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Thermal luminosity degeneracy of magnetized neutron stars with
and without hyperon cores
F. Anzuini,1,2,3★A. Melatos1,4†, C. Dehman5,6, D. Viganò5,6,7, J. A. Pons8
1School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
2School of Physics and Astronomy, Monash University, Victoria 3800, Australia
3OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery, Clayton, Victoria 3800, Australia
4Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), University of Melbourne,
Parkville, Victoria 3010, Australia
5Institute of Space Sciences (IEEC-CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193, Barcelona, Spain
6Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capità 2–4, 08034 Barcelona, Spain
7Institute of Applied Computing & Community Code (IAC3), University of the Balearic Islands, Palma, 07122, Spain
8Departament de Física Aplicada, Universitat d’Alacant, 03690 Alicante, Spain
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The dissipation of intense crustal electric currents produces high Joule heating rates in cooling
neutron stars. Here it is shown that Joule heating can counterbalance fast cooling, making it
difficult to infer the presence of hyperons (which accelerate cooling) from measurements of
the observed thermal luminosity 𝐿𝛾. Models with and without hyperon cores match 𝐿𝛾of
young magnetars (with poloidal-dipolar field 𝐵dip &1014 G at the polar surface and 𝐿𝛾&1034
erg s−1at 𝑡.105yr) as well as mature, moderately magnetized stars (with 𝐵dip .1014 G
and 1031 erg s−1.𝐿𝛾.1032 erg s−1at 𝑡&105yr). In magnetars, the crustal temperature is
almost independent of hyperon direct Urca cooling in the core, regardless of whether the latter
is suppressed or not by hyperon superfluidity. The thermal luminosities of light magnetars
without hyperons and heavy magnetars with hyperons have 𝐿𝛾in the same range and are
almost indistinguishable. Likewise, 𝐿𝛾data of neutron stars with 𝐵dip .1014 G but with
strong internal fields are not suitable to extract information about the equation of state as long
as hyperons are superfluid, with maximum amplitude of the energy gaps of the order ≈1MeV.
Key words: stars: neutron – stars: interiors – stars: magnetic field – stars: evolution
1 INTRODUCTION
Timing measurements of neutron stars show that their inferred mag-
netic field spans a wide range of strengths, from ∼108G in millisec-
ond pulsars (Backer et al. 1982;Boriakoff et al. 1983;Lyne et al.
1987;Manchester 2017;Arzoumanian et al. 2018) up to ∼1015
G in magnetars (Mazets et al. 1979;Mazets & Golenetskii 1981;
Gavriil et al. 2002;Viganò et al. 2013;Vogel et al. 2014;Olausen
& Kaspi 2014;Mereghetti et al. 2015;Kaspi & Beloborodov 2017).
Although the magnetic field configuration of neutron stars at birth
is unknown (Duncan & Thompson 1992;Thompson & Duncan
1993;Spruit 2008), numerous authors have studied the possible ini-
tial magnetic field configuration consistent with MHD-equilibrium
(Braithwaite & Spruit 2006;Ciolfi et al. 2009;Lander & Jones 2009;
Ciolfi & Rezzolla 2013) and the long-term evolution of both crust-
confined or core-threading topologies (Gourgouliatos et al. 2013;
★E-mail: filippo.anzuini@monash.edu
†E-mail: amelatos@unimelb.edu.au
Viganò et al. 2013;Wood & Hollerbach 2015;Gourgouliatos et al.
2016;Elfritz et al. 2016;Igoshev et al. 2021;De Grandis et al. 2021).
These initial configurations are likely over-simplified. For example,
crustal confinement is not guaranteed, and it is unclear how the
twisted torus magnetic configuration (one of the most common ini-
tial topologies employed in numerical studies) would be produced.
As a matter of fact, recent magnetohydrodynamic simulations of the
magnetorotational instability in core-collapse supernovae (Aloy &
Obergaulinger 2021;Reboul-Salze et al. 2021) suggest a different
and more complex picture, in which the magnetic energy of the
protoneutron star spreads over a wide range of spatial scales. Such
simulations find that most of the magnetic energy is contained in
small or medium-scale size magnetic structures, both for the domi-
nant toroidal components, and the weaker poloidal components.
The magnetic evolution of a neutron star is interlinked with its
thermal evolution and hence with its composition, which depends on
the equation of state (Aguilera et al. 2008;Viganò et al. 2013;Pons &
Viganò 2019;Dehman et al. 2020;Viganò et al. 2021;Igoshev et al.
2021;De Grandis et al. 2021;Anzuini et al. 2021). In particular, the
©2021 The Authors
2F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
presence of exotic species such as hyperons accelerates cooling via
direct Urca processes (Prakash et al. 1992;Yakovlev et al. 2001).
In turn, the magnetic field causes anisotropic heat transport across
and along the magnetic field lines, and the internal layers are heated
up by the dissipation of the electric currents that sustain the field
(Joule heating).
In this paper, we show that Joule heating hides the effect of
fast cooling on the observed thermal luminosity 𝐿𝛾. We extend
the results presented in Aguilera et al. (2008) by simulating the
self-consistent evolution of the magnetic field, including Ohmic
dissipation and the generation of small scales by the action of the
Hall drift. Moreover, we consider hyperons in the core, unlike in
Aguilera et al. (2008), and focus on the cooling effect of hyperon di-
rect Urca. Accelerated direct Urca cooling is an important signature
of the presence of hyperons, so Joule heating complicates the link
between cooling curves, internal composition, and ultimately the
equation of state (EoS). We find that the magneto-thermal evolution
of light stars without hyperon cores resembles the evolution of heavy
stars with hyperon cores, if the crustal field is sufficiently strong.
The thermal power produced by Joule heating due to magnetic field
decay dominates the thermal evolution of the crust, with less influ-
ence from neutrino cooling in the core, so that the temperatures of
the crust and core “decouple”, i.e. they evolve approximately inde-
pendently (Kaminker et al. 2006;Kaminker et al. 2007). Part of the
additional heat is dispersed via neutrino emission, and part of it is
transported via thermal conduction to the surface, increasing 𝐿𝛾.
We show that high Joule heating rates affect the interpretation of 𝐿𝛾
data in terms of light models without hyperons and heavy models
with hyperons for young magnetars (𝑡.105yr), with surface value
at the pole of the poloidal-dipolar field 𝐵dip &1014 G, and mature
stars (𝑡&105yr) with 𝐵dip .1014 G.
We study the magneto-thermal evolution of models with or
without concentrations of hyperons in their cores with the updated
version of the two-dimensional, axisymmetric magneto-thermal
code developed by the Alicante group (Aguilera et al. 2008;Pons
et al. 2009;Viganò et al. 2012;Viganò et al. 2013;Pons & Viganò
2019;Dehman et al. 2020;Viganò et al. 2021), recently adapted to
study hyperon stars (Anzuini et al. 2021). We employ the GM1A
EoS (Gusakov et al. 2014), based on the 𝜎𝜔𝜌 𝜙𝜎∗model of nucleon,
lepton and hyperon matter. The model is fitted to hypernuclear data,
and predicts that only Λand Ξ−hyperons appear in dense matter.
Other hyperon species, such as Σ−hyperons, do not appear in the
allowed density range. We extend the results found in Anzuini et al.
(2021) by considering both crust-confined and core-extended initial
magnetic field configurations. Neutron star models obtained with
the GM1A EoS cool down rapidly due to the activation of both nu-
cleonic and hyperonic direct Urca emission. If neutrons are paired
in a large fraction of the stellar core, internal heating is required to
match 𝐿𝛾data (Anzuini et al. 2021). Among the possible heating
mechanisms (Alpar et al. 1984;Shibazaki & Lamb 1989;Fernan-
dez & Reisenegger 2005;Pons & Geppert 2007;Viganò et al. 2013;
Hamaguchi et al. 2019;Pons & Viganò 2019), Joule heating can
supply the necessary thermal power to reconcile theoretical cooling
rates and 𝐿𝛾observations.
This paper is organized as follows. Section 2describes the the-
oretical framework adopted to simulate the magneto-thermal evolu-
tion of neutron stars. It introduces the heat diffusion and magnetic
induction equations, as well as the microphysics input. In Section
3we calculate 𝐿𝛾versus time for a selection of representative
magneto-thermal models. The corresponding surface temperatures
are studied in Section 4.
