Pulsational Pair-Instability Supernova I: Pre-collapse Evolution and Pulsational Mass Ejection

Pulsational Pair-Instability Supernova I: Pre-collapse Evolution and Pulsational Mass Ejection

Shing-Chi Leung Email address: shingchi.leung@ipmu.jp       Ken’ichi Nomoto Email address: nomoto@astron.s.u-tokyo.ac.jp    Sergei Blinnikov Email address: sblinnikov@gmail.com
August 1, 2019
Abstract

We calculate the evolution of stars with the initial masses of 80 - 140 from the main-sequence through the beginning of Fe core collapse for six metallicities of . Our calculations show that these stars undergo pulsational pair-instability (PPI) when the O-rich core is formed for metallicity . Here electron-positron pair production causes a rapid contraction of the O-rich core which triggers explosive O-burning, strong pulsation of the core, and enormous mass ejection. We calculate such a hydrodynamical evolution of He cores for eight masses of to study the mass dependence of the pulsation dynamics, i.e., mass ejection history, thermodynamics, energetics, luminosity and chemical abundances. We find that the resultant H-free circumstellar matter (CSM) around the star is massive enough to reproduce the observed light curve of Type I (H-free) super-luminous supernovae with circumstellar interaction. We also note that the mass ejection sets the maximum mass of black holes (BHs) to be , which is consistent the mass of recently detected massive BH with aLIGO. We compare our results with several other works.

pacs:
26.30.-k,
\move@AU\move@AF\@affiliation

Kavli 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

\move@AU\move@AF\@affiliation

Kavli 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

\move@AU\move@AF\@affiliation

Kavli 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

\move@AU\move@AF\@affiliation

Institute for Theoretical and Experimental Physics, Moscow, Russia

1 Introduction

1.1 Pulsational Pair-Instability (PPI)

The structure and evolution of massive stars depend on stellar mass, metallicity, and rotation [[, e.g.,]]Arnett1996, Nomoto1988, Heger2000, Heger2002, Nomoto2013, Meynet2017, Limongi2017, Hirschi2017. In stars with the zero-age main-sequence (ZAMS) mass of , hydrostatic burning progresses from light elements to heavy elements in the sequence of H, He, C, O, Ne, and Si burning, and finally a Fe core forms and gravitationally collapses to form a compact object including neutron star.

For very massive stars with , effects of the electron-positron pair-production () on stellar structure and evolution are important when the O-rich core is formed (Fowler & Hoyle, 1964). Pair-production causes the dynamically unstable contraction of the O-rich core, which ignites very explosive O-burning. For , the released nuclear energy is large enough to disrupt the whole star, so that the star explodes as pair-instability supernovae (PISN) (Barkat et al., 1967).

For stars with , explosive O burning does not disrupt the whole star, but creating strong pulsations Barkat et al. (1967); Rakavy & Shaviv (1967), which is called Pulsational Pair-instability (PPI). Then these stars undergo distinctive evolution compared to more massive or less massive stars. PPI is strong enough to induce massive mass ejection as in PISN, while the star further evolves to form an Fe core that collapses into a compact object as in the core collapse supernova (CCSN). PPI supernovae (PPISNe) of stars are thus the hybrids of PISN and CCSN.

The exact ZAMS mass range of PPISN depends on the mass loss by stellar wind, thus on metallicity, and also on rotation. For PPISN progenitors, the wind mass loss during H and He burning phases could contribute to the loss of almost a half of the initial progenitor mass. Such mass loss can suppress the formation of a massive He-core. The exact mass of the He core as a function of metallicity and ZAMS mass remains less well constrained compared to the hydrogen counterpart (Vink & de Koter, 2002; Smith & Owocki, 2006). Rotation provides additional support by the centripetal force, which allows PPISN to be formed at an even higher progenitor mass (Glatzel et al., 1985; Chatzopoulos & Wheeler, 2012).

In order to pin down the mass range of PPISN, a mass survey of main-sequence star models is done in Heger & Woosley (2002); Ohkubo et al. (2009) with focus on the zero metallicity stars. Large surveys in other metallicity can be also found in (e.g., Heger & Woosley, 2010; Sukhbold et al., 2016). A large array of stellar models covering also PPISN with rotation has been further explored in Yoon et al. (2012).

The evolution of PPISN is very dynamical in the late phase. During the pulsation, the dynamical timescale can be comparable with the nuclear timescale that hydrostatic approximation is no longer a good approximation. Also, when the star drastically expands after the energetic nuclear burning triggered at the contraction, the subsequent shock breakout near the surface is obviously a dynamical phenomenon. This suggests that during this dynamical but short phase, hydrodynamics instead of hydrostatic is required in order to follow the evolution consistently. Compared to hydrodynamical studies of PISNe (Barkat et al., 1967; Umeda & Nomoto, 2002; Heger & Woosley, 2002; Scammapieco et al., 2005; Chatzopoulos et al., 2013; Chen et al., 2014), systematic hydrodynamical study of PPI has been conducted only recently (e.g., Woosley & Heger, 2015; Woosley, 2017).

1.2 Connections to Observations

The optical aspect of PPI and PPISN might explain some super-luminous supernovae (SLSNe), such as SN2006gy (Woosley et al., 2007; Kasen et al., 2011; Chen et al., 2014). Recent modelling of SLSN PTF12dam (Tolstov et al., 2017) has required an explosion of 40 star with 20-40 circumstellar medium (CSM) with a sum of 6 Ni in the explosion. The shape, rising time and fall rate of the light curves provide constraints on the composition, density and velocity of the ejecta, which provide insights to the modeling of PPISN. It demonstrates the importance to track the mass loss history of a star prior to its collapse. The rich mass ejection can be an explanation to the dense CSM observed in some supernovae, such as SN 2006jc (Foley et al., 2007). Supernova models in the PPISN mass range are further applied to explain some unusual objects, including SN 2007bi (Moriya et al., 2010; Yoshida et al., 2014) and iPTF14hls (Woosley, 2018).

There is also a possible connection to the well observed Eta Carinae, which has demonstrated significant mass loss about 30 (Smith et al., 2007; Smith, 2008).

Furthermore, recent detections of the gravitational waves emitted by the merging of black holes (BH) (Abbott et al., 2016a, b), such as GW150914 and GW170729 imply existence of BHs of masses . In order to study the mass spectrum of BHs in this mass range, the evolutionary path of this class of objects becomes necessary. Such observations have led to the interest in the evolutional origin of massive BHs, including PPI phenomena (e.g., Woosley, 2017; Belczynski et al., 2017; Marchant et al., 2018).

1.3 Present Study

From the above importance of PPI, we re-examine PPI by using the open-source stellar evolution code MESA. We use the MESA code because the recent update of the MESA code (Paxton et al., 2015) has included an implicit energy-conserving (Grott et al., 2005) hydrodynamical scheme as one of its evolution option.

We study a series of the evolution of stars from ZAMS for the masses range of 80 - 140 and various metallicities. This corresponds to the He-core masses from to . We calculate the evolution of such He cores to study the hydrodynamical behavior of PPI including mass ejection.

In Section 2 we describe the code for preparing the initial models and the details of the one-dimensional implicit hydrodynamics code for the pulsation phase.

In Section 3 we examine the evolutionary path of PPISN in the H and He-burning phases and the influence of metallicity on the final He and CO-core masses.

Then in Section 4, we first present the pre-pulsation evolution of our models which include He and C-burning phases. Then we study the dynamics of the pulsation and its effects on the shock-induced mass loss. After that, we present evolution models of He cores with 40 - 64 . We examine their properties from four aspects, the thermodynamics, mass loss, energetic and chemical properties.

In Section 5 we examine the connections of our models to super-luminous progenitors.

