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Neutrino driven neutron star formation

Cernohorsky, J.

Publication date1990

Link to publication

Citation for published version (APA):Cernohorsky, J. (1990). Neutrino driven neutron star formation. Rodopi.

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27 7

Chapterr 4

Flux- l imi te dd Neutr in o Diffusion I n Static Stellar Backgrounds s

J.. Cernohorsky

Institut ee for Theoretical Phytic», University of Amsterdam, Valckenientraat 65, 1018 XE Amsterdam, thee Netherlands, and

Centerr for High Energy Astrophysics, P.O. Box 41882, 1009 DB Amsterdam, the Netherlands

Septemberr 13,1990

Abstract-Th ee numerical implementation of multi-group Levennore-Pomraning Flux-limitedd Neutrino Diffusion Theory (FNDT) is presented. The behaviour of this transport schemee is investigated in five static stellar models. In the calculations the feedback of thee neutrino flow on the stellar matter is neglected. The evolution of the neutrino energy distributio nn function is followed in time, starting from an initia l local thermodynamic equilibriu mm (LTE) distributio n throughout the star, unti l a stationary non-LTE solution iss reached. Spectral and frequency-integrated sources, luminosities and distribution s are presented.. The influence of electron degeneracy on the neutrino transport is highlighted.

Energyy deposition in regions of the stellar models relevant to the delayed-explosion mechanism,, at matter densities broadly between 1012-109

28 8 ChapterChapter 4

11 Introductio n

Neutrin oo transport in a practical stellar application is a difficul t problem. Extended op-ticall yy thick , thi n and semi-transparent regions must be smoothly connected, and the transportt scheme must be able to operate in all three. Steep gradients in density, tem-peratur ee and chemical potentials, as well as the Fermi-statistics of the neutrinos further complicatee the problem, not in the least it s numerical implementation. In two previous paperss I1, 2] the Levermore-Pomraning (LP) Flux-limite d Diffusion Theory!3' (FDT) was extendedd and reformulated for neutrino transport . Thi s Flux-limite d Neutrino Diffusion Theoryy (FNDT) was developed for use in dense stellar environments, notably in the su-pernovaa and neutron star formation context.

Thee FNDT approach to the neutrino transport problem is a practical compromise betweenn physically rigorous '*' but computationally unaffordable approaches on the one hand,, and highly convenient but rather ad hoc ones on the other. The method involves fewestt assumptions in a multigrou p formulation' 1' and is inherently flux-limited . I t retains informatio nn about the neutrino energy distribution , which is in principl e observable wit h earth-basedd detectors.

I nn thi s paper the implementation of the FNDT transport scheme in a computer code is presented.. The final aim is to embed the transport code in a dynamical matter evolution codee and simulate the neutrino-radiatio n driven evolution of the hot proto-neutron star. Suchh a calculation was done previously by Burrow s and Lattimer in their pioneering paper Ref.[5],, but employed a rather crude neutrino transport scheme. Recently, Suzuki and Sato,'6'' simulated neutron star formation using a transport scheme very similar to the one presentedd here. The Kelvin-Helmholt z cooling phase and associated neutrino emission takee place on a relatively long tim e scale and last some 10-20 sec real time. Because of this,, both the matter evolution and the transport code must use an implici t algorithm, as onee cannot afford the numerical time step to be limited by the Courant condition.

Thee transport code must be able to operate in different matter environments and on a finit ee computer budget. I t must be automatically vectorizable, fast, accurate, and robust. I nn order to test it s performance, the FNDT code has been run in five different stellar models.. These have been chosen in such a way as to represent a variety of probable and interestingg physical circumstances. The differences between them are set up systematically soo that the influence of relatively minor variations in their equation of state (EOS) and the electronn fraction Ye can be investigated. In particular the differences in electron degeneracy andd their influence on the transport are highlighted.

Thee spiri t of thi s work is that in order to achieve a complicated final goal, one should go towardss it in small, systematic steps, making the model systematically more complicated andd realistic, and study the effect of each refinement.

Thee aim of thi s paper is to test the transport scheme and code in a set of calibrated andd simple stellar environments, which still contain the essential characteristics of more realisticc settings. In order to do simple things first and to systematically study the effect andd behaviour of the transport , the code is used to simulate vt and vt transport in spher-icallyy symmetric static stellar matter backgrounds. Moreover, if the relaxation time scale onn which the neutrino flows reach a steady state is found to be small compared to the dynamicall evolution tim e scale of the star itself, the matter wil l not change much durin g aa transport step.

Thee stellar models are constructed in hydrostatic equilibrium , so that they can be used ass starting points for dynamical evolution calculations. In order to isolate the behaviour

TransportTransport model 29 9

off the transport from possible complications associated with the EOS of the matter, the latterr is kept as simple as possible. In particular , the baryonic component consists of free nucleonss only. At very high matter densities and in regions of the star through which the bouncee shock has passed this ought to be a good approximation. At densities below some 10loyy cm~* this becomes increasingly less realistic.

Thee neutrinos interact with the matter through the ^-processes (2.8,2.9) and by elastic neutrino-nucleonn scattering (2.7). The calculations presented in this paper do not involve neutrino-electronn scattering, pair and plasmon processes. The reason is twofold. First, thee cross sections of these reactions are two orders of magnitude smaller than those of the /^-reactions.. Second, these processes exchange energy with the matter without removing thee neutrino from the stream. Therefore they couple neutrino energy-bins. This compli-catess the problem conceptually and technically, since the formulation of FNDT as given in Refs.[l,, 2] deals strictl y with monochromatic transport. A self-consistent extension involv-ingg bin-coupling sources has not been made to date. On the technical side, bin-coupling processess change the structure of the matri x (4.8), inverted in the code, from tri-diagonal too tri-block-diagonaL Instead of having single elements along the main diagonal plus and minuss one column, it would have there blocks of size NB x NB with NB the number of energyy bins. The time needed for the calculation with uncoupled bins is linear in NB. Wit hh coupledd bins this is at least (NB)2. Moreover, the relaxation and convergence speeds wouldd be limited by the slowest bin. The solution of a NB times larger coupled system of non-linearr equations may introduce added complications and instabilities. The incorpo-rationn of General Relativity (GB.) introduces the same conceptual and technical problems ass discussed above and is therefore also delegated to a later time. In my view it is not sensiblee to put great effort into the inclusion of 0(10-20%) effects, if their inclusion in a moree or less ad hoc fashion may introduce systematic errors of the same magnitude.

Thee organisation of the paper is as follows. In Sec. 2 the problem is stated. In Sec. 3 backgroundd stellar models are presented and their properties are discussed. In Sec. 4 thee numerical implementation of the transport scheme is presented, and the numerical 'experiments'' are outlined. Section 5 contains the results of the calculations. These are furtherr discussed in Sec. 6. For details of F(N)DT the reader is referred to Refs.[l]-[3]

22 Transport model

Thee technical problem at hand is solving the energy balance equation, which is the first angularr moment of the neutrino transport equation.'1, 3] The energy balance equation givenn below is identical for vt and üt although, of course, the opacities and distribution s aree different for both species. For a spherically symmetric system it reads* in standard notation n

öte+^e-v)-«;, r, t) follow from e and b by multiplicatio n with the factor W*/2T 2,, with ta the particle energy. The equilibriu m energy density is thus given by

BB = (u>>/2w*)f2, (2.2)

'Wee adopt natural units,Kttin g Jl = c = k« = 1.


andd involves the Fermi-Dira c distributio n function

att local matter temperature T(r) = j9" 1(r) 1 wit h

M»00 = Mr ) + M r ) ~ /*n(r) (2.3)

thee local equilibriu m neutrino chemical potential. Thee first Eddington factor ƒ (a>, r) , defined in terms of the normalised second angular

momentt of the distributio n function ƒ„ and equal to the rati o of the spectral flux F(u>) andd spectral energy density U(u>), ƒ = \F/E\ is expressed asl1, 2' 3]

/ = c o t h f l - l / ü ,, (2.4)

i nn terms of the dimensionless quantity121

nn _ —dTe == bK'+e{K,-K./Z)m ( 2 ' 5 )

I nn the absorption opacity K* (W, r ) stimulated emission and absorption are included through thee blocking factor

wit hh ƒ" the LT E electron distributio n function.

Elasticc scattering on nucleons

v + n,p (2.7)

iss accounted for by *c,(w,r ) and kt(uf, r) , the isotropic and anisotropic contribution s respectively.'1,, 2] The expressions for the rates and cross sections used in this paper were takenn from Bruenn.[7] Elastic scattering on nucleons contributes to the total opacity and thee momentum transfer, but because the recoil of the nucleons is neglected, it does not contribut ee to the energy transfer. The scattering opacities are the same for neutrinos and anti-neutrinos. .

Energyy and lepton number are exchanged with the matter through emission and ab-sorptionn by the beta-reaction

v.. + n ^ p + e " , (2.8)

forr neutrinos and vvtt + p ** n -|- e

+ , (2.9)

forr anti-neutrinos. Thee matter background is static. Al l the opacities as well as the equilibrium energy

distributio nn function 6(u>, r ) are 'frozen' in time durin g the transport calculation. The feedbackk of the neutrino flows on the matter exchanging energy and lepton number wit h thee background is neglected. Starting from some initia l neutrino energy distribution , for whichh the LT E distributio n 6(u», r ) is taken here, the distributio n e(uy r , t) wil l evolve int o aa stationary state. The time scale on which t and ƒ reach this steady state defines the relaxationn tim e scale of the neutrino flow.

Matterr background models 31 1

33 Matter background models

I nn order to simulate neutrino transport and test the code one needs to have some quasi realisticc dense stellar matter background. Of course one could resort immediately to the usee of a realistic stellar model as it comes out of a hydrodynamic collapse calculation some tim ee after bounce and the start of the shock propagation through the stellar core. This approachh has been followed e.g. by Janka and Hillebrand t in Ref.[4]. Apar t from the obviouss virtues of such a realistic approach, there are some disadvantages as well. Such aa stellar model is the product of a complicated calculation that involved a plethora of in-putt physics, approximations and the treatment and solution of many numerical problems. Typicall yy it involved a complicated, tabulated EOS and a lot of nuclear physics. Obvi-ouslyy thi s is not an easily reproducible background. Also it may contain many features thatt are poorly understood, but which stil l may influence the behaviour of the neutrino transportt performed on it . I t inhibit s systematic experimentation and the complexity of thee background may obscure some characteristics of the transport .

