Quasi-periodic oscillations during magnetar giant flares in the strangeon star model

Soft gamma-ray repeaters (SGRs) are widely understood as slowly rotating isolated neutron stars. Their generally large spin-down rates, high magnetic fields, and strong outburst energies render them different from ordinary pulsars. In a few giant flares (GFs) and short bursts of SGRs, high-confidence quasi-periodic oscillations (QPOs) were observed. Although remaining an open question, many theoretical studies suggest that the torsional oscillations caused by starquakes could explain QPOs. Motivated by this scenario, we systematically investigate torsional oscillation frequencies based on the strangeon-star (SS) model with various values of harmonic indices and overtones. To characterize the strong-repulsive interaction at short distances and the non-relativistic nature of strangeons, a phenomenological Lennard-Jones model is adopted. We show that, attributing to the large shear modulus of SSs, our results explain well the high-frequency QPOs ($\gtrsim 150\,\mathrm{Hz}$) during the GFs. The low-frequency QPOs ($\lesssim 150\,\mathrm{Hz}$) can also be interpreted when the ocean-crust interface modes are included. We also discuss possible effects of the magnetic field on the torsional mode frequencies. Considering realistic models with general-relativistic corrections and magnetic fields, we further calculate torsional oscillation frequencies for quark stars. We show that it would be difficult for quark stars to explain all QPOs in GFs. Our work advances the understanding of the nature of QPOs and magnetar asteroseismology.

