1937ApJ 85 . . 107H 0 edge, tosaythatthelargestpartofscattered lightcomesfrom particles offairlylarge dimensions (largewithrespecttothewave- by about3mag.ThenebulanearpOphiuchiisdiscussedanditfoundthatthestar material. Thatthisisthecaseascertainedindependentlyfromreddishcolorof the starandnebulafromfaintnessofstar,whichseemstobecutdown bright nebulositynearCD—2412684itisdeducedthatthisstarimmersedinthe the scattering.Itseemsplausible,inpresent stateofourknowl- by atomswhicheventuallyre-emittheenergy. Theothergroupof by theparticlesofnebulaeintospaceorismomentarily absorbed is aboutaparsecinfrontofthematerial. behind, thebrightnebulaappearsringshapedifparticlesinarelarge. inversely asthedistanceofstarfromnebula.Whenisinside an inclinationofthefacesthesenebulaetolinesight.Fromshape metry ofthenebulaeIC1284,1287,andpossiblyBD+8°933canbeascribedto more thanonereflectioninpassingthroughthemedium. nated byastarinfrontofit,theintensitygradientfromcenteroutwardvaries equation oftransfer,similartothecorrespondingintheoryabsorp- and aimedtherebytodiscoverthenatureof particleswhichdo the actualprocessofreflectionorscattering ofthestar’slight, investigations hasconsistedofthecollection evidenceconcerning to thenebulaeandthatlightfromthesestars eitherisreflected medium, thegradientisevenmoresteep.Incaseofathinnebulailluminatedfrom Various importantfeaturesofthesecurvesarepointedout.Whenanebulaisillumi- applicable. Inthiswayitispossibletotakeintoaccounttheenergywhichhasbeen tion andemissionoflightorinthetheoryscatteringbysmallparticles,is dent thatthesourceofthisenergyliesinstarswhich areinproximity Owing totheeffortsofanumberastronomers,todaywefeelconfi- source ofthelight-energywhichcomestousfromtheseobjects. cipally intotwogroups.Onehasbeenconcernedwiththeprimary For amoreopaquenebulathisringshapeisblurredoverbythelightwhichhassuffered which givethedistributionofsurfacebrightnessonfacesseveralidealnebulae. scattered morethanoncebytheparticlesofnebula. It isshownthatasfarlargeparticlescontributetothescatteringofstarlight,an © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The curvesareappliedtoanumberofactualnebulae.Itisshownthattheasym- Investigations ofdiffusenebulaehaveresolvedthemselvesprin- The theoryhasbeentranslatedintographicalform,bypresentingaseriesofcurves The generaltheoryoftheilluminationreflectionnebulaebystarsisdiscussed. THE ILLUMINATIONOFREFLECTIONNEBULAE L. G.HENYEY ABSTRACT 107 I 1937ApJ 85 . . 107H 1 3 2 length oflight),andisproducedbyaprocesssomewhatsimilarto selectively withrespecttothecolorandplaneofpolarization diffuse reflectionbutwiththepossiblepresenceofdiffractioneffects. power. Studiesofthedistributionlightoverapparentdisks of theincidentlight.Weshallseethatacertainamountobserva- Probably asmallamountoflightisscatteredbyparticles, ous, butareinefficientscatterersascomparedwiththeirabsorbing tional evidenceindicatedthatthesesmallparticlesmaybenumer- nebulae havebeendevotedtoquestionsinvolvedinoneofthesetwo groups; photometricobservationsfortheirownsakehavebeenrare. 108 L.G.HENYEY light overtheapparentdisksofseveralactualnebulae. In thispaperweshallbeconcernedwiththeprinciplesinvolvedin to beexpectedinanidealreflectionnebula,andwithapplication has beenformulatedinconsiderabledetailaseriesofpapersby of theseresults,inasemi-qualitativemanner,tothedistribution such aninvestigation,withthedevelopmentofgeneralproperties we receivefromsuchaparticledependsonfourfactors:(i)the particle; (3)thephaseangleunderwhichlightisreflected; other particlesweaken,byoccultations,thelightfallingonour distance oftheparticlefromsource;(2)degreetowhich ticle inthedirectionofobserver.Theamountenergywhich Schoenberg intheHandbuchderAstrophysik. Seeliger. Anaccountofthesepapersmaybefoundinthearticleby from theparticle.Thesurfacebrightnessofnebulaatanypoint of alltheparticlesinacolumnwithunitcross-sectionalarea,and on itsdiskisthenequaltothecontribution,withinunitsolidangle, and (4)theweakening,byoccultations,oflightcomingtous directed alongthelineofsight. receive byreflectionsofthelightcomingfrom otherparticles;that ing withnebulaehavingcontinuousspectra. “lieber kosmischeStaubmassen,etc.,”Sitzung,d.Akad.zu Muenchen,3,1901. Akad., 16,1887;“TheoriederBeleuchtungstaubförmigerMassen, etc.,”ibid.,18,1893; 2 1 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The theoryoftheilluminationadustcloudbypointsource Seeliger considerstheintensityoflightreflectedbyeachpar- Seehger doesnotconsiderthecontributionto energywhichwe “ZurTheoriederBeleuchtunggrossenPlaneten,etc.,” Abhandlungen derBayer. TheworkofKeenan(Ap.84,600,1936)onNGC7023 istheonlyonedeal- 3 2,130ff.andesp.163 1937ApJ 85 . . 