In Section 6 we compare the final stellar mass of our PPISN models with the recently measure black hole masses detected by gravitational wave signals.

In Section 7 we compare our numerical models with those in the literature.

In Section 8 we conclude our results.

We study in Appendix A and Appendix B the effects of some physical inputs in the numerical modeling, including the convective mixing and artificial viscosity.

2 Methods

2.1 Stellar Evolution

To prepare the pre-collapse model, we use the open source code Modules for Experiments in Stellar Astrophysics (MESA) version 8118 (Paxton et al., 2015). It is a one-dimensional stellar evolution code. Recent updates of this code have also included packages for stellar pulsation analysis and implicit hydrodynamics extension with artificial viscosity. We modify the package ccsn to build a He-core or main-sequence star models directly and then we switch to the hydrodynamics formalism according to the global dynamical timescale of the star.

2.2 Hydrodynamics

To understand the behavior of pulsation and runaway burning of the O-core, we use the one-dimensional implicit hydrodynamics code with a nuclear reaction network. This function appears in the third instrument paper of MESA (See also Paxton et al. (2011, 2013)). The energy conserving scheme, coupled with the implicit mass-conserving property of the Lagrangian formalism, allows us to trace the evolution of the star consistently.

We refer the readers to the instrument paper Paxton et al. (2015) (Section 4) where the detailed implementation of this mass- and energy conserving implicit hydrodynamics scheme is documented. Here we briefly outline the specific points which are relevant to our calculation here.

The realization of this scheme relies on the use of artificial viscosity as a substitute to the exact Riemann solver. To capture the shock, the artificial viscosity takes the form

(1)

which has the same unit as the pressure term and it enters the system of different equations by . . We choose . However, it is known that this value of this quantity is chosen by experience. We study the effects on the choice of in the Appendix B.

Although there are also other implementation of implicit hydrodynamics scheme in the literature, we choose the implementation of MESA because of the energy conserving scheme as first reported in Grott et al. (2005). It is shown that by arranging terms in this form one can achieve energy conservation with an accuracy equivalent to that of the hydrodynamics and nuclear reaction solvers. It is known that in all implicit solver the code looks for a solution which satisfies the non-linear equations posed by mass, momentum and energy conservation. However, due to the non-local transport of energy and momentum, it is possible for the solver to obtain irrelevant solution which still satisfies the solution approximately. In order to make sure the solution is a time-evolution of the state we put in as the initial condition, the solution should satisfy the mass- and energy-conservation schemes as time evolves (except when there is mass loss).

We follow the convention for the physics quantities. Density, temperature, isotope mass fractions, specific internal and related thermodynamics quantities are defined on cell center. Position, velocity, acceleration and gravity source terms are defined on the cell boundaries. We impose the innermost boundary conditions as .

The typical timescale during the pulsatoion is comparable to the dynamical timescale. However, after the pulsation phase, it is the KH timescale that dominates the contraction. Even with the implicit nature of the dynamics code, simply using the hydrodynamics formalism to evolve the whole pulsation phase is computationally challenging as the Courant-Friedrich-Levy condition severely limits the maximum possible timescale, although it guarantees the consistency of our calculation. We set conditions for the code to switch back to the hydrostatic approximation. When the star sufficiently expand after bounce so that the evolutionary timescale becomes thermal, we gradually allow larger timesteps in every 100 steps. When the star can evolve continuously with the maximum timesteps, we switch back to the hydrostatic approximation to evolve the star until it becomes dense enough so that dynamical effects emerge. If the star appears to be non-static during the 100-step buffer, the buffer is further extended. After the hydrostatic scheme is used, the convective mixing is also switched on. When the central temperature of the star reaches above K, the implicit hydrodynamics scheme is automatically resumed. Sometimes the star is not fully relaxed even after the buffer phase, the hydrostatic approximation can also yield small steps. In that case, we also switch the code back to the hydrodynamics scheme to further relax the star.

2.3 Microphysics

The MESA code uses the standard Helmholtz equation of state (Swesty, 1999), which contains electron gas with arbitrary relativistic and degeneracy levels, ions in the form of an classical ideal gas, photon gas with Planck distribution and electron-positron pairs. To model the nuclear reactions, we use the ’approx21plusco56.net’ network. This includes the -chain network (He, C, O, Ne, Mg, Si, S, Ar, Ca, Ti, Cr, Fe and Ni), H, He and N for the hydrogen burning and CNO cycle, and Fe and Co to trace the decay chain of Ni. Cr is included to mimic the neutron-rich isotopes formed after electron capture in nuclear statistical equilibrium (NSE).

2.4 Convective Mixing

As indicated in Woosley (2017), convective mixing is important in that it redistributes the fuel and ash in the remnant core. This affects the subsequent nuclear burning when the star contracts again. We choose the Mixing Length Theory (MLT) (Cox & Giuli, 1968) to model the convective process. The MLT approximation is used in the main-sequence phase and also when the star enters the expansion phase. We have attempted to couple the convective mixing in the dynamical phase but it results in impractically small timesteps. Furthermore, the convective velocity during the dynamical phase is in general much smaller than the fluid velocity. This implies that the mixing is important at a much longer timescale (Also see Section 4 the corresponding Kippenhahn Diagrams). Therefore, it becomes numerically manageable while physically consistent to ignore convective mixing in the dynamical phase.

3 PPISN from Hydrogen Main-Sequence Phase

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table \hyper@makecurrenttable

Table 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe 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.

The models assume no mass loss.

Mass
80 34.05 37.40 (27.20) 33.80 (23.93) 30.10 (23.96) 23.60 (21.09) 22.70 (18.66)
100 44.51 49.44 (37.69) 47.16 (34.22) 33.00 (30.65) 31.70 (28.50) 30.30 (24.85)
120 54.87 64.71 (48.40) 59.95 (43.48) 57.10 (41.20) 37.40 (31.73) 15.50 (12.02)
140 65.87 nil 70.85 (56.67) 60.78 (50.36) 20.80 (16.90) 12.80 (9.60)
160 76.50 83.31 (82.40) 89.99 (89.12) 52.93 (46.46) 15.00 (11.63) 11.99 (8.90)

3.1 Kippenhahn Diagram

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefKippenhahn diagram of the main-sequence of at solar metallicity from the H-burning until the core reaches a temperature of K.

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figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 3.1, but for .

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figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 3.1, but for .

In Figures 3.1, 3.1 and 3.1, we plot the Kippenhahn Diagram of the stars with and 120 at . The lines (red, green and blue) correspond to the He-, C- and O- core mass coordinate respectively. Grey shaded regions are the convective zones inside the star.

At solar metallicity the high metal content in the initial composition has largely increased the opacity, which allows strong mass loss during H-burning and He-burning due to its intrinsic high luminosity. It has an extreme large mass loss that half of the stellar mass is lost in the helium burning phase for 80 and 100 . and in the hydrogen burning phase for 120 . The whole H envelope is lost during He-burning, which occurs about year before collapse. The initial He core mass can reach about half of the initial mass, but it gradually decreases due to the later mass loss. Also, in all three models, after the removal of H-envelope or He-burning, the C-core quickly forms with a mass similar to the He-core mass. For 80 and 100 models they have a C core mass about 20 which remains unchanged after it has been formed. The 120 one has a somewhat smaller one due to the previous drastic mass loss.

The convective pattern of the star is consistent with those massive stars. In H-burning phase, the core is mostly convective. The surface is radiative. In the He-burning phase, the core remains convective while some H-envelope becomes convective. But this feature disappears when the mass loss sheds away the H-envelope. Once, the C-core has formed at about year before pulsation, the star begins to contract rapidly. The core becomes radiative. But together with the C-core and C-envelope burning, layers of convective shells appear. They gradually propagates and which the C-core surface. Once the core starts O-burning ( year from the onset of first pulsation), the strong energy generation triggers large scale convection that the whole C-envelope becomes convective. The inner core of O-rich region also becomes convective.