I nn principle , the transport is indifferent to the question whether the matter background iss in hydrostatic equilibrium , and what it s EOS is. Given reasonably smooth profiles of masss density />(r), temperature T(r) , and electron chemical potential /* e(r ) as functions off position r in a spherical model the transport code is capable of calculating the vt and vvee non-LT E energy distribution s e(ur, r, t) as they evolve from a given initia l state for a givenn time dt, or into the stationary solution.

Thee approach to the construction of the matter backgrounds used in this paper is a compromisee between realism and simplicity . The EOS (3.1) that was used to calculate temperaturess and chemical potentials is kept very simple, see below. In particular the modelss do not include any nuclei. Their baryonic composition is made up of free ideal nucleons,, except for the strong-interaction term above nuclear density n ( . However, the modelss are configurations which are in hydrostatic equilibriu m so that in a later stage onee can use them as initia l models for a dynamical evolution calculation including this transportt code.

Thee construction of a matter background model is realised in two stages. In the firs tt stage a 'star ' wit h specified total mass Af* and central density pc is constructed in hydrostaticc equilibrium . The hydrostatic equilibriu m and mass-continuity equations are solved,, using a cold polytropi c EOS, ipoi(p) = Kpy> wit h a given polytropi c index 7. Thi ss yields profiles of the density p{ph), radial position r(mr) and the total polytropi c pressuree Ppoi{mr) as functions of the enclosed mass m, which is transformed using the coordinatee transformation (4.1) int o the parameter f (m, ) , In the second stage the temperaturee T(£) and the electron chemical potential /xe(£) profiles are self-consistently constructedd by specifying a hot EOS and equating the cold polytropi c pressure to it , Jp©f(pU))) — Phot{p(0> ^(O» /*«(£)) I11 order to solve for both T and /*« an electron fraction att each position is specified simultaneously, giving a second equation Ye(£) = Ye(p,T^ne) (3.3).. Wit h the solution of these two equations and the density profil e />(£) already known fro mm the first stage, a temperature T(£) and electron chemical potential profil e /*«(() are constructed.. The four variables p(£), »"(£), T(£) an (* & ( 0 M functions of enclosed mass { (m, )) define the matter background, which is by construction in hydrostatic equilibrium . Heree £e = / i e(£)/T(£) is the electron degeneracy parameter.

Thee second stage of the approach sketched above can in principl e be applied to any EOS.. The simple EOS used in this paper for constructing T and £e as functions of position


o o

C C

id 4 4

id 3 3

id 2 2

id 1 1

id 0 0

1 0 9 9

1 0 8 8

1 0 7 7

1 0 6 6

1 0 5 5

** ^V.

r r

r r

r r

r r

r r

v v > ^ s s

^ * J J

,, , I ,

*Sv v

\ \ s^^ \ \ NN v>>.

r ïN>*-M1 tM1.1 tM2 2

vv ^ N ^ * K K

' ^ K K

vv ^s . M3-£X . .

s M4-»» "N

.. . I . , . I , . . I . . . I . . . I , 12 2 16 6 20 0 24 4

r) )

Figuree 1: Stellar mass-densities as functions of position £(mr ) .

70 0

60 0

50 0

40 0

30 0

20 0

10 0

--

--

: :

' '

>^>^ rr^^ rr*~^~ *~^~

/f''

MatterMatter background models 33 3

iss the following

PhotPhot = %*( 22. )f + n j j T + (*2/45)T« + (2/*>)T*SA{C). (3.1)

911 n,

Thee terms on the rhs of Eq. (3.1) represent the T = 0 contribution11^ to the bary-onicc pressure above nuclear density n,,(with K0 the nuclear incompressibility and ng thee baryonic particle density), an ideal free gas of nucleons, a free photon gas, and an electron-positronn mixture, respectively. The matter is in LTE with itself because it can equilibratee not only through the weak interaction but also electromagnetically. The poly-nomiall Si(x) = (l/4!)[z2(a;2 + 2x2) + (7x4/15)] is the exact Sommerfeld expansion of thee sum of the Fermi-integrals in the expressions for the relativistic e~ and e+ pressure contributions. .

Thee contribution of a veï>c mixture in LTE with the matter, given by

P?P? = ( 1 / * 2 )T 4S4 (W , (3.2)

cann in principle be added to (3.1), but is not included for the models in this paper. Itss inclusion in LTE is only justified in the very dense opaque interior, and moreover, unrealisticallyy raises the electron chemical potentials, because effectively it introduces just anotherr species of 'electrons', be it with half the electron helicity factor. The electron fraction,, defined as Yc = (nc- - ne+)/n.B is given by

YYcc = ( 2 / T T2 ) — 5,(&), (3.3)

ass a function of the matter variables with 5a(z) = ( l /3 ! ) (z3 + T2X).

0.4 4

0.3 3

0.2 2

0.1 1 00 4 8 12 16 20 24

-f(Mj j

Figuree 3: Electron fractions Ye = (ne- - ne +) / n s.

Thee drawback of the use of simplified stellar models such as these is the uncertainty as too whether any 'special effects' are also reproduced in more realistic models. On the other hand,, the occurrence of such effects can be more easily investigated because the models alloww more experimentation. The major drawback of this particular two-step approach to thee construction of a 'hot' proto-neutron star is that the initial polytropic structure with

--

--

: :

--

,.'•--... .

/ /

i i

• •

y y

" ' - - - - .. M4

M1 1

M2 2

. . .. M3

vT-r-rv-;uV. .

34 4 Chapterr 4

centrall densities in the relevant 1014-1015s cm'3 range never yields a star much bigger thann some 30 kilometers. In the hot EOS-fit this compactness leads to artificially high electronn degeneracy.see Fig. 5, even for modest electron fractions Ye = 0.2-0.3.

> > O O

c c 1 1

10 0

- 2 2 10 0

V V \ \

^ ^

i

\ ( ( S>> **•»

>XX '"*-!'*

^ \ ' " -- V x * *

\ N . .. "x'^~'vJ-M4

^\>-- *"***'*• * M11 - > \ \ ~-

II , . , I , , . I , . , l . ,

. .

-- * *

- ,, - . . - , M3

\ X M 2 2 ,, 1 , . > 1 .

12 2 16 6 200 24

Figuree 4: Matter temperatures as functions of position.

50 0

40 0

30 0

20 0

10 0

" "

\ \

** ' V

,, , , i , , . i . , II i " - . 1 ,

122 16 20 24 -?(M,) )

Figuree 5: Electron degeneracy parameters (c = nc/T.

Matterr background models 35 5

Inn this paper five different background models are used, called Ml , Ml.1, M2, M3 and M4.. The first three models, Ml , Ml. 1 an M2 are the simplest, and are related by their polytropi cc structure. They are single polytropes with polytropic index 7 = 4/3, total masss M* = 1.5JW©, and central density pc = 7.10

14 gcm~z. They are put in hydrostatic equilibriu mm by solving the Lane-Emden equation'81 using a stepsize-controlled fourth order Runge-Kuttaa routine.191

Modelss Ml and Ml. 1 have K0 = 0, so that the T = 0 contribution to the baryonic pressure,, the first term in (3.1), vanishes. Models Ml and Ml. 1 differ in the imposed electronn fraction which in Ml is chosen as Ye = 0.33 and in Ml. 1 as Y, = 0.1 throughout thee star, see Fig. 3. The difference in composition translates into differences in electron degeneracyy and temperature, see Pigs. 4 and 5. Model Ml is lepton rich, relatively cold andd contains very degenerate electrons throughout the star. Hence the neutrinos in LTE inn Ml are also very degenerate. Because of their soft cores M l and Ml. 1 have very simple smoothh and monotonie T(£) and £«(£) profiles.

Modell M2 is the same as Ml and Ml. 1 in polytropic structure. The nuclear incompress-ibilit yy in the hot EOS is set to K0 = 110 AfeV" , with nuclear density at n, = 0.12/ro

-3

andd r = 2. The electron fraction is set to Yt = 0.28 throughout. This model has an incompressiblee core. The first component of (3.1) dominates the pressure above nuclear densityy so that there the temperature is low and rises, whereas £e(£) is high and drops dramatically.. The model is quite lepton rich, and contains extremely degenerate electrons inn the dense inner core above nuclear density. Because of the hard core the ensuing T(£) andd £e{£) profiles are no longer monotonie but exhibit some simple structure.

Modell M3 is more elaborate and closer to the physical environment in a hot proto-neutronn star shortly after collapse, bounce and shock-launch. It is a tri-polytrop e in whichh three cold polytropic EOS

P(p)P(p) = KiPr< + Di, t = 1,2,3 (3.4)

aree joined together at certain predefined matter densities pu, P23 such that the pressure is continuouss and differentiable at the interfaces. The Dfs are constants. The constant D3 att the surface of the star is chosen such as to keep the pressure profil e continuous at the surface.. The surface lies at p^t = 107ff cm'3. The total mass of the star is Af* = 1.5M©, andd the central density pe = 4.10

14 g cm~3. The rest of the model parameters are ([/>] = gem-*)gem-*) r x = 2 for 10

14 < p < pet T2 = 1.25 at 1010 < p < 1014 and T3 = 4/3 for p < 10

10. Inn this way the collapsed dense inner core, the shocked outer core and an outer unshocked regionn can be roughly modeled in the first stage. The hydrostatic equilibriu m and mass-continuityy equations are solved directly as a coupled set with a fourth order Runge-Kutta routine.. Because of the composite-polytropic nature of the model, the ensuing profiles are nono longer E-solutions of the Lane-Emden equation.'81 The routine is iterated unti l the K\ iss found which corresponds to the predefined total mass of the star.

Thee parameters of the hot EOS (3.1) for M3 are K0 = 120MeV at nuclear density nntt = 0.14/m~', and T = 2.0. These values correspond to the prescription given in Ref. [10] forr the electron fraction Ye = 0.15, which is set equal to this value throughout the star. It iss a star with a hard core, a lower central density, and it is moderately lepton rich. The temperaturee profil e changes its slope at the edge of the inner core and the gradient of £e(£) flipsflips sign there, see Figs. 4, 5. Maximal electron degeneracy occurs at the beginning of thee 'shocked' outer core, instead of in the centre of the star as in M2. The degeneracy in thee outer regions of the star is lower in M3 than in the previous models.