107H large comparedtothecross-sectionalareaof particles. light undergoeswithinasmallelementofvolume. reflection nebulaecomesfromlargeparticles.However,ourgeneral by dsandtheareaofitsbaseda,latter quantitybeingtaken base isperpendiculartotherayatpoint.Represent itsthickness base. Letthiscylinderbeplacedasshownin Figure1,sothatits mean thatweconsiderthechangewhicharayof lumination ofdustcloudsweusewhatmightbe particles. Itwouldberelativelyeasytouseotherphasefunctions formulae remaincorrectforanytypeofscatteringprocess,anditis in acompletegeneraltheorytheymustbeconsidered.Thisis height issmallcomparedwiththediameterof its rived, andtheirapphcationhasbeenillustratedbymeansofafew function, thatthediscussionbecomeslimitedtocaseoflarge reflection duringtheintervaloftimewhichelapsesafteritleaves is tosay,heneglectstheenergywhichhassufferedmorethanone it seemsbestnottocomplicatetheanalysisandcarrythroughour only onpage119,whereweintroducetheEulerphasefunction, done inthenextsection.Weassumethatscatteredlight source anduntilitagainescapesfromthenebula.Incertaincases element ofvolumeintheshapeacyUnderwhose direction, situatedinthenebula.Letustake an scopic viewpointusedbySeeliger.Bythiswe I havenotsucceededinfindinganebulawhichwouldgiverigorous the zodiacallightissufficientjustificationforourprocedure.While and againonpage121,wherewebegintousetheLambertphase the effectofmultiplereflectionsisnotnegligible,andconsequently scopic” viewpointascontrastedwiththemacro- examples. test ofthephasefunction,methodsforsuchahavebeende- The successofSeeliger’sworkinexplainingtheringsSaturnand computations fortheLambertphasefunction,which,ofcourse,is correct onlyforlargesphericalparticleshavingsmoothsurfaces. (e.g., thosederivedbyBlumerfromtheequationsofMie),but © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In thederivationoffundamentalequations Consider apointandray,havingcertain THE ILLUMINATIONOFREFLECTIONNEBULAE109 II governing theil- called a“micro- Fig. i 1937ApJ 85 . . 107H 2 front surfaceofthiselementvolumeisnotuniformlydistributed over thesurfaceis/,thentotalenergyfallingonfaceinunit some maximumvalue.Iftheaveragevalueofspecificintensity sequently, theilluminationmayvaryfromcompletedarknessto others arereflectinglight,comingfromotherdirections,onit.Con- over theface.Itmayhappenthatparticles,situatedinfrontof surface, arecastingshadowsonit,orthatthesameparticles larities cancelandwefindthatthefightoccultedbyparticlesis radii. IfthenumberofparticlesperunitvolumeisN,total in thevolume.Takeaveragecross-sectionalareaofparticles surface, partofitsenergywillberemovedbyfallingontheparticles time andinunitsolidangleisgivenby form. Sinceweareinterestedonlyinaveragevalues,itispermis- in absorbingfight,sincetheilluminationfallingondaisnotuni- cross-sectional areaexposedtotheray,inelementofvolume,is to beTrp,wherepisthesquarerootofmean Before thelightpassesthroughcylinderandfallsonback negligible. Noteveryfractionofthisareaisaseffectiveanyother Of course,itispossiblethatsomeoftheparticlesarebehindothers sible totakedasuffici^itlylargesothat,ontheaverage,irregu- ciently small,theeffectofsuchoccultationsoverdistancedsis and thetotalareaislessthanthatstated,butifdschosensuffi- no L.G.HENYEY into theoneinwhichweareinterested.We haveseenthatthe is anincreaseduetothedeflectionoffight from otherdirections If therayisinclinedtofaceofcylinder, itisfoundthat the amountofenergyisgivenbysameformula. (per unittimeandsolidangle) © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The lightcominginthedirectionofrayandfallingon In additiontothechangeinintensitydue absorption,there 2 I •TrpNdads.(2) 2 irpNdads . Ida. (i) 1937ApJ 85 . . 107H 2 light orinthetheoryofscatteringbysmallparticles. Supposethat We mustthenreplace(5)bythefollowingequation: scattering particlesarepresentinadditionto reflectingparticles. sponding relationshipinthetheoryofabsorption andemissionof where k,theabsorptioncoefficient,iswrittenfortpN. or, aftersomereduction, element ofvolumeis7+¿7,then,accordingto(1)and(4), If thespecificintensityoflightfallingonbacksurfaceour over allsolidangles. where thenotation,/()dœrepresentsintegralofintegrand volved. Wewilldenoteitby$>.Consequently,theenergydeflectedis on thedirectionsoftworaysandkindparticlesin- is deflectedinthedirectionoffirstray.Thisfractiondepends The totalcontributionisthisquantitysummedoveralldirections,or where Visitsaverageintensity.Acertainfractionofthisenergy time andsolidangle) amount ofenergyabsorbedfromsuchaninclinedrayis(perunit © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem It willbeseenthatequation(5)isentirelysimilartothecorre- Accordingly, thenetgaininenergyis 2f (7 +dl)d
i dl0 + Io = 0 k ds
(7)
i dli fli-Ædœ k ds
Let the boundary values at the entrance surface be assigned to I0 and let the boundary values for the other /’s be equal to zero. Conse- quently, if we add all the equations in (7), we will have (5), provided
/= 70 + Jx + /2 +.... + 7; + (8)
It can be shown that the series (8) converges, if
k > o and if ß^dco < 1 .