Alternatively, involving strange quark may shed light on the physical mechanism of the QPOs in GFs.It is conjectured that the bulk dense matter may be composed of strangeons, which are formerly named strange-quark clusters with nearly equal numbers of , , and  quarks (Xu 2003).Based on phenomenological analysis and comparison with different observations, a strangeon star (SS) model was proposed with very stiff equation of states (EOSs;Xu 2003;Lai & Xu 2009).Moreover, at realistic baryon densities of compact stars, the residual interaction between strangeons could be stronger than their kinetic energy, so strangeons would be trapped in the potential well and the bulk of the dense matter in the compact stars are crystallized into a solid state at low temperature (Xu 2003(Xu , 2009)).SSs can account for many observational facts in astrophysics, such as pulsar glitches, sub-pulse driftings, extremely strong magnetic fields, the transient bursts of GCRT J1745−3009 (see e.g.Xu et al. 1999;Zhou et al. 2004;Xu 2005;Yue et al. 2006;Zhu & Xu 2006;Lai et al. 2023, for details), even for fast radio bursts related to Galactic magnetars (Wang et al. 2022a,b).To characterize the strong-repulsive interaction at short distances and the nonrelativistic nature of strangeons, a phenomenological Lennard-Jones model with two parameters has been adopted to describe the EOS of SSs (Lai & Xu 2009).Besides, the tidal deformability of merging binary SSs, as well as the ejecta and light curves, have been discussed by Lai et al. (2018Lai et al. ( , 2019Lai et al. ( , 2021)).Recently, Gao et al. (2022) have discussed the universal relations between the moments of inertia, the tidal deformabilities, the quadrupole moments, and the shape eccentricity (Gao et al. 2023) of SSs.
If SSs indeed exist, they can release enough gravitational energy during starquakes to allow successful GFs to happen (Xu et al. 2006;Horvath 2007;Xu 2007).In other words, we can adopt the asteroseismological methods to probe the internal structure of compact stars.Some types of oscillation modes strongly couple to the space-time continuum, and can damp on relatively short time scales by emitting gravitational waves (GWs).Li et al. (2022a) have recently studied the oscillation modes and the related GWs of SSs.They discussed the universal relations between the fundamental (  )-mode frequencies and the global properties of SSs, such as compactness and tidal deformability.Moreover, inverted hybrid stars are discussed in Zhang et al. (2023), and extensions to pseudo-Newtonian gravity can be found in Li et al. (2023).
In the present work, for the first time we study the torsional oscillation modes of SSs in detail, and also discuss the effects of the magnetic field on the frequencies of the torsional modes.In the SS scenarios, we attempt to explain the observational QPO frequencies during the GFs of SGR 1806−20, SGR 1900+14 and SGR J1550−5418.Our results suggest that the SS-model can explain the high-frequency (≳ 150 Hz) QPOs, while it meets challenges of some low-frequency QPOs, such as the 18 Hz, 30 Hz and 92 Hz frequencies for SGR 1806−20, and the 40 Hz frequency for SGR J1935+2154.We attribute these difficulties to the large shear modulus of SSs, which could reach 10 32 erg cm −3 (Xu 2003).In view of this, we further consider the interface modes of SSs in the interface between the ocean and crust to explain these low-frequency QPOs.Such an ocean layer could have a width in the range of ∼ 10-50 m, consisting of a plasma of electrons and nuclei (Medin & Cumming 2011).The Coulomb interaction energy between ions is greater than the thermal energy, leading to liquid behaviors.The ocean can influence the transport and release of thermal energy from the surface of SSs.It is in this ocean layer that the burning that produces X-ray bursts takes place.McDermott et al. (1988) have investigated non-radial oscillations of NSs with a fluid core, solid crust, and thin surface fluid ocean.Based on such a three-component model, they proposed a new oscillation mode called the interface mode.This interface mode can be caused by the interface not only between the ocean and the crust but also between the crust and the core.Piro & Bildsten (2005) have discussed the ocean-crust interface wave, exploring its properties both analytically and numerically for a two-component NS envelope model.We follow these ideas and apply them to SSs.We find that the ocean-crust interface mode of SSs can explain well the observed low-frequency QPOs in the GFs.
Furthermore, we also calculate the frequencies of the torsional modes of the quark stars (QSs), which were firstly proposed by Witten (1984).Typically, QSs could not have torsional shear modes due to its ultra-dense quark liquid extending up to the surface (Haensel et al. 1986).However, QSs can have a thin crust that extends to the neutron drip density (Alcock et al. 1986).Jaikumar et al. (2006) suggested that such a crust could be made up of nuggets of strange quark matter embedded in a uniform electron background.Although Watts & Reddy (2007) have calculated the torsional oscillations of QSs and discussed the effects of the magnetic field and temperature on the frequencies of torsional modes, their results should be modified using a more complete model with general-relativistic corrections and magnetic fields.We calculate the torsional oscillations for both the thin crust model and the quark nugget crust model in this work, and discuss the effects of the magnetic field on the frequencies.
The paper is organized as follows.In Sec. 2, we present our equilibrium models for SSs.For non-magnetized and magnetized stars, we discuss in Sec. 3 the numerical setups for solving the perturbation equations of the torsional oscillations with the Cowling approximation.In Sec. 4, we present the frequencies for SSs and QSs, as well as the fitting formulae for the effects of the magnetic field in the oscillation spectrum.Finally, we conclude in Sec. 5. Throughout this paper, we adopt geometric units with  =  = 1, where  and  are the speed of light and the gravitational constant, respectively.

EQUILIBRIUM CONFIGURATION
The general-relativistic equilibrium stellar model is assumed to be spherically symmetric and static, as described by the TOV equations.The line element of spacetime reads where Φ and Λ are functions of .Typically, for magnetars, the magnetic energy ( m ) is a few orders of magnitude smaller than the gravitational energy ( g ) with a ratio, where  and  are the radius and mass of a magnetar, respectively, and  is the surface magnetic field strength.In view of this, deformations on the spherical symmetry induced by the magnetic fields are usually small for magnetars (Colaiuda et al. 2008;Haskell et al. 2008).Note that the general stress-energy tensor for a magnetized relativistic star is given by, where  is the energy density,  is the pressure,   is the 4-velocity of fluid, and   =   / √ 4 is the magnetic field.Based on the above setting, Sotani et al. (2007) investigated torsional oscillations of relativistic stars with dipole magnetic fields.They showed that the magnetic field could exhibit a poloidal geometry, and can be derived by solving the Grad-Shafranov equation, where  1 is the radial component of the electromagnetic fourpotential, and  0 is a constant.We use primes to represent the radial derivatives hereafter.To solve Eq. ( 4), we must specify the boundary conditions at both the center and the surface of a star.Regularity of the solution at the origin requires  1 =  0  2 , where  0 is a constant.At the surface, the internal solutions must be consistent with those above the surface with an external magnetic field.We consider a dipole field in the vacuum in this work.Therefore, the solution above the surface is given by, where  0 is the magnetic dipole moment.The corresponding magnetic field thus has the form (Konno et al. 1999), We plot in Fig. 1 the profiles of the magnetic field components,   and   , against the radial coordinate .The magnetic fields are normalized by  0 / 3 .In this plot, we have used the polytropic EOS,  =    , with  = 2 and a compactness C = 0.2 for the  value.

