# Two-blackbody portraits of radiation from magnetized neutron stars

D. G. Yakovlev1
1 Ioffe Institute, Politekhnicheskaya 26, St Petersburg 194021, Russia
E-mail<EMAIL_ADDRESS>
(Accepted . Received ; in original form)

###### Abstract

We study a simple model describing thermal radiation spectra from magnetized
neutron stars. The model assumes that a star is nearly spherical and
isothermal inside and possesses dipole magnetic fields ($B\lesssim 10^{14}$ G)
near the surface, which make the surface temperature distribution non–uniform.
We assume further that any surface element emits a blackbody (BB) spectrum
with a local effective temperature. We show that such thermal spectra
(including phase–resolved) are accurately approximated by simple equivalent
two–BB (2BB) models. We introduce and study phase–space maps of 2BB parameters
and show that these maps can be useful for interpreting neutron star
observations, in which 2BB spectral fits have been done.

###### keywords:

stars: neutron – stars: atmospheres – stars: magnetic fields

††pagerange: Two-blackbody portraits of radiation from magnetized neutron
stars–Two-blackbody portraits of radiation from magnetized neutron
stars††pubyear: 2010

##  1 Introduction

Thermal radiation of neutron stars is formed in very thin surface layers (of
thickness from few centimeters to a few millimeters, as reviewed, e.g. by
Potekhin 2014). This radiation contains important information on neutron star
parameters (such as mass, raduis, chemical composition and magnetic field in
surface layers). The information can be inferred by interpreting observations
of neutron stars with theoretical atmosphere models or models of radiating
condensed surface. These interpretations can also be useful for exploring
superdense matter in neutron star interiors (see, e.g., Potekhin 2014;
Potekhin et al. 2015a and references therein).

We study thermal radiation, which emerges from stellar interiors of middle-
aged isolated neutron star. We neglect non-thermal radiation component (of
magnetospheric origin) and the effects of rapid rotation. We will mainly
consider simplest emission models of neutron stars which possess purely dipole
magnetic fields in the surface layers with typical field strengths $B\lesssim
10^{14}$ G.

The magnetic field can make the surface temperature distribution anisotropic
and modify the radiation spectrum (e.g. Potekhin et al. 2015a, b; De Grandis
et al. 2021). The anisotropic surface temperature distribution in a middle-
aged neutron star is usually created in the so-called heat blanketing envelope
(e.g. Beznogov et al. 2021), which thermally insulates a warm and nearly
isothermal interior of the star from much cooler surface. This heat blanket is
typically much thicker than the atmosphere. We will take the surface
temperature distrbutions from the models of heat blankets (e.g. Potekhin et
al. 2003; Ho et al. 2008; Potekhin et al. 2015b, and references therein).

The magnetic field effects on thermal emission from the atmosphere are
complicated (see, e.g. Zavlin & Pavlov 2002; Potekhin et al. 2015a; González
Caniulef et al. 2016). The radiation can be strongly polarized, contain
specific spectral and angular features; it can be greatly variable over the
surface. The radiation flux detected by a distant observer must be calculated
by integration over the visible part of the surface, taking into account
gravitational redshift and light bending of photons.

The progress in developing the magnetized atmospheric models of neutron stars
has been reviewed, for instance, by Potekhin (2014); Potekhin et al. (2015a).
Such models can be separated into those with a pure radial magnetic field
constant on the surface and those with more realistic magnetic fields varying
over the surface. The first models are easier to construct but seem less
realistic; we will not discuss them. We focus on the atmosphere models with
the dipole field configuration which are not numerous (e.g. Ho et al. 2008;
Zyuzin et al. 2021). We consider a simplified version of such models, in which
any local element on the stellar surface (with non-uniform temperature
distribution) emits like a blackbody (BB) with its own local surface
temperature $T_{\text{s}}$, meaning an overall multi–BB thermal radiation. The
model was put forward long ago (Greenstein & Hartke, 1983) and analyzed in the
literature (e.g. Page 1995; Page & Sarmiento 1996; Potekhin & Yakovlev 2001;
Potekhin et al. 2003; Geppert et al. 2006; Zane & Turolla 2006). Our main goal
will be to study its approximation by a two-BB (2BB) model, a widely used tool
for investigating neutron-star radiation.

In Sect. 2 we outline a general multi–BB radiation model for a neutron star
with isothermal interiors and dipole surface magnetic field. In Sect. 3 we
study spectral fluxes emitted from such a star. We show that these fluxes can
accurately be fitted by 2BB models and analyse the dependence of fit
parameters on the mean effective temperature of the star, magnetic field
strength, and stellar compactness. Also, we derive accurate 2BB fits for any
observation direction. We summarize the results and conclude in Sect. 4.

##  2 Theoretical multi–BB spectra

###  2.1 Theoretical outline

Let us outline standard calculation of radiation from a spherical neutron star
with any (not necessarily axially symmetric) non-uniform surface temperature
distribution, assuming that any surface element emits a BB radiation with an
effective (local, non-redshifted) surface temperature $T_{\text{s}}$,
resulting in multi–BB emission. Since neutron stars are compact, one needs to
account for gravitational light bending and redshift in Schwarzschild space-
time. These effects are characterized by the neutron star compactness
parameter

$x_{\text{g}}=\frac{r_{\text{g}}}{R},\quad r_{\text{g}}=\frac{2GM}{c^{2}},$
(1)

where $M$ is a gravitational stellar mass, $R$ is a circumferential radius,
$r_{\text{g}}$ is the Schwarzschild radius, $G$ is the gravitational constant
and $c$ is the speed of light. We will use physical quantities either in a
local reference frame near the neutron star surface or in the frame of a
distant observer. The latter will be often marked by the symbol $\infty$. For
instance, $T_{\text{s}}^{\infty}=\sqrt{1-x_{\text{g}}}\,T_{\text{s}}$, and the
‘apparent’ neutron star radius for a distant observer is
$R_{\infty}=R/\sqrt{1-x_{\text{g}}}$.

The observer detects the radiative spectral flux density $F_{\infty}$ [erg
cm-2 s-1 keV-1] at a distance $D\gg R$ per unit interval of the redshifted
photon energy $E\equiv E_{\infty}$. The flux depends on $E$ and observation
direction (along a unit vector $\bm{k}$). Let us take a surface element
$\mathrm{d}S$ of the atmosphere and specify its position either by ordinary
spherical angles $\vartheta$ and $\varphi$ (with respect to the magnetic axis)
or by the unit vector $\bm{n}$ normal to the surface. Then
$\mathrm{d}S=R^{2}\,\mathrm{d}\Omega_{\rm s}$, with $\mathrm{d}\Omega_{\rm
s}=\sin\vartheta\,\mathrm{d}\vartheta\,\mathrm{d}\varphi$ being a solid angle
element measured from the stellar centre. The contribution of this element to
flux density is (e.g. Beloborodov 2002, Poutanen & Gierliński 2003 and
Potekhin 2014)

$\mathrm{d}F_{E}^{\infty}=I(E,\bm{k}_{0})\,{\cal
P}\,\frac{\mathrm{d}S}{D^{2}},\quad{\cal
P}=\cos\theta\left|\frac{\mathrm{d}\cos\theta}{\mathrm{d}\cos\theta_{0}}\right|,$
(2)

$\bm{k}_{0}$ being the unit vector along the propagation of those photons in
the atmosphere, which are registered by the observer. The directions of
$\bm{k}$ and $\bm{k}_{0}$ differ because of gravitational line bending.
Furthermore, let $\theta$ (or $\theta_{0}$) be the angle between $\bm{k}$ (or
$\bm{k}_{0}$) and the line from the stellar centre to the observer. For given
$\bm{n}$ and $\bm{k}$, $\bm{k_{0}}$ is determined by kinematics of light
bending. Next, $I(E,\bm{k}_{0})$ is the radiative intensity emitted in the
atmosphere and redshifted for the observer. Finally, ${\cal P}$ corrects the
intensity of the observed flux due to light bending; it depends on $\theta$,
$\theta_{0}$ and $x_{\rm g}$. Any surface element $\mathrm{d}S$ is vizible by
the observer if $\theta\leq\pi/2$. Neglecting the light bending, one has
$\theta=\theta_{0}$ and ${\cal P}=\cos\theta$.

