# Bardeen-Kiselev black hole with cosmological constant

Manuel E. Rodrigues(1,2)111E-mail address<EMAIL_ADDRESS>Marcos V. de S.
Silva(1)222E-mail address<EMAIL_ADDRESS>Henrique A.
Vieira(1)333E-mail address<EMAIL_ADDRESS>(1)Faculdade de
Física, Programa de Pós-Graduação em Física, Universidade Federal do Pará,
66075-110, Belém, Pará, Brazil
(2)Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará
Campus Universitário de Abaetetuba, 68440-000, Abaetetuba, Pará, Brazil

(August 27, 2024)

###### Abstract

In this work we analyze a solution that mimics the Bardeen solution with a
cosmological constant surrounded by quintessence. We show that this solution
can be obtained by Einstein equations coupled with nonlinear electrodynamics.
We also show that the solution is not always regular and what the conditions
for regularity are. We analyze the thermodynamics associated with this type of
solution by establishing the form of the Smarr formula and the first law of
thermodynamics. We obtain some thermodynamic quantities such as pressure,
temperature, heat capacity and isothermal compressibility. Once we have these
thermodynamic quantities, we check if this solution has phase transitions and
how it behaves at the points where the transitions occur. For some values of
the parameters, we find that the solution exhibits a first-order phase
transition, like a van der Waals fluid.

###### pacs:

04.50.Kd, 04.70.Bw

##  I Introduction

General relativity (GR) is able to predict and describe a number of
astrophysical objects Berti:2015itd . One of the most important predictions of
GR is black holes Barack:2018yly . Due to their causal structure, black holes
have a surface to which any particle or wave passing through this surface
cannot return, this surface is known as the event horizon. Recently, the study
of black holes has become more important. Two events that contributed to this
were the detection of gravitational waves by the LIGO/VIRGO collaboration
Abbott:2016blz ; Abbott:2017vtc ; LIGOScientific:2018mvr ;
TheLIGOScientific:2016pea ; TheLIGOScientific:2016wfe ;
TheLIGOScientific:2016htt ; Abbott:2020khf , and the first image of the
environment of a black hole obtained by the Event Horizon Telescope (EHT)
collaboration Akiyama:2019cqa ; Akiyama:2019brx ; Akiyama:2019sww ;
Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap . Although these
experiments represent a major development in black hole research, the presence
of a shadow or gravitational ringdown are not definitive proof of this type of
solution Herdeiro:2021lwl ; Cardoso:2016rao . In addition to black holes,
there are a number of solutions that can be obtained through GR. Some of these
solutions are, for example, wormholes Morris:1988cz ; Morris:1988tu ;
Visser:1995cc ; Lobo:2017oab ; Bronnikov:2016xvj ; Bronnikov:2016osp ;
Bronnikov:2018vbs ; Blazquez-Salcedo:2020czn ; Bolokhov:2021svp ;
Bronnikov:2021piw , boson stars Colpi:1986ye ; Liebling:2012fv ;
Yoshida:1997qf , black bounce Simpson:2018tsi ; Simpson:2019cer ; Lobo:2020kxn
; Huang:2019arj ; Nascimento:2020ime ; Lobo:2020ffi ; Franzin:2021vnj , and
regular black holes Zaslavskii ; Fan-Wang ; Bronnikov:2017tnz ;
Capozziello:2014bqa ; Rodrigues:2015 ; NED2 ; NED3 ; NED4 ; NED5 ;
Bronnikov:2006fu ; Hollenstein:2008hp ; NED10 ; Rodrigues:2016 ;
Rodrigues:2017 ; Rodrigues:2018 ; bambi ; neves ; toshmatov ; DYM ; ramon ;
berej ; Rodrigues:2019 ; Silva:2018 ; Junior:2020  among others. For some of
these solutions to exist, coupling with auxiliary fields such as scalar,
electromagnetic, or Dirac fields are necessary. The black bounce, for example,
is a type of solution that may arise from the coupling of the gravitational
theory with a scalar field and an electromagnetic field Huang:2019arj .

In the context of black holes, the problem of singularities arises, i.e.,
places in spacetime where the geodesics are interrupted Bronnikov:2012wsj .
Possibly, the existence of singularities is a failure of the theory because it
is a classical theory of gravity. In this sense, a quantum theory of gravity
could solve the problem of the existence of singularities cap2 .

We do not yet have a complete theory of quantum gravity, but there are some
alternatives to eliminate the problem of the singularity. One of these
alternatives is the regular black hole, so named because there are no
singularities. The first regular solution was proposed by Bardeen Bardeen .
Since the Bardeen solution did not satisfy the Einstein equations for the
vacuum, Beato and Garcia showed that this solution can be obtained if one
takes into account the coupling of gravitational theory with nonlinear
electrodynamics Beato1 . In the literature, the study of regular solutions has
become quite extensive, with a large number of proposed solutions and with the
analysis of the properties Macedo:2014uga ; Macedo:2015qma ; Macedo:2016yyo ;
Paula:2020yfr ; Lima:2020wcb .

Another type of solution that avoids the presence of singularities is the
black bounce Simpson:2018tsi . Unlike the regular metrics mentioned above,
this type of solution has a minimal area that is not zero. Depending on the
parameters of this solution, it is possible to obtain the Schwarzschild
metric, a kind of regular black hole where there is a bounce instead of a
singularity, and a Morris-Thorne wormhole Lobo:2020ffi . Another novelty is
that, if we consider GR, these solutions are not described only by coupling
with nonlinear electrodynamics, which requires an additional matter
Franzin:2021vnj ; Huang:2019arj . Bronnikov and his collaborators also
proposed a solution with a structure similar to the black bounce, the black
universe Bronnikov:2016xvj ; Bronnikov:2016osp ; Bronnikov:2006fu . Several
papers have appeared on black bounces, quasinormal modes, shadows, absorption,
and others Tsukamoto:2020bjm ; Zhou:2020zys ; Cheng:2021hoc ;
Tsukamoto:2021caq ; Yang:2021cvh ; Bambhaniya:2021ugr ; Lima:2020auu .

Classically, black holes are objects from whose interior nothing can escape.
Thus, one would expect that no radiation can emanate from them Hawking:1971vc
. By semiclassical analysis, it has been shown that black holes can indeed
emit radiation and that they also have a temperature and entropy
Hawking:1975vcx . In fact, there is a whole thermodynamics associated with the
study of black holes. Just as we have the standard laws of thermodynamics,
there are four laws for black hole thermodynamics Bardeen:1973gs . In
particular, the first law provides us with a kind of energy conservation
equation, and thus it is possible to relate the entropy of the black hole to
its parameters such as mass, charge, and rotation Wald:1999vt . Since it is a
thermodynamic system, it is also possible to analyze the thermodynamic
stability of black holes. For example, the Schwarzschild solution has a
negative heat capacity, so it is a thermodynamically unstable solution
Wald:1999vt .

The entropy of a black hole and the area are related by $S=A/4$
Bekenstein:1973ur . However, for some solutions, if the first law of
thermodynamics is maintained, the relation between area and entropy is no
longer preserved Kumar:2020cve . Since entropy can be determined by Wald’s
formula Wald:1999vt  and we consider general relativity, the relation between
entropy and the area of the black hole is not changed. In this way, the first
law of thermodynamics must be modified depending on the type of solution we
consider Ma:2014qma ; Zhang:2016ilt . This type of modification can affect the
thermodynamic stability of a solution, since its temperature and heat capacity
also change Maluf:2018lyu .

The structure of this paper is as follows. In Sec. II we introduced and
analyze some general properties, such as the spacetime conditions of
regularity. In Sec. III we start analyzing the thermodynamics of this solution
using the temperature, the Smarr formula, and the first law of thermodynamics.
Section IV is devoted to the study of the heat capacity and its importance for
thermodynamic stability. In Sec. V we obtain the equation of state and the
possibility to analyze this solution from the point of view of virtual
micromolecules. In Sec. VI we analyze the Helmholtz free energy and the
isothermal compressibility to compare them with the heat capacity results. In
Sec. VII we see how the thermodynamic functions behave near the critical
points to obtain the critical exponents. Our conclusions and perspectives can
be found in Sec. VIII.

In this paper we consider natural units, where $c=\hbar=G=1$, and the metric
signature $(+,-,-,-)$. We adopt the convention that Greek indices run from $0$
to $3$, so that $x^{0}=t$, $x^{1}=r$, $x^{2}=\theta$, and $x^{3}=\phi$.

