the influence of a new model of progenitor star on supernova explosion

ELSEVIER Chinese Astronomy and Astrophysics 33 (2009) 384–392

CHINESEASTRONOMYAND ASTROPHYSICS

The Influence of a New Model of ProgenitorStar on Supernova Explosion † �

ZHANG Miao-jing� QIU Xiu-qiang PAN Jiang-hong ZHANG Xu-ningCollege of Physics and Electronic Engineering, Guangxi Normal University, Guilin 541004

Abstract By adopting the progenitor model proposed by Woosley in 2002 andusing the program WZYW89, the process of explosion of type II supernovae withmasses of 11−40M� is numerically simulated. The results of computation revealthat this new model can in different degrees affect the collapse and explosion oftype II supernovae as well as the propagation of shock waves. Besides, thedefinitions of the primary energy of shock waves are also discussed.

Key words: star—supernova, general—shock waves

1. INTRODUCTION

Supernova is an explosive phenomenon produced by massive stars (M > 8M�) at the endof their stage of nuclear burning. Supernova explosion is the most violent and spectacularastronomical phenomenon in the starry world. Its high luminosity can be used as the stellarcandle power for the measurement of the distances of remote galaxies. Moreover, it is closelyrelated to the origin and synthesis of heavy elements in the universe, the formation of neutronstars and black holes, the origin and acceleration of cosmic rays, and so forth. Therefore,the research of supernovae plays an exceedingly important role in astrophysics.

The progenitor of supernova, namely a star with an iron core and an envelope composedof silicon, oxygen, magnesium and other intermediate elements, is an essential factor whichaffects the supernova explosion. It has important effect on the subsequent evolution ofsupernova as well as the collapse of its core and the propagation of shock waves. Sincethe progenitor model WZW78[1] was proposed in the late 1970s, many research groups

† Supported by National Natural Science FoundationReceived 2008–07–08; revised version 2008–08–06

� A translation of Acta Astron. Sin. Vol. 50, No. 2, pp. 152–160, 2009� [email protected]

0275-1062/09/$-see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2009.09.004

Chinese Astronomy and Astrophysics 33 (2009) 384–392

ZHANG Miao-jing et al. / Chinese Astronomy and Astrophysics 33 (2009) 384–392 385

have been engaged in the calculations of various progenitor models[2−7] for masses in therange 8 − 25M� . For instance, Hillebrandt[2] established the WZW model of 10M� andobtained a weak explosion. Baron et al.[6] softened the equation of state and in combinationwith the effect of general relativity also achieved a successive explosion for the progenitorswith 12 − 15M�. Wilson[3] utilized a more standard equation of state, but he did notget the result of successful explosion. From the differences of the results yielded by thesecalculations it can be inferred that the model of progenitor star has great influence on theprocess of explosion of supernovae. By using the models of the series of WW93[9] and theirindependently compiled programs, the Chinese scholars Wang Wei-zhong et al.[8] studiedthe effect of progenitor model on the collapse, shock propagation and explosion of type IIsupernovae. In 2002, via the adoption of the new data given by experiment and theory,such as the rate of weak interaction and that of the loss of stellar wind, Woosley et al.[10]

improved the model of evolution of massive stars and acquired new models of the progenitorstar. Nowadays many research groups have applied these new models in their studies. Butin our country the program WZYW89 for the numerical simulation of type II supernovaehas not been used in the investigation of these new models. Hence it is necessary to adopt anew model to probe its effects on the core collapse, shock propagation and explosion of typeII supernovae. For this we utilized the model of progenitor star with mass of 11 − 40M�to carry out calculations. In the investigation of these effects and in analogy with Ref.[8]we selected the four models with progenitor masses of 11M�, 12M�, 13M� and 15M�. (Inthis article they are denoted by S11, S12, S13 & S15, respectively.) Besides, in considerationof the important significance of the primary energy of shock waves for the determination ofthe quantity of primary shock energy and for the judgment of the possibility of the ignitionof explosion and the transfer of energy, we have comprehensively studied 12 models withvarious primary masses and probed which definition of the primary energy of shock wavesis more appropriate.

2. PRIMARY DATA AND METHOD OF COMPUTATION

The masses of the iron cores which are close to the external boundaries of the four modelsselected by us (i.e., S11, S12, S13 & S15) are listed in Table 1. Among them, those ofS11, S12 & S13 are less than the corresponding values in Ref.[8], but that of S15 is larger.Besides, Figs.1 & 2 display the distributions of the density ρ, temperature T , velocity V ,electron abundance Ye and neutrino abundance Yν versus mass (in units of M�) on theprimary cross sections for the four models. In comparison with the primary cross sectiondistributions in Ref.[8], we find that except Yν which is equal to that in Ref.[8] all the otherprimary values for m ≥ 1.2M� differ much from those in Ref.[8]. However, in the case ofm < 1.2M� the results of comparison are as follows:

(1) The values of ρ in models S12 & S12 are basically the same, but they are evidentlylarger than those in Ref.[8]. The ρ in model S13 does not differ much from that in Ref.[8],while the ρ in model S15 is obviously less than that in Ref.[8].