36 6 Chapterr 4

Modell M4 is M3 evolved for one msec of real time using a dynamical stellar evolution code,, coupled to the neutrino transport code described in this paper. It is an aberration ass far as easy reproducibility is concerned. Nevertheless, it is included because it is no longerr a polytrope, i t is twice as big as the other models, and it is evolved from M3, so that comparisonn of the two allows a glimpse into the future of M3. Also the electron fraction iss no longer constant but has a position-dependent profile, see Fig. 3.

Afterr turning on the neutrino transport, the star instantaneously expanded somewhat duee to the sudden additional contribution to the pressure. During the msec evolution the regionn below 1012g cm~3 expanded rapidly as it was heated up by energy deposited by the neutrinoo flow. After one msec the star is nearly twice as big. Most of the deleptonisation onn this short t ime scale has occurred in the semi-transparent region between 1013 and 10u

Matterr background modek 37 7

3 3

200 40 60 0 80 0

M3 3

1000 120 140 a)) in MeV

Figuree 7: Equilibrium i/e energy distribution 6(u>, f) at fixed positions as a function of ui,, M3. Five curves are marked with their corresponding matter densities, 2.4el4 = 2.. 1014ff cm.-3, using standard FORTRAN notation.

3__ 0.028

0.024

0.02 2

0.016 6

0.012 2

0.008 8

0.004 4

0 0

44 1 : 2 .4eH ;-ii 2: 3.3e 13

:: ; i 3: 6.4e11 :: \ ! 4: 3.3e10

5: 4.5e9 !/\ i 6: 1.7e9

:: Ml

'-'-

--

11 N yy \r.i '*'v*^

11 u ****- , A \\ K> *""" -

^iSk.. , K ^ - « ' A . v Q ! ^ » ^ S ! ^ S = r: i , , . , j , , 10 0 200 30

M3 3

40 0 .50 0 CJJ in MeV

Figuree 8: Equilibrium ue energy distribution b(u>, £) at fixed positions as a function of u,

M3. .


I nn the dense inner core above 1013g cm~* the neutrinos are very degenerate. The factt that the peak in the degeneracy lies off-centre is reflected in Fig. 7 where the vt aree slightly less degenerate in the centre of the star marked wit h pe than in the contours furthe rr out. The fact that M3 is quite degenerate causes the ve to he much less abundant inn LT E than vey see Fig. 8. Where (e has it s peak the ve-abundance is lowest. Going out fromm the centre, the LT E ve spectrum first decreases and rises again after the £e-peak.

I nn Table 3 the model parameters are summarised.

Model l M l l Ml. 1 1 M2 2 M3 3 M4 4

Pc Pc 7.1014 4

7.1014 4

7.1014 4

4.1014 4

3.555 1014

7 7 4/3 3 4/3 3 4/3 3

multi i multi i

Ko Ko 0 0 0 0

110 0 120 0 120 0

r r

2 2 2 2 2 2

YYm m 0.33 3 0.1 1 0.28 8 0.15 5

ri,2.3 3 single e single e single e

2,1.25,4/3 3

M/m" 3) )

0.12 2 0.14 4

Pl2jP23 Pl2jP23

1014,, 1010

M33 evolved 1 msec

Tablee 1: Summary of matter background model parameters.

44 Transport scheme

Thee transport code solves Eq. (2.1), while simultaneously imposing Eqs.(2.4,2.5). The energyy balance equation (2.1) is a partia l differential equation of first order in time and secondd order in space. The independent variable is e(a>, r, t). The equation can either be solvedd as a pure differentia l equation on a simple grid or, after being integrated over a shelll volume, as a conservative integro-differentia l equation on a shifted grid consisting of sitess and shells. Both approaches have been implemented. The latter turned out to be superiorr and is described and used here. It s convergence speed and numerical stabilit y comparedd to the solution of the equation as a second order differential equation in space iss about twice as large. It s robustness in dealing with extremely degenerate conditions is considerablyy larger.

Thee integro-differentia l equation is put on a discrete grid, and turned into a difference equation.. The background model has been constructed on a Lagrangean grid , wit h the grid-point ss labeled by the enclosed mass m, withi n a radial distance r from the centre. Thee grid-sites are numbered t = 1,..., JVC? wit h site * = 1 corresponding to the centre of thee star. The number of sites NG is NG = 84 for models Ml , Ml . 1 and M2, and NG = 83 forr M3 and M4. In constructing the matter background, all quantities were defined on thee grid-sites. The underlying grid parameter is the enclosed mass m, transformed to the coordinate e

wit hh M* the star-mass, and i ) < < l a small parameter. This coordinate transformation, basedd on one given in Ref.[ll] , achieves a sufficient resolution of the stellar 'atmosphere' att low densities, labeling shells that contain littl e mass. The parameter ij determines this resolutionn and is best chosen between e~25 < 77 < e- 1 5. The power 1/a wit h a = 3 in the logarith mm ensures that for small nt,, near the centre, r(£) oc £.

I nn the transport code version discussed here, the grid is 'backward' centered, which meanss that the ith shell lies between site * — 1 and t. In the conservative formulation , vectors,, such as F;, R», and the metric quantities m, and r« are defined on the grid-sties.

TransportTransport scheme 39 9

Scalar»,, like e*, 6,-, K,- and tensors of the second grade are defined in between the sites, in thee shells.

Equationn (2.1) is integrated over the volume of each shell, and with Gauss' Law turned into: :

JJ dVidte + ƒ dSi • fe - ƒ dViKl{b - e) = 0. (4.2)

Ass was argued in the introduction, the transport code must be implicit. The implicit conservativee formulation leads to the following difference equation

'''' +1+1~^dVi~^dVi - {Kl)i[bi-e?1)dVi +

St St 4r[r?//+11 < c'+1 >i - r?-i ƒ £ ? < « t+1 >.-!] = 0, (4.3)

withh dVi = (4x/3)(r? - rf^) the volume of shell ». The bracket < e >i puts a quantity definedd in the shell on the site and is denned as

< c > i = , ,

thee average of two scalars in adjacent shells t and t + 1. Thee first Eddington-factor f(R(e,Bre)) is defined on the grid-sites and calculated

separatelyy using Eqs. (2.4) and (2.5). Written out as a difference equation the latter gives

# «« = - A e ^ / A , , i + < C*+l(*. " K./3) >i' (*-V

Thee discrete spatial derivative is defined as

—— - ^ ^ ( ^ ^ ^ (4.5)

withh [£i] = (& + &_i)/2 in the shell-middles. It is important to use this representation off the derivative with the difference taken in terms of the underlying Lagrangean grid-parameterr f instead of taking simply

ArtArt Ti+i - fi

Especiallyy at the edge of the star where the density gradient is the largest, the (-grid remainss well resolved whereas the r tends to become rather singular. When the transport codee is embedded in a dynamical evolution calculation on the Lagrangean grid, r is a dynamicall variable and ( is not.

Becausee of the complicated /(e, dre)-dependence Eq.(2.1) is highly nonlinear. The implicitt equation as a function of its unknowns at each position * is of the form

tf


withh j = i — 1, * ,t+ 1. Alter (k) iterations w ^ = w(fc-i) + ^w(fc)< 1° all iteration sequences,, except for the first one, tD(0) = 0,

Transportt results 41 1

whichh must be assigned an energy scale u>, MeV. The actual energy of bin j is given by u/ju/j = u>tZj MeV. For nearly all calculations presented in this paper u>, — 3 MeV was used. Forr the Pe, energy bins below the reaction threshold (1.805 MeV) of the beta-reaction (2.9) aree left out.

Startingg from the initial condition, the flow-evolution was calculated until the sta-tionaryy state was reached. At predefined times snapshots of the evolution were taken. Thee snapshot-times are t\ = 10-) whichh turned out to be of the order of a few tenths of a millisecond, depending on the energy-bin.. The code must be run in double precision. The time step is limited by the requirementt that the e-profile is not allowed to change more than 2.5% per time step. Thee code vectorizes automatically and completely except for the routine that inverts the tri-diagonall matrix, which uses a recursive algorithm.

Thee code has run on a number of machines. The time-performance of the evolution of 155 bins from initial condition to stationary state on a 84 site spatial grid is listed below. Thee values are normalised on the NP1, and expressed in CPU-seconds of each machine. Thee time performance and robustness of the code is sufficient for a full proto-neutron star

Machine e ENCOREE NP1 NECSX2 2 IBMM 3090-600J/6VF

nominall speed 400 Mflop 1.33 Gflop 1388 Mflop

vector r no no yes s yes s

runn time 100 CPU (sec) = 1

1/130 0 1/20 0

Tablee 2: Code performance on different machines normalised on the NP1.

evolutionn that uses it for the neutrino transport to be completed in a few hours CPU time onn a supercomputer like the SX2.

55 Transport results

Inn the following the results of the five model calculations are presented and discussed. Thee backbone of the discussion is provided by figures. It may at times be difficult to infer detailedd quantitative information from the figures alone. Where quantitative results are discussed,, they have been taken from the numbers and not by eye from these figures.

AA top-down approach is followed. First, frequency integrated, or bulk-quantities are discussed.. All bulk quantities presented are calculated with the steady-state solutions att time t = tfin. Second, the bulk quantities are resolved into their constituent spectral components,, which are the quantities that are directly calculated by the code. Third, the time-dependentt non-equilibrium energy distribution e(ut, r, t), the first Eddington-factor /(w,, r, t) along with some other quantities featuring in FNDT are presented.

5.11 Bulk quantities

Thee bulk quantities presented in the following are all computed in the stationary state. Thee integrations were carried out using a 15-point Gauss-Laguerre quadrature algorithm. Thee Integration accuracy was estimated by doing the integrations with the left and right adjustedd and centered trapezium rule and comparing with the Gauss-Laguerre results. It dependss on the quantity considered, but is always better than 10%. In most cases it is betterr than 1% and the integration errors are invisible to the naked eye.


