN INTRODUCTION TO ASTROPHYSICS
AND COSMOLOGY
BY ANDREW NORTON
Chapter 1 Manipulating numbers and symbols 7
Introduction 7
1.1 Algebraandphysicalquantities 7
1.1.1 Manipulating algebraic expressions 7
1.1.2 Rearranging algebraic equations 9
1.1.3 Solvingsimultaneous equations 10
1.2 Powers,rootsandreciprocals 10
1.2.1 Combiningpowers 11
1.2.2 Solvingpolynomial equations 12
1.3 Imaginary numbers 13
1.4 Unit(dimensional) analysis 14
1.5 Functionnotation 14
1.6 Powersoftenandscientificnotation 15
1.7 Significantfigures 16
1.8 Experimental uncertainties 17
1.8.1 Typesofuncertainty 17
1.8.2 Estimatingrandomuncertainties 19
1.8.3 Uncertainties whencounting randomlyoccurring events 20
1.8.4 Theuncertainty inameanvalue 21
1.8.5 Combininguncertainties inasinglequantity 21
1.9 Logarithmsandlogarithmicfunctions 22
1.10 Graphs 24
1.10.1 Straight-line graphs 25
1.10.2 Makingcurvedgraphs straight 27
1.11 Angularmeasure 28
1.12 Trigonometry 29
1.12.1 Trigonometric ratios 29
1.12.2 Thesineruleandcosinerule 30
1.12.3 Trigonometric functions 31
1.12.4 Inversetrigonometric functions 33
1.13 Vectors 34
1.13.1 Vectorcomponents 34
1.13.2 Additionandsubtraction ofvectors 35
1
Contents
1.13.3 Positionanddisplacement vectors 36
1.13.4 Unitvectors 36
1.13.5 Thescalarproduct 37
1.13.6 Thevectorproduct 37
1.14 Coordinates 38
1.15 Scalarandvectorfields 40
1.16 Matrices 42
1.16.1 Combiningmatrices 42
1.16.2 Specialtypesofmatrices 44
1.16.3 Transposing matrices 45
1.16.4 Thedeterminant ofamatrix 45
1.16.5 Adjointandreciprocal matrices 46
Chapter 2 Stars and planets 51
Introduction 51
2.1 Measuringstarsandplanets 51
2.2 Unitsinastrophysics 53
2.3 Positions, distances andvelocities 54
2.3.1 Observingthepositions ofstars 54
2.3.2 Measuringthevelocities ofstars 58
2.4 Spectraandtemperatures 59
2.5 Luminositiesandfluxes 63
2.6 Astronomicalmagnitudes 63
2.7 Coloursandextinction 66
2.8 TheHertzsprung–Russell diagram 67
2.9 Massesofstars 69
2.10 Lifecyclesofstars 76
2.11 Stellarend-points 81
2.12 Planetarystructure 81
2.12.1 Terrestrialplanets 81
2.12.2 Giantplanets 83
2.13 Extrasolarplanetsandhowtofindthem 86
2.14 Astronomicaltelescopes 89
2.14.1 Telescopecharacteristics 89
2.14.2 Telescopesinotherpartsoftheelectromagneticspectrum 92
2
Contents
Chapter 3 Galaxies and the Universe 97
Introduction 97
3.1 TheMilkyWay–ourgalaxy 97
3.2 Othergalaxies 98
3.2.1 Classification ofgalaxies 98
3.2.2 Originandevolution ofgalaxies 101
3.2.3 Measuring galaxyproperties 101
3.3 Thedistances toothergalaxies 102
3.4 Activegalaxies 105
3.4.1 Thespectraofactivegalaxies 105
3.4.2 Typesofactivegalaxy 110
3.5 Thespatialdistribution ofgalaxies 113
3.6 Thestructure oftheUniverse 115
3.7 Theevolution oftheUniverse 120
3.8 Observational cosmology 123
3.9 Cosmological questions 125
Chapter 4 Calculus 129
Introduction 129
4.1 Differentiation andcurvedgraphs 129
4.2 Differentiation ofknownfunctions 131
4.3 Theexponential function 133
4.4 Thechainrule 135
4.5 Logarithmicdifferentiation 138
4.6 Expansions 139
4.7 Partialdifferentiation 142
4.8 Differentiation andvectors 143
4.9 Differential equations 144
4.10 Integration andcurvedgraphs 146
4.11 Integration ofknownfunctions 147
4.12 Integration bysubstitution 149
4.13 Integration byparts 152
4.14 Multipleintegrals 153
3
Contents
Chapter 5 Physics 159
Introduction 159
5.1 Describingmotion 159
5.1.1 Motioninonedimension 159
5.1.2 Motionintwoorthreedimensions 161