(2) The absolute values of the initial velocities in the four models exhibit the relations:|VS11| ∼ |VS12| ∼ |VS13| > VS15|.

(3) The values of the temperatures in the four models are basically the same, and theydo not differ much from those in Ref.[8].

(4) The values of primary electron abundances in the four models are larger than thoseof Ref.[8] by about 0.01–0.02. Besides, in the range m < 0.8M� there exist the relationsYeS15 ∼ YeS13 ∼ YeS12 ∼ YeS11.

Fig. 1 Initial profiles of parameters at the beginning of collapse (the unit of mass is M�)

Table 1 Mass of iron core (in M�)

Code Mass of iron core Mass from the outer boundary to stellar centerS11 1.3029 1.6000S12 1.3401 1.6000S13 1.3420 1.6000S15 1.470 1.8050

The program of the numerical simulation of supernova WLYW89, which is adopted inour computation, is the program of general relativistic and one-dimensional hydrodynamiccalculations on the basis of the mechanism of instantaneous explosion. The explicit Lagrangedifference method is utilized. During the numerical simulation, we use the equation of statewhich is based on the model of four kinds of particles (i.e., the average heavy nuclides, α

particles, free protons p and free neutrons n). Moreover, in order to reduce the computertime and for the convenience of theoretical research the method of “equivalent thickness”is adopted. Namely, the energy needed for the splitting of α particles in the equation of

386 ZHANG Miao-jing et al. / Chinese Astronomy and Astrophysics 33 (2009) 384–392


state of three kinds of particles (α, n, p) is taken to be an adjustable parameter. Then theeffect, which is equivalent to the adjustment of the thickness of the region outside iron core,is obtained. In our simulative calculations the distance from the center of star to the outerboundary is uniformly divided into 96 envelope layers.

3. RESULTS OF COMPUTATION AND ANALYSIS

The results of calculations for the four models S11, S12, S13 & S15 can be found in Fig.3and Tables 2–5. Among them Fig.3 displays the curves of variations of the shock explosiveenergy Exp with time for various models. (Both Exp and the E11

xp, that will appear in thesubsequent text, satisfy the expression Exp =

∑(c2 + ε)ΓΔμ − c2

∑Δμ. The concrete

meanings of all the physical quantities in this expression can be found in Ref.[8]. It may beseen in the figure that all the explosive energies of the four models are larger than those inRef.[8] by 0.2–0.4 foe (1 foe =1044 J). This clearly implies that the new progenitor modelis more favorable for the explosion of supernovae. Besides, Table 2 lists the times neededfor the core collapse and rebound as well as for the propagation of shock waves. τ

(1)collap

expresses the time of growth of core densiry ρc from 1010g/cm3 to 1012 g/cm3. τ(2)collap is the

time of increase of ρc from 1012 g/cm3 to the maximum ρcmax. τt4−t11 is the time of shock

propagation from t4 to t11 in unit of ms.

Table 2 The times of core collapse, rebound and shock propagation (in ms)

Code τ(1)collap

τ(2)collap

τt4−t11

S11 200.7858 6.4249 10.8035S12 210.4654 6.5798 11.5140S13 251.4866 6.2040 8.7077S15 334.6899 6.0019 7.2607

Table 3 is concerned with the results for the stage of collapse. In the table Ms representsthe mass when ρc attains the maximum ρc

max and the absolute value of collapse velocity isequal to local sound velocity. σ = ρc

max/(2.7× 1014 g· cm−3) is the compression ratio. Y cL ,

Y ce and Y c

ν are, respectively, the abundances of leptons, electrons and neutrinos.

Table 3 Results of core collapse

Code Ms/M� ρcmax(g/cm3) σ Y c

L Y ce Y c

ν

S11 0.7833 1.2986(15) 4.8096 0.4023 0.3213 0.0810S12 0.7833 1.2704(15) 4.7050 0.4022 0.3210 0.0813S13 0.7667 1.2518(15) 4.6364 0.3976 0.3186 0.0791S15 0.7875 1.2075(15) 4.4723 0.3928 0.3141 0.0787

Table 4 lists the average speeds of shock waves in the process of propagation when theyarrive at various fronts. V i→j = (Mj −Mi)/(tj − ti) (i ≥ 4) is the average speed of shockwaves from the front Mi at moment ti to the front Mj at moment tj (unit: M�/ms).