##  II Bardeen-Kiselev solution with cosmological constant

We consider a spherically symmetric spacetime

$ds^{2}=f(r)dt^{2}-\frac{1}{f(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (1)

The solution is magnetically charged and has cosmological constant. The action
that describes this theory is

$S=\int d^{4}x\sqrt{-g}\left[R+2\lambda+L_{NED}\right],$ (2)

where $R$ is the curvature scalar, $\lambda$ is the cosmological constant, and
$L_{NED}$ is the nonlinear Lagrangian of electromagnetic theory. The
Lagrangian is a nonlinear function of the electromagnetic scalar
$\mathcal{F}=F^{\mu\nu}F_{\mu\nu}$, where $F^{\mu\nu}$ is the Maxwell-Faraday
tensor.

For a spherically symmetric spacetime that is only magnetically charged, the
only nonzero component of $F_{\mu\nu}$ is NED4  444In PlebanskiKrasinski , the
authors explicitly show how the nonzero components are obtained for a
spherically symmetric electromagnetic field.

$F_{23}=q\sin\theta,$ (3)

and the scalar $\mathcal{F}$ is

$\mathcal{F}=\frac{2q}{r^{4}},$ (4)

where $q$ is the magnetic charge.

Let us consider the Lagrangian

$L(\mathcal{F})=\frac{24\sqrt{2}Mq^{2}}{\kappa^{2}\left(\sqrt{\frac{2q^{2}}{\mathcal{F}}}+2q^{2}\right)^{5/2}}-\frac{6\omega
c\left(\frac{2\mathcal{F}}{q^{2}}\right)^{\frac{3}{4}(\omega+1)}}{\kappa^{2}}.$
(5)

When $c\rightarrow 0$, the Bardeen solution is recovered. In the limit
$\mathcal{F}\rightarrow 0$

$L(\mathcal{F})\approx-\frac{6c2^{\frac{3}{4}(\omega+1)}\omega\mathcal{F}^{\frac{3(\omega+1)}{4}}}{\kappa^{2}\left(q^{2}\right)^{\frac{3}{4}(\omega+1)}}+\frac{12\sqrt[4]{2}\mathcal{F}\mathcal{F}^{1/4}m}{\kappa^{2}\sqrt[4]{q^{2}}}+O\left(\mathcal{F}\mathcal{F}^{3/4}\right).$
(6)

When $\omega=1/3$, we find

$L(\mathcal{F})\approx-\frac{4c\mathcal{F}}{\kappa^{2}q^{2}}+\frac{12\sqrt[4]{2}\mathcal{F}\mathcal{F}^{1/4}m}{\kappa^{2}\sqrt[4]{q^{2}}}+O\left(\mathcal{F}\mathcal{F}^{3/4}\right),$
(7)

so that for very small values of $\mathcal{F}$ the linear term dominates,

$L(\mathcal{F})\approx\mathcal{F},\mathcal{F}\rightarrow 0.$ (8)

This means that electromagnetism in the Maxwell limit behaves approximately
like Maxwell theory.

In the presence of the cosmological constant the Einstein equations are

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\lambda g_{\mu\nu}=\kappa^{2}T_{\mu\nu}.$
(9)

Using the stress-energy tensor to nonlinear electrodynamics

$T_{\mu\nu}=g_{\mu\nu}L(\mathcal{F})-\frac{dL}{d\mathcal{F}}F_{\mu}^{\
\alpha}F_{\nu\alpha},$ (10)

and solving the Einstein equations, we obtain the metric function

$f(r)=1-2cr^{-3\omega-1}-\frac{2Mr^{2}}{\left(q^{2}+r^{2}\right)^{3/2}}-\frac{\lambda
r^{2}}{3}.$ (11)

For the limit $c\rightarrow 0$ we find the Bardeen-(anti-)de Sitter solution
and for $q\rightarrow 0$ we find the Kiselev-(anti-)de Sitter solution with
cosmological constant Malakolkalami:2015cza .

To obtain the horizons, we need to solve $f(r)=0$. However, we cannot obtain
analytical expressions for the radius of the horizon. The number of horizons
depends on the values of the parameters. In Fig. 1 we see that the number of
horizons can be up to four.

Figure 1: graphic representation of $f(r)$ with respect to the radial
coordinate with different values of $q$, $\lambda$, $c$, and $\omega$.

If we want to check the existence of curvature singularities, since spacetime
is spherically symmetric, we only need to check the Kretschmann scalar
Lobo:2020ffi , $K=R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$, which is given
by

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{4\left(2cr^{-3\omega-1}+\frac{1}{3}r^{2}\left(\lambda+\frac{6M}{\left(q^{2}+r^{2}\right)^{3/2}}\right)\right)^{2}}{r^{4}}+\frac{4\left(2c(3\omega+1)r^{-3\omega-2}+\frac{2Mr\left(r^{2}-2q^{2}\right)}{\left(q^{2}+r^{2}\right)^{5/2}}-\frac{2\lambda
r}{3}\right)^{2}}{r^{2}}$ (12) $\displaystyle+$
$\displaystyle\left(2c(9\omega(\omega+1)+2)r^{-3(\omega+1)}+\frac{2}{3}\left(\lambda+\frac{M\left(-33q^{2}r^{2}+6q^{4}+6r^{4}\right)}{\left(q^{2}+r^{2}\right)^{7/2}}\right)\right)^{2}.$

The spacetime is regular only for some values of $\omega$. When
$\omega\leq-1$, the solution is regular in $r\rightarrow 0$ and to
$\omega\geq-1$ it is regular in $r\rightarrow\infty$ but is not asymptotically
flat. For $\omega=-1$ we have the following limits:

$\displaystyle\lim\limits_{r\rightarrow\infty}K(r)=\frac{8}{3}(6c+\lambda)^{2},$
(13) $\displaystyle\lim\limits_{r\rightarrow
0}K(r)=\frac{8\left(12M\left(q^{2}\right)^{3/2}(6c+\lambda)+q^{6}(6c+\lambda)^{2}+36M^{2}\right)}{3q^{6}}.$
(14)

For this value of $\omega$, the solution is curvature regular everywhere,
since we have only a shift in the cosmological constant.

##  III Thermodynamics

To start the study of black hole thermodynamics, we can obtain the temperature
of a black hole using the surface gravity by Wald:1999vt

$T_{k}=\frac{k}{2\pi},$ (15)

where

$k=\left.\frac{f^{\prime}(r)}{2}\right|_{r=r_{+}},$ (16)

is the surface gravity and $r_{+}$ is the event horizon radius. Considering
Eq. (11), we obtain

$T_{k}=\frac{1}{4\pi}\left(2c(3\omega+1)r_{+}^{-3\omega-2}+\frac{2Mr_{+}\left(r_{+}^{2}-2q^{2}\right)}{\left(q^{2}+r_{+}^{2}\right)^{5/2}}-\frac{2\lambda
r_{+}}{3}\right).$ (17)

The first law of black hole thermodynamics for a charged and static black hole
is given by Gunasekaran:2012dq

$dM=T_{H}dS+\Phi dq+VdP,$ (18)

where $M$ is the total energy of the system, $\Phi$ is the magnetic potential,
$P$ is the thermodynamic pressure, written as

$P=-\frac{\lambda}{8\pi},$ (19)

and $S$ is the entropy of the system. The area law states that
Bekenstein:1973ur

$S=\frac{A}{4}=\pi r_{+}^{2},$ (20)

where $A$ is the area of the black hole.

The first law, as given in Eq. (18), is not valid if we have nonlinear
electrodynamics Ma:2014qma ; Zhang:2016ilt ; Maluf:2018lyu . The temperature
and magnetic potential can be derived from the first law as follows:

$T_{H}=\frac{\partial M}{\partial S},\ \mbox{and}\ \Phi=\frac{\partial
M}{\partial q}.$ (21)

We can obtain $M$ from the condition $f(r_{+})=0$, which leads to

$M=-\frac{1}{6}\left(q^{2}+r_{+}^{2}\right)^{3/2}r_{+}^{-3\omega-3}\left(6c+\lambda
r_{+}^{3\omega+3}-3r_{+}^{3\omega+1}\right).$ (22)

The temperature in (21), with (22), is given by

$T_{H}=\frac{\sqrt{q^{2}+r_{+}^{2}}r_{+}^{-3\omega-5}\left(6c\left(q^{2}(\omega+1)+r_{+}^{2}\omega\right)+r_{+}^{3\omega+1}\left(-2q^{2}-\lambda
r_{+}^{4}+r_{+}^{2}\right)\right)}{4\pi}.$ (23)

The temperature $T_{k}$ is different from $T_{H}$. In addition to temperature,
the others quantities derived from the usual first law of thermodynamics,
derivative of the mass with respect to some parameter, also present problems.
To solve this problem, we need to modify the first law. The usual first law
arises in a context where the Lagrangian of the theory does not explicitly
depend on the mass of the black hole. Thus, when building the first law for
solutions with nonlinear electrodynamics, derivatives of the stress-energy
tensor must be taken into account and these corrections must modify the first
law in such a way that the old thermodynamic quantities must relate to the new
ones through of a correction factor.