5.1.3 Periodicmotion 162
5.2 Newton’slaws 163
5.2.1 Newton’slawsofmotion 164
5.2.2 Newton’slawofgravitation 165
5.3 Relativisticmotion 166
5.4 Predictingmotion 168
5.4.1 Work,energy, powerandmomentum 168
5.4.2 Relativisticmechanics 170
5.5 Rotationalmotion 171
5.6 Propertiesofgases 174
5.7 Atomsandenergylevels 178
5.7.1 Atomicstructure 178
5.7.2 Photonsandenergy levels 180
5.8 Quantumphysics 184
5.8.1 Wavemechanics 185
5.8.2 Quantummechanicsinatoms 188
5.9 Quantumphysicsofmatter 191
5.9.1 Quantumgases 191
5.9.2 Nuclearphysics 194
5.9.3 Particlephysics 199
5.10 Electromagnetism 201
5.10.1 Electricityandmagnetism 202
5.10.2 Electromagnetic waves 203
5.10.3 Spectra 206
5.10.4 Opacityandopticaldepth 210
Solutions 219
4
Introduction
In order to successfully study one or both of the OpenUniversity’s Level3
courses, S382Astrophysics orS383TheRelativistic Universe, youshouldalready
be familiar withvarious topics in cosmology, astronomy, planetary science,
physics andmathematics. Thelevelofskills, knowledge andunderstanding that
weexpect youtohavewhenyouembarkoneither ofthesecourses isequivalent
totheend-points oftheOU’sLevel2courses: S282Astronomy, S283Planetary
Science and theSearch for Life, SXR208Observing theUniverse, S207The
PhysicalWorld andMST209Mathematical MethodsandModels.
ToascertainwhetherornotyoumeettherequiredlevelbeforeembarkingonS382
and/or S383youshould workthrough thedocument entitled AreYouReadyFor
S382 orS383? whichisavailable from the Courses website. If, asaresult of
attemptingthequestions inthatdocument,yourealisethatyouneedtoreviseyour
skills, knowledge andunderstanding incertain areas ofmathematics, physics,
cosmology, astronomy andplanetary science, thenyoushouldstudy therelevant
chapters ofthisdocument carefully.
There are fivemain chapters to this document – one each to introduce the
astronomy andplanetary science, thecosmology andthephysics background,
plustwochaptersofmathematics. Itisimportanttonotethat,becausemostofthis
document revisits concepts andphenomena thatarecovered indetail inLevel2
OpenUniversity courses, the treatment here ismuchlessrigorous than inthe
courses themselves. Forthe mostpart, the subjects covered here aremerely
presented toyouratherthandevelopedgradually throughdetailed argument. This
istoenable youtogetrapidly ‘tothepoint’ andappreciate thekeyinformation
you need inorder to understand whatfollows, and to allow you to progress
quickly tothemainsubstance oftheLevel3courses.
5
Introduction
Acknowledgements
Thematerialinthisdocument hasbeendrawnfromthephysics, mathsand
astronomy thatistaughtinvarious otherOUcourses, including S282,S283,
SXR208,S207,S103andS151. Theauthors oftherelevantparts ofthose
courses: David Adams, John Bolton, David Broadhurst, Jocelyn Bell
Burnell, DerekCapper, AlanCayless, AndrewConway,AlanCooper, Dan
Dubin, AlanDurrant, TonyEvans, Stuart Freake, Iain Gilmour, Simon
Green, IainHalliday, CaroleHaswell, KeithHiggins, KeithHodgkinson,
AnthonyJones,BarrieJones, MarkJones,SallyJordan, UlrichKolb,Robert
Lambourne, RayMackintosh, LowryMcComb,JoyManners,DavidMartin,
PatMurphy, Andrew Norton, LesleyOnuora, John Perring, Michael de
Podesta, Shelagh Ross, DavidRothery, SeanRyan, IanSaunders, Mark
Sephton, Richard Skelding, TonySudbery, Elizabeth Swinbank, John
Zarnecki andStanZochowskiaregratefully acknowledged, alongwiththe
othermembersoftheteamsresponsible forthosecourses.
