19 63MNRAS.125. .461H 4+2 tend tocoolthenebula.Theexcitationratesincreaseveryrapidlywith the excitationenergyisconvertedintohydrogenlineradiation,theseprocesses rates, dominatethecollisionalcooling.However,inouterregion,where with distancefromthestarobtainedinfirstpaperofthisseries. formulation ofthefirstpaperthisseries(HummerandSeaton,1963).Since worthwhile toexaminecollisionalcoolingbyhydrogen. estimated thetemperaturetobeoforder 10 °K.However,since0and ponding tothoseobtainedinPaperI.Later MenzelandAller(1941),indis- hydrogen nebulaneglectingallcollisionalprocesses andobtainedresultscorres- of magnitude,collisionalexcitationhydrogen becomestheprincipalprocessin comparable withthedensityofheavyions,which,havingmuchgreaterexcitation on thenebulaandcompletelycounteractsrapidincreaseofkinetictemperature electron temperature,sothatcollisionalcoolingexertsastrongthermostatingeffect tion ontheionizationandthermalbalancesinapurehydrogennebula,using cussing theelectrontemperaturetakingintoaccount coolingbythe[Oin]lines, density ofneutralhydrogenexceedsthattheheavyionsbythreeorfourorders other heavyionsarenotuniformlydistributed throughoutthenebula,itseems determining theelectrontemperature. THE IONIZATIONSTRUCTUREOFPLANETARYNEBULAE © Royal Astronomical Society • Provided by theNASA Astrophysics Data System . Intheinnerregionofarealnebuladensityneutralhydrogenatomsis Aller, BakerandMenzel(1939)calculatedthe electrontemperatureinpure i. Introduction.—Weexaminetheeffectsofcollisionalexcitationandioniza- 4 from low-lyingstatesofhydrogen;inparticular,itmaybeimportanttoallow value of2x10°K.Collisionalexcitationgreatlyenhancesthelineemission excitation andionizationratesforhydrogen. estimates ofthetotalenergyradiatedbynebula.Therolehydrogen for collisionalexcitationofLyawhenoneobtainsstartemperaturesfrom temperature consideredasafunctionofstarassumesmaximum hydrogen nebulatodeterminetheelectrontemperatureinforawide thermal balanceequationsincludingtheseeffectsaresolvedforapure collisional coolinginrealnebulaeisdiscussedbriefly. range ofstartemperatures.Forreasonablestellarfluxestheelectron lead totheremovalofenergyfromplanetarynebulae.Theionizationand Accurate andconvenientexpressionsareobtainedforthecollisional Collisional excitationandionizationofhydrogenaremechanismswhich II. CollisionalCoolingofPureHydrogenNebulae (Received 1962November14) D. G.Hummer Summary 462 D. G. Hummer Vol. 125 The effects of collisional excitation and ionization of hydrogen in planetary nebulae have been discussed previously by Miyamoto (1938), Sobolev (1947), Chamberlain (1948, 1949, 1953; see also Aller (1956), p. 154), and briefly by Aller (1953). Aller and Minkowski (1956) interpret some features of the observed spectrum of NGC7027 in terms of collisional excitation. Cooling by ionization of hydrogen has also been considered by Mestel (1954) in connection with the growth of stars by accretion. We shall show that the excitation cooling rate exceeds that due to ionization by two orders of magnitude at Te^ 10^ °K. 2. Formulation 2.1. Ionization and thermal balance equations.—Generalizing the ionization and thermal balance equations (3.7) and (3.9) of Paper I to include collisional excitation and ionization, we have in Approximation I,
e 477 f KvJv dv+NeNRq{Te) = NeN+a.B(Te) (2.1) J Vi and
477 p K„J°h(v - V,) dv = Ne{N+kTeßB + NHIH[q(Te) + 0(re)]}, (2.2) J VL where
q(Te) = vQ (ionization), (2-3)
®(Te)=ï(*-^)qn(Te), (2-4) n=2 \ «/ and
(2-5)