2 STELLAR MODEL
In this section we outline the ingredients of the model describing the
star’s magneto-thermal evolution. We introduce the heat diffusion
and the magnetic induction equations in Section 2.1, and the initial
conditions for the magneto-thermal evolution in Section 2.2. The
microphysics input (e.g. superfluid model and neutrino emissivity)
is described in Section 2.3. Section 2.4 contextualizes the mass
range studied in this paper in terms of available neutron star data.
Degeneracies arise when comparing the model output (e.g. cooling
curves) with observations of 𝐿𝛾, as described in Section 2.5.
The magneto-thermal evolution of neutron stars is studied as-
suming that the space-time metric is given by the Schwarzschild
metric. Deviations from spherical symmetry related to the temper-
ature and magnetic field are neglected (Pons & Viganò 2019).
2.1 Heat diffusion, magnetic induction
The internal temperature evolves via the heat diffusion equation
(Aguilera et al. 2008;Pons et al. 2009)
𝑐V𝑒Φ𝜕𝑇
𝜕𝑡 +∇· (𝑒2Φ𝑭)=𝑒2Φ(𝑄J−𝑄𝜈).(1)
In Eq. (1), the heat capacity per unit volume of nucleons, leptons
and hyperons is denoted by 𝑐V. The internal, local temperature and
the dimensionless gravitational potential are 𝑇and Φrespectively,
and the differential operator ∇includes the metric factors. The
heat flux 𝑭reads 𝑭=−𝑒−Φˆ
𝑘·∇(𝑒Φ𝑇), where ˆ
𝑘denotes the
thermal conductivity tensor (Potekhin, A. Y. & Yakovlev, D. G.
2001;Potekhin et al. 2003), while 𝑄Jand 𝑄𝜈denote respectively
the Joule heating rate per unit volume and neutrino emissivity per
unit volume.
Given the high thermal conductivity of the core, the latter be-
comes isothermal a few decades after the neutron star birth. The
crust relaxes slower thermally, during a period that typically lasts
𝑡∼102yr, depending on the thermal conductivity, heat capacity
of the crust layers and whether neutrons are superfluid (Lattimer
et al. 1994;Gnedin et al. 2001). During the thermal relaxation stage
the crust temperature is higher than the core temperature, and the
thermal luminosity of the star does not reflect the thermal evo-
lution of the core. The relaxation stage ends when the “cooling
wave” (Gnedin et al. 2001) propagating from the core reaches the
stellar surface, and the thermal luminosity drops by orders of magni-
tude (depending, among other factors, on the presence of superfluid
phases).
We solve the heat-diffusion equation everywhere in the stellar
interior, except in the outer envelope. The typical time-scales in the
envelope are shorter than in the deeper layers, requiring a smaller
time-step and increasing the computational cost. Instead, we rely on
an effective relation between the internal temperature at the bottom
of the outer envelope 𝑇band the surface temperature 𝑇s. The latter
is obtained from the 𝑇s–𝑇brelation employed in Potekhin et al.
(2015), Viganò et al. (2021) and Anzuini et al. (2021). The 𝑇s–𝑇b
relation depends on the magnetic field (Potekhin et al. 2015); see
also the discussion in Anzuini et al. (2021). In the following, we
assume that the outer envelope is composed of iron.
The magnetic field Bevolves according to the magnetic induc-
tion equation, which in the crust reads
𝜕𝑩
𝜕𝑡 =−∇×h𝑐2
4𝜋𝜎𝑒
∇× (𝑒Φ𝑩) + 𝑐
4𝜋𝑒𝑛𝑒
[∇× (𝑒Φ𝑩)] × 𝑩i,(2)
where 𝑐is the speed of light, 𝜎𝑒is the temperature- and density-
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 3
dependent electrical conductivity, 𝑒is the elementary electric charge
and 𝑛𝑒is the electron number density. The first term is the Ohmic
(dissipative) term and the second is the nonlinear Hall term.
As the temperature drops due to neutrino emission, the ther-
mal and electric conductivities increase and become temperature-
independent for sufficiently low temperatures (Aguilera et al. 2008),
gradually decreasing the Ohmic dissipation rate. At the same time,
the decay of the magnetic field is enhanced by the Hall term in
the magnetic induction equation, because although the Hall term
does not directly dissipate magnetic energy, it produces small-scale
magnetic structures, where Ohmic dissipation is enhanced. Further-
more, the Hall drift tends to push the electric currents toward the
crust-core boundary, where they may be dissipated more efficiently
by the presence of impurities and pasta phases (Pons et al. 2013;
Viganò et al. 2013), producing higher Joule heating rates. As the
magnetic field evolves, the thermal conductivity along and across
the magnetic field lines changes, affecting the local temperature.
Hence, the magnetic evolution influences the thermal evolution and
vice versa.
The evolution of the magnetic field in the core is more un-
certain due to its multifluid nature and the occurrence of proton
superconductivity. There may be regions with protons in the nor-
mal phase or in the superconducting phase, the latter being of type-II
(Baym et al. 1969;Sedrakian & Clark 2019) or type-I (leading to
magnetic field expulsion due to the Meissner effect). Recent calcu-
lations (Wood et al. 2020) predict phase coexistence in mesoscopic
regions (larger than the flux tubes, but smaller than the macroscopic
length-scales), introducing several length and time-scales into the
problem. When superconductivity is neglected, the typical time-
scales for Ohmic dissipation and Hall advection exceed the cooling
time-scales, so that the magnetic field undergoes little change in the
stellar core (Elfritz et al. 2016;Dehman et al. 2020;Viganò et al.
2021). The inclusion of ambipolar diffusion could partially speed
up the dynamics under certain conditions (Castillo et al. 2020). A
more consistent approach including hydrodynamic effects in the
superfluid/superconducting core could substantially accelerate the
evolution, as recently discussed in terms of estimated time-scales
by Gusakov et al. (2020) (see also references therein). Moreover,
as noted above, an initial complex magnetic topology can also re-
duce the typical length- and time-scales, compared to the usually
assumed purely dipolar fields.
In the simulations reported in this work, the induction equation
in the crust includes both the Ohmic and Hall terms. In the core,
only the Ohmic term is included, so that the core magnetic field is
frozen, since the typical diffusion timescale in the core is larger than
the typical age of isolated neutron stars studied here.
2.2 Initial conditions
The evolution is independent of the initial internal temperature (if
the latter is sufficiently high), and we typically adopt a temperature
of 1010 K (Yakovlev et al. 1999;Page et al. 2004,2006;Potekhin
et al. 2015).
The magnetic fields of neutron stars may be sustained by elec-
tric currents both in the crust and in the core. In particular, in the
crust the currents can produce small-scale magnetic fields that en-
hance Joule heating via the Hall cascade (Gourgouliatos & Pons
2019;Brandenburg 2020). In this work we consider various possi-
ble initial magnetic field configurations (listed in Tables 1and 2).
The two main categories are the following. (i) Crust-confined fields.
The radial magnetic field component vanishes at the crust-core in-
terface, while the latitudinal (𝐵𝜃) and toroidal (𝐵𝜙) components
Table 1. Crust-confined initial magnetic configurations for a star with
𝑀=1.8𝑀.𝐵dip is the surface field strength at the pole of the dipolar-
poloidal component. 𝐸T
mag is the magnetic energy stored in the toroidal
component, and 𝐸mag denotes the total magnetic energy. The number of
poloidal multipoles in the crust is denoted by 𝑙pol.
Config. 𝐵dip 𝐸T
mag/𝐸mag 𝑙pol
A1 1.0×1013 G93% 1
A2 5.0×1013 G35% 1
A3 1.0×1014 G35% 1
A4 1.0×1015 G0.5% 1
A5 2.0×1015 G0 1
A1m2,21.0×1013 G77% 2
A1m3,31.0×1013 G51% 3
A1m4,41.0×1013 G29% 4
Table 2. Core-extended initial magnetic configurations. The quantities 𝐵dip,
𝐸T
mag,𝐸mag and 𝑙pol are defined as in Table 1. In the B1, B2 and C1
configurations the initial toroidal field is confined to an equatorial torus
in the core.