TORSIONAL OSCILLATIONS IN THE COWLING APPROXIMATION
Axial and polar perturbations do not couple with each other when we consider the pure axisymmetric perturbations on a spherically symmetric star.Here we consider only the torsional oscillations of a non-magnetized relativistic star with axial perturbations.The density variations in the spherically symmetric star would not be induced.
For this reason, we neglect the perturbations of spacetime using the Cowling approximation.The axial perturbation equations for the elastic solid star in the Cowling approximation is written as (Samuelsson & Andersson 2007;Sotani et al. 2012), where  is the shear modulus,  is the angular frequency,  () describes the radial part of the angular oscillation amplitude, and the integer ℓ is the angular separation constant which enters when  () is expanded in spherical harmonics  ℓ (, ).Now we extend our studies to the torsional oscillation of a magnetized relativistic star.Sotani et al. (2007) derived the perturbation equations of the magnetized relativistic star using the relativistic Cowling approximation.The final perturbation equation is, where the coefficients are given in terms of the functions describing the equilibrium metric, fluid, and the magnetic field of the star, where  = ℓ(ℓ + 1), and To solve Eqs. ( 8) and ( 9) and determine the oscillation frequencies, the boundary conditions require that the traction vanishes at the top and the bottom of the crust.In the next section, we will use perturbation equations ( 8) and ( 9) to study torsional oscillation modes of the SSs and discuss possible effects from the magnetic field.

NUMERICAL RESULTS
We present results of torsional oscillation modes for SSs in Sec.4.1, and make comparisons to QSs in Sec.4.2.Xu (2003) conjectured that cold quark matter with very high baryon density could be in a solid state, and considered a SS at low temperature should be a solid star.The shear modulus of solid quark We first exhibit in Table 1 the frequencies of the SS fundamental modes ℓ  0 for ℓ = 2 to 10.It shows that the frequency of ℓ = 2 mode varies from 145 to 277 Hz, depending on the SS mass.Given the SS mass, the frequencies tend to increase with a higher ℓ.In comparison, for NSs, the frequencies of the fundamental ℓ = 2 mode only range from 17 to 29 Hz (Sotani et al. 2007).Therefore, the frequency range of the ℓ = 2 mode of SSs is much larger than that of NSs.This is introduced by the larger shear modulus of SSs.