Since these conditions are satisfied, (8) gives the solution of (5), and the problem has been reduced to the solution of (7). Now let
dr = kds , and write (7^) in the form
(eTIi) = erfíli-^do).
We shall call r the optical distance. We have, by integrating this equation from the entrance surface (r = o) to the point in question,
T e Ii — lio e' dr $li-ï^dœ j
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1937ApJ 85 . . 107H placed byE,where where thequantitiesr,r,and$mustbeexpressedintermsofr.' which islimitedbyparallelplanesand isilluminatedbya venient. intensity ofthefirst-orderreflectedlight. through aportionofthemedium.ThequantityI*represents of thelightwhichhascomedirectlyfromsourcesandpassed direct solutionofthefundamentalequation(5) maybemorecon- erable methodofsolvingcertainproblems,butinothercasesthe which hassuffered¿-reflections. after leavingthesources,andcorrespondinglyrepresentsthat every pointanddirectioninthemedium. distant source.Itisconvenienttohavethe specific intensityfor This equationisidenticalwiththeofSeeligergiving where ristheopticaldistancefromentrancesurface. since astarispracticallypointsource.Therefore1mustbere- intensity ofthelightwhichhasbeenreflectedoncebyparticles or, sinceI=owheni>o, convenient tousethetotalintensityinsteadofspecificintensity, 0 0 0 i0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Using (10),wehaveforI In mostcasesthesourceofilluminationisastar,anditmore It isseenfromtheserelationsthat/representstheintensity Seehger hascomputedthesurfacebrightnessof auniformnebula, The separationofIintocomponentintensitiesmaybethepref- x 0 THE ILLUMINATIONOFREFLECTIONNEBULAE115 (T/)o /, =e-“-^<$>(T')dT,(ll) r ^ e(j?Ii-zQdu. Ill (9) lió L. G. HENYEY
In Figure 2 let i be the angle of incidence of the light coming from the source. Then
E ^ p—kz' sec i r* 6 for a point which is at a distance zf from the surface facing the star. Since the light from the source is approximately parallel, we have for the intensity in the direction of the arrow
/x = —0 <ï> r e~{T-T')-kz' secidTf . r2 Jo
Since the distance of the star from the nebula is large compared with the thickness of the nebula, the quantities r and $> may be considered constant. When
o < e < - , 2
T = k(h — z') sec € , r = k(h — z) sec e , and
/i = -^ + e)k sec € g—k(z'—z) sec t—kz' sec £
E sec € , > = - <|>(¿ A sec ¿z (sec 6+sec t) g—¿Ä(sec €-|-sec/A (t2 ) r2 K "sec ¿ + sec e v / • v /
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1937ApJ 85 . . 107H When The surfacebrightnessonthesidefacingstarisobtainedby from thesource.FromFigure3wefindthat The surfacebrightnessontheothersideisfoundfrom(13)bysub- substituting 2=oin(12),thatis, E =J+e)hseceI-k(z'-z)€-kz' and ity maybemoreseriousinactualapplications. too seriouslysincetheerrorsintroducedbyassumptionsofuniform- is sufficientlyfarfromthenebula.Thisrestrictionneednotbetaken As wehaveremarked,theserelationshipsarevalidwhenthesource terms oftheapparentorangulardistanceapointonsurface stituting 2=h: inclusion ofthehigher-orderintensities.From the definitionofthese quantity, thealbedo,isgenerallysmall,higher-order intensities quantities weseethatisproportionalto Since this e Sec 6 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In ordertousetheseformulaewemustbeableexpressrandiin The resultsofequations(12)and(13)mustbe completedbythe __j_\ßkzsecei-j-g—kz(secz+sece)| =e SeC 6 J _<£(7_|_)ßkhseccijß—kh(seci+sece)| 1 =e z 2 y; THE ILLUMINATIONOFREFLECTIONNEBULAE117 E C r seci+e /i =—$(i+e) t '^°/Tx/'*j\sec€ 2 yJ1 r seci+e’ 2 tan ¿=-sece—andrxi. r seci“Te f T =—kzsec€,e x 7T 2 < €7T, I jg—kh{sece+secj)| 1937ApJ 85 . . 107H distant source,andneglectallotherterms. the orderofsize/,foraplaneparallelnebulailluminatedby approach zero.Wewillcontentourselveswithanexaminationof As before,wetake(using0ase,inFig.2): when ii8 when be substitutedforztoavoidconfusion.Let0'theanglebetween and The quantityAisgiveninequations(12)and(13),where2'must 2 Also let0bethecorrespondinganglefor1.Then the projectiononfaceofnebula originalincidentray dimension isinvolved,andweintroducetheangles aanda,which In thephasefunctionwecannolongeruse¿ +e,sincethethird coming fromthesource,andprojectionof thedirectionoíI. 