Using equation (2) and integrating over $\mathrm{d}S$, one obtains

$F_{E}^{\infty}=\frac{R^{2}}{D^{2}}\,H_{E}^{\infty},\quad
H_{E}^{\infty}=\int_{\rm viz}I(E,\bm{k}_{0}){\cal P}\,\mathrm{d}\Omega_{\rm
s},$ (3)

where $H^{\infty}_{E}$ is the effective spectral thermal flux (formally
independent of $D$ and $R$) in the direction of $\bm{k}$. The intergration
over $\mathrm{d}\Omega_{\rm s}$ is carried over the visible part of the
stellar surface. In our case, the gravitational redshift and light bending are
decoupled. The light bending enlarges the observable emission area. It allows
one to observe the emission from the opposite side of the star. Note that we
neglect photon absorption by intervening matter along propagation path, which
will be discussed later.

A local radiation intensity $I_{0}(E_{0})$ of any surface element is given by
the Planck function with a local non-redshifted surface temperature
$T_{\text{s}}(\bm{n})$ and non-redshifted photon energy $E_{0}\equiv
E/\sqrt{1-x_{\text{g}}}$. Redshifting $I_{0}$ and $E_{0}$, from equation (3)
we obtain

$H_{E}^{\infty}=\frac{15\sigma_{\rm SB}}{16\pi^{5}k_{\rm B}^{4}}\,\int_{\rm
viz}\mathrm{d}\Omega_{\rm s}\,\frac{(1-x_{\text{g}})^{-1}\,{\cal
P}E^{3}}{\exp(E/\mbox{$k_{\text{B}}$}T_{\text{s}}^{\infty})-1},$ (4)

where $\sigma_{\rm SB}$ is the Stefan–Boltzmann constant. Using accurate
analytic approximations of ${\cal P}$ derived by Poutanen (2020) [his
equations (12) and (13)], one can easily compute $H_{E}^{\infty}$ for any
surface temperature distribution $T_{\text{s}}(\bm{n})$ and observation
direction $\bm{k}$. In addition, one can calculate the total thermal
bolometric luminosity of the star, $L_{\text{s}}$, and introduce an overall
non-redshifted effective surface temperature $T_{\text{eff}}$ according to
(e.g. Potekhin & Yakovlev 2001)

$L_{\text{s}}=\sigma_{\rm
SB}R^{2}\int_{4\pi}\mathrm{d}\Omega_{\text{s}}\,T_{\text{s}}^{4}(\bm{n})\equiv
4\pi\sigma_{\rm SB}R^{2}T_{\text{eff}}^{4}.$ (5)

Here one should integrate over the entire stellar surface.

In addition to the exact effective flux $H_{E}^{\infty}$ in the direction of
$\bm{k}$, which is given by equation (4), one can introduce the flux
$H_{E}^{\rm av\infty}$, averaged over orientations of $\bm{k}$ at a fixed
distance $D$,

$H_{E}^{\rm
av\infty}=\frac{1}{4\pi}\int_{4\pi}\mathrm{d}\Omega_{\bm{k}}\,H_{E}^{\infty},$
(6)

where $\mathrm{d}\Omega_{\bm{k}}$ is a solid angle element of possible
orientations $\bm{k}$ of the detector on the sky. The angle-averaging
simplifies the integration,

$H_{E}^{\rm av\infty}=\frac{15\sigma_{\rm SB}}{16\pi^{5}{k_{\rm
B}^{4}}}\,\int_{4\pi}\mathrm{d}\Omega_{\rm
s}\,\frac{(1-x_{\text{g}})^{-1}\,E^{3}}{\exp(E/\mbox{$k_{\text{B}}$}T_{\text{s}}^{\infty})-1};$
(7)

the angle-averaged bolometric flux determines the total luminosity (5) of the
star, $L_{\rm s}^{\infty}=4\pi D^{2}F_{\rm bol}^{\rm av\infty}=4\pi\sigma_{\rm
SB}R_{\infty}^{2}T_{\text{eff}}^{\infty 4}$.

For a uniform surface temperature distribution, one naturally recovers the
ordinary BB($\equiv$1BB) law,

$H_{E}^{\rm BB\infty}=\frac{15\sigma_{\rm SB}}{4\pi^{4}k_{\rm
B}^{4}}\,\frac{(1-x_{\text{g}})^{-1}\,E^{3}}{\exp(E/\mbox{$k_{\text{B}}$}T_{\text{eff}}^{\infty})-1},$
(8)

and the bolometric effective flux

$H_{\rm bol}^{\rm BB\infty}=\int_{0}^{\infty}H_{E}^{\rm
BB\infty}\,\mathrm{d}E=\frac{\sigma_{\rm SB}T_{\text{eff}}^{\infty
4}}{1-x_{\text{g}}},$ (9)

with $L_{\rm s}^{\infty}=4\pi D^{2}F_{\rm bol}^{\rm BB\infty}=4\pi\sigma_{\rm
SB}R_{\infty}^{2}T_{\text{eff}}^{\infty 4}$.

###  2.2 Illustrative examples

Figure 1:  Logarithm of the surface temperature $T_{\text{s}}$ versus
colatitude $\vartheta$, from magnetic pole ($\vartheta=0$) to equator
($\vartheta=90^{\circ}$) of a neutron star with $M=1.4\,{\rm M}\odot$ and
$R=12$ km at $B_{\text{pole}}=10^{14}$ G. The curves are plotted for seven
effective temperatures, $\log T_{\text{eff}}$ [K]=5.6, 5.8, …6.8. Solid curves
refer to the basic heat-blanket model (Potekhin & Yakovlev, 2001; Potekhin et
al., 2003; Potekhin & Chabrier, 2018), while dots display an artificial model
discussed in Sect. 3.6. See the text for details.

For illustration, Fig. 1 shows (by solid lines) the surface temperature
distribution for a passively cooling magnetized neutron star with pure dipole
magnetic field in the outer layers, assuming $M=1.4\,{\rm M}\odot$ and $R=12$
km [$x_{\text{g}}$=0.344 and the surface gravity $g_{\rm
s}=GM/(R^{2}\sqrt{1-x_{\text{g}}})=1.59\times 10^{14}$ cm s-2]. The surface
field at the magnetic pole is $B_{\text{pole}}=10^{14}$ G. The plots are given
by the standard theory of magnetized heat blanketing envelopes (e.g., Potekhin
& Yakovlev 2001; Potekhin et al. 2003; Potekhin & Chabrier 2018; Beznogov et
al. 2021). The $T_{\text{s}}$ distributions are axially symmetric with respect
to the magnetic axis and equator. It is sufficient to plot $T_{\text{s}}$
versus colatitude $\vartheta$ from $\vartheta=0$ (magnetic pole) to
$90^{\circ}$ (equator). We present the solid curves for seven effective
temperatures $T_{\text{eff}}$. The respective dotted curves are for a toy
model of heat blankets considered in Sect. 3.6.

The star is assumed to be isothermal inside. The main temperature gradient and
the $T_{\text{s}}(\bm{n})$ distribution (solid lines) are mediated by heat
transport through a thin outer heat blanketing envelope, which extends from
the surface to the density $\rho_{\text{b}}\sim 10^{10}$ g cm-3. For
certainty, unless the contrary is indicated, the blanketing envelope is
assumed to be made of iron. The magnetic field makes heat transport
anisotropic and creates variations of $T_{\text{s}}(\bm{n})$, which become
noticeable at $B_{\text{pole}}\gtrsim 10^{11}$ G. In Fig. 1 displays strong
pole-to-equator $T_{\text{s}}$ variations (solid lines) exceeding one order of
magnitude at $\log T_{\text{eff}}$ [K]=5.6. The pole is hotter than the
equator because of higher heat conduction in the polar regions, where the heat
outflows to the surface nearly along magnetic field lines. In contrast, there
is a colder equatorial belt, where the heat outflows nearly across magnetic
lines; such outflow is strongly suppressed. In a hotter star with higher
$T_{\text{eff}}$, the $T_{\text{s}}$ anisotropy is less pronounced due to
reduction of heat transport anisotropy in warmer matter. With increasing
$B_{\text{pole}}$, the polar regions become warmer, while the equatorial belt
colder and thinner.

