1.6 Angular distance between two points on a sphere

73 Pages·2017·2.77 MB·English

by  Edward Brown

EDWARD BROWN
P L A N E T S A N D
T E L E S C O P E S
Aboutthecover: animageofRheaoccultingSaturn,ascapturedbytheCassinispacecraft.
Credit: CassiniImagingTeam,SSI,JPL,ESA,NASA
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Preface
Thesenoteswerewrittenwhileteachingasophomore-levelastronomy
course,“PlanetsandTelescopes”atMichiganStateUniversityduring
SpringSemestersof2015and2016. Thebackgroundrequiredisintro-
ductorycalculusandfreshman-levelphysics.
Inthefirstyear,themaintextwasLissaueranddePater,Bennett
etal.1;inthesecondyear,weswitchedtoRydenandPeterson2andTay- 1JackJ.LissauerandImkedePater.
lor3andincreasedtheamountoftimespentonbasicsofastronomical FundamentalPlanetaryScience:Physics,
ChemistryandHabitability. Cambridge
observationandstatisticalanalysis. Someofthenotesandexerciseson
UniversityPress,2013;andJeffreyO.
statisticsarewrittenintheformofJupyterNotebooks;theseareinthe Bennett,MeganO.Donahue,Nicholas
Schneider,andMarkVoit. TheCosmic
folderstatistics/notebooks.
Perspective. Addison-Wesley,7thedition,
Thetextlayoutusesthetufte-book(https://tufte-latex. 2013
github.io/tufte-latex/)LATEXclass: themainfeatureisalargeright 2BarbaraRydenandBradleyM.Peterson.
FoundationsofAstrophysics. Addison-
margininwhichthestudentscantakenotes;thismarginalsoholds
Wesley,2010
smallfiguresandsidenotes. Exercisesareembeddedthroughoutthetext. 3JohnR.Taylor. AnIntroductiontoError
Theserangefrom“readingexercises”tolonger,morechallengingprob- Analysis. UniversityScienceBooks,
Sausalito,CA,2ndedition,1997
lems. Becausetheexercisesareembeddedwiththetext,alistofexercises
isprovidedinthefrontmattertohelpwithlocatingmaterial.
Inthecourse,aboutthreeweekswerespentcoveringthematerialin
AppendixC,“ProbabilityandStatistics”. Thiswasdonebetweencovering
Chapter2,“LightandTelescopes”andChapter4,“DetectionofExoplan-
ets”. Thisorderingwasdrivenbythedesiretokeepthelecturesandlabs
synchronizedasmuchaspossible. InChapter1,“Coordinates”,severalof
theexercisesrefertothenightskyasviewedfrommid-Michiganinlate
January.
Iamgratefulformanyconversationswith,andcriticalfeedbackfrom,
Prof.LauraChomiuk,whotaughtthelabsectionofthiscourse,graduate
teachingassistantsLauraShishkovskyandAlexDeibel,andundergrad-
uatelearningassistantsEdwardBuieIII,AndrewBundas,ClaireKopen-
hafer,PhamNguyen,andHueiSears.
Thesenotesarebeingcontinuouslyrevised;torefertoaspe-
cificversionofthenotes,pleaseusetheeight-characterstamplabeled
“gitversion”onthefrontpage.

Contents
1 Coordinates: SpecifyingLocationsontheSky 1
2 LightandTelescopes 7
3 Spectroscopy 13
4 DetectionofExoplanets 19
5 BeyondKepler’sLaws 25
6 PlanetaryAtmospheres 31
A ConstantsandUnits 39
B MathematicsReview 41
C ProbabilityandStatistics 45
Bibliography 63