2 x © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem From (9)wehave f r' =k(h—z)sec0,rz)0, T —kz'sec0,~kz0, = e-^-^dT f dm =sinededd. L. G.HENYEY - <07T. 7T o<*<^ 2 Fig. 3 fl&du . Observer 1937ApJ 85 . . 107H 4 0 We havefortheemergentlight,wheno<71-/2, are, respectively,thesupplementsofanglesbetweenIandJï, and and /ï/.Theyarefoundfrom The phasefunctionwewilluseisthatofEuler: higher-order intensitiesarelesssensitivetotheshapeof phasefunctionthanisthe integrating for0', resemblance totheLambertphasefunctionjustifyits usehere.Ingeneral,the and whentt/2<0tt, first-order intensity. since theintegrandisindependentof0'.Now simply considerthecasewheni=oand0or180.Then, To getanestimateoftheordermagnitudeeffectweshall when o
Magnitudes per square degree © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem from thestarandinclinedatanglesshown,(c)Similar to(a),butforinclination nebula. Theordinatesaregiveninmagnitudespersquare degreeasreferredtothe phase functionhasbeenused.Theunitoflengthistheperpendicular distancebetween zero andforvariousopacitiesasgiveninmagnitudes,{d)Similar to(6),butforinclina- nebula observedonthesamesideasstarwithface inclinedtotheplaneof unit, whiletheabscissarepresentsquantityh/x(seeFig. 3).(a)Aninfinitelyopaque tiplied bythefactorgivingobscurationofstarif latterisseenthroughthe star andnebulatheunitofintensityis staratthisdistancemul- nebula illuminatedbyadistantsource.Thealbedoistakentobeunity.Lambert the skyatanglesshown.(6)Anebula1mag.thickobserved ontheoppositeside tion zeroandforvariousopacities. Fig. 4.—Thesecurvesdisplaythedistributionofbrightnessoveraplaneparallel -3 -2-IOI23 h/x 1937ApJ 85 . . 107H 6 we neednothesitatetouseonewhichsatisfiesthegeneralconditions proportion totheproductsofcosinesanglesincidence has beenusedincomputingthecurvesFigure4.Thisfunctionis stated previously,themostcharacteristicfeatureofphasefunc- imposed bythetheoryofreflectionlargeparticles.Ashasbeen applicable tolargesphericalobjectswhosesurfacesscatterlightin are thereasonsforuseofLambertphasefunctioninthis Its agreementwiththisconditionanditsmathematicalsimplicity tion forlargeparticlesisthecondition terial isavailableontheactualnatureofnebularphasefunction, ticles inanebulais,ofcourse,uncertain.Sincenoobservationalma- and ofreflection.Whethersuchafunctionisapplicabletothepar- makes thealbedoequaltoratiooftotalreflectedlight investigated bySeeliger,whoderivedanexpressionforthefirst- we mustmultiplyeachintensitybythisquantity.Thisdefinition order reflections. section, andfortheuseofEulerfunctioninsectiononsecond- order intensity.Thespecificintensityinthiscaseis total incidentlight. berg usesanothervaluewhichgivesthespecificintensityof thelightreflecteddirectly ferred. Hehasusedv=0.4343*'insteadofv. Knowingtheshape function andistabulatedinhisthirdpaper, to whichwehavere- function \p{d)hasbeencomputedbySeeligerfor theLambertphase of thenebulaandpositionstar,onecan evaluatetheangles where v=kh\theotherquantitiesareexplainedinFigure5.The back byasingleaverageparticle. x v 6 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem To findthebrightnessforacertainalbedo7,where The surfacebrightnessofanebulawhichenvelopsitssourcewas Thisdefinitionofthealbedoisinmanyrespectsmost convenient one.Schoen- THE ILLUMINATIONOFREFLECTIONNEBULAE123 2vcotao h =~e{\f/(7r—a)^„(tt)}, vx0 371- ô 7 =jîQdœ, $(7r) =O. 1937ApJ 85 . . 107H which hasanopticalthicknessofonemagnitude.Thevariouscases function ofô,forthecaseaplaneparallelanduniformnebula la, andtheunitofintensityisstarlightat the frontsurfaceofnebula,numberaccompanyingeach distance multipliedbytheexponentialfactor,givingeffectof The unitoflengthisthedistancebetweentwofacesnebu- shown representtheresultsfordifferentdistancesofstarfrom coefficient, determinethevaluesof curve givingtheratioofthisdistancetothicknessmedium. lar pointsconsidered.Hencetheyrepresentthedeviationfroman the absorptionalonglineofsight.Toreducetoagivenalbedo cio andaintermsof5.Theseangles,togetherwiththeabsorption intensity ofthestarlightatthissurface,including theeffectofab- depths givenbythenumbers,inmagnitudes, accompanyingeach inverse squarelawofintensityvariation. squares oftheapparentdistances,5,betweenstarandnebu- another form,theordinatesgivingintensitiesmultipliedby one mustagainmultiplyby7.Figure6bshowsthesameresultsin star fromthefrontsurface,andunitofintensity istheactual to beinfinitelyopaque,andthestarisplacedat thedifferentoptical 124 L.G.HENYEY curve. Inthesecasestheunitoflengthis actual distanceofthe sorption. x © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The curvesgiveninFigure6ashowthesurfacebrightnessasa We havenowtoexaminetheimportanceof higher-order reflec- A similarsetofcurvesisgiveninFigure6c.Thenebulataken o r-