Figure 2:  Effective spectral flux density $H^{\infty}_{E}$ versus redshifted
photon energy $E$ at (from bottom to top) $\log T_{\text{eff}}$ [K]=5.6, 5.8,
…6.8 from a $1.4\,{\rm M}\odot$ neutron star with $R=12$ km and
$B_{\text{pole}}=10^{14}$ G (as in Fig. 1). Solid lines refer to pole
observations (inclination angle $i=0$), dashed lines to equator observations
($i=90^{\circ}$); dotted lines are for the 1BB model with given
$T_{\text{eff}}$.  Figure 3:  Effective spectral flux density $H^{\infty}_{E}$
in units of $H^{\parallel\infty}_{E}$ versus redshifted photon energy $E$ at
$\log T_{\text{eff}}$ [K]=6.4 for a $1.4\,{\rm M}\odot$ neutron star with
$R=12$ km and $B_{\text{pole}}=10^{14}$. Solid lines refer to pole
observations, dashed lines to equator observations; dot-dashed line shows the
angle-averaged flux, while the dotted line is for the 1BB
($T_{\text{s}}=T_{\text{eff}}$) model. Grey solid and dashed lines are the
same fluxes $H^{\parallel\infty}_{E}$ and $H^{\perp\infty}_{E}$, as the black
lines, but neglecting the gravitational light bending of photons.

Fig. 2 shows the thermal spectral flux density $H_{E}^{\infty}$ emitted by
same star as in Fig. 1 (for the same seven values of $T_{\text{eff}}$ diplayed
by solid lines). The flux is plotted in logarithmic scale as a function of
decimal logarithm of the redshifted photon energy $E$. For each value of
$T_{\text{eff}}$ we present three curves. The solid curves give the flux
$H_{E}^{\parallel\infty}$, as measured in pole observations (marked as
$\parallel$, with the inclination $i=0$ to the magnetic axis). The dashed
curves present the flux $H_{E}^{\perp\infty}$ for the equator observations
(marked with $\perp$, at $i=90^{\circ}$). The dotted curves demonstrate the
plain 1BB model, equation (8), with given $T_{\text{s}}=T_{\text{eff}}$
constant over the surface. The angle-averaged curves $H_{E}^{\rm av\infty}$,
given by equation (6), always lie between $H_{E}^{\parallel\infty}$ and
$H_{E}^{\perp\infty}$ and would be indistinguishable from the dotted curves in
the logarithmic format.

All four fluxes for each $T_{\text{eff}}$ look very close in the logarithmic
format of Fig. 2. To demonstrate their difference, Fig. 3 plots spectral
fluxes $H^{\infty}_{E}$ in natural format. Here we take $\log
T_{\text{eff}}=6.4$ and express all $H^{\infty}_{E}$ in units of
$H^{\parallel\infty}_{E}$. This normalization gives the unit relative flux
(solid black line) for pole observations. The flux for the equator
observations (long–dashed black line) is always lower because the observer
detects radiation emitted from overall colder regions. The
$H^{\perp\infty}_{E}/H^{\parallel\infty}_{E}$ ratio is seen to decrease with
photon energy. The black dash–dot line refers to $H^{\rm av\infty}_{E}$, while
the dotted black line to $H^{\rm BB\infty}_{E}$. One can see that these lines
become noticeably different at higher $E$. Finally, the grey solid and
long–dash curves refer to the same $H^{\parallel\infty}_{E}$ and
$H^{\perp\infty}_{E}$ fluxes, as the black curves, but calculated neglecting
gravitational light bending. These grey curves are systematically lower than
their black counterparts because the neglect of light bending decreases
observable surface area of the star. This confirms the importance of light
bending. Note also that the light bending greatly reduces the difference of
fluxes in the pole and equator observations, making the thermal neutron star
radiation more isotropic (Page, 1995).

##  3 2BB approximation

###  3.1 2BB fits

Here we show that the multi–BB fluxes discussed in Sect. 2.2 can be accurately
fitted by a simple 2BB model. The idea is not new (e.g. Popov et al. 2017)
but, to the best of our knowledge, it has not been studied in great details.
Let us take the same model of a spherical neutron star with a dipole magnetic
field near the surface. We have calculated the spectral flux $H_{E}^{\infty}$
from equation (4) for a number of values of $M$, $R$, $B_{\text{pole}}$,
$T_{\text{eff}}$ and inclination angles $i$. We have mainly considered photon
energies $E$ from 0.064 keV to $E_{\rm max}=42.49$ keV (overlapping thus
typical observable X-ray range) on a grid of 142 energy points placed
logarithmically equidistant within this range. The spectra have been fitted by
the 2BB model,

$H_{E}^{\infty}=s_{\text{c}}H_{E}^{\rm BB\infty}(T_{\rm
effc})+s_{\text{h}}H_{E}^{\rm BB\infty}(T_{\rm effh}).$ (10)

Here $H_{E}^{\rm BB\infty}(T_{\text{eff}})$ is given by equation (8), with
$T_{\rm effc}=p_{\text{c}}T_{\text{eff}}$ and $T_{\rm
effh}=p_{\text{h}}T_{\text{eff}}$; the subscripts ‘c’ and ‘h’ refer to the
colder and the hotter BB components, respectively. Four positive dimensionless
fit parameters $s_{\text{c}}$, $s_{\text{h}}$, $p_{\text{c}}$ and
$p_{\text{h}}$ define relative contributions of the two BB components to the
total thermal flux $H_{E}^{\infty}$. The parameters $p_{\text{c}}$ and
$p_{\text{h}}$ determine the effective temperatures of the components, while
$s_{\text{c}}$ and $s_{\text{h}}$ are effective fractions of their surface
areas (without imposing $s_{\text{c}}+s_{\text{h}}=1$). Since the first
component is colder, we have $p_{\text{c}}<p_{\text{h}}$. We treat
$s_{\text{c,h}}$ and $p_{\text{c,h}}$ just as formal fit parameters (not the
parameters of real colder and hotter uniform areas on the neutron star
surface). Very roughly, one can visualize $R_{\rm c}=R\,\sqrt{s_{\rm c}}$ and
$R_{\rm h}=R\,\sqrt{s_{\rm h}}$ as the effective radii of the colder and
hotter emission regions, respectively. All four fit parameters $s_{\text{c}}$,
$s_{\text{h}}$, $p_{\text{c}}$ and $p_{\text{h}}$ depend on $M$, $R$,
$B_{\text{pole}}$, $T_{\text{eff}}$ and $i$. Actually, these fit parameters
are not independent. They should guarantee the required bolometric luminosity
of the star, that is determined by given $T_{\text{eff}}$ in accord with
equation (5). We used this equation to check the quality of our 2BB fits and
it always worked well.

###  3.2 2BB fits: pole and equator observations

We start with the two basic models for pole and equator observations
($H_{E}^{\parallel\infty}$ and $H_{E}^{\perp\infty}$, $i=0$ and 90∘), leaving
the case of arbitrary $i$ to Sect. 3.3.

For example, Table 1 presents the fit results for the star with $M=1.4$ ${\rm
M}\odot$ and $R=12$ km. The fitting has been performed on the indicated grid
of 142 energy points $E$. Before fitting, we have removed the energy points,
at which $H_{E}^{\infty}<1$ erg s-1 cm-2 keV-1 and the spectral flux is
negligible. Table 1 gives the best fit parameters $p_{\text{c}}$,
$p_{\text{h}}$, $s_{\text{c}}$ and $s_{\text{h}}$ for magnetic pole
($\parallel$) and equator ($\perp$) observations at four values of
$B_{\text{pole}}=10^{11}$, $10^{12}$, $10^{13}$ and $10^{14}$ G. For each
$B_{\text{pole}}$, we consider $6\times 2$ atmosphere models with $\log
T_{\text{eff}}$ [K]=5.6, 5.8,…,6.6 (6 lines with eight fit parameters for any
$T_{\text{eff}}$ in Table 1). In each line, we present also decimal logarithm
of the internal temperature $\log T_{\text{b}}$ and two root mean squared
relative fit errors $\delta_{\rm rms}$ (for the two observation directions,
$\parallel$ and $\perp$). One can see that in all 48 fit cases $\delta_{\rm
rms}$ does not exceed 2.5 per cent. The maximum relative fit error is
typically $\approx(2-2.5)\,\delta_{\rm rms}$. Such fits can be regarded as
nearly perfect (‘too accurate’), taking into account simplified nature of the
initially calculated fluxes. If plotted on Figs. 2 and 3, these fits would be
indistinguishable from corresponding solid and long–dashed curves.

