Pulsational Pair-Instability Supernova I: Pre-Collapse Evolution and Pulsation Shing-Chi Leung1, Ken’Ichi Nomoto1, Ming-Chung Chu3, Sergei Blinnikov1,21,2,3

Draft version June 15, 2018 Typeset using LATEX twocolumn style in AASTeX62

Pulsational Pair-Instability Supernova I: Pre-collapse Evolution and Pulsation Shing-Chi Leung1, Ken’ichi Nomoto1, Ming-Chung Chu3, Sergei Blinnikov1,21,2,3

11Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 22Institute for Theoretical and Experimental Physics, Moscow, Russia 33Physics Department, the Chinese University of Hong Kong, Hong Kong S. A. R., China

(Dated: June 15, 2018)

ABSTRACT Pulsational Pair-Instability Supernova (PPISN) is the explosion of main-sequence stars of masses from 80 - 140 M⊙. The electron-positron pair production in the stellar core is believed to make strong contraction of the carbon-oxygen after the depletion of He core, which triggers the explosive O-burning and then enormous mass ejection. The ejected mass then becomes the circumstellar material around the star, which has strong optical effects when the star finally explodes as its core collapses. In this article we reexamine this idea by using the one-dimensional stellar evolution code MESA to model PPISN. Using the recent implicit hydrodynamics formalism we are able to follow the whole evolution from the main-sequence to the onset of collapse in single runs. We study the influence of metallicity on the progenitor. Then we examine in details the dynamical and chemical properties of pulsations as a function of He core mass. At last, we discuss its connection to superluminous supernova and to the massive black hole formation. 1. INTRODUCTION causes ejection of high velocity matter on the surface Stars with a progenitor mass 80 - 160 solar masses and dissipates the energy. After that, the core becomes have very distinctive paths of evolution compared to bounded again. The pulsation restarts after it has lost more massive or less massive stars. For more mas- most of its previously produced energy by radiation and 56 sive stars, they directly explode as pair-instability su- neutrinos. The whole process repeats until the Fe de- 56 pernovae (PISN); for less massive stars, they collapse cayed from Ni exceeds the Chandrasekhar mass that after the iron core of Chandrasekhar mass has been it collapses by its own gravity before the compression 16 formed. This class of stars experiences strong pulsations heating can reach the further outgoing O-rich enve- before their collapse, thus referred to as Pulsation Pair- lope. instability supernova (PPISN) in the literature. The The origin of massive star has been a matter of debate. pulsation phase occurs due to electron-positron pair in- One major uncertainty is the mass loss rate. For PPISN, stability (γ → e−+e+). The core is mostly supported by owing to its high luminosity during its H-burning main- the radiation pressure. With the catastrophe in pair pro- sequence phase, almost the whole hydrogen envelope is duction, the supporting pressure suddenly drops, where shed off before advanced burning takes place. In par- the core softens with corresponding equation of state ticular, for PPISN, the mass loss by stellar wind can adiabatic index γ < 4/3 in the core. However, unlike the contribute to more than half of the initial progenitor mass. Such mass loss can suppress the build up of the stars with 15 - 40 M⊙ which have rich iron cores at the moment of their collapse, in PPISN the core is mostly helium core, leaving a light He core instead. The ex- made of 16O when contraction starts. The softened core act mass of the He core as a function of metallicity allows a very strong contraction and the 16O-rich core and progenitor mass remains less well constrained com- can reach the explosive temperature which releases a pared to the hydrogen counterpart (Vink & de Koter large amount of energy, sufficient to unbind the star. 2002; Smith & Owocki 2006). 28Si and 56Ni can be produced during the contraction, The effects of pair-production induced instability have where the central temperature can reach beyond 109.5 been studied since a few decades ago, starting from the K. As a result, the star stops its contraction and starts treatise for pair-formation in massive stars and super- its expansion. The rapid expansion causes strong com- novae (Fowler & Hoyle 1964). The possibilities of pul- pression to the matter on the surface, which efficiently sation prior to its collapse in massive star is suggested 2 in Barkat et al. (1967). The evolution of unstable mas- their evolution scenario. As indicated in Woosley et al. sive He- and CO cores maintained by rotation can be (2007), the origin of 40 - 60 M⊙ He cores can be from found in Glatzel et al. (1985); Chatzopoulos & Wheeler stars about 95 − 130M⊙, which is the mass range of (2012). The optical aspect of pair-instability induced PPISN. Therefore, in order to study the mass spectrum pulsation and explosion is connected to super-luminous of BH in this mass range, the evolutionary path of this supernovae, such as SN2006gy (Woosley et al. 2007; class of objects becomes necessary. Kasen et al. 2011; Chen et al. 2014). In order to pin For these reasons, we want to re-examine the concept down the mass range of PPISN, a mass survey of main- of PPISN by using the open-source stellar evolution code sequence star models is done in Heger & Woosley (2002) MESA to study PPISN. We use the MESA code for two with focus on the zero metallicity stars. In Woosley reasons. First, its open-source nature allows future val- (2017) the hydrodynamics of PPISN pulsation is studied idation and confirmation from other users in the com- in details with further connection to the well observed munity. Second, the recent update of the MESA code Eta Carinae, which has demonstrated significant mass (Paxton et al. 2015) has included an implicit energy- loss about 30 M⊙ (Smith et al. 2007; Smith 2008). conserving (Grott et al. 2005) hydrodynamics scheme as Compared to pair-instability supernovae (PISN) one of its evolution option. Such dynamical treatment (Heger & Woosley 2002; Scammapieco et al. 2005; in the fluid motion can naturally address the dynami- Chatzopoulos et al. 2013; Chen et al. 2014), which are cal perspective of PPISN in the stellar evolution con- thermonuclear explosion of very massive stars, PPISN text, where conventionally hydrostatic approximation has received much less study in the literature. Woosley with acceleration correction is frequently used. (2017) is the first work extensively devoted to PPISN systematically using the KEPLER code. A series of 1.1. Structure main-sequence star models with masses from 70 - 140 In Section 2 we describe the code we use for prepar- M⊙ is studied. This corresponds to the helium core ing for the initial models and the details of the one- mass from ≈ 35 to 70M⊙. The pulsation history is ex- dimensional implicit hydrodynamics code for the pul- plored with a focus on the possible optical signals from sation phase. Then in Section 3 we present the pre- the pulsation. The connection to different subclasses in bounce stellar evolution and its thermodynamics prop- Type II supernova is also discussed. erties. We first present the pre-pulsation evolution of The pulsation of the PPISN is a dynamical phe- our models which include the He- and C- burning. Then nomenon. During the pulsation, the dynamical we study the dynamics of the pulsation and its effects timescale can be comparable with the nuclear timescale on the shock-induced mass loss. After that, we present that hydrostatic approximation is no longer a good ap- evolution models of He cores with 40 - 64 M⊙. We proximation. Also, when the star drastically expands examine their properties from four aspects, the thermo- after the energetic nuclear burning triggered at the con- dynamics, mass loss, energetic and chemical properties. traction, the subsequent shock breakout near the surface In Section 4 we refer to the connections of our models to is obviously a dynamical phenomenon. This suggests super-luminous progenitors. We then further study the that during this dynamical but short phase, hydrody- effects of some physical input in the numerical modeling, namics instead of hydrostatic is required in order to including the convective mixing and artificial viscosity. follow the evolution consistently. The expected neutrino detection rates by the existing Recent modelling of super-luminous supernovae and proposed neutrino detectors are discussed. At last PTF12dam (Tolstov et al. 2017) has required an ex- we conclude our results. plosion of 40 M⊙ star with 20-40 M⊙ circumstellar 56 medium (CSM) with a sum of 6 M⊙ Ni in the explo- 2. METHODS sion. The shape, rising time and fall rate of the light 2.1. Stellar Evolution curves provides constraints on the composition, density To prepare the pre-collapse model, we use the open and velocity of the ejecta, which provides insights to the source code Modules for Experiments in Stellar Astro- modeling of PPISN. It demonstrates the importance to physics (MESA) version 8118 (Paxton et al. 2015). It track the mass loss history of a star prior to its collapse. is an one-dimensional stellar evolution code with realis- Furthermore, recent detections of the gravitational tic microphysics input including the OPAL opacity ta- waves emitted by the merging of black holes (BH) ble. The modular structure of the code allows attach- (Abbott et al. 2016a,b), such as GW150914 imply ex- ing extra physics components to the main structure of istence of BHs of masses about 30 M⊙. This has led to the code. Recent updates of this code have also in- the interest of the origin of BHs in this mass range and cluded packages for stellar pulsation analysis and im- 3 plicit hydrodynamics extension with artificial viscosity. which uses the information of both the next step (step We modify the package ccsn to build a helium core or n+1) and the current step (step n) with equal weight. In main-sequence star models directly and then we switch Grott et al. (2005); Paxton et al. (2015) it is shown that to the hydrodynamics formalism according to the global by arranging terms in this form one can achieve energy dynamical timescale of the star. conservation with an accuracy equivalent to those of the hydrodynamics and nuclear reaction solvers. 2.2. Hydrodynamics We follow the convention for the physics quantities. To understand the effects of pulsation and runaway Density, temperature, isotope mass fractions, specific burning of the oxygen core on the dynamics and neutrino internal and related thermodynamics quantities are de- signals, we use the one-dimensional implicit hydrody- fined on cell centers. Position, velocity, acceleration namics code with a nuclear reaction network. This func- and gravity source terms are defined on the cell bound- tion appears in the third instrument paper of MESA. aries. We impose the innermost boundary conditions as The dynamical formalism of the fluid motion is impor- r0 = 0. tant because the system is not in quasi-static equilib- Due to the semi-implicit nature of the dynamics code, rium, especially when the bounce shock has formed. The the typical timescale is comparable to the dynamical energy conserving scheme, coupled to the implicit mass- timescale. However, after the pulsation phase, it is the conserving property of the Lagrangian formalism, allows KH timescale that dominates the contraction. Sim- us to trace the evolution of the star consistently. For the ply using the semi-implicit hydrodynamics formalism hydrodynamics, the code solves the equations of motion to evolve the whole pulsation phase is computation- for the fluid in the Lagrangian form, namely ally challenging as the Courant-Friedrich-Levy condi- Dr tion severely limits the maximum possible timescale, al- i = v , (1) Dt i though it guarantees the consistency of our calculation. Dvi Gmenc,i We set conditions for the code to switch back to the = − 2 , (2) Dt ri hydrostatic approximation. To make sure the star suf- Dǫi Pi ficiently expand after bounce so that the evolution be- = − (∇ · ~v)i +ǫ ˙nuc,i, (3) Dt ρi comes mostly thermal transport, we gradually relax the DX maximum timestep for about 100 steps. When the star ij = X˙ + X˙ . (4) Dt ij,nuc ij,mixing can evolve continuously with the maximum timesteps, we switch back to the hydrostatic approximation to Here, r, v, ρ, P , ǫ and m are the position, velocity, enc evolve the star until it becomes dense enough so that density, pressure, specific internal energy and enclosed dynamical effects emerges. If the star appears to be mass.ǫ ˙ is the heat gain/loss by nuclear reaction in- nuc non-static during the 100-step buffer, the buffer is fur- cluding the thermal neutrino contributions Itoh et al. ther extended. After the hydrostatic scheme is used, the (1989). The subscript i stands for the physical quanti- convective mixing is also switched on. When the central ties of the ith cell and j stands for the jth isotope. X ij temperature of the star reaches above 109.3 K, the semi- (j = 1, 2, ...) are therefore the mass fraction of the jth i implicit hydrodynamics scheme is resumed. Sometimes isotope. m = dm . enc,i 1 i the star is not fully relaxed even after the buffer phase, To capture the shock,P the artificial viscosity takes the the hydrostatic approximation can also yield small steps. form 6 In that case, we also switch the code back to the hydro- 4πri vi+1 vi Qi = −Cavρi − , (5) dynamics scheme to further relax the star. dmi ri+1 ri which has the same unit as the pressure term and it We include the list of ingredients we have used for enters the system of different equations by P → (P +Q). building the model and also for the dynamical evolution of the stellar pulsation in the Appendix. (∆r)i = ri −ri−1. We choose Ca =0.002−0.02. We also study the effects on the choice of Ca in the appendix. To ensure energy conservation of the system, the grav- 2.3. Microphysics itational potential force is defined as We use the standard Helmholtz equation of state Gm Gm (Swesty 1999), which contains electron gas with arbi- enc = enc . (6) 2 n n+1 trary relativistic and degeneracy levels, ions in the form r ri ri of an classical ideal gas, photon gas with Planck distribu- The source term of the dynamical equations is defined tion and electron-positron pairs. To model the nuclear implicitly, reactions, we use the ’approx21 plus co56.net’ network. vn + vn+1 v = i i , (7) This includes the α-chain network (4He, 12C, 16O, 20Ne, i 2 4