Gratefulacknowledgement isalsooffered toCarolinCrawfordforcritically
reading andAmandaSmithforproof-reading thisdocument, although any
remaining errorsaretheresponsibility oftheeditor.
Grateful acknowledgement ismadetothe following sources offigures:
Figure1.4(Photograph ofJupiter): NASA/SciencePhotoLibrary; Figure
1.4(Photograph oftheEarth): NASA;Figure1.4(Photograph ofagalaxy):
TheRegents, UniversityofHawaii;Figure2.2: TillCredner, Allthesky.com;
Figure 2.29: Observatoire deParis; Figure2.31: SKorzennik, Harvard
UniversitySmithsonianCenterforAstrophysics; Figure3.10: Lee,J.C.etal.
(2002) ‘Theshape ofthe relativistic Iron KαLinefrom MCG6-30-15
measured withtheChandra highenergy transmission grating spectrometer
and theRossi X-Raytiming explorer’, Astrophysical Journal, Vol 570.
c TheAmerican Astrophysical Society; Figure3.13: LeGrandAtlasde
(cid:13)
l’Astronomie 1983. Encyclopaedia Universalis; Figure3.14: W.N.Colley
and E. Turner (Princeton University), J. A. Tyson (Bell Labs, Lucent
Technologies) and NASA;Figure 3.17: Adapted from Landsberg, P.T.
and Evans, D.A.Mathematical Cosmology. 1977, Oxford University
Press;Figure3.21: AdaptedfromSchwarzschild, B.(1998) ‘VeryDistant
Supernovae suggest that thecosmic expansion isspeeding up’, Physics
Today, June 1998. American Institute ofPhysics; Figures 3.22and3.23:
Bennett, C.L.etal. ‘FirstYearWilkinson Microwave Anisitrophy Probe
(WMAP)Observations’, Astrophysical Journal SupplementSeries,Volume
148,Issue1.
Every effort has been made to contact copyright holders. Ifany have
been inadvertently overlooked wewillbepleased tomakethenecessary
arrangements atthefirstopportunity.
6
Chapter 1 Manipulating numbers
and symbols
Introduction
Inthischapter, wewillconcentrate onthevariousrulesformanipulating numbers
and algebraic symbols, including how to manipulate equations containing
fractions, powers,logarithms andtrigonometric functions, andhowtodealwith
vectorsandmatrices.
1.1 Algebra and physical quantities
Physical quantities, such asmassandposition, are commonly represented by
algebraic symbolssuchasmorx. Wheneversuchsymbolsareused,itshouldbe
recalledthattheycomprisetwoparts: anumericalvalueandanappropriate unitof
measurement, such asm =3.4kgor x=6.0m. Theunits willgenerally be
internationally recognized SIunits, although inastrophysics andcosmology, cgs
unitsandotherlessconventional unitsareusedwhereconvenient.
Quantities maybecombined using the standard operations of addition (+),
subtraction ( ),multiplication ( )ordivision (/or ). Notethat theorderof
− × ÷
addition ormultiplication isnotimportant; i.e. a+b= b+a,anda b= b a,
× ×
buttheorderofsubtraction anddivisionis;i.e. a b = b a,anda/b = b/a.
− 6 − 6
Wheneverquantities arecombined, theirunitsarecombined inthesameway. For
example, ifp = mv,wheremassmismeasured inkgandspeedv ismeasured
in metres persecond (m/s orm s−1), then their product p willhave units of
kilograms timesmetrespersecond, orkgms−1,pronounced ‘kilogram metres
persecond’. Iftwoquantities aretobeaddedorsubtracted, thentheymusthave
thesameunits. (Unitanalysisisdiscussed inSection1.4.)