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefKippenhahn diagram of the main-sequence of at 0.1 from the H-burning until the core reaches a temperature of K.

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figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 3.1, but for .

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figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 3.1, but for .

In Figures 3.1, 3.1 and 3.1 we plot similar to the previous three figures but at . Different from the solar metallicity models, the low metallicity has lowered the opacity in the matter, and thus lowered the mass loss during the H- and He-burning phase. There is a clear signature of massive He core from 30 - 50 . Due to the smaller mass loss, the He core mass remains constant after it has formed. Near the occur of first pulsation, a massive CO core about 10 is also formed. A generally larger O-rich core is formed at the end of simulation.

Due to the preservation of H-envelope after He-burning, the star consists of a rich structure of convection activities before its collapse. The convective core has a similar structure to the higher metallicity case, due to the extended H-envelope remained after He-burning, there is also a second convective zone which gradually move inward to the stellar core from its initial 60 to . After that the He-core is fully convective during He-burning and the convection zone extends into the H-envelope. The surface is also convective. During C-burning, the convective layer propagates outwards from the core to the C-core surface. The outer layers of He- and C- envelopes are convective. During contraction before the onset of O-burning, the core returns to be radiative dominated. Similar to the high metallicity case, near the onset of pulsation, the core becomes convective.

3.2 He Core and CO Core Mass Relations

To study the effects of metallicity and rotation, we perform pre-pulsation stellar evolution models for different metallicity from up to for non-rotating main-sequence star model using the MESA code. In Table 3 we tabulate the pre-pulsation configurations of the main-sequence stars for their He- and CO- core masses when the core exhausts all H and He respectively. The CO core masses are written in brackets.

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe He core mass against progenitor mass when the core exhausts its hydrogen for stellar models at different metallicity. For , the models assuming no mass loss is assumed because of numerical instability encountered during He-burning in the asymptotic red-giant branch.

For the He core mass, it can be seen that He core mass grows monotonically with progenitor mass when . For star models with a higher metallicity, the mass loss rate, which is proportional to the metallicity, makes the He core mass drops at the high mass end. This transition starts at a lower mass for models with a higher metallicity. Notice that the change and the transition mass is not linearly proportional to the star metallicity due to the non-linear dependence of mass-loss rate. Also, the mass loss affects the gravity, which changes the equilibrium structure of the star even in the H-burning phase.

In Figure 3.2 we also plot the relations He core mass against progenitor mass for different metallicity. On one hand, at low mass, the He core mass is not so sensitive to metallicity, that the He core mass approaches its asymptotic value when . On the other hand, at high mass, the He core mass is very sensitive to metallcity that from to the He core mass can drop by 90 % at the star model , about 15 . At such low mass, the He core already leaves the pulsation pair-instability regime, and will evolve as a normal core collapse supernova. Furthermore, one can see that the maximum He core mass for models at solar metallicity only barely reaches the transition mass 40 .

For models which completely covers the whole PPISN mass range, we require stellar model with a metallcity at most . This shows that the PPISN is very sensitive to the progenitor metallicity, while stars with solar metallicity are less likely to form PPISN owing to its mass loss.

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe C core mass against progenitor mass when the core exhausts its He for stellar models at different metallicity. Again, for the models assume no mass-loss due to numerical instability.

Then we examine the CO core mass. Before the CO core mass can be defined, the massive CO core has already started its contraction, which increases the CO core mass. The metallicity effect is a similar trend to the He core mass. In all models, at the lower mass branch CO core mass in general increases with progenitor mass, but it drops at the high mass end. The CO core mass also shows a monotonic decreasing relation against metallicity for the same progenitor mass. The mass loss effect in near solar metallicity models are more significant that the CO core mass contributes to less than 10 % of the stellar mass, while those with a lower metallicity can be about one third of the progenitor mass.

In Figure 3.2 we also plot the relation CO core mass against progenitor mass at different metallicty. The significance of the metallicity of mass loss rate can be seen. By increasing models from 0.5 to 0.75 , the CO core mass can drop by 75 % at . The CO core mass shows a much clearer relation than the He core mass. The mass scaled clearly with metallicity, except at , where the model without mass loss () has a lower mass than its counterpart of .

4 PPISN from He Stars

In this section we study the evolution of PPISN using the He-core as the initial condition. We do not evolve PPISN from H-burning because during the main-sequence stage, because during the strong mass loss, the star develops to have an extended but very low density H-envelope. Because the H-envelope is not tightly bounded by the gravitational well of the star, it is easily disturbed and obtains high velocity in the dynamical approach. The situation becomes more difficult when the shock from the pulsation reaches the H-envelope. Thus, to keep the H-envelope while evolving the whole star becomes computationally difficult. However, as the H-envelope does not couple strongly to the inner core, the pulsation dynamics is not significantly changed, when we do not consider the effects of H-envelope. Therefore, in this section, we consider the dynamics, energetics, mass loss and chemical properties of the PPISN by using the He-core as the initial condition.

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table \hyper@makecurrenttable

Table 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe main-sequence star models prepared by the MESA code. and are the initial and final masses of the star. , , are the hydrogen, He- and CO-mass before the hydrodynamics code starts. All masses are in units of solar mass.


Model
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

4.1 Kippenhahn Diagram

In this part we examine the overall evolution of the PPISN evolved from He core until the onset of Fe-core collapse.

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefKipperhahn Diagram for the Model He40A () until the onset of final collapse. The lines correspond to the inner boundary where the mass fractions of the respective elements drop below . By this definition, the surface mass coordinate of the star, if it does not experience strong mass ejection, is the He-core mass since we start from a He-star.

\H@refstepcounter

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 4.1, but for .

\H@refstepcounter

figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 4.1, but for .

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefSimilar to Figure 4.1, but for .

In Figures 4.1, 4.1, 4.1 and 4.1 we plot the Kippenhahn diagram of Models He40A, He50A, He60A and He62A. The coloured zone is again the convective zone while the lines (solid, dotted, dashed, long-dash, dot-dash) are the He-, C-, O-, Si- and Fe-core mass coordinate. The x-axis is the time counting backward from its collapse. We define the core boundary to be the inner boundary of the mass fraction for that corresponding element to drop below . Therefore, since we start from a He core, the He-rich surface, which is also the total mass of the star, stands for the He-core. Notice that for the cases with strong mass ejection, the whole He-rich surface can be shredded off.

For Model He40A, after the strong pulsation, the star expands and the whole star becomes convective. Also, the star established its O- and Si-cores at and its Fe-core at . After that, when the star contracts, due to the rapid contraction, the star fails to reach an equilibrium state, which disfavours the convection. Thin layers of convection shells can be found in most part of the He-envelope and C-envelope. At about year, the Si and O core can reach as far as . This is because during the propagation of the acoustic wave near surface, the density gradient accelerates the wave into a shock, which heats up the matter around there. As a result, in such He-rich material, it facilitates the He-burning and gives product including C, O, Ne and Si. However, accompanied with the extended convection during the expansion-contraction phase, the outer O- and Si-rich zones disappear and the values correspond to the inner layers, which come from previous hydrostatic burning.