Table 4 Average velocity of shock propagation (in 107 m/s)

Code v̄4→5 v̄5→6 v̄6→7 v̄7→8 v̄8→9 v̄9→10 v̄10→11 v̄11→12

S11 1.8315 0.4843 0.2251 0.0894 0.0370 0.0232 0.0070 0.0055S12 1.6026 0.4794 0.2192 0.0855 0.0352 0.0219 0.0058 0.0062S13 1.2121 0.4550 0.2096 0.0891 0.0429 0.0296 0.0109 0.0117S15 0.4112 0.5903 0.1992 0.1066 0.0438 0.0299 0.0154 0.0076


Table 5 is concerned with the energy loss in the process of shock propagation. ΔEν =Eν(t4)−Eν(t11) represents the energy loss of neutrinos when the shock waves arrive at thefront M11. Eν(t4) and Eν(t11) are, respectively, the energies carried by neutrions at momentst4 and t11. EH

b is the total energy of combination in the inner core. EHb = EH

b (t4)−EHb (t11)

expresses the energy transferred from inner to outer core in the time interval from t4 to t11.E

(11)xp is the energy of explosion when shock waves arrive at M11 and is not the energy when

shock waves arrive at the boundary of iron core (unit: foe).

Table 5 Energy transfer in the process of shock propagation (in foe)

Code Eν(t4) Eν(t11) ΔEν EHb

(t4) EHb

(t11) ΔEHb

E(11)xp

S11 -1.7358 -3.1919 1.45611 -7.3587 -15.3418 7.9831 0.9795S12 -1.7580 -3.2063 1.4483 -8.9103 -16.2982 7.3878 0.8127S13 -1.7688 -3.2413 1.4725 -5.7651 -14.5722 8.8071 0.9402S15 -1.9130 -3.5362 1.6231 -8.1325 -15.6760 7.5435 0.7231

In the following we carry out an analysis of the two stages of core collapse and shockwave rebound.

(1) For the stage of core collapse, it may be found from the τ(1)collap in Table 2 that

in comparison with Ref.[8] the times of collapse in the three models S11, S12 & S12 haveshortened, while that in the model S15 has lengthened. (In Ref.[8] the value for S15 is 238.34ms.) From the above-stated analysis of primary data it can be found that the primarydensities in the models S11 & S12 have increased, while their times of collapse τ

(1)collap’s have

decreased. Similarly, the primary density in the model S15 has decreased (the maximum ismerely 6.16×109 g/cm3) and is less than the value in Ref.[8] by about 3×109 g/cm3), whilethe value of its τ

(1)collap has increased . This correlation implies that the quantity of primary

density can affect the time of collapse of iron core. This is because if the initial density ofiron core is small, then in the process of collapse the time needed for the iron core densityto grow to a certain value is rather long. Besides, in the model S13 the initial density isslightly too large, but the time of collapse decreases much. In consideration of the influenceof the variations of primary velocity, temperature and abundance of electrons, we get thefollowing results. The changes of primary temperature and velocity are small and may beignored. The growth of electron abundance can lead to to the reaction of electron captureand promote the collapse of iron core, hence the time of collapse is shortened. This impliesthat the variation of electron abundance can also affect the collapse of iron core.

Moreover, it can be seen in Fig.1 that when m < 1.2M� the value of primary densityρ and the absolute value of primary velocity are minima in the model S15 (the smaller theprimary velocity, the slower the velocity of collapse), but the primary temperature basicallyagrees with those in the other three models. The primary electron abundance of model S15is larger than those of the other models by about 0.01, and the value of τ

(1)collaps of model

S15 is the largest. This means that the magnitudes of initial density and velocity have greatinfluence on the time of collapse, especially in the range of density from 1010 to 1012 g/cm3,yet the values of τ (2) do not differ much. Being analogous to the conclusion of Ref.[8], therises of density from 1012 g/cm3 to ρc

max are prompt and are basically the same.When the central density of iron core ρc attains at the maximum ρc

max, it can be knownfrom the results of collapse of stellar core listed in Table 3 that all the values of the threequantities Ms, ρc

max and σ are obviously larger than those in Ref.[3], especially ρcmax and σ


become twice larger than the original values. The increase of Ms implies that the outer partof stellar core becomes thinner, namely the path that should be traversed by shock wavesis shortened. This signifies that the energy loss in the process of shock wave propagationcan be reduced. But the increases of ρc

max and σ make the inner core to be harder and therebound of shock waves to be stronger. All these are favorable for the shock propagation.As for the abundances of leptons, electrons and meutrinos, their values are close to those inRef.[8].