Bothlowercase and upper case letters areused asalgebraic symbols, and in
general willrepresent different quantities withdifferent units. Forinstance, g
is often used to represent the acceleration due togravity near tothe Earth’s
surface (9.81 m s−2), whilst G is the universal gravitational constant
(6.67 10−11 Nm2 kg−2). Notealsothat(upper andlowercase)Greekletters
×
arefrequently usedassymbols forphysical quantities. Youwillsoon become
familiarwiththelettersthatarecommonlyused.
1.1.1 Manipulating algebraic expressions
Themostimportantruletonotewhenmanipulating algebraic expressions is:
Algebraic symbols aremanipulated inthesamewayaspurenumbers and
algebraic fractions aremanipulated inexactly thesamewayasnumerical
fractions.
7
Chapter 1 Manipulating numbers and symbols
Thefollowing examples illustrate someoftherules ofmanipulating algebraic
expressions. Tomultiply one bracket byanother, multiply each term in the
right-hand bracket byeachtermintheleft-hand one,takingcareful accountofthe
signs,asinthefollowingtwocases:
(a+b)(c+d)= a(c+d)+b(c+d) = ac+ad+bc+bd
(a+b)(a b) = a(a b)+b(a b) = a2 ab+ba b2 = a2 b2
− − − − − −
Exercise1.1 Multiplyoutthefollowing expressions toeliminate thebrackets.
(a) t[2 (k/t2)] (b) (a 2b)2
− −
n
Tomultiply fractions, multiply thenumerators (top lines) together and then
multiplythedenominators (bottomlines)together. Forexample, 2 3 = 6 = 1.
3 × 4 12 2
Soingeneral,
a c ac
= (1.1)
b × d bd
Todivide byafraction, multiply byitsreciprocal (i.e. bythefraction turned
upsidedown). Forexample, 1 1 = 1 6 = 2. Soingeneral,
3 ÷ 6 3 × 1
a/b a c a d ad
= = = (1.2)
c/d b ÷ d b × c bc
Inordertoaddorsubtract twofractions, itisnecessary forthembothtohavethe
samedenominator. Innumericalwork,itisusuallyconvenient topickthesmallest
possible number forthis denominator (thelowest commondenominator), for
example,
1 1 2 1 2 1 1
= = − =
3 − 6 6 − 6 6 6
Ifthelowestcommondenominator isnoteasytospot, youcanalwaysmultiply
thetopandbottom ofthefirstfraction bythedenominator ofthesecond fraction,
andthetopandbottom ofthesecond fraction bythedenominator ofthefirst,for
example,
1 1 1 6 1 3 6 3 3 1
= × × = = =
3 − 6 3 6 − 6 3 18 − 18 18 6
× ×
Thisisthemethodtoapplytoalgebraic fractions:
1 1 b a b+a
+ = + = (1.3)
a b ab ab ab
1 1 b a b a
= = − (1.4)
a − b ab − ab ab
Exercise 1.2 Simplifythefollowing expressions: (a) 2xy z (b) a2−b2
z ÷ 2 a+b
(c) 2 + 5 (d) a c
3 6 b − d
n
Evenifthenumerical valuesofalgebraic quantities areknown,itisadvisable to
retainthesymbolsinanyalgebraic manipulations untiltheverylaststepwhenthe
numerical values canbesubstituted in. Thisallowsyoutoseetheroleofeach
quantityinthefinalanswer,andgenerally minimizeserrors.
8
1.1 Algebraand physical quantities
1.1.2 Rearranging algebraic equations
Physicallawsareoftenexpressed usingalgebraic equations whichmayhavetobe
rearranged toobtain expressions forthequantity orquantities ofinterest. For
example, theequation relating thepressure, volumeandtemperature ofagasis
usually writtenPV = NkT,butitmaybethatwewishtoobtainanexpression
forT intermsofotherquantities. Thebasicruleformanipulating anequation is
thattheequality mustnotbedisturbed. Thatis,
Whateveryoudoononesideofthe‘equals’ sign, youmustalsodoonthe
otherside,sothattheequalityofthetwosidesismaintained.
Thisisillustrated bythefollowingexamples. (Donotbeconcerned atthisstage
withthemeaningofthevarioussymbols.)