The new first law is written as Singh:2020xju

$d\mathcal{M}=TdS+\Phi dq+PdV,$ (24)

where

$d\mathcal{M}=dM\left(1+4\pi\int_{r_{+}}^{\infty}r^{2}\frac{\partial T^{0}_{\
0}}{\partial M}dr\right)=W(r_{+},q)dM,$ (25)

where $W(r_{+},q)$ is the correction factor, given by

$W(r_{+},q)=\left(1+4\pi\int_{r_{+}}^{\infty}r^{2}\frac{\partial T^{0}_{\
0}}{\partial M}dr\right),$ (26)

and $T^{0}_{\ 0}$ is one of the components of the stress-energy tensor.

We find that the relation between the temperatures is

$T_{k}=W\left(r_{+},q\right)T_{H}=W\left(r_{+},q\right)\frac{\partial
M}{\partial
S}=\frac{r_{+}^{-3\omega-2}\left(6c\left(q^{2}(\omega+1)+r_{+}^{2}\omega\right)+r_{+}^{3\omega+1}\left(-2q^{2}-\lambda
r_{+}^{4}+r_{+}^{2}\right)\right)}{4\pi\left(q^{2}+r_{+}^{2}\right)},$ (27)

and $W(r_{+},q)$ is

$W(r_{+},q)=\frac{r_{+}^{3}}{\left(q^{2}+r_{+}^{2}\right)^{3/2}}.$ (28)

When we have an extreme black hole, as we see in the first image of Fig. 1,
the temperature arising from surface gravity is zero. To understand this we
just need to realize that the radius of the horizon is just where we have a
minimum of the function $f(r)$. So the derivative at that point is zero. The
correction factor is not zero for an extreme black hole. So, both $T_{k}$ and
$T_{H}$ are zero for extreme black holes.

The temperatures for Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter are
given by

$\displaystyle T_{kB}$ $\displaystyle=$ $\displaystyle-\frac{2q^{2}+\lambda
r_{+}^{4}-r_{+}^{2}}{4\pi q^{2}r_{+}+4\pi r_{+}^{3}},$ (29) $\displaystyle
T_{kK}$ $\displaystyle=$ $\displaystyle\frac{6c\omega
r_{+}^{-3\omega-1}-\lambda r_{+}^{2}+1}{4\pi r_{+}}.$ (30)

Figure 2: Graphic representation of the temperature associated to the solution
(11) with $M^{2}\lambda=-0.08\pi$ and $\omega=-2/3$.

It is not so obvious what the differences are between this temperature and the
temperature of Eq. (17). To better analyze this situation, let us see how the
temperature behaves when we change certain parameters in Fig 2. If we assign a
value to $r_{+}$, the temperature decreases as the charge and the parameter
$c$ increase. These results show that the solution(11) is colder than the
Bardeen- (anti-)de Sitter solution and the Kiselev-(anti-)de Sitter solution.

To obtain the complete first law, we must first find the Smarr formula, a
relation that connects the mass (energy) to the other parameters through a
homogeneous function. We can derive the Smarr formula from the properties of a
homogeneous function Homogenea .

In terms of the entropy, mass can be written as

$M(S,q,c,\lambda)=\frac{1}{6}\left(q^{2}+\frac{S}{\pi}\right)^{3/2}S^{-\frac{3}{2}(\omega+1)}\left((3\pi-\lambda
S)S^{\frac{3\omega}{2}+\frac{1}{2}}-6c\pi^{\frac{3(\omega+1)}{2}}\right).$
(31)

To determine the degree of homogeneity, we write

$M(l^{a}S,l^{b}q,l^{d}c,l^{h}\lambda)=\frac{1}{6}\left(l^{2b}q^{2}+\frac{l^{a}S}{\pi}\right)^{3/2}l^{-\frac{3a}{2}(\omega+1)}S^{-\frac{3}{2}(\omega+1)}\left(\left(3\pi-l^{h}\lambda
l^{a}S\right)l^{\frac{a}{2}(3\omega+1)}S^{\frac{1}{2}(3\omega+1)}-6\pi^{\frac{3(\omega+1)}{2}}l^{d}c\right).$
(32)

To isolate $l$, we assume $h=-a$, $d=a(3\omega+1)/2$, $b=a/2$, and $a=1$,
which leads to

$M(lS,l^{1/2}q,l^{(3\omega+1)/2}c,l^{-1}\lambda)=-\frac{1}{6}\sqrt{l}\left(q^{2}+\frac{S}{\pi}\right)^{3/2}S^{-\frac{3}{2}(\omega+1)}\left(6\pi^{\frac{3(\omega+1)}{2}}c+\left(\lambda
S-3\pi\right)S^{\frac{1}{2}(3\omega+1)}\right),$ (33)

such that the mass is a homogeneous function with degree of homogeneity
$n=1/2$ Homogenea .

Euler’s identity states that for a homogeneous function with degree $n$, we
have Homogenea

$n.f(x_{1},x_{2}...x_{m})=a_{1}x_{1}\frac{\partial f}{\partial
x_{1}}+a_{2}x_{2}\frac{\partial f}{\partial x_{2}}+a_{3}x_{3}\frac{\partial
f}{\partial x_{3}}+...+a_{m}x_{m}\frac{\partial f}{\partial x_{m}}.$ (34)

Thus we obtain

$\frac{1}{2}M(S,q,c,\lambda)=T_{H}S+\frac{1}{2}\Phi
q+\frac{1}{2}\left(1+3\omega\right)A_{c}c-A_{\lambda}\lambda,$ (35)

where $T_{H}$ and $\Phi$ are given by (21), and $A_{c}$ and $A_{\lambda}$ are

$A_{c}=\frac{\partial M}{\partial c}\ \mbox{and}\ A_{\lambda}=\frac{\partial
M}{\partial\lambda}.$ (36)

With Eq. (21) and (36) we get that (35) and (31) are equal.

The first law is written as

$d\mathcal{M}=W(S,q)dM=T_{k}dS+W(S,q)\Phi
dq+W(S,q)A_{c}dc+W(S,q)A_{\lambda}d\lambda.$ (37)

For regular black holes, it is normal that the Lagrangian depends explicitly
on some parameters, such as charge and mass. Fluctuations in the matter sector
are not a problem because they are compensated by fluctuations in the geometry
sector by the first law, (37).

##  IV Heat Capacity

From thermodynamics, we know that the stability of a thermodynamic system
requires $C_{P}\geq C_{V}\geq 0$ and $k_{T}\geq k_{S}\geq 0$, where $C_{P}$
and $C_{V}$ are the heat capacity at constant pressure and volume,
respectively, and $k_{T}$ and $k_{S}$ are the isothermal compressibility and
isentropic compressibility, respectively Stanley .

In black thermodynamics, compressibility and heat capacity also give us
information about the stability of the system. The isothermal compressibility
and heat capacity at constant pressure are given by Davies:1977bgr

$\displaystyle C_{P}=\left.T_{k}\frac{\partial S}{\partial T_{k}}\right|_{P},$
(38) $\displaystyle k_{T}=-\left.\frac{1}{V}\frac{\partial V}{\partial
P}\right|_{T}.$ (39)

We will focus on the heat capacity. We get that the heat capacity at constant
pressure is

$\displaystyle C_{P}$ $\displaystyle=$ $\displaystyle\left\\{2S\left(\pi
q^{2}+S\right)\left(S^{\frac{3\omega}{2}+\frac{1}{2}}\left(2\pi^{2}q^{2}+\lambda
S^{2}-\pi S\right)-6c\pi^{\frac{3(\omega+1)}{2}}\left(\pi
q^{2}(\omega+1)+S\omega\right)\right)\right\\}$ (40) $\displaystyle\times$
$\displaystyle\Big{\\{}6c\pi^{\frac{3(\omega+1)}{2}}\left(\pi
q^{2}S(\omega(6\omega+7)+4)+\pi^{2}q^{4}(\omega+1)(3\omega+2)+S^{2}\omega(3\omega+2)\right)$
$\displaystyle+$ $\displaystyle S^{\frac{3\omega}{2}+\frac{1}{2}}\left(\pi
S^{2}\left(3\lambda q^{2}+1\right)-7\pi^{2}q^{2}S-2\pi^{3}q^{4}+\lambda
S^{3}\right)\Big{\\}}^{-1}$

This result is valid only if we derive temperature from the surface gravity.
If we use (21), we get

