Strong effects of time-dependent ionization in early SN 1987A

0.5 MB·English

by  V. P. Utrobin

Astronomy&Astrophysicsmanuscriptno.Bochum February2,2008
(DOI:willbeinsertedbyhandlater)
Strong effects of time-dependent ionization in early SN 1987A
VictorP.Utrobin1,2 andNikolaiN.Chugai3
1 Max-Planck-Institutfu¨rAstrophysik,Karl-Schwarzschild-Str.1,D-85741Garching,Germany
2 InstituteofTheoreticalandExperimentalPhysics,B.CheremushkinskayaSt.25,117259Moscow,Russia
3 InstituteofAstronomyofRussianAcademyofSciences,PyatnitskayaSt.48,109017Moscow,Russia
5
0 Received23December2004/accepted16March2005
0
2
Abstract. Westudyatime-dependenthydrogenionizationintheatmosphereofSN1987Aduringthefirstmonthaftertheex-

n plosion.Themodelincludeskineticsofhydrogenionizationandexcitation,molecularhydrogenkinetics,andatime-dependent
a energybalance.Theprimarystrongeffectofthetime-dependentionizationistheenhancedhydrogenionizationcomparedto
J
thesteady-statemodel.Thetime-dependent ionizationprovidesasufficientpopulationofexcitedhydrogenlevelstoaccount

4 fortheobservedHαwithoutinvokingtheexternal56Ni.WefindthattheBaII6142ÅlineinSN1987A canbereproduced

fortheLMCbariumabundance. Thisresolvesthelong-standingproblemoftheunacceptablyhighbariumoverabundance in
1
SN 1987A. The key missing factor that should be blamed for the ”barium problem” is the time-dependent ionization. The
v
6 modellingoftheHαprofileonday4.64indicatestheratioofthekineticenergytotheejectedmass≈0.83×1050ergM⊙−1.
3
0 Keywords.stars:supernovae:individual:SN1987A–stars:supernovae:typeIIPsupernovae
1
0
5 1. Introduction model has ignored a time-dependent energy balance; instead
0
simplelawshavebeenusedtomimictheevolutionoftheelec-
/ Spectra of type IIP supernovae (SN) combined with photo-
h trontemperatureintheatmosphere.
metric data are the primary source of our knowledge about
p
- the mass, energy, chemical composition, mixing, asymmetry
o of ejecta that are of a vital importance for the verification of Keepinginmindtheimportanceofthenon-steadystateion-
r
t the explosion mechanism. Generally, to recover the informa- izationeffectsfortheunderstandingofthephenomenaseenin
s
a tionimprintedinthespectraoneneedstouseratherelaborated earlySN1987A,werevisitthisissue.Tothisendweupgrade
: models.Unfortunately,theultimatemodelwithallthephysics the atmosphere model that apart from the time-dependentki-
v
i includedisbeyondreach,sosomesimplificationsareunavoid- netics includes also the time-dependent energy balance. Our
X able.Oneoftheassumptionsacceptedintheatmospheremod- maingoalistocomputetheionizationandexcitationofhydro-
r elsisthesteady-stateapproximationfortheionizationkinetics. genintheatmosphereofSN1987Afordifferentepochs,andto
a
Yet,thatthetime-dependenteffectofthehydrogenioniza- comparethecalculatedHαlinewiththeobservations.Wealso
tion may play a role in the SN envelope has been recognized address a highly intriguing problem of tremendous Ba over-
by Kirshner & Kwan (1975); they applied this effect to ac- abundancethathasbeenrecoveredbyseveralstudies(Mazzali
countforthehighHαluminosityofthetypeIISN1970G.The et al. 1992; and references there). Our preliminary guess was
time-dependent ionization has been exploited for SN 1987A that the Ba overabundanceproblemwas caused by the strong
to accountforthe highexcitationofhydrogenin theouterat- underestimationoftheBaIIionizationfraction.
mosphere(v > 7000kms−1) duringthe first 40days (Chugai
1991a). The crucial role of the time-dependent ionization at We beginwith a question,whytime-dependenteffectsare
the late nebular epoch of SN 1987A has been emphasized by important (Sect. 2) and then describe in detail the proposed
Claytonetal.(1992)andFransson&Kozma(1993). model(Sect. 3). The Hα strength on day 4.64is used to con-
Recently, using a time-dependent model of the hydrogen strainthekineticenergyofthehydrodynamicmodelSect.4.1,
ionizationwiththekineticsofatomicandmolecularhydrogen thecomputationsoftheionizationandtemperaturestructureof
wehavefoundthattime-dependenteffectsinSN 1987Awere theatmospherealongwiththeHαprofilesforselectedphases
strongduringthefirstmonthandthattheycouldprovidethehy- are presented in Sect. 4.2, while in Sect. 4.3 we address the
drogenexcitationrequiredtoaccountfortheHαlinestrength problemofhighBaoverabundance.
(Utrobin&Chugai2002)whichwasunderproducedinsteady-
statemodelsofSN1987Aatmosphere.However,ourprevious
We adopt here the radial velocity of SN 1987A of 286.5
Sendoffprintrequeststo:V.Utrobin,e-mail:[email protected] kms−1(Meaburnetal.1995)andthedistanceof50kpc.
2 V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A
2. Whytime-dependent?
A widely shared view that the steady-state statistical equilib-
riumisareasonableapproximationforSNIIPatmospheresre-
lies on the claim thatthe recombinationtime is muchsmaller
thantheSNage
(αN )−1 ≪t, (1)
e
whereαisthecoefficientofrecombinationtotheexcitedstates,
N is the electron number density. Adopting α ∼ 3 × 10−13
e
cm3s−1 and a typical SN age at the photospheric epoch t ∼
106 − 107 s one finds that the requirement (1) is satisfied
for N ≫ 107 cm−3. This condition is actually met at the
e
photosphere thus supporting the steady-state approximation.
Meanwhile, it is violated in the outer, high velocity layers,
where N may dropaslow as107 cm−3. Therefore,generally,
e
thetimedependentapproachisneededatleast,ifoneaddresses
thesituationintheouterlayers.