Worked Example 1.1
Essentialskill:
(a) Rearrange PV = NkT toobtainanexpression forT. Rearranging equations
(b) Rearrangev = u+attofindanexpression fort.
(c) Rearrange E = 1mv2 toobtainanexpression forv.
2
(d) Rearrangeω = g toobtainanexpression forL,whereω istheGreek
L
letteromega. q
Solution
(a) Toisolate T on theright-hand side, divide both sides by Nk. So
PV/Nk = NkT/Nk = T,i.e. T = PV/Nk.
(b) Firstsubtractufrombothsides,togivev u= u+at u= at. Then
− −
dividebothsidesbyatogive(v u)/a = at/a= t. Hencet = (v u)/a.
− −
(c) The easiest route isfirst toisolate v2 onone side of the equation.
Sofirst multiply both sides by 2and divide both sides by m to give
2E/m = 2mv2/2m = v2. Thentakingthesquarerootofbothsides,
2E
v =
± m
r
Thesquare rootofanumberhastwovalues, onepositive andonenegative.
Thesquarerootsymbol(√)denotes onlythepositivesquareroot,hencethe
needfor the‘ ’sign, whichisreadas‘plus orminus’. Thisreflects the
±
mathematics oftheproblem. Sometimesthephysics oftheproblem allows
youtorule outone ofthese twovalues. Forexample, ifv represented a
speed, thenitwouldhavetobegreater thanorequaltozero, andthusonly
thepositivesquarerootwouldberetained.
(d) Thefirststepistosquarebothsidesoftheequation: ω2 = g/L,andthe
nextstepistomultiplybothsidesbyL/ω2,togiveL = g/ω2.
9
Chapter 1 Manipulating numbers and symbols
Havingseensomeexamples,trythefollowingforyourself.
Exercise1.3 Rearrange eachofthefollowingequations togiveexpressions for
themassmineachcase.
(a) E = GmM (b) E2 = p2c2+m2c4 (c) T = 2π m
− r k
n
p
1.1.3 Solving simultaneous equations
Twodifferent equations containing thesametwounknown quantities arecalled
simultaneous equations if both equations must be satisfied (hold true)
simultaneously. Itispossible tosolvesuchequations byusingoneequation to
eliminateoneoftheunknownquantities fromthesecond equation. Anexample
shouldmaketheprocedure clear.
Worked Example1.2
Inacertainbinarystarsystemitisdetermined thatthesumofthemassesof
Essentialskill:
thetwostarsism +m = 3.3timesthemassoftheSun,whilsttheratioof
Solvingsimultaneous equations 1 2
thetwomassesism /m = 1.2. Whataretheindividual massesofthetwo
1 2
stars?
Solution
Ifwerewritethefirstequation togiveanexpression form intermsofm ,
1 2
thenwecaninsertthisresultintothesecondequation togiveanexpression
form alone.
2
Rearrangement ofthefirstequation givesm = 3.3 m ,thensubstituting
1 2
−
form inthesecondequationgives(3.3 m )/m = 1.2. Multiplyingboth
1 2 2
−
sidesoftheequation bym ,wehave(3.3 m ) = 1.2m ;thenaddingm
2 2 2 2
−
toboth sidesgives3.3 = 2.2m ;fromwhichclearly m = 3.3/2.2 = 1.5
2 2
timesthemassoftheSun. Substitution form into eitherofthefirsttwo
2
equations showsthatm = 1.8timesthemassoftheSun.
1
Notetheimportantfactthatinordertofindtwounknowns,twodifferentequations
relating themarerequired. Byextension, itisalwaysnecessary tohaveasmany
equations asthereareunknowns. Youwillfindyourself constantly applying this
principleasyousolvenumerical problemsinastrophysics andcosmology.
Exercise1.4 Solvethefollowing pairsofequations tofindthevaluesofaand
b. (a) a b= 1anda+b= 5 (b) 2a 3b = 7anda+4b = 9
− −
n
1.2 Powers, roots and reciprocals
Thepowertowhichanumberisraised isalsocalled itsindex orexponent. So
24 = 2 2 2 2 = 16canbesaidas‘twotothepoweroffour’orsimply‘two
× × ×
10