$\bar{C}_{P}=\frac{2S\left(\pi
q^{2}+S\right)\left(S^{\frac{3\omega}{2}+\frac{1}{2}}\left(2\pi^{2}q^{2}+\lambda
S^{2}-\pi S\right)-6c\pi^{\frac{3(\omega+1)}{2}}\left(\pi
q^{2}(\omega+1)+S\omega\right)\right)}{6c\pi^{\frac{3(\omega+1)}{2}}\left(2\pi
q^{2}S\left(3\omega^{2}+5\omega+2\right)+\pi^{2}q^{4}\left(3\omega^{2}+8\omega+5\right)+S^{2}\omega(3\omega+2)\right)+S^{\frac{3\omega}{2}+\frac{1}{2}}\left(\pi
S^{2}-4\pi^{2}q^{2}S-8\pi^{3}q^{4}+\lambda S^{3}\right)}.$ (41)

These two heat capacities are connected by

$C_{P}=W(S,q)\frac{\partial T_{H}}{\partial T_{k}}\bar{C}_{P}.$ (42)

The heat capacity to the Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter
solutions are

$\displaystyle C_{PB}=\frac{2S\left(\pi
q^{2}+S\right)\left(2\pi^{2}q^{2}+\lambda S^{2}-\pi S\right)}{\pi
S^{2}\left(3\lambda q^{2}+1\right)-7\pi^{2}q^{2}S-2\pi^{3}q^{4}+\lambda
S^{3}},$ (43) $\displaystyle
C_{PK}=\frac{2S\left(-6c\pi^{\frac{3(\omega+1)}{2}}\omega+\lambda
S^{\frac{3(\omega+1)}{2}}-\pi
S^{\frac{3\omega}{2}+\frac{1}{2}}\right)}{6c\pi^{\frac{3(\omega+1)}{2}}\omega(3\omega+2)+\lambda
S^{\frac{3(\omega+1)}{2}}+\pi S^{\frac{3\omega}{2}+\frac{1}{2}}}.$ (44)

In Fig. 3 we see how the heat capacity to the solution (11) behaves. There are
both positive and negative values for $C_{P}$, so the solution is
thermodynamically stable for some values of entropy. This behavior is
different from the Schwarzschild solution where the heat capacity is always
negative, $C_{P}=-2S$. We also see that in the range of small values of
entropy and negative heat capacity, the temperature is also negative and
therefore this range has no physical meaning. In the Bardeen-(anti-)de Sitter
solution, the first phase transition occurs at a higher temperature value than
in the other cases, while the second phase transition occurs at a lower
temperature value than in the other cases. In this way, the Bardeen solution
has a smaller temperature range for the intermediate phase. From Fig. 4 we see
that the Kiselev-(anti-)de Sitter solution has only one phase transition,
while the solution (11) has two phase transitions.

Figure 3: Heat capacity at constant pressure and temperature associated to the
solution (11) with $q=0.2M$, $M^{2}\lambda=-0.08\pi$ and $\omega=-2/3$.

Figure 4: Heat capacity at constant pressure and temperature associated to the
solution (11) with $M^{2}\lambda=-0.08\pi$, $\omega=-2/3$, and $Mc=0.2$.

##  V Equation of State

From the modified first law (37), the thermodynamic volume is given by

$V=\frac{\partial\mathcal{M}}{\partial P}=\frac{4\pi r^{3}_{+}}{3}.$ (45)

With (17) and (45) we obtain the equation of state $P(T,V)$, i.e.

$P(T,V)=\frac{4q^{2}\left(6\pi^{4/3}T\sqrt[3]{V}+(6\pi)^{2/3}\right)+6(6\pi)^{2/3}TV-3\sqrt[3]{6}V^{2/3}}{36\sqrt[3]{\pi}V^{4/3}}-\frac{4^{\omega}\pi^{\omega+\frac{2}{3}}c}{3^{\omega+\frac{2}{3}}V^{\omega+\frac{5}{3}}}\left(2\sqrt[3]{2}q^{2}(\omega+1)+\left(\frac{3V}{\pi}\right)^{2/3}\omega\right).$
(46)

The pressure to the Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter
solutions are

$\displaystyle P_{B}(T,V)$ $\displaystyle=$
$\displaystyle\frac{\sqrt[3]{\frac{\pi}{6}}\left(2q^{2}+3TV\right)}{3V^{4/3}}+\frac{2\pi
q^{2}T}{3V}-\frac{1}{2\ 6^{2/3}\sqrt[3]{\pi}V^{2/3}},$ (47) $\displaystyle
P_{K}(T,V)$ $\displaystyle=$
$\displaystyle-c\left(\frac{4\pi}{3}\right)^{\omega}\omega
V^{-\omega-1}+\frac{\sqrt[3]{\frac{\pi}{6}}T}{\sqrt[3]{V}}-\frac{1}{2\
6^{2/3}\sqrt[3]{\pi}V^{2/3}}.$ (48)

The critical points result from the conditions Dehyadegari:2017flm ;
Dayyani:2017fuz ; Wei:2020poh

$\left(\frac{\partial P}{\partial V}\right)_{T}=0,\ \mbox{and}\
\left(\frac{\partial^{2}P}{\partial V^{2}}\right)_{T}=0.$ (49)

Unfortunately, we are not able to solve this equation analytically, but if we
fix some values for parameters $q$, $\omega$, and $c$, we get numerical values
for $T_{c}$, $P_{c}$, and $V_{c}$. In Fig. 5 we show the isotherm considering
three cases, $T<T_{c}$, $T>T_{c}$, and $T=T_{c}$, where $T_{c}$ is the
critical temperature. It is clear that the pressure is higher for higher
temperature values. At a fixed temperature, the pressure increases as the
charge and $c$ increase. For the Kiselev-(anti-)de Sitter solution it is not
possible to solve the condition (49) because there is no inflection point in
$P(V)$. So we are not able to find the critical points.

Figure 5: Graphic representation of the $P-V$ behavior of the solution (11)
with $q=0.2M$, $Mc=0.2$ and, $\omega=-2/3$. There is also the behavior at a
fixed temperature and different values of $q$ and $c$.

Since we have the equation of state $P\left(T,V\right)$, we can obtain the
compressibility factor $Z$, which is defined as Stanley

$Z=\frac{PV}{T}.$ (50)

For an ideal gas, i.e., a gas in which there is no interaction between the
particles that compose it, we have $Z=1$ Stanley . For a real gas where there
is interaction between the particles that make it up, $Z$ depends on factors
such as pressure and temperature Stanley . Thus, the compressibility factor
basically tells the deviation of the behavior of a real gas from that of an
ideal gas. Since $P\left(T,V\right)$ is a function of temperature and volume,
$Z$ is also a function of $T$ and $V$ and is written as

$Z(T,V)=\frac{4q^{2}\left(6\pi^{4/3}T\sqrt[3]{V}+(6\pi)^{2/3}\right)+6(6\pi)^{2/3}TV-3\sqrt[3]{6}V^{2/3}}{36\sqrt[3]{\pi
V}T}-\frac{c}{T}\left(\frac{4\pi}{3V}\right)^{\omega+\frac{2}{3}}\left(q^{2}(\omega+1)+\left(\frac{3V}{4\pi}\right)^{2/3}\omega\right).$
(51)

To Bardeen-(anti-)de Sitter e Kiselev-(anti-)de Sitter we find

$\displaystyle Z_{K}(T,V)$ $\displaystyle=$
$\displaystyle-\frac{c\left(\frac{4\pi}{3}\right)^{\omega}\omega
V^{-\omega}}{T}-\frac{\sqrt[3]{V}}{2\
6^{2/3}\sqrt[3]{\pi}T}+\sqrt[3]{\frac{\pi}{6}}V^{2/3},$ (52) $\displaystyle
Z_{B}(T,V)$ $\displaystyle=$
$\displaystyle\frac{\sqrt[3]{\frac{\pi}{6}}\left(2q^{2}+3TV\right)}{3T\sqrt[3]{V}}+\frac{2\pi
q^{2}}{3}-\frac{\sqrt[3]{V}}{2\ 6^{2/3}\sqrt[3]{\pi}T}.$ (53)

In Fig. 6 we see how $Z$ behaves when we increase the pressure for different
values of the temperature. The compressibility factor diverges to small values
of the pressure. As said, for a real gas $Z\neq 1$, but for small values of
pressure some gasses behave like an ideal gas. For a van der Waals fluid, the
compressibility factor at the critical point is $Z_{c}=0.375$ Dayyani:2017fuz
. For the solution (11) we find $Z_{c}\approx 0.75$, which is twice as much as
in the van der Waals case.

Figure 6: Compressibility factor in terms of the ratio $P/P_{c}$ to $q=0.2M$,
$Mc=0.2$, and $\omega=-2/3$.

In the context of fluids, the compressibility factor, when analyzed
microscopically, provides information about the interaction of the molecules
that make up the fluid James . The most important point is this interpretation
of the molecules in the context of black holes. According to Wei and Liu, it
is possible to analyze a black hole microscopically through the idea of
virtual molecules Wei:2015iwa . Thus, it is interesting to use the number
density of virtual micromolecules of the black hole, i.e.