Magnitudes per square degree 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem parallel nebulaandfori=o°.TheresultsareshowninFigures7a from thenebula:(a)
" 126 L. G. HENYEY
% For any reasonable value of y (J or less) it will be seen that 12 may S be neglected. When the star is behind the nebula, the situation is entirely dif- ferent. Using as the value of the albedo y, we find for a nebula hav-
ing an optical thickness of 0.75 mag. that I2 corresponds to about
16.7 mag. per square degree. The maximum of I1 is for x equal to about i, and it corresponds to 16.8 mag. per square de- gree. The two intensities are nearly the same, while for
denser nebulae I2 increases rapidly with increasing op- tical thickness, and becomes much more important than
I1. While no computations have been made for the third-order intensity, it is Opacity in magnitudes probable that this, too, Fig. 7.—The second-order intensity for a would be large. plane parallel nebula. The albedo is taken to be unity. The phase function is the Euler. The IV units of length and intensity are the same as in The curves in Figures 4-6 Fig. 4. The intensity is shown for a point at the foot of the perpendicular from the star to the display a number of features nebula and for the direction at right angles to which, though depending on the face. The intensity is shown for various the simplifying assumptions opacities of the nebula. we have made, are applica-
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1937ApJ 85 . . 107H it becomessteeper.Forapositionofthestaronfrontsurface pressed inthedirectionofabscissae.Itwillbeseenthatnot decreases veryrapidlywithincreasingdistance.Nearthestar only doesthesurfacebrightnessincreasebutcurverepresenting that, asthestarapproachesnebula,thesecurvesmustbecom- predict forthinnebulaedependstoacertainextentonthephase ma arefartherapart.Ifthesecond-orderintensityisextremely mum andwillsomewhatmodifytheconclusions.Thesurfacein- minimum value.Higher-orderreflectionswilltendtofillinthismini- intensity issimilar,asweseefromFigure6.Asthestarapproaches function wehavechosen.Weassumedthatthetypeofscatter- strong, nominimummaybeobserved. reaching amaximum,decreasesagain(seeFig.4).Forpositionsof tensity nearthestarbecomesgraduallysteeper,butwhenitas- the backsurfaceofnebula,distributionin- intensity variesasi/d. but theyinvolveparticleswhichareofthesameordersizeas,or ing whichismostcommonthatofdiffusereflectionbylargepar- sumes apositionjustoutsidethenebulacentralintensityhas the starfartheraway,centralminimumisbroaderandmaxi- tensity increaseswithincreasingdistancefromthestarand,after (see Fig.6)thesurfacebrightnessisverygreatnearstarand particles arelarge.However,inallthecasesconsidered, itissimple viously, observationalevidencedefinitelyfavorstheviewthat function inordertosatisfytheconditionsofa problem.Ingeneral, There areprocessesofscatteringwhichdonotsatisfythiscondition, to replacetheLambertphasefunctionbyany appropriatephase ticles, forwhich very littlechangewillbeproducedintheshape ofanythecurves, function issuchthati>(o)approacheszero.This isthecasewhen except Figure4¿,bysuchasubstitution,unless thenewphase smaller than,thewave-lengthoflight.Ashasbeenpointedoutpre- the scatteringparticlesareextremelysmalland scatterthelightfor- es) American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For positionsofthestarfartherintonebuladistribution It mustbeemphasizedthatthesmallcentralintensitywhichwe THE ILLUMINATIONOFREFLECTIONNEBULAE127 $(7r) =O. 1937ApJ 85 . . 