Strangeon stars
We show in Table 2 the frequencies of the first overtone ℓ  1 for SSs.We find that the frequencies range from 300 Hz to 1700 Hz.Table 2 shows that the frequency of the first overtone decreases as the SS mass increases, which is the same as the fundamental modes in Table 1.However, such an anti-correlation differs from those of NSs, where the frequency of the first overtone would increase with a higher NS mass (Sotani et al. 2007).Note that such anti-correlation can also be found in higher overtones (see Table 3).
In order to compare with the torsional mode frequency of NSs, we calculate the torsional mode frequency of SSs using the same mass range of 1.2 ⊙ -2.8 ⊙ .However, the 2.8 ⊙ value is not the SSs' maximum mass.As described in the Introduction, in a reasonable model, the EOS of SSs is completely determined by the depth of the potential and the number density of baryons at the surface of the star.Because the strangeons are nonrelativistic there is a very strong repulsion at a short inter-cluster distance (Lai & Xu 2009, 2017;Gao et al. 2022;Li et al. 2022a), which could lead to the maximal mass of SSs over 3  ⊙ .
However, there appears to have difficulty when using the SS model to interpret the low-frequency QPOs (e.g., 18 Hz and 29 Hz for SGR 1806−20).For this reason, we propose that SSs may have a thin surface ocean with density and temperature in the range of 10 6 -10 9 g cm −3 and 10 8 -10 9 K, respectively.With such an ocean layer, we can use the interface modes of the ocean-crust interface in order to explain the low-frequency QPOs.The frequency of the interface mode can be analytically approximated by (Piro & Bildsten 2005), where  8 ≡ /10 8 K,  is the baryon number, and Γ is a dimensionless parameter that determines the liquid-solid transition via, where  B is the Boltzmann's constant, and  is the proton number.
The crystallization point occurs at Γ = 173 (Farouki & Hamaguchi 1993).Using Eq. ( 13), we calculate the frequencies of the interface modes with different ℓ.The SS mass and radius are fixed at  = 1.4  ⊙ and  = 10 km, respectively.According to Eq. ( 13), we see that the frequency is independent of the number of the overtone.It is not difficult to verify that the ocean-crust interface modes could interpret the recorded low-frequency QPOs, such as 1  = 16.3Hz, As already emphasized by Sotani et al. (2007), the shift in the frequencies would be significant when the magnetic field exceeds ∼ 10 15 G.Following Sotani et al. (2007), now we discuss the effects of the magnetic fields.In the presence of magnetic fields, frequencies are shifted as, where ℓ   is a coefficient depending on the structure of the star, and ℓ  (0) is the frequency of the non-magnetized star.Note that the typical magnetic field strength is defined as   ≡ (4)1/2 .More details can be found in Messios et al. (2001).
In Figs. 2 and 3, we show the effects of the magnetic field on the frequencies of the torsional modes.The magnetic field strength is normalized by   = 4 × 10 16 G.Different dashed lines in Figs. 2 and 3 are our fits to the calculated numerical data with a high accuracy.For  >   , we find that the frequencies follow a quadratic increase against the magnetic field, and tend to become less sensitive to the SS parameters.NSs could have similar behaviors, but the turning-point value of the magnetic field strength is much lower (∼ 4 × 10 15 G; Sotani et al. 2007).