$n=\frac{1}{v}=\frac{1}{2l_{p}^{2}r_{+}},$ (54)

where $v$ is the specific volume of the black hole fluid and
$l_{p}=\sqrt{\hbar G/c^{3}}$ is the Planck length. Since we are considering
natural units, $l_{p}=1$ and $n=1/v=1/2r_{+}$. Note that the specific volume
is linear with the radius of the event horizon. In terms of the number
density, the temperature is

$\displaystyle
T(n)=\frac{3c\left(2n\right)^{3\omega+2}n\left(4n^{2}q^{2}(\omega+1)+\omega\right)-8n^{4}q^{2}+2\pi
P+n^{2}}{2n\pi\left(4n^{2}q^{2}+1\right)}.$ (55)

For Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter we find

$\displaystyle T_{B}(n)$ $\displaystyle=$
$\displaystyle\frac{-8n^{4}q^{2}+n^{2}+2\pi P}{8\pi n^{3}q^{2}+2\pi n},$ (56)
$\displaystyle T_{K}(n)$ $\displaystyle=$
$\displaystyle\frac{6c2^{3\omega}\omega
n^{3\omega+2}}{\pi}+\frac{P}{n}+\frac{n}{2\pi}.$ (57)

In Fig. 7 see how the temperature behaves when we change the number density
for different values of the pressure. There is a maximum value of $n$ and,
depending on the value of $P$, there are other limits in the number density.
For example, for small values of pressure, there is a range where the
temperature is negative, so these values of $n$ are not allowed. Towards
larger values of charge, the number density increases. As $c$ increases, the
number density decreases.

Figure 7: Behavior of ratio $T_{k}/T_{c}$ in terns of number density to
$\omega=-2/3$.

##  VI Helmholtz free energy and isothermal compressibility

To verify the phase transition, we should analyze the Helmholtz free energy
Dehyadegari:2017flm ; Dayyani:2017fuz ; Wei:2020poh ; Liu:2014gvf ;
Nam:2018tpf ; Dehyadegari:2020tau ; NaveenaKumara:2020biu , $F$, given by

$F=M-TS.$ (58)

With (22), (17), and (20), we obtain

$F=\frac{1}{6}\left(\frac{\left(q^{2}+r_{+}^{2}\right)^{3/2}\left(\left(8\pi
Pr_{+}^{2}+3\right)r_{+}^{3\omega+1}-6c\right)}{r_{+}^{3(\omega+1)}}+\frac{3r_{+}^{2}\left(6cr_{+}^{-3\omega}-8\pi
Pr_{+}^{3}-3r_{+}\right)}{2\left(q^{2}+r_{+}^{2}\right)}-\frac{6c}{r_{+}^{3\omega}}-3c(3\omega+1)r_{+}^{-3\omega}+3r_{+}\right).$
(59)

To Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter we find

$\displaystyle F_{B}$ $\displaystyle=$
$\displaystyle\frac{r_{+}}{2}-\frac{r_{+}^{3}\left(8\pi
Pr_{+}^{2}+3\right)}{4\left(q^{2}+r_{+}^{2}\right)}+\frac{\left(8\pi
Pr_{+}^{2}+3\right)\left(q^{2}+r_{+}^{2}\right)^{3/2}}{6r_{+}^{2}},$ (60)
$\displaystyle F_{K}$ $\displaystyle=$
$\displaystyle-\frac{c}{r_{+}^{3\omega}}\left(1+\frac{3\omega}{2}\right)-2\pi
Pr_{+}^{3}+\frac{r_{+}}{6}\left(8\pi Pr_{+}^{2}+3\right)-\frac{r_{+}}{4}.$
(61)

In Fig. 8 we show the behavior of the Helmholtz free energy. For small values
of the temperature, we have only small black holes. As we increase the
temperature, there is more than one possible state. There is a temperature
value for which there are three different states, but two of them have the
same energy. Interestingly, there is a black hole, the intermediate one, for
which the heat capacity is negative, that is, it is thermodynamically
unstable. When we increase the pressure, the intermediate black hole
disappears and the phase transition occurs from a small black hole to a large
black hole. The Kiselev solution has only two phases and as the charge
increases, the temperature required for the phase transition decreases. In the
Bardeen solution, there are three phases, but as $c$ increases, one of the
phases disappears at positive temperatures.

Figure 8: Helmholtz free energy as a function of temperature to $\omega=-2/3$.

Another way to analyze the phase transition is through isothermal
compressibility. If we use (39) with (46), we get

$\displaystyle k_{T}$ $\displaystyle=$ $\displaystyle
24\pi\left(r_{+}^{2}+q^{2}\right)r_{+}^{3\omega+5}\left[r_{+}^{3\omega+1}\left(q^{2}r_{+}^{2}\left(24\pi
Pr_{+}^{2}+7\right)+8\pi Pr_{+}^{6}+2q^{4}-r_{+}^{4}\right)\right.$ (62)
$\displaystyle-$
$\displaystyle\left.6c\left(q^{2}r_{+}^{2}(\omega(6\omega+7)+4)+q^{4}(\omega+1)(3\omega+2)+r_{+}^{4}\omega(3\omega+2)\right)\right]^{-1}.$

To Bardeen-(anti-)de Sitter and Kiselev-(anti-)de Sitter we get

$\displaystyle k_{TB}$ $\displaystyle=$ $\displaystyle\frac{24\pi
r^{4}\left(q^{2}+r^{2}\right)}{q^{2}r^{2}\left(24\pi Pr^{2}+7\right)+8\pi
Pr^{6}+2q^{4}-r^{4}},$ (63) $\displaystyle k_{TK}$ $\displaystyle=$
$\displaystyle\frac{24\pi
r^{3\omega+3}}{6c\omega-(18c\omega^{2}+18c\omega)+\left(8\pi
Pr^{2}-1\right)r^{3\omega+1}}.$ (64)

When the isothermal compressibility is positive, the solution is
thermodynamically stable Stanley . In Fig. 9 we see that the solution is
stable for a small and large black. As we already know from the heat capacity,
the intermediate black hole is thermodynamically unstable.

Figure 9: Isothermal compressibility associated to the solution (11) with
$q=0.2M$, $M^{2}P=0.01$, $\omega=-2/3$, and $Mc=0.2$.

##  VII Critical exponents

As we have seen, some physical quantities such as isothermal compressibility
and heat capacity diverge during a phase transition. Even though these
quantities diverge, we can analyze their behavior near the transition point.
For this purpose, we use the critical exponents Dayyani:2017fuz ; Wei:2020poh
. To obtain the critical exponents, we use

$t=\frac{T-T_{c}}{T_{c}},\ v=\frac{V-V_{c}}{V_{c}},\ \mbox{and}\
p=\frac{P}{P_{c}}.$ (65)

The critical exponents $\alpha$, $\beta$, $\gamma$, and $\delta$ are obtained
from Stanley

$\displaystyle C_{V}=T\left(\frac{\partial S}{\partial
T}\right)\propto\left|t\right|^{-\alpha},$ (66)
$\displaystyle\eta=V_{1}-V_{2}\propto\left|t\right|^{\beta},$ (67)
$\displaystyle k_{T}=-\frac{1}{V}\frac{\partial V}{\partial
P}\propto\left|t\right|^{-\gamma},$ (68)
$\displaystyle\left|P-P_{c}\right|\propto\left|V-V_{c}\right|^{\delta},$ (69)

where $\eta$ is the difference in volume between two phases.

Before proceeding, we have analyzed the phase transition graphically for
certain parameter values. Thus, in the next analyses, although it is not
obvious, we consider the same values as before, $\omega=-2/3$, $q=0.2M$, and
$Mc=0.2$.

If we use (65) and rewrite (46) and expand it for small values of $t$ and $v$,
we get

$p=1+t(a-bv)-dv^{3}+O\left(v^{4},tv^{2}\right),$ (70)

where $a$, $b$, and $d$ are just constants that are combinations of $\omega$,
$q$, and $c$. If we consider $t$ as constant and derive $p$ to $v$, we get

$dp=-\left(bt+3dv^{2}\right)dv.$ (71)

This relation is important to apply Maxwell’s area law, which states
Dayyani:2017fuz ; Wei:2020poh

$\int VdP=0.$ (72)

Thus, considering Maxwell’s area law and the fact that the pressure is
constant at the phase transition, we obtain

$\displaystyle p$ $\displaystyle=$ $\displaystyle
1+t(a-bv_{1})-dv_{1}^{3}=1+t(a-bv_{2})-dv_{2}^{3},$ (73) $\displaystyle 0$
$\displaystyle=$
$\displaystyle\int_{v_{1}}^{v_{2}}\left(v+1\right)\left(bt+3dv^{2}\right)dv.$
(74)

Solving these equations yields the nontrivial solution as follows:

$v_{1}=-v_{2}\propto\sqrt{t}\ \mbox{and}\ \eta\propto\sqrt{t}.$ (75)

This means that one of the critical exponents is $\beta=1/2$.

