# Asymptotically (A)dS dilaton black holes with nonlinear electrodynamics

S<EMAIL_ADDRESS>and A.
<EMAIL_ADDRESS>1 Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran

###### Abstract

It is well-known that with an appropriate combination of three Liouville-type
dilaton potentials, one can construct charged dilaton black holes in an
(anti)-de Sitter [(A)dS] spaces in the presence of linear Maxwell field.
However, asymptotically (A)dS dilaton black holes coupled to nonlinear gauge
field have not been found. In this paper, we construct, for the first time,
three new classes of dilaton black hole solutions in the presence of three
types of nonlinear electrodynamics, namely Born-Infeld, Logarithmic and
Exponential nonlinear electrodynamics. All these solutions are asymptotically
(A)dS and in the linear regime reduce to the Einstein-Maxwell-dilaton black
holes in AdS spaces. We investigate physical properties and the causal
structure, as well as asymptotic behavior of the obtained solutions, and show
that depending on the values of the metric parameters, the singularity can be
covered by various horizons. Interestingly enough, we find that the coupling
of dilaton field and nonlinear gauge field in the background of (A)dS spaces
leads to a strange behaviour for the electric field. We observe that the
electric field is zero at singularity and increases smoothly until reaches a
maximum value, then it decreases smoothly until goes to zero as
$r\rightarrow\infty$. The maximum value of the electric field increases with
increasing the nonlinear parameter $\beta$ or decreasing the dilaton coupling
$\alpha$ and is shifted to the singularity in the absence of either dilaton
field ($\alpha=0$) or nonlinear gauge field ($\beta\rightarrow\infty$).

PACS numbers: 04.70.Bw, 04.20.Ha, 04.20.Jb

##  I Introduction

The investigations on the black holes solutions in the background of AdS
spacetimes have got renewed attentions in the two past decades. This is mostly
due to the Maldacena’s conjecture which corresponds a gravity theory in an AdS
space with a conformal field theory (CFT) living on the boundary of this
space, known as AdS/CFT correspondence maldacena . Based on this duality,
thermodynamics of black holes in $n$-dimensional AdS space can be identified
with that of a certain dual CFT in the high temperature limit on the
$(n-1)$-dimensional boundary of this spacetime. In fact, the AdS/CFT
correspondence provides an ideal tool to study strongly coupled field
theories. For example, many authors have investigated condensed matter systems
such as superconductors via this correspondence SC . Moreover, it was shown
that a relevance exists between thermodynamic variables and the stress-energy
tensor of large, rotating black holes and fluids on its conformal boundary
Bhattacharyya . This fact further motivates theoretical physicists to study
black holes in the background of (A)dS spaces.

The motivation of perusing dilaton field comes from the fact that this field
appears in the low energy limit of string theory where the Einstein action is
supplemented with other fields like axion, gauge fields and scalar dilaton
field. The dilaton field changes the causal structure of the black hole and
leads to the curvature singularities at finite radii. In the absence of
dilaton potential, exact solutions of charged dilaton black holes have been
constructed by many authors CDB1 ; CDB2 . These black holes are all
asymptotically flat. It was shown that the presence of the dilaton potential,
which can be regarded as the generalization of the cosmological constant, can
change the asymptotic behavior of the solutions to be neither flat nor (A)dS.
Indeed, it was shown that, no dilaton (A)dS black hole solution exists with
the presence of only one Liouville-type dilaton potential MW . In the presence
of one or two Liouville-type potential, black hole spacetimes which are
neither asymptotically flat nor (A)dS have been investigated (see e.g. CHM ;
Cai1 ; Clem ; Shey0 ; DHSR ). The studies where also generalized to dilaton
black holes with nonlinear electrodynamics. Physical properties,
thermodynamics and thermal stability of the dilaton black objects in the
presence of Born-Infeld (BI) nonlinear electrodynamics have been investigated
Tam1 ; Tam2 ; YI ; yaz ; Clement ; yazad ; SRM ; Shey1 ; Shey2 . When the
gauge field is in the forms of Exponential, Logarithmic and Power-Maxwell
nonlinear electrodynamics, dilaton black holes which are neither
asymptotically flat nor (A)dS have been constructed in ShKa ; ShN ; Kord .

It is important to note that all these solutions (MW ; CHM ; Cai1 ; Clem ;
Shey0 ; Tam1 ; Tam2 ; YI ; yaz ; Clement ; yazad ; SRM ; Shey1 ; Shey2 ; DHSR
; ShKa ; Kord ; ShN ), however, are neither asymptotically flat nor (A)dS. A
question then arises: Is it possible to construct exact analytical black hole
solutions of dilaton gravity in the background of (A)dS spaces? Gao and Zhang
were the first who answered this question and derived exact analytical black
hole solutions of Einstein-Maxwell-dilaton gravity in the background of (A)dS
spacetime gao1 ; gao2 . For these purpose, they combined three Liouville-type
dilaton potentials, in an appropriate way. Following gao1 ; gao2 , a lot of
investigations have been done on asymptotically AdS dilaton black
holes/branes. A class of charged rotating dilaton black string solutions in
the background of AdS spaces was presented in SheAdS . Thermodynamic
instability of a class of $(n+1)$-dimensional charged dilatonic spherically
symmetric black holes in the background of AdS universe have been explored in
SheAdS2 . Topological AdS black branes of $(n+1)$-dimensional Einstein-
Maxwell-dilaton theory and their thermodynamic properties were also
investigated in SheAdS3 . Three-dimensional static and circularly symmetric
solution of the Einstein-Born-Infeld-dilaton system with (A)dS asymptotic was
constructed in Ida . Other studies on the charged dilaton black holes in the
background of AdS spaces were carried out in Other .