List of Figures
1.1 Thecelestialsphere 1
1.2 Theecliptic 2
1.3 Rightascensionanddeclination 3
1.4 ThemovementoftheEarthfromnoontonoon 4
1.5 Theparallaxangleofastar 5
1.6 Angulardistancebetweentwopointsonasphere 5
1.7 Angulardistancebetweenlinesofconstantrightascension 6
2.1 Theelectricforceinalightwave 8
2.2 Top: reflectionoflightfromasurface. Bottom: refractionoflightasit
passesfromamediumwithindexn intoamediumwithindexn . 9
1 2
2.3 Changeinangularsizeofanobjectinwater. 9
2.4 Aplanewaveincidentonadetector 10
2.5 Additionofvectorswithphasedifferences 11
2.6 Illustrationofairmass 12
3.1 Spectrallinesofneutralhydrogen. 14
3.2 Adiffractiongrating. 15
3.3 Takingaspectrumofanastronomicalobject. 16
3.4 Schematicofthedopplereffect 18
4.1 Centerofmass 20
4.2 Orbitalelements 21
4.3 Schematicoftheinclinationofaplanetaryorbit 21
4.4 Schematicofaplanetarytransit 22
4.5 Schematicoftheprobabilitydistributionoforbitalinclination 22
5.1 Fourfreelyfallingbodies 25
5.2 TidalforceontheEarth 26
5.3 Componentsofthetidalforce 26
5.4 TorqueontheEarth’stidalbulge 27
5.5 Polarcoordinates 27
5.6 Changeinunitvectorsunderrotation 27
5.7 Movementonamerry-go-round 28
viii
5.8 LagrangepointsforasystemwithM =0:1M . 29
2 1
6.1 Afluidelementinhydrostaticequilibrium 31
6.2 Themassofacolumnoffluid 32
6.3 Motioninahorizontallayerinasmallregionatlatitudeλ. 36
6.4 Forcesonaparcelofaircirculatingaboutalow. 36
B.1 Constructionfromtheunitcircle 41
B.2 Schematicoftheadditionoftwoangles 42
C.1 Sets 47
C.2 Thecomplementofaset 47
C.3 Theunionoftwosets 47
C.4 Theintersectionoftwosets 47
C.5 ThePoissondistribution 53
C.6 Normaldistributionswithdifferentmeans 57
C.7 Normaldistributionswithdifferentstandarddeviations 57
C.8 Proabilityregionsforoneandtwostandarddeviations 58
List of Exercises
1.1 AltitudeofBetelgeuse 2
1.2 CurrentrightascensionoftheSunandBetelgeuse 3
1.3 Coordinatesystems 3
1.4 Currentsiderealtime 4
1.5 AngularsizeofthePleiades 6
2.1 Relationbetweenmagnitudeandflux 8
2.2 TheB(cid:0)Vindex 8
2.3 Magnificationofanobjectinwater 9
2.4 Diffractionofimageofapointsource 11
2.5 Resolvingpowerofvariousinstruments 11
2.6 Angularsizeofstar,planet 12
3.1 Lightsensitivityofhumaneye 13
3.2 Diffractionbyacompactdisk 15
3.3 DispersionoftheGoodmanspectrograph 16
3.4 Dopplershiftofradarbeam 18
4.1 Meandistancetostarinasample 19
4.2 Ratiooffluxfromplanet,star 19
4.3 Resolvingplanets 19
4.4 Center-ofmassforSun-Jupiter 20
4.5 OrbitalspeedoftheSun 21
4.6 DopplershiftofSun 21
4.7 Inclinationrequiredtoobservetransit 22
4.8 Mass-functionanddistributionofmasses 23
4.9 Fractionoforbitintransit 23
5.1 Demonstrationoftidalacceleration 25
5.2 Radialcomponentoftidalforce 26
5.3 Coriolisaccelerationonmerry-go-round 28
5.4 VanishingaccelerationatL 29
4
5.5 HillradiusoftheSun-Jupitersystem 29
5.6 OverflowofRochelobe 29
x
6.1 Pressureincreaseinwater 31
6.2 Scaleheightfordryair 33
6.3 Adiabaticlapserate 35
6.4 Coriolisvs.centripetalaccelerationaroundariverbend 36
6.5 Scaleofmid-latitudeweathersystems 36
6.6 Whythestrongeststormsareassociatedwithlow-pressuresystems 37
C.1 Meaningof“probability” 46
C.2 Probabilityofvariousdrawsfromadeckofcards 47
C.3 Probabilityoftwoeventsthatarenotmutuallyexclusive 48
C.4 RecurrencerelationofPascal’striangle 49
C.5 Probabilityofdrawingaflush 50
C.6 Probabilitydistributionforacoinflip 51
C.7 Scoreforrandomguessingonamultiple-choiceexam 51
C.8 Probabilityofmatchingbirthdays 54
C.9 RecurrencerelationsforPoissondistribution 56
C.10Expectationvalueofstandarddeviation 59
C.11Propagationofuncertainties 60
C.12Illustrationoferrorinthemean 61