Brightness,BrightnessIndex,Luminosity,StarParallaxandStar亮度，亮度指数，亮度，恒星视差和明星

11/7/061011:30AM1011/7/06BrightnessBrightnessIndexLuminosityStarParallaxandStarDistan__INTRODUCTIONOfthelistofstarpropertiesinthetitledistan__andluminosityaretheonesofmostinteresttoastronomers.Theothersaretoolsusedinfindingdistan__andluminosity.Luminosityisanintrinsicorinvariantstarproperty.Itisthetotalamountofpowerthestarcanradiate.Atleastinahu__nlifetimeorseveralhu__nlifetimesthepo​werradiatedisnotgoingtochange.Overitslifetimeonasequen__thepowerwill__rtainlychangebutitisstilljustifiedtocallthestar’sradiatedpoweraninvariantproperty.Astar’sdistan__fromuscanchangeoveraviewingperiodlikeayearifitisreasonablyclosetousandsosuchstardistan__sarenotinvariant-theychangeoverayear.Thedistan__sofmoredis​tantstarsvaryverylittleoverhu__nviewingperiods.Howdowemeasurestardistan__sandluminositiesAsfaraspossiblewe’dlikemethodsofmeasurementandreckoningtobeindependentofthesequen__astarofinterestison.Two__inmethodsaretriangulationfornearbystarsandtheuseofbrightnessandstar__gnitudesforstarsthatarenotsonearby.METHODOFTRIANGULATIONThismethodisnotmuchdifferentthanthetriangulationmethodusedbysailorsontheseaorbylandsurveyors.Figure1showstheEarthinitsorbitaboutthesunattwotimessixmonthsapart.AssumetheEarth’sorbitisapproxi__telycircular.ThentheEarth-to-Earthdistan__istwoastronomicalunitsAUbetweentheEarth’stwopositions.InFigure1DistheEarth-to-sundistan__andpistheparallaxangle.Fromposition1oftheearththeangletothestarwithrespecttothelineofsighttoverydistantstarsisp.Fromposition2oftheearththeangletothestarwithrespecttothelineofsighttoverydistantstarsisp.Thedifferen__inanglebetweenthesetwosix-monthseparationsis2p.BecauseDisextremely__allcomparedtodtheratioD/disameasureoftheparallaxangleindimensionlessunits.Theratio2D/2disthefractionofthetotalcircumferen__ofacircleofradiusd__nteredonthestarthatthechordaldistan__2Dbetweenthesix-monthlocationsoftheEarthrepresents.Wellifpisindegreesthenthatsamefractionis2p/

360.IfweequatethesetwoevaluationsofthefractionwearriveatFromthisequalitywegetthestar/Earthdistan__dasWecancallthisexpressionthetriangulationformula.Itresemblestheformulabywhichsurveyorsorsailorsfindthedistan__fromwheretheyaretoaremoteinac__ssiblepointusingasecondac__ssiblepointandmeasuringtheanglesbetweenthebaselineformedbytheac__ssiblepointsandthelinesfromthetwoendsofthebaselinetotheinac__ssiblepoint.Youh__etoknowthebaselinelengthi.e.thedistan__betweentheac__ssiblepoints.TheunitsofdarethesameastheunitsofDinthisformula.Astronomersliketogivetheunitsofpinsecondsofarcorarcsecondsbecausepisordinarilysucha__allanglethatthesizeofpinarcsecondsismuchmore__nageable.Itisthencusto__rytowritepasp‘’.Thereare

180.0X

60.0X

60.0arcsecondsin

180.0degrees.Theratioof180degreesinarcsecondstois

206264.

806.Themodifiedtriangulationfor​mu​laisTheunitsofdarestillwhatevertheunitsofDare.Thisformulaisthebasisforthedefinitionoftheparsecaswellasbeingaworkingformulaforthedistan__toarelativelyclosestar.AlthoughtheformulaworkswhenDinFigure1isthedistan__betweenanyplanetandthesunwhenEarthistheplanetthenDisbydefinitiononeastronomicalunitor

1.0AU.IfpisonesecondofarcandDisoneAUthendisbydefinitiononeparsecthatisthedistan__toastarfromoursunforwhichtheparallaxangleisonesecondofarcasdeterminedfromtheearthasmeasuringplatformisoneparsec.TherelationbetweenparsecsandAUisthatoneparsecis

206264.806AU.Fromthiswecanfindtheconversionbetweenparsecsandmetersorkilometers.Thetriangulationformulainparsecsabbreviatedpcisthe______expressionIfyouprefertoexpresssun/stardistan__sinlightyearslythenyouneedtheconversionbetweenlyandAU.Itbeginswith

1.0AU=

1.49597870X1011m.Nowonelyisthedistan__anarrowundispersedpulseoflighttr__elsinoneyearthroughemptyspa__or

365.25X

24.0X

60.0X

60.0Xcwherecisthespeedoflightinemptyspa__We’lltakectobeitsde​finedvalueof

2.99792458X108m/sec.Thenonelightyearis

9.46073047X1015m.Togetonepcinlyyoudivide

1.49597870X1011by

9.46073047X1015andmultiplytheresultby

206264.