In this paper, we would like to continue the investigations on the
asymptotically AdS dilaton black holes by considering the gauge field in the
form of nonlinear electrodynamics. As far as we know, exact analytical (A)dS
dilaton black holes coupled to nonlinear electrodynamics in $n\geq 4$
spacetime dimensions have not been constructed. We shall consider three kind
of nonlinear electrodynamics as our gauge field, namely, Born-Infeld,
Exponential and Logarithmic nonlinear electrodynamics. Since the asymptotic
behavior of these three Lagrangian are the same as BI case, they are well-
known as the BI-type nonlinear electrodynamics BI ; Soleng ; Hendi . With the
combination of three Liouville type dilaton potentials, we are able to
construct three new classes of dilaton black hole solutions corresponding to
three type of BI-type nonlinear electrodynamics.

The organization of this paper is as follows. In the next section, we review
the structure of the Einstein-dilaton gravity coupled to nonlinear
electrodynamics and introduce the Lagrangian of the BI-type nonlinear
electrodynamics coupled to the dilaton field. Sections III, IV and V are
dedicated to constructing asymptotically (A)dS dilaton black hole solutions in
the presence of Born-Infeld (BI), Exponential nonlinear (EN) and Logarithmic
nonlinear (LN) electrodynamics, respectively. In section VI we analyze and
discuss the physical properties of the obtained solutions. We conclude the
paper with closing remarks in section VII.

##  II Action and Lagrangian

We consider an action in which gravity is coupled to nonlinear electrodynamic
and dilaton field as

$S=\frac{1}{16\pi}\int{d^{4}x\sqrt{-g}\left(\mathcal{R}\text{
}-2g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-V(\Phi)+L(F,\Phi)\right)},$
(1)

where we display the Ricci scalar curvature and dilaton field with
$\mathcal{R}$ and $\Phi$, respectively. $V(\Phi)$ is the potential for $\Phi$
and $L(F,\Phi)$ is the Lagrangian of nonlinear electrodynamics coupled to the
dilaton field given by Shey2 ; ShKa ; ShN

$L(F,\Phi)=\left\\{\begin{array}[]{ll}$$4\beta^{2}e^{2\alpha\Phi}\left(1-\sqrt{1+\frac{e^{-4\alpha\Phi}F^{2}}{2\beta^{2}}}\right)\quad\quad\quad\quad$$\rm
BID,&\\\ &\\\
$$4\beta^{2}e^{2\alpha\Phi}\left[\exp\left(-\frac{e^{-4\alpha\Phi}F^{2}}{4\beta^{2}}\right)-1\right],\quad\quad\quad$$\rm
END,&\\\ &\\\
$$-8\beta^{2}e^{2\alpha\Phi}\ln\left(1+\frac{e^{-4\alpha\Phi}F^{2}}{8\beta^{2}}\right),\quad~{}\quad\quad\quad$$\rm
LND,\end{array}\right.$ (2)

The parameter $\beta$ represents the strength of the electromagnetic field and
has the dimension of mass. In fact, $\beta$ determines the strength of the
nonlinearity of the electrodynamics. In the limit of large $\beta$
($\beta\rightarrow\infty$), the systems goes to the linear regime. This
implies that the nonlinearity of the theory disappears and the nonlinear
electrodynamic theory reduces to the linear Maxwell electrodynamics. On the
other hand, as $\beta$ decreases ($\beta\rightarrow 0$), the system tends to
the strong nonlinear regime of the electromagnetic. Here,
$F^{2}=F_{\mu\nu}F^{\mu\nu}$, with
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor. The constant $\alpha$ illustrates the strength
of coupling between the scalar and electromagnetic fields. In the limit of
large $\beta$, all these Lagrangian have similar expansion, namely

$\displaystyle L_{{}_{\rm
BID}}(F,\Phi)=-e^{-2\alpha\Phi}F^{2}+\frac{e^{-6\alpha\Phi}F^{4}}{8\beta^{2}}-\frac{e^{-10\alpha\Phi}F^{6}}{32\beta^{4}}+O\left(\frac{1}{\beta^{6}}\right),$
$\displaystyle L_{{}_{\rm
END}}(F,\Phi)=-e^{-2\alpha\Phi}F^{2}+\frac{e^{-6\alpha\Phi}F^{4}}{8\beta^{2}}-\frac{e^{-10\alpha\Phi}F^{6}}{96\beta^{4}}+O\left(\frac{1}{\beta^{6}}\right),$
$\displaystyle L_{{}_{\rm
LND}}(F,\Phi)=-e^{-2\alpha\Phi}F^{2}+\frac{e^{-6\alpha\Phi}F^{4}}{16\beta^{2}}-\frac{e^{-10\alpha\Phi}F^{6}}{192\beta^{4}}+O\left(\frac{1}{\beta^{6}}\right).$
(3)

For $\beta\rightarrow\infty$, all three BI-type Lagrangian reduce to the
standard linear Maxwell Lagrangian coupled to the dilaton field, namely,
$L=-e^{-2\alpha\Phi}F^{2}$ CHM . It is convenient to use the following
simplification,

$\displaystyle L_{{}_{\rm
BID}}(F,\Phi)=4\beta^{2}e^{2\alpha\Phi}\mathcal{L}(Y),~{}~{}\qquad\mathcal{L}(Y)=1-\sqrt{1+Y},\qquad~{}~{}Y=\frac{e^{-4\alpha\Phi}F^{2}}{2\beta^{2}},$
(4) $\displaystyle L_{{}_{\rm
END}}(F,\Phi)=4\beta^{2}e^{2\alpha\Phi}\mathcal{L}(Y),\qquad~{}~{}\mathcal{L}(Y)=\exp(-Y)-1,\qquad
Y=\frac{e^{-4\alpha\Phi}F^{2}}{4\beta^{2}},$ (5) $\displaystyle L_{{}_{\rm
LND}}(F,\Phi)=-8\beta^{2}e^{2\alpha\Phi}\mathcal{L}(Y),\qquad\mathcal{L}(Y)=\ln\left(1+Y\right),~{}~{}\qquad
Y=\frac{e^{-4\alpha\Phi}F^{2}}{8\beta^{2}}.$ (6)

In order to construct (A)dS dilaton black hole solutions, we examine the
potential in the form gao1

