Constraining the Nuclear Equation of State of a Neutron Star via High-frequency Quasi-periodic Oscillation in Short Gamma-Ray Bursts

Abstract

The determination of the equation of state (EOS) of a neutron star (NS) and its maximum mass is very important for understanding the formation and properties of NSs under extreme conditions, but they remain open questions. Short-duration gamma-ray bursts (GRBs) are believed to originate from the merger of binary NSs or giant flares (GFs) of soft gamma repeaters (SGRs). Recently, the high-frequency quasi-periodic oscillations (QPOs) have been claimed to be identified from two short GRBs (GRB 931101B and GRB 910711). In this paper, we propose that the observed high-frequency QPOs in these two short GRBs result from torsional oscillations in the GFs of SGRs associated with cold NSs, or from radial oscillations of hypermassive NSs as the hot remnants of binary NS mergers, and then to constrain the EOS of NSs. For torsional oscillations, the six selected EOSs (TM1, NL3, APR, SLy4, DDME2, and GM1) of NSs suitable for the zero-temperature condition exhibit significant overlap in mass ranges, suggesting that we cannot constrain the EOS of NSs. For radial oscillations, the six selected EOSs (IUF, TM1, TMA, FSG, BHBLp, and NL3) of NSs suitable for the high-temperature condition cannot be ruled out when redshift is considered. However, it is found that the EOS can only be constrained if the redshift and temperature of the remnant can be measured.

Full text

Constraining the Nuclear Equation of State of a Neutron Star via High-frequency Quasi-
periodic Oscillation in Short Gamma-Ray Bursts
Jun-Xiang Huang , Hou-Jun Lü , and En-Wei Liang
Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, People’s Republic of
China; lhj@gxu.edu.cn
Received 2024 September 8; revised 2025 January 12; accepted 2025 January 15; published 2025 February 18
Abstract
The determination of the equation of state (EOS)of a neutron star (NS)and its maximum mass is very important
for understanding the formation and properties of NSs under extreme conditions, but they remain open questions.
Short-duration gamma-ray bursts (GRBs)are believed to originate from the merger of binary NSs or giant flares
(GFs)of soft gamma repeaters (SGRs). Recently, the high-frequency quasi-periodic oscillations (QPOs)have been
claimed to be identified from two short GRBs (GRB 931101B and GRB 910711). In this paper, we propose that
the observed high-frequency QPOs in these two short GRBs result from torsional oscillations in the GFs of SGRs
associated with cold NSs, or from radial oscillations of hypermassive NSs as the hot remnants of binary NS
mergers, and then to constrain the EOS of NSs. For torsional oscillations, the six selected EOSs (TM1, NL3, APR,
SLy4, DDME2, and GM1)of NSs suitable for the zero-temperature condition exhibit significant overlap in mass
ranges, suggesting that we cannot constrain the EOS of NSs. For radial oscillations, the six selected EOSs (IUF,
TM1, TMA, FSG, BHBLp, and NL3)of NSs suitable for the high-temperature condition cannot be ruled out when
redshift is considered. However, it is found that the EOS can only be constrained if the redshift and temperature of
the remnant can be measured.
Unified Astronomy Thesaurus concepts: Gamma-ray bursts (629)
1. Introduction
Neutron stars (NSs)are one of the most intriguing compact
objects in astrophysical studies, due to their extreme properties.
The maximum mass and equation of state (EOS)of NSs play
an important role in understanding their structure and proper-
ties (S. L. Shapiro & S. A. Teukolsky 1983). The first estimate
of the maximum mass of an NS (M
max
)for a static equilibrium
solution in early studies was M
max
∼0.72 M
e
by simplifying
the NS as a cloud of noninteracting cold Fermi gas (J. R. Opp-
enheimer & G. M. Volkoff 1939). However, from an
observational point of view, subsequent precise measurements
of NS mass from binary systems have revealed maximum
masses that exceed the theoretical predictions (R. A. Hulse &
J. H. Taylor 1975; S. B. Anderson et al. 1990; A. Wolszc-
zan 1991; M. Burgay 2003), and this suggests that the
interactions between particles in the core cannot be ignored.
Due to the poorly known and uncertain composition and
interactions in the core of NSs, as well as the EOS of NSs, the
maximum mass M
max
of an NS remains highly uncertain,
ranging from 1.46 to 2.48 M
e
for different EOSs (P. Haensel
et al. 2007). So far, the maximum mass of an NS that we have
accurately observed is -
+M
2
.08 0.07
0.07 (68.3% credibility)from
PSR J0740+6620, determined by relativistic Shapiro delay
(E. Fonseca et al. 2021).