5.1.11 Dens i t i es

Thee bulk non-equil ibrium vc energy and number (particle) densities are defined as (c.f.(2.2) andd above)

Ebuik{r,tfiEbuik{r,tfinn)) = / du>E(u>,r,tfin) Jo Jo

nbulk{r,tjnbulk{r,tjinin)) = / duE(u,r,tfin)/c Jo Jo

(5.1) )

(5.2) )

andd are shown as functions of position in Figs. 9 and 10 together with their LTE counter-parts.. The bulk vc energy and particle densities are presented in Figs. 11 and 12, and are definedd analogically. The particle densities are plotted in units per nucleon. In all models threee regions in the star can be distinguished. The first is the region deep inside the star wheree the neutrinos are in LTE with the matter. Hence -Etuifc = E ûk and n^ik = n ^J t-Thee LTE regions for ue li e at matter densities p > 10

12 gcm~3, —{ < 6 for the most degeneratee models M l and M2, and at p > 5.1012gcm~3, (—f < 5) for the rest.

en n

CO CO

c c

LJJ J

ic?4 4

1C?3 3

1(7 2 2

1C?1 1

1 0 s 0 0

i 3 9 9

1

TransportTransport results 43 3

>--1 0 * *

1 0 3 3

1 0 2 2

10 0

1 1

-1 1 10 0

- 2 2 10 0

rr / m // //'M2

M44 - * / / . M3

/ „ - • ' '' , - ' / M 1 . 1

'.'. .'V'

rr ,/ -•'.'•

rr /?.' / M4

! -- - - - v ^ ^ ^ r / - - - -M2

^^ ~^


E E o o en n (D D

c c

1U U

1


> > 2 2 3 3

240 0

200 0

160 0

120 0

80 0

40 0

7MI I

-- i> "" t»

,, 11

VA- M3 VÖÖ M4

i - - ry?r? t s ^^ i , 12 2 166 20 24 4

-!(Mr) )

Figuree 13: Bulk average vt particle energies.

> > 2 2 3 3

120 0

100 0

80 0

60 0

40 0

20 0

-\ \

11 \

\ \\ VvM.1

"%>h "%>h :;M22 ' ; ^ T ^ > ^ M4 . '' . , , i ^-r—,-^r-.--y:^^z-r1i-'^'-(^-r^--7-:i-=i-1?-T- i ,

12 2 16 6 20 0 24 4 -«Mr ) )

Figuree 14: Bulk average ve particle energies.


iss shown in Figs. 13 and 14 for i> c and ^respectively.

Thee inner cores down to 1012gcm~3 are dominated by high particle energies, in the 100-2000 MeV range, corresponding to the electron chemical potentials, cf. Figs. 4, 5. Neutrinoss in models M l and M2 have the highest w and correspondingly the vt have the lowestt average energy. In all models at low matter densities w levels out at energies between 100 MeV (Ml.1) and 16 MeV (Ml ) . Although the differences in electron degeneracy in the modelss lead to large differences in average particle energies deep inside the stars, the w's off the escaping neutrinos in the atmospheres are not very sensitive to this.

5.1.22 Sources

Thee bulk energy, lepton number and momentum sources, denned as

SQ(0SQ(0 = (l/nB(())JdwKl(w,()[B{u>,t)-E(w,()],

SI({)SI({) = ( l / n B ( 0 ) / d w ^ Ü [ B ( a , , 0 - £ K O ],

SA(t)SA(t) = - ( l / n f l ( f ) ) Jdu>Ktot(u,,t)F(w,t)

(5.3) )

(5.4) )

(5.5) )

respectively,, in units per baryon, are shown in Figs 15- 20. The spectral flux is given byy F(w,£) = f(u>,()E(u>,$). The momentum source SA({) features in the momentum balancee (hydrostatic equilibrium) equation. These sources account for the exchange of

c c o o >, , D D _o o i _ _ CD D Q_ _

F F o o en n i _ _ 0} 0}

c c

. .

O O Ul Ul

1 1

C C

- 1 1

- 2 2

- 3 3

- 4 4

.. M1 Ml ,, M2 «1.E-12 M1.1,, M3, M4«1.E-13

M1.1 1

'M2 2

,'M3 ,'M3

12 2 16 6 20 0 24 4

Figuree 15: Neutrino energy source term SQ((). Different models have différents scales, ass depicted in the figure. In this way e.g. the peak in M l lies at 1.2 l O ' ^ e r p c m -1 per baryon. .

energy,, lepton number and momentum between neutrinos and matter. In the equations thatt govern the matter evolution they enter as sources into the energy conservation, lepton


c c o o

o o -Q Q L> > > Q . .

E E o o c c

on n

0.4 4

0 0

0.4 4

0.8 8

1.2 2

1.6 6

- 2 2

2.4 4

7 7

: :

LJy\ LJy\

ii , .

- V ^ \ M 44 M1.M2«1.E-7 \ \ M 11 \ M1.1.M3*1.E-8 \ \\ I M4«1.E-9

\ \ \\ I _._.M2

'.. M V - . n - " 7 ' * " "

I I M 3 \ . . r '' -^

11 . . . 1 .

12 2 16 6 20 0 24 4 -f(Mr) )

Figuree 16: Neutrino lepton number source term SI({).

c c O O

o o

k. .


c c o o

o o -O O

k_ _

O) ) Q--

E E u u

t/ ) )

2 2

1 1

0 0

1 1

2 2

--

--

/. /.

Ml.,1. .

•• A

ff\ ff\ C>> 1

11 , I

3 3

\ \

ii M4',

\\ \

\ \

(\\ \ :: ""tz"

,, 1

\ \

MÏV--

\ 'M2 2

._-;—--

11 i i

M11 «1.E-17 M1.11 1 M2»1.E-15 5 M3*1.E-11 1 M4»1.E-10 0

:.r^-r^-.%^>^=r»rX^C^:-z'-::.r^-r^-.%^>^=r»rX^C^:-z'-: r

122 16 20 24

-m) -m) Figuree 18: Anti-neutrino lepton number source term SI(£).

numberr conservation and hydrostatic equilibrium equations, respectively. They establish thee feedback of the neutrino-flow on the stellar background which drives the proto-neutron starr evolution.

Wheree SQ (SI) is positive there is a net transfer of energy (lepton number) from the mat terr into the neutrino fluid. Where it is negative, energy (lepton number) is deposited intoo the matter. The energy and lepton number sources SQ and SI are pure non-LTE quantit ies,, as their integrands are functions of the deviation from LTE (b(u) — e(w)).

Thee division of the stellar models into three regions that was sketched above in the discussionn of the energy and number densities is reflected in the sources. In the inner LTE-coress at p > 1013 - 5.1Q12gem'3, ( - £ < 3) the neutrinos are in LTE with the matter,, hence the energy and lepton number sources are zero.

Betweenn the LTE-cores and the atmospheres at 3 < —£ < 9 corresponding to 1013 -5.10122 > p > 3 . -5 .101 1 g cm'3, there are emitting regions where energy and lepton number aree transported out of the matter. Electron capture on free protons (2.8) is responsible for thee ivemission. In a dynamical situation the electron fraction Yt would decrease rapidly here.. This is i l lustrated in Fig. 3, in the comparison of M3 and M4. The i/„-emission is drivenn by e+ capture on neutrons (2.9) which creates free protons that in turn become availablee for electron capture. A situation is feasible in which there is only energy but noo net lepton number emission in this region. The emitting region in M4 is anomalous inn comparison to the other models. It is a net lepton number emitter, but a net energy absorber.. High-energy neutrinos coming from the interior, see the small peak around —ff « 4 in Fig. 15 which corresponds to the Ye gradient there, (cf. Fig. 3) are absorbed andd reemitted again at lower energies, while the emission of low-energy neutrinos goes on unabated.. As a net effect the total energy carried away from this region is smaller than thee energy deposited by incoming high-energy ve.

Thee atmospheres at - £ > 8 absorb energy and lepton number. Dynamically, the vt-


absorptionn on neutrons (2.8) wil l raise the electron and proton fractions in the atmospheres, cf.. M3 and M4 in Fig. 3. In turn, the capture of vc on protons (2.9) wil l decrease Ye. A dynamicall equilibrium in which both processes proceed at equal rates wil l try to establish itself,, creating an nearly stable Yc. In that case almost no net lepton number deposition wil l occur.. However, the energy deposition wil l be enhanced because the opposing reactions wil l keepp the abundances of the absorbing particles at optimal levels, keeping both absorption channelss open. This effect is il lustrated to some degree by M4 when compared with its 'parent'' M3. In M4 the Pe emission and absorption, Figs. 17-18, is much higher than inn the other models. The i/e-flow is being turned on by the local changes in the electron fraction,, Fig. 3, while these choke off the vc-exchange.

I nn the most degenerate models M l and M2, which also had the highest vt-densities, the i/ e-LTE-coree is biggest. Also the vt emission and absorption rates are highest. For the ve thee situation is also consistent with the discussion of the densities. The i/e-LTE-cores are muchh smaller in M l and M2, and their vt emission and absorption rates are respectively aboutt 7 and 5 orders of magnitude smaller than in the rest of the models. Although in M l . 11 the relatively high ve exchange is due to the low electron degeneracy, the magnitude off its i/e-LTE-core is in the same category with M l and M2. Their common polytropic structuree and high central density is responsible for this.

-3.5 5

- 4 4

M1,, M2, M3, M4«1.E-11 Ml.11 «1.E-12

'M1 1

12 2 16 6 20 0 24 4 -l(Mr ) )

Figuree 19: Neutrino momentum source term SA(£).

Thee bulk momentum sources SA(r,tfin) (5.5) are shown in Figs. 19 and 20. Both the LT EE and the total sources are shown, always in the same line-type. The curves depicting thee total sources level out at nonzero values in the atmospheres, whereas the LTE branches goo rapidly to zero.