Moreover,itturnsoutthatthesteady-stateapproximationis
violatedalso intheinneratmosphere,wherethecondition(1)
holds.Thepointisthatthisinequalityhaslittleincommonwith
theconditionofthesteadystateformultilevelatomirradiated
Fig.1. The combined UV (IUE) (Pun et al. 1995) and opti-
bytheopticalcontinuum,becauseoftheefficientreionization
cal (CTIO) (Phillips et al. 1988) spectra of SN 1987A (thin
ofrecombinedatom.Theimportanceofthisphenomenonhas
solid line), the black-body continuum at the calculated effec-
been first recognizedfor the problemof the hydrogenrecom-
tive temperature of the high-energy model (dotted line), and
bination in the early universe, where Lα trapping along with
the observed continuum (thick solid line) between days 4.64
the reionization from the second level results in the signifi-
and44.21.
cant increase of the recombination time t = −N /(dN /dt)
rec e e
(Zeldovichetal.1968).
The net recombinationrate dN /dt refers to transitions to Thisshowsthattheeffectiverecombinationtimeisbyafactor
e
thegroundstate thateitherdonotemitLαorfollowedbythe of1/w21 ∼ 500greaterthantherecombinationtimetoexcited
escape of the emitted Lα quantum without resonance scatter- states 1/(αNe). For Ne ∼ 108 cm−3 one obtains trec > 107 s
ing.Lettheprobabilityofsuchatransitionisw .Theninthe >t.Wethusconcludethatthesteady-stateapproximationmay
21
”twolevelpluscontinuum”approximationthenetrecombina- breakdowninthedenseinneratmosphereaswell.
tionrate(neglectingtheexpansioneffect)is These qualitative considerations demonstrate that the hy-
drogenrecombinationintheatmosphereofSNIIPatthepho-
dN
e =−αN2w . (2) tosphericepochisessentiallyatime-dependentphenomenon.
dt e 21
With all the major processesfromthe secondleveltaken into
3. Themodelandinputphysics
accounttheprobabilityw is
21
A self-consistenttreatmentoftheSN atmosphererequiresthe
A +N q +A β
w = 2q e 21 21 21 , (3) hydrodynamicmodellingwithtime-dependentradiationtrans-
21 P2+A2q+Neq21+A21β21 port, energy balance, and ionization kinetics. At present such
a general approach is beyond reach of computation possibil-
whereP isthephotoionizationratefromthesecondlevel;A
2 2q
ities. To study time-dependent effects in the atmosphere of
is the probability of the two-photon decay evaluated for the
SN1987Aandyettomaketheproblemsolvable,wesacrifice
equipartition of 2s and 2p levels; q is the collisional deex-
21
theradiationtransferintheatmosphereusing,instead,asimple
citationcoefficient;A β istheLαescapeprobability.
21 21
descriptionofthecontinuumradiationfield.
Let us estimate w forthe neutralhydrogennumberden-
21
sity N > 109 cm−3 andtheelectronnumberdensityN < 108
1 e
cm−3, when the two-photon transition dominates over colli- 3.1.Continuumradiation
sional transition and Lα escape, i.e., w ≈ A /P . This sit-
21 2q 2
The SN atmosphere is very opaque in the Lyman continuum
uationis typicalfor the inner atmosphereof SN IIP. With the
photospherictemperatureofT ≈ 5000KonegetsP ≈ 104W (ν > ν1), so the photoionization by the photospheric Lyman
2
s−1, where W is the dilution factor. Close to the photosphere continuumcanbeneglected.Onthe otherhand,thehydrogen
(forW >0.1)onefindsw <2×10−3.Theeffectiverecombi- ionization by the recombination Lyman continuum in the at-
21
mosphere can be easily taken into account via cancelling the
nationtimeisthen
photoionizationofhydrogenfromthegroundlevelandthere-
t =−N /(dN /dt)=1/(αN w )∼500/(αN ). (4) combinationtothegroundlevelintherateequations.Intheat-
rec e e e 21 e
V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A 3
mospheretheradiationfieldincontinuumatthesefrequencies
isconsideredasbeinginthethermalequilibriumwithmatter.
Theimportantsimplificationweadoptisthatthecontinuum
intherangeofν < ν correspondstothefree-streamingcase,
1
i.e.,theaverageintensityofthecontinuumintheatmosphereis
J =WI ,whereI isthespecificintensityofthephotospheric
ν ν ν
radiation. The photosphericradius R and effective tempera-
ph
tureT areprovidedbythehydrodynamicmodel.
eff
Thefree-streamingapproximationisagoodapproximation
inthevisualband,wheretheopticaldepthoftheatmosphereis
quitelow.However,fortheultraviolet(UV)radiationbetween
Lyman and Balmer edges this approximationis indeed rather
artificial,becauseofthelargeopticaldepthduetotheUVline
opacity.Asaresult,theaverageintensityoftheUVradiationin
theatmospheregenerallyshoulddifferfromthefree-streaming
approximation.
Nevertheless,thisapproximationisnotasbadasonewould
think at first glance. It predicts behavior of J ∝ r−2 in the
most of the atmosphere. In the case of pure scattering with
the constant scattering coefficient (χ(r) = const) one obtains
forthe extendedatmosphere J ∝ [1+1.5Rχ(R/r−1)]where
Ristheouterradiusoftheatmosphere(Chandrasekhar1934). Fig.2. Comparison of the low and high-energy models of
AccordingtothisexpressionthebehaviorofJ(r)issteeperthan SN 1987A.Top: (a) bolometriclight curve of the low-energy
r−2 forouterlayers(r ∼ R)and J(r) ∝ r−1 forr ≪ R.So,our model (thick solid line) compared with the observations of
approximationcorrespondsto some intermediate range in the SN1987AobtainedbyCatchpoleetal.(1987,1988)(opencir-
atmosphere.Moreover,since the scatteringof UV metal lines cles)andHamuyetal.(1988)(opentriangles),and(b)theHα
isliabletophotonsplitting,theinteractionoftheUVradiation lineprofileonday4.64computedonthebasisofthishydrody-
withmatterwillbeclosetotheabsorptionratherthanscatter- namicmodelfortheblack-bodycontinuum(modelA)(dotted
ing. We expect, therefore, more steep behavior of J(r) in the line)andtheobservedcontinuum(modelB)(thicksolidline).