107H illuminated willnotbeaffectedgreatly.Butifthestellarsourceis needed. Thisproblemisatpresentbeinginvestigatedbythewriter portant caseofthestarsituatedinfrontmaterial,first- liminary character.Evenwhenaccurateobservationsareavailable, phase functionadopted.Theyarearesultoftheintensitydistribu- important pointwhichpresentsitselfistheshiftofmaximum order intensitymaynolongerbethelargestpartoftotalin- outside ofthemedium,situationisdifferent.Inmoreim- ward insteadofreflectingitbackasinthecaselargeparticles. larities alwaysdistortandmaskthegeneralfeatureswhichhave front ofthenebula.Thesegeneralfeaturesareindependent tensity distributionwhenanebulaisobservedatanangle.Themost been discussed.Actualgalacticnebulaearecertainlynottheideal surface brightness,dependingontheinclination,whenstarisin and willbediscussedinanewpaper. tensity. Itisobviousthatadifferenttreatmentoftheproblem Even thentheintensitydistributionoveranebulawhichisinternally tometric dataareavailable,sothatourconclusionswillbeofapre- tion ofthestarlightfallingonnebula. it isinsideoronthefartherside,aknowledgeofitsdistance,lumi- derived. Inthefirstplace,itispossibletoascertainwhether objects whichwehavetheoreticallyconsidered. care mustbeexercisedinapplyingthetheory,sincestructuralirregu- given above,tosomeactualnebulae.Inmostcasesnoaccuratepho- nosity, andapparentbrightnesswilldetermine theopticalthickness shape oftheintensityprofileasgiveninany ofthesetscurves extent ofthenebulaor,ifparallaxisknown, inabsoluteunits. the frontsurfaceofnebulamaybedetermined intermsofthe of thematerialinfrontit. 128 L.G.HENYEY This quantitymaybefoundwithreasonable certainty,sincethe source isornotwithintheboundariesofnebularmaterial.If © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem An interestingfeatureofthecurvesisasymmetryin- With accurateobservationsagreatdealofinformationmaybe In thissectionweshallapplythecurves,whichhavebeen Furthermore, whenthestarisinforeground, itsdistancefrom V 1937ApJ 85 . . 107H losity. Itcanbeseenthatthenebulosityis,nevertheless, one-sided, fessor E.BarnardandusedinhisAtlasof Selected Regionsofthe line perpendiculartotheaxisofsymmetryandpassingthrough mined, itispossible,inmostcases,toderivethisquantityfromthe immediately behindit.Thisdistanceis will bedetermined.Ifthestarisimmersedinaveryopaquenebula, and themaximumintensityseemstobesouthwest ofthestar.The Milky Way. relative brightnessofthestarandnebula. namely, thealbedo.Oncescaleofobjecthasbeendeter- best bedonebyfirstconsideringtheintensitydistributionalonga I shalldiscussindetail.Thesearecopiedfrom platestakenbyPro- tive orabsoluteunits,betweenthestarandpointonnebula star. Wecandeterminefromthisdistributionthedistance,inrela- from thefrontsurfaceinananalogousmanner.Whenstaris line ofsight,anestimatecanbemadethisinclination.Thismay tion coefficientisgivenbytheratioofopticaldistanceto termined fromtheopticalthickness.Inthesetwocasesabsorp- a thinlayerofmaterial,thetruethicknessmaybede- one can,iftheobscurationofstarisknown,getitstruedistance served case,and,sincetheunitoflengthisquantitysought,it The procedureissimplytofitthescalealongabscissaob- From thesetwonumbersemaybecomputed. true distance. approximately The distancetothepointofmaximumintensityis,insameunits, (Fig. 4)isremarkablysimilarforvaryingthicknessesofthemedium. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Plates IandIIreproducephotographsofreflection nebulae,which When astarisinfrontofnebulawithitssurfaceinclinedtothe The nebulaIC1287fallsinaregionoffairlyuniform darknebu- One moreimportantquantityisderivablefromphotometricdata, THE ILLUMINATIONOFREFLECTIONNEBULAE129 X CSC€. # sine. VI 1937ApJ 85 . . 