Constraints on the nuclear symmetry energy via asteroseismology

We examine the fundamental torsional oscillations of the crust in the neutron star. Especially, we focus on the crystalline properties obtained from macroscopic nuclear models. As a result, we find that the frequencies of fundamental torsional oscillations are sensitive to the density dependence of the symmetry energy, but almost independent of the incompressibility of symmetric nuclear matter. Identifying the lowest frequency among the quasi-periodic oscillations observed in the giant flares as the fundamental ℓ = 2 torsional oscillations, we put constraints on the density derivative of the symmetry energy as L ≊ 50 MeV.


Introduction
One of the most promissing evidences about the oscillations of neutron stars are the quasiperiodic oscillations (QPOs) in giant flares observed in soft gamma repeaters (SGRs). The SGRs are considered as the candidate of magnetars, which are neutron stars with strong magnetic fields [1]. In addition to the sporadic X-and gamma-ray bursts, SGRs rarely emit much stronger gamma-rays called "giant flares". Up to now, at least three giant flares have been detected, i.e., SGR 0526-66, SGR 1900+14, and SGR 1806-20. Through the timing analysis of the decaying tail of giant flares, the existence of QPOs has been discovered, whose frequencies are in the range from tens Hz up to a few kHz [2]. To explain the observed QPO frequencies theoretically, many attempts have made with the torsional oscillations in the crust region and/or the magnetic oscillations (e.g., [3,4,5,6,7,8,9,10,11]). Associated with the QPOs in giant flares, there also exist the suggestions of the importance to consider the microscopic properties of crust region [12] and the possibility to obtain the information about nonuniform nuclear structure in the crust [13].
The structure of neutron stars can be described as follows. The ocean of melted iron exists in the vicinity of stellar surface for density lower than 10 6 − 10 8 g/cm 3 . Subsequently, the elastic crust region extends up to a density of the order of saturation density of nuclear matter, ρ s ∼ 3 × 10 14 g/cm 3 . At last, the inner region where the density is higher than ∼ ρ s corresponds to the fluid core of neutron stars. The nuclei in the greater part of the crust form a body center cubic (bcc) lattice due to Coulomb interactions. But the recent studies suggest the existence of a nonuniform nuclear structure in the bottom of the crust, i.e., the shape of nuclear matter could be changing from sphere (bcc lattice) to cylinder, slab, cylinder hole, spherical hole, and uniform matter (inner fluid core) with increasing the density [14,15]. This variation in the shape of nuclear matter is known as "nuclear pasta". It should be noticed that not only the density region of pasta phase but also the charge number of spherical nuclei depend strongly on the symmetry energy density derivative coefficient, L (see below for the definition of L) [16]. However, to see the properties of crust in neutron stars is not so easy. One way to solve this difficulty might be the constraint via the observations of the neutron star oscillations, which is well-established as "asteroseismology". In such a way, one could be able to see the stellar properties and/or to restrict on the stellar equation of state (EOS) (e.g., [17,18,19,20]). In this article, we will examine the torsional oscillations in the neutron star crust and set make a restriction on the curst EOS via observations. The more details of this study can be found in Ref. [21].