Several previous studies have attempted to constrain the
EOS of NSs, such as through the observed internal plateau
together with the collapse time in the X-ray emission of short-
duration gamma-ray bursts (GRBs; P. D. Lasky et al. 2014; H.-
J. Lü et al. 2015), gravitational-wave (GW)radiation from
hypermassive NSs (HMNSs; M. Shibata 2005), GW radiation
from supramassive NSs (L. Lan et al. 2020), GW radiation
from the merger of NS binaries (K. Takami et al. 2014), and the
threshold mass for prompt black hole (BH)formation from
double NS mergers (A. Bauswein et al. 2020). However, the
constraints from these results are not yet sufficient. Thus,
understanding the EOS of NSs remains an open question in
astrophysics and particle physics (J. M. Lattimer 2021).
On the other hand, by considering the NS as a good
resonator, numerous quasi-periodic oscillation (QPOs)modes
of the NS would be excited. Based on their restoring forces, the
oscillation modes can be categorized into radial and different
nonradial modes (such as fundamental mode, pressure mode,
gravity mode, Rossby mode, and shear mode; P. N. McDermott
et al. 1988; N. Stergioulas 2003). A. L. Watts & T. E. Strohm-
ayer (2007)discovered QPOs with frequencies ranging from
tens of Hz up to a few kHz by analyzing the X-ray light curves
of SGR 0526–66, SGR 1900+14, and SGR 1806–20. The
discovery of QPOs in the three cases is believed to be the first
direct detection of magnetar oscillations, and it is naturally
interpreted as crustal shear modes triggered by starquakes,
which are associated with giant flares (GFs;
R. C. Duncan 1998; A. L. Piro 2005; U. Lee 2007; L. Samue-
lsson & N. Andersson 2007; H. Sotani et al. 2007,2012;
A. L. Watts & T. E. Strohmayer 2007; A. W. Steiner &
A. L. Watts 2009). The oscillations are strongly dependent on
the density distribution of the stellar interior and the shear
modulus within the elastic region. By comparing the observed
frequencies with theoretical models, precise constraints can be
placed on the properties of ultradense matter that cannot be
replicated to check in terrestrial laboratories (J. M. Lattimer &
M. Prakash 2001; H. Sotani et al. 2012).
Recently, C. Chirenti et al. (2023)claimed that kHz quasi-
periodic signals were identified in two short-duration GRBs
(GRB 931101B and GRB 910711)from BATSE data (see
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 https://doi.org/10.3847/1538-4357/adaceb
© 2025. The Author(s). Published by the American Astronomical Society.
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of the Creative Commons Attribution 4.0 licence. Any further
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1

Table 1). They proposed that the physical origin of these two
GRBs may originate from the merger of binary NSs, and such a
merger results in a hot remnant, such as an HMNS. If the QPOs
of those two short-duration GRBs are indeed from the quasi-
periodic modulation of gamma rays caused by the global
oscillations, then it may provide a new opportunity to explore
the properties of NSs by employing seismology. In this paper,
we consider two potential mechanisms that may produce the
observed high-frequency QPOs in two short-duration GRBs.
One is torsional oscillation with crustal shear modes in cold
NSs (Section 2), and the other one is radial oscillation in the hot
merger remnants by considering the effect of temperature
(Section 3). Then, we constrain the EOS of neutron stars (NSs)
by adopting the observed high-frequency QPOs in those two
short-duration GRBs (Section 3.2). Conclusions are drawn in
Section 4, with some additional discussions.
2. Torsional Oscillation
Soft gamma repeaters (SGRs)are young, slowly spinning
magnetars, and strong flaring activities exhibiting durations
from milliseconds to seconds with QPOs can be produced
during the active outburst phase (R. C. Duncan & C. Thomp-
son 1992). Because of the intense γ-ray luminosity and the
spectral characteristics, the emission of GFs and that of short-
duration GRBs are quite similar, and GFs of magnetars were
also detected as disguised short-duration GRBs from nearby
galaxies. Recently, two short GRBs (GRB 200415A and GRB
051103A)have been interpreted as giant magnetar flares
(J. Yang et al. 2020; O. J. Roberts et al. 2021; D. Svinkin et al.
2021). If this is the case, the GFs with QPOs are driven by
elastic shear modes in the crust of a cold NS that is believed to
consist of a fluid core with a radius r
c
, and an elastic crust with
a thickness of ∼1 km accounts for a mass of ∼0.01 M
e
.As
strong magnetic fields and/or rotational effects evolve with
time, the stresses will build up in the crust. When these stresses
reach a critical threshold, the crust will eventually fracture to
trigger oscillations. Because the velocity field of torsional
oscillations (i.e., the restoring force supported by shear force)is
divergence-free with no radial component, it is not related to
the varying density and pressure of the star (B. L. Schumaker &
K. S. Thorne 1983). Thus, it is more easily excited with a
fundamental frequency of ∼30 Hz after a crustquake of a
magnetar with a strong magnetic field (B. L. Schumaker &
K. S. Thorne 1983; P. N. McDermott et al. 1988;
R. C. Duncan 1998).
One needs to note that C. Chirenti et al. (2023)excluded an
SGR origin, primarily based on energetics estimates from the
redshift zinferred using distances to galaxies with moderately
active star formation, which are consistent with the burst
localizations. However, this scenario cannot be entirely ruled
out as a possibility in a low-redshift context. In this section, we
hypothesize that the observed short-duration GRBs with high-
frequency QPOs (GRB 931103B and GRB 910711)are caused
by GFs of magnetars at low redshift. Specifically, we consider
z<0.015, corresponding to a maximum isotropic energy
E
iso
=5.3 ×10
46
erg in the GF sample (see below for details).