24Mg, 28Si, 32S, 36Ar, 40Ca, 44Ti, 48Cr, 52Fe and 56Ni), that the thermodynamics properties of the core before 1H, 3He and 14N for the hydrogen burning and CNO pulsation depend on only its mass. cycle, and 56Fe and 56Co to trace the decay chain of In Figure 1 we plot the Hertzsprung-Russell diagram 56Ni. 56Cr is included to mimic the neutron-rich iso- for the Models 40HeA, 45HeA, 50HeA, 55HeA and topes formed after electron capture in nuclear statistic 60HeA. The qualitative features of their paths are sim- equilibrium (NSE). ilar, which consist of an initial jump in luminosity and a near horizontal move towards a higher Teff . A Model 2.4. Convective Mixing with a higher mass gives a higher luminosity at the same effective temperature. The moments where the core He As indicated in Woosley (2017), convective mixing is and C are exhausted for each models are linked by the important in that it redistributes the fuel and ash in black lines. Unlike low-mass stars, all the helium and the remnant core. This affects the subsequent nuclear carbon burning occurs in such a short time scale that burning when the star contracts again. We choose the the star does not vary much in luminosity. Mixing Length Theory (MLT) (Cox & Giuli 1968) to In Figure 1 we plot the profiles we used as the ini- model the convective process. The MLT approximation tial conditions for the dynamical phase of some mod- is used in the main-sequence phase and also when the els, including Models 40HeA, 50HeA and 60HeA. The star enters the expansion phase. We have attempted to three models have very similar initial profiles. Due to couple the convective mixing in the dynamical phase large scale convection the whole star has very flat tem- but it results in impractically small timesteps. Fur- perature and density profiles for most part of the star. thermore, the convective velocity during the dynami- These quantities drop around q = m(r)/M ≈ 0.8. Mi- cal phase is in general much smaller than the fluid ve- nor differences can be seen among models from the den- locity. This implies that the mixing is important at a sity profiles that the models with a lower He mass has much longer timescale. Therefore, it becomes numeri- a slightly higher density in the inner part. cally manageable while physically consistent to ignore In the lower right panel we plot the chemical abun- convective mixing in the dynamical phase. dance profiles similar to those for Models 40HeA and 60HeA. Due to the sensitivity of nuclear burning to 3. HYDRODYNAMICS temperature and density, the chemical abundance shows 3.1. Pre-pulsation evolution more varieties. For example, in the lighter mass model, there is no remaining 12C. From q =0.1−0.8, the differ- We first present the results for the pre-collapse evolu- ence between 12C and 20Ne is larger, showing that there tion of the helium core and main-sequence star in Table is less 20Ne in this layer. The 12C-burning envelope is 1. The pre-pulsation evolution uses the hydrostatic ap- extended to almost the surface of the star, leaving very proximation and it is done until the central temperature little mass of pure He. On the contrary, in the more reaches 109.1 K, where the dynamical timescale begins massive He core model, 12C is not yet burnt at the cen- to be comparable with the nuclear reaction timescale. tral temperature. Also there is a comparatively higher From the table we can see that the initial He core mass 20Ne in the zone q =0.1−0.8. The shell carbon burning affects the pre-pulsation carbon and oxygen core. We beyond q =0.8 has just started. The He surface remains choose the helium core mass from 40 to 64 M⊙, which unchanged. produces a CO core from 30.82 to 50.42 M⊙, with the To study the effects of metallicity and rotation, we remaining unburnt helium in the envelope. perform pre-pulsation stellar evolution models for dif- In Figure 1 we plot the central temperature against −3 ferent metallicty from Z = 10 Z⊙ up to Z = 1Z⊙ for central density in log scale, together with the pair- non-rotating main-sequence star model using the MESA instability zone. Models 40HeA, 45HeA, 50HeA, 55HeA code. In Table 1 we tabulate the pre-pulsation configu- and 60HeA are included. Below 109.3 K the star evolves rations of the main-sequence stars for their He- and CO- in a quasi-static manner. We include also the lines indli- core masses when the core exhausts all H and He respec- cating the model with He and C exhausted (defined by tively. The CO core masses are written in brackets. their central abundance being below 10−3). When the For the He core mass, it can be seen that He core initial He mass increases, the exhaustion of helium and mass grows monotonically with progenitor mass when carbon end with a lower central density and tempera- Z < 0.75Zodot. For star models with higher metallicity, ture. Also, a model with a higher mass starts and ends the mass loss rate, which is proportional to metallicity, with a lower density for a given temperature, with the makes the He core mass drops at the high mass end. Model 60HeA being the closest to the pair-instability This transition starts at lower mass for model with a zone. There is no intersection among models, showing 5

Table 1. The main-sequence star models prepared by the MESA code. Mini and Mﬁn are the initial and ﬁnal masses of the star. MH, MHe, MCO are the hydrogen, helium and carbon-oxygen mass before the hydrodynamics code starts. All masses are in units of solar mass.

Model Mini Mﬁn MH MHe MC MO Z remark 40HeA 40 40 0 6.79 3.13 27.5 / only He core 45HeA 45 45 0 7.38 4.03 31.3 / only He core 50HeA 50 50 0 7.82 4.16 35.2 / only He core 55HeA 55 55 0 8.27 4.30 39.0 / only He core 60HeA 60 60 0 8.69 4.43 42.9 / only He core 62HeA 62 62 0 8.77 4.59 44.6 / only He core 63HeA 63 63 0 8.89 4.64 45.3 / only He core 64HeA 64 64 0 8.96 4.63 46.1 / only He core

6.6 Pair-instability region C He 9 ) -1 6.4 (K) c T 10 L (erg s 10 log

M(He) = 40 log 40HeA M(He) = 45 45HeA M(He) = 50 8.5 6.2 50HeA M(He) = 55 55HeA M(He) = 60 60HeA

1 2 3 4 5 5.5 5.4 5.3 5.2 5.1 ρ -3 log10 c (g cm ) log10 Teff (K) 10 1 Model 40HeA 5

) 0 4 -3 Model 40HeA 0.1 He -5 12 C 10 16 (g cm O ρ 5 20 10 0.01 Ne 0 1 Model 60HeA

Model 50HeA mass fraction -5 10 T (K), log 10 5 0.1 log 0

-5 Model 60HeA 0.01 0.001 0.01 0.1 1 0.001 0.01 0.1 1 m(r)/M m(r)/M

Figure 1. (upper left) The temperature evolution of Model He60A around the second pulse. The profiles are chosen in the way that (upper right) Similar to the upper left panel, but for the density profiles. (middle left) Similar to the upper left panel, but for the velocity profiles. (middle right) Similar to the upper left panel, but for 16O mass fraction. (lower left) Similar to the upper left panel, but for 28Si mass fraction. (lower right) Similar to the upper left panel, but for 56Ni mass fraction.