Crust Equilibrium Configuration
The bulk energy per nucleon near the saturation point of symmetric nuclear matter can be written as a function of number density, n, and the neutron excess, α [22]; where w 0 , n 0 , and K 0 are the saturation energy, saturation density, and incompressibility of symmetric nuclear matter, respectively. Additionally, the parameters L and S 0 are associated with the symmetry energy coefficient, S(n), which corresponds to the coefficient of α 2 in Eq.
(1). That is, S 0 is the symmetry energy coefficient at n = n 0 , while L = 3n 0 (dS/dn) n=n 0 is the symmetry energy density derivative coefficient. Three parameters of them, w 0 , n 0 , and S 0 , can be comparatively easier to determine via experiments in the laboratory [23]. On the other hand, the remaining two parameters, L and K 0 , are more difficult to determine experimentally. Now, instead of L, we introduce a new parameter, y, defined as y = −K 0 S 0 /(3n 0 L), which denotes the slope of the saturation line in the vicinity of α = 0 [16,23]. This is because y can express the characteristic of EOS, although y is not a parameter determined from experiments directly. Hence, the curst EOS can be parametrized by two parameters, y (or L) and K 0 . Similar to [16], in this article, we adopt that 0 < L < 160 MeV and 180 MeV ≤ K 0 ≤ 360 MeV, as long as y < −200 MeV fm 3 . In practice, these parameter ranges are consistent with the mass and radius data for stable nuclei and effectively cover even extreme cases [16]. With such parameters, using a Thomas-Fermi model to agree with the experimental results of stable nuclei, the EOS in the crust region can be obtained. The adopted values of parameters are shown in Table 1, where n 1 corresponds to the baryon number density at which the spherical nuclear matter changes to cylindrical one, while n 2 corresponds to the density at which the nuclear matter becomes uniform. Namely, the interval between n 1 and n 2 is corresponding to the nuclear pasta phase. The equilibrium structure of nonrotating neutron star becomes the static, spherically symmetric solution of the Tolman-Oppenheimer-Volkoff (TOV) equations. In this case, the metric can be described as where Φ and Λ are functions with respect to r. In order to be closed the equation system, one needs to prepare the additional equation, i.e., EOS. Then, if knowing the EOS inside the neutron star, one can construct the neutron star models with one parameter, such as the central density of stars. However, the EOSs for inner core are still quite uncertain, in spite of many suggestions  based on the assumptions of nuclear interactions. In order to avoid such an uncertainty of EOSs for inner core, as in Ref. [24], we construct the crust region with the crust EOS mentioned the above by integrating from the stellar surface inward to the boundary between the curst and core regions. In this way, one needs two parameters to determine the stellar model, where we adopt the stellar mass and radius, M and R, as those parameters.
In the limit of zero temperature, the shear modulus can be described as where n i and +Ze are corresponding to the ion number density and the ion charge, while a is defined as a 3 = 3/(4πn i ), which corresponds the average ion spacing [25,26]. It should be noticed that this approximate formula for the shear modulus is derived with the assumption of bcc crystalline state, which is averaged over all directions. So, if one takes into account the effect of nonuniform nuclear structure known as nuclear pasta, the shear modulus might be modified [13]. Actually, it is suggested that the elastic properties in pasta phase could be softer than that in bcc lattice [27]. However, since the realistic shear modulus in pasta phase is not revealed yet, in this article we will deal with the pasta phase as liquid, i.e., µ = 0 for n ≥ n 1 . With this assumption, the shear modulus may be effectively estimated smaller. As a result, the corresponding frequencies of torsional oscillations also become smaller, i.e., the obtained frequencies in this article could be corresponding to lower limit. But, as mentioned later, the effect of pasta phase might not be so important in the torsional oscillations.