Within this assumption, we discuss the torsional oscillations of
NSs and adopt the high-frequency QPOs observed in these
bursts to constrain the EOS of NSs.
2.1. Physical Model
K. S. Thorne & A. Campolattaro (1967)first derived the
equations governing the nonradial oscillation of NSs in general
relativity. Similarly, we consider a spherical NS under the
hypothesis of a nonrotating and nonmagnetic field. The marked
feature of the torsional oscillation mode is that it does not
involve radial displacement or induce any change in internal
density and pressure. Consequently, within the frame of
spherical coordinates (r,θ, and f), the defined crystalline
matter element moves only along the angle f. The magnitude
of proper displacement in the f-direction of a material element
during oscillation can be denoted as dl, which can be expressed
as
() ( ) () () ( ) ()wq q qq==
¶
¶
dl rY r i t b b Pexp , cos . 1
ℓℓ ℓ
Here b
ℓ
(θ)is related to the Legendre polynomial (
)
qPcos
ℓ, and
it is the only θ-dependent function that governs the angular
behavior of displacement. Hence, ℓis the angular index that
determines the angular nodes of the oscillation. The compre-
hensive mathematical framework for torsional oscillations of
nonrotating stars in the context of general relativity was
established by B. L. Schumaker & K. S. Thorne (1983). The
function Y(r)governs the radial behavior of the oscillation and
can be formulated as (B. L. Schumaker & K. S. Thorne 1983;
H. Sotani et al. 2012)
/
()
()()
w
++F-L+
+- =
m
m
r
m
¢¢ ¢ ¢ ¢
+-F +- L
¢
YY
eeY0.
r
Pc ℓℓ
r
4
22 212
2
2
⎡
⎣⎤
⎦
Here Φand Λare radial functions of the Schwarzschild metric
()qqf=- + + +
FL
ds e dt e dr r d r dsin . 2
2 222222222
Neglecting spacetime perturbations, Φand Λcan be mathe-
matically expressed as
/
() ( ())
() ( )
()F= -L =-
-
rr
Gm r rc
exp exp 1
12
.3
2
The shear modulus, denoted as μ, plays a crucial role in the
oscillations of the crust by acting as the necessary restorative
force. The crust is treated as an isotropic body-centered cubic
Coulomb solid, and a detailed calculation of the shear modulus
Table 1
The Median Frequencies and Widths of QPO for Two GRBs, along with Their ±1σValues of Central Frequencies
GRB T
90
ν
1
Δν
1
τ
1
ν
2
Δν
2
τ
2
(s)(Hz)(Hz)(s)(Hz)(Hz)(s)
931101B 0.034 -
+
877 8
0-
+
1
52
7-
+
0.067 0.021
0.010 -
+
2
612 8
9-
+
1
43
7-
+
0.071 0.024
0.019
910711 0.014 -
+
1
113 8
7-
+
2
57
9-
+
0.040 0.011
0.01
6
-
+
2
649 7
6
-
+
2
67
9-
+
0.038 0.009
0.015
Note. We also quote T
90
and estimate the damping times τbased on the widths for each GRB.
2
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

is presented in S. Ogata & S. Ichimaru (1990). For the cold-
catalyzed crust with temperature T=0, μis given as
()m=nZ e
a
0.1194 , 4
i22
where n
i
is the total number density of ions in the plasma
matter, //
()p=
a
n34
i13is the average ion spacing, Zis the
charge number of a single ion, and eis the elementary charge.
The equilibrium nuclides (A,Z)present in the cold-catalyzed
matter at different densities are listed in P. Haensel & B. Pichon
(1994)and J. W. Negele & D. Vautherin (1973). The
discontinuity of μoccurs at the point where a change in
nuclide composition takes place. To facilitate numerical
calculations, we adopt the smooth composition model to
simulate the shear modulus (F. Douchin & P. Haensel 2000).
Finally, by giving the mass and EOS of an NS, the
spherically symmetric density ρand pressure Pof the NS can
be solved by the Tolman–Oppenheimer–Volkoff equation.
Then, the eigenfrequency ωcan be obtained by solving
Equation (2.1), supplemented by the boundary conditions
()¢=
Y
r0
cand ()¢=
Y
R0at the core–crust interface and the
surface of the NS, respectively.