Config. 𝐵dip 𝐸T
mag/𝐸mag 𝑙pol
B1 1.0×1013 G50% 1
B2 1.0×1014 G50% 1
C1 1.0×1014 G42% 1
C2 2.0×1014 G0 1
C1m2,01.0×1014 G0 2
are different from zero. (ii) Core-threading fields. At the crust-core
interface the radial component of the magnetic field is 𝐵𝑟≠0,
and the magnetic field lines penetrate into the core. In both cases,
at the surface the magnetic field is matched continuously with the
potential solution of a force-free field (i.e. the electric currents do
not leak into the magnetosphere).
In Table 1we list the crust-confined magnetic field configura-
tions considered in this work. From a computational point of view,
there is limited capability to follow numerically the rich dynamics
of small-scale magnetic fields in the crust; however, the configura-
tions studied here may reproduce typical Joule heating rates of more
realistic, small-scale crustal fields. We vary the ratio of the magnetic
energy stored in the toroidal component (𝐸T
mag) and the total mag-
netic energy (𝐸mag), as well as the number of poloidal multipoles
𝑙pol. Crust-confined magnetic fields generate typically higher Joule
heating rates than core-extended fields, for similar values of the total
magnetic energy. As a matter of fact, in the crust-confined case all
currents are forced to circulate in the crust, where the resistivity is
orders of magnitude larger than in the core.
We consider two families of core-extended configurations
(listed in Table 2). The first family includes the B1 and B2 configura-
tions (studied for example by Dehman et al. (2020) and Viganò et al.
(2021)), where the electric currents that sustain the magnetic field
reside exclusively in the core. The corresponding Joule heating is
typically lower than crust-confined configurations for two reasons.
First, the resistivity in the core is low, so that Joule heating is lower
than in the crust. Second, any additional heat produced via Joule
heating in the core is carried away by neutrinos. The second family
includes configurations with both crustal and core electric currents.
To mimic the total currents in both the crust and core in neutron
stars, in the C1, C2 and C1m2,0configurations we assume the exis-
tence of large-scale poloidal-dipolar fields threading the core, plus
MNRAS 000,1–15 (2021)
4F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
crustal fields. For example, in the C1 configuration, a large-scale
dipolar-poloidal field threads the star, sustained by currents in the
core. Additionally, there is a crustal dipolar-poloidal field, sustained
by crustal currents. The azimuthal field is almost entirely confined
to a torus in the core. In the C1m2,0configuration, there is a core-
threading large-scale poloidal dipole sustained by electric currents
in the core, plus an additional poloidal field with two multipoles in
the crust. The core-extended configurations with crustal and core
electric currents reproduce qualitatively similar Joule heating rates
to the ones expected from small-scale, crustal fields, attained by the
crust-confined configurations in Table 1.
2.3 Microphysics input
Neutron star models calculated with the GM1A EoS include concen-
trations of nucleons, leptons and hyperons (𝑛𝑝 𝑒𝜇𝑌 matter, where
𝑌denotes hyperonic species). In particular, the EoS is fitted to
modern hypernuclear data (e.g. Millener et al. (1988); Schaffner
et al. (1994); Takahashi et al. (2001); Weissenborn et al. (2012), see
Gusakov et al. (2014)), and predicts that only the Λand Ξ−hyper-
ons appear in dense matter, while Σ−hyperons are absent because
their potential in dense nuclear matter is repulsive. Below we list
concisely the microphysics input in our simulations, such as heat
capacity, neutrino emissivity, and thermal and electric conductivi-
ties, emphasizing the novelties compared to the last version of the
code (Viganò et al. 2021;Anzuini et al. 2021).
•Heat capacity. We include the contribution to the heat capacity
of 𝑛𝑝𝑒𝜇𝑌 matter (Yakovlev et al. 1999) as well as the contribution
of ions in the crustal rigid lattice.
•Neutrino emission. In the core, neutrinos are produced via nu-
cleonic and hyperon direct Urca reactions, Cooper pair breaking and
formation processes, neutrino bremsstrahlung and modified Urca.
We implement the in-medium corrections to the modified Urca pro-
cess emission rates in Shternin et al. (2018), where the enhancement
factors are calculated only for the neutron branch (Eqs. (8) and (9)
in Shternin et al. (2018)). We apply the same formulae to the proton
branch as well, which we interpret as upper limits to the in-medium
corrections1. Such corrections make a negligible impact on the
cooling curves in our case, given the superfluid model adopted (see
below) and the activation of direct Urca cooling processes. For the
crust, we match the GM1A EoS with the SLy4 EoS (Douchin &
Haensel 2001), including the crustal neutrino emission processes
considered in Anzuini et al. (2021). We note that neutron star models
obtained with the GM1A EoS cool down fast via nucleonic direct
Urca (which is active for 𝑀≥1.1𝑀). For 𝑀&1.49 𝑀the
hyperon direct Urca involving protons and Λhyperons is triggered,
and for 𝑀&1.67 𝑀the hyperon direct Urca involving Ξ−and Λ
hyperons activates.
•Electric and thermal conductivities. The conductivities depend
on density and temperature and vary by orders of magnitude in the
crust and core regions. In our simulations, the thermal conductivity
is a tensor because of anisotropic heat transport caused by the mag-
netic field (Potekhin, A. Y. & Yakovlev, D. G. 2001;Potekhin et al.
2003), with components parallel and perpendicular to the magnetic
field lines. We do not include contributions of nucleons, muons or
hyperons to the electrical conductivities due to their lower mobility
with respect to electrons.
1We check that the correction factors implemented in our code reproduce
the results in Shternin et al. (2018) for the BCPM EoS (Sharma et al. 2015).
•Superfluid phases. Nucleons and hyperons can be superfluid,
suppressing both the heat capacity and most channels of neutrino
production (only partially compensated by the Cooper pair breaking
and formation neutrino channel). As in Anzuini et al. (2021), we
assume that neutrons pair in the singlet channel in the crust (“SFB”
model in Ho et al. (2015)) and in the triplet channel in the core (“c”
model in Page et al. (2004), see also Yanagi et al. (2020)). Protons
pair in the singlet channel throughout the stellar core (“CCDK”
model (Ho et al. 2015)). We use the parameters reported in Ap-
pendix A in Anzuini et al. (2021) for singlet pairing of hyperon
species, reproducing similar gaps to the ones calculated by Raduta
et al. (2018). We also neglect the occurrence of nucleon-hyperon
superfluid phases arising from the interaction of nucleons and hy-
perons (Zhou et al. 2005;Nemura et al. 2009;Haidenbauer et al.
2020;Sasaki et al. 2020;Kamiya et al. 2022). The study of the
magneto-thermal evolution with different hyperon superfluid gaps,
obtained for example from lattice quantum chromodynamics simu-
lations (Aoki et al. 2008,2012;Hatsuda 2018;Sasaki et al. 2020;
Kamiya et al. 2022) is left for future work.
2.4 Mass models
We study the thermal luminosity of hyperon and non-hyperon stars
by comparing the magneto-thermal evolution of light-mass models
(𝑀=1.3𝑀) and massive models with hyperon concentrations in
the core (𝑀=1.8𝑀). Given the large nucleon and hyperon gaps,
the thermal luminosity of low-mass stars with 𝑀=1.3𝑀is similar
to models with masses in the range 1.1𝑀.𝑀.1.4𝑀(see
Anzuini et al. (2021)). The same applies to the thermal luminosities
of high-mass stars with 𝑀=1.8𝑀, which are similar to models
with masses in the range with 1.5𝑀.𝑀.1.8𝑀(Anzuini
et al. 2021).
