The neutrino emission from thermal processes in very massive stars in the local universe

The important neutrino processes in stellar environment discussed in the literature are pair annihilation, photoneutrino production, plasmon decay, bremsstrahlung and recombination processes. Itoh et al. (1996) and in the series of papers produced by the same group have provided the details of the total emission for a broad range of temperatures and densities based on the well known Weinberg-Salam theory in the Standard Model of particle physics. These include neutrino emissivity that can be read from tables and approximation (fitting) equations that can be utilized directly in stellar models or apply in post-processing work. The total neutrino energy loss or neutrino emissivity, $Q_{\nu}$ is the sum all of the losses from the five different neutrino processes:

where $Q_{\mathrm{pair}},Q_{\mathrm{photo}},Q_{\mathrm{plasma}},Q_{\mathrm{brems}}$ and $Q_{\mathrm{recomb}}$ are the neutrino emissivities from the pair annihilation, photoneutrino production, plasmon decay, bremsstrahlung and recombination processes respectively. In the following sections, we summarize the formalism found in Itoh et al. (1996) that is relevant in our calculations.

In our calculations, VMS models were calculated with the GENEC stellar evolution code by Yusof et al. (2013). Full details of the very massive stars can be obtained from the same publication. It was suggested that for models that were predicted to end as a PISN are three models from the SMC metalicity (Z=0.002) which are 150, 200 and 300 M_⊙ rotating models and the LMC metallicity (Z=0.006), 500 M_⊙ rotating model. We include the SMC 300 M_⊙ model to complete the grids and for this model, it is expected to end up as a core-collapse supernova (CCSN) or massive black hole (BH). The physics ingredients in calculating these models are the same as in Ekström et al. (2012) and Yusof et al. (2013). We summarise below some of the few important ones used in these models:

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  The initial abundances of mass fractions of ¹H, ³He, ⁴He and metal (Z) are listed in Table 1. The mixture of heavy elements, Z is taken from Asplund et al. (2005) except for the Ne abundance which was adopted from Cunha et al. (2006) and the isotropic ratios are taken from Lodders (2003).

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  For the treatment of convection, the Schwarzschild criterion is used and a modest overshooting with an overshoot parameter, $d_{\text{over}}/H_{P}$ = 0.10, is used for core hydrogen and helium burnings only.

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  Mass loss affected strongly the evolution and fate of the PISN progenitors. Prescription by Vink et al. (2001) is used for the main sequence stars. For the domain not covered by Vink et al. (2001), we implement the prescription by De Jager et al. (1988) to the models with log $T_{\mathrm{eff}}>3.7$ while for log $T_{\mathrm{eff}}\leq 3.7$, a linear fit is performed to the data from Sylvester et al. (1998) and van Loon et al. (1999).

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  The neutrino prescriptions are taken from Itoh et al. (1989) and Itoh et al. (1996).

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  The treatment of rotation in the GENEC code has been described in Maeder & Meynet (2012).

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  For the nuclear reaction rates, we implement the basic 21 nuclear isotopes in the nuclear network.

Note that we do not include the $e^{+}e^{-}$ equation of state (EOS) in the GENEC code. This can be seen in Figure 2 where all the selected VMS models enter the instability region instead of stopping before entering the adiabatic region. For the final fate estimation, we use the carbon-oxygen core mass (M${}_{\text{CO}}$) instead of the mass of the helium core. This estimate shows similar fate for stars with the same mass Heger et al. (2003) of the CO core found in various investigations of the early Universe (Bond et al., 1984).

Since the models chosen for this calculation focus on very massive stars, particularly the aforementioned progenitors that are expected to explode as PISN supernova, we carefully examine the temperature and density diagram to check the possibility of the stars entering the stability phase indicated by $\Gamma\geq 4/3$. Figure 3 represents the central temperature-density ($T_{c}$-$\rho_{c}$) diagram with the burning stages of all four models with the start and end of burning stages labelled in the graph. All four models selected in this work have had stopped their evolution at the end of oxygen burning. Since we implement the mass loss mechanism in these models, our stars become a Wolf–Rayet (WR) star when the value of the surface hydrogen mass fraction, $X_{s}$ is below 0.3 and the effective temperature, $\log T_{\mathrm{eff}}$ would be greater than 0.4. Neutrino losses constitute a critical aspect in the stellar evolution of massive stars especially during the end of the He-burning phase (Woosley et al., 2002). Since neutrino losses are very sensitive with temperature and density, the rates of the neutrino losses are expected to increase when the temperature increases due to the heavier fuel that burn at the core of the stars. For very massive stars, this might be different since the initial temperature and density are higher than normal massive stars, thus we would expect to see neutrinos from thermal processes produced earlier than in massive stars since $T_{c}\approx 10^{8}$ K at the end of the H-burning phase (see Figure 3).