(2) In the stage of the propagation of rebounded shock waves, the time of propagationτt4−T14 does not differ much from that stated in Ref.[4] (see Table 2), but the diiferenceof average velocities (see Table 4) is rather large. Taking v̄7→8 as the dividing line, all thevalues before v̄7→8 (including v̄7→8) are larger than those in Ref.[8], while those behind v̄7→8

are rather small. Namely, in these models the propagation of shock waves is at first fast andthen slow. Besides, as shown by Table 4, in the models S13 and S15 the values after v̄7→8

are larger than the corresponding values in the models S11 and S12. This is so especiallyfor the values of v̄10→11. This implies that the shock waves in the models S13 and S15 stillretain rather large intensities in their later periods. Moreover, this may explain why theirvalues of τt4−t11 are slightly smaller than those of S11 and S12.

In the probe of the explosive energy of shock waves (see Fig.3 and Table 5) we findthat the energy carried away by neutrinos ΔEν is slightly less than the value in Ref.[8]by approximately 0.05 foe, but for the photodisintegration energy Edis only that of S15is slightly large and those of all the other models are somewhat smaller. Moreover, it isnoteworthy that the value of ΔEH

b increases by about 2 foe. Namely, in the new model theenergy transferred from the inner to the outer stellar core increases much, and this providesthe important energy for the outward transfer of shock waves. By synthesizing the above-stated three items, all these changes are favorable for the shock propagation. Hence thevalue of explosive energy E

(11)xp is relatively larger.

4. DEFINITIONS OF THE PRIMARY ENERGY OF SHOCK WAVES

From the results of numerical simulation we may obtain the energy of the final explosion ofstar, i.e., the energy Exp of shock waves when they arrive at the boundary of the iron coreof star. However, for the primary energy of shock waves there are various understandings.Nowadays there are five different definitions of the primary energy of shock waves. They arethe following ones:

(1) Bruenn’s[11] definition—the maximal possible energy transferred from the innerstellar core to the rebounded shock waves: E

(1)shock.

(2) Muller’s[12] definition—the kinetic energy of inner stellar core when the reboundedshock waves have just been formed: E

(2)shock.

(3) The shock energy is provided in the form of stairs, and it is not the kinetic energyof the newly formed stellar core that is provided once. In the process of outward shockpropagation the energy of inner region is continuously transferred to the outer region viaconvective motion. This is the definition E

(3)shock.

(4) The shock energy is transferred outward just after the formation of shock waves.After excluding the excessive kinetic energy at the final moment and the kinetic energy

transferred to the envelope, the remaining part is the total amount of shock energy in outerregion. This is the definition E

(4)shock.

(5) The maximum of the kinetic energy in outer region when the shock waves have justbeen formed and are propagated outward. This is the definition E

(5)shock.

Fig. 2 Primary abundance of neutrinos Fig. 3 Energy of explosion

As for which of these five definitions is more appropriate, this question was probedby Wang Wei-zhong et al.[13,14]. But until peresent there is not yet a definite conclusion.Therefore, we like to synthetically investigate the definitions of primary shock energy withthe results given by twelve models, i.e., S11, S12, S13, S15, S16, S17, S20, S22, S30, S31,S35 & S37. The results of calculations and analyses are listed in Tables 6, 7 & 8. Table 6shows the energy dissipated by photodisintegration. Moc is the mass of outer stellar core.M ef

oc is the equivalent mass of outer stellar core. Edis is the energy of photodisintegrationin outer stellar core. Exp is the energy of explosion. Eoc

ν = Eν(t4) − Eν(t) is the energycarried away by neutrinos (unit: foe). Table 7 gives the calculated values of the primaryenergy of shock waves according to the above-stated five definitions. Table 8 indicates themodels which are best suited to respective definitions of the primary energy of shock waves.