Table 1: Fit parameters $p_{\text{c}}$, $p_{\text{h}}$, $s_{\text{c}}$ and $s_{\text{h}}$ in equation (10), together with rms relative fit errors $\delta_{\rm rms}$, for a neutron star with $M=1.4$ ${\rm M}\odot$, $R=12$ km at two observation directions ($\parallel$ and $\perp$), four values of $\log B_{\text{pole}}$ [G]=11, 12, 13, 14, and six values of $\log T_{\text{eff}}$ [K]=5.6, 5.8, …, 6.6 (with corresponding logarithms of the internal temperature, $\log T_{\text{b}}$ [K]).  $\log B_{\text{pole}}$ | $\log T_{\text{s}}$ | $\log T_{\text{b}}$ | $p_{\rm c\parallel}$ | $p_{\rm h\parallel}$ | $s_{\rm c\parallel}$ | $s_{\rm h\parallel}$ | $\delta_{\rm rms}^{\parallel}$ | $p_{\rm c\perp}$ | $p_{\rm h\perp}$ | $s_{\rm c\perp}$ | $s_{\rm h\perp}$ | $\delta_{\rm rms}^{\perp}$
---|---|---|---|---|---|---|---|---|---|---|---|---
11 | 5.6 | 7.3854 | 0.9708 | 1.1262 | 0.5126 | 0.3669 | 0.011 | 0.9624 | 1.1239 | 0.5620 | 0.3002 | 0.014
11 | 5.8 | 7.7477 | 0.9552 | 1.1014 | 0.5641 | 0.3898 | 0.009 | 0.9493 | 1.0991 | 0.6258 | 0.3218 | 0.011
11 | 6.0 | 8.0979 | 0.9591 | 1.0750 | 0.5793 | 0.4062 | 0.005 | 0.9553 | 1.0729 | 0.6453 | 0.3384 | 0.006
11 | 6.2 | 8.4447 | 0.9685 | 1.0506 | 0.5696 | 0.4269 | 0.002 | 0.9663 | 1.0488 | 0.6360 | 0.3601 | 0.002
11 | 6.4 | 8.7918 | 0.9775 | 1.0313 | 0.5426 | 0.4569 | 0.001 | 0.9761 | 1.0297 | 0.6046 | 0.3950 | 0.001
11 | 6.6 | 9.1410 | 0.9860 | 1.0191 | 0.5309 | 0.4690 | 0.000 | 0.9853 | 1.0180 | 0.5931 | 0.4069 | 0.000
12 | 5.6 | 7.3488 | 1.0057 | 1.1628 | 0.4465 | 0.3312 | 0.011 | 0.9952 | 1.1605 | 0.4757 | 0.2667 | 0.014
12 | 5.8 | 7.7410 | 0.9845 | 1.1572 | 0.4682 | 0.3526 | 0.014 | 0.9743 | 1.1546 | 0.5055 | 0.2863 | 0.018
12 | 6.0 | 8.1185 | 0.9622 | 1.1490 | 0.4945 | 0.3734 | 0.017 | 0.9523 | 1.1463 | 0.5406 | 0.3060 | 0.021
12 | 6.2 | 8.4868 | 0.9435 | 1.1359 | 0.5281 | 0.3913 | 0.017 | 0.9348 | 1.1331 | 0.5836 | 0.3231 | 0.021
12 | 6.4 | 8.8435 | 0.9398 | 1.1158 | 0.5582 | 0.4022 | 0.013 | 0.9330 | 1.1132 | 0.6209 | 0.3335 | 0.016
12 | 6.6 | 9.1898 | 0.9485 | 1.0895 | 0.5735 | 0.4108 | 0.007 | 0.9440 | 1.0872 | 0.6397 | 0.3423 | 0.009
13 | 5.6 | 7.2925 | 1.0032 | 1.1590 | 0.4312 | 0.3461 | 0.011 | 0.9933 | 1.1567 | 0.4616 | 0.2799 | 0.013
13 | 5.8 | 7.6673 | 0.9901 | 1.1646 | 0.4613 | 0.3434 | 0.014 | 0.9794 | 1.1621 | 0.4955 | 0.2776 | 0.018
13 | 6.0 | 8.0568 | 0.9767 | 1.1642 | 0.4818 | 0.3500 | 0.017 | 0.9658 | 1.1616 | 0.5203 | 0.2840 | 0.021
13 | 6.2 | 8.4502 | 0.9621 | 1.1590 | 0.4923 | 0.3659 | 0.019 | 0.9515 | 1.1563 | 0.5360 | 0.2989 | 0.023
13 | 6.4 | 8.8348 | 0.9476 | 1.1527 | 0.5025 | 0.3824 | 0.020 | 0.9375 | 1.1499 | 0.5513 | 0.3146 | 0.024
13 | 6.6 | 9.2082 | 0.9347 | 1.1452 | 0.5176 | 0.3968 | 0.019 | 0.9253 | 1.1422 | 0.5719 | 0.3284 | 0.024
14 | 5.6 | 7.1730 | 1.0225 | 1.1914 | 0.4613 | 0.2819 | 0.013 | 1.0095 | 1.1891 | 0.4807 | 0.2230 | 0.016
14 | 5.8 | 7.6315 | 0.9956 | 1.1718 | 0.4702 | 0.3268 | 0.015 | 0.9840 | 1.1692 | 0.5017 | 0.2629 | 0.018
14 | 6.0 | 7.9913 | 0.9742 | 1.1603 | 0.4688 | 0.3629 | 0.016 | 0.9637 | 1.1576 | 0.5084 | 0.2958 | 0.020
14 | 6.2 | 8.3648 | 0.9650 | 1.1644 | 0.4896 | 0.3589 | 0.019 | 0.9539 | 1.1616 | 0.5315 | 0.2922 | 0.023
14 | 6.4 | 8.7685 | 0.9561 | 1.1632 | 0.5022 | 0.3636 | 0.020 | 0.9449 | 1.1603 | 0.5469 | 0.2968 | 0.025
14 | 6.6 | 9.1668 | 0.9456 | 1.1580 | 0.5064 | 0.3774 | 0.020 | 0.9350 | 1.1550 | 0.5547 | 0.3099 | 0.025

###  3.3 2BB fits: any observation angle

In case of arbitrary inclination $i$, the spectral flux
$H_{E}^{\infty}=H_{E}^{i\infty}$ can be computed from equation (4). We have
performed such computations for a range of values of $M$, $R$,
$B_{\text{pole}}$, $T_{\text{eff}}$ and $i$. In all the cases the computed
fluxes $H^{i\infty}_{E}$ are nicely approximated by

$H^{i\infty}_{E}=H_{E}^{\parallel\infty}\cos^{2}i+H_{E}^{\perp\infty}\sin^{2}i,$
(11)

where $H_{E}^{\parallel\infty}$ and $H_{E}^{\perp\infty}$ are the fluxes for
the pole and equator observations, discussed above. The fit accuracy of
equation (11) is of the same order of magnitude as the 2BB fit accuracy of
equation (10). Therefore, any thermal spectral flux $H^{i\infty}_{E}$ observed
at any direction $i$ is determined by the two basic pole and equatorial
spectral fluxes, $H_{E}^{\parallel\infty}$ and $H_{E}^{\perp\infty}$, which
are accurately described by 2BB fits with easily computable fit parameters
(Table 1). This gives practical solution for the assumed model of magnetized
neutron star emission. Note that equation (11) is the same as the equation for
the flux produced by a classic electric dipole emission in flat space.

###  3.4 2BB phase–space maps

Figure 4:  Map of 2BB fit parameters versus $\log T_{\text{eff}}$ for a
$1.4\,{\rm M}\odot$ neutron star with $R=12$ km and $B_{\text{pole}}=3\times
10^{13}$ G for pole observation ($i=0$). The left–hand panel shows the
fractional surface areas $s_{\rm h}$ and $s_{\rm c}$ for the hotter and colder
BB components as lengths of segments of vertical lines at given
$T_{\text{eff}}$ in the upper and lower shaded zones, respectively. The blank
intermediate zone gives ‘non-emitting’ fractional surface area. The
short–dashed and dotted grey lines show the boundaries of the shaded zones for
the equator observation ($i=90^{\circ}$). The right–hand panel demonstrates
characteristic temperatures. The shaded zone shows range of
$T_{\text{s}}$–variations over the surface, from the highest temperature
$T_{\rm pole}$ at the pole to the lowest one $T_{\rm eq}$ at the equator. The
dotted line is the total effective temperature $T_{\text{eff}}$, while the
dashed and solid lines are the effective temperatures $T_{\rm effh}$ and
$T_{\rm effc}$ for the hotter and colder BB components (which are almost
insensitive of $i$). See the text for details.