In general, the oscillation of an NS has an infinite number of
discrete eigenfrequency solutions. The different modes of
torsional oscillation can be labeled as
ℓ
t
n
, which correspond to
the frequency
ℓ
ω
n
=2π
ℓ
ν
n
. Here the indices n=0, 1, 2, . . .
and ℓ=2, 3, 4, Krepresent the nodes of oscillation in radial
and angular directions, respectively. The specific excitation of
oscillation modes is related to the properties of the crust and the
processes of coupling and damping. However, some oscillation
modes may be excited but not detected. The reason may be
related to the modulation modes of the gamma-ray light curve,
which remain unclear.
2.2. Constraint on the EOS of NSs
For the fundamental oscillations without radial nodes
(n=0), the momentum transfer of shear perturbations in the
θ-direction is significantly greater than that in the r-direction. If
this is the case, it will result in the formation of standing waves
during the propagation (with a typical length scale of ∼πR)of
shear perturbations along the θ-direction. In contrast, the length
scale of propagation in the r-direction is ∼0.1R(M. Gabler
et al. 2012). Consequently, this process takes a longer time to
result in lower oscillation frequencies
ℓ
ν
0
∼20–100 Hz, which
are much lower than those of observed QPOs in the two short-
duration GRBs GRB 931103B and GRB 910711 (C. Chirenti
et al. 2023). Hence, it is believed that the origin of QPO
frequencies observed in those two short-duration GRBs is not
likely to relate to fundamental oscillations. On the other hand,
by considering the eigenfrequency from ordinary oscillations
with n1(one or more radial nodes)as a function of mass,
one can obtain the “fine splitting”structure, which is similar to
a spectrum. The frequencies change slightly with increasing
values of ℓcompared to the change with n. Therefore, we adopt
ℓ=2 to replace the other modes with ℓ>2. Initially, we
consider only cold-catalyzed (zero-temperature), nonmagnetic,
nonrotating spherically symmetric models. The given mass of
an NS is strongly dependent on the frequency spectrum. In this
case, we adopt a 0.5% error in the frequency spectrum lines,
which may be caused by the greater complexity of the actual
NS (e.g., temperature, magnetic fields, rotation, and
nonspherically symmetric features), but our knowledge about
these complexities is limited.
Then, we investigate the relationship between torsional
oscillation frequency and mass for different given EOSs of NSs
(see Figures 1and 2). In our calculations, we selected six
unified EOSs that are suitable for the condition of zero
temperature: GM1 (N. K. Glendenning & S. A. Moszkow-
ski 1991), TM1 (Y. Sugahara & H. Toki 1994), NL3
(G. A. Lalazissis et al. 1997), APR (A. Akmal et al. 1998),
SLy4 (F. Douchin & P. Haensel 2001), and DDME2
(G. A. Lalazissis et al. 2005). In Figures 1and 2, the horizontal
dashed lines and shading correspond to the frequencies and
error ranges of the two QPOs listed in Table 1, respectively.
The vertical dashed lines and shading indicate the ranges of
mass that are capable of producing torsional oscillations at
those frequencies. Two high-frequency QPOs in the same GRB
correspond to two ranges of mass, and it is expected that the
ranges of mass for those two high-frequency QPOs can overlap
within the margin of error.
By considering the effects of the magnetic field, rotation, and
nonspherical symmetry of NS, but with limited knowledge
about these factors, we adopt a 0.5% error in the frequency
spectrum lines. For example, the magnetic field confined to the
crust may affect the frequencies of various torsional modes.
One can make a rough estimate of such an effect of the
magnetic field Bby using Newtonian physics, which is
expressed as (R. C. Duncan 1998; N. Messios et al. 2001)
/
()
()
n
n
=+´
B
1410 ,5
ℓn
ℓn
015
212
⎡
⎣
⎢⎛
⎝⎞
⎠⎤
⎦
⎥
where ()
n
ℓn
0is the frequency in the nonmagnetic case. Based on
the numerical simulation of torsional and magneto-elastic
oscillations of a realistic magnetar model, M. Gabler et al.
(2011)found that a ∼10
14
G magnetic field can achieve a
similar damping time on the order of tens of milliseconds. For
the given value of magnetic field of ∼4×10
14
G, it will result
in a correction closer to 0.5%. This allows us to believe that the
0.5% error in the frequency spectrum we adopt is reasonable.
On the other hand, the nonmeasured redshift (z)of those two
short-duration GRBs also introduces significant uncertainty.
The observed frequency of QPOs is directly related to the
frequency in the rest frame, namely,
()nn
=+z1.6
obs rest
The frequency in the rest frame is higher than that in the
observed frame with a factor (1+z). However, the observed
maximum fluxes of GRB 910711 and GRB 931101B are
1.5 ×10
−4
erg cm
−2
s
−1
and 2.6 ×10
−5
erg cm
−2
s
−1
,
respectively. On the contrary, the maximum isotropic energy
E
iso
in the GF sample can only reach to 5.3 ×10
46
erg
(E. Burns et al. 2021). By assuming that the isotropic energy
can reach to the maximum isotropic energy of GFs from SGRs,
one can derive z<0.015, which corresponds to a frequency
error of less than 0.15%. However, it should be emphasized
that the error of 0.5% is only a simple assumption for an
uncertainty originating from various complexities. In more
extensive error range cases, the ranges of mass required to
generate a lower-frequency QPO may overlap simultaneously
with multiple ranges of mass (i.e., the intersection of the
3
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

higher-frequency QPO and spectrum lines with different n).
