Determination of the symmetry energy from the neutron star equation of state

We analyze the uncertainties introduced in the determination of the neutron star matter proton fraction, in a range of densities close to the saturation density, if the cold $\beta$-equilibrium neutron star matter equation of state (EOS) is known. In particular, we discuss the effect of neglecting the muon contribution and of considering that the energy density of nuclear matter is well described by taking only terms until second order in the proton-neutron asymmetry. It is shown that two types of uncertainties may be associated with the extraction of the symmetry energy from the $\beta$-equilibrium EOS: an overestimation if terms above the parabolic approximation on the asymmetry parameter are neglected, or an underestimation if the muon contribution is not considered. The effect of the uncertainty on the symmetric nuclear matter EOS on the determination of the proton fraction is discussed. It could be shown that the neutron star mass-radius curve is sensitive to the parabolic approximation on the asymmetry parameter.

Figure 1

Relative deviation between the proton fraction $x_p$ calculated using Eq. (23) and the exact one $x$ given by the DD2 model, considering in Eq. (23): no contribution from $S_4$ (black), $S_4$ calculated with the effective mass $m^{*}$ (blue), and $S_4$ calculated with the vacuum mass $m_N$ (red). The line type indicates which leptons are considered in the Eq. (23), when extracting $x_p$, and in the reference model (DD2) to calculate $x$: ${ϵ}_{\mu }=0$ and ${ϵ}_{\beta }^e$ (solid), ${ϵ}_{\mu }=0$ and ${ϵ}_{\beta }^{e,\mu }$ (dashed), ${ϵ}_{\mu }\ne 0$ and ${ϵ}_{\beta }^{e,\mu }$ (dotted).

Figure 2

Relative deviation between the proton fraction $x_p$ [Eqs. (23) and (24)] and the exact one $x$ given by the DD2 model, for the different approximations defined in Eq. (4). Muons were not considered, i.e., ${ϵ}_{\mu }=0$ and ${ϵ}_{\beta }^e$. Top panel: the $S_4$ term and higher orders were neglected, middle panel: the $S_4$ term was included using the vacuum mass, bottom panel: the $S_4$ term was included using the saturation effective mass.

Figure 3

The same as in Fig. 2 but now considering that muons were included (${ϵ}_{\beta }^{e,\mu }$) when determining $x$ with DD2 model but not in the determination of $x_p$ (${ϵ}_{\mu }=0$). See Eqs. (23) and (24).

Figure 4

The same as in Fig. 2 but now considering that muons were included both when determining $x$ (${ϵ}_{\beta }^{e,\mu }$) with DD2 model and in the determination of $x_p$ (${ϵ}_{\mu }\ne 0$). See Eqs. (23) and (24).

Figure 5

The exact DD2 symmetry energy (black line) is compared with the symmetry energy calculated from Eq. (23) with ${ϵ}_{\mu }=0$ and with different approximations for the symmetric nuclear matter EOS, $E_{SNM}/A$ (see text for details). The brown line in the top panel was obtained taking the DD2 expansion terms up to second order in Eq. (6).

Figure 6

Symmetry energy (top panel) and relative deviation from the exact symmetry energy (bottom panel) predicted by Eqs. (17) and (23) with ${ϵ}_{\mu }=0$ and using for $E_{SNM}$ a Taylor expansion up to second order with ${\rho }_0=0.164{\mathrm{fm}}^{-3}$, $E_0=15.86\mathrm{MeV}$, and $K_0=215\mathrm{MeV}$ (middle line) and ${\rho }_0=0.171(0.158){\mathrm{fm}}^{-3}$, $E_0=16.43(15.29)\mathrm{MeV}$, and $K_0=175(255)\mathrm{MeV}$ define the bottom (top) limits [50]. The blue (red) bands considers ${ϵ}_{\beta }^{e,\mu }$ (${ϵ}_{\beta }^e$).

Figure 7

Top panel: Symmetric nuclear matter energy per particle $E_{SNM}/A$; bottom panel: symmetry energy extracted from the Eq. (23) with ${ϵ}_{\mu }=0$. For $E_{SNM}/A$ a Taylor expansion functional with terms until fourth order fitted to the DD2 $E_{SNM}/A$ in the range of densities $0.08<\rho <0.8{\mathrm{fm}}^{-3}$ was considered, see Table 1. The dashed green line represents a fourth order expansion for $S(\rho )$ [Eq. (6)] with the parameters defined in Table 1. The yellow dashed-dotted and solid red lines show $S(\rho )$ when muons are or are not considered (${ϵ}_{\beta }^{e,\mu }$ or ${ϵ}_{\beta }^e$), respectively, in Eq. (23).

Figure 8

Mass-radius curve for the DD2 model (full black line) and (left panel) taking the fits using Eq. (25) (fourth and sixth order) and (6) (fourth order), (right panel) taking the fourth order fits and including the $S_4$ term in the energy calculated with the vacuum mass or the effective mass at saturation. No muons have been included in the $\beta$-equilibrium EOS, i.e., ${ϵ}_{\beta }^e$.