Figure 5:  2BB fit parameters versus $\log T_{\text{eff}}$ for a $1.4\,{\rm
M}\odot$ neutron star with $R=12$ km and $B_{\text{pole}}=10^{12}$ [panel (a)]
or $10^{14}$ G [panel (b)]. Black and grey lines refer, respectively, to pole
and equator observations. Black solid lines show $p_{\rm c\parallel}$ and
$s_{\rm c\parallel}$ (for the colder BB components), while black long–dashed
lines show $p_{\rm h\parallel}$ and $s_{\rm h\parallel}$ (for the hotter BB
components). Short–dashed grey lines refer to $p_{\rm c\perp}$ and $s_{\rm
c\perp}$, whereas dotted grey lines to $p_{\rm h\perp}$ and $s_{\rm h\perp}$.

Figure 6:  Same 2BB fit parameters as in Fig. 5 for a $1.4\,{\rm M}\odot$
neutron star with $R=12$ km but versus $B_{\text{pole}}$ at $\log
T_{\text{eff}}$[K]=6.4 [panel (a)] and $5.8$ [panel (b)]. See the text for
details.

Many observed thermal X-ray spectra of neutron stars have been fitted by 2BB
models. Since 2BB models accurately approximate a wide class of multi–BB
models, it seems attractive to develop a technique for studying properties of
magnetized stars, as multi–BB emitters, using already available 2BB fits of
the observed spectra. This can be done by comparing theoretical 2BB maps (2BB
parameters calculated in a wide phase–space of input parameters, such as $M$,
$R$, $B_{\text{pole}}$, $T_{\text{eff}}$, $i$), with observations and
selecting then most suitable phase–space domains.

Some maps are demonstrated in Fig. 4. The figure refers to a $1.4{\rm M}\odot$
star with $R=12$ km and $B_{\text{pole}}=3\times 10^{13}$ G, observed from the
pole. The fit parameters are displayed versus $T_{\text{eff}}$. The left–hand
side plots emitting fractions of the cold and hot BB surface areas, $s_{\rm
c}$ and $s_{\rm h}$ (lower and upper shaded areas, respectively). For any
$T_{\text{eff}}$ we have $s_{\rm c}+s_{\rm h}<1$, which may be viewed as the
presence of ‘non-emitting’ fraction of the stellar surface shown as blank.
This effect supports common knowledge that one should be careful in using
$s_{\rm c}$ and $s_{\rm h}$ for estimatimg true neutron star radius from
observations. Notice that the presence of the ‘dark’ surface does not affect
excellent quality of the 2BB fits. With increasing $T_{\text{eff}}$, the
fraction of BB ’dark’ surface decreases.

The right–hand panel of Fig. 4 shows characteristic temperatures of the
emitting star. The shaded zone is restricted by the maximum temperature
$T_{\rm pole}$ at the magnetic pole and the minimum temperature $T_{\rm eq}$
at the equator. We see that temperature variations over the surface of a
magnetized star are large. The dotted curve plots the mean effective
temperature $T_{\text{eff}}$ (to guide the eye). The dashed and solid courves
are the effective BB temperatures, $T_{\rm effh}$ and $T_{\rm effc}$,
respectively.

Although Fig. 4 is designed for the pole observations, the grey lines on the
left-hand panel show the deformations of the emitting zones for the equator
observations; these deformations are not dramatic but noticeable.
Corresponding deformations on the right–hand panel are not plotted; they would
be almost invisible.

Figure 7:  2BB fit parameters versus $\log T_{\text{eff}}$ for a neutron star
with $B_{\text{pole}}=10^{14}$ G (as in Fig. 5b at $M=1.4\,{\rm M}\odot$ and
$R=12$ km) but for different $M$ and $R$. Panel (a) refers to a less compact
star with $M=1.4\,{\rm M}\odot$ and $R=16$ km. Panel (b) is for a more compact
star with $M=1.6\,{\rm M}\odot$ and $R=12$ km.

Phase–space maps like those in Fig. 4 are convenient but take much space. We
will use a more economic format, which allows plotting all four fit parameters
on one panel. Fig. 5 demonstrates the behavior of fit parameters (Table 1)
versus $T_{\text{eff}}$ of the colder and hotter BB components from the star
with the same $M$ and $R$ at $B_{\text{pole}}=10^{12}$ and $10^{14}$ G [panels
(a) and (b), respectively]. The fits for the pole observations are plotted in
black, while those for the equator observations in grey. The upper solid and
long–dashed black curves show the parameters $p_{\rm c\parallel}$ and $p_{\rm
h\parallel}$, which specify the temperatures $T_{\rm effc\parallel}$ and
$T_{\rm effh\parallel}$, respectively. The lower solid and long–dashed black
curves show the relative surface areas $s_{\rm c\parallel}$ and $s_{\rm
h\parallel}$ of the colder and hotter surface regions. For the equator
observations, $p_{\rm c\perp}$ and $p_{\rm h\perp}$ are plotted by the
short–dashed and dotted upper grey lines, while $s_{\rm c\perp}$ and $s_{\rm
h\perp}$ by similar lower grey lines.

For both panels in Fig. 5 we have $T_{\rm effh\parallel}\approx T_{\rm
effh\perp}$ and $T_{\rm effc\parallel}\approx T_{\rm effc\perp}$. This is
because the spectral fluxes are generally close to isotropic 1BB with given
$T_{\text{eff}}$ (Page 1995; also see Fig. 2). However, $T_{\rm effh}$ is
noticeably higher than $T_{\rm effc}$, indicating that hotter and colder
surface regions are distinctly pronounced in the spectra. Also, we have
$s_{\rm c\parallel}\approx s_{\rm c\perp}$ and $s_{\rm h\parallel}\approx
s_{\rm h\perp}$.

The dependence of fit parameters on $T_{\text{eff}}$ is seen to be smooth and
not too sensitive to $B_{\text{pole}}$. Systematically, $T_{\rm effh}$ is from
10 to 20 per cent higher than $T_{\rm effc}$. In a colder star ($\log
T_{\text{eff}}\sim 5.5$) one has $T_{\text{eff}}\approx T_{\rm effc}$, but for
a hotter star ($\log T_{\text{eff}}\gtrsim 6.5$) $T_{\text{eff}}\approx T_{\rm
effh}$. The depencence of the fractional areas $s_{\text{c}}$ and
$s_{\text{h}}$ on the observation angle is stronger than for $p_{\text{c}}$
and $p_{\text{h}}$.

Fig. 6 shows the behavior of the same 2BB fit parameters, as in Figs. 5
($M=1.4\,{\rm M}\odot$, $R=12$ km), but versus $B_{\text{pole}}$ at $\log
T_{\text{eff}}$[K] =6.4 [panel (a)] and 5.8 [panel (b)]. The dependence of fit
parameters on $B_{\text{pole}}$ is weak. For instance, $B_{\text{pole}}$
varies over 3.5 orders of magnitude in Fig. 6 but the fit parameters vary much
less.

Fig. 7 shows the same 2BB fit parameters versus $T_{\text{eff}}$ at
$B_{\text{pole}}=10^{14}$ G, as in Fig. 5b, but for other values of $M$ and
$R$. Fig. 7a refers to a less compact star ($M=1.4\,{\rm M}\odot$, $R=16$ km,
smaller $M/R$), while Fig. 7b is for a more compact star, with $M=1.6\,{\rm
M}\odot$ and $R=12$ km. The overall behavior of the curves in panel (b) is
similar to that in panel (a). Generally, the parameters $p$ on panel (b) are
less affected by the stellar compactness than the parameters $s$. The stronger
the compactness, the smaller the difference between the fit parameters for the
pole and equator observations, making the thermal radiation more uniform due
to stronger general relativisitc (GR) effects. This was shown by Page (1995)
by direct calculations of some spectral fluxes.