This results in two overlapping areas in the mass range, or even
the range of masses required for lower-frequency QPOs may be
wholly covered. Thus, in this case, the mass range is roughly
determined by the QPO with the lower frequency.
For the six given EOSs of NSs in Figures 1and 2, it is found
that all selected EOSs exhibit significant overlapping mass
ranges for both GRB 931101B and GRB 910711. This suggests
that an NS with a maximum mass of ∼1.38 M
e
(GM1),
∼1.88 M
e
(TM1),∼2.01 M
e
(NL3),∼1.70 M
e
(APR),
∼1.73 M
e
(SLy4), and ∼2.28 M
e
(DDME2)would satisfy
the criteria to be considered as a potential source for the QPOs
observed in GRB 931101B, and similarly for a maximum mass
of ∼1.79 M
e
(GM1),∼2.15 M
e
(TM1),∼2.52 M
e
(NL3),
∼2.09 M
e
(APR),∼2.01 M
e
(SLy4), and ∼2.48 M
e
(DDME2)
for GRB 910711 (see Table 2). In other words, we cannot
constrain the EOS of NSs via the high-frequency QPOs in these
two short-duration GBRs if we believe that they originated
from the GFs of magnetars.
3. Radial Oscillation
The first direct evidence of short GRB origin, which comes
from an NS–NS merger, is the detection of GW170817 by
Advanced LIGO and Virgo associated with short GRB GRB
170817A and an optical/infrared transient known as kilonova
AT 2017gfo (B. P. Abbott et al. 2017; A. Goldstein et al. 2017;
B. B. Zhang et al. 2018). From a theoretical point of view, there
are four different types of remnants of binary NS mergers, such
as a BH, an HMNS, a supramassive NS supported by rigid
rotation with a lifetime of hundreds of seconds before
collapsing into a BH, and a stable NS (Z. G. Dai &
T. Lu 1998b,1998a; S. Rosswog et al. 2000; Z. G. Dai et al.
2006; Y.-Z. Fan & D. Xu 2006; L. Rezzolla et al. 2010;
B. Giacomazzo & R. Perna 2013; B. Zhang 2013; P. D. Lasky
et al. 2014; H. Gao et al. 2017; H.-J. Lü et al. 2017). If the
remnant is a newborn NS, the temperature is expected to reach
several MeV, or even exceed tens of MeV in the immediate
aftermath of the merger (R. Oechslin et al. 2007; A. Bauswein
et al. 2010). Such high temperatures already exceed the point at
which a crystalline lattice is stable, and the initial state of the
remnant from NS–NS should be fluid filled. The cooling
timescale is expected to range from thousands of seconds to
days before the crystal forms. Therefore, torsional oscillations,
which are shear forces from a crystalline crust, are unlikely to
occur, but radial modes are more suitable (N. Stergioulas et al.
2011). The radial oscillation is driven by pressure and gravity,
producing a radial displacement that is directly affected by the
density and pressure profiles of NSs (S. Chandrasekhar 1964;
G. Chanmugam 1977; D. Gondek et al. 1997). D. Gondek et al.
(1997)found that the frequency spectrum of the lowest modes
differs significantly from that of cold NSs. In this section, we
will discuss the radial oscillation of NSs and constrain the EOS
of NSs by adopting the high-frequency QPO of short-duration
GRBs GRB 931103B and GRB 910711, assuming that these
two short GRBs are indeed from an NS–NS merger.
Figure 1. The frequency spectrum nlog
n
vs. the mass Mof an NS. The spectrum lines from the bottom to top correspond to n=1, 2, 3, 4, 5, 6, 7. The pink and violet
horizontal dashed lines correspond to the two high-frequency QPOs from GRB 931101B (see Table 1), and the vertical dashed lines mark the masses corresponding to
these two high-frequency QPOs. The shading indicates the error range.
4
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

3.1. Physical Model
Within the framework of general relativity, the study of
infinitesimal radial adiabatic oscillations in stars was initially
proposed by S. Chandrasekhar (1964)and applied to enhance
its utility for numerical calculation by G. Chanmugam (1977)
later. The study focused on two essential quantities, one being
the radial displacement (Δr), which represents the radial
motion of a matter element relative to its equilibrium position,
and the other being the Lagrangian pressure perturbation (ΔP),
which quantifies the change in pressure associated with the
displacement. These quantities are determined by a pair of
ordinary differential equations, which are expressed as follows:
() ()
xxx
r
=- + D
G-+
d
dr r
P
P
dP
dr P c
13,7
2
⎛
⎝⎞
⎠
{}
{}
{}
()
()
()
() ()
()
()
xr
xr
r
=+-
+-+
+D - +
w
r
p
r
p
DL-F
+
L
+
L
ePcr
eP cPr
PPcre
4
.8
dP
dr c
dP
dr
dP
dr
r
Pc
G
c
dP
dr P c
G
c
2
282
14 2
2
2
24
24
Here ξ=Δr/ris the normalized radial displacement. To avoid
the singularity in ξat the stellar center and ensure a well-
behaved normalization for the eigenfunctions, we adopt ξ
(0)=1 as the boundary condition. Parameter ω
n
=2πν
n
is the
angular frequency of radial oscillation with the eigenfrequency
ν
n
; the stability condition of oscillations requires ω
2
0. Γis
the adiabatic index that is related to the baryon number density
Figure 2. Similar to Figure 1, but for GRB 910711.