For Model He50A, after the pulsation, the C-, O- and Si-cores are produced simultaneously. But the O- and Si-cores quickly retreat from 30 to 5 and 10 respectively. The early formation of O- and Si-cores is because when the shock reaches the surface, the shock heating is capable in producing O- and Si-rich material around that region. However, away from the shock-heated zone, no significant O- and Si-productions take place. However, after the production, the mixing and mass loss by pulsation quickly remove these material. As a result, the O- and Si-core mass coordinates return to the corresponding inner values, where the real O-core and Si-core locate. At year before the final collapse, the contraction of star allows the central density to be high enough for burning until NSE. The Fe- and neutron-rich core forms almost simultaneously at . Different from Model 40HeA, the inner core is no longer convective, except after pulsation. There is an extended period of time at year before its final collapse, the star continues to hold a fragmented convective structure.

For Model He60A, there is also no inner convective core after its pulsation. Again, the shock heating creates a temporary outer O- and Si-core outside surface, but they return to the true values after mass ejection and mixing, to 5 and 15 . Different from the previous two models, the expanded star after pulsation does not reach any convective state before its second pulse or final collapse. An outward propagating convective structure can be seen from year before collapse. It moves from to 40 . The convection zone is small that it does not contribute in bringing the fuel from outer layer to the actively burning layer. Similar to Model He50A, the Fe- and neutron-rich cores appear at at year before collapse.

For Model He62A, it is different from the previous three models because of its extensive mass loss after pulsation. After the first pulse, the star reaches a very extended period years of fully convective state. Again, the convection washes away the external C- and O-envelope. The first pulse creates a final C-, O-cores which locate at 20 and 5 respectively. In the second pulsation, the Fe-core is also produced which has a mass 3 . During its contraction at 1 year before its final collapse, the core reaches the third fully convective state. During contraction, the outer extended convective zone also moves outward from 20 to 40 . The convective structure is again fragmented. A 2 Fe-core is formed only near year before the final collapse.

4.2 Pre-pulsation evolution

We first present the results for the pre-collapse evolution of the He-core and main-sequence star in Table 4. The pre-pulsation evolution uses the hydrostatic approximation and it is done until the central temperature reaches K, where the dynamical timescale begins to be comparable with the O-burning timescale. Below K the star evolves in a quasi-static manner but with the velocity flag turned on. From the table we can see that the initial He core mass affects the pre-pulsation C- and O- core. We choose the He-core models with a mass from 40 to 64 , which produce CO cores from 30.82 to 50.42 , with the remaining unburnt He in the envelope.

In Figure 4.2 (upper-left) we plot the evolution of the central temperature against the central density, together with the pair-instability zone. Models 40HeA, 45HeA, 50HeA, 55HeA and 60HeA are included. We include also the lines indicating the model when the central He or C is exhausted (defined by their mass fraction being below ). When the initial He mass increases, the exhaustion of He and C end with a lower central density and temperature. Also, a model with a higher mass evolves with a lower density for a given temperature, with Model 60HeA being the closest to the pair-instability zone. There is no intersection among models, showing that the thermodynamics properties of the core before pulsation depend on only its mass.

In Figure 4.2 (upper-right) we plot the evolution in the Hertzsprung-Russell diagram for Models 40HeA, 45HeA, 50HeA, 55HeA and 60HeA. The qualitative features of their paths are similar, which consist of an initial jump in luminosity and a near horizontal move towards a higher . A Model with a higher mass has a higher luminosity at the same effective temperature. The moments where the core He and C are exhausted for each models are linked by the black lines. Unlike low-mass stars, all the He- and C- burning occurs in such a short time scale that the star does not much vary in luminosity.

In Figure 4.2 (lower-left) we plot the profiles of the initial conditions for the dynamical phase of Models 40HeA, 50HeA and 60HeA. The three models have very similar initial profiles. These quantities drop around . Minor differences can be seen among models from the density profiles that the models with a lower He mass has a slightly higher density in the inner part.

In the lower right panel we plot the chemical abundance profiles similar to those for Models 40HeA and 60HeA. In the lighter mass model, there is no remaining C. From , the difference between C and Ne is larger, showing that there is less Ne in this layer. The C-burning region is extended to almost the surface of the star, leaving very little mass of pure He. On the contrary, in the more massive He core model, C is not yet burnt in the central region. Also there is a comparatively higher Ne in the zone . The shell C-burning beyond has just started. The He surface remains unchanged.

4.3 Pulsation

We first study the time evolution of the pulsation. To do so, we examine the second pulse of Model He60A, which is a strong pulse (with mass ejection) of mass . We choose this particular pulse because it is strong enough to create global change in the profile so that we can understand the changes during the contraction (before maximum of central temperature in the pulsation) and expansion (after minimum of that in the pulsation) phases.

The core is mostly supported by the radiation pressure. With the catastrophe in pair production, the supporting pressure suddenly drops, where the core softens with corresponding equation of state adiabatic index in the core. However, unlike the stars with a mass 10 - 80 which have rich Fe-cores at the moment of their collapse, in PPISN and PISN the core is mostly made of O when contraction starts. The softened core allows a very strong contraction and the O-rich core can reach the explosive temperature which releases a large amount of energy, sufficient to disrupt the star. Si and Ni can be produced during the contraction, where the central temperature can reach beyond K. As a result, the star stops its contraction and starts its expansion. The rapid expansion causes strong compression to the matter on the surface, which efficiently causes ejection of high velocity matter on the surface and dissipates the energy. After that, the core becomes bounded again. The pulsation restarts after it has lost most of its previously produced energy by radiation and neutrinos. The whole process repeats until the Fe core, formerly Ni, exceeds the Chandrasekhar mass that it collapses by its own gravity before the compression heating can reach the further outgoing O-rich envelope.

In Figure 4.3 we plot in the upper left, upper right and middle left panels the temperature, density and velocity evolution at selected time respectively. We pick the profiles when the core temperature reaches , and K before the core reaches its peak temperature during the pulse for Profile 1-3, at its peak temperature for Profile 4, and after the core has reached its peak temperature for Profile 5 - 7 for the same central temperature interval. In the middle right, lower left and lower right panels we plot the chemical abundance profiles for isotopes O, Si and Ni respectively.

First we study the hydrodynamics quantities. For the temperature, in the contraction (expansion) phase the star shows a global heating (cooling) due to the compression (expansion) of matter, and no temperature discontinuity can be observed. This shows that the whole star contracts adiabatically, where the nuclear reactions take place evenly inside the star. By comparing the temperature profiles at the same central temperature (Profiles 1 and 7 for K, profiles 2 and 6 for K and profiles 3 and 5 for K), the net effect of nuclear burning can be extracted. The part outside has a higher temperature after the pulse. Similar comparison can be carried out for the density profile. The inner core within is unchanged after pulsation, while the density in the outer part increases. The velocity profiles show more features during the pulse. Before the star reaches its maximally compressed state, the velocity everywhere is much less than cm s. At the peak of the pulse, the envelope has the highest infall velocity of cm s. After that, in Profile 5, the core starts the homologous expansion phase, with a sharp velocity discontinuity peak near the surface between the outward going core matter and the infalling envelope. Beyond Profile 6, the discontinuity reaches the surface and creates a shock breakout. The surface matter can freely escape from the star.

For the chemical composition, the effects of the pulse becomes clear. Since the second pulse, part of the core O is already consumed in the first pulse, which is converted to Si already. During the compression, before the core reaches its maximum temperature, O is significantly consumed and forming Si. When the core reaches the peak temperature, the O within is completely burnt, where intermediate mass elements, such as Si, is produced. However, Fe-peak elements, such as Ni are not yet produced. On the other hand, during the expansion phase, most O-burning ceased, making the O and Si unchanged after the central temperature reaches , while advanced burning still proceeds slowly to form Fe-peak elements.

4.4 Global Properties of Pulse

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Table 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe 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. is the current mass in units of solar mass. is the amount of energy released by nuclear reaction in units of erg. , , , , , and 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 K.

Model bounce 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

Here we study some representative models of He core with a from 40 to 62 . They show very different pulsation history, by their number of pulses and their corresponding strengths. In Table 4.4 we tabulate the stellar mass and the element mass in the star after each of the pulse.