Let us stress that we have fitted unabsorbed computed spectral fluxes by
unabsorbed 2BB models. In this way we have established one-to-one theoretical
correspondence between the two models. On the other hand, observational data
allow observers (in principle) to infer 2BB emission parameters, which are
corrected for interstellar absorption, for a possible existence of non-thermal
radiation component and similar effects. If so, we can compare these
observational data with the parameters given by our unabsorbed 2BB models
considered here.

So far we have studied 2BB maps for neutron stars with heat blankets made of
iron (e.g. Potekhin et al. 2003; Beznogov et al. 2021). As a test, we have
constructed a 2BB map for the star with $M=1.4\,{\rm M}\odot$, $R=12$ km and
$B_{\text{pole}}=10^{14}$ G versus $\log T_{\text{eff}}$ using the blanket
composed of accreted matter. We do not present the plot; it looks similar to
that in Fig. 5b.

###  3.5 Phase-resolved spectroscopy

Phase-resolved spectroscopy from magnetized spherical neutron stars with
dipole magnetic fields in the surface layers was extensively studied by Page
(1995) by calculating a number of phase-resolved spectra.

We could do the same using our 2BB fits to the spectral fluxes for the pole
and equator observations and composing the fluxes for any observation
direction (Sect. 3.3). The results would be similar. Some differences would
stem from using somewhat more refined surface temperature distribution
(Potekhin et al., 2007) than in Page (1995). We do not describe the details
but refer to Page (1995) for an analysis of pulse fractions. For a star with
$M=1.4\,{\rm M}\odot$, $R=12$ km, $B_{\text{pole}}\sim 10^{12}$ G and $\log
T_{\text{eff}}$[K]$\sim 5.6$ at $E\sim 3$ keV the maximum pulse fraction is
about 10 per cent. It increases with the growth of $B_{\text{pole}}$, $E$, and
$R/M$ but decreases with the growth of $T_{\text{eff}}$. It could be
significantly higher if one neglected the effects of light bending above the
neutron star surface. Other models of emission from magnetized neutron stars
including pulse resolved spectroscopy, more complicated field geometry and a
possible presence of condensed surface, have been considered in many
publications (e.g. Page & Sarmiento 1996; Pérez-Azorín et al. 2005; Geppert et
al. 2006; Zane & Turolla 2006; Hambaryan et al. 2011; Popov et al. 2017; De
Grandis et al. 2021).

###  3.6 Testing cold equatorial belt

So far we have studied spectral fluxes and their 2BB maps for one particular
model (Sect. 2.2) of the surface temperature distribution
$T_{\text{s}}(\vartheta)$, corresponding to a pure dipole magnetic field in
the surface layer of the star. Obviously, realistic $T_{\text{s}}(\bm{n})$
distributions can be more complicated. For simplicity, we restrict ourselves
to sufficiently weak axially symmetric variations of
$T_{\text{s}}(\vartheta)$, which are symmetric also with respect to the
magnetic equator.

The most doubtful place in our $T_{\text{s}}(\vartheta)$ distribution is a
thin cold equatorial surface belt (Sect. 2.2, Fig. 1). The temperature is much
lower there, than in other places of the surface. Because this temperature
distribution was calculated (e.g. Beznogov et al. 2021) taking into account
only the radial heat transport, the equatorial belt is the first place, where
this $T_{\text{s}}(\vartheta)$ can be questioned. The heat can spread along
the surface of the cold belt to the equator, reducing surface temperature
variations there.

We have checked this effect on phenomenological level. We have taken a narrow
equatorial surface strip of angular half-size $\delta\vartheta_{\rm belt}$ and
left the $T_{\text{s}}(\vartheta)$ dependence untouched outside this strip
($|\vartheta-\vartheta_{\rm eq}|\geq\delta\vartheta_{\rm belt}$, with
$\vartheta_{\rm eq}=90^{\circ}$). Inside the strip, we have artificially
assumed that $T_{\text{s}}(\vartheta)=(A-C\sin^{2}\vartheta)^{1/4}$, where
constants $A$ and $C$ have been chosen to ensure continuity of
$T_{\text{s}}(\vartheta)$ and $\partial T_{\text{s}}/\partial\vartheta$ at
$\vartheta=\vartheta_{\rm belt}=\delta_{\rm eq}-\delta_{\rm belt}$. For
$\delta\vartheta_{\rm belt}=30^{\circ}$, these profiles are shown by dotted
lines in Fig. 1 (for the same values of $T_{\text{eff}}$ as the standard solid
lines, but using modified calculation of $T_{\text{eff}}$ for the new
profiles).

We have calculated modified spectral fluxes and 2BB maps for the same
conditions, as in Fig. 5b, with two values $\delta\vartheta_{\rm
belt}=10^{\circ}$ and $30^{\circ}$. In the first case the maps exactly
coincided with the initial ones but the surface temperature still varied
strongly over the belt. In the second case the $T_{\text{s}}$ variations are
greatly smoothed out (Fig. 1) but the maps appeared almost the same as the
initial ones.

Therefore, variations of $T_{\text{s}}$ in the cold equatorial belt seem to
have almost no effect on the thermal surface emission (as has been anticipated
earlier, e.g. Beznogov et al. 2021).

###  3.7 Warmer polar spots

Figure 8:  2BB fit parameters for a neutron star with $M=1.4\,{\rm M}\odot$,
$R=12$ km, $\log T_{\text{eff}}$[K]=5.8 and $B_{\text{pole}}=3\times 10^{13}$
G with extra hotspots of angular size $\vartheta_{0}=10^{\circ}$ on magnetic
poles [described by equation (12)] versus extra relative surface temperature
increase $\delta$ at the pole. See the text for details.

Here we introduce two symmetric slightly warmer phenomenological spots on
magnetic poles. Let $T_{\mathrm{s0}}(\vartheta)$ be the local effective
surface temperature given by our basic model (Sect. 2.2) and $\vartheta_{0}$
be a small angle that deteminies the size of hotspots. We assume that

$T_{\text{s}}(\vartheta)=T_{\mathrm{s0}}(\vartheta)\,\left[1+\delta\,\cos^{2}\left(\frac{\pi\vartheta}{2\vartheta_{0}}\right)\right]\quad\mathrm{at}\leavevmode\nobreak\
\vartheta\leq\vartheta_{0},$ (12)

and that $T_{\text{s}}(\vartheta)=T_{\text{s}}(\vartheta_{0})$ is not
disturbed outside hotspots. The parameter $\delta$ determines a small extra
temperature enhancement in the spot’s center. This enhancement smoothly
disppears as $\vartheta\to\vartheta_{0}$. The presence of spots renormalizes
(slightly increases) the total effective temperature $T_{\text{eff}}$,
equation (5).

Otherwise computations of spectral fluxes are the same as in Sect. 2.2. These
fluxes can also be approximated by 2BB fits as in Sect. 3.1, and can be
analyzed via phase–space maps. For the examples presented below the relative
the fit accuracy becomes worse (reaching sometimes 10 per cent) but the fit is
still sufficiently good and robust.

Fig. 8 plots the illustrative 2BB fit parameters versus $\delta$ for a star
with 1.4 ${\rm M}\odot$, $R=12$ km, $B_{\text{pole}}=3\times 10^{13}$ G, $\log
T_{\text{eff}}$[K]=5.8 and hotspot angular size $\vartheta_{0}=10^{\circ}$. In
the absence of hotspots ($\delta=0$), the results are very close to those in
Fig. 5b calculated at the same $\log T_{\text{eff}}$[K]=5.8 (but for slightly
higher $B_{\text{pole}}=10^{14}$ G). However, when $\delta$ grows up, the fit
parameters become different. The temperature $T_{\rm effh}$ of the hotter BB
component and the effective emission surface area $s_{\rm c}$ of the colder BB
component noticeably increase, whereas the emission surface area of the hotter
component dramatically falls down. Even with really small hotspot temperature
enhancement $\delta\gtrsim 0.15$ (equivalent to 0.6 per cent luminosity
enhancement), we obtain qualitatively new phase–space portrait with $s_{\rm
h}\ll s_{\rm c}$. Such 2BB fits have been often obtained from observations of
cooling isolated neutron stars (see Sect. 4); these sources are usually
interpreted as neutron stars with small hotspots.