Table 2. The pre-pulsation He core mass at the exhaustion of H in the core. The numbers in brackets are the CO core mass at the exhaustion of He in the core. All masses are in units of solar mass. −3 −2 Mass(⊙) Z = 10 Z⊙ Z = 10 Z⊙ Z = 0.1Z⊙ Z = 0.5Z⊙ Z = 0.75Z⊙ Z = 1Z⊙ 80 34.05 34.18 32.71 30.82 (23.96) 30.37 (21.09) 29.67 (18.66) 100 44.51 44.65 34.20 40.46 (30.65) 38.50 (28.50) 38.23 (24.85) 120 54.87 54.81 43.48 50.47 (41.20) 47.40 (31.73) 43.73 (12.02) 140 65.87 65.34 62.79 59.20 (50.36) 54.79 (16.90) 34.91 (9.60) 160 76.50 75.92 73.18 69.03 (46.46) 50.95 (11.63) 28.69 (8.90) 6

-3 contraction, which makes the For models where the CO Z = 10 Zsun -2 core mass can be extracted, the metallicity effect is more Z = 10 Zsun Z = 0.1 Z sun significant. In all models, at the lower mass branch CO Z = 0.5 Z 60 sun Z = 0.75 Zsun core mass in general increases with progenitor mass, but Z = Z sun it drops at the high mass end. The CO core mass also shows a monotonic increasing relation against metallic- 40 ity for the same progenitor mass. The mass loss effect in

He core mass (solar mass) near solar metallicity models are more significant that the CO core mass contributes to less than 10 % of the stellar mass, while those in lower metallicity model can 20 80 100 120 140 160 progenitor mass (solar mass) be about one third of the progenitor mass. In Figure 3 we also plot the relation CO core mass against progeni- Figure 2. The He core mass against progenitor mass when tor mass at different metallicty. The significance of the the core exhausts its hydrogen for stellar models at different metallicity of mass loss rate can be seen. By increasing metallicity. models from 0.5 Z⊙ to 0.75 Z⊙, the CO core mass can drop by 75 % at M = 160M⊙. 60

Z = 0.1 Zsun Z = 0.5 Z sun 3.2. Pulsation Z = 0.75 Zsun Z = Z We ﬁrst study the time evolution of the pulsation. To 40 sun do so, we examine the second pulse of Model He60A, which is a strong pulse (with mass ejection) of mass ≈ 10M⊙. We choose this particular pulse because it

20 is strong enough to create global change in the proﬁle CO core mass (solar mass) so that we can understand the changes during the contraction (before maximum of central temperature in the

80 100 120 140 160 pulsation) and expansion (after minimum of that in the progenitor mass (solar mass) pulsation) phases. Figure 3. The C core mass against progenitor mass when In Figure 4 we plot in the upper left, upper right and the core exhausts its helium for stellar models at different middle left panels the temperature, density and veloc- metallicity. ity evolution at selected time respectively. We pick the profiles when the core temperature reaches 109.3, 109.4 higher metallicity. In Figure 2 we also plot the relations and 109.5 K before the core reaches its peak temperature He core mass against progenitor mass for different metal- during the pulse for Profile 1-3, at its peak temperature licity. On one hand, at low metallicity, the He core mass for Profile 4, and after the core has reached its peak is not so sensitive to metallicity, that the He core mass temperature for Profile 5 - 7 for the same central tem- −2 approaches its asymptotic value when Z ≤ 10 Z⊙. On perature interval. In the middle right, lower left and the other hand, at high metallicity, the He core mass lower right panels we plot the chemical abundance pro- 16 28 56 is very sensitive to metallcity that from Z =0.75Z⊙ to files for isotopes O, Si and Ni respectively. Z = Z⊙ the He core mass can drop by half at the star First we study the hydrodynamics quantitites. For the model M = 160M⊙, about 30 M⊙. At such low mass, temperature, in the contraction (expansion) phase the the He core already leaves the pulsation pair-instability star shows a global heating (cooling) due to the compres- regime, and will evolve as a normal core collapse su- sion (expansion) of matter, and no temperature discon- pernova. Furthermore, one can see that the maximum tinuity can be observed. This shows that the whole star He core mass for models at solar metallicity only barely contracts adiabatically, where the nuclear reactions take reach the transition mass 40 M⊙. This shows that the place evenly inside the star. By comparing the temper- PPISN is very sensitive to the progenitor metallicity, ature profiles at the same central temperature (Profiles 9.3 9.4 while stars with solar metallicity is less likely to form 1 and 7 for Tc = 10 K, profiles 2 and 6 for Tc = 10 9.5 PPISN owing to its mass loss. K and profiles 3 and 5 for Tc = 10 K), the net effect For the CO core mass, we only list the results for of nuclear burning can be extracted. The part outside mostly high metallicity because the code fails to proceed q ∼ 0.3 becomes hotter after the pulse. Similar compar- after the He exhaustion. Before the CO core mass can ison can be carried out for the density profile. The inner be defined, the massive CO core has already started its core within q ∼ 0.3 is unchanged after pulsation, while 7

8 4×10 6

9 × 8 4 2 10 ) -1

T (K) T (K) 2 0 10 Profile 1 10 Profile 1

Profile 2 Profile 2 v (cm s log 8 log Profile 3 Profile 3 Profile 1 Profile 4 Profile 4 Profile 2 0 Profile 3 Profile 5 Profile 5 × 8 Profile 6 Profile 6 -2 10 Profile 4 Profile 7 Profile 7 Profile 5 Profile 6 -2 Profile 7 7 × 8 0 0.5 1 0 0.5 1 -4 10 0 0.5 1 m(r)/M m(r)/M m(r)/M 1 1 1 Profile 1 Profile 2 Profile 3 Profile 4 Profile 5 Profile 6 Profile 7 O) Si) Ni) 16 28 0.1 0.1 56 0.1 X( X( X( Profile 1 Profile 1 Profile 2 Profile 2 Profile 3 Profile 3 Profile 4 Profile 4 Profile 5 Profile 5 Profile 6 Profile 6 Profile 7 Profile 7 0.01 0.01 0.01 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 m(r)/M m(r)/M m(r)/M

Figure 4. (upper left) The temperature evolution of Model He60A around the second pulse. The profiles are chosen in the way that the central temperature reaches 109.3 (Profile 1), 109.4 (Profile 2), 109.5 (Profile 3) before the pulse, during the peak (Profile 4), and the central temperature returns to 109.5 (Profile 5), 109.4 (Profile 6) and 109.3 K (Profile 7) respectively. (upper right) Similar to the upper left panel, but for the density profiles. (middle left) Similar to the upper left panel, but for the velocity profiles. (middle right) Similar to the upper left panel, but for 16O mass fraction. (lower left) Similar to the upper left panel, but for 28Si mass fraction. (lower right) Similar to the upper left panel, but for 56Ni mass fraction. the density in the outer part increased. The velocity Here we study some representative models of He core profiles show more features during the pulse. Before the mass from 40 - 62 M⊙. They show very different pulsa- star reaches its maximally compressed state, the velocity tion history, by their number of pulses and their corre- everywhere is much less than 108 cm s−1. At the peak sponding strengths. of the pulse, the envelope has the highest infall velocity In Table 3.3 we tabulate the stellar mass and the ele- of ≈ 2 × 108 cm s−1. After that, in Profile 5, the core ment mass in the star after each of the pulse. starts the homologous expansion phase, with a sharp ve- For Model He40A, most of the pulses are weak, how- locity discontinuity peak near the surface between the ever, following each of the pulse, mass of 16O is gradu- outward going core matter and the infalling envelope. ally consumed and produce 28Si. At late pulses, where Beyond Profile 6, the discontinuity reaches the surface the core reaches beyond 107 g cm−3, NSE elements are and creates a shock breakout. The surface matter can also produced. In the last pulse, the core is sufficiently freely escape from the star. compressed such that an iron core beyond 1.4 M⊙ is pro- For the chemical composition, the effects of the pulse duced, which is accompanied with the later mass loss. becomes clear. Since the second pulse, part of the core Most of the ejected mass is He. 16O is already consumed in the first pulse, which is con- For Model He45A, again most of the pulses are weak. verted to 28Si already. During the compression, be- With the number of pulses increased, not only Si, but fore the core reaches its maximum temperature, 16O also 56Ni are produced. The last pulse, which is the 28 is significantly consumed and forming Si. When the strongest overall, produces about 0.56 M⊙ Ni, while the core reaches the peak temperature, the oxygen within generated heat creates a shock to eject about 6 M⊙ mat- q ≈ 0.06 is completely burnt, where intermediate mass ter before the final collapse. elements, such as 28Si, is produced. However, iron-peak For Model 50HeA, the number of pulses becomes elements, such as 56Ni are not yet produced. On the smaller and again only the last pulse is a strong pulse other hand, during the expansion phase, most O-burning which can eject mass. Compared to previous models, in ceased, making the 16O and 28Si unchanged after the each pulse more 16O is consumed, which produces Si. central temperature reaches 109.4, while advanced burn- At the final strong pulse, less Ni is produced, while the ing still proceeds slowly to form iron-peak elements. accompanying mass loss eject the He in the envelope. It should note that its lower mass ejection compared to 3.3. Global Properties of Pulse Model He45A comes from the difference that the O in 8

Table 3. The masses and chemical compositions of the models. ”bounce” means the number of pulse in the chronological order, where ”E” stands for the model at the end of the simulation. Msum is the current mass in units of solar mass. Enuc is the 51 amount of energy released by nuclear reaction in units of 10 erg. MHe, MC, MO, MMg, MSi, MIME and MNSE are the masses of He, C, O, Mg, Si, intermediate mass elements and and elements of nuclear statistical equilibrium in the star. For weak pulse, the moment is deﬁned by the minimum temperature reached between pulses. For strong pulse, the composition is determined when the core cools down to a central temperature of 109.3 K.