$V(\Phi)={\frac{2\Lambda}{3\left({\alpha}^{2}+1\right)^{2}}\left[8\,{\alpha}^{2}{{\rm
e}^{\Phi\left(\alpha\,-{{1}/{\alpha}}\right)}}+{\alpha}^{2}\left(3\,{\alpha}^{2}-1\right){{\rm
e}^{-\,{{2\Phi}/{\alpha}}}}-\left({\alpha}^{2}-3\right){{\rm
e}^{2\,\alpha\,\Phi}}\right]},$ (7)

where $\Lambda$ is the cosmological constant. It is clear the cosmological
constant is coupled to the dilaton in a very nontrivial way. The above
potential reduces to $V(\Phi=0)=2\Lambda$ in the absence of dilaton field
($\Phi=\alpha=0$). This type of potential can be obtained in the
cementification of the higher dimensional theory to the four dimensions,
including various super-gravity models Radu . In particular, by choosing
$\alpha=\pm 1,\pm\sqrt{1/3},\pm\sqrt{3}$, this potential becomes the SUSY
potential gao1 .

In order to construct static and spherically symmetric black holes, we assume
the following form for the metric

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}R^{2}(r)\Big{(}d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\Big{)},$
(8)

where $f(r)$ and $R(r)$ are functions of $r$ which should be determined. In
the next three sections, we try to construct asymptotically (A)dS dilaton
black holes in the presence of three nonlinear electrodynamics (2).

##  III BID black holes in (A)dS spacetime

In order to build asymptotically (A)dS-BID black hole, one should obtain
equations of motion by varying action (1) with respect to all fields by
considering the BID Lagrangian $L(F,\Phi)$ given in (4). The variation with
respect to the gravitational field $g_{\mu\nu}$, dilaton filed $\Phi$ and
gauge field $A_{\mu}$ yields the following field equations

${\cal
R}_{\mu\nu}^{{}^{BID}}=2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)-4e^{-2\alpha\Phi}\partial_{Y}{{\cal
L}}(Y)F_{\mu\eta}F_{\nu}^{\text{
}\eta}+2\beta^{2}e^{2\alpha\Phi}\left[2Y\partial_{Y}{{\cal L}}(Y)-{{\cal
L}}(Y)\right]g_{\mu\nu},$ (9) $\displaystyle\nabla^{2}\Phi$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\partial
V}{\partial\Phi}+2\alpha\beta^{2}e^{2\alpha\Phi}\left[2{Y}\partial_{Y}{{\cal
L}}(Y)-{\cal L}(Y)\right],$ (10)
$\nabla_{\mu}\left(e^{-2\alpha\Phi}\partial_{Y}{{\cal
L}}(Y)F^{\mu\nu}\right)=0.$ (11)

Note that in case of linear electrodynamics with ${\cal L}(Y)=-Y/2$, the
system of equations (9)-(11) reduce to the well-known equations of Einstein-
Maxwell-dilaton (EMd) gravity CHM . First of all, the electromagnetic filed
equation (11) can be integrated immediately, where all components of
$F_{\mu\nu}$ are zero expect $F_{tr}$,

$F_{tr}=X^{1/2}\beta\,{{\rm e}^{2\alpha\Phi}},$ (12)

where $q$ is the integration constant which is related to the electric charge
of the black hole, and

$X=\frac{q^{2}}{{\beta}^{2}\,r^{4}R^{4}(r)+{q}^{2}}$ (13)

By calculating the flux of the electromagnetic field at infinity (Gauss’s
theorem), the electric charge of the black hole is obtained as

$Q=\frac{1}{4\pi}\int
e^{-2\alpha\Phi}\,{}^{*}F\,d\Omega=\frac{q\omega}{4\pi}.$ (14)

where $\omega=4\pi$ represents the area of a unit $2$-sphere. It is worthwhile
to note that the electric field is finite at $r=0$. This is expected in BI
theories. It is interesting to consider three limits of Eq. (12). First, for
large $\beta$ (where the BI action reduces to Maxwell case) we have
$F_{tr}=qe^{2\alpha\Phi}/(rR)^{2}$ as presented in CHM . On the other hand, if
$\beta\rightarrow 0$ we get $F_{tr}=0$. Finally, in the absence of the dilaton
field ($\alpha=0$), it reduces to the electric field of BI theory Dey ; Cai2

$F_{tr}=\frac{\beta q}{\sqrt{\beta^{2}r^{4}+q^{2}}}.$ (15)

It is worth mentioning that $F_{tr}$ expression given in Eq. (12) for BID
theory in (A)dS space, is the same as the case of non-(A)dS space Shey2 .
However, as we shall see shortly, the explicit form of the electric field
obtained from these two theory have complectly different behaviour. This is
due to the fact that the behavior of the dilaton field $\Phi(r)$ differs in
these two cases. We will back to this issue at the end of this section.
Substituting metric (8) and the electromagnetic field (12) in the system of
equations (9) and (10), we can write the components of the fields equations as

$\displaystyle
Eq_{{}_{\Phi}}=\frac{rfR}{2}\Phi^{\prime\prime}+\left(rfR^{\prime}+\frac{R}{2}\left[2f+rf^{\prime}\right]\right)\phi^{\prime}-\frac{rR}{8}\frac{dV(\Phi)}{d\Phi}+r\alpha\beta^{2}R(1-\frac{1}{\sqrt{X+1}})e^{2\alpha\Phi(r)}=0,$
, (19) $\displaystyle
Eq_{tt}=rRf^{\prime\prime}+[2rR^{\prime}+2R]f^{\prime}+r\,R\,V(\Phi)-4rR\,\beta^{2}\,e^{2\alpha\Phi}\Big{(}1-\frac{r^{4}R^{4}\beta^{2}\,X}{q^{2}\sqrt{X+1}}\Big{)}=0,$
$\displaystyle
Eq_{rr}=4rf\,R^{\prime\prime}+[2rf^{\prime}+8f]R^{\prime}+\left(rf^{\prime\prime}+2f^{\prime}+r\Big{[}4f{\Phi^{\prime}}^{2}+V(\Phi)\Big{]}\right)$
$\displaystyle-4rR\,\beta^{2}\,e^{2\alpha\Phi}\Big{(}1-\frac{r^{4}R^{4}\beta^{2}\,X}{q^{2}\sqrt{X+1}}\Big{)}=0,$
$\displaystyle
Eq_{\varphi\varphi}=Eq_{\theta\theta}=\Big{[}rf^{\prime}+\frac{r^{2}}{2}V(\Phi)+f\Big{]}R^{2}+\Big{[}r^{2}f\,R^{\prime\prime}+\Big{(}rf^{\prime}+4f\Big{)}r\,R^{\prime}\Big{]}R+r^{2}f\,{R^{\prime}}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2r^{2}R^{2}\beta^{2}e^{2\alpha\Phi}\left(1-\frac{1}{\sqrt{X+1}}\right)-1=0,$