Although the models were experiencing mass loss throughout their evolution and even when the rotating is included, the SMC models are able to retain their mass during the main sequence. This phenomenon can be observed in the Kippenhahn diagram as illustrated for 150 M_⊙ SMC model (see Figure 4). We also presented the abundance profile for 150 M_⊙ with respect to the mass fraction in Figure 5 at the oxygen core burning stage. In the abundance profile, our model contains 80% of helium and around 20% of carbon at the surface of the star. The hydrogen is totally depleted in our model. Since all the models are rotating models, the convective core size is higher than their corresponding non-rotating models convective core (for more detail discussion refer to Yusof et al. (2013)).

For 150, 200, 300 M_⊙ SMC and 500 M_⊙ LMC models, we calculated and generated the total neutrino emissivity, $Q_{\nu}$ and neutrino luminosity, $L_{\nu}$ for five thermal neutrino processes at the end of five different nuclear burning stages, hydrogen (H-burning), helium (He-burning), carbon (C-burning), neon (Ne-burning) and oxygen (O-burning). The reason we choose the end of the burning stages is we want to investigate how much the neutrino energy loss is produced before the final carbon burning stage since for very massive stars, the temperature and density are high enough to produce neutrinos as early as during the H-burning stage. The temperature and density are taken from the structure of the models, which are at the chosen burning stages where we took at that particular time step, the values of the temperature, density and the isotropic nuclei from the center to the surface of the star.

We plot the structure profile for temperature-density versus mass fraction for 150 M_⊙ Z=0.002 (SMC) and 500 M_⊙ Z=0.006 (LMC) from center to the surface of the stars (see Figure 6) as an indicator for the cross-check for the prediction of neutrino energy loss. These two variables are important as input to test the neutrino energy loss. The major difference in the profiles can be seen at H-burning when 500 M_⊙ has higher central temperature and density compared to the less massive VMS model.

For very massive stars, their lifetimes are relatively shorter than its lighter siblings. More massive the star is, the temperature and density would be higher inducing faster nuclear reaction rates leading to shorter evolution time. For our selected progenitors, their lifetimes are around 2.3 - 3.0 Myrs. The H-burning (and total) lifetimes of VMS are lengthened by rotations as in lower mass stars (Yusof et al., 2013). As the stars go more massive, it evolves more rapidly where the lifetimes for each stage of nuclear burning depends on the progenitor mass. Neutrino energy loss, if not included in the evolution as early as helium burning stage, the carbon burning will start as soon as helium in the core is depleted (Rose, 1969). Our reason to investigate the neutrino energy loss starting at the early until advanced burning stages is to see how sensitive are density and temperature dependent towards the production of thermal neutrinos since the initial temperature of VMS is relatively high compared to its less massive siblings. From our progenitor models, the lifetime of each burning stages for different masses are relatively homogeneous (see Table 3). Small differences between metallcities are due to the different rate of mass loss at different metallicities. Figure 7 shows the radial distribution of $Q_{\nu}$ at each burning stages for each progenitor models. Most of the time, $Q_{\nu}$ is maximum at $\log R/R_{\odot}\approx-4$ which is located at the central region and start declining rapidly when the radius becomes larger (i.e towards the surface). The same pattern of the $Q_{\nu}$ is observed in massive stars progenitors (Patton et al., 2017a) even the radius of their massive stars is smaller than our models. We note that there is a sharp discontinuity at the surface of the stars at hydrogen and helium burning phases which reflects the shell structure of the star. During H-burning, we observed the neutrino emissivity varies from one model to another due to the difference in the temperature range when the stars start to enter the main sequence.

Since our models are pre-supernova models, at the end of O-burning, the total emissivity, $Q_{\nu}$ from our VMS models are three orders of magnitude higher than the massive star models except for 500 M_⊙ LMC model since it has the same magnitude of neutrino emissivity at the core of the star (see Patton et al. (2017a) Table 2). Stars with lower metalicity (in this case SMC) are more luminous with higher effective temperature. This is reflected in the central temperature where the location of 500 M_⊙ LMC track is at the most bottom in the $T_{c}-\rho_{c}$ diagram (refer to Figure 3).