Therefore, the theory predicts two types of neutron stars whose spectra are
described by the 2BB models. The first ones are those with smooth surface
temperature distributions, created by nonuniform surface magnetic fields and
considered mainly thoughout this paper. We will call these spectral models as
2BB with smooth magnetic atmospheres. The sources of the second type are those
with distinct hotspot BB component. We will call them as 2BB with hotspots.
Obviously, the sources of the two types are different but there is a smooth
transition between them (for instance, by increasing $\delta$ in Fig. 8)
meaning possible existence of the sources of intermediate type. Such a
transition would require some extra heating in the hotter place of the
magnetic atmosphere. According to Sect. 3.6, an extra heating of the colder
equatorial belt would be less efficient.

##  4 Discussion and conclusions

###  4.1 Candidates

Let us mention several candidates, among cooling isolated middle-aged neutron
stars (not magnetars), which may demonstrate their multi–BB spectra as 2BB
ones. We do not pretend to be complete, but give some examples. We base on the
recent catalog (Potekhin et al., 2020) available also on the Web: //www.
ioffe.ru/astro/NSG/thermal/cooldat.html.

1\. XMMU J172054.5$-$372652 is a neutron star that is probably associated with
the SNR G350.1$-$0.3 (as suggested by Gaensler et al. 2008). There is no
direct evidence of pulsations. Lovchinsky et al. (2011) presented arguments
that the SNR G350.1$-$0.3 is freely expanding and estimated its age as $\sim
1000$ years. Potekhin et al. (2020) used archival Chandra data and fitted the
X-ray spectrum with a neutron star (nsx) atmosphere model assuming $M=1.4{\rm
M}\odot$ and $R=13$ km. They obtained $T_{\text{eff}}\approx 2$ MK, but did
not perform two-component fits which would be interesting.

2\. PSR B1055$-$52 (J1057$-$5226) is a well known moderately magnetized
middle–aged pulsar. Its effective magnetic field, as determined from the
standard model of magnetic dipole braking and reported in the ATNF pulsar
catalog (Manchester et al., 2005), is $B_{\text{eff}}=1.1\times 10^{12}$ G (at
the equator of an imaginary star with $M=1.4\,{\rm M}\odot$ and $R=10$ km).
This value does not allow one to accurately determine $B_{\text{pole}}$ but
indicates that $B_{\text{pole}}$ is roughly equals a few times of $10^{12}$ G.
Potekhin et al. (2020) presented the value
$\mbox{$k_{\text{B}}$}T_{\text{eff}}^{\infty}\approx 70$ eV
($T_{\text{eff}}^{\infty}\approx 0.8$ MK). It is based on the 2BB spectral fit
by De Luca et al. (2005) which included also a power-law (PL) non-thermal
radiation component. Potekhin et al. (2020) have corrected the results by De
Luca et al. (2005) for more plausible distance estimate to the source obtained
by Mignani et al. (2010). The fit gives $T_{\text{effh}}/T_{\text{effc}}\sim
2.3$ and $s_{\text{h}}\ll s_{\text{c}}$. It agrees with a 2BB model containing
hotspots.

3\. PSR J1740+1000 has $B_{\text{eff}}=1.8\times 10^{12}$ G. The 2BB spectral
fit was done by Kargaltsev et al. (2012). Taking the same version of the fit
as selected by Potekhin et al. (2020), we again have
$\mbox{$k_{\text{B}}$}T_{\text{eff}}^{\infty}\approx 70$ eV
($T_{\text{eff}}^{\infty}\approx 0.8$ MK),
$T_{\text{effh}}/T_{\text{effc}}\sim 2.8$ and $s_{\text{h}}\ll s_{\text{c}}$,
with the same conclusion as for the PSR B1055$-$52.

4\. PSR B1823$-$13 (J1826$-$1334) is located in the SNR G18.0$-$00.7 and has
$B_{\text{eff}}=2.8\times 10^{12}$ G. Its X-ray emission is mostly non-thermal
(Pavlov et al., 2008; Zhu et al., 2011) but the PL fits suggest the presence
of some thermal component. The 1BB+PL fit gives the radius of thermally
emitting region $R_{\mathrm{eff}}^{\infty}\approx 5$ km, that is smaller than
a realistic radius of a neutron star. Adding the second BB component does not
seem statistically sound with the present data but might be possible in the
future.

5\. RX J1605.3+3249 (RBS 1556) is a neutron star studied by many authors (e.g.
Motch et al. 2005; Posselt et al. 2007; Tetzlaff et al. 2012; Pires et al.
2019; Malacaria et al. 2019) with contradictionary conclusions on its distance
and other properties (see Potekhin et al. 2020, for details). Its spin period
has been been found but later disproved by Pires et al. (2019). The timing and
spectral analysis of X-ray emission have been performed under different
assumptions using BB and neutron star atmosphere models. Recently Pires et al.
(2019) have analysed the XMM-Newton observations. Also, Malacaria et al.
(2019) performed a joint analysis of the NICER and XMM-Newton data. Both teams
report improving of 2BB fits if one adds a broad Gaussian absorption line. In
this case, they obtain $\mbox{$k_{\text{B}}$}T_{\rm effc}^{\infty}\sim 60$ eV
($T_{\rm effc}^{\infty}\sim 0.7$ MK), and $T_{\rm effh}/T_{\rm effc}\sim 2$.
This can be the 2BB case with not very warm spots, where the effects of smooth
magnetic atmosphere are not negligible.

6\. RX J1856.5$-$3754 is a neutron star with nearly thermal spectrum. It was
discovered by Walter, Wolk & Neuhäuser (1996). It is slowly rotating, with the
sipn period of about 7 s; its effective magnetic field is $B_{\text{eff}}\sim
1.5\times 10^{13}$ G (although maghetic field properties are still highly
debated; e.g. Popov et al. 2017; De Grandis et al. 2021 and references
therein). Its spectrum has been measured in a wide range of wavelengths,
including X-rays, optical and radio. It has been interpreted with a number of
spectral models, particularly, with the model of thin partially ionized
magnetized hydrogen atmosphere on top of solidified iron surface (e.g., Ho et
al. 2007; Potekhin 2014). Note alternative 2BB, 2BB+PL, and 3BB fits
constructed by Sartore et al. (2012) and Yoneyama et al. (2017). In
particular, the 2BB fits give $\mbox{$k_{\text{B}}$}T_{\rm effc}^{\infty}\sim
40$ eV ($T_{\rm effc}^{\infty}\approx 0.46$ MK), $T_{\rm effh}/T_{\rm
effc}\sim 1.6$, $R_{\rm effh}\sim 0.5R$ and $R_{\rm effc}\sim R$. These 2BB
fits look closer to the 2BB spectral models with smooth magnetic atmosphere,
than other 2BB fits described above. Introducing warmer spots simplifies this
interpretation.

In summary, if 2BB spectral fits are done, they can give a hint on a possible
importance of localized hotspots or smooth variations of the surface
temperature. For refining this interpretation, it would be better to use more
advanced models of neutron star atmospheres (e.g. Ho et al. 2008; Zyuzin et
al. 2021).

###  4.2 Conclusions

We have studied simple models (Sect. 2) of thermal spectra emitted from
surfaces of spherical isolated neutron stars with dipole surface magnetic
fields $10^{11}\lesssim B_{\text{pole}}\lesssim 10^{14}$ G. Such fields make
the surface temperature distribution noticeably non-uniform. The model assumes
BB radiation with a local temperature from any surface element, meaning
multi–BB emission as a whole. The model properly treats the GR effects as well
as variations of temperature and magnetic fields over the surface (such as
colder equatorial belts and hotter magnetic poles).

We have shown that these multi–BB radiation spectra are accurately fitted by
2BB models for all observation directions by specifying eight fit parameters
(temperatures and effective emitting areas of colder and hotter BB components
for pole and equator observations) at any given set of values of
$B_{\text{pole}}$, $T_{\text{eff}}$, $M$ and $R$. For example, Table 1
presents fit parameters for a star with $M=1.4\,{\rm M}\odot$ and $R=12$ km on
a grid of $B_{\text{pole}}$ and $T_{\text{eff}}$ values. We can easily
generate such tables for any values of input parameters. We have studied the
so called 2BB portraits of neutron star emission, which give the dependence of
fit parameters on input phase–space ones ($B_{\text{pole}}$, $T_{\text{eff}}$,
etc.). Comparing these theoretical 2BB maps with 2BB fits, inferred from the
observed spectra, gives a method of estimating the magnetic field properties
on neutron star surface.