Thee momentum source SA (5.5) is a function of the spectral flux F(u) = f(ui) E(w) andd is therefore nonzero even in LTE as long as there are gradients in the matter. Where


c c o o

o o -Q Q _̂ _

CD D Q_ _

u u

to to c c

< < on n

0 0

0.5 5

- 1 1

1.5 5

- 2 2

99 Bj

- 3 3

'' /'"> = ^

•• : ',M2 i i

1"" i'M3 '. ,• T:: ; >*

?ii / ,;< 2:22 M1/1 / '

I ' II / It " I f .. »< - 1 ' '' ' / '

l ll '-* * l>> ,1

-- l l /* r »*

ll\\ I '\'\ 1 -"" V/M1

, , , i

LM4 4 44 '•>•*' /^~

\.\. £-~fa ::T--' '*'* • r i / / •' '

I I II '

' '

,, , I , , , I

M£--^ ^ — —

M11 *1.E—21

M1.11 «1.E-14

M22 «1.E-19

M33 «1.E-15

M 4 * 1 . E - 1 4 4

ii , . . i , ,

M3 3

,, i .

12 2 16 6 20 0 24 4 -KM,) )

Figuree 20: Anti-neutrino momentum source term S J 4 ( £ ).

1 1

o o

r r o o 0) )

O O -1 1 c c

'FF " u u

c c

3 3

o o

V V

0 0

-0.2b b

- 0 . 5 5

-0.75 5

- 1 1

-1.25 5

- 1 . 5 5

-1.75 5

\\\\\\ '\ / /

^VV ƒ \ /

'*' \ --

i : 2 . 4 e 1 4 4

LL 2 :8 .9e13 3:: 1.2e13 4 !! 6.4e11

II . . . . 1 . .

t t

V V

"3"' '

,, , 1 ,

'' \

V V A A

\ \ ** / \\ /

, , . ! . , , ,

-- ;>N / /

/ /

/ /

ii I , I „ i . i , ! i, 1 1 1 1 1 1 1 1 , 20 0 40 0 600 8 0

M3 3

1000 12 0 14 0 ww i n MeV

Figuree 21: Integrand of vt-SA as function of w at fixed positions in the star (M3).

TransportTransport results 51 1

itt is negative, momentum is transferred to the matter, acting as a positive contribution to thee total pressure, see Figs. 19 and 20. Because of the occurrence of inward vm fluxes in M2,, M3 and M4 the effective contribution of their momentum sources to the pressure can bee negative in the inner core.

Thee momentum sources are largest in the inner cores. There they are LTE quantities, dominatedd by high-energy neutrinos, see Fig. 21. Deviations from LTE become apparent inn the emitting region 1011 < p < I012g cmr*, decreasing the total momentum transfer. Thee LTE contribution, with the equilibrium flux ¥**[&) = fe*(b(u))B(w) in (5.5), rapidly vanishess at densities below 1011 gem-3. In the atmospheres SA is dominated by its non-LTEE contribution.

5.1.33 Luminosities

Inn the preceding two sections it was established that the stars can be distinguished into threee contiguous regions characterized by LTE and no energy and lepton number transfer, emissionn and hence positive SQ and SI (5.3,5.4), and absorption with negative sources inn the atmosphere, respectively. In this section it will be shown how this stratification translatess into bulk energy and particle luminosities, both inside the stars and as seen by ann observer at infinity. The bulk luminosities offer the simplest handle for establishing wheree and how much energy and lepton number is being deposited into the matter by the neutrinoo stream passing through it. The bulk energy and particle luminosities are shown inn Figs. 22-25, and are defined respectively as

LLEE(Ufin)(Ufin) = 4xr2(0 Jdwf{u>,Utin)E{u,^t tin)y (5.6)

NNLL{t,t{t,tfinfin )) = 4xr2(e)|rfu;/(W,f,t / in)E(a,,e,i / in)/u;. (5.7)

Inn all models and for both vt and ve the luminosities behave globally the same. In thee LTE-cores and for most of the emitting region they rise. They reach a peak at thee transition from the emitting into the absorptive region around p « 5. lO11^ cm - 3 , (2.510111 for Ml) and decrease between 5.1011 > p > 10 l o5cm_3 where the energy and numberr absorption rates are highest. They level out on constant values further out in the atmospheress at p « 1010 - 109, - £ « 12, and assume their output values observable at infinity.. Although the luminosities level off, the absorption of energy and lepton number inn units per nucleon never actually stops. However, the nucleon density is so low in the atmospheress that the exchange no longer makes a visible dent in the luminosities. The spectraa still change slightly at densities lower than this, modifying the 'colour'-composition off the luminosities, but for the bulk luminosities the free-streaming region sets in at pp a 1010.

Inn the v, luminosities also inward (negative) fluxes occur at the edge of the dense inner coree in models M2, M3 and M4. These can be attributed to the temperature inversion in M2,, and to the £c-bump (and therefore ^-trough with corresponding gradients) in M3 andd M4.

Alll models are very luminous in v„ putting out in energy between 3.810" (M4) andd 4.410Merffsec-1 (Ml) and with a particle output between 2.11057 (M4) and 1.66 10M vmsec~

x (Ml). The v, luminosities are some 2-4 orders of magnitude lower than thee v, luminosities in Ml.1, M3 and M4, and 9 and 7 orders respectively in Ml and M2.

52 2 Chapterr 4

o o co o rj> > L --CU U

_ï ï

b b

5 5

4 4

3 3

2 2

1 1

n n

~ ~

--

_ _

--

M4 4

./.,*•'•:'' i'

'\M1.1 1

) y ^ M 2 2

1* *

;; i' ;; f •• f

tl tl

;; ,.-~.M3

ii > i I i i i I

M1,, M2, M3 «1.E53 M1.1.. M4 «1.E52

•• > I i •

122 16 20 24 -l(Mr) )

Figuree 22: Bulk vc energy luminosities. Curves corresponding to different models have beenn scaled with a different scale factor.

o o 00 0

4 4

3 3

2 2

1 1

n n

--

--

*&£. *&£.

,-.M3 3

; ; i i

i i

i i

,--.M1.1

M4 4

^rr M1

/ // ' ' ' . M2 / . ' :: t

ii l ;:;: 1

11 1 i l 1 1 1 1

Ml,, M2 *1.E58 M1.1.. M3, M4*1.E57

ii i l i i i i . i

122 16 20 0 24 4 - | (M r ) )

Figuree 23: Bulk vc number luminosities.

file://'/M1.1


Thee fact that the vt luminosities are so high and the vt luminosities so much lower, indi-catess that all the models must be considered to be 'early*. They are more representative forr the neutrino flash associated with rapid electron capture than for the long-time cooling phase.. In the cooling phase the t^-flow is expected to turn on, and develop a luminosity comparablee to the that of vt. Moreover, if the luminosities were to remain as high in the coursee of time during the cooling phase the latter could never last 10-20 sec. Consistently withh this classification the luminosities are most evenly divided in the more evolved model M4,, ve luminosities being just a factor 20 higher than the i?e luminosities.

Thee ve number luminosities peak slightly further out, at matter densities of about aa factor two lower than the energy luminosities. The ve energy luminosities peak in all models,, and drop much more after reaching their peak values than the number luminosities. Thereforee the ^e-flows deposit more energy than lepton number into the semi-transparent region.. On average high-energy neutrinos are being absorbed through the inverse /?-reactionn (2.8) (the absorption rate depends on e(u>, r, t)). Subsequently they are (partly) reemittedd with the LTE distribution (2.2). This energy 'downgrading' of the flow is further illustratedd in the next section where the bulk quantities are resolved into their constituent spectrall components. The downgrading is also the reason that the number luminosities doo not peak in all models, but rise continuously in Ml and M4. The energy deposition att matter densities broadly between 1012 > p > 109 g cm"3 is quite substantial, ranging fromfrom 6.81052 (M2) to 9 . 10" erg sec'1 (M4). This corresponds to 17% and 19% of their peakk luminosities respectively, and per second amounts to several times a total supernova explosionn energy in kinetic energy and light. The maximal lepton number deposition occurss in M3 at a rate of 7.1056 sec-1. However, it must be stressed at this point that thesee are 'flash'-values, which in a dynamical situation will decrease rapidly, c.f. M3 and M4. .

Thee energy luminosity peak in model M4 lies deeper inside the star than in the other models,, at p as 7.1012jf cm~z (—£ « 4), corresponding to the first peak in its emitting region,, see Fig. 15. The luminosity peak corresponds to the maximal slope of the drop in electron-fractionn Ye, see Fig. 3. The primary emission peak is driven by the steep gradient inn the electron degeneracy, but lies at high matter density. Neutrinos that are emitted there withh the high average particle energy corresponding to the local LTE chemical potential, aree promptly reabsorbed where the electron degeneracy is lower. The neutrino energy re-emissionn at lower w in the rest of the emitting region cannot keep up with the energy absorptionn because the chemical potentials and temperatures are much lower there than att the primary emission peak. The number luminosity does not drop at all in the emitting region,, which indicates that the absorbed high-energy ve are indeed promptly reemitted, makingg the number luminosity rise continuously. This extreme energy flow downgrading iss consistent with the anomalous behaviour of the sources SQ and 5 7 in M4, as discussed inn the previous section.

Thee vt luminosities shown in Figs. 24 and 25, peak slightly deeper inside the stars att p « 1012 g cm~*, quite consistently with the behaviour of their sources and densities discussedd in the previous sections. Most energy in v€ is deposited in Ml .1 , 5.21O

40 erg I sec, whichh is 8% of the peak luminosity. The vt luminosities in M3 rise monotonically. Energy flowflow downgrading and energy deposition in the matter occurs for Pe in the same way as forr ve, at slightly higher matter densities.

