Evolution of a proto-neutron star with a nuclear many-body equation of state: Neutrino luminosity and gravitational wave frequencies

In a core-collapse supernova, a huge amount of energy is released in the Kelvin-Helmholtz phase subsequent to the explosion, when the proto-neutron star cools and deleptonizes as it loses neutrinos. Most of this energy is emitted through neutrinos, but a fraction of it can be released through gravitational waves. We model the evolution of a proto-neutron star in the Kelvin-Helmholtz phase using a general relativistic numerical code, and a recently proposed finite temperature, many-body equation of state; from this we consistently compute the diffusion coefficients driving the evolution. To include the many-body equation of state, we develop a new fitting formula for the high density baryon free energy at finite temperature and intermediate proton fraction. We estimate the emitted neutrino signal, assessing its detectability by present terrestrial detectors, and we determine the frequencies and damping times of the quasinormal modes which would characterize the gravitational wave signal emitted in this stage.

Figure 1

$T=0$ mass-radius diagrams for the three EOSs considered in this work.

Figure 2

The pressure (upper panels), the energy density (middle panels), and the speed of sound (bottom panels) are plotted versus the baryon number density for the EOSs considered in this paper, and for different values of selected parameters [cases (i)–(iii) described in Sec. 2d]. The black solid line refers to the GM3 EOS determined by solving numerically the mean-field equations, the black crosses to the GM3 EOS determined through the fit and the procedure described in Sec. 2c, the blue dashed line to the LS-bulk EOS, and the red dot-dashed line to the CBF-EI EOS.

Figure 3

Comparison among the three EOSs considered in this paper in the three cases described in Sec. 2d. In the left and central plots we show the temperature and in the right plot we show the entropy per baryon. Colors and line styles are as in Fig. 2.

Figure 4

Comparison between the diffusion coefficient $D_2$ (upper panel), the electron neutrino scattering mean-free paths [middle panel, the neutrino incoming energy is $E_{{\nu }_e}=\max ({\mu }_{{\nu }_e},\pi T)$], and the baryon effective masses, $m^{*}/m_n$ (lower panel) for the three EOSs considered in this paper in the three cases described in Sec. 2d. For the GM3 EOS, the effective masses of proton and neutron are identical [22, 23]. We do not show the LS-bulk EOS effective masses, since we have set them equal to the bare ones, $m_{\{p,n\}}^{*}/m_n=1$. Colors and line styles are as in Fig. 2, apart for the line styles in the lower panel, where the CBF-EI proton (neutron) effective masses are dotted (double dashed).

Figure 5

The initial profiles (at $t=200\mathrm{ms}$) of the baryon density (left), entropy per baryon (center), and lepton fraction (right) are plotted versus the enclosed baryonic mass for a $M_{\mathrm{B}}=1.60M_{\odot }$ star. The initial entropy per baryon and lepton fraction profiles are the same for the three EOSs adopted, whereas the baryon density depends on the EOS (colors and line styles are as in Fig. 2).

Figure 6

Time dependence of the maximum and central temperature (left and central upper plots), central entropy per baryon (right upper plot), neutrino and proton fraction (left and central lower plots) and central baryon density (right lower plot), for a star with total baryon mass $M_{\mathrm{B}}=1.60M_{\odot }$ evolved using the three EOSs. Colors and line styles are as in Fig. 2. The three vertical lines in the temperature plots mark the end of the Joule-heating phase (see text).

Figure 7

Time dependence of the total neutrino luminosity (upper panels), gravitational mass (middle panels), and stellar radius (lower panels) of a PNS evolved with the three EOSs considered in this paper and the baryon stellar masses $M_{\mathrm{B}}=(1.25,1.40,1.60)M_{\odot }$. The black solid lines correspond to the GM3 EOS determined through the fit and the procedure described in Sec. 2c, the blue dashed lines to the LS-bulk EOS, and the dot-dashed red lines to the CBF-EI EOS. The gravitational masses at the end of the simulations are: for $M_{\mathrm{B}}=1.25M_{\odot }$, $M_{\mathrm{GM}3}=1.1554M_{\odot }$, $M_{\mathrm{LS}-\mathrm{bulk}}=1.1492M_{\odot }$, $M_{\mathrm{CBF}-\mathrm{EI}}=1.1548M_{\odot }$; for $M_{\mathrm{B}}=1.40M_{\odot }$, $M_{\mathrm{GM}3}=1.2824M_{\odot }$, $M_{\mathrm{LS}-\mathrm{bulk}}=1.2750M_{\odot }$, $M_{\mathrm{CBF}-\mathrm{EI}}=1.2806M_{\odot }$; and for $M_{\mathrm{B}}=1.60M_{\odot }$, $M_{\mathrm{GM}3}=1.4478M_{\odot }$, $M_{\mathrm{LS}-\mathrm{bulk}}=1.4386M_{\odot }$, $M_{\mathrm{CBF}-\mathrm{EI}}=1.4442M_{\odot }$.

Figure 8

Signal in the Super-Kamiokande III Cherenkov detector, for the three EOSs considered in this paper. In the top panels, electron antineutrino detection rate; in the bottom panels, electron antineutrino cumulative detection. In the left plots, we consider a star with $M_{\mathrm{B}}=1.25M_{\odot }$, in the central plots $M_{\mathrm{B}}=1.40M_{\odot }$, and in the right plots $M_{\mathrm{B}}=1.60M_{\odot }$. Colors and line styles are as in Fig. 7.

Figure 9

Time dependence of the PNS quasinormal mode frequencies and damping times for the three EOSs and for $M_{\mathrm{B}}=1.40M_{\odot }$.

Figure 10

Ratio between the first $p$-mode period and the fundamental period as a function of the quantity $Q_0$, for a PNS in the configurations studied in this paper. The points at the top-left correspond to late evolutionary times, whereas the initial configurations are in the bottom right. See text for details.

Figure 11

Total energy $E_{\text{total}}$ and total lepton number $N_{\text{total}}$ conservation for a PNS with the CBF-EI EOS and $1.60M_{\odot }$ baryon mass, normalized with the stellar initial energy and lepton number (our simulations start at 200 ms, see Sec. 4b). The time step is changed during the evolution in such a way that the relative variation in a time step of the profiles of entropy per baryon and lepton fraction is approximately equal to ${10}^{-4}$.

Figure 12

Total energy $E_{\text{total}}$ (normalized with the stellar initial energy) and instantaneous energy fractional conservation for a PNS with the CBF-EI EOS and $1.60M_{\odot }$ baryon mass. The time step $\mathtt{dt}$ is kept fixed during the evolution and $\mathtt{n}$ is the number of grid points. The plots begin at 200 ms because this is the initial time of our simulations (see Sec. 4b).

Figure 13

Dynamical time scale (solid lines) and beta equilibrium time scale (dashed lines) profiles at different times for a PNS with the CBF-EI EOS and $1.60M_{\odot }$ stellar mass.

Figure 14

Stellar temperature (solid lines) and critical temperature (dashed lines) for the formation of alpha particles at different times, for a PNS with the CBF-EI EOS and $1.60M_{\odot }$ baryon mass. When the baryon density reaches the nuclei density, $0.155{\mathrm{fm}}^{-3}$, alpha particles could not form and we do not plot the critical temperature anymore.