where the prime indicates derivative with respect to $r$. Eqs.(III)-(19)
contain three unknown functions $R(r)$, $\Phi(r)$ and $f(r)$. Subtracting Eq.
(19) from Eq. (19), we arrive at

$\displaystyle 2R^{\prime}+rR^{\prime\prime}+rR\Phi^{\prime 2}=0.$ (20)

Then we make the ansatz drans

$R(r)=e^{\alpha\Phi(r)}.$ (21)

Inserting ansatz (21) in Eq. (20), we can find the following equation for
$\Phi$

$r\alpha\Phi^{\prime\prime}+2\alpha\Phi^{\prime}+r(1+{\alpha}^{2})\Phi^{\prime
2}=0,$ (22)

It is easy to show that Eq. (22) has a solution in the form

$\Phi(r)={\frac{\alpha}{{\alpha}^{2}+1}\ln\left(1-{\frac{b}{r}}\right)},\\\ $
(23)

where $b$ is integration constant. From (23), we see that the solution only
exist for $r\geq b$. Besides, since $\alpha>0$, we have always $\Phi<0$, and
$\Phi\rightarrow-\infty$ as $r\rightarrow b$. Further, in the asymptotic
region where $r\rightarrow\infty$ or ($r\gg b$), we have $\Phi=0$, which is an
expected result, since we are looking for the asymptotically (A)dS solutions.
The fact that we have no dilaton field at the asymptotic region is in contrast
to the case of non asymptotically (A)dS dilaton black hole solutions Shey0 ,
where the dilaton field does not vanish as $r\rightarrow\infty$ Shey2 .
Indeed, as we argued already, the presence of the dilaton field in those cases
changes the asymptotic behaviour of the solution to be neither flat nor (A)dS.
Now, we use Eq. (19) to obtain the only remaining unknown function $f(r)$. We
find

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{\Lambda}{3}r^{2}\left(1-\frac{b}{r}\right)^{\gamma}+\frac{1}{r(\alpha^{2}+1)-b}\left(1-\frac{b}{r}\right)^{1-\gamma}$
(24)
$\displaystyle\times\Bigg{\\{}C_{1}+r(\alpha^{2}+1)+2\beta^{2}(\alpha^{2}+1)\int
r^{2}\left(1-\frac{b}{r}\right)^{2\gamma}\left(1-\sqrt{1+\eta}\right)dr\Bigg{\\}},$

where $C_{1}$ is a integration constant, $\gamma=2\alpha^{2}/(\alpha^{2}+1)$
and

$\eta=\frac{q^{2}}{\beta^{2}r^{4}}\left(1-\frac{b}{r}\right)^{-2\gamma}.$ (25)

In the absence of dilaton field $(\alpha=0)$ the metric function (24) reduces
to the metric function of BI-(A)dS black holes Dey ; Cai2

$f(r)=1+\frac{C_{1}}{r}-\frac{r^{2}\Lambda}{3}+\frac{2\beta^{2}r^{2}}{3}-\frac{2\beta}{r}\int\sqrt{\beta^{2}r^{4}+q^{2}}\,dr.$
(26)

For large $\beta$, one can expand (26), to arrive at

$f(r)=1+\frac{C_{1}}{r}+{\frac{{q}^{2}}{{r}^{2}}}-\frac{\Lambda
r^{2}}{3}-{\frac{{q}^{4}}{20\,{r}^{6}{\beta}^{2}}}+{\frac{{q}^{6}}{72\,{r}^{10}{\beta}^{4}}}+O\left(\frac{1}{{\beta}^{6}}\right).$
(27)

For $\beta\rightarrow\infty$, it has the form of charged black hole solutions
in (A)dS spaces Bril1 ; Cai3 . The last term in the right-hand side of the
above expression is the leading BI correction to the topological black hole in
the large $\beta$ limit. Next, we should check whether the obtained solutions
(23) and (24) are satisfied all components of the field equations or not. Our
calculations show that a condition to fully satisfy all equations of motion is
required. After a number of manipulations, we find that this condition induces
a relation between $q$, $\beta$, $C_{1}$ and the integral part of Eq. (24)in
the following form

$\displaystyle\frac{b+C_{1}}{2(\alpha^{2}+1)}-\frac{q^{2}}{r\,b\,\eta}\Big{(}r(\alpha^{2}+1)-b\Big{)}\left[1-\sqrt{1+\eta}\right]+\beta^{2}\int
r^{2}\left(1-\frac{b}{r}\right)^{2\gamma}\left[1-\sqrt{1+\eta}\right]dr=0.$
(28)

In the limiting case where $\beta\rightarrow\infty$, condition (28) reduce to
the corresponding case of AdS dilaton balck holes given in gao1 , namely

$\displaystyle
C_{1}=-\frac{b^{2}+(1+\alpha^{2})q^{2}}{b}+O\left(\frac{1}{\beta^{2}}\right),$
(29)

which is independent of $r$. Combining condition (28) with solution (24), we
find the final form of the solution which is fully satisfied all components of
the field equations as