Model bounce Msum MHe MC MO MMg MSi MIME MNi MNSE remark 40HeA 1 40.00 6.77 2.65 26.70 0.69 0.96 3.89 0.00 0.00 Weak 40HeA 2 40.00 6.65 2.34 24.70 0.81 1.99 6.30 0.00 0.01 Weak 40HeA 3 40.00 6.59 2.14 23.16 0.89 2.95 7.70 0.11 0.40 Weak 40HeA 4 40.00 6.57 2.07 22.52 0.91 3.07 7.83 0.24 1.01 Weak 40HeA 5 40.00 6.54 1.94 21.77 0.92 3.13 7.79 0.24 1.91 Weak 40HeA 6 40.00 6.51 1.85 20.06 0.92 3.43 7.78 1.69 3.76 Strong 40HeA E 37.78 4.68 1.79 20.06 0.91 3.43 7.77 0.07 3.42 Final 45HeA 1 45.00 7.19 3.00 30.87 0.83 0.62 3.94 0.00 0.00 Weak 45HeA 2 45.00 7.00 2.48 28.05 1.13 2.41 7.47 0.00 0.01 Weak 45HeA 3 45.00 6.92 2.32 26.60 1.16 3.43 9.04 0.00 0.11 Weak 45HeA 4 45.00 6.91 2.20 25.36 1.19 4.11 9.78 0.56 0.75 Strong 45HeA E 39.26 1.74 1.95 25.32 1.17 3.43 8.44 0.06 1.80 Final 50HeA 1 50.00 7.59 2.95 34.61 1.10 0.84 4.85 0.00 0.00 Weak 50HeA 2 50.00 7.38 2.41 31.16 1.37 3.13 9.03 0.02 0.02 Weak 50HeA 3 50.00 7.29 2.15 28.38 1.35 5.06 11.62 0.47 0.57 Strong 50HeA E 47.39 5.21 2.17 28.38 1.33 4.08 9.80 0.09 1.73 Final 55HeA 1 55.00 7.96 2.83 38.00 1.47 1.27 6.20 0.00 0.00 Weak 55HeA 2 55.00 7.87 2.42 35.06 1.62 3.46 9.63 0.02 0.03 Strong 55HeA 3 53.55 6.35 1.92 31.70 1.72 4.64 11.17 1.99 2.40 Strong 55HeA E 48.22 1.75 1.59 31.66 1.53 4.50 10.72 0.01 2.49 Final 60HeA 1 60.00 8.43 2.75 41.71 1.72 1.67 7.11 0.00 0.00 Strong 60HeA 2 59.52 7.91 2.22 36.78 1.64 5.42 12.46 0.13 0.15 Strong 60HeA E 51.48 0.75 1.92 36.75 1.61 4.21 10.32 0.09 1.64 Final 62HeA 1 62.00 8.52 2.44 41.91 1.85 3.11 9.13 0.00 0.00 Strong 62HeA 2 58.34 4.85 1.75 37.17 1.84 5.63 12.88 1.49 1.68 Strong 62HeA E 49.15 0.07 0.09 34.52 1.60 4.88 10.96 0.04 2.66 Final 64HeA 1 64.00 8.69 2.39 42.87 1.90 3.63 10.05 0.00 0.00 Strong 64HeA E 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Final

Model 45HeA is burnt in a much compressed state. This For Model 62HeA, it has a similar pulse pattern as creates a much stronger shock wave when the expansion Model 60HeA but is stronger. Each pulse can consume approaches the surface, which extends the mass loss. about 3 − 4M⊙ of O. Different from previous models, For Model 55HeA, it starts to have two strong shocks Model 62HeA has an abundant amount of O even during compared to lower mass models. Its pulses are qualita- its contraction towards collapse, and O continues to be tively similar to Model 50HeA. consumed before it collapses. For Model 60HeA, it no longer has weak pulses. The For Model 64HeA, which is a pair-instability super- contraction always makes a significant mass of O to be nova instead of PPISN, there is only one pulse before its burnt to produce the thermal pressure to support the total destruction. Due to its much lower density when softened core against its contraction. Due to the strong large-scale O-burning occurs, even when about a few so- mass ejection, at the end of the simulation the star al- lar mass of O is burnt during the pulse, the energy is most runs out of He. However, one difference of this sufficient to eject all mass when the pulse reaches the model from others is that it has a much lower Ni mass surface. after pulsation. Most of the Fe, which leads to the col- 3.4. Thermodynamics lapse, is created during the contraction towards collapse. 9

Fe photo-disintegration 10 Fe photo-disintegration 10 (K) (K) c c

T T 9.5

10 9.5 10 PPI instability PPI instability log log

9 E-cap / GR e-cap / GR 9 5 10 5 10 ρ -3 ρ -3 log10 c (g cm ) log10 c (g cm ) Fe photo-disintegration Fe photo-disintegration 10 10

(K) (K) 9.5 c 9.5 c

T PPI instability T PPI instability 10 10 log log

9 9

E-cap / GR e-cap / GR

5 10 5 10 ρ -3 ρ -3 log10 c (g cm ) log10 c (g cm ) Figure 5. (cont′d) (left panel) The central temperature Figure 6. (cont′d) (left panel) Similar to the upper left and density against time for Model 40HeA. The zones for panel, but for Model 50HeA. (right panel) Similar to the PPI, photo-disintegration of 56Ni into 4He, and the dynami- upper left panel, but for Model 55HeA. cal instability due to general relativistic effects and electron capture are included for comparison. (right panel) Similar scale O-burning in the outer core, which leads to a dras- to the upper left panel, but for Model 45HeA. In each plot, the PPI instability zone is enclosed by the red line and the tic drop in the central density and temperature, showing 9.2 Fe photo-disintegration zone is enclosed by the blue line (see that the star is expanding, until the Tc reaches 10 K. online version for the colour figure) Then the core resumes its contraction. Since most O in the core is burnt, there is no extra energy input during In Figures 5, 6 and 7 we plot the central density and the contraction. The star directly collapses. temperature against time for Models 40HeA ,45HeA, For Model 45HeA, it shows a fewer number of pulses 50HeA, 55HeA, 60HeA and 62HeA in the six panels re- than Model 40HeA. It has three small pulses and one big spectively. To show the rapid contraction comes from pulse. The initial path is closer to the PPI instability 9.7 the PPI, we show in each plot the zones where electron- zone. The last pulse is triggered at Tc = 10 K and 8.9 positron pair creation, the dynamical instability induced has a lowest Tc of 10 K when it is fully expanded. by photo-disintegration of 56Ni into 4He and the dy- For Model 50HeA, it has only two small pulses and one namical instability induced by general relativistic effects big pulse. The evolution shows less structure compared and/or rapid electron capture. to the previous two models because of the earlier trigger For Model 40HeA, at the beginning the central den- of large-scale O-burning in the core. The core starts the 9.6 sity is the highest among the all six models. It has big pulse when Tc = 10 K and its expansion makes Tc 8.8 thus weaker pulses because the core is more compact reaching 10 K at minimum. Before its collapse, there and degenerate. It has five small pulses and one big is a small wiggle along its trajectory. We notice that at pulse (indicated by arrows in the figure) where each of this phase the core has a small pulsation when the core the small pulses only leads to a small drop of the central becomes degenerate. The pulse is similar to the previous density and temperature. Then the core quickly resumes pulse, but it is not contributed by nuclear reaction. its contraction again. Only at the final pulse, when the For Model 55HeA, it has one small pulse and two big 9.5 core begins to reach the Fe photo-disintegration zone, pulses. The two big pulses start when Tc reaches 10 9.7 the softened core leads to a fast contraction and reaches and 10 K respectively, with a minimum temperature 8.8 8.6 9.8 after relaxation at 10 and 10 K. a central temperature Tc = 10 K. This triggers a large 10

photo-disintegration 10 closer to the PPI instability as mass increases. Fourth, the big pulse strength increases with time. Energetic 9.5 3.5. PPI instability (K) c In Figures 8 and 9 we plot the energy evolution T 10 for Models 40HeA, 45HeA, 50HeA, 55HeA, 60HeA and log 9 62HeA, including the total energy Etotal, internal energy Eint, gravitational energy Egrav and kinetic energy Ekin.