For Model He40A, most of the pulses are weak, however, following each of the pulse, mass of O is gradually consumed and produce Si. At late pulses, where the core reaches beyond g cm, NSE elements are also produced. In the last pulse, the core is sufficiently compressed such that an Fe core beyond 1.4 is produced, which is accompanied with the later mass loss. Most of the ejected mass is He.

For Model He45A, again most of the pulses are weak. With the number of pulses increased, not only Si, but also Ni are produced. The last pulse, which is the strongest overall, produces about 0.56 Ni, while the generated heat creates a shock to eject about 6 matter before the final collapse.

For Model 50HeA, the number of pulses becomes smaller and again only the last pulse is a strong pulse which can eject mass. Compared to previous models, in each pulse more O is consumed, which produces Si. At the final strong pulse, less Ni is produced, while the accompanying mass loss ejects the He in the envelope. It should be noted that its lower mass ejection compared to Model He45A comes from the difference that the O in Model 45HeA is burnt in a much compressed state. This creates a much stronger shock wave when the expansion approaches the surface, which extends the mass loss.

For Model 55HeA, it has two strong pulses. The lower mass models have only one strong pulse. Its pulses are qualitatively similar to Model 50HeA.

For Model 60HeA, it has no weak pulse. The contraction always makes a significant mass of O to be burnt to produce the thermal pressure to support the softened core against its contraction. Due to the strong mass ejection, at the end of the simulation the star almost runs out of He. However, one difference of this model from the others is that it has a much lower Ni mass after pulsation. Most of the Fe, which leads to the collapse, is created during the contraction towards collapse.

For Model 62HeA, it has a similar pulse pattern as Model 60HeA but is stronger. Each pulse can consume about of O. Different from previous models, Model 62HeA has an abundant amount of O even during its contraction towards collapse, and O continues to be consumed before it collapses.

For Model 64HeA, which is a pair-instability supernova instead of PPISN, there is only one pulse before its total destruction. Due to its much lower density when large-scale O-burning occurs, even when about a few solar mass of O is burnt during the pulse, the energy is sufficient to eject all mass when the pulse reaches the surface.

4.5 Thermodynamics

In Figures 4.5, 4.5 and 4.5 we plot the central density and temperature against time for Models 40HeA ,45HeA, 50HeA, 55HeA, 60HeA and 62HeA in the six panels respectively. To show that the rapid contraction comes from the PPI, we show in each plot the zones where electron-positron pair creation, the dynamical instability induced by photo-disintegration of Ni into He and the dynamical instability induced by general relativistic effects and/or rapid electron capture.

For Model 40HeA, at the beginning the central density is the highest among the all six models. It has thus weaker pulses because the core is more compact and degenerate. It has five small pulses and one big pulse (indicated by arrows in the figure) where each of the small pulses only leads to a small drop of the central density and temperature. Then the core quickly resumes its contraction again. Only at the final pulse, when the core begins to reach the Fe photo-disintegration zone, the softened core leads to a fast contraction and reaches a central temperature K. This triggers a large scale O-burning in the outer core, which leads to a drastic drop in the central density and temperature, showing that the star is expanding, until the reaches K. Then the core resumes its contraction. Since most O in the core is burnt, there is no extra energy input during the contraction. The star directly collapses.

For Model 45HeA, it shows a fewer number of pulses than Model 40HeA. It has three small pulses and one big pulse. The initial path is closer to the PPI instability zone. The last pulse is triggered at K and has a lowest of K when it is fully expanded.

For Model 50HeA, it has only two small pulses and one big pulse. The evolution shows less structure compared to the previous two models because of the earlier trigger of large-scale O-burning in the core. The core starts the big pulse when K and its expansion makes reaching K at minimum. Before its collapse, there is a small wiggle along its trajectory. We notice that at this phase the core has a small pulsation when the core becomes degenerate. The pulse is similar to the previous pulse, but it is not contributed by nuclear reaction.

For Model 55HeA, it has one small pulse and two big pulses. The two big pulses start when reaches and K respectively, with a minimum temperature after relaxation at and K.

For Model 60HeA, there is no small pulse and two big pulses, where the stellar core intersects with the PPI instability zone during its expansion. The two pulses start when reaches and K. The core finishes its expansion when it reaches and K.

For Model 62HeA, the star model becomes very close to the PPI instability where the core enters the zone for a short period of time during its expansion. It is similar to Model He60A that there are two big pulses. The two peaks start at and K while both pulses end at a minimum temperature of K, showing that the two pulses are of similar strength. After that, the core starts collapsing similar to all other five models.

By comparing all six models, we can observe the following trend for the pulse structure as a function of progenitor mass. First, when the progenitor mass increases, the number of small pulses decreases while the number of big pulses increases. Second, the strength of the big pulses increase with the progenitor mass, which leads to a lower central temperature and density during its expansion. Third, the path during its early pulses becomes closer to the PPI instability as mass increases. Fourth, the big pulse strength increases with time.

4.6 Energetics

In Figures 4.6, 4.6, and 4.6 we plot the energy evolution for Models 40HeA, 45HeA, 50HeA, 55HeA, 60HeA and 62HeA, including the total energy , internal energy , gravitational energy and kinetic energy . The energy is scaled in order to make the comparison easier.

In all six models, it can be seen that the energy evolution does not depend on the stellar mass strongly, 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 pulse leads to a denser and hotter core, where neutrino emission continuously draw energy from the system. At 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 again. Similar jumps in and show that the core is strongly heated due to contraction heating and nuclear reactions. After that, the star reaches a quiescent state with very mild increases of total energy due to the Ni decay, then followed by a quick drop when it contracts again.

4.7 Luminosity

In Figures 4.7, 4.7 and 4.7 we plot the luminosity evolution for the six models similar to Figure 4.6. During the pulse, the extra energy from nuclear reactions allows the luminosity to grow by 3-4 orders of magnitude. For a short period of time ( year), then the star becomes dim suddenly. After that the star resumes its original luminosity quickly and remains unchanged until the next pulse or final collapse.

The neutrino luminosity is more sensitive to the structure of the star. The neutrino luminosity can also jump by 3 - 10 orders of magnitude from its typical luminosity in the hydrostatic phase to the maximally compressed state. After the star has relaxed, the neutrino luminosity drops drastically. Depending on the strength of the pulse, neutrino cooling can become unimportant in the quiescent phase.

4.8 Mass Loss History

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Table 0. \Hy@raisedlink\hyper@@anchor\@currentHrefEnergetic 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. is temperature range of the ejecta in units of K. is the ejecta energy in units of erg. (He), (C), (O), (Ne), (Mg), (Si) are the masses of He, C, O, Ne, Mg and Si in the ejecta in units of solar mass.

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

During the pulsation, when the bounce can create an large-scale burning in the core and inner envelope, sufficient energy is produced that it can create an outgoing shock, which ejects the outermost matter away. Such 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 creates the second shockwave, which interacts with the CSM. The chemical and hydrodynamics properties of the CSM thus become important, which influence the formation of the light curve of the explosion.

In Figures 4.8, 4.8, 4.8 and 4.8 we plot the ejecta profiles of Models 40HeA, 45HeA, 50HeA, 55HeA, 60HeA and 62HeA respectively. Three patterns can be observed in the mass ejection.

The first group is the strong pulse in the lower mass branch. In Models 40HeA and 45HeA the last pulse is the pulse which ejects mass. It shows wiggles in its density profiles, showing that the thermal expansion creates the first wave of mass ejection, while the following shock as the velocity discontinuity approaching the surface creates the second wave of mass ejection. In both cases, only the He layer is affected, but as the He layer becomes thin the interface near the CO layer is also ejected.