We emphasize that neutron stars are commonly found with
masses of the order of 𝑀≈1.3𝑀, while heavy stars are less
common. Massive stars may form in merger events (if the remnant
does not collapse into a black hole) (Fryer et al. 2015;Mandel &
Müller 2020;Ruiz et al. 2021), or due to matter accreted over long
time-scales (up to ≈0.1𝑀in roughly 10 Gyr, in the optimal
scenario) (Chevalier 1989;Kiziltan et al. 2013) for example. Some
mass measurements obtained via X-ray and optical observations of
neutron stars in binary systems with white dwarfs fall in the range
𝑀&1.6𝑀(Kiziltan et al. 2013;Alsing et al. 2018). Heavy
stars formed via merging or accretion may have inhomogeneous
internal temperatures and complex magnetic field configurations,
far from the initial conditions commonly employed in the literature
of neutron star cooling. For the purpose of this work (i.e. the study
of the 𝐿𝛾degeneracy between low-mass and high-mass models),
we adopt the standard initial conditions employed by several authors
(Yakovlev et al. 1999;Page et al. 2004;Potekhin & Chabrier 2018;
Raduta et al. 2018,2019), bearing in mind that the magneto-thermal
evolution of heavy stars likely requires more realistic temperature
and magnetic field configurations initially.
2.5 Internal heating
In principle, it should be possible to constrain the internal com-
position of a neutron star and hence the EoS of dense matter by
comparing the output of cooling simulations, specifically 𝐿𝛾as a
function of the stellar age, with optical and X-ray measurements of
𝐿𝛾(Viganò et al. 2013;Potekhin et al. 2020). In practice, there are
several scenarios where the task is complicated by internal heating.
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 5
Consider an internal heating mechanism in the crust. When the
thermal power supplied by the source is high enough, the local tem-
perature increases and becomes almost independent of the influence
of the neutrino emission processes deep in the core, so that the crust
and core are thermally decoupled (Kaminker et al. 2006;Kaminker
et al. 2007;Kaminker et al. 2009,2014). We emphasize that de-
coupling in this context means that the crust and core temperatures
evolve approximately independently. It does not mean that the crust
and core are thermally insulated; there is still a heat flux between
the crust and core. One key role is played by the location where the
additional thermal power is supplied (Kaminker et al. 2006;Anzuini
et al. 2021). If the additional heat is supplied at the bottom of the
crust, it is transported via thermal conduction to the core, where it is
easily lost via neutrino emission processes. As a result, the crust and
core are thermally coupled, and the star cools down faster. On the
other hand, if the heater deposits heat close to the outer envelope, it
increases the local temperature, a fraction of the heat is transported
via thermal conduction to the surface (increasing the surface tem-
perature and hence 𝐿𝛾), and a fraction makes its way into the core,
where it is lost by neutrino emission processes. If the heating rate
is sufficiently high, the local temperature in the crust is dominated
by the heater, and the thermal evolution of the crust decouples from
the core. In this scenario, it is challenging to ascertain whether the
observed 𝐿𝛾is the result of fast cooling counteracted by internal
heating, or if fast cooling is not active at all.
In this paper we focus on Joule heating. Joule heating rates
may be sufficiently high to hide the cooling effect of nucleonic
direct Urca, as discussed in Aguilera et al. (2008). Here we extend
those results to include hyperon species and to consider stars with
𝐵dip .1014 G and strong internal fields. Although 𝐵dip can be
inferred from timing properties, there is no direct method to infer
the strength of the internal field, which may decay and keep the
star hot via Joule heating. Other internal heating mechanisms are
also plausible but are not modelled in this paper, such as vortex
creep (Shibazaki & Lamb 1989;Page et al. 2006) or rotochemical
heating (Reisenegger 1995;Hamaguchi et al. 2019) (see Gonzalez
& Reisenegger (2010) for a concise review).
3 THEORETICAL COOLING CURVES AND 𝐿𝛾
DEGENERACY
In this section we compare the theoretical cooling rates of some of
the initial magnetic configurations reported in Tables 1and 2with
the available data of isolated, young magnetars with 𝐵dip &1014 G
and of older neutron stars with 𝑡&105yr and 𝐵dip .1014 G. For
magnetars, we use the data corresponding to 16 objects with ages
.107yr reported in Viganò et al. (2013). Typically, the sources
have inferred magnetic fields with 𝐵dip &1014 G (some 𝐵dip values
have been updated2). For stars with weaker fields, we are mostly
interested in ages &104yr, and we use the data reported in Potekhin
et al. (2020). We consider two typical masses, namely a low-mass
model with 𝑀=1.3𝑀(without hyperons in the stellar core) and
a high-mass model with 𝑀=1.8𝑀(with hyperons). We use the
nucleon and hyperon gap models specified in the previous section.
The magneto-thermal evolution of models with 𝑀=1.8𝑀and
hyperon cores is studied in detail in Appendix A.
2We refer the reader to the McGill online magnetar catalogue http://
www.physics.mcgill.ca/~pulsar/magnetar/main.html (Olausen
& Kaspi 2014)
3.1 Magnetars
We first focus on magnetars, which typically have an inferred dipolar
poloidal field strengths at the polar surface in the range 1014 G.
𝐵dip .1015 G and ages 𝑡.105yr.
In Figure 1we display the cooling curves corresponding to
the A4, A5 and C2 configurations (red, blue and orange curves
respectively). Magnetars are represented by red dots; stars with
lower 𝐵dip are represented by black dots. Figure 1(a) studies the
model with 𝑀=1.3𝑀, which cools mainly via nucleonic direct
Urca, and Figure 1(b) reports the cooling curves corresponding
to the model with 𝑀=1.8𝑀, cooling via both nucleonic and
hyperonic direct Urca.
In Figure 1(a), the A5 configuration (blue, dotted curve) main-
tains higher 𝐿𝛾with respect to the A4 configuration (red, solid
curve) and the C2 configuration (orange, dotted-dashed curve).
The blue curve matches some of the most luminous sources with
𝐿𝛾&1035 erg s−1. The model maintains 𝐿𝛾&1034 erg s−1up
to 𝑡≈2×105yr. The red cooling curve attains lower values of
𝐿𝛾, and at later times (𝑡&104yr) it maintains a similar thermal
luminosity to the blue curve (𝐿𝛾&1034 erg s−1). The C2 config-
uration matches lower thermal luminosities of both magnetars and
stars with lower fields.
In Figure 1(b) we study the same magnetic configurations as
in Figure 1(a), but for a star with 𝑀=1.8𝑀and superfluid hyper-
ons. The blue dotted and red solid lines (A5 and A4 configurations
respectively) attain similar thermal luminosities to the correspond-
ing low-mass models in Figure 1(a) for 103yr .𝑡.105yr. For
105yr .𝑡.2×105yr, both curves fall below 𝐿𝛾≈1034 erg s−1,
contrarily to the corresponding curves in Figure 1(a). The orange
curve (C2 configuration) attains lower 𝐿𝛾with respect to the cor-
responding curve in Figure 1(a) for 𝑡.104yr. At later times, the
orange curves in Figure 1(a) and Figure 1(b) reach similar 𝐿𝛾.
The comparison between the cooling curves displayed in
Figure 1(a) and Figure 1(b) shows that the thermal luminosity
observations and age estimates of magnetars can be explained
equally by stars cooling mainly via nucleonic direct Urca emis-
sion (𝑀=1.3𝑀) and stars cooling mainly via both nucleonic and
hyperonic direct Urca emission (𝑀=1.8𝑀). The decay of the
strong magnetic field produces sufficient thermal power to decouple
the thermal evolution of the crust and the core, and the measured 𝐿𝛾
of magnetars is dominated by Joule heating in the crust, regardless
of the neutrino emission mechanisms active in the core involving
hyperons.
Similar conclusions hold if hyperons are not superfluid, and
hyperon direct Urca operates without being suppressed by superfluid
effects (Figure 2). The cooling curves in Figure 2(a) correspond to
models with the A4 initial magnetic configuration. The red curve
shows the case 𝑀=1.3𝑀, and the orange dotted-dashed curve
the case 𝑀=1.8𝑀with hyperon superfluidity. The blue, dotted
curve is obtained for 𝑀=1.8𝑀without hyperon superfluidity.