inner layer compared to the pure scattering case. In that case The calculated spectra are overplotted on the observed spec-
thelawJ ∝r−2maybeasensibleapproximationinthedeeper trum (Phillips et al. 1988) (thin solid line). Bottom: (c) bolo-
layersaswell. metriclightcurveofthehigh-energymodeland(d)thecorre-
To further allow for the uncertainty of our approximation spondingHαprofileonday4.64.
weconsidertwoextremecases:(i)thephotosphericbrightness
isblack-bodywiththeeffectivetemperature(modelA);(ii)the
photosphericbrightnesscorrespondstotheobservedspectrum
(model B) (Fig. 1). The real situation is somewhere between
the evolutionary model of non-rotating single star. This cor-
thesetwocases.
rection was needed to fit the observed bolometric light curve
ofSN 1987A(Fig.2a)with amoderatemixingofradioactive
3.2.Hydrodynamicmodel 56Niinthevelocityrange≤ 2500kms−1 (Utrobin2004).The
finalmodelisdubbedasthelow-energymodel.
The density distribution, chemical composition, radius of the
We consider also the high-energy version of the above
photosphereandeffectivetemperatureareprovidedbythehy-
model.Thismodelwith thekineticenergy1.5×1051 ergdif-
drodynamicmodelofSN1987A(seefordetailsUtrobin2004).
fersfromthelow-energymodelbythepresupernovaradiusof
We considerthenon-evolutionarymodelN of Utrobin(2004)
35.0R⊙,56Nimassof0.075M⊙,andtheadoptedcolorexcess
with18M⊙ejecta,kineticenergyof1051erg,and56Nimassof of E(B−V) = 0.17mag(Michaelet al. 2003). Thechemical
0.073 M⊙.ForthedistancemodulustotheLMCofm−M =
compositionoftheouterlayersistypicaltotheLMCchemical
18.5 mag, color excess of E(B− V) = 0.15 mag, and an in-
composition, X = 0.743, Y = 0.251,and Z = 0.006(Dufour
terstellarextinctionof A = 3.1E(B−V)magaprogenitorof
V 1984). This model agrees with the observed bolometric light
SN 1987A, Sanduleak −69◦202, had the radius of 46.8 R⊙. curveofSN1987Afor56Nimixedinthevelocityrange≤3000
To produce the presupernova density distribution the evolu- kms−1(Fig.2c).
tionary model l20n2ae of Woosley et al. (1997) was rescaled
to adopted radius and ejecta mass. We assumed for the outer
layers the circumstellar helium abundance Y = 0.441 (Wang 3.3.Linetransfer
1991; Lundqvist & Fransson 1996) and the LMC metallicity
Z = 0.004 (Dufour 1984). The structure of the presupernova Starting the first day the envelope of SN 1987A expands ho-
outer layers was modified to increase density compared with mologously (Utrobin 2004), so the Sobolev approximation is
4 V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A
applicable.Inthisapproximationthelinetransferisdescribed collisional processes for the atoms and ions with a detailed
bytheescapeprobability(Sobolev1960;Castor1970) level structure (Appendices A.1–A.4), all bound-free, radia-
tiveandcollisionalprocessesforthetwo-stateatomsandions
1−exp(−τ )
β = lu , (5) (AppendixA.5). We consider kinetics of hydrogenmolecules
ul τ
lu thatmayaffectthehydrogenionizationdegree.Sevenradiative
where and 37 collisional processes for the molecules are taken into
account(AppendicesB.1–B.2).
πe2 g
τ = f λ t N − lN (6) Forone-dimensional,sphericallysymmetricflowtheequa-
lu lu lu l u
m c g !
e u tionofcontinuitysimplifiesto
is the Sobolev optical depth in a transition from level l to ∂ρ ρ 1 ∂lnv v
levelu.HereN,g, N ,g arenumberdensitiesandstatistical =− , = 2+ , (10)
l l u u ∂t t t ∂lnr!r
exp exp
weightsofatomsinthelowerandupperlevels,respectively, f
lu
where∂/∂tistheLagrangeantime derivative;ρisthedensity
istheoscillatorstrength,andλ isthelinewavelength.Inthe
lu
ofthematter;randvareradiusandvelocity,respectively.The
Sobolevapproximationthefrequency-averagedmeanintensity
net rate of transitions between level k and all other levels of
ofthelineis
neutralatomzbyatomicprocessesis
J =(1−β )S +β J(ν ), (7)
lu ul lu ul lu ∂N J
z0,k = N A 1− lu +N N α
where " ∂t # z0,u uk S ! z+ e k
Xu>k ku
Slu = 2hcν23lu ggluNNul −1!−1 (8) − Nz0,kXl<k Akl 1− SJlluk!−Nz0,kPk
isthelinesourcefunctionandνulisthelinefrequency.Thenet + Ne Nz0,uquk−Nz0,kqku +Nz+Ne2qc,k
frequency-integratedlineemissivityis Xu>k(cid:16) (cid:17)
− N N q −N q −N N q
1 J(ν ) e z0,k kl z0,l lk e z0,k k,c
ηul = 4πhνluNuAulβul 1− S lu !, (9) Xl<k (cid:16) (cid:17)
lu + Rγ−Rγ , (11)
k k,c
whereA istheEinsteinspontaneousemissionprobabilityfor
ul whereN isthenumberdensityofneutralatomzatthelevel
thetransitionfromlevelutolevell. z0,k
k; N is the number density of singly ionized element z; N
FortheveryopaqueLαlinetheSobolevapproximationis z+ e
is the number density of free electrons; J is the frequency-
violated,ifoneconsidertheLαradiationtransfer,butremains lu
averaged mean intensity of the line (7); S is the line source
valid,ifoneisinterestedinthecomputationsoftheescaperate lu
function (8); α and P are the total radiative recombination
(Chugai1980). k k
coefficient, including the spontaneousand induced processes,
and the photoionization rate of neutral atom z in level k, re-
3.4.Gamma-raysandpositronsdeposition spectively (Mihalas 1978); q is the electron collisional ex-
lu
citation rate from level l to level u; q = (N /N )∗q is
The gamma rays with energy of about 1 MeV from the de- the correspondingdeexcitationrate andulthe ratzi0o,l(Nz0,u/Nlu )∗
cay chain 56Ni →56Co →56Fe deposit their energy through z0,l z0,u
forthelocalthermodynamicequilibrium(LTE)isgivenbythe
Comptonscatteringbyfreeandboundelectrons.TheCompton
Boltzmann formula; q is the electron collisional ionization
electronslosetheirenergythroughtheCoulombheatingoffree rate from level k; q k=,c (N /N N )∗q is the correspond-
electronsandthe ionizationandexcitationof atomsand ions. ingrecombinationcco,kefficienzt0,akndzt+heeLTEk,cratio(N /N N )∗
Theratesoftheheating,non-thermalionizationandexcitation isgivenbytheSahaequation;Rγ andRγ arethenz0o,kn-thze+rmeal
foratomsandionsare takenaccordingto Kozma& Fransson k k,c
excitationandionizationratesofneutralatomzinlevelk, re-
(1992).Thegamma-raystransportiscalculatedintheapprox-
spectively(Kozma&Fransson1992).