and the rest of the notation is as in Paper I. The additional term in (equation 2.1) represents collisional ionization; those in (2.2) represent loss of electron kinetic energy by ionization and collisional excitation respectively. Two additional assumptions have been made : (1) Collisional excitation has no direct effect on the ionization balance, since the life-times of the excited states are so short. There will be some atoms trapped temporarily in the zs state, but the total number of atoms in excited state at any instant will be negligible compared with the number in the ground state. (2) Collisional de-excitation of the excited states is negligible because of their short life-times, so that the electron kinetic energy expended in collisional excitation is never recovered by the electrons. Collisional de-excitation of zs via the zp state will certainly occur, but the energy transferred to the electron is completely negligible. The thermostating character of the collisional processses is apparent from
equation (2.2), since the function 0(Te)+ #(7^) increases very rapidly with
temperature, while the recombination term TßB increases very slowly. 2.2. Plane parallel nebula.—We want to solve equations (2.1) and (2.2), for the fraction of neutral hydrogen
è = NHlN (2.6)
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 19 63MNRAS.125. .461H where wehavedefined and fortheelectrontemperature,whichwewriteas notation ofsections4,4.1and4.3PaperI, For aplaneparallelnebulaitisconvenienttorewrite(2.1)and(2.2)inthe and where Q{E)isthecross-sectionforparticulartransitionandEthreshold q(Tf) aredefinedas and No. 5,1963Theionizationstructureofplanetarynebulae,II can immediatelyintegrate(3.1)intermsofthegammafunctiontoobtain where Qandaareparameterstobedetermined.Thisformisnotbasedonany by theform the useof(3.3)totemperaturessuchthatkT %E,althoughforionizationthe where theoretical thresholdlawbutispurelyempirical.AssumingQandaknown,one energy. WefindthatEQ(E)canbefittedforaconsiderablerangeabovethreshold normalized theirresults totheBornApproximationathighenergies, andby results shouldbeaccurateuptokT~2l. In general(3.3)willtendtoover- and by FiteandBrackmann (1958)andbyBoksenberg(1961),both ofwhom estimate q(T). absolute determination. WeadoptBoksenberg’sresults,renormalized tothe Born exchangecalculations ofPeterkop(1962)andGeltman, Rudge and Rothe, Marino,Neynaber andTrujillo(1962),whoclaimtohave madean n en eE e © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 3.1. Ionization,—Theionizationcross-sectionfor hydrogenhasbeenmeasured 3. Excitationandionizationcross-section,—Thecollisionalratecoefficients The useof(3.2),whichrepresentsEQaccurately onlynearthreshold,limits 2 9<£>exp(7)(3i> Ku{t)w{t) =ii^(i)+(i-0[?(O1>(O],(2.9) y =V5L",(á)-^''<á)’ =E Ku(r) =(t)-(i£)q(t)(2.8) XB T ?( e)=Cexp[-+(!a)InAJ, -4 t —ioT(Tin°K).(2.7) C EQ{E) =QE(f^y,(3.2) e n =0/—r(«+i) X y() =^ß(t.(2.II tB EkT K =Je- K=A1¡N (2.10) V 7Tm e 33 (3-5) (3-4) (3-3) 463 19 63MNRAS.125. .461H where tisdefinedbyequation(2.7). Following asuggestionofDrM.J.Seaton,wefitEQtowithinfewpercent has beenmeasuredbyFite,StebbingsandBrackmann(1959)thatforis-^zs n =zstatesinhydrogenisatpresentunsettled.Thecross-sectionforis-^zp in TableI. used theminpreferencetotheexperimentalresults.Ourresultsaretabulated giving cross-sectionswellabovethoseobtainedbyFite’sgroup.Sinceitis by LichtenandSchultz(1959)Stebbings,Fite,HummerBrackmann 464 those foris->2s. difficult toseehowthesetheoreticalresultscouldbesomuchinerrorwehave