Next we show (Sect. 3.6) that a possible extra heating of cold equatorial
surface belt by heat flows from neighboring warmer surface regions does not
have strong impact on theoretical 2BB maps. In constrant, even slight extra
heating of magnetic poles can strongly affect the radiation spectra but does
not violate the 2BB approximation (Sect. 3.7). It increases the effective
temperature $T_{\rm teffh}$ of the hotter BB component, strongly reduces the
fractional emission area $s_{\rm h}$ of this component, drastically changes
2BB maps and realizes a transition to 2BB emission with hotspots. The latter
regime is quite known from observations. We have mentioned some cooling
neutron stars which emit as 2BB with hotspots or as 2BB transiting between hot
spots and smooth magnetic atmospheres (Sect. 4.1.)

Our consideration is restricted by studying dipole–like surface magnetic
fields. Actual fields can be more complicated and the assumption that any
surface element radiates as a BB can be violated, but such effects go beyond
the scope of this paper. However, the present consideration can be extended to
more sophisticated field geometries and the 2BB approximation can be a proper
tool for theoretical interpretations of these observations as well.

## Acknowledgments

I am grateful to the anonymous referee for constructive critical remarks. This
research was partly supported by the grant 14.W03.31.0021 of the Ministry of
Science and Higher Education of the Russian Federation and by the travel
grants 316932 and 332666 of the Academy of Finland.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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11institutetext: Depto. de Computación, Facultad de Cs. Exactas y Naturales,
Universidad de Buenos Aires
11email: {jmperez<EMAIL_ADDRESS>22institutetext: Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET) 33institutetext: Facultad de
Matemática, Astronomía y Física, Universidad de Córdoba 33email:
<EMAIL_ADDRESS>

#  _pysentimiento_ : A Python Toolkit for Sentiment Analysis and SocialNLP
tasks

Juan Manuel Pérez 1122 Juan Carlos Giudici 11 Franco Luque 22 3 3

###### Abstract

Extracting opinions from texts has gathered a lot of interest in the last
years, as we are experiencing an unprecedented volume of user-generated
content in social networks and other places. A problem that social researchers
find in using opinion mining tools is that they are usually behind commercial
APIs and unavailable for other languages than English. To address these
issues, we present _pysentimiento_ , a multilingual Python toolkit for
Sentiment Analysis and other Social NLP tasks. This open-source library brings
state-of-the-art models for Spanish and English in a black-box fashion,
allowing researchers to easily access these techniques.

##  1 Introduction

Extracting opinions and states-of-mind from user-generated context has drawn a
lot of attention in the past years, particularly since the eclosion of Social
Networks. Applications for these techniques come in many flavors, such as
commercial uses, political campaigns, and even studying the changing patterns
of emotions during the COVID-19 pandemics [8].

An issue that inhibits users of these opinion-mining technologies is that
there is not a big spectrum of libraries for these tasks; mainly, one has to
resort paid APIs provided by companies, or rely on models that are very out-
of-the-date or even unavailable for a given language different from English.

In order to foster research using sentiment and other opinion-mining analysis
as black-box tools, we present _pysentimiento_ , a multilingual toolkit for
these tasks. _pysentimiento_ provides state-of-the-art transformed-based
models for Sentiment Analysis and Emotion Analysis in an out-of-the-box
fashion, with current support of Spanish and English. This library is released
as free and open-source software 111Code is available at
http://github.com/pysentimiento/pysentimientofor anyone interested in using it
for research purposes. We now describe the datasets we used, the models we
trained, the results, and the selected default models in our library.

###  1.1 Previous Work

Appart from the commercial APIs, there are not many options of libraries
providing out-of-the-box models for Sentiment Analysis. _VADER_[7] is a
lexicon and rule-based library for Sentiment Analysis in English, specially
crafted for Social Media. It provides multilingual support through translation
from the target language to English.

_Textblob_ is a library providing pretrained models for many NLP tasks using
classic machine learning models such as Naive Bayes. Support is primary in
English, with some adapters for other languages.

_Transformers_[14] is a library providing pretrained language models such as
_BERT_[3] for classification and language-generation tasks. It also provides
pipelines for Sentiment Analysis and other NLP tasks but –as far as we know–
only English models are provided.

##  2 Data

We tackled two tasks: Sentiment Analysis and Emotion Analysis, both on Twitter
datasets. For the former, we used two datasets: _TASS 2020_ Task 1 [4] and
_SemEval 2017_ Task 4 Subtask 1[12]. Both datasets were labeled with general
polarity using positive, negative and neutral outcomes. We merged together the
Spanish subsets for each dialect, summing up to 6,000 tweets (CHECK THIS!).
The English dataset has around 50k tweets for training and around 12k for
testing.

Regarding Emotion Analysis, we used _EmoEvent_ [1], a multilingual emotion
dataset labelled with the six Ekman’s basic emotions (_anger_ , _disgust_ ,
_fear_ , _joy_ , _sadness_ , _surprise_) and also a “neutral” emotion.

Table 1 summarizes this info along the number of instances used for train and
test.

Table 1: Statistics of the datasets used in this work Task | Language | Source | Train | Test | Total
---|---|---|---|---|---
Sentiment | es | TASS Task 1 | 7,245 | 7,264 | 14,509
en | SemEval 2017 Task 4 | 49,645 | 12,284 | 61,929
Emotion | es | TASS Task 2 (EmoEvent) | 6,732 | 1,677 | 8,409
en | TASS Task 2 (EmoEvent) | 5,477 | 1,826 | 7,303

##  3 Method

Transformer-based models have become state-of-the-art in NLP, both for
classification and generation tasks, displacing models based on recurrent
networks. BERT [3] and GPT [11] are flagships of these models.

We performed experiments with several models. For English, we tested _BERT
base_[3], _RoBERTa base_[9], _BERTweet_[10] and multilingual models, namely
_DistilBERT_[13] and _mBERT_[3]. Spanish has lesser availability of models: we
used _BETO_[2], a Spanish-trained version of BERT, and the aforementioned
multilingual models.

Models were trained in a fairly standard way, using small triangular learning
rates ($\sim 10^{-5}$) [6] for 5 epochs (10 in the case of SemEval dataset as
it is big enough). For the emotion datasets we used class weights in the
cross-entropy loss as it is somehow imbalanced.

Training (and posterior delivering) is done using _transformers_[14] library.

##  4 Results

Table 2: Classification results. First lines show the models  |  | Sentiment | Emotion
---|---|---|---
| Model | Micro f1 | Macro f1 | Micro f1 | Macro f1
en | distilbert | 0.649 | 0.642 | 0.503 | 0.383
mbert | 0.645 | 0.643 | 0.516 | 0.394
bert | 0.686 | 0.684 | 0.559 | 0.439
roberta | 0.686 | 0.684 | 0.563 | 0.445
bertweet | 0.697 | 0.696 | 0.584 | 0.476
es | distilbert | 0.602 | 0.599 | 0.600 | 0.463
mbert | 0.609 | 0.604 | 0.610 | 0.474
beto | 0.672 | 0.667 | 0.688 | 0.548

Table 2 contains the results of the experiments for both tasks and languages.
In English, the best performing model is _BerTweet_ but just with a slight
difference with _RoBERTa_. Both are powerful models known to have better
performance than _BERT_ ; particularly _BERTweet_ is very suited to these
SocialNLP tasks as it is entirely trained on tweets.

We can observe that in Spanish _BETO_ yields the best results for a large
margin (around 6 F1 points) against _mbert_ and _distilbert_ , something
expected as it is well known that monolingual models greatly outperform
multilingual ones. Although there are models similar to _BERTweet_ in Spanish
[5] we were not able to test them as they were not available in _huggingface_
’s model hub.

##  5 Conclusion and future work

In this work we presented _pysentimiento_ , a multilingual toolkit for
Sentiment Analysis and Emotion Analysis. We provide state-of-the-art models
and an easy-to-use interface in Python, expecting this will help researchers
interested in opinion mining from social networks.

For the time being, we support these two tasks. We plan to expand to more
tasks, such as hate speech detection, irony detection, and others. Also, we
plan to provide more powerful pretrained models for other languages (mainly
Spanish) to enhance performance in other languages than English.

## References

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