We can also use (70) to calculate the isothermal compressibility and we find
that

$k_{T}\propto\frac{1}{t},$ (76)

such that $\gamma=1$.

We can rewrite the condition
$\left|P-P_{c}\right|\propto\left|V-V_{c}\right|^{\delta}$ as
$\left|p-1\right|\propto\left|v\right|^{\delta}$. From (70) we get

$p-1\propto t(a-bv)-dv^{3}.$

However, in the critical isotherm we have $t=0$, so that,

$p-1\propto v^{3}.$ (77)

With this result, we find the critical exponent $\delta=3$.

To determine $\alpha$, we need to analyze the heat capacity at constant
volume. To obtain $C_{V}$, we need to rewrite the entropy as a function of
volume, which is given by

$S(V)=\sqrt[3]{\frac{9V^{2}\pi}{16}}.$ (78)

If the volume is constant, the heat capacity at constant volume is

$C_{V}=\left.T\left(\frac{\partial S}{\partial T}\right)\right|_{V}=0.$ (79)

This means that the critical exponent is $\alpha=0$. We see that these
exponents satisfy the Griffiths, Rushbrooke and Widom equalities Wei:2020poh ;
Stanley ; Griffiths ;

$\displaystyle\alpha+\beta\left(1+\delta\right)=2,\ \mbox{Griffiths}$ (80)
$\displaystyle\gamma\left(\delta+1\right)=\left(2-\alpha\right)\left(\delta-1\right),\
\mbox{Griffiths}$ (81) $\displaystyle\alpha+2\beta+\gamma=2,\
\mbox{Rushbrooke}$ (82) $\displaystyle\gamma=\beta\left(\delta-1\right),\
\mbox{Widom}$ (83)

which says that there are only two independent exponents.

##  VIII Conclusions

In this article, we have obtained and analyzed a magnetically charged solution
that mimics the Bardeen regular black hole, magnetically charged, with
cosmological constant, and surrounded by quintessence. The Lagrangian of this
solution has as a limit in the Bardeen case at $c=0$. We find a metric
function that can have up to four horizons depending on the parameters. By
analyzing the Kretschmann scalar, we find that the solution is regular: in the
whole spacetime up to $\omega=-1$, at infinity of the radial coordinate when
$\omega\leq-1$, in $r=0$ up to $\omega\geq-1$.

It was also possible to analyze the solution from the point of view of black
hole thermodynamics. We obtained two different temperatures $T_{k}$ and
$T_{H}$ reflecting the use of nonlinear electrodynamics, and therefore we
modified the first law by the correction factor (28). Moreover, analyzing
(33), we found that the mass is a homogeneous function with a degree of
homogeneity $n=1/2$, and thus we found the Smarr formula (35).

We obtain two values for the heat capacity connected by (42). Analyzing Figs.
3 and 4, we conclude that $C_{P}$ can take both positive and negative values
as a function of entropy, i.e., there are regions where the solution is
thermodynamically stable $(C_{P}\geq 0)$ and in others it is unstable
$(C_{P}<0)$. The equation of state (46) behaves like a van der Walls fluid for
$q=0.2M$, $Mc=0.2$ and $\omega=-2/3$, and at the critical point the
compressibility factor is twice as large as in the van der Walls case. We also
construct the temperature as a function of the number density of virtual
micromolecules of the black hole, as shown in Fig. 7. To obtain a positive
temperature, $n$ cannot take any arbitrary value, so there must be some bounds
on this variable.

By analyzing the Helmholtz free energy and the isothermal compressibility of
the system, we show that the solution has two thermodynamically stable states,
namely small and large black holes. During the phase transition, a third
intermediate phase occurs, but it is thermodynamically unstable. Finally,
fixing the values $q=0.2M$, $Mc=0.2$, and $\omega=-2/3$, we calculated the
critical exponents $\beta=1/2$, $\gamma=1$, $\delta=3$, and $\alpha=0$, but
only two of them are independent since they are connected by the Griffiths,
Widom, and Rushbrooke equalities.

In future work, we hope to explore some topics that are missing in this paper,
such as geodesic analysis, shadows, causal structure, maximum analytical
extension, and the analysis of the interaction with bosonic and fermionic
fields.

Acknowledgements: M. E. R. thanks Conselho Nacional de Desenvolvimento
Científico e Tecnológico - CNPq, Brazil, for partial financial support. This
study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES) - Finance Code 001.

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aff1]Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D
12489 Berlin, Germany aff2]Max-Planck-Institut für Kernphysik, P.O. Box
103980, D 69029 Heidelberg, Germany aff3]Univ. Grenoble Alpes, IPAG, F-38000
Grenoble, France
CNRS, IPAG, F-38000 Grenoble, France aff5]Aix Marseille Université,
CNRS/IN2P3, CPPM UMR 7346, 13288 Marseille, France aff6]Université Bordeaux,
CNRS/IN2P3, Centre d’Études Nucléaires de Bordeaux Gradignan, 33175 Gradignan,
France aff7]Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3,
F-91128 Palaiseau, France aff9]Dublin Institute for Advanced Studies, 31
Fitzwilliam Place, Dublin 2, Ireland aff10]DSM/Irfu, CEA Saclay, F-91191 Gif-
Sur-Yvette Cedex, France [cor1]Corresponding author<EMAIL_ADDRESS>berlin.de

# Observations of Binary Systems with the H.E.S.S. Telescopes

P. Bordas  G. Dubus  P. Eger  J.-P. Ernenwein  H. Laffon  C. Mariaud  T.
Murach  M. de Naurois  C. Romoli  F. Schüssler  R. Zanin  for the H.E.S.S.
Collaboration  [  [  [  [  [  [  [  [

###### Abstract

Observations of binary systems obtained recently with the High Energy
Stereoscopic System (H.E.S.S) of Cherenkov telescopes are reported. The
outcomes of a detailed observation campaign on PSR B1259-63 during its
periastron passage in 2014 will be presented. This system was observed for the
first time with H.E.S.S. II, providing spectra and light curves down to 200
GeV, which will be compared with observations conducted during previous
periastron passages and with results from an analysis of contemporaneously
taken Fermi-LAT data. Also long-term observations of LS 5039 with H.E.S.S in
phase I and phase II are reported. This source was monitored at very high
energies (VHEs) in a period of time spanning more than ten years. Its spectral
energy distribution measured with H.E.S.S. II extends down to
$120\text{\,}\mathrm{GeV}\text{/}$. Spectral results from the Fermi-LAT
observations are shown as well, and the compatibility with H.E.S.S. results in
the overlapping energy range is discussed. The identification of the new
gamma-ray binary candidate HESS J1832-093 will also be presented. Furthermore,
the search for VHE emission from the microquasars GRS 1915+105, Circinus X-1
and V4641 Sgr based on data from H.E.S.S. observations conducted
contemporaneously with the _RXTE_ satellite experiment will be reported. These
data provide constraints on the integral gamma-ray flux at different X-ray
states of the three sources.

##  1 Introduction

A majority of $70\text{\,}\mathrm{\char 37\relax}\text{/}$ of all stars in our
galaxy are part of binary systems [1]. Depending on the orbital parameters,
the nature of the individual objects and properties of the emission, these
systems can be grouped into several categories. The following sections will
focus on results from observations of gamma-ray binaries and microquasars
(MQs) in the VHE (very high energy, with photon energies above
$\sim$100\text{\,}\mathrm{GeV}\text{/}$$) gamma-ray domain. The former are
binary systems whose non-thermal radiative output is dominated by gamma rays,
i.e. photons with energies above $\sim$1\text{\,}\mathrm{MeV}\text{/}$$. They
consist of a compact object, either a neutron star or a stellar-mass black
hole, and a massive O- or B-type star. Only seven of such systems are known to
date. MQs are binary systems which emit X-rays and- exhibit extended radio
emission. High-energy (HE,
$$100\text{\,}\mathrm{MeV}\text{/}$<E<$100\text{\,}\mathrm{GeV}\text{/}$$)
gamma-ray emission has been reported from at least two such systems, Cyg X-1
and Cyg X-3. However, a detection of VHE gamma-ray emission at the 5σ level is
still lacking.

Both gamma-ray binaries and microquasars exhibit flux variability and
sometimes, especially in case of the former, periodicity at all wavelengths.
Details depend on the parameters of the systems, for example on the orbital
periods which can last from a few days up to a few years, influencing the
location of the interaction zone and the energy and particle densities in this
area. Hence these systems provide a large variety of observables, making them
interesting targets for observations and subjects of theoretical models.