8.5 e-cap / GR The energy is scaled in order to make the comparison easier. 5 10 ρ -3 In all six models, it can be seen that the energetic log10 c (g cm ) evolution does not depend on the stellar mass strongly, photo-disintegration 10 except for the energy scale. In all these models, the small pulses do not make observable changes in the energy except for very small wiggles. The contraction before a 9.5 PPI instability

(K) pulse leads to a denser and hotter core, where neutrino c T

10 emission continuously draw energy from the system. At

log 9 a big pulse, the total energy shows a rapid jump which increases close to zero, then the ejection of mass quickly removes the generated energy, making the star bounded 8.5 e-cap / GR again. Similar jumps in Eint and Egrav shows that the 5 10 core is strongly heated due to contraction heating and -3 log ρ (g cm ) 10 c nuclear reactions. After that, the star reaches a quies- Figure 7. (cont′d) (left panel) Similar to the upper left cent state with very mild increases of total energy due 56 panel, but for Model 60HeA. (right panel) Similar to the to the Ni decay, then followed by a quick drop when upper left panel, but for Model 62HeA. In each plot, the it contracts again. PPI instability zone is enclosed by the red line and the Fe photo-disintegration zone is enclosed by the blue line (see 3.6. Luminosity online version for the colour figure) In Figures 10 and 11 we plot the luminosity evolution for the six models similar to Figure 8. During the pulse, For Model 60HeA, there is no small pulse and two big the extra energy from nuclear reactions allows the lumi- pulses, where the stellar core intersects with the PPI nosity to grow by 3-4 orders of magnitude. For a short instability zone during its expansion. The two pulses period of time (6 10−2 year), then the star becomes 9.4 9.6 starts when Tc reaches 10 and 10 K. The core fin- dim suddenly. After that the star resumes its original ishes its expansion when it reaches 109.0 and 108.3 K. luminosity quickly and remains unchanged until the next For Model 62HeA, the star model becomes very close pulse or final collapse. to the PPI instability where the core enters the zone for The neutrino luminosity is more sensitive to the struc- a short period of time during its expansion. It is similar ture of the star. The neutrino luminosity can also jumps to Model He60A that there are two big pulses. The two by 3 - 10 orders of magnitude from its typical luminosity peaks start at 109.5 and 109.7 K while both pulses end in the hydrostatic phase to the maximally compressed at a minimum temperature of 108.4 K, showing that the state. After the star has relaxed, the neutrino luminos- two pulses are of similar strength. After that, the core ity drops drastically. Depending on the strength of the starts collapsing similar to all other five models. pulse, neutrino cooling can become unimportant in the By comparing all six models, we can observe the fol- quiescent phase. lowing trend for the pulse structure as a function of pro- 3.7. Mass Loss History genitor mass. First, when the progenitor mass increases, the number of small pulses decreases while the number During the pulsation, when the bounce can create an of big pulses increases. Second, the strength of the big large-scale burning in the core and inner envelope, suffi- pulses increase with the progenitor mass, which leads to cient energy is produced that it can create an outgoing a lower central temperature and density during its ex- shock, which ejects the outermost matter away. Such pansion. Third, the path during its early pulses becomes ejected matter later cools down and becomes the circumstellar medium (CSM). Such CSM will be important in the scenario when the later explosion of the star 11

51 E / 10 erg 51 total 53 Etotal / 10 erg 52 E / 10 erg E / 10 erg total 52 int 53 Eint / 10 erg 5 52 E / 10 erg -E / 10 erg int 5 52 grav 52 -Egrav / 10 erg 50 -E / 10 erg E / 10 erg 5 grav 51 kin 53 Ekin / 10 erg Ekin / 10 erg 0 energy energy 0 energy

-5 0 -5

0.1 0.2 0.3 0.4 0.01 0.1 1 10 100 0.01 0.1 1 10 100 time (year) time (s) time (year)

Figure 8. Total Etotal, internal Eint, net gravitational —Egrav— and kinetic Ekin energies against time for Model 40HeA (left panel), Model 45HeA (middle panel) and Model 50HeA (right left) respectively.

10 51 E / 10 erg 10 51 total E / 10 erg 52 total E / 10 erg 52 int E / 10 erg 52 int E / 10 erg 0 52 5 grav -E / 10 erg 51 5 grav 51 Ekin / 10 erg Ekin / 10 erg

51 energy energy energy Etotal / 10 erg 0 53 0 Eint / 10 erg -5 53 -Egrav / 10 erg 51 E / 10 erg -5 -5 kin

0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 time (year) time (year) time (year)

′ Figure 9. (cont d) Total Etotal, internal Eint, net gravitational —Egrav— and kinetic Ekin energies against time for Model 55HeA (left panel), Model 60HeA (middle panel), and Model 62HeA (right panel) respectively.

15 15 )

-1 L ) ) L sun 10 sun 10 ν 10 (L (L (erg s ν ν ν L L L 10 10 10 L, log L, log

5 L, log 10 10

L 10 5

log L log ν log L Lν 0 0 0 0.1 0.2 0.3 0.4 0.01 0.1 1 10 100 0.01 0.1 1 10 100 time (year) time (s) time (year)

Figure 10. Total luminosity and neutrino luminosity against time for Model 40HeA (upper left panel), Model 45HeA (upper middle), Model 50HeA (upper right).

15 15 15 L Lν

) L L -1 Lν Lν 10 10 10 ) ) (erg s ν sun sun L (L (L 10 ν ν L, L L, L L, log

10 5 5 5 log

0 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 10000 time (year) time (year) time (year)

Figure 11. Similar to Figure 10, but for Model 55HeA (lower left) Model 60HeA (lower middle), and Model 62HeA (lower right). 12

Table 4. Energetic and chemical composition of the ejecta. ’Pulse’ stands for the sequence of pulses in its evolution. ’Time’ is the occurrence time in units of year. Tej is temperature range of the ejecta in units of K. Eej is the ejecta energy in units of 1050 erg. M(He), M(C), M(O), M(Ne), M(Mg), M(Si) are the masses of He, C, O, Ne, Mg and Si in the ejecta in units of solar mass.

Model Pulse time Mej Eej Tej M(He) M(C) M(O) M(Ne) M(Mg) M(Si) 40HeA 6 9.9 × 10−2 1.0 1.0 6.3-6.8 1.0 0.0 0.0 0.0 0.0 0.0 45HeA 1 7.4 × 10−2 4.0 6.6 6.5-6.9 3.8 0.2 0.0 0.0 0.0 0.0 50HeA 2 2.0 × 10−1 4.0 2.5 6.7-7.2 3.9 0.1 0.0 0.0 0.0 0.0 55HeA 1 6.2 × 10−2 0.3 1.8 6.8-7.1 0.3 0.0 0.0 0.0 0.0 0.0 55HeA 2 1.9 10.0 13.1 6.0-6.7 7.5 1.0 0.8 0.2 0.3 0.2 60HeA 1 1.7 × 10−1 10.6 5.1 5.3-6.4 8.6 2.4 0.8 0.6 0.2 0.2 60HeA 2 7.4 × 103 38.7 59.0 6.0-6.5 0.0 1.1 32.5 1.3 1.3 2.0 62HeA 1 5.5 × 10−2 0.6 0.1 6.8-7.4 0.6 0.0 0.0 0.0 0.0 0.0 62HeA 1 3.5 × 103 55.4 29.6 4.6-7.2 7.8 1.7 33.2 1.6 1.8 5.8 64HeA 1 4.9 × 10−2 21.8 29.4 4.4-7.1 8.5 1.8 9.9 0.9 0.3 0.6 ) ) × -4 ) × -4 × -4 -3 -3 1 10 1 10 -3 1 10 -6 -6 -6 1×10 1×10 1×10 (g cm (g cm (g cm

ρ -8 ρ -8 ρ -8 1×10 1×10 1×10

10 10 10 ) ) 1×10 ) 1×10 1×10 -1 -1 9 -1 9 9 1×10 1×10 1×10 8 8 8 (cm s (cm s 1×10 (cm s 1×10 1×10 esc esc 7 esc 7 7 1×10 1×10 1×10

v, v 6 v, v 6 v, v 6 1×10 1×10 1×10 1 1 1 4 4 4 He 12 He 12 He 0.1 C 0.1 C 0.1 16 16 0.01 O 0.01 O 0.01 mass fraction mass fraction mass fraction 0 0.0002 0.0004 0 0.0001 0.0002 0.0003 0.0004 0 0.0001 time (year) - 0.3275 year time (year) - 0.0757 year time (year) - 0.3275 year

Figure 12. (left panel) The mass ejection history of Model 40HeA, including in the upper panel the ejecta density, in the middle panel the ejecta velocity and the escape velocity and in the lower panel the ejecta chemical composition. (middle panel) Similar to the upper left panel, but for Model 45HeA. (right panel) Similar to the upper left panel, but for Model 50HeA. ) ) -3 ) -3 -3

-3 × -3 1×10 -3 1×10 1 10

-4 -4 × -4 × × (g cm (g cm 1 10 (g cm 1 10 1 10 ρ ρ ρ

10 10 10 ) × ) × ) × 1 10

1 10 1 10 -1 -1 9 -1 9 9 1×10 1×10 1×10 8

8 8 (cm s × (cm s 1×10 (cm s 1×10 1 10 esc esc 7 esc 7 × 7 1×10 1×10 1 10 v, v v, v 6 v, v 6 × 6 1×10 1×10 1 10 1 1 1 4 4 4 He He He 12 0.1 0.1 C 0.1