Itt must be stressed that the models considered here may not be typical of a proto-neutronn star or a supernova after shock launch and during shock stall. First, they con-stitutee merely one 'snapshot' in the evolution of the matter background. Second, the


o o in in

03 3

3.5 5

3 3

2.5 5

2 2

1.5 5

1 1

0.5 5

0 0

-- * *. ^ •• ' 1 '

Ml,-, . . /.'V;.M 2 2

/ '' / M3

ii ' it

.'' / li '

iV ,, , , i . , . i

M11 »1.E4 4

M1.11 «1.E5 1 M22 *1.E4 6 M3*1.E5 0 0 M44 *1.E5 1

,, , , i , , . i ,

M4 4

,, I ,

12 2 16 6 20 0 24 4 -£(Mr) )

Figuree 24: Bulk vt energy luminosities. Curves corresponding to different models have beenn scaled with a different scale factor.

o o ID ID 5 5

4 4

3 3

2 2

1 1

0 0

--• •

• •

ƒ/ /

9 9 i i

I I II ' II ,

ii ' t t

rr '

r-r"> N N

/' ' / /

I I

,-• •

' A — i — i — 1 — i — ii i _

M11 *1.E4 9 M1.11 *1.E5 5 M22 «1.E5 1

M3*1.E5 5 5 M44 »1.E5 6

II . , , i , . . i ,

M2 2

...M 3 3

M4 4

•• i l i

12 2 166 20 24 4 -?(M,) )

Figuree 25: Bulk vt number luminosities.


factt that the baryonic component of the matter does not include nuclei at densities below 10100 g cm~3 is unrealistic and enhances the energy deposition in the matter at low den-sities.. Also, the constant low Ye in the outer atmospheres, and hence the high neutron abundancee aids the deposition. Moreover, it is impossible to predict here how much energy wouldd be deposited in the course of time, as this is a question involving the matter dynam-ics.. This will be the subject of a subsequent paper.'191 Nevertheless, in realistic collapse modelss the bounce shock stalls roughly in the density region 1010 < p < 5.10ugcm~3

wheree we found that the neutrino flows deposit most of their energy. Where the shock hass ploughed through it has dissociated nuclei into a-particles and nucleons, so that per-hapss at least in some part of the main absorptive region the free nucleon gas composition mayy be a reasonable approximation. Hence, the neutrino flow energy downgrading and thee associated energy deposition in the matter in this region may be an indication of the delayed-explosionn mechanism in its infancy.

5.22 Spectral quantities

Thee following presentation of spectral quantities and the resolution of bulk quantities at fixedfixed positions in their frequency-dependent constituents is not exhaustive but concen-tratess on M3. The individual curves in the figures below depict spectral quantities as functionss of w at different positions in the star. Each curve corresponds to one position.

3 3

1000 120

M3 3

140 0 CJJ in MeV

Figuree 26: Energy ve distribution spectrum e(u>, £ = constant) of M3. The curves are at fixedfixed positions in the star.

Inn Fig. 26 the ve spectrum e(u>) in M3 is shown at fixed positions in the star, some off which are labeled with their matter density. At densities above 101* gcmr3 the ve are highlyy degenerate and in LTE. First the low-energy gap appears between 1014 > p >


IO133 g cm"3, (cf. Fig. 7, showing the LTE spectrum b{w)). Going further out the peak inn the vc spectrum becomes lower and shifts towards higher frequencies because the low frequencyy gap widens. This trend is reversed and the peak starts shifting to lower energies att the beginning of the emitting region, corresponding to the peak in bulk emissivity and luminosityy cf. Figs. 15, 16 and 22. In the emitting region the matter becomes increasingly transparentt to the lower frequencies, while high-energy neutrinos are being absorbed and part lyy re-emit ted again with lower energies. This is further illustrated in Fig. 27, depicting thee spectral composition of the source SQ (5.3). Between 5.1012 > p > 1 0u gcm~3 the

c c o o CU U

u u C C

E E u u

3 3

I I m m

1 1

0 0

1 1

I I

3 3

:: A :: P\V\-\ -.%-.% \È^~ \\ \|\f

VV \S:-II IF "" \È I f f

'' . . . . 1 . . . . 1 . . . . I ,

1: : 2 : : 3: : 4 : : 5: : 6: :

,, , 1 , , , , 1 ,

4.6e12 2 1.7e12 2 6.4e11 1 8.8e10 0 3.3e10 0 4.5e9 9

,, , . 1 , , , , 1 , . , ,

20 0 40 0 60 0 80 0 1000 120 140 0 a>> in MeV

M3 3

Figuree 27: Integrand of vc-SQ as function of u> at fixed positions in the star (M3).

mat terr is absorptive at high energies and an emitter at low energies. This flow downgrading accountss for the rise in the luminosity at lower frequencies around 10 MeV in this region, seee Fig. 28 showing the spectral composition of the luminosity at different positions. In thee atmosphere the spectrum shifts slightly more to lower frequencies until at densities betweenn IO10 > p > 109 gcm~3 the stable output spectrum is reached in the truly free-streamingg region.

Thee vc-spectrum is presented in Fig. 29, and behaves very differently. In the very densee inner core it is relatively high and in LTE, cf. Fig. 8, but decreases as £e increases, cf.. Fig 5. After the peak in (c is passed the vt abundance rises, peaks at p as 5.10

12 gem,-3

andd afterwards decreases in the emitting and absorptive regions shifting its peak to lower frequencies,, due to flow downgrading.

I nn this atmosphere only the (very) high end of the spectrum (o> > 30 MeV) is still depleted.. See for this Fig. 30 where spectral luminosities are shown as functions of position forr a number of energy bins. The luminosities of the high-energy bins are being depleted untill relatively far out in the star, and finally level out completely between 10 < - £ < 12, correspondingg to densities between IO10 > p > 109 gem-3. For the bulk luminosity this


o o O) ) (n (n

3 3 Ms Ms

6 6

5 5

4 4

3 3

2 2

1 1

n n

"" /A 4 '77 ' \

"" br''\

-- Il \:'A

1// / f: •

111 / K '

" / - . - * • ' ' . w w

\\ \ 3

ii \ \ . > — .

1 :: 8.9e13 211 3.3e13 3:: 4.6e12 4 !! 6.4e1 1 5:: 2.4e11 6:: 3.3e10

1 1

200 4 0 6 0 8 0 10 0 12 0 14 0 ww i n MeV

M3 3

Figuree 28: Spectral c, energy-luminosity as function of a; at fixed positions.(M3).

3 3

(L) )

0.016 6

0.014 4

0.012 2

0.01 1

0.008 8

0.006 6

0.004 4

0.002 2

0 0 0 0

1:: 2.4e14 2:: 1.2e13 3:: 4.6e12 4 :: 2.4e11

10 0 400 50 coco in MeV

200 30

M3o o

Figuree 29: Energy ve distribution spectrum e{u,£ = constant) of M3.

58 8 Chapterr 4

U U

w w

3 3

- 2 | | 10 0

- 4 4 10 0

- 6 6 10 0

10 0

- 1 0 0 10 0

- 1 2 | | 10 0

- 1 4 4

10 0

- 1 6 | | 10 0 - 1 8 8 10 0

rr ^-s.' * -

rr 11'/j/C \

FWW \ \ 'v-p'' ' i / \ \

rr y \ \ rr \ \ rr \ ^v

\ \ rr 3.65 MeV \ rr * 16.3 MeV \ rr V 30.4 MeV \ ff A 50.0 MeV \

ff 77.0 MeV V ff «115 .6 MeV \ rr \ rr , , , I . , , i . . 7—J_ .

if— if—

•— —

..... i , ... . i . . . i

12 2

M3 3

16 6 20 0 24 4 -f(Mr) )

Figuree 30: Spectral i/e luminosities as functions of position (M3) for 6 energy bins

iss not significant. However, for the detection of neutrino signals on earth this depletion at loww density is not without importance. Cross sections rise quadratically with the particle energyy so that high-energy neutrinos are more readily detectable.

I nn Fig. 31 the first Eddington factor f(w, £) (2.4) is presented as a function of w. The energyy dependence of the opacities causes low-energy (< 10 MeV) neutrinos to decouple firstfirst from LTE, at matter densities of about 1013 gcm~3. high-energy neutrinos remain diffusivee and in LTE for p > lO11*/cm - 3.

I nn Fig. 32 ƒ is shown as a function of position for a selection of energy bins. The definitionn of the neutrinosphere as the position where ƒ reaches some predefined value, e.g.. ƒ = 1/2, such as proposed in Ref.[12] is meaningful only spectrally. For different binss the spatial spread in this position encompasses the whole emitting region between 233 < r < 33 km ( 5 < —( < 9) as bins of increasing u> decouple further out at lower density.. The thickness of the neutrinosphere amounts to 30% of the total radius of the star. .

Alsoo included in Fig. 32 is the energy averaged Eddington factor < ƒ > = FbulkiO/EbulkU)-FbulkiO/EbulkU)- The spatial spread of < ƒ > shown in Fig. 33 for all models is up too a factor two comparable to the spread for different energy-bins per model, cf. Fig. 32. Notee in Fig. 34 the small region of negative < ƒ > for ve in M2. This corresponds to the inwardd flux that causes the non-LTE vt energy and particle densities to be above their LT EE values in the dense inner core above nuclear density.

Thiss spectral analysis completes the picture sketched in the discussion of the bulk quantit ies.. The semi-transparent region is characterized by energy downgrading of the neutrinoo flow. The energy is deposited in the matter and wil l heat it up. The downgrading andd deposition mechanism is caused by the ^-dependence of the opacities and redistributes


3 3

0. 8 8

0. 6 6

0. 4 4

0. 2 2

:=^\~~~-~i^r"" - - " ~

t"~~ \ ^ ^

U',, , 1 : 6.4e11 III \ . 2 : 8.8e10 II \ \ . 3: 3.3e10 !! \ * \ A: 6.2e8 !\\ \

5\\\ \ ""••--... i'i\\ \ '̂ '"'•-. t',, \ \ l "-„ - : '' \ >v "* - ^

200 4 0 6 0 8 0 10 0 12 0 14 0 CJJ i n MeV

M3 3

Figuree 31: First ue Eddington factor f(u>) as a function of w at fixed positions (M3).

3 3

0.8 8

0.6 6

0.4 4

0.2 2

' '

' ' . .

// 7

// V'

JJ V^\ /** _JI : ; 1

// I///J

y ii ^ Ü i ^ r i

77 ' ^'m// PP sit

) as a function of position for a selection of 66 energy bins. The solid curve marked with the 'swiss cross' depicts the bulk Eddington factor.. (M3)


0.88 -

0.6 6

0.4 4

0.2 2

Figuree 33: Bulk ue first Eddington factors F/E as function of position for all models. Theyy are depicted from left to right in the order as given on the figure. The solid line correspondss to M4.

1 1

0.8 8

0.6 6

0.4 4

0.2 2

0 0

--

--

'' ,

i i

//..' ..'

' '

i'.';/.'' M1. M2, M1.1, M4, M3

''' :' /' ':'' /.'

. .. 1 . . . 1 . . . 1 . . . 1 . . . 1 .