and to alinearfunctionoiE—IfromthresholdEgreaterthan50ev.Weobtain but theseresultsarenotexpectedtogive reliableresultsnearthreshold. The resultsforis->2paremostaccurate,thenthoseis->2s+2pandfinally by HummerandSeaton(1961). A slightlymorecompactformis be fairlyreliable,butwillnotusedhere since asimplerapproximationis where fall backonanempiricalprocedure,based on theassumptionthatallis-^n Approximation, butagaintheresultsarenotreliable atlowenergies.Thuswe adequate forthispurpose.McCarrol(1957) gives theBornApproximationfor excitation functionshavethesamefunctional dependenceonthevariable (i960). Theseexperimentsandthetheoreticalsituationtothattimearediscussed Peterkop (1962)havepublishedsolutionsofthecoupledis—2s2pequations Seaton (1962),givingacross-sectionslightlyhigherthanthatofRotheetal. Hence wewrite B is-^n andSeaton(1962)animpactparameter treatmentfori->nptransitions, Schey andSmith(1962)havecalculatedtheis-^^pcross-sectionthisshould Sommerville (1962)hascalculatedthe3s,3^, 3¿/cross-sectionsintheBorn—II (E £2/^),andcanbenormalizedathighenergies totheBornApproximation. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 3.2. Excitationofn=2.—Thesituationwithregardtotheexcitation 3.3. Excitationofn¿z3states.—Noexperimentalresultsareavailable.Burke, Recently Burke,ScheyandSmith(1962)independentlyDamburg F(E) =Q(E) Transition 15 15 15 25 2S +2p 2p x 10-9112exi6 ?(0 =5’31P(-5789/0.(3-) 2s +2p 2 -83 2 Q O«») Q(E) =AF(EE/E), n2 C= 2*ixiocm/sec. 0 =o-97mz>a=I, o 2*30 3*04 0*82 Burke, ScheyandSmith, E<4/^ McCarrol, E^>4l. H D. G.Hummer Table I 0*50 0-70 0-62 a 3 C (cm/sec) 3-9 x10- 5'i xio' i*4 xio' Vol. 125 (3-7) 19 63MNRAS.125. .461H 2 to determinetheconstantsin(3.2)foreachn.Wefindthataisindependentofn That F(E)isnotdefinedimmediatelyaboveE=4/#doesconcernushere.We and equaltothevalueforrc=2,oc0*62.InTableIIwegiveresults^3. determine AathighenergiesfromMcCarroFscross-sectionsandthenuse(3.7) where may beexplicitlycarriedout,weobtain expanding eachfactorinthehighertermspowersofw~,sothatsummations No. 5,1963Theionizationstructureofplanetarynebulae,II465 defined byequation(2.4).Treatingthetermsforw=2,3,4and5exactly Using equation(3.3)andTablesIIIwecanevaluatethefunction tabulated inTablesIIIandIVofPaperI.Wenowsolveequations(2.8) radiate asablackbodyattemperatureT,thefunctionsu{r)andw{t)arethose between (2.8)and(2.9)weobtain where n solution t.Workingout throughthenebulaonecanestimatetfairly wellfor values oftobtainedin this wayarequiteaccurate. the nextvalueofr,so that theleftsideof(4.1)needbeevaluatedfor onlytwo on agraph.GivenKandr,theleftsideof (4.1)isevaluatedforafewvalues (2.9) byasimplegraphicalmethodfor|andtasfunctionsofr.Eliminating£ or threevaluesoftfor each r.Sincetheleftsideof(4.1)variesslowly witht of tandplottedonthesamegraph,intersection ofthetwocurvesgiving except atthelowesttemperatures andF(£)increasesveryrapidly with t,the 8 3 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 4. Numericalresultsforblackbodyradiation,—Ifthecentralstarisassumedto The functions-F*canbetabulatedonceandfor allandF^(t)plottedaccurately (T') =exp(—o-izlnA)[39—o*75À)+7*7exp—o-SSgÀ) e -9 + 2*8exp(—o*938À)i*4exp—o*96À)3*8exp—À)]xio, 6 4 n 5 3 F(t) +zc(r)i 0 Ku(t) F(t) =®lq ^2(0 = 0 z2 p (\_a(
4 Fig. i.—Electron temperature t= io~ Te and neiitral hydrogen density plotted vs. distance from 4 star for five plane-parallel nebulaey illuminated by stars with T$ = 2, 4, 8, 12 and 20 X 10 °K. The total flux j2 =2*002 X 1012 in all cases.