The second group is the weaker pulse of the more massive branch. In Models He50A, He55A He60A and He62A the first strong pulse occurs after the core starts to consume O collectively. Since it burns much less O than other strong pulses, the ejection comes from the rapid expansion of the star, which includes matter in the He envelope.

The third group is the strong pulse of the more massive branch. In Models He55A and Model 60HeA, the second pulse is stronger so that the ejecta density gradually decreases. A continuous ejection of mass in terms of smooth density profile is found. The mass ejection is sufficient deep that at the end of pulsation, traces of C and O can be found. We remark that the inclusion of massive elements (compared with H and He) will be important for the future light curve modelling because they contribute as the main source of opacity.

One of the pulsations needs to be discussed separately because of its very massive mass ejection, which involves very unique chemical composition in its ejecta. In Model 62HeA (right plot), the second pulse becomes strong enough that, besides its decreasing density profile, the later ejected material contains a significant amount of heavier elements including C, O, Mg and Si, showing that the He envelope is completely exhausted before the star is sufficiently relaxed.

4.9 Chemical Properties

In Figures 4.9, 4.9, 4.9 and 4.9 we plot the isotope profiles at different moments of Models He40A, He45A, He50A, He55A, He60A and He62A. We selected moments before and after each strong pulse to extract the nuclear burning history. By comparing the isotope distribution, we can understand which part of burning contributes to the evolution of pulsation.

In all models, it can be seen that the star is simply a pure O-star with a minute amount of Si in the core or C in the envelope, covered by a pure He surface. However, their changes can be very different depending on the progenitor mass.

In Model 40HeA, after the strong pulse, due to its previous weak pulses which continue to burn matter in the core, a range of elements are produced including Fe, Fe, Fe and also Ni. There is a clear structure for each layer, which comes in the order of Fe, Fe, Ni, Ca, O and then He. After that, the core relaxes and becomes quiescent until it completely loses its thermal energy produced during the pulse, while at the same time convection re-distributes the matter for a flat composition profile. It can be seen that most convection occurs at , where the convective shells of different sizes make a staircase like structure.

Models 45HeA and He50A (upper middle and upper right panels) share a similar nuclear reaction pattern. The strong pulse provides the required temperature and density to make Ni in the center and Si in the outer zone. The Si-rich zone extends to . During the quiescent phase, convection not only mixes the material in the envelope, but also in the core, which is seen by the stepwise distribution of Fe and Fe.

In Models 55HeA, He60A and He62A (see the middle left, middle right and lower middle panels), the first pulse makes the core, which is mostly O, and then Si and some Ca. Again, the convective mixing during the quiescent 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. Ni forms in the innermost part, with a small amount of Fe isotopes like Fe and Fe. Then it is the Si and Ca middle layer and at last the He envelope. During the quiescent phase, the convection occurs in a deeper layer compared to the lower mass models.

5 Connection to Super-luminous Supernovae

PPISN has been frequently used to illustrate the physical origin of extremely luminous supernovae such as SN2006gy (Woosley et al., 2007) and PTD12dam (Tolstov et al., 2017). The PTD12dam gives a more challenging model since there is no model so far demonstrating the explosion history required. In particular, it requires a 20-40 CSM prior to the explosion of the star. Furthermore, it needs a composition with the presence of He, C and O in order to give a high opacity of surrounding matter to sustain the light curves.

By comparing with our models, it can be seen that 64 He core, the first pulse is strongly enough to produce an ejecta of mass . Our model gives an ejecta with He, C and O masses of 8.5, 1.8, 9.9 . The corresponding ratio of C:O is therefore 1:5.5. This is close to the optimal value in their models of C:O = 1:5. This has further confirmed the necessity of applying the pulsation pair-instability supernovae as the model for PTD12dam. Whether the following collapse of the remnant can explode energetically with an energy of erg is uncertain. This may require multi-dimensional hydrodynamics simulations with a realistic neutrino transport in order to model the collapse phase, which is beyond the scope of this article.

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe final density profile for the Model 50HeA including the core and ejecta matter (CSM) at the onset of collapse.

In Figure 5 we plot the density profile of the Model 50HeA at the onset of collapse ( g cm). Both the CSM and core are included. The CSM is constructed from the mass ejection history. The material is followed in a post-process manner until the time the core begins its collapse. We can see that the core consists of a compact core within the innermost cm. Outside that a smooth envelope up to cm can be seen. At the outermost part there is a surface which extends up to cm. We remark that the outermost envelope is mostly the remaining matter which cannot be ejected near the end of mass ejection event. They are mostly decoupled gravitationally from the core. The origin stellar envelope is the middle envelope of the final density profile. As discussed in Section 4.8, the mass ejection of He50A is rather smooth. Therefore, there is no significant mass shell collision, which may create some density discontinuities. The CSM profile in general follows the scaling, which extends from to cm. We note that in the calculation, there is a mass gap between the outer envelope of the core to the inner boundary of the CSM from to cm. In the reality, there should be some fallback due to gravitational tidal force on the ejecta matter. However, due to resolution and timestep constraints, such process is not included. When this fallback mechanism is included, the density gap between the core and CSM will be much smaller.

6 Connection to Massive Black Hole

The gravitational wave detectors LIGO and Virgo has recently detected gravitational wave signals from black hole-black hole merger and neutron star-neutron star merger. Some massive black holes, for example in GW 150914, the black hole binary of masses 35.4 and 29.8 are measured (Abbott et al., 2016b). Another massive black hole merger even is GW 170104, where the binary consists of black holes of mass 31.4 and 19.4 respectively. A recent statistics has further pushed the maximum pre-merger black hole mass to about 55 (The LIGO Scientific Collaboration et al., 2018). It has been a matter of debate, whether the massive black hole forms directly from the collapse of massive star, or has experienced many black hole merger prior to the black hole-black hole merger the gravitational wave detectors observed.

From our model, it becomes clear that the single star scenario has a maximum black hole mass to be formed from the single star scenario. For the He core with a mass more massive than 64 , the star does not experience any collapse, but explode as a pair-instability supernova. The collapse only appears in He core with a mass larger than 260 (for zero metallicity) (Heger & Woosley, 2002). The corresponding black hole mass is .

To connect PPISN with the measured black hole mass spectra, we plot in Fig. 6 the remnant mass against progenitor mass, together with the mass range of the black hole implied by the gravitational wave signals. But for He core mass between 40 - 64 , a mass correction is included to account for the pulsation-induced mass loss. Beyond the star enters the pair-instability regime and no compact remnant object is left. Near the remnant mass is the maximum at . Some of the events can be explained by the current PPISN picture. This includes the primary black hole in the events GW150914, GW170104, GW170729, GW170809, GW170818 and GW170823 and the secondary black hole in the event GW170729. We remark that the actual PPISN black hole mass range can have a wider range, both higher upper limit and lower lower limit. For the higher side, it is because the current spherical model assumes the pulsation and shock propagate in a perfectly spherical manner. However, in reality, such pulse may trigger hydrodynamics instabilities, in particular Rayleigh Taylor instabilities. Therefore, the shock will tend to be dissipated by the mixing process. For the lower limit, it is because during formation of neutron star and black hole, a significant amount () of mass will be lost through neutrino emission and mass ejection. They both remove part of the final black hole mass.