0.01 0.01 16 0.01 mass fraction

mass fraction mass fraction O -5 0 0.0001 0 0.0002 0 5×10 time (year) - 0.04182 year time (year) - 75.9362 year time (year) - 0.03569 year

Figure 13. (cont′d) (left panel) The mass ejection history of Model 55HeA during the ﬁrst major mass ejection. (middle panel) Similar to the left panel, but for Model 55HeA during the second major mass ejection. (right panel) Similar to the left panel, but for Model 60HeA during the ﬁrst major mass ejection. 13 ) -3 ) -3 ) -3 -3 1×10 -3 1×10 -3 1×10

× -4 × -4 × -4 (g cm (g cm (g cm 1 10 1 10 1 10 ρ ρ ρ

10 10 10 ) 1×10 ) 1×10 ) 1×10 -1 9 -1 9 -1 9 1×10 1×10 1×10 8 8 8 (cm s 1×10 (cm s 1×10 (cm s 1×10

esc 7 esc 7 esc 7 1×10 1×10 1×10

v, v 6 v, v 6 v, v 6 1×10 1×10 1×10 1 1 1 4 4 4 16 He He He O 12 0.1 0.1 0.1 C 28 Si 0.01 0.01 0.01 24

mass fraction mass fraction mass fraction Mg -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 0 1×10 2×10 3×10 4×10 0 1×10 2×10 0 1×10 2×10 3×10 4×10 time (year) - 0.7078 year time (year) - 0.05984 year time (year) - 2473.2795 year

Figure 14. (cont′d) (left panel) The mass ejection history of Model 60HeA during the second major mass ejection. (middle panel) Similar to the left panel, but for Model 62HeA during the first major mass ejection. (right panel) Similar to the left panel, but for Model 62HeA during the second major mass ejection. creates the second shockwave, which interacts with the profile, the later ejected material contains a significant CSM. The chemical and hydrodynamics properties of amount of heavier elements including C, O, Mg and Si, the CSM thus becomes important, which influence the showing that the He envelope is completely exhausted formation of the light curve of the explosion. before the star is sufficiently relaxed. In Figures 12, 13 and 14 we plot the ejecta profiles of Models 40HeA, 45HeA, 50HeA, 55HeA, 60HeA and 3.8. Chemical Properties 62HeA respectively. Three patterns can be observed in In Figures 15, 16 and 17 we plot the isotope profiles the mass ejection. at different moments of Models He40A, He45A, He50A, The first group is the strong pulse in the lower mass He55A, He60A and He62A. We selected moments be- branch. In Models 40HeA and 45HeA the last pulse fore and after each strong pulses to extract the nuclear is the pulse which ejects mass. It shows wiggles in its burning history. By comparing the isotope distribution, density profiles, showing that the thermal expansion cre- we can understand which part of burning contributes to ates the first wave of mass ejection, while the following the evolution of pulsation. shock as the velocity discontinuity approaching the sur- In all models, it can be seen that the star is simply a face creates the second wave of mass ejection. In both pure O-star with a minute amount of Si in the core or C cases, only the He layer is affected, but as the He layer in the envelope, covered by a pure He surface. However, becomes thin the interface near the CO layer is also their changes can be very different depending on the ejected. progenitor mass. The second group is the weaker pulse of the more In Model 40HeA, after the strong pulse, due to its pre- massive branch. In Models He50A, He55A He60A and vious weak pulses which continue to burn matter in the He62A the first strong pulse occurs after the core starts core, a range of elements are produced including 52Fe, to consume O collectively. Since it burns much less O 54Fe, 56Fe and also 56Ni. There is a clear structure for than other strong pulses, the ejection comes from the each layer, which comes in the order of 56Fe, 54Fe, 56Ni, rapid expansion of the star, which includes matter in 40Ca, 16O and then 4He. After that, the core relaxes the He envelope. and becomes quiescent until it completely loses its ther- The third group is the strong pulse of the more mas- mal energy produced during the pulse, while at the same sive branch. In Models He55A and Model 60HeA, the time convection re-distributes the matter for a flat com- second pulse is stronger so that the ejecta density grad- position profile. It can be seen that most convection ually decreases. A continuous ejection of mass in terms occurs at q > 0.1, where the convective shells of differ- of smooth density profile is found. The mass ejection is ent sizes make a staircase like structure. 12 sufficient deep that at the end of pulsation, trace of C Models 45HeA and He50A (upper middle and upper 16 and O can be found. We remark that the inclusion right panels) share a similar nuclear reaction pattern. of massive elements (compared with H and He) will be The strong pulse provides the required temperature and important for the future light curve modelling because density to make Ni in the center and Si in the outer they contribute as the main source of opacity. zone. The Si-rich zone extends to q ≈ 0.2. During the One of the pulsation needs to be discussed seperately quiescent phase, convection not only mixes the material because of its very massive mass ejection, which in- in the envelope, but also in the core, which is seen by volves very unique chemical composition in its ejecta. the stepwise distribution of 52Fe and 54Fe. In Model 62HeA (lower right plot), the second pulse be- In Models 55HeA, He60A and He62A (see the mid- comes strong enough that, besides its decreasing density dle left, middle right and lower middle panels), the first 14

1 1 1 16 16 16 12 O O C O 0.1 0.1 20 0.1 20 Ne Ne 28 20 28 28 40 Si 12 Ne Si Si 12 0.01 Ca C 0.01 0.01 C 1 56 1 1 Ni 56 56 0.1 Fe 0.1 0.1 Ni

52 40 0.01 56 0.01 40 0.01 54 mass fraction mass fraction Ni mass fraction Ca Fe Fe Ca 1 1 1 56 0.1 54 0.1 0.1 Fe Fe 54 56 0.01 0.01 Fe Fe 0.01 52 52 Fe Fe 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 m(r)/M m(r)/M m(r)/M

Figure 15. (upper left) The chemical composition of Model 40HeA before the first pulse, after the first pulse and before the final pulse in the upper middle and lower plot. Here we define the star entering the pulsation phase when the core reaches 109.3 K. (upper middle) Similar to the upper left panel, but for Model 45HeA before the first pulse, after the first pulse and before the final pulse in the upper middle and lower plot. (upper right) Similar to the upper left panel, but for Model 50HeA before the first pulse, after the first pulse and before the final pulse in the upper middle and lower plot.

1 1 1 16 16 16 56 20 O Ni Ne O O 0.1 20 0.1 54 0.1 20 Ne Fe Ne 28 28 12 28 Si Si C Si 12 0.01 12 0.01 52 40 0.01 C C Fe Ca 1 1 56 1 Fe 0.1 0.1 0.1

56 40 0.01 Ni 40 0.01 0.01 Ca mass fraction mass fraction Ca mass fraction 1 1 1

0.1 0.1 0.1

56 4 0.01 Fe 0.01 He 0.01

0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 m(r)/M m(r)/M m(r)/M

Figure 16. (cont′d) (left panel) The chemical composition of Model 55HeA before the first pulse, after the first pulse and before the second pulse in the upper middle and lower plot. (middle panel) The chemical composition of Model 55HeA after the second pulse, before the final pulse and at the end of simulation in the upper middle and lower plot. (right panel) The chemical composition of Model 60HeA before the first pulse, after the first pulse and before the second pulse in the upper middle and lower plot.

1 1 1 16 16 20 16 56 56 28 O 12 O Ni O 20 Si Ne C Ne 0.1 Ni 0.1 20 0.1 Ne 28 28 Si 12 Si 12 C 0.01 40 0.01 C 0.01 52 40 Ca Fe Ca 1 1 1 56 0.1 0.1 0.1 Fe 40 52 0.01 0.01 0.01 Fe mass fraction Ca mass fraction mass fraction 1 1 1 56 Fe 0.1 0.1 0.1 4 He 0.01 0.01 54 0.01 Fe 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 m(r)/M m(r)/M Figure 17. (cont′d) (left panel) Similar to the upper left panel, but for Model 60HeA after the second pulse, before the final pulse and at the end of simulation in the upper middle and lower plot. (middle panel) Similar to the upper left panel, but for Model 62HeA before the first pulse, after the first pulse and before the second pulse in the upper middle and lower plot. (right panel) Similar to the upper left panel, but for Model 62HeA after the second pulse, before the final pulse and at the end of simulation in the upper middle and lower plot. pulse makes the core, which is mostly O, to be Si and quiescent phase, the convection occurs in a deeper layer some Ca. Again, the convective mixing during the qui- compared to the lower mass models. escent state redistributes the matter near the surface. In the second strong pulse, the nuclear reaction is very similar to the late pulses of Model He40A and He45A. 4. DISCUSSION Ni forms in the innermost part, with a small amount of 4.1. Comparison with Models in the Literature 52 54 Fe isotopes like Fe and Fe. Then it is the Si and Ca In this section we compare our results with some rep- middle layer and at last the He envelope. During the resentative PPISN models in the literature. 15