imationoftheeffectiveabsorptionopacityof0.06Y cm2g−1,
e The rate equation for the number density N of neutral
whereY is the numberofelectronsperbaryon.Positronsare H0,2
e hydrogeninlevel2isdeterminedbythenetrateofflowofpar-
assumedtodeposittheirenergylocally.
ticlesintotheunitvolumewith acharacteristictime t (10),
exp
the net rate (11), hydrogen two-photon decays, and other de-
3.5.Rateequations populationprocesses:
∂N N ∂N
Thefollowingelementsandmoleculesarecalculatedintime- H0,2 = − H0,2 + H0,2 −A2q N
dependentnon-LTEchemicalkinetics:H,He,C,N,O,Ne,Na, ∂t texp " ∂t # 2s,1s H0,2s
Mg,Si,S,Ar,Ca,Fe,Ba,H−,H ,H+,andH+.Allelementsbut − k N N −k N N
2 2 3 5 H0 H0,2s 6 H0 H0,2p
Haretreatedwiththethreeionizationstages.Neutralhydrogen − k N N −k N N
8 H0,1 H0,2s 9 H0,1 H0,2p
andionsMgII,FeII,andBaIIaremodelledwith15,9,30,and
− k N N , (12)
17levels,respectively,whileotheratomsandionsareassumed 14 H2 H0,2
toconsistofthegroundstateandcontinuum.Thereactionnet- whereN = 0.25N andN = 0.75N arethenum-
H0,2s H0,2 H0,2p H0,2
work includes all bound-boundand bound-free,radiative and ber density of hydrogenlevels2s and 2p calculatedassuming
V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A 5
15 and for other elements from He to Ba with three ionization
onlycollisionaltransitionbetweenthesestates;N = N
H0 H0,k
k=1 stages
is thenumberdensity ofneutralhydrogen;N isthePnumber
density of H2; A22qs,1s is the probability of theH2hydrogen two- Nz0 +Nz+ +Nz2+ = mρXAz , (21)
photon decay; k , k , k , k , and k are the rate coefficients u z
5 6 8 9 14
forthecorrespondingreactionsfromTableB.1.Therateequa- andthechargeconservation
tionsforthenumberdensityofneutralhydrogeninlevel3and
Ba
higherlevelsk>3are
NH+ −NH− +NH+ +NH+ + (Nz+ +2Nz2+)= Ne , (22)
2 3
∂NH0,3 = −NH0,3 + ∂NH0,3 −k N N zX=He
∂t t " ∂t # 7 H0 H0,3 whereX isthemassfractionofelementz,andA isitsatomic
exp z z
− k N N (13) weight.
14 H2 H0,3
and
3.6.Gasenergyequation
∂N N ∂N
∂Ht0,k =− tH0,k +" ∂Ht0,k#−k14NH2NH0,k , (14) Theequationofstateofaperfectgascanbewrittenas
exp
P =(N +N )kT , (23)
respectively.Similarly,therateequationforthenumberdensity g t e e
N ofionsz+ =Mg+,Fe+,Ba+ inlevelksimplyis
z+,k whereP isgaspressureandN isthetotalnumberdensityof
g t
atoms, molecules, and ions of all types. The specific internal
∂N N ∂N
z+,k =− z+,k + z+,k . (15) energy of particles (per gram) is the sum of the translational,
∂t t " ∂t #
exp excitation,andionizationenergies:
The rate equations for ionized hydrogen, negative hydrogen
3N +N
ion, and molecular hydrogen H , H+ and H+ are given in E = t ekT +E +E , (24)
2 2 3 g 2 ρ e exc ion
AppendixC.
Forthetwo-stateatomsandionsthenetrateoftransitions, where
forexample,fromneutralatomzintoionz+ byionizationand
Ba+
recombinationprocessesisgivenby E = ǫ N (25)
exc z,k z,k
∂N Xz=HXk>1
z+ = N (P +N q +Rγ )
" ∂t # z0 z0 e z0,c z0,c istheexcitationenergyand
− Nz+Ne(αz0 +Neqc,z0). (16) Eion = IHNH+ −IH−NH− +IH2NH+2 +IH3NH+3
Using the net rate (16) the rate equations for elements with Ba
threeionizationstagesareeasilygeneralized: + [Iz0Nz+ +(Iz0 +Iz+)Nz2+] , (26)
zX=He
∂N N ∂N ∂N
z+ =− z+ + z+ − z2+ (17) is the ionization energy. Here ǫz,k is the excitation energy of
∂t t " ∂t # " ∂t #
exp state k forthecorrespondingatomsandions,and IH, IH−, IH2,
and IH3, Iz0, Iz+ are the ionization potentials of the corresponding
atoms,molecules,andions.