107H // o probable thatthefaceofnebulaineachcaseisinclinedslightly have tolieinfrontofthematerial,andthisseemsreasonablesince to thelineofsightandproducesobservedeffect.Thestarswould toward thesouthinaheldofuniformdarkmaterial.Itseemshighly nebula IC1284showsasimilarcharacteristic;itextendsfarther We getfromtheserelationships,veryroughly,that fact thatmostreflectionnebulaearenearlyorapproximatelysym- the intensitygradientis,inbothcases,veryflat. rived quantitiesareadmittedlyveryrough,but theirreasonableness parent magnitudeofthestaratnebula may beestimatedat la IC1287tothestarandhavefoundthatxsineisabout2oo.The ly nearoractuallyinsideofthenebula. metrical indicatesthatthestarsilluminatingmatteraregeneral- is unquestionableandleadstotheconclusion that theassumptions an angle,sincetheprobabilityofoccurrenceagiveneissine.The and the methodofanalysis,thatthisdistanceisroughly1.5Xesce.Then the starisaround7so".Letusassume,forsakeofillustrating width ofthenebulaperpendiculartolinejoiningcenterand degree, whichishalfamagnitudefainterthan thesky.Thesede- distance ofthestarandnebulatobeabout300parsec.Thisgives From thespectraltype,B,ofBD—io°47i3Ihaveestimated as thesurfacebrightnessofcenternebula 5mag.persquare Taking theabsolutemagnitudeofaB3staras —1.0mag.,theap- are consistent. I3O 3 — 7.5.WithanalbedoofaboutJ,thecurvefor 30inFigure4gives © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Indeed, weshouldexpecttoobservethemajorityofnebulaeat I havemeasuredroughlythedistancefromcenterofnebu- x =100,000astronomicalunits. X CSC€=500". æ sin€=200", L. G.HENYEY x =300". € 35 o 1937ApJ 85 . . 107H © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem PLATE I
ber of Yerkes spectrograms. No giant characteristics were found for ß, o', and v Scorpii and for p Ophiuchi. A second check is found in the color of the nebulosity. Two plates, one photographic and the other photovisual, which were taken with a Schmidt camera by Struve, Elvey, and Roach for their study of this region of nebulosity, were examined for color effects. It was found that, while the very bright filaments around 22 Scorpii were almost absent on the photo visual plate, the nebulosity in question was quite bright. Measurements with a recording microphotometer verified this observation and showed that there exists a difference in color index between these nebulae of about 0.2 mag. No correction was made for the color of the sky and the color of the diffuse illumi-
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1937ApJ 85 . . 107H 11 10 corrected fortheintensityofsky. one near—24°i2684. Struve statesthathehasvisuallyfoundthenebulanear22 Scorpii tobebluerthanthe tially correctanddonotconflictwiththestatementsmade inthispaper.Infact, total redlightaround22ScorpiiequaltothatCD —24°i2684.Thisexplains measurements ontheseplates.Thegivenin theirTable1aresubstan- why noeffectofreddeningwasdetectedbyStruve,Elvey, andRoachintheiroriginal curve alreadyused,is star atthenebulais—6.0.Thebrightnessofnebula,from For anabsolutemagnitudeof—0.6theapparent proximately, star by0.7^.Takingtheparallaxofas0T01,weget,ap- Using themeasuresofStruve,Elvey,andRoach,Ifindthatthis puted itsdistancefromthefrontsurface.Thebrightnessatapoint chi. FromtheuppercurveinFigure4cthispointisdistantfrom is 0.7mag.fainterthanthenebulosityimmediatelyaroundpOphiu- be redeterminedwithgreateraccuracy. about 30'northofpOphiuchiisthesameasthataroundAntares. for —24°i2684.Itisextremelydesirablethatthesecolorexcesses conclusions. ThatforpOphiuchiisapproximatelyone-halfofthat The smallvalueofthecolorexcessfor22Scorpiialsoagreeswithour This conclusionisverifiedinthemostunexpectedmannerby large colorexcessof—24°i2684,asfoundbyScaresandHubble. merely increasethedifferencefound;thereforenonewasapplied. The rednessofthenebulaCD—24°!2684maybeexplainedby wave-lengths. Correctionsfortheseeffectsareuncertainandwould fact thatthesourceisshiningthroughthicklayersofnebulosity. nation emanatingfromAntares,whichissouthof22Scorpiiand strongly illuminatesthenebulosityaroundthislatterstarinlong 11 10 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Op.cit.yTable1.ThereferencesImaketothesemeasures refertotheirvalues ThestrongredlightcomingfromAntaresandscattered by thenebulamakes Assuming thatpOphiuchiisinfrontofthenebula,Ihavecom- THE ILLUMINATIONOFREFLECTIONNEBULAE135 x =250,000astronomicalunits. ii.2 —6.02.5log7. 1937ApJ 85 . . 