In Yoshida et al. (2016), the PPISN model of mass Models He40, He50, He60 and He62 respectively. In from about 54 M⊙ to 60 M⊙ (corresponding to a pro- our models, we have 6, 3, 2 and 2 pulses with a total genitor mass from 140M⊙ to 250 M⊙ in zero metallicity) mass loss of 2.22, 2.61, 9.52 and 12.85 M⊙ for Models are computed. In that work, the calculation is separated He40A, He50A, He60A and He62A respectively. Again, into two parts. During the quiescent and pre-pulsation our code tends to produce fewer pulses and the pulses in phases, the hydrostatic stellar evolution code is used. general eject fewer matter. The large discrepency can be During the pulsation phase, the star model is transferred related to the nature of the instability of PPISN. Since to the dynamical code PPM, which follows the expan- the trigger of the explosive O-burning comes from the sion of the star until the mass ejection has ended (∼ 104 pair-instability, which is very sensitive to the initial con- s). Then they map the results to the stellar evolution dition (e.g. how we evolve the stellar evolution model code again until the next pulsation. before the pulsation and between pulses) and numerical Their 140 M⊙ and 250 M⊙ have similar configura- treatment (e.g. how convection and mass ejection are tions as our Models 55HeA and 60HeA. First, in their treated). For example, a stronger contraction can lead 140 M⊙ model (250 M⊙ model), they observe a total to more O-burning in the core, which gives much ener- of six (three) pulses which ejected 3.99 (7.87) M⊙ of getic pulsation and hence more mass loss. In fact, such matter before collapse. The Models 55HeA (60HeA) ex- dependence can also be seen in other field. For example, hibit three (two) pulses before collapse, which ejects 6.78 in the propagation of flame, since it is unstable towards (8.52) M⊙ of matter. Our models show a smaller num- hydrodynamics instability, in Glazyrin et al. (2013) the ber of pulses, but give similar ejecta mass. This means burning history can be highly irregular in the unstable our models can capture the energetic pulse well, but not regime. the smaller pulses. Next we compare the timescale of the pulsation. In Then we compare the ejection timescales. The 140 this work, the whole pulsation until collapse last for 0.38, M⊙ (250 M⊙) model show all pulses within a period 61.3, 2806 and 6610 years for the four models, while in of 0.92 (1434) years, while the Models 55HeA (60HeA) Woosley (2017) they are 2.48 × 10−3, 0.38, 2695 and show all pulses within a period of 1341 (2806) years. 6976 years. It shows that for massive He cores, our re- There is a huge difference in the pulsation period in our sults agree with their work but there are large differences Model 55HeA and their 140 M⊙ model. We notice that when the He core becomes less massive. In that case, the difference comes from the strengths of the pulses. In our final pulse is always strong enough to re-expand the particular, our second pulse leads to a transition about star again before the final collapse, which significantly 100 years while ejecting 1.45M⊙. The most similar event lengthens the pulsation period. in their model is the fourth pulse, but with a transition Then we compare the Fe core mass. In Woosley of only 0.279 year. (2017) the core has 2.92, 2.76, 1.85 and 3.19 M⊙ Fe. At last we compare the final core composition. The In our models, we have 3.42, 1.73, 1.64 and 2.66 M⊙. 140 M⊙ (250 M⊙) model has an iron (CO) core mass We can see there is a dropping trend from Model at 2.57 (43.51) M⊙, while in our model, we have 2.49 He40A to He60A, which corresponds to the trend that (38.60) M⊙ for the Fe (CO) mass. This shows that, the pair-instability occurs at lower density when the despite the difference in the mass ejection history, our mass increases. On the other hand, near the pair- models can still capture the major mass ejection events, instability regime, the pulsation becomes sufficiently which results in a similar mass ejection and core com- vigorous which enhances the NSE-burning. position. However, there is a strong pulse in our He55A At last we compare the explosion energy. We com- model, which is not seen in 140 M⊙ their model. pare the Model He62A, which has the largest explosion Next, we compare our models with the models from energy. In our model, in the second big pulse, the star Woosley (2017). We have chosen the PPISN close to has its total energy increased by 2.0 × 1051 erg while the that work; in particular, ours Models He40A, He50A, maximum kinetic energy achieved is 2.8×1051 erg. This He60A and He62A can be compared directly with the is very similar to the result in Woosley (2017), where the He40, He50, He60 and He62 models. In Woosley (2017), pulse is observed to have a kinetic energy of 2.8 × 1051 the Kepler code, which consists of both hydrostatic and erg. hydrodynamics components, is used to follow the whole One major difference we notice is in the pair- evolution of PPISN. instability limit, for the Model He64A, our model shows First we compare the mass ejection history. In a higher explosion energy. Across the strongest pulse, Woosley (2017), there are 9, 6, 3 and 7 pulses with there is a change of total energy by 1.6×1052 erg, where 52 a total mass loss of 0.97, 6.31, 12.02 and 27.82 M⊙ for the maximum kinetic energy of the system is ∼ 1.7×10 16 erg. In Woosley (2017) the kinetic energy is reported to The gravitational wave detectors LIGO and Virgo has be 4 × 1051 erg. We observe that the difference comes recently detected gravitational wave signals from black from the number of pulsation, where our Model 64HeA hole-black hole merger and neutron star-neutron star has two big pulses but only one in their work. The first merger. Some massive black holes, for example in GW pulse has incinerated the 16O in the core while ejecting 150914, the black hole binary of masses 35.4 and 29.8 some surface matter. This means the star has to reach M⊙ are measured Abbott et al. (2016b). Another mas- a more compact state before the star can explode. As a sive black hole merger even is GW 170104, where the result, the amount of energy produced in the exploding binary consists of black holes of mass 31.4 and 19.4 M⊙ pulse is much larger. respectively. It has been a matter of debate, whether Our results show a systematically lower number of the massive black hole forms directly from the collapse of pulses with slightly lower ejecta mass. The pulsation massive star, or has experienced many black hole merger periods qualitatively agree with each other except for prior to the black hole-black hole merger the gravita- models with a final strong pulse, which may significantly tional wave detectors observed. lengthens our pulsation period. Also, in our explosion From our model, it becomes clear that the single models, the system tends to store the energy in terms star scenario has a maximum black hole mass to be of internal energy instead of kinetic energy, as a result, formed from the single star scenario. For the He core the star tends to expand globally, where the excess en- with a mass more massive than 64 Modot, the star ergy and momentum of the star is transferred mostly does not experience any collapse, but explode as a pair- to the surface. This ejects the low density matter and instability supernova. The collapse only appears in He leaves a bounded and hot massive remnant. Despite the core with a mass larger than 260 M⊙ (for zero metallic- differences in the pulsation, globally the nucleosynthe- ity) (Heger & Woosley 2002). The corresponding black sis agrees with each other because most of the heavy hole mass will be about 100 M⊙. elements are produced by the strong pulses, where our To connect PPISN with the measured black hole mass results are consistent with those in the literature. spectra, we plot in Fig. 18 the remnant mass against progenitor mass, together with the mass range of the 4.2. Connection to Super-luminous Supernovae black hole implied by the gravitational wave signals. PPISN has been frequently used to illustrate the The remnant mass is approximated as the He core mass physical origin of extremely luminous supernovae such as reported in Table 1. But for He core mass between as SN2006gy (Woosley et al. 2007) and PTD12dam 40 - 64 M⊙, a mass correction is included to account for (Tolstov et al. 2017). The PTD12dam gives a more the pulsation-induced mass loss. challenging model since there is no model so far demon- We remark that the use of He core mass as the rem- strating the explosion history required. In particular, nant mass provides not the exact black hole mass, but an upper limit of the black hole mass. This is because it requires a 20-40 M⊙ CSM prior to the explosion of the star. Furthermore, it needs a composition with the before all matter is accreted into black hole after the col- presence of He, C and O in order to give a high opacity lapse, two explosions are possible to occur. First, during of surrounding matter to sustain the light curves. the formation of neutron star, the bounce shock when By comparing with our models, it can be seen that the core matter reach nucleon density, can eject part of the matter. But whether the bounce shock can success- ≈ 64 M⊙ He core, the first pulse is strongly enough to fully reach the surface remains a matter of debate due produce an ejecta of mass ∼ 22M⊙. Our model gives to the complication of neutrino. When the shock can in- an ejecta with He, C and O masses of 8.5, 1.8, 9.9 M⊙. The corresponding ratio of C:O is therefore 1:5.5. This is deed propagate to the surface, it can efficiently eject the close to the optimal value in their models of C:O = 1:5. low density matter, mostly He envelope, away. This may This has further confirmed the necessity of applying the significantly lower the final mass of the black hole. The pulsation pair-instability supernovae as the model for second explosion is the accretion disk jet. Depending on PTD12dam. Whether the following collapse of the ≈ the accretion scenario, it is possible that the accretion disk forms around the black hole. The magnetodynam- 40M⊙ remnant can explode energetically with an energy of 2 − 3 × 1052 erg is uncertain. This may require multi- ics instability of the accretion disk can easily fragment dimensional hydrodynamics simulations with a realistic the disk and send high-speed jet to the stellar envelope. neutrino transport in order to model the collapse phase, This may also easily send away partially the infalling which is beyond the scope of this article. envelope, again lowing the mass. However, for both explosions further input physics are necessary. Therefore, 4.3. Connection to Massive Black Hole 80 -3 Z = 10 Z sun ply the black hole information in PPISN to constrain the -2 70 Z = 10 Z sun mass loss, population of black hole mass will become im- Z = 0.1 Zsun Z = 0.5 Z 60 sun portant, which can directly constrain the current mass Z = 0.75 Zsun Z = Z 50 sun loss model, when combined with suitable stellar initial Primary BH, GW150914 mass function. 40