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma}+\Bigg{\\{}1-\frac{2(\alpha^{2}+1){\beta}^{2}{r}^{3}}{b}\left[\sqrt{1+\eta}-1\right]\left(1-{\frac{b}{r}}\right)^{2\gamma}\Bigg{\\}}\left(1-{\frac{b}{r}}\right)^{1-\gamma},$
(30)

Combining $\Phi(r)$ in (23) and ansatz (21) with relation (12), we can obtain
the explicit form of the electric field for asymptotically (A)dS-BID black
holes as

$F_{tr}(r)=E(r)=\frac{q}{r^{2}}\frac{1}{\sqrt{1+\eta}},$ (31)

where $\eta$ is given by relation (25). Let us note that the electric field
given in (31) differs from electric field of BID black holes in non-(A)dS
spaces given in Shey2 . As we mentioned this is due to the difference in the
dilaton field of these two cases. Indeed, in case of asymptotically (A)dS
solution, the dilaton field vanishes as $r\rightarrow\infty$ as it can be seen
from relation (23). While for asymptotically non-(A)dS solution, the dilaton
field diverges in the large $r$ limit Shey2 . In the case of $\alpha=0$,
electric field will be

$E(r)\Big{|}_{\alpha=0}=\frac{q\beta}{\sqrt{q^{2}+\beta^{2}r^{4}}}=\frac{q}{r^{2}}\frac{1}{\sqrt{1+q^{2}/(\beta^{2}r^{4})}},$
(32)

which is finite when other metric parameters have finite values. Also at $r=0$
it does not diverges and has a finite value Dey . Expanding the electric field
for large $\beta$, we arrive at

$E(r)=\frac{q}{r^{2}}-\frac{q^{3}}{2r^{6}\beta^{2}}\left(1-\frac{b}{r}\right)^{-2\gamma}+O\left(\frac{1}{\beta^{4}}\right).$
(33)

In the limiting case where $\beta\rightarrow\infty$ or ($r\rightarrow\infty$),
the electric field does not depend on the dilaton field and reduces to the
electric field of dilaton black holes in (A)dS spaces SheAdS .

It is important to note that due to dilaton field (23), our solutions do not
exist for $0<r<b$ for $\alpha\neq 0$. In order to solve this problem we
exclude this region from the spacetime. One can use the following radial
coordinate

$r^{2}=\rho^{2}+b^{2}\Longrightarrow
dr^{2}=\frac{r^{2}}{r^{2}+b^{2}}\,d\rho^{2}.$ (34)

So, we can introduce our metric duo to new coordinate

$ds^{2}=-f(\rho)dt^{2}+\frac{\rho^{2}\,d\rho^{2}}{(\rho^{2}+b^{2})f(\rho)}+(\rho^{2}+b^{2})R^{2}(\rho)(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(35)

where $f(\rho)$ is given by Eq. (30) with replacing $r=\sqrt{\rho^{2}+b^{2}}$.
We find that the coupling of dilaton field and nonlinear BI field in the
background of (A)dS spaces leads to a strange behaviour for the electric
field.

(a) $\beta=0.5$

(b) $\alpha=0.6$

Figure 1: The behavior electric field $E(\rho)$ for BID black holes in (A)dS
spaces versus $\rho$ for $q=b=1$

As one can see from Fig.1, in the presence of dilaton field ($\alpha\neq 0$),
the electric field has zero value at $\rho=0$. For $\rho>0$, it increases
smoothly until reaches a maximum value, then it decreases smoothly until goes
to zero as $\rho\rightarrow\infty$. It is worth mentioning that the maximum of
the electric field increases with increasing $\beta$ or decreasing $\alpha$.
Besides, for $\alpha>0$, as $\beta\rightarrow\infty$, the maximum value of the
electric field goes to $\rho=0$ and we have no relative maximum for $\rho>0$.
In the absence of dilaton field ($\alpha=0$), the electric field has a finite
value at $\rho=0$ for a finite value of $\beta$ as one may see from Eq. (32)
that

$E(\rho)\Big{|}_{\alpha=0,\,\rho=0}=\frac{q\beta}{\sqrt{q^{2}+\beta^{2}b^{4}}}$
(36)

.

##  IV END black holes in AdS spacetime

In this section we examine exponential nonlinear electrodynamics coupled to
the dilaton field (END) with Lagrangian (5). In this case the gravitational
field equation is written as ShKa

$\displaystyle{\cal R}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)-2e^{-2\alpha\Phi}\partial_{Y}{{\cal
L}}(Y)F_{\mu\eta}F_{\nu}^{\text{
}\eta}+2\beta^{2}e^{2\alpha\Phi}\left[2Y\partial_{Y}{{\cal L}}(Y)-{{\cal
L}}(Y)\right]g_{\mu\nu},$ (37)

while the dilaton and electromagnetic field equations are still the same as
those in the BID case and given by Eqs. (10) and (11), respectively.
Integrating the electromagnetic field equation (11) for END Lagrangian (5),
the only nonzero component of $F_{\mu\nu}$ is obtained as

$F_{tr}=\frac{qe^{2\alpha\Phi(r)}}{r^{2}R^{2}(r)}\exp\left[-\frac{1}{2}L_{W}\left(\frac{q^{2}}{\beta^{2}r^{4}R^{4}(r)}\right)\right],$
(38)

where $L_{W}(x)=LambertW(x)$ is the Lambert function Lambert  which is a
special function with definition

$L_{W}(x)e^{L_{W}(x)}=x,$

where $x$ is a compelex number. It is clear that $L_{W}(0)=0$, $L_{W}(e)=1$
and $L_{W}(\infty)=\infty$. The Taylor expansion of the Lambert function is

$L_{W}(x)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}x^{n}=x-x^{2}+\frac{3}{2}x^{3}-\frac{8}{3}x^{4}+....$
(39)

The series expansion (39) converges for $|x|<1$. The derivative of $L_{W}(x)$
can be calculated as