Up to 𝑡≈104yr, the red, orange and blue cooling curves are similar,
and are compatible with the 𝐿𝛾data of the same magnetars. The
orange and blue curves are degenerate up to 𝑡≈104yr. Only at
later times Joule heating becomes weaker (e.g. 𝑡&104yr), and
one can distinguish between low-mass and high-mass stars, with
or without hyperon superfluidity. Figure 2(b) reports a scenario
similar to Figure 2(a), but for the A5 initial configuration. The
red and orange curves (𝑀=1.3𝑀and 𝑀=1.8𝑀models
respectively, the latter including hyperon superfluidity) and the blue
curve (𝑀=1.8𝑀, without hyperon superfluidity) are similar up
to 𝑡≈2×104yr. As in Figure 2(a), the three cooling curves are
MNRAS 000,1–15 (2021)
6F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
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32
34
36
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(a) M= 1.3M
A4
A5
C2
12345
log10(t/yr)
32
34
36
log10(Lγ/erg s−1)
(b) M= 1.8M
A4
A5
C2
Figure 1. Cooling curves for models with 𝑀=1.3𝑀and 𝑀=1.8𝑀, with nucleons and hyperons in the superfluid phase. Overlapped are the data points
corresponding to magnetars (red dots, (Viganò et al. 2013)) and data corresponding to moderately magnetized stars (black dots, (Potekhin et al. 2020)). (a)
𝑀=1.3𝑀.(b) 𝑀=1.8𝑀. The legends report the initial magnetic field configurations (see Tables 1and 2for details).
12345
log10(t/yr)
32
34
36
log10(Lγ/erg s−1)
(b) A5
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
12345
log10(t/yr)
32
34
36
log10(Lγ/erg s−1)
(a) A4
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
Figure 2. Effect of hyperon superfluidity on the cooling curves of models with magnetar-likefields. (a) A4 initial magnetic configuration. (b) A5 initial magnetic
configuration. In both panels, the blue dotted curves (𝑀=1.8𝑀, with hyperons in the normal phase) and orange dotted-dashed curves (𝑀=1.8𝑀, with
superfluid hyperons) are degenerate for 𝑡.104yr; the red solid curves (𝑀=1.3𝑀) and orange curves are similar for 𝑡.104yr. Nucleons are superfluid
in both panels.
distinguishable only for 𝑡&2×104yr, which exceeds most of the
age estimates of the magnetar population (Viganò et al. 2013).
Some magnetar sources lie above the blue curves in Figure 1.
We remind the reader that in this work we consider magnetized, iron-
only outer envelopes. The data points with 1035 .𝐿𝛾/erg s−1.
1036 may be explained by invoking accreted envelopes and/or higher
initial magnetic fields (Potekhin, A. Y. & Yakovlev, D. G. 2001;
Potekhin et al. 2003;Viganò et al. 2013). Furthermore, the thermal
luminosity of these magnetars may be higher because of the pres-
ence of small hot spots, produced by the inflow of magnetospheric
currents on the stellar surface. The latter process is not modeled in
our simulations. We also note that our simulations do not include
Joule heating in the highly resistive layer of the outer envelope,
which may help to increase the thermal luminosity and match the
data of the brightest magnetars.
Above we consider low-mass and high-mass models obtained
with the same EoS. However, there are several more scenarios in
which the cooling curves become nearly indistinguishable. Con-
sider for example two models with the same (high) mass, the first
obtained with the GM1A EoS (hosting 𝑛𝑝 𝑒𝜇𝑌 matter and cooling
via nucleonic and hyperonic direct Urca) and the second with a
different EoS, for example hosting only 𝑛𝑝 𝑒𝜇 matter and cooling
only via nucleonic direct Urca. If both stars are born with strong
magnetic fields, their magneto-thermal evolution can lead to similar
observed thermal luminosities, making it hard to infer the presence
of hyperons in the stellar core. We also emphasize that our results
depend unavoidably on the superfluid model adopted, in particular
on the nucleon energy gaps in the core. Smaller nucleon gaps lead to
higher nucleon direct Urca emissivity, widening the 𝐿𝛾difference
between light and heavy models. In this case, stronger Joule heating
may be required to obtain the 𝐿𝛾degeneracy discussed above.
In summary, thermal luminosity data of magnetars with 𝐿𝛾&
1034 erg s−1are not suitable to infer whether the core contains
hyperons or not and hence infer the internal composition. We note
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 7
also that it is not possible to constrain the properties of hyperon
superfluid phases, since 𝐿𝛾is degenerate for stars with cores hosting
normal or superfluid hyperons with large energy gaps. Our results
show that low-mass models composed of 𝑛𝑝𝑒 𝜇 matter and high-
mass models composed of 𝑛𝑝𝑒𝜇𝑌 matter have a similar magneto-
thermal evolution due to crust-core decoupling.
3.2 Low-𝐵dip neutron stars
Magnetars are only a subset of the observable neutron star popu-
lation. Timing measurements reveal that most neutron stars have
inferred fields satisfying 𝐵dip .1014 G. We study the evolution
of such stars in Figure 3, where we display the cooling curves
corresponding to the A2, A1m2,2and A1m3,3configurations (blue
dotted, red solid and orange dotted-dashed curves respectively). The
superfluid energy gaps are the same as in Figure 1.
Figure 3(a) reports the case 𝑀=1.3𝑀. For 𝑡.104yr,
the blue, red and orange lines attain similar thermal luminosi-
ties. The A1m2,2and A1m3,3configurations produce almost de-
generate cooling curves, which are compatible with X-ray emit-
ting isolated neutron stars (XINS), such as RX J1856.5–3754 and
RX J1605.3+3249, and ordinary pulsars such as PSR J0357+3205
(Potekhin et al. 2020).
The case 𝑀=1.8𝑀with superfluid hyperons is presented
in Figure 3(b). The blue, red and orange curves correspond to the
A2, A1m2,2and A1m3,3initial magnetic configurations respec-
tively. They differ from the curves in panel (a) for 𝑡.105yr,
attaining lower values of 𝐿𝛾. However, at later times they match
the same sources as in Figure 3(a), e.g. RX J1605.3+3249 and PSR
J0357+3205. It is not possible to distinguish at 𝑡&105yr between
the cooling curves of low-mass stars cooling via nucleonic direct
Urca (Figure 3(a)) and high-mass stars cooling via both nucleonic
and hyperonic direct Urca (Figure 3(b)). The cooling due to the
appearance of hyperons is masked by the high Joule heating rate
caused by the decay of the unobserved internal field. One may ask
why the cooling curves in Figure 3(a) differ from the corresponding
ones in Figure 3(b) for 𝑡.105yr, and attain similar values of 𝐿𝛾
for 𝑡&105yr. The reason is that the thermal power supplied by
Joule heating for the A2, A1m2,2and A1m3,3initial configurations
is insufficient to counterbalance the power lost due to nucleonic
and hyperonic direct Urca emissivity when the star is relatively hot.
The crust-core decoupling is incomplete. However, at later times
the direct Urca emissivity is weaker due to the lower internal tem-
perature, and Joule heating dominates the thermal evolution of the
star. Consequently, the cooling curves of low-mass and high-mass
stars are similar for 𝑡&105yr.
Below we investigate further the consequences of the incom-
plete crust-core thermal decoupling in stars with 𝐵dip .1014 G.
We show that if hyperons are in the normal phase, the cooling
curves of high-mass hyperon stars are clearly distinguishable from
the curves of stars without hyperons in their core. Figure 4displays
the cooling curves corresponding to the A1m3,3and A2 magnetic
configurations. In Figure 4(a) (A1m3,3initial configuration) the red
(solid) and orange (dotted-dashed) lines correspond to models with
𝑀=1.3𝑀and 𝑀=1.8𝑀(the latter assuming that hyperons
are superfluid). The blue, dotted line corresponds to a model with
𝑀=1.8𝑀, but with hyperons in the normal phase. The blue
curve falls below 𝐿𝛾=1033 erg s−1already for 𝑡.103yr, and
matches the data corresponding to PSR J0357+3205 for example.
There is no degeneracy between the orange and the blue curve.
Similar results are found in Figure 4(b) (A2 initial configura-
tion), where the red and orange curves become almost degenerate
for 𝑡&105yr. However, the blue curve (𝑀=1.8𝑀model
without hyperon superfluidity) is clearly distinguishable, attaining
𝐿𝛾.1032 erg s−1for 𝑡&105yr.