Torsional Oscillations and Numerical Results
Since the torsional oscillation on the spherically symmetric star is incompressible, this type of oscillation does not induce the stellar deformation and density variation. Thus, one can determine the frequencies of torsional oscillations with satisfactory accuracy even if the metric perturbations would be neglected by setting δg µν = 0, which is known as the relativistic Cowling approximation. With this approximation, the torsional oscillations can be described with one perturbation variable, i.e., the angular displacement of the stellar matter, Y, which is associated with the φ-component of the perturbed 4-velocity of fluid, δu φ , as δu φ = e −Φ ∂ t Y(t, r)∂ θ P (cos θ)/ sin θ. Here, ∂ t and ∂ θ denote the partial derivative with respect to t and θ, while P is the Legendre polynomial of order . Assuming that Y(t, r) = e iωt Y(r), one can obtain the perturbation equation from the linearlized equation of motion [28]; where and p correspond to the energy density and pressure, and the prime denotes the derivative with respect to r. In order to determine the eigenfrequencies, ω, we adopt the following boundary conditions; a zero-traction condition at n = n 1 and zero-torque condition at the stellar surface. The both conditions can reduce to Y = 0 [6,28]. First, to see the dependence of the fundamental torsional oscillations on the EOS parameters, we calculate their frequencies on the typical stellar model with M = 1.4M and R = 12km. Fig.  1 shows the frequencies of = 2 fundamental torsional oscillations as a function of L, where the lines with circles, diamonds, and squares are corresponding to the results for K 0 = 180, 230, and 360 MeV. In addition to the numerical results, for comparison, the dot-dashed line denotes the lowest observed frequency in the SGR 1806-20, which is 18 Hz. From this figure, one can observe that the frequencies of fundamental torsional oscillations are almost independent from the value of K 0 at least in the parameter range of our calculations. We make sure of this dependence with different stellar models, such as R = 10, 12, and 14km for M = 1.4M and 1.8M . It could be because the charge number of spherical nuclei, Z, which contributes largely to the shear modulus as in Eq. (3), depends strongly on not K 0 but L [16]. As a consequence, one can derive the fitting formula of the frequencies of = 2 fundamental torsional oscillations as where c 0 , c 1 , and c 2 are constants depending on the stellar properties. For comparison with the observational data, we also add the horizontal dot-dashed line, which is the lowest QPO frequency observed from SGR 1806-20 [2]. Additionally, the solid thick line denotes the fitting formula (5).
With the obtained fitting formula, one can draw Fig. 2, where the stellar mass is fixed to be M = 1.4M . It is found from this figure that the dependence of the frequency of = 2 fundamental torsional oscillation on the stellar radius is relatively small. Namely, the frequency with the fixed value of L is determined within the accuracy of ∼ 20%, if the stellar radius is in the range of 10 ≤ R ≤ 14km. Additionally, we find that the dependence of 0 t 2 on the stellar mass becomes smaller than that on the radius, i.e., the frequencies for R = 10, 12, and 14 km decrease ∼ 14, 10, and 9% if the stellar mass increases from 1.4M up to 1.8M . In practice, the frequencies of = 2 fundamental torsional oscillations for the stellar models with 10 km ≤ R ≤ 14 km and 1.4M ≤ M ≤ 1.8M can be confined in the colored region in Fig. 3. Thus, almost independent of the stellar models, one could be possible to restrict on the value of L via the observations of torsional oscillations.
It has been suggested that the observed QPOs in giant flares can be due to the torsional oscillations in neutron star crust. On the other hand, among many frequencies of torsional oscillations, the frequency of = 2 fundamental oscillations is the lowest one. So, if the observed QPO frequencies would be corresponding to the torsional oscillations in neutron star crust, the frequency of = 2 fundamental torsional oscillation should become lower than the lowest frequency in the observed QPOs. Based on this statement, one can restrict on L as L > ∼ 50 MeV independent of the stellar models (see Fig. 3). It should be noticed that as mentioned the above, the calculated frequencies might be underestimated because we neglect the effect of pasta phase in our calculations. In other words, the restriction that L > ∼ 50 MeV might be severer than that including the effect of pasta phase. However, the region of past phase become narrower as L increases, and that region could vanish for L > ∼ 100 MeV [16]. Thus, even if the effect of pasta phase will be taken into account, the restriction that L > ∼ 50 MeV may not be modified.

Conclusion
To verify completely the EOS of nuclear matter with the ground experiments is not so easy and many uncertainties still exist. In contrast to this, the asteroseismology might provide the additional constraints in the EOS. This is an approach from totally another direction. In fact, the QPOs observed in the giant flares, which are considered as the results of oscillations in neutron stars, could become the appropriate evidence to reveal the interior properties. In this  article, we show that the frequency of fundamental torsional oscillation depends only on L. Compared with the QPO frequencies observed in the giant flares, one can expect that L > ∼ 50 MeV independent of the stellar models. We point out that our restriction on L is fairly stringent because experimental constraints on L have yet to converge (e.g., [31,32]). Additionally, in this article, we omit the effect of neutron superfluidity, which can affect on the enthalpy density and the torsional oscillations [29,30]. Including such an effect, one can make further constraint in the EOS parameters. Moreover, since the existence of nonuniform structure in the core of neutron star is also suggested, the torsional oscillations in such phase might be important to theoretically explain the observed QPO [33].
This work was supported in part by Grants-in-Aid for Scientific Research on Innovative Areas through No. 23105711 and No. 24105008 provided by MEXT and in part by Grant-in-Aid for Young Scientists (B) through No. 24740177 and for Research Activity Start-up through No. 23840038 provided by JSPS.