$L_{W}^{\prime}(x)=\frac{dL_{W}(x)}{dx}=\frac{L_{W}(x)}{x(1+L_{W}(x))},\qquad\qquad\qquad\rm
for\ \ \emph{x}\neq 0.$

In order to get metric and dilaton functions for this type of black holes, we
use the previous ansatz (21) again. It is a matter of calculations to show
that the components of the field equations (10) and (37) can be written

$\displaystyle
Eq_{{}_{\Phi}}=\frac{rfR}{2}\Phi^{\prime\prime}+\left(rfR^{\prime}+\frac{R}{2}\left[2f+rf^{\prime}\right]\right)\phi^{\prime}-\frac{rR}{8}\frac{dV(\Phi)}{d\Phi}+r\alpha\beta^{2}R$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{q\big{(}1+2Z\big{)}}{r^{2}R^{2}\beta\sqrt{-2Z}}-1\right)e^{2\alpha\Phi(r)}=0,$
(40) $\displaystyle
Eq_{tt}=rRf^{\prime\prime}+(2rR^{\prime}+2R)f^{\prime}+r\,R\,V(\Phi)+4\beta^{2}rRe^{2\alpha\Phi}\left[1-\frac{q}{r^{2}R^{2}\beta}L_{W}\left(\frac{q^{2}}{\beta^{2}r^{4}R^{4}}\right)^{-\frac{1}{2}}\right]=0,$
(41) $\displaystyle
Eq_{rr}=4rR^{\prime\prime}f+\Big{[}2rf^{\prime}+8f\Big{]}R^{\prime}+\Big{[}rf^{\prime\prime}+2f^{\prime}+4rf{\Phi^{\prime}}^{2}+rV(\Phi)\Big{]}R+4rR\,\beta^{2}e^{2\alpha\Phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\left[1-\frac{q}{r^{2}R^{2}\beta}L_{W}\left(\frac{q^{2}}{\beta^{2}r^{4}R^{4}}\right)^{-\frac{1}{2}}\right]=0,$
(42) $\displaystyle
Eq_{\varphi\varphi}=Eq_{\theta\theta}=\Big{[}rf^{\prime}+\frac{r^{2}}{2}V(\Phi)+f\Big{]}R^{2}+\left[r^{2}fR^{\prime\prime}+\Big{(}rf^{\prime}+4f\Big{)}rR^{\prime}\right]R+r^{2}f{R^{\prime}}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2r^{2}R^{2}\beta^{2}e^{2\alpha\Phi}\Big{[}e^{-Z}(2Z+1)-1\Big{]}+1=0,$
(43)

where

$Z=-\frac{1}{2}L_{W}\left(\frac{q^{2}}{\beta^{2}r^{4}R^{4}}\right).$ (44)

Subtracting Eq. (41) from Eq. (42), and using the ansatz (21) we arrive at Eq.
(22). Thus the dilaton field is again given by Eq. (23). This implies that the
nonlinearity does not affect the dilaton field. Having ansatz (21) and the
dilaton field (23) in hand, we can obtain the metric function $f(r)$ by
solving Eq. (43). We find

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma}+\frac{1}{r(\alpha^{2}+1)-b}\left(1-\frac{b}{r}\right)^{1-\gamma}$
$\displaystyle\times\left[C_{2}+r(\alpha^{2}+1)-2\beta^{2}(\alpha^{2}+1)\int
r^{2}\left(1-\frac{b}{r}\right)^{2\gamma}\left[1+\sqrt{\eta}\left(\sqrt{L_{W}(\eta)}-\frac{1}{\sqrt{L_{W}(\eta)}}\right)\right]dr\right],$

where $C_{2}$ is an integration constant related to the mass of the black
hole. In the absence of dilaton field ($\alpha=0$), the above solution reduces
to

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1+\frac{C_{2}}{r}-\frac{\Lambda
r^{2}}{3}-\frac{2\beta^{2}r^{2}}{3}-\frac{2\beta
q}{r}\bigintsss\Bigg{[}\sqrt{L_{W}\left(\frac{q^{2}}{r^{4}\beta^{2}}\right)}-\frac{1}{\sqrt{L_{W}\left(\frac{q^{2}}{r^{4}\beta^{2}}\right)}}\Bigg{]}\,dr,$
(46)

which is the metric AdS black holes in in the presence of EN electrodynamics
AsympDrHendi . The series expansion of (46) for large $\beta$ becomes

$f(r)=1+\frac{C_{2}}{r}+{\frac{{q}^{2}}{{r}^{2}}}-\frac{\Lambda
r^{2}}{3}-{\frac{{q}^{4}}{20\,{r}^{6}{\beta}^{2}}}+{\frac{5{q}^{6}}{216\,{r}^{10}{\beta}^{4}}}+O\left(\frac{1}{{\beta}^{6}}\right).$
(47)

For $\beta\rightarrow\infty$, the above solution reduces to RN-AdS black hole.
Now, we substitute the metric function (IV) in the all components of the field
equations (IV)-(43). We find that in order to fully satisfy all field
equations, we should have the following conditions for $q$, $\beta$ and
$C_{{}_{2}}$

$\displaystyle\frac{b+C_{2}}{2(\alpha^{2}+1)}+\frac{q^{2}}{r\,b\,\eta}\left[1+\sqrt{\eta}\left(\sqrt{L_{W}(\eta)}-\frac{1}{\sqrt{L_{W}(\eta)}}\right)\right]\Big{(}r(\alpha^{2}+1)-b\Big{)}$
$\displaystyle-\beta^{2}\int
r^{2}\left(1-\frac{b}{r}\right)^{2\gamma}\left[1+\sqrt{\eta}\left(\sqrt{L_{W}(\eta)}-\frac{1}{\sqrt{L_{W}(\eta)}}\right)\right]dr=0.$
(48)