In summary, we find two trends. If the star is born with a
magnetar-like magnetic field with 𝐵dip &1014 G and/or a strong
internal field that stores a large fraction of the total magnetic energy,
the crustal temperature is regulated by Joule heating and is almost
independent of neutrino cooling in the core, causing crust-core
thermal decoupling. Magnetar data can be explained by both low-
mass and high-mass models, regardless of the presence of hyperons.
If the star has a lower field at birth (𝐵dip .1014 G) but the internal
field stores most of the magnetic energy, the crust-core thermal
decoupling is incomplete. In the latter scenario, if hyperons are
superfluid, the magneto-thermal evolution is similar for low-mass
and high-mass stars for 𝑡&105yr. This raises the question of how
to distinguish between stars that cool via nucleonic and hyperonic
direct Urca heated by magnetic field decay, and stars that cool
down only via nucleonic direct Urca (or even stars where direct
Urca is not active at all), given that the internal field configuration
and strength are unknown. On the contrary, if hyperons are not
superfluid, the cooling curves are clearly distinguishable also for
𝑡&105yr. We emphasize that we do not claim that neutron stars
with low inferred values of 𝐵dip always have strong internal fields,
nor that the available data of thermal emitters with low 𝐵dip must
be interpreted in terms of strong internal heating. Such magnetic
configurations may be characteristic of a subset of the neutron star
population, rather than a common feature of thermally emitting
stars.
4 SURFACE TEMPERATURE
We now calculate the surface temperature of some of the models
reported in Figures 2and 4. We study the similarities between
the redshifted surface temperature (𝑇∞
S) of low- and high-mass
models in Figure 5for the A5, A2 and A1m3,3initial magnetic
configurations.
The top row in Figure 5displays snapshots of 𝑇∞
Sversus the
colatitude 𝜃taken at 𝑡=103,104,105yr for the A5 configuration.
At 𝑡=103yr, the values of 𝑇∞
Sfor the 𝑀=1.3𝑀model (red
curve), the 𝑀=1.8𝑀model with hyperons in the superfluid
phase (orange curve) and the 𝑀=1.8𝑀model with normal hy-
perons (blue curve) are similar, being higher at the equator than at
the poles. At later times (𝑡&104yr), the 𝑀=1.8𝑀model with
hyperons in the normal phase cools more quickly due to the higher
neutrino emissivity, increasing the difference between the blue and
the orange and red curves at the poles. At the equator, the curves
remain similar up to 𝑡=105yr. In line with the results of 𝐿𝛾re-
ported in Figure 2, the three models have a similar magneto-thermal
evolution, producing almost indistinguishable cooling curves up to
𝑡≈2×104yr, and a similar surface temperature map.
For initial configurations with weaker fields (middle and bot-
tom rows in Figure 5, A2 and A1m3,3configurations respectively),
we find the opposite trend for the red and orange curves with respect
to the A5 configuration. For example, for the A2 configuration we
find that the red and orange curves become increasingly similar as
the star cools. This evolution reflects the trend of the cooling curves
displayed in Figure 4: as the internal temperature decreases, the
cooling effect of direct Urca in the core weakens, and Joule heating
in the crust dominates the thermal evolution and hence 𝐿𝛾and 𝑇∞
S.
On the contrary, the blue curve shows that 𝑇∞
Sremains substantially
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8F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
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33
34
35
36
log10(Lγ/erg s−1)
(a) M= 1.3M
A2
A1m2,2
A1m3,3
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31
32
33
34
35
36
log10(Lγ/erg s−1)
(b) M= 1.8M
A2
A1m2,2
A1m3,3
Figure 3. Cooling curves for (a) 𝑀=1.3𝑀and (b) 𝑀=1.8𝑀models with the A2, A1m2,2, A1m3,3initial magnetic configurations (see Table 1). As in
Figures 1and 2, the red data points correspond to magnetars (taken from Viganò et al. (2013)). The black dots corresponds to stars with 𝐵dip .1014 G (taken
from Potekhin et al. (2020)). Nucleon and hyperon species are superfluid.
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31
32
33
34
35
36
log10(Lγ/erg s−1)
(a) A1m3,3
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
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31
32
33
34
35
36
log10(Lγ/erg s−1)
(b) A2
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
Figure 4. Effect of hyperon superfluidity on the cooling curves of stars with 𝐵dip .1014 G. (a) A1m3,3initial magnetic configuration. The solid red
(𝑀=1.3𝑀) and dotted-dashed orange (𝑀=1.8𝑀with hyperon superfluidity) cooling curves are similar for 𝑡&105yr. The blue curve (𝑀=1.8𝑀
without hyperon superfluidity) is clearly distinguishable from the orange and red ones. (b) As in panel (a), but for the A2 initial magnetic configuration. The
red solid and orange dotted-dashed curves are similar for 𝑡&105yr. There is no degeneracy between the blue and orange curves, unlike in Figure 2. In both
panels, nucleons are in the superfluid phase.
lower if hyperons are in the normal phase, as Joule heating is not
sufficient to control the thermal evolution due to the high emis-
sivity of hyperon direct Urca. The bottom row in Figure 5shows
the presence of three hot spots above and below the equator. The
surface temperature of light and heavy models becomes similar for
𝑡&104yr, if hyperons are superfluid. Note that𝑇∞
Sis not symmetric
with respect to the equator because of the north-south asymmetric
magnetic field configuration (see Appendix Afor a detailed study
of the corresponding magneto-thermal evolution). Contrarily to the
A5 configuration, the values of 𝑇∞
Sfor a star with hyperon concen-
trations in its core are clearly distinguishable from low-mass stars
only if hyperons are not superfluid, producing a difference in the
thermal luminosity of approximately one order of magnitude up to
𝑡≈105yr (cf. Figure 4).
5 CONCLUSION
Measurements of 𝐿𝛾as a function of age are one means of probing
the composition of neutron star interiors (Yakovlev et al. 2001;Page
et al. 2004,2006;Potekhin et al. 2015;Potekhin et al. 2020), at least
in principle. For example, if it is discovered that 𝐿𝛾is lower than
predicted theoretically for 𝑛𝑝𝑒 𝜇 matter, one possible scenario is that
accelerated direct Urca cooling caused by hyperons is responsible
(Prakash et al. 1992;Haensel & Gnedin 1994;Raduta et al. 2018,
2019). In this paper, we show that the situation is more complicated,
because Joule heating can mask the cooling effects of direct Urca
emission (Aguilera et al. 2008). Specifically, cooling curves of both
low-mass stars without hyperon cores and high-mass stars with
hyperon cores can explain thermal luminosity data of magnetars
and stars with 𝐵dip .1014 G equally well.
We study the magneto-thermal evolution of hyperon stars with
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 9
0123
θ(rad)
6.10
6.20
6.30
6.40
6.50
6.60
log10(T∞
S/K)
(a) t= 103yr
A5
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
6.05
6.10
6.15
6.20
6.25
6.30
6.35
6.40
log10(T∞
S/K)
(b) t= 104yr
A5
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
6.00
6.05
6.10
6.15
6.20
6.25
6.30
log10(T∞
S/K)
(c) t= 105yr
A5
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.70
5.80
5.90
6.00
6.10
log10(T∞
S/K)
A2
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.70
5.80
5.90
6.00
log10(T∞
S/K)
A2
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.60
5.70
5.80
5.90
6.00
log10(T∞
S/K)
A2
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.20
5.40
5.60
5.80
6.00
6.20
log10(T∞
S/K)
A1m3,3
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.40
5.50
5.60
5.70
5.80
5.90
6.00
6.10
log10(T∞
S/K)
A1m3,3
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
0123
θ(rad)
5.50
5.60
5.70
5.80
5.90
log10(T∞
S/K)
A1m3,3
M= 1.3M
M= 1.8M
M= 1.8M, NO Y SF
Figure 5. Snapshots of the redshifted surface temperature 𝑇∞
Sversus the colatitude 𝜃, taken at times 𝑡=103,104,105yr (left, middle and right columns).
The top row reports the evolution corresponding to the A5 configuration; the middle row studies the A2 configuration; the bottom row studies the A1m3,3
configuration.
crust-confined and core-extended magnetic field configurations.