The results presented here are derived from observations performed with the
High Energy Stereoscopic System (H.E.S.S.) telescopes. In phase I of the
H.E.S.S. experiment, regular observations were conducted with four telescopes
with $107\text{\,}{\mathrm{m}}^{2}\text{/}$ reflectors each arranged in a
square with a side length of $120\text{\,}\mathrm{m}\text{/}$ started in 2004.
In 2012, a much larger fifth telescope with a reflector area of
$596\text{\,}{\mathrm{m}}^{2}\text{/}$ called CT 5 was placed into the centre
of the array, marking the beginning of phase II of the H.E.S.S. experiment. In
these proceedings, results from observations conducted with both the
H.E.S.S.-I and the H.E.S.S.-II configurations are reported. For more
information about the experimental setup, see e.g. Aharonian  _et al._ [2].

In the following sections, results from observations of PSR B1259-63/LS 2883
around the 2014 periastron in are presented. Furthermore, results from long-
term observations of LS 5039 are reported. Lastly, the discovery of a gamma-
ray binary candidate is discussed and results from observations of three
microquasars are presented.

##  2 Observations of PSR B1259-63/LS 2883

PSR B1259-63/LS 2883 is a binary system comprising a massive O9.5Ve star with
a circumstellar disk and a pulsar with a rotational period of
$48\text{\,}\mathrm{ms}\text{/}$ and a spin-down luminosity of
$8\text{\times}{10}^{35}\text{\,}\mathrm{erg}\text{/}\mathrm{s}$. The orbit of
the pulsar around the central star is very eccentric, with an eccentricity of
$e=0.87$, and has a period of $3.4\text{\,}\mathrm{yr}\text{/}$. The system is
located at a distance of $2.3\text{\,}\mathrm{kpc}\text{/}$. It is the only
gamma-ray binary system for which the nature of the compact object is known
[3, 4].

At VHEs, PSR B1259-63/LS 2883 was discovered with the H.E.S.S. telescopes
around the 2004 periastron [5]. Since then, observations with the H.E.S.S.
telescopes were conducted around all periastron passages since then [6, 7]. In
case of the 2014 periastron passage, data equivalent to
$60\text{\,}\mathrm{h}\text{/}$ of live time were collected with the five-
telescope H.E.S.S.-II array. Both the pre- and post-periastron phases were
covered as well as, for the first time, the time of periastron itself. Gamma
rays from this system were detected with a statistical significance of $>30σ$.

The differential photon spectrum obtained from a monoscopic analysis of the
2014 data performed with the MonoReco algorithm [8] is shown in Fig. 1
together with results from an extension of the Hillas reconstruction presented
in Aharonian  _et al._ [2] enabling analyses of data recorded with the
H.E.S.S. II array.

\begin{overpic}[width=216.81pt]{spectra-crop.pdf}
\put(12.0,8.0){{{\color[rgb]{0.828125,0.10546875,0.08984375}H.E.S.S.
Preliminary}}} \end{overpic}

Figure 1: Differential photon spectra obtained from monoscopic and
stereoscopic analyses of the 2014 data set. Also spectra resulting from re-
analyses of archival data from previous periastra are shown. Spectral indices
are given in the legend.

Spectral indices as given in the figure are compatible with each other. Also
the flux normalisations at $1\text{\,}\mathrm{TeV}\text{/}$ are compatible
within systematic uncertainties assumed to be $20\text{\,}\mathrm{\char
37\relax}\text{/}$ of the flux. For comparison, also spectra from re-analyses
of archival data from observations around previous periastra are shown. These
spectra are obtained from analyses with the ImPACT algorithm [9]. While all
spectral indices are compatible with each other within errors, the flux
normalisation differs from one year to the other. This is expected, as
different parts of the orbit were sampled each year. It should be noted that
the safe energy threshold of the re-analyses of archival data are signifiantly
lower than the published thresholds due to refined analysis techniques. The
monoscopic safe energy threshold is expected to be lowered by several tens of
GeV in the future, since the current value of approximately
$200\text{\,}\mathrm{GeV}\text{/}$ is chosen arbitrarily to guarantee robust
results.

Fluxes above $200\text{\,}\mathrm{GeV}\text{/}$ are shown in the left panel of
Fig. 2 with a night-wise and a monthly binning. In the night-wise light curve,

\begin{overpic}[width=212.47617pt]{fluxes_200_2014-crop.pdf}
\put(10.0,59.0){{{\color[rgb]{0.828125,0.10546875,0.08984375}H.E.S.S.
Preliminary}}}
\end{overpic}\begin{overpic}[width=212.47617pt]{fluxes_1000GeV_allYears-
crop.pdf}
\put(10.0,59.0){{{\color[rgb]{0.828125,0.10546875,0.08984375}H.E.S.S.
Preliminary}}} \end{overpic}

Figure 2: Light curves above $200\text{\,}\mathrm{GeV}\text{/}$ (left) and
$1\text{\,}\mathrm{TeV}\text{/}$ (right) with night-wise and monthly binnings.
The times of the disk crossings and of the periastron are marked by grey
dashed and dotted lines, respectively.

several interesting features can be identified. A rising flux well before the
first disk crossing is observed, with maximum fluxes compatible to the flux
level observed around the actual disk crossings. After the first disk
crossing, the flux decreases towards periastron. Finally a high flux, again
with fluxes similar to the fluxes observed during the disk crossings, is seen
at the time of the GeV flare occurring between $32\text{\,}\mathrm{d}\text{/}$
and approximately $60\text{\,}\mathrm{d}\text{/}$ after periastron. For
comparison also light curves obtained from all years are shown on the right
panel of Fig. 2 for energies above $1\text{\,}\mathrm{TeV}\text{/}$. The
general features previously discussed remain the same. It should be noted,
though, that the high flux before the first disk crossing is followed by a low
flux at the time of the nominal disk crossing. This behaviour is still under
investigation. Furthermore it can be seen that also above
$1\text{\,}\mathrm{TeV}\text{/}$ the flux is high at the central part of the
GeV flare, which may constrain current theoretical models of the origin of the
periodic HE flare.

Data from measurements with Fermi-LAT corresponding to the release version
_Pass 8_ were analysed as well. The obtained light curve is shown in Fig. 3.

\begin{overpic}[width=260.17464pt]{fermi_lc_units-crop.pdf}
\put(10.0,47.0){{{\color[rgb]{0.828125,0.10546875,0.08984375}H.E.S.S.
Preliminary}}} \end{overpic}

Figure 3: Fermi-LAT light curve with a weekly binning obtained from a re-
analyses of the 2010 and 2014 data sets based on the _Pass 8_ release. Upper
limits are indicated by arrows.

Apart from the fact that the prominent flares observed approximately
$30\text{\,}\mathrm{d}\text{/}$ after periastron in both 2010 and 2014 can be
reproduced it should be noted that, for the first time, several flux points
with a significance of at least 3σ were obtained for the times around the disk
crossings in both years. There is no indication of a high flux state at the
time of the high flux state at VHEs before the first disk crossing.

##  3 Observations of LS 5039

The gamma-ray binary LS 5039 consists of an O6.5Ve star with a mass of
$23\text{\,}\mathrm{\mbox{$M_{\odot}$}}\text{/}$ and a compact object of
unknown nature with a mass of
$3.7\text{\,}\mathrm{\mbox{$M_{\odot}$}}\text{/}$. These two objects are in a
very close orbit with a period of only $3.9\text{\,}\mathrm{d}\text{/}$. In
this section, results from ten years of H.E.S.S. observations are presented.
The data set comprises $104\text{\,}\mathrm{h}\text{/}$ and
$19\text{\,}\mathrm{h}\text{/}$ of live time recorded with the H.E.S.S. array
in phase I and phase II, respectively.

Analyses of all available data yield the spectral energy distributions (SEDs)
shown on the left side of Fig. 4. Both an SED resulting from a monoscopic
analysis of the data taken with the H.E.S.S. II array and an SED obtained from
a stereoscopic analysis of H.E.S.S. I data are shown. The analyses were
performed with the _Model Analysis_ [10].

Figure 4: Left: SEDs obtained from monoscopic and a stereoscopic analyses of
the H.E.S.S.-II and H.E.S.S.-I data sets, respectively. Results of fits with
power-law functions are given in the inset. Also an SED obtained from a re-
analysis of Fermi-LAT data is shown. Right: SEDs resulting from H.E.S.S.-I
analyses for parts of the orbit corresponding to the inferior or superior
conjunction. The corresponding orbital phase ranges are given for reference.
Fit results are given in the main text.

In case of the monoscopic analysis, the energy threshold is approximately
$120\text{\,}\mathrm{GeV}\text{/}$. Together with the H.E.S.S. curves, also an
SED obtained from a re-analysis of Fermi-LAT data in the _Pass 8_ release is
shown. For the first time the energy ranges of LS 5039 SEDs resulting from the
two experiments significantly overlap. The data points in this region are
compatible with each other within statistical uncertainties.