∂N N ∂N
z2+ =− z2+ + z2+ . (18) The gas energyequation,the first law of thermodynamics
∂t texp " ∂t # for the material, including radiative losses in continuum and
lines,theComptoncooling,andnon-thermalheatingthusreads
Inparticular,therateequationforsinglyionizedoxygentaking
accountofthechargetransferprocessesincollisionsofH+with ∂E ∂ 1 4π ∞
OandO+ withHis ∂tg + Pg∂t ρ!= ρ Z κνJν−ηtν−η2νq dν
0 (cid:16) (cid:17)
∂N∂tO+ = −NteOxp+ +"∂N∂tO+#−"∂N∂Ot2+# − 4ρπBa+ ηul+ηC+εγ , (27)
+ kHONH+NO0 −kOHNO+NH0 , (19) Xz=HXl,u 
where
whereN isthenumberdensityofionizedhydrogen.Thenet
H+
rate(16)fortheionswithadetailedlevelstructure(Mg+,Fe+, ηC = 4kTeσ N J 1− Tr (28)
andBa+)iscalculatedincludingallrelevantatomicprocesses. mec2 e e Te!
To close the system of the above rate equations (12–15,
isthenetemissivityoftheComptoncooling(Weymann1966).
C.1–C.5, 17–19) we use the particle conservation for hydro-
Here κ and ηt are the monochromatic true absorption coef-
gen ν ν
ficient corrected for stimulated emission and the monochro-
ρX matic thermal emissivity, including bound-free and free-free
NH0 +NH+ +NH− +2NH2 +2NH+2 +3NH+3 = muAHH (20) processesforallatomsandions,respectively(Mihalas1978);
6 V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A
η2q is the two-photon emissivity of hydrogen; η is the net 4. Results
ν ul
frequency-averagedemissivityofline(9);εγ istherateofen-
4.1.EarlyHαandthehydrodynamicmodel
ergydepositionfromtheradioactivedecays;σ istheThomson
e
scattering cross section; J is the integrated mean intensity of
Preliminary modelling of Hα has revealed that at the early
continuum;T = (πJ/σ )1/4 is the radiation temperature and
r R epochthe Hα line strengthwas sensitive to the density of the
σ istheStefan-Boltzmannconstant.Theenergyequationde-
R outerlayers.AsimilarsensitivityoftheHαstrengthtotheden-
terminestheelectrontemperatureintheSNatmospherewhich
sitywasfoundinthesteady-statemodel(Eastman&Kirshner
isthesameforneutralsandions.
1989). We use therefore the Hα on day 4.64 to constrain the
kineticenergyofthehydrodynamicmodel.
Assuming the initial electron temperature equal to the ef-
3.7.Computationalmethod fective temperature Te = Teff we solved the system of rate
and energy equations and then computed Hα profile for the
low-energy model, i.e., for the model N with the kinetic en-
A hydrodynamicmodel with 300 mass zones providesus the ergy E = 1051 erg (Utrobin 2004). The corresponding bolo-
velocityv(r,t)anddensityρ(r,t) profiles,thephotosphericra- metric light curve and the calculated Hα profile on day 4.64
dius R (t) and effective temperature T (t). The simplified are shown in Figs. 2a and 2b. The computed Hα line has too
ph eff
description of the continuum radiation field as diluted photo- weak absorption at high velocities compared to the observa-
sphericradiationpermitsustosolveforagivenmasszonethe tions(Phillipsetal.1988).Thesituation,however,ismarkedly
rateequations(12–15, C.1–C.5, 17–19),theparticleconserva- improvedfor the high-energymodelwith E = 1.5×1051 erg
tionequations(20, 21),thechargeconservationequation(22), (Figs. 2c and2d).Thelargerstrengthofthe Hαabsorptionin
andthegasenergyequation(27)independentlyofothermass thehigh-energymodelisthedirectoutcomeofthehigherden-
zones. sity in the outer layers. Thus, the strength of the Hα absorp-
tionathighvelocitiesindicatesthefavoringenergy-to-massra-
Asthesupernovaenvelopeexpandsthephotospheremoves tio E/M ≈ 0.83×1050ergM⊙−1. We adoptthe modelwith the
inward the envelope. In a steady state the physical values in highenergyasthestandardmodelforourcomputationsofthe
the atmosphereat certain momentt are determinedby the in- SN1987Aatmosphere.
stantdensityandradiationfield.Inatime-dependentapproach
onehastostartfromamomentt atwhichagivenmasszone
ph 4.2.EvolutionofhydrogenionizationandHα
crossesthephotosphere,andthentofollowthetimeevolution
until t. To obtain the distribution of physicalvalues in the at- Withtheadoptedphotosphericcontinuumandtheinitialelec-
mospherethisprocedureshouldbeappliedtoeachmasszone. trontemperatureweareabletocomputeforgivenmomentsthe
distributionoftheessentialphysicalvaluesintheatmosphere,
The simultaneoussolutionof the aboveequations(12–14, including the hydrogenlevel populations. The initial electron
C.1–C.5, 17–22, 27)isachievedbyreducingthemtoasystem temperature at the photosphere is adopted equal to the effec-
ofordinarydifferentialequations.Inparticular,thegasenergy tive temperatureTeff whichis taken fromthe hydrodynamical
equation (27) is reduced to the ordinary differential equation calculations(Utrobin2004).
for electron temperature T . To properly formulate the prob-
e
lemoftimeevolutionofphysicalvaluesforagivenmasszone,
4.2.1. Day4.64
initialdatamustbespecifiedatthephotosphericlevel.Thisis
donebysolvingtheequations(12–14, C.1–C.5, 17–22)atthe On day 4.64 the Hα profile computed in the time-dependent
photosphericlevelassuming the steady state for a givenelec- model A fits the observed one fairly well (Fig. 3a), while the
trontemperatureTe.Thisisthesimplestwaytomodelthenon- model B underproduces the emission component (Fig. 3b).
equilibriumconditionsattheSNphotosphere.Thederivedsys- ThissuggeststhattheaverageintensityoftheUVradiationin
tem of ordinarydifferentialequationsis stiff and is integrated theatmosphereatthisandearlierepochsisclosertothatofthe
bytheimplicitmethodofGear(1971)withanautomaticchoice modelA.Thesteady-statemodelAproducesratherstrongHα
of both the time integrationstep and the order of accuracy of linealthoughboththeabsorptionandemissioncomponentsare
themethod. markedlyweakerthanintheobservedline,whileinthesteady-
statemodelBtheHαlineisextremelyweak.Theweaknessof
Therecoveredtime-dependentstructureoftheatmosphere the Hα line in the steady-state model reflects the simple fact
isused thentocalculatespectra atselectedepochs.Thespec- that the steady-state ionization is lower than that in the time-
traaremodelledbyMonteCarlotechniquewithrelativisticef- dependentmodel.