Fields sustained by both crustal and core electric currents produce
sufficient Joule heating to explain the observed luminosities of both
young magnetars (𝐵dip &1014 G and 𝐿𝛾&1034 erg s−1for 𝑡.105
yr) and stars with lower fields (𝐵dip .1014 G and 𝐿𝛾.1034 erg
s−1). The internal temperature in the crust is inhomogeneous due
to anisotropic electronic transport across and along the field lines
and localized Ohmic dissipation (Pons & Viganò 2019). If multi-
polar structures are present, several hot regions appear in the crust,
producing inhomogeneous surface temperature maps (Viganò et al.
2013;Dehman et al. 2020).
We find that the thermal luminosities of light stars composed
of 𝑛𝑝𝑒𝜇 matter (𝑀=1.3𝑀) and heavy stars composed of 𝑛𝑝𝑒 𝜇𝑌
matter (𝑀=1.8𝑀) become degenerate due to Joule heating. Joule
heating causes crust-core thermal decoupling in magnetars born
with 𝐵dip &1014 G and/or strong internal fields for 𝑡.2×104yr.
The cooling effect of hyperon direct Urca is masked by the thermal
power generated by the dissipation of electric currents in the crust,
and the cooling curves corresponding to models with or without
hyperons match the same sources. The comparison between high-
mass models with hyperons in the superfluid and normal phases
shows that Joule heating is sufficient to counterbalance the losses
MNRAS 000,1–15 (2021)
10 F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
due to hyperon direct Urca processes, even if the latter are not
suppressed by superfluid effects. Consequently, most magnetars may
not be suitable candidates to infer information regarding the internal
composition of the star, or to constrain hyperon superfluidity. In stars
with inferred fields satisfying 𝐵dip .1014 G that harbour strong
internal fields, the crust-core thermal decoupling is incomplete.
For 𝑡.105yr, the cooling curves of low-mass and high-mass
stars (with superfluid hyperons) are distinguishable. At later times
however, the thermal power supplied by Joule heating dominates
the thermal evolution, and the distinction between low- and high-
mass stars lessens. If hyperons are superfluid, it remains an open
question whether the core composition can be inferred using 𝐿𝛾
data, given that the internal field configuration and strength are
unknown and Joule heating in the crust may dominate the evolution.
Such degeneracy can be broken if hyperons are not superfluid, as
Joule heating is unable to supply sufficient thermal power to reduce
the cooling effect of hyperon direct Urca, when the latter is not
suppressed by superfluid effects.
We stress that the observational degeneracy discussed in this
work concerns only 𝐿𝛾and 𝑇∞
S. In principle, accurate mass and
radius measurements of thermal emitters can clearly distinguish be-
tween light and heavy stars, but they are currently unavailable. Yet,
even with sufficiently accurate mass and radius estimates, inferring
the internal composition may still be a difficult task, when so many
𝑛𝑝𝑒𝜇 and 𝑛𝑝 𝑒𝜇𝑌 EoSs lead to similar macroscopic properties of
neutron stars.
We conclude with some cautionary remarks. In this work we
focus on stars with strong internal fields, and we show that if cer-
tain conditions are met, thermal luminosity data cannot be uniquely
interpreted in terms of the internal composition. However, it is not
clear whether strong internal fields are ubiquitous across the neutron
star population or not; for example, stars with 𝐵dip .1013 G may
not necessarily contain strong internal fields. Furthermore, the goal
of this paper is not to assess whether hyperons are present or not in
neutron stars, but rather to determine whether one key signature of
their appearance (i.e. hyperon direct Urca emission) has a “distin-
guishable” effect on the cooling curves in the presence of high Joule
heating rates. We emphasize that in order to draw definitive conclu-
sions about neutron stars with hyperon cores, several microphysical
details (such as the EoS (Schaffner-Bielich et al. 2002;Rikovska
Stone et al. 2007;Fortin et al. 2015;Raduta et al. 2018;Motta et al.
2019;Motta & Thomas 2022), superfluid model or the neutrino
emissivity for example) and evolutionary details (e.g. typical initial
conditions for isolated and binary neutron stars) must be ascertained
more accurately than they are at present. For example, for smaller
neutron triplet and proton superfluid energy gaps, nucleon direct
Urca emission is stronger, and the Joule heating rate required for
the cooling curves of light and heavy stars to become degenerate
may be higher than the one calculated in this work. Additionally,
in our study we focus on massive stars containing hyperons. Stars
with 𝑀&1.6𝑀are often found in binary systems (Kiziltan et al.
2013;Alsing et al. 2018), and may experience accretion at different
epochs. Accretion alters the magnetic field configuration, heats the
surface and internal layers and modifies the chemical composition
of the crust (Payne & Melatos 2004;Haensel & Zdunik 2008;Priy-
mak et al. 2011;Fantina et al. 2018;Potekhin et al. 2019;Gusakov
& Chugunov 2020;Gusakov & Chugunov 2021). Alternatively,
massive stars may be remnants of merger events (provided that the
remnant does not collapse into a black hole), and their initial con-
ditions are likely to be more complicated than the standard ones
employed in this work and in the literature of neutron star cooling.
More realistic initial conditions for massive stars will be studied in
future work.
ACKNOWLEDGEMENTS
We thank Prof. Xavier Viñas and Prof. Mario Centelles for pro-
viding the BCPM nuclear energy density functional used to test the
in-medium correction factors for the modified Urca emissivity. FA is
supported by The University of Melbourne through a Melbourne Re-
search Scholarship. AM acknowledges funding from an Australian
Research Council Discovery Project grant (DP170103625) and the
Australian Research Council Centre of Excellence for Gravitational
Wave Discovery (OzGrav) (CE170100004). DV is supported by
the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (ERC Starting
Grant "IMAGINE" No. 948582, PI DV). CD is supported by the
ERC Consolidator Grant “MAGNESIA” (No. 817661, PI Nanda
Rea) and this work has been carried out within the framework
of the doctoral program in Physics of the Universitat Autònoma de
Barcelona. JAP acknowledges support by the Generalitat Valenciana
(PROMETEO/2019/071), AEI grant PGC2018-095984-B-I00 and
the Alexander von Humboldt Stiftung through a Humboldt Research
Award.
DATA AVAILABILITY
The data corresponding to the EoSs employed in this work are
taken from the Web page http://www.ioffe.ru/astro/NSG/
heos/hyp.html, and are presented in “Physics input for modelling
superfluid neutron stars with hyperon cores” (Gusakov et al. 2014).
The thermal luminosity and age data of moderately magnetized
neutron stars are reported in the paper “Thermal luminosities of
cooling neutron stars” by Potekhin et al. (2020) and are accessible
at the Web page http://www.ioffe.ru/astro/NSG/thermal/
cooldat.html. The magnetar data are taken from the paper “Unify-
ing the observational diversity of isolated neutron stars via magneto-
thermal evolution models” by Viganò et al. (2013).
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12 F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
Figure A1. Magnetic and thermal evolution of models with 𝑀=1.8𝑀and different initial magnetic configurations. The top, middle and bottom panels
correspond to the A1, A2 and A3 configurations respectively (see Table 1for details). The left hemisphere in each polar plot displays the contours of the
toroidal magnetic field component 𝑩tor =𝐵𝜙ˆ
𝜙in units of 1012 G. Overplotted are the poloidal field lines projected on the meridional plane. The map of the
internal redshifted temperature 𝑇i(in units of 108K) is displayed in the right hemispheres. The thickness of the crust here and in all the following figures is
enlarged by a factor of 8 for visualization purposes.
APPENDIX A: MAGNETO-THERMAL EVOLUTION
In this Appendix we study the magnetic field and internal temper-
ature evolution of a neutron star model with representative mass
𝑀=1.8𝑀hosting nucleons, electrons, muons, Λand Ξ−hyper-
ons, assuming that nucleon and hyperon species are superfluid. We
first consider the case of crust-confined magnetic configurations
(listed in Table 1) in Section A1. Core-threading magnetic field
configurations (listed in Table 2) are presented in Section A2.