Table 6 The energy dissipated in photodisintegration in outer core (in foes)

Code Moc Mefoc Mef

oc × 20foe/M�(foe) Edis Eocν Exp

S11 0.5196 0.3002 6.0040 5.0851 1.4590 0.9742S12 0.5568 0.3217 6.4348 5.4491 1.4549 0.8461S13 0.5753 0.3324 6.6480 5.6302 1.4849 0.8355S15 0.6826 0.3944 7.8870 6.6803 1.6550 0.6878S16 0.6330 0.3658 7.3160 6.1949 1.6146 0.6824S17 0.6370 0.3681 7.3620 6.2340 1.6256 0.6160S20 0.6893 0.3983 7.9660 6.7458 1.5320 0.5244S22 0.6621 0.3826 7.6520 6.4796 1.6459 0.4417S30 0.6879 0.3975 7.9500 6.7321 1.7405 0.2900S31 0.5552 0.3208 6.4160 5.4335 1.7017 0.3443S35 0.7033 0.4064 8.1280 6.8828 1.7754 0.2277S37 0.7091 0.4097 8.1940 6.9396 1.8096 0.2608



Table 7 Initial energy of shock waves

Code E(1)shock

E(2)shock

E(3)shock

E(4)shock

E(5)shock

S11 6.6233 0.4296 9.9640 7.9759 8.7667S12 6.0513 1.1936 9.5948 6.4811 7.3714S13 7.4132 0.2864 7.4975 8.3912 8.5424S15 5.9807 1.1282 8.3243 7.2652 7.9189S16 7.3900 0.3483 10.451 8.4416 8.5493S17 6.9445 0.3475 8.9954 8.6165 8.6430S20 6.9684 0.3224 9.9480 8.5102 8.4458S22 6.0035 0.9306 11.414 8.1586 8.6628S30 5.8649 0.7243 11.639 8.0684 8.6218S31 6.1293 0.9165 6.8340 8.2410 8.7505S35 5.7055 1.1411 8.3051 7.9593 8.3754S37 6.9272 0.5502 13.048 8.6728 8.3691

Table 8 The models fitted to various definitions

Definition Models fitted to this definition

E(1)shock

(none)

E(2)shock

(none)

E(3)shock

S13, S15

E(4)shock

S11, S13, S16, S17, S20, S31, S37

E(5)shock

S12, S13, S16, S17, S20, S22, S30, S35

For a successful explosion, the energy of shock waves Eshock must satisfy the followingsufficient conditions[14]:

Eshock >MFe −Ms

M�× 20(foe) , (1)

Eshock > Edis + Eocν . (2)

For this, we at first examine whether a model satisfies the sufficient conditions requiredby every definition (see Tables 6 & 7). From the data listed in these tables the followingresults are inferred:

(1) When E(1)shock is taken, only three models (i.e., S11, S13 & S16) satisfy the condition

(1), and all the other models do not. But all the above-mentioned three models do notsatisfy the condition (2). E

(2)shock is evidently too small, so it cannot satisfy the sufficient

conditions. Hence both E(1)shock and E

(2)shock are not suitable options.

(2) When E(3)shock is taken, only except S11 all the other models can satisfy the conditions.

(3) When E(4)shock is taken, the models which do not satisfy the conditions are S12, S15,

S30 & S35.(4) When E

(5)shock is taken, the models S15 & S35 do not satisfy the conditions.

In the case that the explosive energy is combined, i.e., the condition Eshock ∼ Edis +Eoc

ν + Exp is required, the value of Eshock cannot be either too small or too large. TakingS11 as an example, its Edis + Eoc

ν + Exp = 7.5183 (foe). When E(4)shock is adopted this is

rather appropriate, while if E(5)shock and E

(3)shock are used, this seems to be too large. Via our

analysis, the suitable models for various definitions are listed in Table 8. As shown by thistable, when E

(4)shock is adopted, there are 7 such models; when E

(5)shock is adopted, there are 8

such models. This implies that for the choice of the definition of the primary shock energy,E

(4)shock and E

(5)shock are comparatively appropriate.


5. SUMMARY

By analyzing the results yielded by the four models S11, S12, S13 & S15 we find that in thestage of collapse and due to the variations of the initial density, temperature, and electronabundance the times of collapse τ

(1)collap’s of the models S11, S12 & S13 are shortened, while

that of S15 is lengthened. At the same time, the increase of Ms shortens the distance ofshock propagation and that of the central density ρc

max makes the rebound of shock wavesstronger. All these are favorable for the propagation of shock waves. In the process of shockpropagation and in comparison with the original model, for the new model the propagationat first is fast and later it becomes slow. But the times of propagation do not differ much.However, owing to the increase of ΔEH

b as well as the decreases of Edis and ΔEν , the shockenergy is effectively enhanced. Therefore, the value of explosive energy E

(11)xp in the new

model is rather large. In the investigation of the definition of the primary energy of shockwaves, we have examined twelve models and the conclusion drawn by us is as follows. E

(4)shock

and E(5)shock are rather good choices, yet they are not appropriate for any one model. Thus

we have to further seek for a still more suitable definition.

ACKNOWLEDGEMENT The authors thank the network station http://www. stel-larevolution.org for providing the data of the progenitor model.

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the influence of a new model of progenitor star on supernova explosion

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