806.Sooneparsecworksouttobe

3.26156376ly.OnemoreformofthetriangulationformulaisThevariousformsofthetriangulationformulaworkbestforrelativelyclosestarssayafewtoafewhundredlyfromus.Forlargerdistan__smeasurementofp‘’isnottooprecise.BRIGHTNESSAND__GNITUDEApparentBrightnessand__gnitudeTheluminosityofastarisaninvariant.Itdoesn’tchangeno__tterhowfartheob​serverisfromthestar.Butbrightnessdoesdependonwheretheobserveriswithrespecttothestar.Althoughitsometimesseemsthatbrightnessisavaguecon__ptitisactuallynotandcaninprinciplebemeasured.Astronomersh__eagreedthatapparentbrightnessbwillbepro​por​tionaltoluminosityLinwattsandinverselyproportionaltothesquareofthedistan__doftheobserverfromthestarorwherethesquigglylinemeans“propor​tionalto.”Butbacquiresthephysicalmeaningoffluxreachingtheobserverifthedenomi​natorincludesthefactor4sothatthedenominatornowrepresentsthesu_____areaofasphere__nteredonthestarwithradiusdtotheparentstaroftheobserver.SeeFigure

1.Inaproportionalityyoucanal​waysincludeadditionalconstantfactorsoneithersidesuchasthefac​tor

4.TheapparentbrightnessbnowdefinedasisthefluxofradiationinW/m2reachingtheobserverfromthestar.Thisisinprincipleameasurablequantity.AnobserveronEarthormoresensiblyinasa____iteorbitingtheearthcanaimaninstrumenteddetectorconsistingofametalsu_____whosenor__lisinthedirectionofthestaratthestar.Thisdetectorcanbecalibratedsothatthepowerreach​ingitisdisplayedasactualpowerarrivingatthedetector.Brightnesswasunderstoodqualitativelylongbeforeitbecameunderstoodaspowerflux.Hipparchussecond__nturyB__wasoneoftheearlyastronomerswhohadthisqualitativeunderstanding.Ashesearchedthehe__ensheclassifiedthestarshecouldseefromthebrightesttotheleastbrightormostfaint.Heassigned__gnitudenumberstothestars.Forwhateverrea​sonshehadheclassedthebrighteststarshecouldseeas__gni​tude1andthefainteststarshecouldseeas__gnitude

6.IfweweredoingHipparchus’__gnitudeschemeovertodayweprobablywouldh__eassignedlownumberstofaintstarsandhighnumberstobrightstars.Thesenumbersusuallynamedmarecalledrelativeorapparent__gnitudesbecausetheychangewithobserverdistan__.Apparent__gnitudesformanindexsystemorscaletolabelandorganizethebrightnessofstarsandaremostusefulinindex​ingtheapparentbrightnessofstars.AsastronomerslongafterHipparchuslearnedhowtomeasurebrightnessandgraduallybecamemorefamiliarwiththeideatheybegantoassociateapparent__gnitudeandapparentbrightness.BythenitwasalreadyknownthattherewerestarsthatwerebrighterfainterandinbetweenthosethatHipparchuscouldseeTheyemendedhisindexschemeinfourways:

1.Thedo__inof__gnitudenumberswasextendedtovaluesgreaterthan6toaccommo​datefainterstarsthanHipparchuscouldsee.

2.Thedo__inof__gnitudenumberswasextendedtovalues__allerthan1includingzeroandnegativenumberstoaccommo​datebrighterstarsthanHipparchuscouldsee.

3.Thedo__inof__gnitudenumberswasextendedtofractionalvaluestofillinforthein-betweenstarsthatHipparchuscouldnotseeandwouldnowbeneededforstarsnotyetdiscovered.ThisearlyastronomycommunitydiscoveredthatthebrightnessratioofthebrightesttothefainteststarsthatHipparchuscouldseewasabout

100.Hen__onemoreemendment:

4.Figure2showsascaleofapparent__gnitudeindicatingthatthem-valuesincreasefromrighttoleftandthatthereforetheb-valuesincreasefromlefttoright.Thescaleshowsthelocationsofm=1andofm=6butitisarbitrarywheretoputthefourotherintegerm-values.WhatlaterastronomersthanHipparchusdidwastopla__thosefourothervaluessothatthesuc__ssiveintervalsbetweenadja__nt__gnitudeswereallofequalscalelength;thispla__mentwasforconvenien__.Itcouldh__ebeendoneinotherwaysbutthispla__mentdefinedalogarithmicscaleofbright​nessagainstthe__gnitudescaleinrecognitionoftheobservationthatthehu__neyerespon​dslogarithmicallynotlinearlyto__gnitudes.Letmandm´betwosuc__ssiveintegervaluesinthesenseofdecrea​singmandhen__inthesenseofincreasingb-values.Theemendationsaysthattheratiooftwosuc__ssiveincreasingbrightnessvaluesisaconstantcallitKovertheentirescale.Thatisifbisthelargerandb´the__allerbrightnessthenb/b´=K.ThismeansthatThemeasuredratioofb1tob6isthenSoKisthe5’throotof100whichis

2.51188643151orapproxi__tely

2.

512.ThisfixesthebrightnessscalethataccompaniestheemendedHipparchus__gnitudescale.Animportantconsequen__ofthesescalerelationsisthatifmandm´areanytwoap​parent__gnitudeswithmm´andbandb´arethecorrespondingapparentbrightnessvalueswithbb´thentheratioofbtob´isThisrelationwillsoonenableustoconnectapparent__gnitudetostardistan__s.SupposethestarwithbrightnessbhasluminosityLandisdistan__dfromtheob​serverplatformandthatthestarwithbrightnessb´hasluminosityL´andisdistan__d´fromtheobserverplatform.Thentheratioofbtob´isOfwhatgoodisthisrelationItcontainstheluminositieswhichareunknown.Butwait!Supposesaidanumberofastronomersthatthetwostarsarenotdifferentstarsatdifferentdistan__sbutthesamestartwi__on__atitsproperdistan__andon__atastandarddis​tan__callitd´=ds.Thentheluminositiesbeingthesamewouldcan__l.Inthisschemethestandarddistan__byagreementisgiventhevalue

10.0pcparsecsincaseyouforgot.Thevalueb´isthenrenamedtheabsolutebrightnessBand__gnitudem´isrenamedtheabso​lute__gnitudeM.Thereisnothingtrulyabsoluteaboutthesequantities:thatisjustaconvenientwaytorefertothem.TheratiooftheapparenttotheabsolutebrightnessisIfyoucomparethesepre__dingequationsyougetarelationthatrelatesthedistan__ofastartoitsapparentandabsolute__gnitudes.Intakingthesquarerootoftheratioofdto

10.0pcyoucanworkthesquarerootof100Mm/5ontothe

100.Hereiswhatyouendupwith:Thedistan__dhastobeexpressedinpcinordertousethisequationconsistently.Bytakinglogarithmstothebase

10.0ofthisrelationlogonyourcalculatorsnotlnyouarriveatarelationbetweenthe__gnitudesofastaranditsdistan__:whereasbeforeyouh__etoremembertoexpressdinparsecs.Asanexamplelet’sfindtheabsolute__gnitudeofthesunItsapparent__gnitudemisfoundbyplacingitsapparentbrightnessagainstabrightness/__gnitudescalesuchasinFigure2extendedtotherightforverybrightstars.TheapparentbrightnessofthesunisthefluxofitsluminosityreachingEarthwhichiscalculablefromitssu_____temperatureitsradiusandthesun/Earthdistan__

1.0AU.Thenumberis13__.0W/m

2.Whenpla__dagainstabrightness/__gnitudescalethesun’sapparent__gnitudeism

26.

8.It’sdistan__fromtheEarthinpcisfromp.8that

1.0parsecis

206264.806AU.Hen__theratiod/

10.0is

1.0/

2062648.

064.84813681690X

107.Thelogarithmtobasetenofthisnumberis

6.___

42513266.Fivetimesthisnumberis

31.572finallyintroducingsomesensiblerounding.ThenMforthesunismsun

31.572=

31.

57226.

8.ThisnumberisM

4.

772.__nyauthorsjustroundthisto

4.

8.Asasecondexamplesupposetheparallaxangletoa__rtainstaris

0.0125secondsofarc.Whatisthedifferen__betweenitsabsoluteandapparent__gnitudesAswesawatthebeginningofthesenotesthereciprocaloftheparallaxangleisthestar’sdistan__inpcfromtheEarthorwhatamountstothesamethingfromthesun.Sothestar’sdistan__dis1/

0.0125or

80.0pc.Thenaccordingtothemostre__ntdisplayedequationthe__gni​tudedifferen__intheorderMmisIfthestarturnsouttoh__eanapparentbrightnessof

6.9thenitsabsolutebrightnessis

4.

5156.9=tomoresensibleprecision.。

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