Table 2
Overlapping Mass Ranges of Short-duration GRBs GRB 931101B and GRB
910711 for Different EOSs in Units of Solar Mass (M
e
)
EOS 931101B 910711
GM1 1.193−1.272 1.536−1.684
1.284−1.379 1.702−1.786
TM1 1.707−1.717 2.034−2.080
1.737−1.883 2.117−2.145
NL3 1.737−1.772 2.224−2.320
1.791−2.008 2.435−2.521
APR 1.543−1.704 1.914−1.969
2.057−2.090
SLy4 1.621−1.728 1.910−1.946
2.001−2.014
DDME2 2.185−2.276 2.437−2.478
5
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

n
b
and pressure (G. Baym et al. 1971), namely,
()
r
r
G= = +n
P
dP
dn
Pc
Pc
dP
d.9
b
b
2
2
By solving Equations (7)and (8), it is necessary to apply
boundary conditions at the stellar center and surface, namely,
()xD=-G =PPr3,for 0. 10
()D= =PrR0, for . 11
If this is the case, Equations (7)and (8)can be reduced to a
second-order differential equation for ξ(r)and switched to the
form of a Sturm–Liouville eigenvalue problem. Similar to the
case of torsional oscillation, the eigenvalues ω
2
can also yield a
discrete spectrum
w
ww<<<
n0
21
22
, and each eigenvalue
corresponds to an eigenfunction ξ
n
that has nnodes within
0rR. The fundamental mode (n=0), also known as the f-
mode, corresponds to the lowest eigenvalue, while the higher
modes with n=1, 2, 3, Kare referred to as p-modes
(V. Sagun et al. 2020). The eigenfrequencies not only describe
the nature of stellar oscillations but also serve as critical
indicators of the system’s stability under small perturbations.
Hence, all eigenvalues should satisfy
w
0
n
2for every nand
ensure that a static stellar model is dynamically stable under
small, radial, and adiabatic perturbations.
3.2. Constraint on the EOS of NSs
An HMNS, supported by differential rotation that survives
10–100 ms before collapsing into a BH, is a possible remnant
of a binary NS merger. Initially, the HMNS is expected to have
an extremely high temperature, potentially reaching tens of
MeV. Such high temperatures significantly affect the physical
properties of HMNSs, such as the EOS of NSs. By assuming
that the high-frequency QPOs in the short-duration GRBs GRB
931101B and GRB 910711 are caused by the radial oscillation
of an HMNS from a binary NS merger, we employ six EOSs
for NSs (NL3, IUF, TM1, TMA, FSG, and BHBLp), which are
suitable for the condition of high temperature. The interpolated
data for the selected EOS models were obtained from the
publicly available CompOSE database
1
(S. Typel et al. 2015;
M. Oertel et al. 2017).
Figures 3and 4show the eigenfrequencies (ν
n
)of various
EOS models as functions of the central density (log ρ
c
)with
different temperatures for GRB 931101B and GRB 910711,
respectively. The maximum temperature of five EOSs (i.e.,
IUF, TM1, TMA, FSG, and BHBLp)can reach 40 MeV, while
for NL3 it extends up to as high as 60 MeV. It is found that the
frequencies of n=1 modes at zero temperature are
significantly higher than those of observed values for those
two short GRBs. However, the frequency of the n=1 mode is
rapidly decreased when the temperature is increased. The
shaded vertical regions in Figures 3and 4correspond to the
ranges of ρ
c
where the eigenfrequencies of n=0 and n=1
modes match the observed lower and higher frequencies of the
QPO, respectively. If this is the case, the temperature at which
the observed lower and higher frequencies of the QPO
correspond to the same ρ
c
needs to be determined and can be
used to constrain the EOS of NSs.
To find the temperature at which the vertical bands overlap,
we define a metric: Metric =ρ
gap
−ρ
cross
, which quantifies the
spatial relationship between the two vertical bands. Here the
ρ
gap
represents the distance between the closest edges of the
two regions, namely,
[( )
()]
r=
-
xx
xx
max 0, max ,
min , .
gap 0,min 1, min
0,max 1, max
Here [x
0,min
,x
0,max
]and [x
1,min
,x
1,max
]are the ρ
c
ranges
obtained by comparing the n=0 mode with the observed
lower-frequency QPO and the n=1 mode with the higher-
frequency QPO, respectively. If the bands overlap, ρ
gap
=0.