We remark that the use of He core mass as the remnant mass is not necessary the exact black hole mass, but an upper limit of the black hole mass. This is because before all matter is accreted into black hole after the collapse, two explosions are possible to occur. First, during the formation of neutron star, the bounce shock when the core matter reach nucleon density, can eject part of the matter. But whether the bounce shock can successfully reach the surface remains a matter of debate due to the complication of neutrino. When the shock can indeed propagate to the surface, it can efficiently eject the low density matter, mostly He envelope, away. This may significantly lower the final mass of the black hole. The second explosion is the accretion disk jet. Depending on the accretion scenario, it is possible that the accretion disk forms around the black hole. The magnetodynamics instability of the accretion disk can easily fragment the disk and send high-speed jet to the stellar envelope. This may also easily send away partially the infalling envelope, again lowing the mass. However, for both explosions further input physics are necessary. Therefore, for the first estimation of our result, we use the He core mass as a optimistic estimate of the black hole mass.

We note that in a single observation, the solution for matching the black hole mass with our remnant mass is degenerate for both mass and metallicity. To further apply the black hole information in PPISN to constrain the mass loss, population of black hole mass will become important, which can directly constrain the current mass loss model, when combined with suitable stellar initial mass functions.

\H@refstepcounter

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe pre-collapse mass of the PPISN against progenitor mass with the measured black hole masses obtained from binary black hole merger events (Abbott et al., 2016a, b, 2017; The LIGO Scientific Collaboration et al., 2018). The left and right data point corresponds to the primary and secondary black holes respectively. The error bars for the pre-merge neutron star for GW170817 is too small to be seen in the current scale.

In Figure 6 we plot the final stellar mass, C- and O-core masses for all the He-core models. We define the boundary of the C- and O-cores to be the inner boundaries where the local He and C mass fractions drop below . We can see three layers appear. For , 45 and 50 , there are explicit He-envelope, C- and O-layers. For and 60 , the huge mass loss completely eject the pure He layer, which exposes the C-rich layer (combined with He). At , the mass ejection further shreds off the C-rich layer, exposing the O-layer. The whole star has everywhere the mass fraction of C below . Therefore, the He-core and C-core masses coincide with the stellar total mass. From this we can see to what level the mass ejection takes place for the PPISN models. However, we also remark that the definition of He- and C-core masses can be ambiguous at the end of simulations because the matter becomes O-rich before the C is exhausted. Similarly on the surface there can be non-zero C instead of pure He.

\H@refstepcounter

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Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe pre-collapse mass of the PPISN, C- and O- core mass against progenitor He-mass. Here we define the C- and O-core to be the inner boundaries where the local He and C mass fractions drop below .

7 Comparison with Models in the Literature

7.1 Yoshida et al. (2016)

In this section we compare our results with some representative PPISN models in the literature.

In Yoshida et al. (2016), the PPISN model of mass from about 54 to 60 (corresponding to a progenitor mass from 140 to 250 in zero metallicity) are computed. In that work, the calculation is separated into two parts. During the quiescent and pre-pulsation phases, the hydrostatic stellar evolution code is used. During the pulsation phase, the star model is transferred to the dynamical code PPM, which follows the expansion of the star until the mass ejection has ended ( s). Then they map the results to the stellar evolution code again until the next pulsation.

Their 140 and 250 models have similar configurations as our Models 55HeA and 60HeA. First, in their 140 model (250 model), they observe a total of six (three) pulses which ejected 3.99 (7.87) of matter before collapse. Model 55HeA (60HeA) exhibits three (two) pulses before collapse, which ejects 6.78 (8.52) of matter. Our models show a smaller number of pulses, but give similar ejecta mass. This means our models can capture the energetic pulse well, but not the smaller pulses.

Then we compare the ejection timescales. The 140 (250 ) model show all pulses within a period of 0.92 (1434) years, while Model 55HeA (60HeA) shows all pulses within a period of 1341 (2806) years. There is a huge difference in the pulsation period in our Model 55HeA and their 140 model. We notice that the difference comes from the strengths of the pulses. In particular, our second pulse leads to a transition about 100 years while ejecting . The most similar event in their model is the fourth pulse, but with a transition of only 0.279 year.

At last we compare the final core composition. The 140 (250 ) model has an Fe (CO) core mass at 2.57 (43.51) , while in our model, we have 2.49 (38.60) for the Fe (CO) mass. This shows that, despite the difference in the mass ejection history, our models can still capture the major mass ejection events, which results in a similar mass ejection and core composition. However, there is a strong pulse in our He55A model, which is not seen in their 140 model.

7.2 Woosley (2017, 2019)

Next, we compare our models with the models from Woosley (2017). We have chosen the PPISN close to that work; in particular, ours Models He40A, He50A, He60A and He62A can be compared directly with the He40, He50, He60 and He62 models. In Woosley (2017), the Kepler code, which consists of both hydrostatic and hydrodynamics components, is used to follow the whole evolution of PPISN.

First we compare the mass ejection history. In Woosley (2017), there are 9, 6, 3 and 7 pulses with a total mass loss of 0.97, 6.31, 12.02 and 27.82 for Models He40, He50, He60 and He62 respectively. In our models, we have 6, 3, 2 and 2 pulses with a total mass loss of 2.22, 2.61, 9.52 and 12.85 for Models He40A, He50A, He60A and He62A respectively. Again, our code tends to produce fewer pulses and the pulses in general eject fewer matter. One of the differences can be how shock is treated. For a shock-capturing scheme with a larger dissipation, the kinetic energy will be partly dissipated into thermal energy, such that the star is globally thermalized instead of ejecting matter through kinetic pulses. Another origin of the differences can be related to the nature of the instability of PPISN. Since the trigger of the explosive O-burning comes from the pair-instability, which is very sensitive to the initial condition (e.g. how we evolve the stellar evolution model before the pulsation and between pulses) and numerical treatment (e.g. how convection and mass ejection are treated). For example, a stronger contraction can lead to more O-burning in the core, which gives much energetic pulsation and hence more mass loss. In fact, such dependence can also be seen in other field. For example, in the propagation of flame, since it is unstable towards hydrodynamics instability, (Glazyrin et al., 2013). the burning history can be highly irregular in the unstable regime.

Next we compare the timescale of the pulsation. In this work, the whole pulsation until collapse last for 0.38, 61.3, 2806 and 6610 years for the four models, while in Woosley (2017) they are , 0.38, 2695 and 6976 years. It shows that for massive He cores, our results agree with their work but there are large differences when the He core becomes less massive. In that case, our final pulse is always strong enough to re-expand the star again before the final collapse, which significantly lengthens the pulsation period.

Then we compare the Fe core mass. In Woosley (2017) the core has 2.92, 2.76, 1.85 and 3.19 Fe. In our models, we have 3.42, 1.73, 1.64 and 2.66 . We can see there is a dropping trend from Model He40A to He60A, which corresponds to the trend that the pair-instability occurs at lower density when the mass increases. On the other hand, near the pair-instability regime, the pulsation becomes sufficiently vigorous which enhances the NSE-burning.

At last we compare the explosion energy. We compare the Model He62A, which has the largest explosion energy. In our model, in the second big pulse, the star has its total energy increased by erg while the maximum kinetic energy achieved is erg. This is very similar to the result in Woosley (2017), where the pulse is observed to have a kinetic energy of erg.

One major difference we notice is in the pair-instability limit, for Model He64A, our model shows a higher explosion energy. Across the strongest pulse, there is a change of total energy by erg, where the maximum kinetic energy of the system is erg. In Woosley (2017) the kinetic energy is reported to be erg. We observe that the difference comes from the number of pulsation, where our Model 64HeA has two big pulses but only one in their work. The first pulse has incinerated the O in the core while ejecting on the surface. This means the star has to reach a more compact state before the star can explode. As a result, the amount of energy produced in the exploding pulse is much larger.