In Figure 8, we present the result for all neutrino processes (pair, plasma, photo, bremsstrahlung and recombination) and the comparison with the sum of all processes represented by the purple line for 150 M_⊙ using the structure profile at O-burning stage. We particularly choose this stage since we expect our model to stop evolving due to the star is about to enter the PISN stage. The grid points for the x-axis represent the density with range $10^{0.1}\mathrm{gcm^{-3}}<\rho<10^{7}\mathrm{gcm^{-3}}$ and temperature (right Figure 8) of the model from surface to center which spans from $10^{7.5}\mathrm{K}<T<10^{9.5}\mathrm{K}$. It is interesting to note that the pair neutrino process completely dominates the neutrino loss in the inner half of the star before it contributes the minimal towards the total loss after that and then photo-neutrino process takes over until the surface. Active neutrino productions can be seen clearly at the core or centre of the star when the most dense and hot region located. In the hot dense core of very massive stars, when the temperature $T\gtrsim 10{{}^{9}}$ K, the electron-positron pairs are produced because they are in equilibrium with radiation. The interchanging of $e^{-}+e^{+}+\leftrightarrow{\gamma+\gamma}$ occur and this can be annihilated via weak interaction too, $e^{-}+e^{+}\leftrightarrow{\nu+\bar{\nu}_{e}}$. This contributes to the pair annihilation process that is important in both massive and very massive stars. Degenerate and non-degenerate environments affect the thermal processes that occur in the VMS. For pair annihilation, this process declines as $\rho^{-1}$ in non-degenerate regions and causes the star to contract. In degenerate situations, the electron-positron pair creation suppresses the pair neutrino production and the energy loss rate of the neutrinos drops.

Photoneutrino from the analog of Compton scattering dominates at the non-degenerate electron gas at lower energy which can be describe as

Photoneutrino process plays an important role in removing the thermal energy from degenerate core during the early stage of H and He nuclear burnings. During the early stage of evolution the dominant process that involve is photoneutrino for all models of VMS (see Figure 1). This process is severely suppressed at very high density because of the absence of unoccupied electron states and pair production also similarly suppressed at very high density and causes the pair neutrino emission greatly reduced (Festa & Ruderman, 1969).

Based on the relation of neutrino emissivity and temperature-density region, Figure 1 shows that bremsstrahlung, plasma and recombination processes are the least important in neutrino production in very massive stars and hence the contribution to the neutrino energy loss. In our result, neutrinos produced from bremsstrahlung overwhelmed the neutrino production from plasma decay process. At the region where the electrons are strongly degenerate, plasma decay is suppressed and it leads to the dominance of bremsstrahlung neutrino emission (Guo & Qian, 2016).

When a star becomes dense, photons in the dense gas have an effective mass and are termed as plasmons and can decay into neutrino-antineutrino pairs. This physical process creates the plasma neutrinos. In the plasma decay, photons and longitudinal plasma may decay into neutrino-antineutrino pairs $\gamma\rightarrow{\nu_{x}+\bar{\nu_{x}}}$ of all flavors $x$. As mentioned in Eq. ($3$) and Eq. ($4$), for the $\nu_{\mu}$ and $\nu_{\tau}$ spectra, the value of axial part contribution is very small. This suppresses these neutrino flavors.

Recombination neutrino process starts at $\rho\lesssim 10^{3.7}$ g cm^-3 and temperature $T\lesssim 10^{8.9}$ K and this happens at mass shell $\gtrsim 0.8$. This is consistent with the temperature-density region shown in Figure 3 where this process is dominant. At values of density and temperature greater than these limits, electrons are relativistic and the recombination treatment becomes invalid (Itoh et al., 1996). As this process also depends strongly on the heavy element abundances $\thicksim$ Z¹⁴, oxygen being the most abundant element drops drastically at this mass shell (Figure 5) resulting in the reduction of the recombination neutrino emissivity towards the surface.

From Table 3, we presented the summary for each burning stages in VMS including lifetime, central temperature, central density, neutrino luminosity for all thermal processes (plasma, bremsstrahlung, pair, photo and recombination) and the energy loss, $Q_{\nu}$. From our calculation, at the early burning stages (H and He), photoneutrino process contributed the most in the production of thermal neutrinos. The value of luminosity is lower than the value predicted by Odrzywolek & Heger (2010) for H-burning phase since we only considered thermal processes and do not include any contribution from $\beta$ decay processes. However, the neutrino luminosity at He-burning phase from photoneutrino process and not from plasma process that contributed around $10^{39}$ erg/s which is higher from other previous studies (see for example (Odrzywolek & Heger, 2010). This is due to VMS has higher density and temperature ($10^{2}$ gcm^-3 and $10^{8}$ K respectively) that falls within the photoneutrino region. This is also confirmed by our illustration in Figure 1. Carbon burning phase is an important phase in the life of massive or very massive stars. During this phase, extreme temperature regime is required for C-burning and this induced the production of $e^{+}e^{-}$ pairs from thermal distribution. These pairs can annihilate into $\nu\bar{\nu}$ pairs. At this stage, the central temperature of VMS is $T\sim 10^{9}$ K and this leads to the powerful neutrino energy loss in the stars. Neutrinos from thermal processes have overtaken nuclear energy in the energy production that make the star shift to a $\nu$-cooled star. The burning process continues to the Ne and O-burning stages. At these two phases, the duration of the burning in very massive stars are relatively very short, $\sim 78$ and $\sim 26$ days respectively where the core of oxygen burning becomes a classical neutrino-cooled stage. We also observed like in the C-burning phase, the pair neutrino is the most dominant process at Ne and O-burning phases given the high temperature and density at these phases. For our progenitors models, all models fall within the PISN explosion based on the estimate of M_CO (see Yusof et al. (2013) for details). From our luminosity calculation, the orders of magnitude for all neutrino processes are consistent with calculation done by Wright et al. (2017) and Leung et al. (2020) although their progenitor models have lower metallicity than our models. During the neon/oxygen burning phase, the VMS will experience instability due to the production of $e^{+}e^{-}$ pairs from photons. The creation of $e^{+}e^{-}$ pairs in the oxygen rich core soften the equation of states, leading to further contraction and this runaway collapse is predicted to produce a powerful explosion, with energy $10^{53}$ erg, that would break entirely the structure of stars without remnants (Kasen et al., 2011; Whalen et al., 2014; Gilmer et al., 2017).

