Effects of Nuclear Equation of State on Type-I X-ray Bursts and Implication for Clocked Burster GS 1826–24

. We study the e ﬀ ects of nuclear equation of state (EOS) on Type-I X-ray bursts, focusing on the Clocked burster GS 1826–24. According to the shell-ﬂush model, the surface gravity of the neutron stars (NSs) strengthens burst activities, where the recurrence time and peak luminosity are higher for larger-radius EOSs. We show that such a prediction matches with the full multi-zone model, in which whole NS regions are considered in burst calculation. We also show that the large-radius EOSs are not preferred to account for the observed light curves of GS 1826–24.


Introduction
Type-I X-ray bursts, which are caused by unstable H or He burning onto a neutron star (NS), are valuable phenomena to understand the various properties of X-ray binaries such as the accretion rate, compositions of accreted matter, and nuclear reaction rates relevant to the rp process. Few observed bursters show the constant shape of light curves in a burst sequence. They are called clocked bursters, which are useful for constraining the model parameters of low-mass X-ray binaries. Several works have been done for modeling of a representative clocked burster, GS 1826-24, and finally constrained model parameters. These studies, however, mostly focused on the outer layers of the NS (e.g., [1]).
The physical properties inside the NS, such as the nuclear equation of state (EOS), crustal heating, and neutrino cooling processes, can affect the X-ray-burst light curves. Thanks to the recent developments of NS observations, the nuclear EOS has been rigidly constrained. In fact, the observation of the highest NS mass beyond 2M [2][3][4] ruled several out softer EOSs. For the NS radius, several experimental and observational constraints have been established (for a recent review, see [5]), but there remains significant uncertainties.
In this study, we investigate the dependence of light curves for X-ray bursts on the NS EOS, focusing on the uncertainties of the NS radius. We compare our calculated burst models with the observations of GS 1826-24.   To calculate X-ray-burst light curves considering the central NS, we use a onedimensional general-relativistic stellar-evolution code [6] (hereinafter called full multi-zone model). Namely, we follow the thermal evolution of accreting NSs in quasi-hydrostatic equilibrium. To reduce the costs of numerical calculations, we adopt a reduced nuclear-reaction network with 88 nuclei for H or He mixed burning (APRX3 in [7]). For time evolution of H and He mass fractions and energy generation rates, APRX3 reproduces the resulting energy generation rates with a large reaction network with 897 nuclei within the 40% error.

The methods of multi-zone X-ray burst calculations
We implement sets of realistic nuclear EOSs of Togashi [8], LS220 [9], TM1e [10], and TM1 [11]. The information of NS EOS such as the pressure and energy density comes from the energy per nucleon w(ρ B , Y p ), where ρ B is the baryon density and Y p is the proton fraction, which can be expanded around the saturation density ρ 0 : where w 0 , K 0 , E sym , and L are, respectively, energy per nucleon, incompressibility, symmetry energy, and its slope at ρ 0 . Thus, any EOS can be mostly characterised by the five parameters. For our adopted EOSs, we show their values in Table 1. The important parameter for the NS radius is L, being still uncertain (e.g., [12]). In our EOSs, the TM1 has the largest L value and radius of R   The relation between the mass-radius and burst behavior is roughly expressed by the shell-flush model, assuming the constant pressure during a burning onto the NS [13]: where σ is the column depth and g s is the surface gravity, which considers the general relativistic effect. We show the value of g s in Figure 1. Suppose that P ign is independent of the NS mass and radius, σ is higher with a large-radius EOS. This implies that the observable recurrence time ∆t and the peak luminosity L peak should also be higher. The EOS uncertainties affect burst behavior, which should be clarified in full multi-zone X-ray burst models. Since P ign comes from the heating and cooling processes inside a NS, it should not be a free parameter, unlike the shell-flush model. Many previous studies with multi-zone Xray-burst models, which cover the only accreted layers, introduce a similar artificial heating parameter Q b to impose the envelope and crust boundary conditions (e.g., [1]). To treat the internal energy of NS consistently, we consider the conventional crustal heating process [14] and slow neutrino cooling processes, including the modified Urca and bremsstrahlung (e.g., see [15]). Such cooling processes effectively increase P ign , because they decrease the overall temperature before the NS becomes very old ( 10 5 yrs). Therefore, they change the burst behavior from Eq. (2). In the proceeding, however, we simply focus on the EOS dependence, which does not significantly change neutrino luminosity without any enhanced cooling processes.
To focus on the EOS uncertainties, we fixṀ −9 = 2.5 and Z CNO = 0.01 in the proceeding, whereṀ −9 is the accretion rate normalized by 10 −9 M yr −1 and Z CNO is the initial metalicity. Note that the influence on light curves is crucially important, but the trend of EOS dependence does not change even if other parameters related to the companion are chosen. As the initial model for calculations, we use the steady-state model of accreting NSs (see [16]).

Results
In Figure 2, we show the dependence of ∆t and L peak on the EOS. As we see, if the radius is smaller, ∆t and L peak are lower because the amount of accreted matter is increased due to a stronger gravitational field. It finally shorten the ignition time, and therefore this trend is  [17]. There is no model matching with observed ∆t, but ifṀ −9 , Z CNO and NS mass change, it could be reproduced (see the model of "L2n20Z1" in [7];Ṁ −9 = 2.0, Z CNO = 0.01, LS220 EOS, and M NS = 1.58 M .). We show that the comparison with the observed light curve of GS 1826-24 in 2007 in Figure 3. For the TM1e, the peak luminosity is higher than the Togashi due to the fuel amount difference. Thus, the Togashi agrees with the shape of observed light curves, while the TM1e disagrees. The luminosity is too high in large-radius NSs and sometimes reaches the Eddington luminosity. However, GS 1826-24 do not show photospheric radius expansion. Therefore, large-radius EOSs (e.g., TM1) tend to be rejected, although the model comparison highly depends on other input parameters, e.g.,Ṁ −9 and Z CNO .

Conclusion
We study the dependence of the NS EOS on Type-I X-ray burst using multi-zone X-rayburst models, with an emphasis on the observation of a clocked burster GS 1826-24. We find that ∆t and L peak are higher with larger-radius EOSs, whose trend are consistent with the shell-flush model. As a result of the comparison with the observation of GS 1826-24, we also find that larger-radius EOS is unpreferred due to higher luminosities. Since the EOS is related to internal heating and cooling processes, self-consistent burst calculations including the evolution of a NS are necessary. In the burst calculations presented here, we adopt the conventional heating and cooling processes, which should be updated. The more detailed analysis which are done with a series of X-ray burst calculations are shown in the paper [18].