On the other hand, collisional ionization is completely negligible compared with 4 12 photo-ionization ; for TÄ = 8 x 10 °K and «â = 2-002 x 10 the ratio of collisional to photo-ionization never exceeds 4X io~6. These results cannot be applied to a real nebula since the addition of 20 per cent helium alters the picture completely. Although the mean energy of the photo-electrons is considerably reduced, the effect of ionization of He+ is to greatly increase ionization of H, since all stellar quanta with v > 4^ are transformed into Hen Ly a which is much more effective in ionizing hydrogen. The rate of collisional cooling is then substantially reduced and we cannot draw any quanti- tative conclusions about the temperature in regions where heavy ions do not
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 19 63MNRAS.125. .461H 468 Theionizationstructureofplanetarynebulae,IIVol.125 cooling thehydrogenmechanismwillcomeintoplaywithconsequent where theheavyionsdonotexistinproperstages^ofionizationforefficient temperatures byabsorbingthehighenergyquantawhichproducethem. by Zanstra(1931)andotherstodeterminethetemperatureofcentralstar. the otherhandinouterregionofnebulaeandanypartinterior exist intheproperstagesofionizationtoregulatetemperature.Theaddition produced Lyawillbeconsiderablyreducedfromtheestimatesgivenabove.On of heliumalsodestroystheextendedtransitionregionsweobtainforhighstar publication. ThisworkwassupportedbytheUnitedStatesOfficeofNaval nebulae andforhisadviceonthehydrogencross-sectionsusedinthispaper.I increase inLyaemission.Thiscouldinfluencetheenergybalancemethodused next paperofthisseries. The electrontemperatureinrealnebulaewillbediscussedsomedetailthe also wishtothankDrP.G.Burkeforgivingmehishydrogencross-sectionsbefore discussions ontheconsequencesofcollisionalexcitationhydrogeninreal Aller, L.H.,1953,Ap.J.,118,547;1956,GaseousNebulae,(ChapmanandHall,London). Research underContractN62558—3207. Aller, L.H.,Baker,J.G.,andMenzel,D.1939,Ap.J.,90,601. Aller, L.H.,andMinkowski,R.,1956,Ap.J.,124,no. Boksenberg, A.,1961,Thesis,London. Burke, P.G.,Schey,H.M.,andSmith,K.,1962,Phys.Rev.,129,1258. Damburg, R.,andPeterkop,1962,Proc.Phys.Soc.,80,563. Chamberlain, J.W.,1948,Ap.J.,108,142;1949,109,480;1953,117,387. Hummer, D.G.,andSeaton,M.J.,1961,Phys.Rev. Letters,6,471;1963,M.N.,125,437. Fite, W.L.,andBrackmann,R.T.,1958,Phys.Rev.,112,1141. Rothe, E.W.,Marino,L.L.,Neynaber,R.H.,and Trujillo,S.M.,1962,Phys.Rev.,125, Lichten, W.,andSchultz,S.,1959,Phys.Rev.,116, 1132. Fite, W.L.,Stebbing,R.F.,andBrackmann,T.,1959,Phys.Rev.,116,356. Peterkop, R.,igbz,J.E.T.P.,14,1377. McCarroll, R.,1957,Proc.Phys.Soc.,70,460. Geltman, S.,Rudge,M.R.H.,andSeaton,J.,1963,Proc.Phys.Soc.,81,875. Zanstra, H.,1931,Publ.Dom. Astr.Obs.Victoria,4,209. Miyamoto, S.,1938,Mem.Coll.Sc.Kyoto,A21,173. Seaton, M.J.,1955,M.N.,115,279;i960,Reports onProgressinPhysics(Phys.Soc., Menzel, D.H.,andAller,L.1941,Ap.J.,94,30. Mestel, L.,1954,M.N.,114,437. Sobolev, V.V.,1947,Moving EnvelopesofStars,trans.S.Gaposchkin(Harvard University Stebbings, R.F.,Fite,W. L.,Hummer,D.G.,andBrackmann,R.T.,i960, Phys.Rev., Somerville, W.B.,1962,Proc. Phys.Soc.,80,806. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Dept, ofPhysics, 1962 November. Acknowledgments.—I wishtothankDrM.J.Seatonforseveralinteresting Because ofthecoolingfromcollisionalexcitationheavyions,collisionally University College, 582. Press, Cambridge,Mass., i960), Chapter3. London), 23,313;1962,Proc.Phys.Soc.,79,1105. 119, 1939. London, W.C.i\ References