fectsincluded(Mihalas1978;Jeffery1993,1995).Wesuggest As expected, the electron temperature in the model A is
that the photosphere diffusively reflects the incident photons largerthaninthemodelBsincetheradiativeheatingrateinthe
(Chugai&Utrobin2000).Thelinescatteringisgenerallynon- modelAwiththeblack-bodyUVcontinuumishigherthanin
conservative and is described in terms of the line scattering themodelB(Figs.3cand3d).Theremarkablepropertyofthe
albedo.TheThomsonscatteringonfreeelectronsandRayleigh distributionofelectrontemperatureintheatmosphereforboth
scatteringonneutralhydrogenaretakenintoaccount. modelsisafastdropfromtheinitialvalueatthephotosphereof
V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A 7
Fig.3. Hα line profile and the structure of the atmosphere Fig.4. Hα line profile and the structure of the atmosphere on
on day 4.64. Panel (a): Hα profile computed with the time- day8.69.SeeFig.3legendfordetails.
dependentmodelA(thicksolidline)andwiththesteady-state
modelA(dottedline),andoverplottedontheobservedprofile
obtained by Phillips et al. (1988) (thin solid line). Panel (b):
4.2.2. Day8.69
the same as panel (a) but for the model B. Panel (c): the cal-
culatedelectronnumberdensity(thicksolidline)andelectron On day 8.69 the both A and B models reproduce satisfacto-
temperature (thin solid line) in the time dependent model A. rily the absorption component of Hα (Figs. 4a and 4b). The
The dashed line shows the initial electron temperature Tph at strengthofobservedemissionliesbetweentwomodelswhich
e
thephotosphereasafunctionofvelocity,whilethedottedline impliesthattheactualradiationfieldintheUVbandisinterme-
is the distribution of the electron temperature Tad evaluated diatebetweenAandBcases.Thesteady-stateapproachinboth
e
with the purely adiabatic cooling. Arrow indicates the photo- modelspredictsveryweakHαlineatthisepoch.Thisisqual-
sphericlevel.Panel(d):thesameaspanel(c)butforthemodel itativelyconsistentwiththesteady-statemodellingofMitchell
B. etal.(2002).
The electron concentration shows a kink at about 14000
T to∼(0.75–0.80)T .Thisdropresultsfromasimplefact kms−1(Figs.4cand4d)thatwasonlybarelyseenattheprevi-
eff eff
ousdate(Figs.3cand3d).Thepositionofthiskinkcorrelates
thattheelectrontemperaturecalculatedinthedilutedradiation
withthetransitionfromthefullionizationtopartialionization
fieldislowerthantheeffectivetemperature.Thesamephysics
atthephotospherethatoccurredaftert≈2days.
leadstoawellknownrelationfortheboundarytemperatureof
grayatmosphereT ≈0.81T (see,e.g.,Mihalas1978).
0 eff
Althoughtheelectrontemperatureisrelativelyhighinthe 4.2.3. Day14.68and19.68
most of the atmosphere, T > 4000 K, (Figs. 3c and 3d) it
e
isinsufficienttomaintainthelargeopticaldepthofHαdemon- ThecalculatedHαprofilesonday14.68(Figs.5aand5b)and
stratedbythetime-dependentmodels.Theprimarymechanism 19.68(Figs.6aand6b)showsimilarbehavior.ThemodelBfits
ofthepopulationofthesecondlevelistherecombinationtoex- totheobservedprofilefairlywell,whereastheemissioncom-
citedlevels. ponent in the model A is unacceptably strong. This indicates
Noteworthy, the populations of excited levels are small thataroundtheseepochstheUVcontinuumradiationiscloser
numbers that are controlled by the relatively fast transition tothatofthemodelB.Thesteady-statemodelatthisphaseis
rates.Asaresulttherelaxationtimesforthelevelpopulations veryweakwithpracticallyzeroabsorptionandemissioncom-
of hydrogenare small: N /(dN /dt) ≪ t. So the popula- ponents.
H0,k H0,k
tions of excited levels may be considered in the steady-state AttheseepochstheadiabaticcoolinginthemodelBdom-
approximation. We conclude that the population of the sec- inates at the low velocities as indicated by the close run of
ondhydrogenlevel,andtheHαopticaldepthaswell,isdeter- temperatures T and Tad (Figs. 5d and 6d), and the most of
e e
minedmainlybythe currentvaluesoftheelectronconcentra- theatmosphereiscooledbelow∼ 2000K.InthemodelAthe
tionandthephotoionizationrates, whichareessentiallytime- heatingduetothephotoionizationisstrongenoughtomaintain
dependent. ratherhightemperature∼4000Kthroughouttheatmosphere.
8 V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A
Fig.5. Hα line profile and the structure of the atmosphere on Fig.7. Hα line profile and the structure of the atmosphere on
day 14.68. Observed profile is given by Hanuschik & Dachs day 29.68. Observed profile is given by Hanuschik & Dachs
(1988).SeeFig.3legendfordetails. (1988).SeeFig.3legendfordetails.
Fig.8.TheHαopticaldepthonday29.68inthemodelA.The
local minimum at 5000 kms−1 is a characteristic feature re-
quiredfortheexplanationofthebluepeak.
ThebehaviorofelectrontemperatureT andelectroncon-
e
centration N is very different for both models (Figs. 7c and
e
7d)thatemphasizesthesensitivityoftheresulttothebehavior
of the UV radiation field in the atmosphere. Yet, it is highly
remarkable that even the time-dependent model B, in which
Fig.6. Hα line profile and the structure of the atmosphere on theUVcontinuumintensityispossiblyunderestimated,repro-
day 19.68. Observed profile is given by Hanuschik & Dachs duces the strength of the Hα line in the whole range of the
(1988).SeeFig.3legendfordetails. radialvelocities.Thismeansthattheadmixed56Niintheouter
layersisnotneededtoaccountfortheHαlineinSN1987Aat
theageoft≤30days.