Comparisons with quark stars
The frequencies of torsional oscillations depend sensitively on the property of the crust.Considering there is no solid region for bare QSs, they could not account for the torsional shear oscillations.However, Alcock et al. (1986) suggested that QSs could have a thin nuclear crust that extends to the neutron drip density (i.e. ≈ 4 × 10 11 g cm −3 ).Jaikumar et al. (2006) proposed another possible model that a crust is made up of nuggets of strange quark matter embedded in a uniform electron background.In this subsection, we present detailed calculations of both models.
Following Watts & Reddy (2007), we use the standard general relativistic algorithm with a shear speed   = (/) 1/2 and an alfvén speed   = /(4) 1/2 .We show in Fig. 4 the depth of the crust for different crust models. 1 We find that the shear speed in the nugget crust is smaller than that in the thin nuclear crust, which is consistent with the results at a constant pressure  S ∼ √︁  5/3 / in Watts & Reddy (2007), where  denotes the baryon number.In particular, Fig. 4 shows that both  and / of the nuggets decrease rapidly with the depth.In our calculation, for the thin nuclear crust models, the EOS of QSs is described by the MIT bag model, and the Frequencies of different overtones as a function of the normalized magnetic field for ℓ = 2.The dashed lines are our fits using Eq. ( 15).The SS mass is  = 1.4  ⊙ .The coefficients ℓ   are 0.3, 0.42 and 0.48 for  = 0,  = 1, and  = 2, respectively.EOS of the crust is given by Baym et al. (1971).The shear modulus  is, where   is the ion number density,  = [3/(4  )] 1/3 is the average ion spacing, and + is the ion charge.For the thin nuclear crust models, the frequency of the fundamental torsional mode with ℓ = 2, 2  0 , is 64 Hz for a given mass  = 1.4  ⊙ .We find that the frequency of the fundamental torsional mode ranges from 26 to 64 Hz.To obtain a frequency ≤ 30 Hz requires a high mass ( ≥ 2.4  ⊙ for all radii).The overtone frequencies are even higher for thin nuclear crust models.When  >   = 4 × 10 13 G the modes change the character and become dominated by the magnetic field.
In Fig. 5, we show the effects of the magnetic field on the torsional mode frequencies.The frequencies of the fundamental modes have increased by up to 35%, which is similar to NSs.The frequencies of the first overtone have increased by up to 24%.The typical value for magnetic field is   ≡ (4) 1/2 , which depends on the crust EOS and the shear modulus.In particular, for the crust EOS that was described by Baym et al. (1971), the shear modulus could be ∼ 10 28 erg cm −3 .
For the nugget crust models, we show in Fig. 6 the relations of the pressure and the shear modulus against energy density.Figure 6 shows that the EOS and shear modulus are sensitive to the strange quark mass,  s .The number  and / of nuggets decrease rapidly as the energy density increases.Using typical quark model parameters, i.e., the MIT bag constant  = 65 MeV fm −3 and  s = 150 MeV, we find that QSs can have a crust width of Δ = 40 m for a given mass  = 1.4  ⊙ with a radius  = 10 km.We find that the frequency of the fundamental mode and the first overtone is 5.11 Hz and 4282 Hz, respectively.Additionally, for  s = 250 MeV, we find that the thickness Δ is 217 m, and the frequency of the fundamental mode and the first overtone is 7.49 Hz and 2678 Hz, respectively.These results could explain some QPOs in the range of 18-150 Hz but it appears to be difficult to explain the high frequencies of 625 Hz and 1837 Hz.
In the left panel of Fig. 7, we show the relations between the torsional mode frequencies in the nugget crust model.We adopt  s = 150 MeV and a normalized magnetic field   = 4 × 10 11 G.For this case when  =   the frequencies of fundamental mode have increased by up to 25%, compared to the frequencies of nonmagnetized models.Using the empirical formula (15), we fit the coefficients for the effects of the magnetic field.The fitting coefficient values are 0.52, 0.49, and 0.07 for 2  0 , 7  0 , and 2  1 , respectively.In the right panel of Fig. 7, we show the results with  s = 250 MeV and   = 4 × 10 14 G.The fitting coefficient values are 0.6, 2.3, and 5.6 for 2  0 , 7  0 , and 2  1 , respectively.In both panels of Fig. 7, we find that, as expected, when the magnetic field strength is close to   = 4 × 10 14 G, the frequencies of the fundamental mode and the first overtone significantly increases.
Using a plane-parallel geometry, Watts & Reddy (2007) calculated the torsional oscillation of QSs, and discussed the effects of the magnetic field and temperature on the torsional mode frequencies.Similarly, in our work, we adopt the same EOS of the quark star and thin crust.However, the magnetic field is a constant in the work of Watts & Reddy (2007).Differently, we consider a relativistic star with dipole magnetic fields, calculate the frequencies of torsional modes, and study the effects of magnetic fields.Compared with earlier results, when considering the magnetic field strength   , the frequencies have increased by 25%-35% compared to the frequencies of non-magnetized models (see Figs.