On the right side of Fig. 4, two other SEDs resulting from an analysis of the
H.E.S.S.-I data set are shown. The two SEDs correspond to either the inferior
or the superior conjunction phase of the orbit. While the latter SED is well-
fit by a power law, the former one exhibits a curvature and is therefore fit
with a power law with an exponential cut-off at approximately $6.6\pm
1.6\text{\,}\mathrm{TeV}\text{/}$. This cut-off energy is approximately 1000
times the cut-off energy of $5.3\pm 0.6\text{\,}\mathrm{GeV}\text{/}$ observed
in the HE gamma-ray spectrum shown on the left side of the figure. The indices
of the spectra are $Γ=$2.41\pm 0.06$$ and $Γ=$1.84\pm 0.07$$ for the superior
and inferior conjunctions, respectively. At $1\text{\,}\mathrm{TeV}\text{/}$,
the normalisations of the differential photon fluxes are $(1.01\pm
0.05)\text{\times}{10}^{-12}\text{\,}\mathrm{p}\mathrm{h}\mathrm{/}\mathrm{T}\mathrm{eV}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{s}$
and $(2.58\pm
0.1)\text{\times}{10}^{-12}\text{\,}\mathrm{p}\mathrm{h}\mathrm{/}\mathrm{T}\mathrm{eV}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{s}$.

For more information, the reader is referred to Mariaud  _et al._ [11].

##  4 Observations of HESS J1832-093

HESS J1832-093 was detected at VHEs during the galactic plane scan performed
with the H.E.S.S. experiment [12]. The observed gamma-ray excess in the
direction of object source is compatible with the excess from a point-like
source of VHE gamma rays. The differential photon spectrum is well described
by a power law with an index of $2.6\pm 0.3$. Several scenarios about the
nature of this object are discussed in the literature. Here the tentative
identification as one of the seven aforementioned gamma-ray binaries is
presented.

Recent measurements with the Chandra satellite experiment reveal that the
apparent angular distance of the star 2MASS J18324516−0921545 and the X-ray
source XMMU J183245−0921539 is only $None$, suggesting an association of these
two objects due to the low chance coincidence of $0.3\text{\,}\mathrm{\char
37\relax}\text{/}$. Also the high column density inferred from a spectral fit
of the X-ray data, which is approximately ten times higher than the total
galactic value, supports the hypothesis that the objects form a binary system.
The fact that HESS J1832-093 is compatible with a point-like source at VHEs
disfavours the identification as a pulsar wind nebula or a supernova remnant,
since such objects appear, in general, as extended sources. Furthermore, the
flux observed at X-ray energies is variable, with a factor of approximately
six between the low and high flux states, which is similar to the value
obtained from the gamma-ray binary HESS J0632+057. Another similarity to the
latter system is the fact that no significant HE gamma-ray emission could be
detected with current-generation instruments. Despite the lack of such
emission, the spectral energy distribution is dominated by the VHE emission,
which is a necessary requirement for the identification as a gamma-ray binary.

Based on the above arguments, HESS J1832-093 has been catalogued as a gamma-
ray binary system. For further information and the discussion of associations
with different object classes, the reader is referred to Eger  _et al._ [13].

##  5 Observations of Microquasars

As mentioned earlier, HE gamma rays have been detected from MQs already. VHE
gamma rays are predicted to be emitted by MQs during transient outbursts by
various theoretical models, but have not been detected yet. In this section,
results from H.E.S.S. observations of three MQs conducted contemporaneously to
observations with the _RXTE_ satellite experiment are presented. All
observations were performed with the H.E.S.S. I array in 2004, aiming to cover
times of expected broadband flaring events. One target of the observations was
GRS 1915+105. Observations of this object were performed after a flux decrease
in the radio band at frequencies of approximately
$15\text{\,}\mathrm{GHz}\text{/}$ between April 28 and May 3, 2004 as shown in
Fig. 5.

Figure 5: Top panel: Integrated flux between $2\text{\,}\mathrm{keV}\text{/}$
and $20\text{\,}\mathrm{keV}\text{/}$ as a function of time as measured by
_RXTE_. Central panel: Hardness, defined as the ratio of the fluxes in the
higher and lower energy band of the X-ray flux as indicated on the ordinate as
a function of time. Bottom panel: Radio flux at
$15\text{\,}\mathrm{GHz}\text{/}$ as a function of time as measured by the
Ryle experiment. Colours denote the times during which H.E.S.S. observations
were performed. Adapted from Abdalla  _et al._ [14].

Cir X-1 was observed around the time of periastron between June 18 and June
20. V4641 Sgr was observed after a rapid increase of the intensity was
measured in the radio, optical and X-ray domains between July 7 and July 9.
None of the observations yielded the detection of a significant flux in the
VHE gamma-ray regime. Thus flux upper limits at the $99\text{\,}\mathrm{\char
37\relax}\text{/}$ confidence level were calculated after an analysis of the
data with the Hillas reconstruction algorithm. The results are displayed in
Tab. 1.

For more information, the reader is referred to Abdalla  _et al._ [14].

Table 1: Flux upper limits at the $99\text{\,}\mathrm{\char 37\relax}\text{/}$ confidence level. Limits were calculated assuming a power-law shape of the spectrum with an index of 2.6 or 2.0 for the std and hard cut configurations defined in Aharonian  _et al._ [2], respectively. The live time $T_{L}$, the energy threshold $E_{\textrm{thr}}$ and the flux $I$ are listed in the table. Adapted from Abdalla  _et al._ [14]. |  1cbCuts |  1cb$T_{L}/[$\mathrm{h}\text{/}$]$ |  1cb$E_{\textrm{thr}}/[$\mathrm{G}\mathrm{e}\mathrm{V}$]$ |  1cb$I(>E_{\textrm{thr}})/[$\mathrm{p}\mathrm{h}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}\mathrm{/}\mathrm{s}$]$
---|---|---|---|---
GRS 1915+105 | std | 6.9 | 562 | $<$7.338\text{\times}{10}^{-13}$$
| hard | 6.9 | 1101 | $<$1.059\text{\times}{10}^{-13}$$
Circinus X-1 | std | 5.4 | 562 | $<$1.172\text{\times}{10}^{-12}$$
| hard | 5.4 | 1101 | $<$4.155\text{\times}{10}^{-13}$$
V4641 Sgr | std | 1.8 | 237 | $<$4.477\text{\times}{10}^{-12}$$
| hard | 1.8 | 422 | $<$4.795\text{\times}{10}^{-13}$$

##  6 Summary

In these proceedings, results from observations of binary systems with the
H.E.S.S. telescopes have been presented. The gamma-ray binary PSR B1259-63/LS
2883 was observed around the 2014 periastron, exhibiting a high flux state at
the time of the $\mathrm{GeV}\text{/}$ flare. A high flux state before first
disk crossing is currently under investigation. There are no clear signatures
for inter-orbital variation of the spectral index or the flux level at VHEs.
The spectrum could be extended down to an energy of
$200\text{\,}\mathrm{GeV}\text{/}$. In case of LS 5039, results from ten years
of observations with both the H.E.S.S.-I and H.E.S.S.-II arrays have been
presented. Significant spectral variation is observed between orbital phases
corresponding to the inferior and superior conjunctions, respectively. The
phase-averaged SEDs obtained from analyses of both H.E.S.S.-I and H.E.S.S.-II
data agree well with each other and also with the phase-averaged SED resulting
from a re-analysis of Fermi-LAT data. The identification of the VHE gamma-ray
emitter HESS J1832-093 as a gamma-ray binary is discussed as well. This
identification is mostly based on features of the spectral energy
distribution, flux variability and the source morphology. Lastly, also
observations of MQs are presented. These observations yield no significant
detection of VHE gamma ray photons, but competitive upper limits for the
emission of such photons from microquasars have been computed.

##  7 Acknowledgements

The support of the Namibian authorities and of the University of Namibia in
facilitating the construction and operation of H.E.S.S. is gratefully
acknowledged, as is the support by the German Ministry for Education and
Research (BMBF), the Max Planck Society, the German Research Foundation (DFG),
the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle
Interdisciplinary Programme of the CNRS, the U.K. Science and Technology
Facilities Council (STFC), the IPNP of the Charles University, the Czech
Science Foundation, the Polish Ministry of Science and Higher Education, the
South African Department of Science and Technology and National Research
Foundation, the University of Namibia, the Innsbruck University, the Austrian
Science Fund (FWF), and the Austrian Federal Ministry for Science, Research
and Economy, and by the University of Adelaide and the Australian Research
Council. We appreciate the excellent work of the technical support staff in
Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia
in the construction and operation of the equipment. This work benefited from
services provided by the H.E.S.S. Virtual Organisation, supported by the
national resource providers of the EGI Federation.

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