A1 Crust-confined fields
A1.1 Dipolar-poloidal, quadrupolar-toroidal fields
In Figure A1 we study a star with 𝑀=1.8𝑀, whose initial
crust-confined magnetic field has dipolar poloidal and quadrupolar
toroidal components. In the following, the left hemispheres of the
polar plots display the contours of the toroidal component 𝑩tor =
𝐵𝜙ˆ
𝜙. Overplotted are the meridional projections of the poloidal
field lines. The right hemispheres display the internal redshifted
temperature (𝑇i=𝑇𝑒Φ) maps. We allow the 𝐵𝜙and 𝑇iscales to
vary for all the configurations and snapshots, in order to preserve a
high level of detail in the magnetic and temperature maps.
The top panels in Figure A1 show the magneto-thermal evolu-
tion of the A1 initial magnetic configuration. At 𝑡=5×102yr, the
magnetic field configuration (left hemisphere) is almost identical
to the one at birth. The corresponding 𝑇imap (right hemisphere)
shows significant inhomogeneities in the crust (the bottom of the
outer envelope is placed at ≈1010 g cm−3, corresponding to the
outermost boundary in the plots). This effect is related to anisotropic
heat transport and localized Joule heating. At 𝑡=5×103yr, the
poloidal field lines bend above and below the equator. At later times
(𝑡=105yr), the poloidal field lines form two large closed merid-
ional loops just below and above the equator. The right hemisphere
shows that the equatorial region is hotter than the rest of the star.
The middle panels display the A2 configuration. The evolution
of the poloidal field lines is similar to the top panels for 𝑡.5×103
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 13
Figure A2. As in Figure A1, but for the A1m2,2, A1m3,3and A1m4,4configurations (top, middle and bottom panels respectively). See Table 1for details.
yr. At 𝑡=5×103yr the maximal values of 𝐵𝜙are higher with
respect to the snapshot at 𝑡=5×102yr, revealing a redistribution
of magnetic energy between the poloidal and toroidal components
due to the Hall term in the induction equation. The right hemisphere
shows that at the equator a hotter region forms. By comparing the
snapshot at 𝑡=105yr in the middle row with the corresponding
one for the A1 configuration, one finds two main differences: (1)
the closed poloidal loops are absent in the A2 configuration; and
(2) the toroidal field in the A2 configuration is more compressed at
the bottom of the inner crust, where its dissipation rate is enhanced
due to the presence of impurities in the crustal lattice and by pasta
phases (Pons et al. 2013;Viganò et al. 2013;Anzuini et al. 2021).
The bottom panels report the evolution of the initial A3 con-
figuration. The magnetic field evolution is similar to the one of the
A2 configuration (middle panels), but Joule heating is higher than
both the A1 and A2 configurations. We note that, as found for the
A2 configuration, the snapshot at 𝑡=105yr does not display closed
poloidal loops. These are present only in the A1 configuration,
where the toroidal field stores most of the total magnetic energy.
A1.2 Multipolar topologies
We consider multipolar magnetic fields in Figure A2, where we sim-
ulate the magnetic and thermal evolution of neutron stars assuming
the presence of two, three or four poloidal and toroidal magnetic
multipoles (Dehman et al. 2020). In reality one may expect multi-
poles of higher orders; however, currently it is not feasible to evolve
such configurations numerically.
As a general feature, the presence of a given number of mag-
netic multipoles leads to the appearance of an equal number of “hot
regions” inside the stellar crust. The top panels report the evolution
of the A1m2,2configuration. As a consequence of the north-south
asymmetry in the magnetic field configuration, the temperature dis-
tribution is asymmetric with respect to the equator for 𝑡.5×102
yr. The northern hemisphere is characterised by a thick, hot layer
in the crust, whose temperature is higher than the hot layer below
the equator. The electric currents are asymmetric with respect to
the equatorial plane; they are more intense in the northern hemi-
sphere and hence produce higher Joule heating rates than in the
southern hemisphere. At later times (𝑡=5×103yr) the shorter
visible poloidal field line in the northern hemisphere shifts slightly
towards the equator, and so does the corresponding hot region in
MNRAS 000,1–15 (2021)
14 F. Anzuini, A. Melatos, C. Dehman, D. Viganò, J. A. Pons
Figure A3. Evolution of neutron star models with magnetic fields sustained by core and crustal electric currents. The top panels correspond to the C1 initial
configuration, the middle panels to the C2 configuration and the bottom panels to the C1m2,0configuration. Details about the magnetic configurations are
specified in Table 2.
the temperature map. At 𝑡=105yr, both the northern and southern
hot regions in the right hemisphere extend to denser layers of the
crust.
Upon increasing the number of initial multipoles (middle pan-
els, A1m3,3configuration), the magneto-thermal evolution becomes
more complex. Three corresponding hot regions appear in the lay-
ers beneath the outer envelope in the temperature map (snapshot at
𝑡=5×102yr). The hottest one is again located in the northern
hemisphere. As in the top panels, the snapshot at 𝑡=5×103yr
(right hemisphere) shows that the hot regions remain in roughly the
same locations shown in the snapshot at 𝑡=5×102yr. At 𝑡=105
yr, while the northern and the equatorial hot regions move closer,
the southern one drifts further towards the south pole. As for the
A1m2,2configuration, all three hot regions extend to deeper parts
of the crust. Interestingly, we note that at 𝑡=105yr both positive
and negative values of 𝐵𝜙are enclosed within the plotted poloidal
field lines, contrarily to what is found for the A1m2,2configuration.
The bottom panels display the A1m4,4configuration. In gen-
eral, the hot regions remain approximately in their original locations,
and expand progressively towards deeper layers with increasing 𝑡.
At 𝑡=105yr, the toroidal field assumes both positive and negative
values of 𝐵𝜙in each region enclosed by the plotted poloidal field
lines, similarly to what is seen in the corresponding panel for the
A1m3,3configuration.
A2 Core-threading fields
The top panels in Figure A3 display the snapshots corresponding to
the magneto-thermal evolution of the C1 initial configuration. The
kinks of the poloidal field lines at the crust-core interface are due to
the presence of electric currents both in the crust and the core. At
𝑡=5×102yr the magnetic field lines bend in the crust, where the
Hall term in the induction equation redistributes magnetic energy
between the poloidal and toroidal components, forming a toroidal
field and twisting the magnetic field lines. The right hemisphere
shows that two equatorially-symmetric colder layers form in the
north and southern hemispheres, while a hotter region resides at
the equator. At later times (𝑡=5×103yr), the poloidal field lines
are warped in the crust due to the Hall term, and 𝐵𝜙peaks in
denser regions of the crust. The internal, redshifted temperature is
MNRAS 000,1–15 (2021)
Thermal luminosity degeneracy of neutron stars 15
inhomogeneous near the equator, and the two cold layers above and
below the equator become thicker. At 𝑡=105yr, the crustal toroidal
field is stronger at the crust-core interface, where the dissipation
rate of electric currents is enhanced by the presence of impurities
in the crustal lattice and pasta phases. We find that𝑇iis comparable
to the crust-confined configurations considered in Section A1.
The middle panels in Figure A3 display the evolution of the C2
initial topology. The difference with respect to the C1 configuration
is that the field does not include an initial toroidal component con-
fined to an equatorial torus, and the initial value of 𝐵dip is higher. In
general, the magneto-thermal evolution of this model is similar to
the top panels in Figure A3. However, due to the stronger initial field,
the star maintains higher temperatures than the C1 configuration.
In the bottom panels we display results for the C1m2,0config-
uration. At 𝑡=5×102yr, the Hall term in the induction equation
generates a strong toroidal field. Given the equatorial asymmetry
of the initial crustal field, the temperature map at 𝑡=5×102yr
is asymmetric as well. A hot layer in the crust below the outer
envelope is contained within the shortest plotted poloidal field line
visible in the northern hemisphere. Below the equator, a colder layer
forms, spanning a smaller region than its northern counterpart. At
𝑡=5×103yr, the crustal poloidal field lines bend, while the toroidal
field peaks in the middle of the crust. The temperature map shows
that the hot regions extend to deeper regions than earlier. At 𝑡=105
yr the region where 𝐵𝜙switches from negative to positive values
drifts towards the equator and is located at the crust-core interface.
This paper has been typeset from a T
EX/L
A
T
EX file prepared by the author.
MNRAS 000,1–15 (2021)