4.2.4. Day29.68
Between day 20 and 29 the additionalblue and red peaks
On day 29.68 the model A leads to the unacceptably strong emerge in Hα profile, a phenomenon known as ”Bochum
Hαemission(Fig.7a),whilethemodelBreproducestheover- event” (Hanuschik & Dachs 1988). We believe that the red
allstrengthoftheabsorptionandemissioncomponentsofHα peak is related with the high velocity 56Ni clump in the far
(Fig.7b).TheHαlineinthesteady-statemodelsisextremely hemisphere, whereas the blue peak reflects a minimum in
weak,althoughslightlystrongerthanonday14.68and19.68. thespherically-symmetricradialdistributionoftheHαoptical
V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A 9
4.3.Bariumproblem
The ”Ba problem” is the disparity between the high Ba over-
abundancefactorζ =(Ba/Fe) /(Ba/Fe) ∼ 5derivedfrom
87A LMC
BaII6142Åline(Mazzalietal.1992)andthetheoreticalpre-
dictionsofs-processcalculations.Thes-processnucleosynthe-
sis in the massive stars evolving to the presupernova star of
SN1987A(Prantzosetal.1990)yieldstheBaoverabundance
ofζ ∼1.3–1.4inthehydrogen-richenvelope,assumingthata
totalejectamassis15–18M⊙,amass-cutisat2M⊙(Woosley
&Weaver1995),andthesynthesizedBaiscompletelymixed
overtheejecta.Infact,itishighlydifficulttoimaginethatBa
produced at the He-burning stage can be significantly mixed
with the hydrogenenvelope. More realistic would be a situa-
tionwiththeLMCabundanceofBainthehydrogenenvelope.
Inourtime-dependentapproachwerecoverthedistribution
oftheelectronconcentrationthatfitstotheHαprofileonday
14.68 and 19.68 in the model B. Therefore we hope that our
model can recover a more realistic abundance of Ba from Ba
II 6142 Å line. We computethe Ba ionization and excitation,
using the multilevel Ba II model described in Appendix A.4,
for the two choicesof the far UV (10.0 < hν < 13.6 eV) ra-
Fig.9.EvolutionofBaII6142Ålinefromday8.69till19.68
diation in the SN envelope.In the first versionthe continuum
in SN 1987A. The Ba II 6142 Å profiles, computed with the
inthisenergyrangeisthedilutedphotosphericradiation—our
time-dependentmodelsA(dottedline)andB(thicksolidline),
standardassumptionformodelsAandB(Fig.1).Inthesecond
areoverplottedonthoseobservedbyPhillipsetal.(1988)and
versionwefollowthesuggestionofMazzalietal. (1992)that
Hanuschik & Dachs (1988) (thin solid line). Left panel (a):
thefarUVcontinuumshouldbeessentiallysuppressedbythe
models for the relative Ba overabundanceof 1.5. Right panel
line-blocking (in fact Rayleigh scattering also contributed to
(b):modelswiththesuppressedfarUVcontinuum.
theblockingeffect).Thisassumptionessentiallyincreasesthe
ionization fraction of Ba II and thus decreases the overabun-
dance factor. We, however,modify this assumption making it
more realistic: we adopt that the far UV (10.0 < hν < 13.6
eV) radiation is in equilibrium with matter and is character-
ized by the local electron temperature of our time-dependent
model. This is the same approximationas accepted abovefor
depth at ≈ 5000 kms−1 (Chugai1991b; Utrobin et al. 1995).
hν>13.6eV.Unfortunately,theIUEobservationscannotpro-
The mechanism responsible for such a minimum is still not
vide a confident verification of our assumption about the far
identifiedconfidently.Both A and B modelsdo notshow any
UVfluxinSN1987Abecauseofthedominantcontributionof
signatureofthebluepeakinHαprofile.Remarkably,however,
the lightfromthe stars2and3to thisband(Punetal.1995).
themodelArevealsanessentiallynon-monotonicbehaviorof
WefindthattheacceptedsuppressionofthefarUVcontinuum
the Hα optical depth τ(Hα) (Fig. 8) with the position of the
doesnotaffectthecomputedHαlineforAandBcasesatall.
minimum at 5000 kms−1, just at the place that corresponds
The Ba II 6142 Å line profile calculated for three phases
to the blue peak.Unfortunately,the optical depthat this min-
preceding the Bochum event are shown in Fig. 9. In the first
imum is too high to produce the blue peak in the Hα profile.
versionofthecontinuumtheBarelativeabundanceofζ = 1.5
Therequiredvalueshouldbe30timeslower.Howeverthevery
isrequiredtoreproducetheBaII6142Åline(Fig.9a),while
possibility that the non-monotonicbehaviorof τ(v) may arise
inthe secondcasetheLMCabundanceofBa (ζ = 1)isquite
in a model with monotonic evolution of the photospheric pa-
enough(Fig. 9b). We findthatfor ζ = 5 the bluewing ofthe
rametersandmonotonicdensitydistributionisverypromising
linecanbereproducedinthemodelBonday14.68and19.68
for the explanation of the blue peak in the Hα profile at the
withthephotosphericfarUVradiation.NotethatMazzalietal.
Bochumeventphase.
(1992)obtainedasimilaroverabundanceinthemodelwiththe
Previously,usingthetime-dependentmodelwefoundthat essentiallysuppressedfarUVradiation.Inourmodelwiththe
the enhanced hydrogen neutralization at about 4000 kms−1 suppressedfarUVradiationtheLMCabundanceofBasuffices
governedbyH−andH+mightberesponsibleforthelocalmin- to account for the Ba II 6142 Å line on day 14.68 and 19.68
2
imumofN andthus,fortheminimumoftheHαopticaldepth (Fig.9b).NotethatinthiscasethemodelBproducestoostrong
e
τ(Hα) (Utrobin & Chugai 2002). However, in that model we BaII6142Ålineonday8.69,whilethemodelAproducestoo
adopted rather simple behavior of electron temperature T in weakline.Thisisconsistentwithourpreviousconclusionthat
e
theatmosphereandatthe photosphere.Asa resultthe roleof atthisepochtheHα linerequiresthe intermediatecontinuum
themolecularprocesseshasbeensignificantlyexaggerated. betweenmodelAandB.