It is interesting to check the limiting case of (IV) for large $\beta$,

$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\frac{b^{2}+(1+\alpha^{2})q^{2}}{b}+O\left(\frac{1}{\beta^{2}}\right),$
(49)

which is again the condition in case of AdS dilaton balck holes coupled to the
linear Maxwell gauge field gao1 . Combining condition (IV) with solution (IV),
we find the final form of the metric function, which fully satisfy the field
equations, for END black holes in the background of (A)dS spaces as

$\displaystyle f(r)=-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma}+\Bigg{\\{}1-\frac{2q^{2}(\alpha^{2}+1)}{r\,b\,\eta}\left[1+\sqrt{\eta}\Bigg{(}\sqrt{L_{W}(\eta)}-\frac{1}{\sqrt{L_{W}(\eta)}}\Bigg{)}\right]\Bigg{\\}}\left(1-{\frac{b}{r}}\right)^{1-\gamma}$
(50)

In order to calculate the explicit form of the electric field for END-AdS
black hole, we substitute the dilaton field (23) in the electromagnetics
tensor (38). We find

$E(r)=\frac{q}{r^{2}}\sqrt{\frac{L_{W}(\eta)}{\eta}}.$ (51)

where $\eta$ is given by (25). When $\alpha=0$, the electric field of black
holes in the presence of EN electrodynamics is restored AsympDrHendi . It is
worthy to note that the electric field (51) differs from the case of non-
asymptotically (A)dS dilaton black holes in the presence of EN electrodynamics
ShKa . Also, the expansion of Eq. (51) for large $\beta$, yields

$E(r)=\frac{q}{r^{2}}-\frac{q^{3}}{2r^{6}\beta^{2}}\left(1-\frac{b}{r}\right)^{-2\gamma}+O\left(\frac{1}{\beta^{4}}\right).$
(52)

when $\beta\rightarrow\infty$, the electric field does not depend on the
dilaton field and reduces to the electric field of AdS dilaton black hole
SheAdS .

(a) $\beta=0.5$

(b) $\alpha=0.6$

Figure 2: The behavior of electric field $E(\rho)$ for END black holes in
(A)dS space versus $\rho$ for $q=b=1$

In order to have better understanding on the behavior of the electric field,
we plot Fig. 2. Again, our solution does not exist in the range $0<r<b$, so we
use the new radial coordinate (34) to exclude this region from the spacetime.
In Fig. 2, $E(\rho)$ is plotted versus $\rho$ for different values of $\alpha$
and $\beta$. From this figure we see a new behaviour for the electric field of
END black holes in AdS spaces. As one can see the electric field vanishes at
$\rho=0$ and has a maximum value at some $\rho=\rho_{0}>0$. The maximum point
of the electric field depends on the parameters. Indeed, for a fixed value of
$\beta$ the maximum of electric field increase as $\alpha$ decrease (fig.
2(a)). Also, in the presence of the dilaton field ($\alpha>0$), by increasing
the nonlinear parameter $\beta$, this maximum increases, while its position
shifted to the point $\rho_{0}\rightarrow 0$ (fig. 2(b)).

##  V LND black holes in AdS spacetime

In this section we would like to construct black hole solution for Logarithmic
nonlinear electrodynamics coupled to dilaton field in the background of (A)dS
spaces. Variation with respect to both gravitational and dilaton fields yield
the following field equations ShN

$\displaystyle{\cal R}_{\mu\nu}^{LND}$ $\displaystyle=$ $\displaystyle
2\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)+2e^{-2\alpha\Phi}\partial_{Y}{{\cal
L}}(Y)F_{\mu\eta}F_{\nu}^{\text{
}\eta}-4\beta^{2}e^{2\alpha\Phi}\left[2Y\partial_{Y}{{\cal L}}(Y)-{{\cal
L}}(Y)\right]g_{\mu\nu},\qquad$ (53) $\displaystyle\nabla^{2}\Phi$
$\displaystyle=$ $\displaystyle\frac{1}{4}\frac{\partial
V}{\partial\Phi}-4\alpha\beta^{2}e^{2\alpha\Phi}\left[2{Y}\partial_{Y}{{\cal
L}}(Y)-{\cal L}(Y)\right].$ (54)

Note that the equation of motion for the electrodynamics still obeys Eq. (11)
with Lagrangian given in (6), and considering metric (8), one can show that it
has the following solution

$F_{tr}=\frac{2\,q\,e^{2\alpha\Phi(r)}}{r^{2}R^{2}(r)}\left(1+\sqrt{1+\frac{q^{2}}{\beta^{2}\,r^{4}\,R^{4}(r)}}\right)^{-1}.$
(55)

Substituting the metric (8) and the electric field (55) in the field equations
(53) and (54), we can write these equations as

$\displaystyle
Eq_{{}_{\Phi}}=\frac{rfR}{2}\Phi^{\prime\prime}+\left(rfR^{\prime}+\frac{R}{2}\left[2f+rf^{\prime}\right]\right)\phi^{\prime}-\frac{rR}{8}\frac{dV(\Phi)}{d\Phi}-2r\alpha\beta^{2}R$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigg{\\{}\ln\Big{(}1+W\Big{)}-\left(1+\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}\right)W\Bigg{\\}}e^{2\alpha\Phi(r)}=0$
(56) $\displaystyle
Eq_{tt}=rRf^{\prime\prime}+\Big{[}2rR^{\prime}+2R\Big{]}f^{\prime}+r\,R\,V(\Phi)+8\beta^{2}rRe^{2\alpha\Phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigg{\\{}\ln(W+1)-2\left(\frac{\frac{W\beta^{2}r^{4}R^{4}}{1+W}\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}+\frac{q^{2}}{2}}{\beta^{2}r^{4}R^{4}\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}+\frac{q^{2}}{2}}\right)\Bigg{\\}}=0,$
(57) $\displaystyle
Eq_{rr}=4rR^{\prime\prime}f+\Big{[}2rf^{\prime}+8f\Big{]}R^{\prime}+\Big{[}rf^{\prime\prime}+2f^{\prime}+4rf{\Phi^{\prime}}^{2}+rV(\Phi)\Big{]}R-16\beta^{2}rRe^{2\alpha\Phi}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigg{\\{}-\frac{1}{2}\ln(W+1)+\left(\frac{\frac{W\beta^{2}r^{4}R^{4}}{1+W}\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}+\frac{q^{2}}{2}}{\beta^{2}r^{4}R^{4}\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}+\frac{q^{2}}{2}}\right)\Bigg{\\}}=0,$
(58) $\displaystyle
Eq_{\varphi\varphi}=Eq_{\theta\theta}=\Big{[}rf^{\prime}+\frac{r^{2}}{2}V(\Phi)+f\Big{]}R^{2}+\left[r^{2}fR^{\prime\prime}+\Big{(}rf^{\prime}+4f\Big{)}rR^{\prime}\right]R$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r^{2}f{R^{\prime}}^{2}+4r^{2}R^{2}\beta^{2}e^{2\alpha\Phi}\left(\ln(W+1)-\frac{2W}{W+1}\right)-1=0,$
(59)