30 4.4. Conclusion Primary BH, GW170104 remnant mass (solar mass) 20 In this article, we present our study of pulsational 10 pair-instability supernova (PPISN) for the helium core 0 50 100 150 from 40 - 64 M⊙ using the one-dimensional stellar evolu- progenitor mass (solar mass) tion code MESA. We applied the implicit hydrodynam- Figure 18. The remnant mass against progenitor mass for ics module implemented in the version 8118. We fol- models at different metallicity. The remnant mass is ap- low the evolution of the He core throughout the main- proximated by the He core mass but with the mass correc- sequence phase, pulsation until the iron-core collapse. tion from the pulsation. The observed black hole masses are We have investigated the energetic, thermodynamics included for comparison Abbott et al. (2016a, 2017). The and mass loss history of the pulsation. We have also upper (lower) curves of the same symbol correspond to the final remnant with (without) the dynamical mass ejection. studied the evolution of chemical abundance in the star during the pulsation. We show that our results are qualitatively consistent with the results in the litera- for the first estimation of our result, we use the He core ture, although some minor differences can be found. We mass as a optimistic estimate of the black hole mass. also discuss the possible connections of PPISN, espe- We can see that in the low mass regime, the remnant cially the ones with massive mass ejection, with the re- mass is not sensitive to the metallicity. The remnant cently observed super-luminous supernovae SN 2006gy mass monotonically increases with the progenitor mass. and PTF12dam. Then we also discuss the possible con- Deviations appear for higher mass. For stellar model nection between the massive black holes detected from with low metallicity (Z ≤ 0.5Z⊙), due to their lower the gravitational wave signals from black hole merger mass rate in H- and He- hydrostatic burning phase, its and PPISN. He core can easily reach beyond 64 M⊙. As a result, they In the next work, we will focus on the observables of end up as a pair-instability supernova, where the whole the PPISN in terms of neutrino and light curve. Us- star explodes and no remnant leaves behind. Therefore, ing our hydrodynamics model calculated from MESA, the remnant mass drastically drops to zero at that end. the expected neutrino signals detected by terrestrial and For higher metallicity, the remnant mass also drops, but the expected light curve will be calculated. The results its mass is significantly higher than those in lower metal- will provide a more fundamental understanding to the licity. but because of the main-sequence stage mass loss, properties of PPISN, which may be constrained from the which strongly prohibits the formation of massive He observables of one the the PPISN candidates, the Eta core. As a result, the core does not experience any pul- Carinae. sation before the iron-core collapses. From the figure we observe that the black hole mass has a strong limit at 5. ACKNOWLEDGMENT about 50 M⊙. For all metallicity and masses, no remnant mass can exceed 50 M⊙. Such limit can be con- This work has been supported by the World Premier fronted with the maximum black hole detected in the International Research Center Initiative (WPI Initia- future. tive), MEXT, Japan, and JSPS KAKENHI Grant Num- We note that in a single observation, the solution for bers JP26400222, JP16H02168, JP17K05382 We thank matching the black hole mass with our remnant mass is the developers of the stellar evolution code MESA for degenerate for both mass and metallicity. To further ap- making the code open-source.

REFERENCES

Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, Barkat, Z., Rakavy, G., & Sack, N. 1967, Phys. Rev. Lett., Astrophys. J., 818, L22 18, 379 —. 2016b, Phys. Rev. Lett., 116, 241103 Chatzopoulos, E., & Wheeler, J. C. 2012, Astrophys. J., —. 2017, Phys. Rev. Lett., 118, 221101 748, 42

17 Chatzopoulos, E., Wheeler, J. C., & Couch, S. M. 2013, Paxton, B., Marchant, P., Schwab, J., et al. 2015, Astrophys. J., 776, 129 Astrophys. J. Suppl., 220, 15 Chen, K.-J., Woosley, S., Heger, A., Almgren, A., & Scammapieco, E., Madau, P., Woosley, S. E., Heger, A., & Whalen, D. J. 2014, Astrophys. J., 792, 28 Ferrara, A. 2005, Astrophys. J., 633, 1031 Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Smith, N. 2008, Nature, 455, 201 Structure (New York: Gordon and Branch) Smith, N., Li, W., & Foley, R. D. 2007, Astrophys. J., 666, Fowler, W. A., & Hoyle, F. 1964, Astrophys. J. Suppl., 9, 1116 210 Smith, N., & Owocki, S. P. 2006, Astrophys. J., 645, L45 Glatzel, W., Eid, M. F. E., & Fricke, K. J. 1985, Astron. Swesty, F. X. T. F. D. 1999, Astrophys. J. Suppl., 126, 501 Astrophys., 149, 413 Tolstov, A., Nomoto, K., Blinnikov, S., et al. 2017, Glazyrin, S. I., Blinnikov, S. I., & Dolgov, A. 2013, Mon. Astrophys. J., 835, 266 Not. R. astr. Soc., 433, 2840 Vink, J. S., & de Koter, A. 2002, Astron. Astrophys., 393, Grott, M., Chernigovski, S., & Glatzel, W. 2005, Mon. Not. 543 R. astr. Soc., 360, 1532 Woosley, S. E. 2017, Astrophys. J., 836, 244 Heger, A., & Woosley, S. E. 2002, Astrophys. J., 567, 532 Woosley, S. E., Blinnikov, S., & Heger, A. 2007, Nature, Itoh, N., Adachi, N., Nakagawa, M., Kohyama, Y., & 450, 390 Munakata, H. 1989, Astrophys. J., 339, 354 Yoshida, T., Umeda, H., Maeda, K., & Ishii, T. 2016, Mon. Kasen, D., Woosley, S. E., & Heger, A. 2011, Astrophys. J., Not. R. astr. Soc., 457, 351 734, 102

18 10 with convective mixing without convective mixing (K) c T 10 log

10 ) -3 5 (g cm c ρ 10 0 log

0.1 1 10 100 1000 10000 time (year)

Figure 19. (upper panel) The central temperature against time for the Model 60HeA with and without convection. (lower panel) Similar to the upper panel, but for the central densities.

1 O

Si 0.1

0.01 1 56

mass fraction Fe

Ne C 0.1

54 Fe Mg 0.01 0.01 0.1 1 m(r)/M

Figure 20. The chemical abundance proﬁles for Model 60HeA prior to its second contraction at a central temperature ≈ 109 K with convection (upper panel) and without convection (lower panel).

APPENDIX

A. EFFECTS OF CONVECTIVE MIXING In Woosley (2017) the PPISN is prepared for models with convective mixing. It is mentioned that the convective mixing is essential to evolve the star correctly to readjust the chemical composition of the remnant. It is unclear how much the convective mixing can change the evolutionary path of the PPISN. Here we compare the model of He60A by treating the convective mixing as an adjustable parameter. In Figure 19 we plot the central temperature (upper panel) and central density (lower panel) against time for the Model 60HeA for both choices. It can be seen that the effects of convective mixing are huge. In the model with mixing switched on, in the second pulse it leads to a large amplitude expansion, which leads to significant mass loss afterwards before its third contraction to its collapse. On the other hand, the model without convective mixing has a faster growth of central temperature and central density, where the star collapses without any pulsation. To understand the difference, we plot in Figure 20 the chemical composition of the star before the second contraction takes place. We pick both star models when it has a central temperature of 109 K. It can be seen that the role of convective mixing is clear that the mixing not only re-distribute the energy of the matter, the composition in the large-scale is modified. A considerable amount of fuel is re-inserted into the core, which contains O and Si from the unburnt envelope, and some remained 54Fe and 56Fe produced in the first contraction. This shows that the convection during the expansion is important for the future nuclear burning to correctly predict the strength of the pulse, which affects the nucleosynthesis as well as the mass loss.

19 10

9 (K) c

T 8 10

log 7

6 -3 6 Cav = 2x10 (default) -2 C = 2x10 c 4 av ρ -1

10 C = 2x10 2 av log 0 -2 0.1 1 10 100 1000 10000 time (year)

Figure 21. (upper panel) The central temperature against time for the Model 60HeA with diﬀerent levels of artiﬁcial visocity. (lower panel) Similar to the upper panel, but for the central densities.

B. EFFECTS OF ARTIFICIAL VISCOSITY Another important parameter in numerical hydrodynamics modeling is the artificial viscosity. Owing to the lack of Riemann solver (exact or approximate) for the spatial derivative, artificial increases of pressure is needed to prevent the shock from over-clumping the mass shells. However, the artificial viscosity formula contains one free parameters −2 Cav. The default value from the package ’ccsn’ in the MESA test suite is Cav =2 × 10 . To probe the effects of this parameter, we carry out a control test by varying Cav. In Figure 21 the time dependence of the central temperature (upper panel) and central density (lower panel) are −3 −2 −1 −3 plotted for Model He60A with Cav = 2 × 10 , 2 × 10 (default value) and 2 × 10 . Results with Cav =2 × 10 −2 and 2 × 10 are almost identical. This shows that the default choice of Cav can maintain the shock propagation −1 and produce convergent results. On the other hand, when Cav =2 × 10 , very different outcome appears. The first expansion has reached to a lower central temperature and density. Furthermore, the two quantities are in general lower than the cases with lower Cav during the expansion. The second contraction also takes place a few thousand years before the other two cases. This shows that if a too large artificial viscosity is chosen, the pressure heating also alters the shock heating and its associated nuclear burning in the star, thus affecting the consequent configurations.