10 V.P.Utrobin&N.N.Chugai:Time-dependenteffectsinSN1987A
We come to the remarkable result: the model B with the AppendixA: Atomicdata
suppressedfarUVradiationthatwasalsoassumedinthemodel
A.1.Hydrogen
of Mazzali et al. (1992) recovers the LMC abundance of Ba
from the Ba II 6142 Å line. Therefore, we conclude that at
For hydrogen we use a model atom consisting of 15 levels
presentthereisnoseriousreasonanymoretosuggesttheover-
all of them being treated as single levels. The energy levels
abundanceofBainthehydrogen-richenvelopeofSN1987A.
and oscillator strengths for the all transitions are taken from
The ”Ba problem” in SN 1987A was mainly a by-productof
Wiese et al. (1966). The radiative probability of the hydro-
the steady-state approximation used for the determination of
gen 2s–1s two-photon transition and the distribution of the
hydrogenionization.
hydrogentwo-photoncontinuumare givenby Nussbaumer&
Schmutz(1984).Thephotoionizationcrosssectionsofhydro-
gen are computed with the formulae and tables of Karzas &
5. Discussionandconclusions Latter (1961). The temperature-averagedfree-free Gaunt fac-
tor is interpolated from the table and accurate extrapolations
Ourgoalhasbeentoexploretheroleoftime-dependenteffects givenbySutherland(1998).Theelectroncollisionalexcitation
intheionizationandexcitationofhydrogenintheatmosphere anddeexcitationratesforatomichydrogenarecalculatedwith
ofSN1987A.Wehavedevelopedthemodelthatcalculatedthe theeffectivecollisionstrengthstakenfromScholzetal.(1990),
time-dependentkineticsofrelevantprocessestogetherwiththe Callaway (1994), Aggarwal et al. (1991), and Giovanardi et
time-dependent energy balance. The computations have con- al. (1987).The electroncollisionalionizationratesforatomic
firmedourpreviousconclusion(Utrobin&Chugai2002)that hydrogen are calculated using the approximate formulae of
theeffectofthetime-dependentionizationinSN1987Aatthe Johnson(1972).Theratecoefficientsforchargetransferincol-
photosphericepochisstrong:thehydrogenionizationintheat- lisionsofO+ withHandH+ withOaregivenbyStanciletal.
mosphereis significantlylargerand,as a result, the hydrogen (1999).
excitation is higher as well. The Hα formation at the photo-
spheric epoch during the first month is essentially controlled
A.2.Singlyionizedmagnesium
bythe time-dependentionizationeffectandthe additionalex-
citation by the external 56Ni is not needed to account for the
For singly ionized magnesiumwe use a modelion consisting
strengthofHαline.
of9singlelevels.Theenergylevelsandoscillatorstrengthsfor
Although we have failed to reproduce the additional blue thealltransitionsarecompiledfromtheAtomicSpectroscopic
peakintheHαline(Bochumevent)thatemergedbetweenday Database (ASD) at the National Institute of Standards and
20and29(Hanuschik& Dachs1988),wehavefoundthaton Technology(NIST)andfromKurucz&Bell(1995).Thepho-
day29.68oneofthemodelsrevealedanon-monotonicbehav- toionization cross sections of ionized magnesium are taken
ioroftheHαopticaldepthwiththepositionsofmaximumand fromtheOpacityProjectatomicdatabase(Cuntoetal.1993).
minimumthatweresuggestedbytheHαprofileatthisepoch. The effective collision strengths for the electron excitation
This result raises a hope that the blue peak will be explained of ionized magnesium were computed by Sigut & Pradhan
inamoreadvancedtime-dependentmodelwithacorrecttreat- (1995).Theelectroncollisionalionizationratesforsinglyion-
mentoftheUVradiationtransfer. ized magnesium are evaluated by fits of Voronov (1997), the
At the epoch preceding the Bochum event the time- approximate formula of Milkey & Mihalas (1974), and the
dependent model produces the realistic distribution of the semi-empiricalformulaofLotz(1969).
hydrogen fractional ionization in the atmosphere. With this
structure we are able to reproduce the Ba II 6142 Å line in
A.3.Singlyionizediron
SN1987Aonday14.68and19.68withtheLMCabundanceof
Ba.We believe,therefore,thattheBa problem,i.e.,unaccept-
ForthespecificcaseoftheFeIIiona30-term,multilevelatom
ably high Ba overabundance in SN 1987A, is resolved. The
modelisemployed.Atotalnumberof212transitionsbetween
missing physics of the former models is identified: the time-
thesetermsfrom2150Åto55µm arecalculated.Theenergy
dependentionizationthatprovidesahigherelectronconcentra-
levels and oscillator strengths for the all transitions in singly
tion in the atmosphere and thus more efficient recombination
ionizedironarecompiledfromASDatNISTandfromKurucz
ofBaIIIintoBaIIcomparedtothesteady-statemodels.
& Bell (1995). The photoionization cross sections of ionized
iron are taken from Nahar & Pradhan (1994) and Bautista &
Pradhan (1998), and averaged over resonance structures ac-
Acknowledgements. V.P.U. is grateful to Wolfgang Hillebrandt and
cording to Bautista et al. (1998). The Maxwellian averaged
Ewald Mu¨ller for hospitality during staying at the MPA, and also
rate coefficients for the infrared, optical and ultraviolet tran-
wouldliketothankManuelBautistaandKeithButlerforkindlypro-
sitionsinionizedironaretakenfromZhang&Pradhan(1995).
vidinghimwiththephotoionization crosssectionsof singlyionized
The collision strengths for some transitions are calculated by
ironandbarium,respectively.N.N.C.thanksRogerChevalierforhos-
pitality in Department of Astronomy of the University of Virginia, using an empiricalformuladerivedfromthe available experi-
where a part of this work has been done. This work has been sup- mentalandtheoreticaldataforFe II(Lietal. 1993).Therate
portedinpart bytheRussianFoundation forFundamental Research ofdirectcollisionalionizationforsinglyionizedironfromthe
(01-02-16295). ground state is computed using the fits from Voronov (1997)