Parameter ρ
cross
is defined as
[( )
()]
r=
-
xx
xx
max 0, min ,
max , .
cross 0,max 1, max
0,min 1, min
If the bands do not overlap, ρ
cross
<0; otherwise, it is the size
of the overlapping region between the two bands. Metric =0
implies that there exists a temperature at which the observed
lower and higher frequencies of the QPO correspond to the
same ρ
c
at a specific temperature. Figure 5shows the
relationship between the Metric and temperature (T)for
different EOSs. It is found that most EOSs allow the Metric
to reach a negative value at a specific temperature for those two
short GRBs, except for the stiffest EOS, NL3. Hence, except
NL3, we cannot rule out the other five EOSs based on the
observed lower and higher frequencies of the QPO for those
two short GRBs.
On the other hand, the effect of redshift in the above
calculation is ignored due to the lack of redshift measurements.
In fact, one needs to consider the uncertainty caused by the
redshift of these two short GRBs. Figure 6shows the
relationship between the temperature Tand the redshift zfor
different EOSs. Here temperature Tvalues correspond to the
points where the Metric reaches its minimum value. For
example, the minimum value of temperature in Figure 5for a
given EOS corresponds to z=0. Then, we perform the
calculations by varying the redshift range from 0 to 1 based on
the observations.
2
We find that the temperature decreases as the
redshift increases, and it is due to the relativistic effect of
redshift that increases the frequency in the rest frame.
Moreover, it is found that the NL3 EOS can reach the overlap
of ρ
c
ranges at a higher temperature compared to other EOSs
when z0.5. For example, Figure 7shows the eigenfrequency
as a function of the NS mass for n=0(solid lines)and n=1
(dashed lines)oscillation modes for different EOSs at z=0 and
z=1, respectively.
In any case, it is difficult to constrain the EOS of NSs based
solely on the observed high-frequency QPO in those two short
GRBs, and we cannot rule out any EOS by considering the
effect from both temperature and redshift. However, it is found
that the different EOSs of NSs correspond to different
temperatures for a given redshift with Metric =0. In other
words, if we believe that the observed high-frequency QPOs in
short GRBs are indeed caused by radial oscillations of HMNSs
from binary NS mergers, one can constrain the EOS only when
the redshift and temperature of the remnant can be measured.
1
https://compose.obspm.fr/table.
2
H.-J. Lü et al. (2015)found that the redshift distribution for short GRBs
with redshift measured is from 0 to 1, and the average redshift is
about z=0.58.
6
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

4. Conclusion and Discussion
NSs offer unparalleled opportunities for investigating
extreme physics phenomena in the universe. Gaining insights
into the composition of matter and the structure of NSs would
be a significant advancement in nuclear physics and astro-
physics. The application of seismology in QPOs of SGRs has
demonstrated immense potential in this regard, particularly in
Figure 3. Eigenfrequencies (ν
n
)of various EOS models as a function of the central density ((
)
rlog c)for different temperatures. The solid and dashed lines correspond
to n=0 modes (i.e., f-mode)and n=1 modes (i.e., p-mode), respectively. The color bar is the temperature in units of MeV. The horizontal lines represent the two
observed high-frequency QPOs for GRB 931101B. The shaded vertical regions correspond to the ranges of ρ
c
where the eigenfrequencies of n=0 and n=1 modes
match with the observed lower and higher frequencies of the QPO, respectively.
Figure 4. Similar to Figure 3, but for GRB 910711.
7
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

Figure 5. Metric obtained for different EOS models with varying temperatures. The Metric quantifies the spatial relationship between the ρ
c
ranges corresponding to
the n=0 and n=1 modes.
Figure 6. The relationship between the temperature Tand the redshift zfor different EOS models. The temperature values correspond to the points where the
Metric <0(from Figure 5)and reaches its minimum, ensuring overlap between the ρ
c
ranges of the n=0 and n=1 modes.
Figure 7. Eigenfrequencies ν
n
of the n=0(solid lines)and n=1(dashed lines)modes as functions of NS mass for different EOS models. The left and right panels
correspond to a redshift of z=0 and z=1, respectively. The temperatures corresponding to the minimum Metric values at z=0 and z=1 for each EOS are taken
from Figure 6. Horizontal lines are the observed QPO frequencies for GRB 931101B. The vertical shaded mass ranges highlight regions where the frequencies of both
modes align with the observed QPOs.
8
The Astrophysical Journal, 980:220 (10pp), 2025 February 20 Huang, Lü, & Liang

providing constraints on the EOS and the thickness of the crust
of NSs. The recent identification of QPOs from two short-
duration GRBs (GRB 931101B and GRB 910711)may provide
a new opportunity to explore the nature of NSs by employing
seismology.