107H 12 13 cause ofthelargecoefficientabsorptionnebularmaterial, whichcausesbothstar and nebulatoappearfaint. wise, theresultsderivedinthispaperaregoodagreementwith losities wehavediscussedshowsthatHubble’sstatisticalmethod those obtainedbyStruveandStoryfromanapplicationofHubble’s Rayleigh scattering,wouldleadtoatotalphotographic absorption nebula forwhichthesourceiswelloutsideofmedium.Other- differently, withvaryingexposuretime,thanwillthediameterofa method. diameter ofthenebulaassociatedwithCD—24°!2684willbehave cannot beappliedtoindividualobjects.Forinstance,thelimiting dividing theabsorptioninstarlightbydistancejustfound; photometric standardizationisavailableforanyoftheplates doing this,weget main centralpartofthedarknebulainScorpiusandOphiuchus,not ness ofthenebula.Incomparisonwiththis,diameter interesting becauseitgivesanapproximatelowerlimittothethick- compute anotherinterestingquantity,theabsorptioncoefficient,by found whenthedistanceofstarbeyondfrontsurfacewas intensity andgalvanometerdeflection.Afairlysatisfactoryfitwas counting thearms,isabout5parsec.Usingthisdistance,wecan taken as100,000astronomicalunits,or0.5parsec.Thisresultis Barnard, andalthoughthemeasureswereveryrough,itisinterest- ing toseewhatisobtainedbyassumingalinearrelationshipbetween in Figure6foranopticaldepthofthesource3mag.Althoughno fainter thanthesky,orabout5.7mag.Equatingandsolvingfor7, we find From themeasuresofStruve,Elvey,andRoachthismustbe1.2mag. 136 L.G.HENYEY 13 12 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem However,diffusereflectionnebulaewiththesourceinside areapparentlyrare,be- The colorexcessofCD—24°!2684,ifinterpreted asaresultof Mt.W.Contr.,No.250;Ap.J.,56,1922. This markeddifferenceintheintensitydistributionsofnebu- The measurementsofCD—2412684havebeenfittedtothecurve k =6mag./parsec. 7 =0.6. VII 1937ApJ 85 . . 107H scattered lightonlytoaslightextent,thelarger particlesbeingthe particles whichabsorblightbutonlyscattera verysmallamountof fraction oftheincidentenergyinOhmiccurrents generatedby it. Suchsmallparticlesmayconsistofmetalswhich dissipatealarge served polarizationinreflectionnebulaedoes not precludethispos- the electricfieldofradiation.Byitself, smallamountofob- would notproduceexcessivevelocitiesinthestarsitsvicinity. results. Ourpreferenceforlargeparticlesrests,asbefore,merely sibility sincethesesmallparticleswouldgovern thepropertiesof interesting toinvestigatethepossibilityofexistencesmall where disthedensityofmediumandgspecificweight we findfromRussell’sformula be verylargesinceotherwisethemassofnebulawouldexces- upon theevidencefromcolorsanddegreeofpolariza- sider thisresultasfinal.Ourobservationshavefailedtogiveusa having diametersoftheorder0.1mmwouldgiveamassthat It isobviousthatthismassnotexcessiveandevenparticles we havefoundnoinconsistencywithLambert’sfunction,itistrue definite clueconcerningtheformofphasefunction,and,while of 2mag.tolargeparticles.Itwouldbepremature,however,con- by smallparticlesoftheorder1mag.Sinceobservedabsorp- the particle,thattotalmassofnebula,forAw=6mag.,is consistent withtheseobservations.Obviously,theparticlescannot tion. Itisdifficulttostatethelowerlimitforsizeofparticles that otherphasefunctionswouldhavegivenequallysatisfactory tion isatleast3mag.,wecouldattributetheremainingabsorption sive. Ifweassumethattheradiusofanaverageparticle © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Professor B.Strömgrenhaspointedouttomethatitwouldbe THE ILLUMINATIONOFREFLECTIONNEBULAE137 23 Area ofnebulaincmXhd~10. -4 p =10X5X10cm, Aw =0.8—, Pg 1937ApJ 85 . . 107H various suggestionsconcerningtheforegoingpresentation,toMiss most importantsourcesofthenebularradiation.Itisthroughtheir Annie J.Cannonforherkindre-examinationofvariousspectra, for hisvaluablecriticismofthework,toProfessorKuiper and smallparticles,bothabsorbinglightbutonlythelargerones and theweaknessofpolarization,onotherhand—allsupport in thereflectedlightofnebulaeassociatedwithunobscuredstars, absorption, ontheonehand;absenceofstrongselectiveeffects reveal theirpresence.Thecoexistenceofselectiveabsorptionin and inparticulartoProfessorStruveforhiswillinggenerous the hypothesisthatwearedealingwithanagglomerationoflarge absorbing power,whichmaybeselective,thatsuchparticlesshould co-operation duringeveryphaseoftheworkandforinspiration which hehasprovided. scattering it. CD —24T2684anditsnebulawithalargevalueofthephotographic 138 L.G.HENYEY © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem I wishtorecordmysincereappreciationProfessorStrömgren Yerkes Observatory December 1936