Our results show a systematically lower number of pulses with slightly lower ejecta mass. The pulsation periods qualitatively agree with each other except for models with a final strong pulse, which may significantly lengthens our pulsation period. Also, in our explosion models, the system tends to store the energy in terms of internal energy instead of kinetic energy, as a result, the star tends to expand globally, where the excess energy and momentum of the star is transferred mostly to the surface. This ejects the low density matter and leaves a bounded and hot massive remnant. Despite the differences in the pulsation, globally the nucleosynthesis agrees with each other because most of the heavy elements are produced by the strong pulses, where our results are consistent with those in the literature.

In Woosley (2019), the He star models are further evolved with mass loss. The final black hole mass is in general lower compared to his previous work.

7.3 Marchant et al. (2018)

This work is one of the recent work which uses the MESA code (Version 11123) (Paxton et al., 2017) to evolve the evolutionary path of PPISN. Their work has a similar setting to this work. Here we briefly compare their results with our results.

They have computed an array of single star models from 40 - 240 with semi-convection, Riemann solver using the HLLC solver and the nuclear reaction network. They treat the mass loss of the star by considering the average escape velocity.

Their work agrees qualitatively with our works. For a lower mass He core model, some distinctive differences can be seen. For example, in their models, multiple pulsations are observed. They observe a total of 4 pulses for the 54 stars (corresponds to 39.73 at He depletion). On the other hand, our 40 He star model gives a total of 6 pulses. They observe in total 0.63 mass ejection before collapse while ours is about 2.2 . The duration in their model is shorter ( year) but ours is longer ( year) after the onset of pulsation. For a higher mass He core model such as (corresponds to 60.04 at He depletion). They show only 2 pulses, which is the same as our 60 He star model. The duration also agrees with each other (their model shows a duration of years while ours is shorter at years. A total of 4.6 mass loss is found in their model (the pre-He depletion mass loss is excluded) while ours is at a higher value .

We notice that their models and our models do not completely agree with each other. We notice that there are some critical differences in the implementation of this work from their work. First, they consider the star from the H main-sequence, with a metallicity at 0.1 . The He-core mass is therefore a function of the progenitor mass, instead of a direct model parameter as controlled in our models. Also, due to the low metallicity, their models consist of a significant amount of H-envelope at the moment of the beginning of CO burning. Furthermore, they use the Riemann solver (HLLC) in the newer version instead of the artificial viscosity scheme. How the pulse transfers into shock at the near-surface area can be different.

8 Conclusion

In this article, we present our study of pulsational pair-instability supernova (PPISN) for the He-core from 40 - 64 (corresponding to 80 - 140 main-sequence stas) using the one-dimensional stellar evolution code MESA. We applied the implicit hydrodynamics module implemented in the version 8118.

We first compute the main-sequence model from pre-main-sequence star until the central temperature reaches K at different metallicity. We studied how the initial metallicity affects the pre-pulsation mass loss through stellar wind and its effects on the He- and CO-core mass.

We then follow the evolution of the He core throughout the main-sequence phase, pulsation until the Fe-core collapse for the He-core from 40 to 64 . We have investigated the energetic, thermodynamics and mass loss history of the pulsation. We have also studied the evolution of chemical abundance in the star during the pulsation. We have examined explicitly how each pulsation changes the chemical composition of the star and how the later convection alters the post-pulsation star.

We show that our results are qualitatively consistent with the results in the literature, although some minor differences can be found. We also discuss the possible connections of PPISN, especially the ones with massive mass ejection, with the recently observed super-luminous supernovae SN 2006gy and PTF12dam. We show that the PPISN model can be a robust candidate for producing massive CSM outside the star, which may be able to explain some super-luminous supernovae based on the shock-CSM interaction. Then we also discuss the possible connections between the massive black holes detected from the gravitational wave (GW) signals from black hole merger and PPISN. We examine the possible mass range from the gravitational wave events of both neutron star-neutron star merger and black hole-black hole mergers. We classify which GW events can be explained by the black holes formed by PPISN as the progenitor.

In the future work, we will focus on the observables of the PPISN in terms of neutrino and light curve. Using our hydrodynamics model calculated from MESA, the expected neutrino signals detected by terrestrial and the expected light curve will be calculated. The results will provide a more fundamental understanding to the properties of PPISN, which may be constrained from the observables of one the the PPISN candidates.

9 Acknowledgment

This work has been supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI Grant Numbers JP26400222, JP16H02168, JP17K05382 We acknowledge the support by the Endowed Research Unit (Dark Side of the Universe) by Hamamatsu Photonics K.K. We thank the developers of the stellar evolution code MESA for making the code open-source. We also thank Raphael Hirschi for the insightful discussion in the stellar evolution of PPISN and his critical comments. We at last thank Ming-Chung Chu for his assistance in editing the manuscript.

\onecolumngrid

APPENDIX

A Effects of convective mixing

\H@refstepcounter

figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHref(upper panel) The central temperature against time for Model 60HeA with and without convection. (lower panel) Similar to the upper panel, but for the central densities.

\H@refstepcounter

figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe chemical abundance profiles for Model 60HeA prior to its second contraction at a central temperature K with convection (upper panel) and without convection (lower panel).

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 A we plot the central temperature (upper panel) and central density (lower panel) against time for 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 A the chemical composition of the star before the second contraction takes place. We pick both star models when it has a central temperature of 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 Fe and Fe 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.

\H@refstepcounter

figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHrefThe central temperature against central density in log scale for two models, the He60A with standard mixing scheme (See also Section 2) and with artificial enforced mixing scheme. The region on the left, above and right are the pair-, photo-disintegration induced and electron capture induced instabilities. The arrows (black solid arrows and red dashed arrow) are the moments of the pulsation for both models and the tested model.

To further demonstrate the importance of convective mixing to the strength and number of pulsations, we perform some contrasting study of two models, one is Model He60A the other is similar to He60A, but with enhanced mixing. We have shown in Section 4 from the Kipperhahn diagram that the convection mixing in Model 60HeA is less strong that during its quiescent phase after pulsation, the star does not exhibit the global convective phase, unlike other models like He40A, He50A and He62A. So, this model becomes a good candidate to demonstrate the effects of convective mixing between pulsation. To provide the enhanced mixing, we enforce the whole star to undergo mixing process during its expansion and when it is fully relaxed. We defined the critical temperature be K below that the star is fully relaxed for convective mixing.

In Figure A we plot the central temperature against central density (both in logarithmic scale) for the two models. The evolution of He60A is exactly the same as that presented in previous section. Here, we look into more details for the model with artificial mixing. Before the second pulse, the two models exhibit exact the same trajectories. It is because the central temperature has barely reached below to trigger the mixing. But after the second pulse, which has mass ejection, its central temperature goes below K. The one with enhanced mixing, because it involves mixing material with the outer elements, which has in genearl lower temperature and lower atomic mass, it can reach a low central temperature during its expansion. Also, the mixing process brings in the C- and O-rich material into the core. In the third contraction, unlike the ”standard” model presented in the main text, the core exhibits the third pulsation. However, the strength is not strong enough to trigger mass loss on the surface. Then, although the core reaches one more time below the for the hand-made convective mixing, the O-abundance of the star becomes too low that the core becomes massive enough to collapse directly, without triggering the fourth pulsation.

B Effects of artificial viscosity

\H@refstepcounter

figure \hyper@makecurrentfigure

Figure 0. \Hy@raisedlink\hyper@@anchor\@currentHref(upper panel) The central temperature against time for Model 60HeA with different levels of artificial visocity. (lower panel) Similar to the upper panel, but for the central densities.

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 . The default value from the package ’ccsn’ in the MESA test suite is . To probe the effects of this parameter, we carry out a control test by varying .

In Figure B the time dependence of the central temperature (upper panel) and central density (lower panel) are plotted for Model He60A with , (default value) and . Results with and are almost identical. This shows that the default choice of can maintain the shock propagation and produce convergent results. On the other hand, when , 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 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.

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