In this paper, we investigate the potential of torsional and
radial oscillation models to explain the observed high-
frequency QPOs in two short-duration GRBs (GRB 931101B
and GRB 910711), and then we try to constrain the EOS of
NSs. The following interesting results can be obtained:
1. Torsional oscillation. By assuming that the observed
short-duration GRBs (GRB 931103B and GRB 910711)
with high-frequency QPOs originated from GFs of
magnetars with torsional oscillation, the selected six
EOSs (i.e., TM1, NL3, APR, SLy4, DDME2, and GM1)
of NSs that are suitable for the condition of zero
temperature (i.e., cold crust)exhibit significant over-
lapping in mass ranges, and this suggests that we cannot
constrain the EOS of NSs via the high-frequency QPOs in
those two short-duration GBRs.
2. Radial oscillation. By assuming that the observed short-
duration GRBs (GRB 931103B and GRB 910711)with
high-frequency QPOs originated from the radial oscilla-
tion of an HMNS from a binary NS merger, the selected
six EOSs (i.e., IUF, TM1, TMA, FSG, BHBLp, and NL3)
of NSs that are suitable for the condition of high
temperature (i.e., hot remnant)cannot be ruled out with
redshift considered. However, it is found that the different
EOSs of NSs correspond to different temperatures for a
given redshift with Metric =0. This means that the EOS
can only be constrained if the redshift and temperature of
the remnant can be measured.
On the other hand, the origin of these two short GRBs may
be related to the inspiral phase of the binary NS merger. This
phase could also explain the observed high-frequency QPOs,
which caused by torsional oscillation. If this is the case, the γ-
ray emission of short GRBs could result from the interaction of
the NS magnetospheres during the inspiral phase
(M. Vietri 1996; B. M. S. Hansen & M. Lyutikov 2001;
S. T. McWilliams & J. Levin 2011; C. Palenzuela et al. 2013;
J.-S. Wang et al. 2018), or from the cracking of the tidal crust
in the last moments of a compact binary merger (E. Troja et al.
2010). In this framework, the GRB could be identified as an
isolated precursor rather than being associated with relativistic
outflows from post-merger remnants (D. Burlon et al. 2008; S.-
Q. Zhong et al. 2019; P. Coppin et al. 2020; J.-S. Wang et al.
2020). Unlike the fluid-dominated, high-temperature environ-
ment of an HMNS formed post-merger, the inspiral phase
involves relatively cold NSs with intact crusts, where shear
stresses can excite torsional modes. The maximum luminosity
of the precursor from the interaction of NS magnetospheres is
proportional to B
2
. In fact, this requires introducing an
additional assumption, namely, the binary system must contain
at least one NS with a magnetar-like magnetic field
(B>10
15
G)to achieve the observed luminosity (J.-S. Wang
et al. 2020).
Recently, numerical relativity studies have suggested that the
quasi-periodic kHz substructure observed in electromagnetic
outflows from HMNSs may be dominated by magnetohydro-
dynamic (MHD)shearing processes rather than being related to
internal stellar oscillations (E. R. Most & E. Quataert 2023).
This indicates that QPOs in relativistic outflows may not be
necessarily related to internal NS oscillations, making it
difficult to constrain the EOS of NSs. This is consistent with
the results obtained in this work. However, asteroseismology
remains a potential method to constrain the EOS of NSs when
more precise information of the NS temperature can be
obtained, such as the redshift and magnetic field strength.
Moreover, V. Guedes et al. (2024)adopted numerical relativity
simulations of post-merger remnants and their relationship to
constrain the binary tidal coupling constant, and then they
constrain the EOS of NSs by invoking the observed high-
frequency QPOs in those two short GRBs. They claim to have
found tight constraints on the mass–radius relation, which
differs significantly from what we have obtained in this paper.
The possible reason is that the adopted method and the physical
conditions are different from each other. Our method can
constrain the redshift based on the presence of two distinct
QPO frequencies (ν
1
and ν
2
)in these bursts, with certain strong
assumptions (e.g., temperature). In contrast, the approach from
V. Guedes et al. (2024)adopted ν
2
and the ratio ν
2
/ν
1
to
constrain the redshift, and the ratio between ν
2
and ν
1
is
redshift independent. We hope that multimessenger observa-
tions, combining electromagnetic signals and GWs, will offer
more precise constraints on the physical properties of NS
remnants in the future, such as their temperature, redshift, and
magnetic field strength.
Acknowledgments
This work is supported by the Guangxi Science Foundation
(grant No. 2023GXNSFDA026007), the National Natural
Science Foundation of China (grant Nos. 12494574,
11922301, and 12133003), the Program of Bagui Scholars
Program (LHJ), and the Guangxi Talent Program (“Highland of
Innovation Talents”).
ORCID iDs
Jun-Xiang Huang https://orcid.org/0000-0002-0798-8846
Hou-Jun Lü https://orcid.org/0000-0001-6396-9386
En-Wei Liang https://orcid.org/0000-0002-7044-733X
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