DISCUSSIONS AND CONCLUSIONS
In this work we first studied the torsional oscillation modes of SSs using the Cowing approximation with no magnetic field on the equilibrium configuration.According to our results, we find that SSs can explain well the high-frequency QPOs in the GFs of some SGRs.We further discuss the effects of magnetic field on the torsional oscillation frequencies.The typical value of the magnetic field strength is adopted as   = 4 × 10 16 G, which is much larger than the ordinary NS models (Sotani et al. 2007).
To explain the observed low-frequency QPOs in the GFs of some SGRs, we consider that SSs may have a thin surface ocean with a density in the range of 10 6 -10 9 g cm −3 .The depth is considered to be in the range of 10 − 50 m.We estimate the frequencies of the ocean-crust interface modes and find that the interface modes can interpret well the observed low-frequency QPOs in GFs for some SGRs.Watts & Reddy (2007) have also investigated the thin nuclear crust model and the quark nugget crust model.They calculated the frequencies of the torsional oscillation modes using plane-parallel approximation and discussed the effects of the magnetic field and the temperature on the frequencies of the torsional modes.Compared with their results, when considering the magnetic field strength   , the frequencies have increased by up to 25%-35%, compared to the frequencies of non-magnetized models.For the thin nuclear crust model, the typical magnetic field value is   ∼ 4 × 10 13 G.The frequencies of the first overtone could increase up to 24% at   ∼ 4 × 10 13 G.The nugget crust model has a wider range of frequencies due to its uncertainty in the strange quark mass  s .Our results show that both the thin nuclear crust model and the nugget crust model are difficult to reproduce well the recorded QPO frequencies.
Analysis of the magnetar QPOs in GFs could enable us to look for a correct interpretation of their origin and the physical nature of the oscillations (Abbott et al. 2009;Kalmus et al. 2009;Abadie et al. 2011;Abbott et al. 2022).In the future, the strong couplings of magnetar oscillations to GWs will provide an excellent opportunity to apply the asteroseismological methods to compact star studies, and eventually uncover the nature of astrophysical compact objects.The lines correspond to our fits using the empirical formula (15) with different coefficient values.

Figure 1 .
Figure1.Profiles of the magnetic field components against the radial coordinate .  and   are evaluated at  = 0 and  = /2, respectively.We set the compactness C = 0.2.The purple dotted-dashed line denotes the surface of the star.

Figure 2 .
Figure2. of the fundamental  = 0 modes with ℓ = 2, ℓ = 3, and ℓ = 4 as a function of the magnetic field.We set the SS mass to be  = 1.4  ⊙ .Individual numerical results are denoted with different marks in different colours.The dashed lines correspond to the empirical formula (15) with different coefficient values.The fitting coefficients ℓ   are 0.3, 0.18, and 0.12 for ℓ = 2, ℓ = 3, and ℓ = 4, respectively.

Figure 4 .
Figure 4. Shear speed  S and alfvén speed  A in the crust of stars for a QS with a fixed mass  = 1.4  ⊙ , a radius  = 12 km , and a magnetic field  = 10 14 G.The upper panel is for the thin nuclear crust, and the lower panel is for the crust with nuggets.Note that in the lower panel, we consider  s = 150 MeV and  s = 250 MeV denoted with different lines in different colors.

Figure 5 .
Figure5.Torsional mode frequencies in the QS thin nuclear crust model as a function of the magnetic field.The dashed lines correspond to our fits using the empirical formula (15) with different coefficient values.The fitting values are 1.1, 1.5, and 2.5 for 2  0 , 7  0 , and 2  1 , respectively.

Figure 6 .
Figure 6.The upper panel shows the EOS of QSs in the nugget crust model.The lower panel shows the relationship between the shear modulus and the energy density.
5 and 7 in this paper and the left panels of Figs. 2 and 3 in Watts & Reddy 2007).

Figure 7 .
Figure 7. Frequencies of the fundamental  = 0 modes with ℓ = 2 , 7 and the first overtone in the nugget crust model.The left panel shows the results with strange quark mass  s = 150 MeV and the normalized magnetic field strength   = 4 × 10 11 G, while for the right panel  s = 250 MeV and   = 4 × 10 14 G.The lines correspond to our fits using the empirical formula (15) with different coefficient values.

Table 1 .
Frequencies (in the unit of Hz) of the fundamental torsional modes ℓ  0 for SSs without the magnetic fields (i.e. = 0 and  = 0).The subscript in the models denote the SS mass; taking SS 12 as an example, the subscript "12" represents  = 1.2  ⊙ .Values in bold are close to the QPO observations (within 2%) for SGR 1806−20, SGR 1900+14 and SGR J1550−5418.

Table 2 .
Same as Table1, but for the first overtone ( = 1) of the torsional modes ℓ  1 of SSs.

Table 3 .
Same as Table1, but for the higher overtones 3   of the torsional modes of SSs for ℓ = 3.