The stars are assumed to be in local thermodynamic equilibrium and thus the time derivative would vanish leaving the neutrino luminosity, $L_{\nu}$ production described by the differential equation

where $M_{r}$ is the mass shell at the radius $r$ of the star. The luminosity differential equation is integrated over the models to obtain the total luminosity at the surface of the stars. We use the neutrino emissivity $Q_{\nu}$ to calculate the rate of the neutrino energy $\epsilon_{\nu}(\rho,T)$ released per unit stellar matter from the thermal processes. This integration is done for all burning stages of each VMS.

We presented the total luminosity $L_{\nu}$ of neutrino production at different burning stages with respect with the initial mass in Figure 9. From the graph, we verify that neutrino productions from thermal processes increased rapidly when the stars start to experience the C-burning phase. At the O-burning stage, where the luminosity $\approx 10^{51}$ erg/s, we found that the models at SMC metallicty have higher luminosity compared to the model at the LMC metallicity even when the mass is higher in LMC. This result is consistent with the total luminosity calculated for pulsation pair instability supernovae (PPISN) by Leung et al. (2020) and PISN (Wright et al., 2017). For 500 M_⊙ model, their lifetime span is shorter than the lighter VMS SMC models and less hotter at O-burning phase due to initial temperature difference during ZAMS between these two metallicities. Although there are work (see Moriya (2014)) that show mass loss of massive stars near Eddington luminosity has impacted on the neutrino luminosity where mass loss has in fact increased the neutrino luminosity; this impact didn’t effect much for VMS stars.

The evolution of neutrino luminosity $L_{\nu}$ with age for each nuclear burning stages is shown in Figure 10. From the graph, the evolution of neutrino luminosity is consistent with mass at the same metallicity. Comparing with the neutrino luminosity of 20 M_⊙ ($L_{\nu}=7.4\times 10^{39}$ erg/s) calculated by Odrzywolek et al. (2004), $L_{\nu}$ for the 150 M_⊙ model ($L_{\nu}\sim 10^{45}$ erg/s) is six orders of magnitude larger at the C-burning stage meanwhile at O-burning, the 150 M_⊙ neutrino luminosity ($L_{\nu}\sim 10^{45}$ erg/s) is three orders of magnitude larger than the 20 M_⊙ ($L_{\nu}\sim 10^{43}$ erg/s) at the same stage. Our neutrino luminosity calculation is in agreement with the total PISN neutrino luminosity for 250 M_⊙, $Z=10^{-3}$ (Wright et al., 2017) where the progenitor model was generated from the same stellar evolution code. It is well known that photon luminosity is always greater than neutrino luminosity of stars at the main sequence (Masevich et al., 1965; Shi et al., 2020; Farag et al., 2020). It would be an interesting test to estimate the difference between photon luminosity, $L$ with neutrino luminosity for each stages. In Table 4, we demonstrate the comparison between these two different luminosities. For photon luminosity, photon energy is released by $\sim 10^{40}$ erg/s throughout its evolution. For neutrino luminosity, the value is smaller at the main sequence which confirms our hypothesis with a difference around seven orders of magnitude lower than the photon luminosity. Even in He-burning, photon luminosity is higher since energy generated from the stars is dominated by nuclear reactions. The neutrino luminosity increases rapidly when its enter the neutrino cooling phase which is C-burning stage and it has overtaken the photon luminosity by six orders of magnitude indicating that large number of neutrinos are escaping from the stars.

From our result, pair annihilation process is dominant during the advanced stages of of evolution, C-burning, Ne-burning and O-burning for $150$ M_⊙, $200$ M_⊙ and $300$ M_⊙ models as shown in Figure 1. Although we investigated the emissivity of all types of neutrinos from the progenitor phase, it is beyond the scope of this paper to discriminate the contribution of each neutrino flavor on the emissivity.