122 16 20 0 24 4 -f(Mr) )

Figuree 34: Bulk vc first Eddington factors F/E as function of position for all models. Theyy are depicted from left to right in the order as given on the figure. The solid line correspondss to M4.


thee composition of the spectrum. Because its working depends on the local non-LTE spectraa it is difficul t to see how it could be reliably reproduced by a non-multigroup transportt approach, without putt ing it in by hand.

Betweenn 1010 > p > 109 g cm~3 the spectrum reaches its value observable at infinity . In orderr to get a complete picture of its high energy tail and to observe the energy deposition inn the mat ter it is necessary to follow the transport down to these densities.

5.2.11 Thermal fit

3 3 70 0

6Ü Ü

50 0

30 0

20 0

10 0

0 0

:: 1: 8.9e13 2!! 3.3e13 3!! 1.7el2

:: 4: 3.3e10 5: 1.2e10

j * r r

j ^ ^ . -- " ' ^ — — " ^ \ .

k i s ^ ^ = ^ ^

v^V-"" " "

^ - ^ T ^ - - ; . : : . : . ' . " "

i . . , ,, i .

-- -' " ^

. 2 2

! . . . . « . . . . . 200 40 60 0 800 100 120 140

CJJ in MeV

Figuree 35: 4>(w) = log(e_1 - 1) at fixed positions in the star (M3). For vt.

Assumee that the non-equilibrium energy distribution e(u>) is thermal of the Fermi-Dirac form,, and involves two parameters that could be associated with a distinct neutrino tem-peraturee T„ and a non-LTE neutrino degeneracy parameter £„, such as was proposed in Ref.[12].. In that case the function (p(w) = log(e-1(u>) - 1) should be a linear function of u>. u>.

Inn Fig. 35 this function is shown as calculated for vt on M3, the curves being at fixed positionss in the star. Obviously the spectrum as a whole cannot be well represented this way,, except for perhaps in the very dense interior above 1014ff cm~3. This conclusion was alsoo drawn by Janka and Hillebrandt, [13] and Myra and Burrows, [14]. Although the high-energyy tail at energies w > 20 MeV can be reproduced quite well by

e/i((w)) = a+bu a+bu ++ 1'

(5.8) )

thee spectrum as a whole resists such a fit. However, it can be parametrized in two parts. Inn addition to the parametrization of the high-energy tail as given above, the low-energy partt of the spectrum for w < 20 MeV, can be fitted quite well with three parameters and thee form

JaMM = (5.9) )

Inn Fig. 36 the complete fit in two parts combined, for w < 20 and u > 20MeV is shown. Suchh a combined 5 parameter fit is the best and involving the least fitting parameters


Spectro ll Fit M3

Figuree 36: Combined 5 parameter fit to vt-e(u, f ) . The low and high-energy ends are fittedfitted separately (M3).

found,, that represents the total spectrum with statistically meaningful accuracy, but it is nott very simple and does not add much to one's physical insight.

I tt is tempting to interpret the parameters a and i in the high-energy tail fit as a non-equilibriumm neutrino degeneracy parameter a = — £„ and an inverse neutrino temperature 66 = X" 1 . We yield to this temptation in Figs. 37 and 38, where T„ an £„ are plotted with theirr LTE mat ter counterparts.

ii I_J ' i L _ I , i . i i i i i . t _ i i ; — i i i_ 00 4 8 12 16 20 24

Temperature ss M3

Figuree 37: vt temperature, obtained from high-energy tail fit (M3).

Inn the dense inner core and emitting region where the high-energy neutrinos are diffu-sive,, this identification is obviously quite good. Outside their diffusive region at densities beloww 1011 gcm~3 the high-energy neutrinos to which the fit was made finally decouple fromm the mat ter and their temperature and chemical potential start to deviate from LTE significantly. .


»» 10

166 20 24 -*(M,) )

Figuree 38: ve non-equilibrium degeneracy parameter, obtained from high-energy tail fit (M3). .

o o 0) ) in n

1 1

10 0

- 2 2 10 0

- 3 3 10 0

- 4 4 1Ü Ü

-5 5 1Ü Ü

-6 6 10 0

•• ƒ /

:: /

V V \\ \\

j j i i

! !

: :

r r

,, - " " ,*' ,*'

,' '

,, , i i

""•--;-. . **•


Thee neutrino signal from SN 1987A largely sampled the high-energy tail of the spec-t rum.. Certainly this was true for the 1MB detections, with the 1MB detector threshold at 200 MeV. Most, if not all analyses of the neutrino signal assumed, imposed or concluded thatt the observed spectrum was thermal of the form (5.8), at zero or nonzero chemical potential,, and identified the fitting parameters as above. It was then assumed that the wholee spectrum, the low-energy part included, can be represented with these parameters. Thee fitted spectrum was used in the calculation of the total energy and particle luminosi-ties.. These were then in turn used in arguments e.g. on the neutrino magnetic moment, thee existence of neutrino oscillations, and axion emission.

s s

2

JTT 1.25

1 1

0.75 5

0.5 5

0.25 5

00 — 00 4 8 12 16 20 24

-f(M,) ) M3 3

Figuree 40: Bulk vc energy luminosity (dashed), and the one fitted with the high-energy taill parameters (dotted).

Lett us pretend for the moment that the spectra calculated here are 'real', and identify thee fitting parameters of the thermal fi t to the high-energy tail with Tv and f„ . Assume thatt the whole spectrum is thermal and represented by these parameters. Calculate the bulkk luminosity using this fitted quasi-thermal spectrum. This approach results in Fig. 399 where the luminosity spectrum as i t leaves the star is given together with its fit , and inn Fig. 40 in which the real and fitted bulk energy luminosities throughout the star are plotted.. Although the high-energy tail of the luminosity is in very good agreement with thee thermal fit , the low-energy part and peak are not. This can be understood in terms off the low-energy gap. In fact the fitted luminosity overestimates the real one by a factor off nearly three.

5.2.22 T i m e d e p e n d e n ce

Untill now the time-dependence of the neutrino flow has not been discussed and all results weree presented in the steady state. For the situation at hand the stationary state is of higherr physical interest than the time-evolution of e(w, £, t). The latter strongly reflects thee initial condition, which was to some degree arbitrary. A different initial condition, taking,, e.g., e(u>, f, t = 0) = b(u, £,t = 0)/2 results in a different evolutionary track. In contrast,, the stat ionary solution is unique, because in the steady state (2.1) reduces to a boundaryy value problem. In fact, the time evolution in a static background constitutes

u-fu u

--L , ,


OO 1.2 * *

Too 1 CD D CO O

en n

»» 0.8

0.6 6

0.4 4

0.2 2

0 0

' '

; ;

:: j

tt i

ii i . . i

• •

'ii i

* *

\ l 2 2

vv '-

ii 1 i ,, i 1

t+. .

. . ! . • •

ii ii t-stat t

ii 1

12 2

M3 3

16 6 200 24

Figuree 41: Time-evolution of the bulk v, energy luminosity in M3. Snapshot-times: tltl = 10~6,t2 = 10-5,t3 = 10-4, t4= 5.10"4 sec.

0.6 6

ëë 5 en n

en n a33 0.4

0.3 3

0.22 h

0.1 1

0 0 0 0

--

--

\ \ ] ] 1.. i—tó ' '


3 3

88 12 16 20 3.655 MeV M3

24 4 f(M,) )

Figuree 43: Time evolution of the 3.65 MeV vc bin energy distribution in M3, as function off position.

3 3

0.8 8

0.66 -

0.44 -

0.2 2

--

--t3, , -stott Ijl

1'1' I

ii'h ii'h

ii . . . i

j^~Aj^~A _

/ V 1 1

.. , , i , , , i , , , i . . . i

88 12

3.655 MeV M3

166 20 24 -*(M r ) )

Figuree 44: Time evolution of the 3.65 MeV vc bin Eddington factor ƒ in M3, as function off position.


noo more than the relaxation of a transient, and is of littl e dynamical importance. Never-theless,, for a full evolution calculation in which the matter is also dynamical, being able too solve (2.1) with the time dependence included is important. In that case the initial conditionn for each transport step is no longer arbitrary but is the state of the system at thee previous time. In Figs. 41 and 42 the time evolution of the bulk ve and ve energy luminositiess in M3 is plotted.

10 0

1Ü Ü -3 -3

1U U

10 0

,0-7 7 1Ü Ü

10 0

1Ü Ü

10 0

fd d 10 0 - 1 3 3

id d - 1 4 4 10 0

iff iff

ff \ rr *

r r

r r r r r r r r r r r r r r

r r

\\ %s. \\ **-„

\ " "

ii , . . i , . . \ )

t3.. t-stot

, i , l , , .. i 66 12 16

30.366 MeV M3

24 4

-f(Mr ) )

Figuree 45: Time evolution of the 30.36 MeV vc bin energy distribution in M3, as function off position.

24 4 f(Mr) )

Figuree 46: Time evolution of the 30.36 MeV v, bin Eddington factor ƒ in M3, as function off position.

Thee evolution takes place on a 10- s-10- 4 sec time scale. This relaxation time scale iss the same for all models. Individual (lower) energy bins evolve on a somewhat longer timee scale, but after 0.1 msec bulk quantities no longer change very much. Individual bins reachh the stationary state after at most 0.5-1 msec, see Figs. 43-46, when we apply the

68 8 Chapterr 4

rigorouss local requirement for stationarity mentioned in Sec. 4. Thee relaxation time scale of the neutrino flow is much shorter than the dynamical

evolutionn time scale of the matter in the proto-neutron star, which is of the order of at leastt milliseconds. If the proto-neutron star evolution can be calculated with discrete timee steps of ) (2.4) of the high-energy bins u; > 30 MeV, develops a 'shoulder'' in its position dependent profile at the transition from the semi-transparent intoo the free-streaming region. This effect was previously reported in Ref. [15]. In the ƒ -££ plot (Fig. 32) the steep rise in ƒ levels off somewhat, a plateau develops, which is followedd again by a steeper rising region towards ƒ = 1. This behaviour only occurs in the high-energyy bins. The value of ƒ in this plateau depends relatively weakly on the energy off the bin, and lies at ƒ « 0.78. This value does depend on the matter background model, inn particular on the value of Yt at the position of the plateau.

Withh these observations in mind an analysis of this behaviour can be made in an attemptt to understand it. All assumptions made in the following analysis were afterwards checkedd and found to be correct. The e(£) profile of a high-energy bin that exhibits a / ( f ) -plateauu in the steady state is a solution of the time-independent energy balance equation,, (2.1) with the dte term dropped. Splitting out the gradient term and dividing thee equation by e leads to

^ r ( r V )) + ^ = - < ( 1 - hft). (5.10)

Becausee the gradient of ƒ is small in the plateau, and it occurs in the outer regions of thee star, the first term on the lhs can be neglected with respect to the others. The plateauu develops in the atmosphere only for high-energy bins, for which in this region 6(u/)/e(u>)) ) < e(t*>), the medium must be a pure absorber at that particlee energy. Second, in the region where it occurs, phase space inhibitions must be absent,, i.e. 0 = 1 see Fig. 47. These two requirements are only met by high-energy bins. Third,, Ye must be constant in this region, because otherwise with (5.12) it is clear that ƒƒ (£) could not form a plateau.


Thee prediction (5.11) can be interpreted on a somewhat more fundamental level by notingg that the ratio of the lhs and the rhs of Eq. (5.11) is the ratio of the microscopic andd macroscopic length scales, given by the total mean free path / = K^t) and energy scalee height L = |(—dre/e)~

1\ respectively, and can be interpreted as the local Knudsen numberr (Kn) of the system. Apparently, the e(f) profile develops in such a way that in thee stationary state Kn is constant and of order unity in this shoulder plateau region. Thatt this is indeed the case is clear from Fig. 48, where i t is shown for a number of energy binss in M3.

3 3

0.8 8

0.6 6

0.4 4

0.2 2

-- 1 4j j

" "

" "

3131

11 . . . 1 ,

// 1 : 3.65 MeV // 2 : 16.3 MeV

VV 3: 50.0 MeV // 4 : 94.2 MeV

12 2

M3 3

16 6 20 0 24 4 -f(M,) )

Figuree 47: The blocking factor 6, which is included in K^(CÜ). It is plotted for four energy bins.. Note that only for the two highest energy bins 8 = 1 in the region where the Knudsen-plateauu develops.

Figuree 48: The Knudsen number as a function of position for four selected energy bins. Notee the plateau forming at Kn s» 1 between 9 < - £ < 11 in the high-energy bins.

Thee reason for this behaviour at the transition from the hydrodynamic regime, Kn =


I/LI/L < 1, to the streaming regime, Kn > 1, is unclear. However, the resulting «(f) profilee is a perfectly good stationary solution of Eq.(2.1), and therefore unique. This behaviourr is exhibited independently of the numerical details of the solution method, and wass reproduced in all models by all versions of the transport code, irrespective of whether thesee were formulated as a conservative scheme or as an explicit differential equation,'16' andd with several different OBC's implemented. It is therefore not likely that the effect is aa purely numerical artefact, nor that it is caused by the OBC. Although unexpected and remarkable,, the 'shoulder' effect is little more than of academic interest. It is probably aa peculiarity of FNDT as it has not been reported in other transport schemes. However, thiss may be due in part to the fact that in the realistic model» usually used Yt is not constant,, so that a shoulder would not appear.

5.2.44 Anuotropy

Thee angular dependence of the distribution function is averaged over, and this information iss lost as a consequence of the use of an angular moment method such as this. Nevertheless itt is instructive to investigate the angular dependence if> of the intensity I„ — (u>/2x)if i> whichh was defined131 as

iKw,«,, r, *) = ƒ„ (w, O, r, t)jE(uy r, t), (5.13)

andd features prominently in the derivation of F(N)DT, see Refs.[l]-[3]. Within the for-malismm of F(N)DT it is expressed as

^'•"4UM'-R.«-- Thee angular dependence of rp lies in the combination R. • ft = Ru with u = cos(0). From thiss it immediately follows that f • R = fR. Of course, calculating iff with values ƒ and R calculatedd in the transport code does not yield ^ as a solution of the Boltzmann equation.

Takingg the values of ƒ and R in the stationary state of the 16.65 MeV bin of M3, and usingg these to calculate if> as a function of u at different positions in the star, we obtain thee plot shown in Fig. 49. In FNDT it is natural for R (2.5) to become large when the meann free paths of the neutrinos become large compared to the macroscopic scale height. Inn Ref.[l] it was shown that in this limit

if>if> fa 4rR{l4rR{l + 6(R)-

Conclusions Conclusions 71 1

ZJ J

3 3

1 0 2 2

10 0

1 1

- 1 1 10 0

.o ' 2 2

ID" 3 3

- 4 4 10 0

io" s s

- 6 6 10 0

-11 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 U U

M33 u = 16.28 MeV

Figuree 49: The angular part of the vt distribution in M3 at different positions. The distri-butionn is isotropic (ip = l/4ir) at densities above 2.1012gcm~3, and becomes increasingly forwardd peaked in the semi-transparent and free-streaming regions.

Thee forward peaking as presented here does not prove anything because rp is not a solutionn of the Boltzmann equation. Still it suggests that the anisotropy of the neutrino flowflow may occur easily and may be quite extreme. This would strongly suppress the effi-ciencyy of i/i/-annihilation as a heating source'1 ' in the delayed explosion scenario, as was previouslyy argued by Cooperstein et al. in Ref. [18].

66 Conclusions

Thee semi-transparent region in all models lies broadly between 5.1012 and 1010


ons.. Inclusion of neutrino-electron scattering, pair and plasmon processes and GR may modifyy the deposition picture further. Nevertheless, in view of the relative magnitude off the cross sections involved, it seems reasonable to assume that the ^-reactions are responsiblee for the bulk of the behaviour.

Itt is impossible to predict at this point how the luminosity and energy deposition willl develop in time. This question involves the dynamical evolution of the matter, and willl be the subject of a subsequent paper.'19' The time-integrated energy deposition over thee whole cooling phase will certainly be much lower than a linear extrapolation of the depositionn rates found here would suggest. However, the initial 'flash1 energy deposition ratee bears some promise. It occurs in the right place, and if it remains relatively high, it wouldd be certainly more than sufficient to drive a delayed explosion.

Highh electron, and hence neutrino, degeneracy (Ml and M2) causes the decoupling of thee neutrino flow from the matter to occur at lower matter densities than in less degenerate settingss (Ml. l ,M3,M4), for obvious reasons. High electron degeneracy produces higher ve, andd very much lower vt luminosities.

Spectraa still change at densities as low as 1O1O-1O90 c m - 3 , mainly in the high-energy taill at w > 40 MeV. Because it is the high-energy neutrinos that are most easily detectable, itt is necessary to extend neutrino transport to these densities in order to calculate the spectrumm at infinity. This has been previously pointed out by Janka and Hillebrandt.'41

Thee outcoming stationary spectra are 'pinched', showing a deficit in both the low and high-energyy ends of the spectrum. This feature has been reported by many authors, in particularr Janka and Hillebrandt'4,13) from Monte-Carlo (MC) simulations, Myra and Burrows,'1411 and Suzuki.'6'

Thee emergent neutrino spectra are without exception non-thermal, mainly due to thee low-energy gap. It is impossible to represent the entire spectrum with a thermal Fermi-Diracc distribution even if it involves a nonzero chemical potential. Such a fit can bee made only on the high—energy tail of the spectrum. In practice, neutrino detections onn earth heavily sample this part of the spectrum. The 19 events from SN 1987A were noo exception. When neutrino temperatures and chemical potentials derived from such a thermall fit to the high-energy tail are used to represent the spectrum, the calculated bulk energyy luminosity leaving M3 is overestimated by a factor of three.

Becausee of the u?-dependence of the opacities, neutrinos of different energies decouple fromm the matter background at very different positions. The concept of a neutrinosphere ass a geometrical entity for the neutrino flow as a whole is therefore meaningless. It is difficultt to envision a realistic bulk transport scheme. First, such a scheme always involves ann assumption on the functional form of the non-equilibrium spectrum, like in Ref.[12]. Becausee the spectrum is intrinsically non-thermal, it is problematic to construct a realistic Ansatz.. In addition, some prescription'11 is needed to calculate a bulk first Eddington factorr < ƒ >. No obvious simple candidate for such a prescription which would not requiree a spectral calculation, springs to mind.

Apartt from these conceptual objections, it is unlikely that a bulk transport scheme couldd in a natural way reproduce the downgrading of the neutrino flow. The downgrading requiress knowledge of the non-LTE spectrum e(u>) at each position in the star.

Itt is no wonder that bulk and leakage schemes which simply declare the neutrinos to bee free-streaming below 1012-1011 j c m " 3 fail to produce delayed explosions. They throw thee effect a priori away in the region where it occurs. Fancier bulk transport schemes, evenn flux-limited ones, will probably fail as well, or succeed by construction where FNDT displayss the effect naturally. Stopping the calculation at 10Ugcm~3 misses most of the

Conclusions Conclusions 73 3

energyy deposition and therefore misrepresents the output spectra. Thee neutrino flow evolves with a relaxation time scale of 10_B-10~4 sec. After maxi-

malijj 0.5-1 msec the stationary state is reached. This time scale is much shorter than the dynamicall matter evolution timee scale. Treating the matter as a static background during onee transport-step is therefore justified. If in a fully dynamical calculation the evolution cann be followed with time steps of 0(1O~4-1O~3 sec), the neutrino flows will evolve from onee stationary state into the next.

Thee results produced by FNDT are qualitatively consistent with other transport approaches,'4,, ••1 S- l 4 ] including physically more rigorous ones, like the MC simulations of Jankaa and Hillebrandt.'4*1S' A quantitative comparison is complicated by the differences inn matter backgrounds. However, the FNDT transport scheme is orders of magnitude faster,, and is therefore suitable for use in dynamical calculations.

Thee transport code is sufficiently robust to simulate the neutrino transport in a hot neutronn star in a wide range of environments. The code transports vt with the same ease andd using the same time step and accuracy control as ve. It can deal with steep gradients, inwardd fluxes, inverse temperature and chemical potential gradients and high electron andd neutrino degeneracy. The accuracy in e and ƒ is better than one percent. This has beenn estimated by using different grids, different differencing schemes and

UvA-DARE (Digital Academic Repository) Neutrino driven neutron … · 300 ChapterChapter 4...

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