where

$W=-\frac{q^{2}}{r^{4}R^{4}\beta^{2}}\left(1+\sqrt{1+\frac{q^{2}}{r^{4}R^{4}\beta^{2}}}\right)^{-2}.$
(60)

Following our approach in the previous sections, we substrate Eq. (V) from Eq.
(V) and again arrive at equation (20). After using ansatz (21), we find Eq.
(22), which admits a solution in the form (23). Inserting ansatz (21), the
dilaton field (23) and the potential (7) in Eq. (V), we obtain the following
solution

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma}+\frac{1}{r(\alpha^{2}+1)-b}\left(1-\frac{b}{r}\right)^{1-\gamma}$
(61)
$\displaystyle\times\Bigg{\\{}C{{}_{3}}+r(\alpha^{2}+1)-4q^{2}(\alpha^{2}+1)\int\frac{1}{r^{2}\eta}\left[\frac{\eta}{1+\sqrt{1+\eta}}+\ln\left(\frac{2}{1+\sqrt{1+\eta}}\right)\right]dr\Bigg{\\}}.$

Here $\eta$ is again given by (25) and $C_{3}$ is a constant of integration.
When $\alpha=0$, the above solution reduces to

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1+\frac{C_{3}}{r}-\frac{\Lambda
r^{2}}{3}-\frac{4}{r}\bigintsss\Bigg{\\{}\frac{q^{2}}{r^{2}}\frac{1}{1+\sqrt{1+\frac{q^{2}}{r^{4}\beta^{2}}}}+r^{2}\beta^{2}\ln\Bigg{(}\frac{2}{1+\sqrt{1+\frac{q^{2}}{r^{4}\beta^{2}}}}\Bigg{)}\Bigg{\\}}\,dr,$
(62)

which is the metric of AdS black holes in in the presence of LN
electrodynamics AsympDrHendi . The series expansion of (62) for large $\beta$
becomes

$f(r)=1+\frac{C_{3}}{r}+{\frac{{q}^{2}}{{r}^{2}}}-\frac{\Lambda
r^{2}}{3}-{\frac{{q}^{4}}{40\,{r}^{6}{\beta}^{2}}}+{\frac{{q}^{6}}{216\,{r}^{10}{\beta}^{4}}}+O\left(\frac{1}{{\beta}^{6}}\right).$
(63)

In the liming case where $\beta\rightarrow\infty$, the above solution reduces
to RN-AdS black hole. Inserting the metric function (61) in the field Eqs.
(V)-(V), after a time-consuming calculations, we find a relation between
constants $q$, $\beta$, $C_{3}$ and the integration part of solution (61) as

$\displaystyle\frac{b+C_{3}}{4(\alpha^{2}+1)}+\frac{q^{2}}{r\,b\,\eta}\left[\frac{\eta}{1+\sqrt{1+\eta}}+\ln\left(\frac{2}{1+\sqrt{1+\eta}}\right)\right]\Big{(}r(\alpha^{2}+1)-b\Big{)}$
$\displaystyle-q^{2}\int\frac{1}{r^{2}\eta}\left[\frac{\eta}{1+\sqrt{1+\eta}}+\ln\left(\frac{2}{1+\sqrt{1+\eta}}\right)\right]dr=0,$
(64)

Again expanding the above condition for large $\beta$, we find

$\displaystyle
C_{3}=-\frac{b^{2}+(1+\alpha^{2})q^{2}}{b}+O\left(\frac{1}{\beta^{2}}\right),$
(65)

Combining condition (V) with solution (61), we arrive at the final form of the
metric function for LND black holes in (A)dS spaces as

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma}+\Bigg{\\{}1-\frac{4q^{2}(\alpha^{2}+1)}{r\,b\,\eta}\left[\frac{\eta}{1+\sqrt{1+\eta}}+\ln\left(\frac{2}{1+\sqrt{1+\eta}}\right)\right]\Bigg{\\}}\left(1-{\frac{b}{r}}\right)^{1-\gamma}.$
(66)

Now we back to the calculate the explicit form of the electric field. One can
show that the combination of dilaton field, Eq. (21) and Eq. (55) results the
electric field for this type of black hole as

$E(r)=\frac{2\,q}{r^{2}}\frac{1}{1+\sqrt{1+\eta}}.$ (67)

In the absence of dilaton field ($\alpha=0$), this expression for the electric
field reduces to one presented in AsympDrHendi . Let us note that in the large
limit of nonlinear parameter $\beta$, the electric field becomes

$E(r)=\frac{q}{r^{2}}+\frac{q^{3}}{4\,r^{6}\,\beta^{2}}\left(1-\frac{b}{r}\right)^{2\gamma}+O\left(\frac{1}{\beta^{4}}\right).$
(68)

It is clear that the first term in the above relation is the electric field of
EMd black holes in AdS spaces SheAdS  which is independent of dilaton coupling
$\alpha$, and the second term is the leading order correction term which
appears due to the nonlinearity of the electrodynamic field.

Let us note that at $\rho=0$ the electric field becomes zero like the two
previous nonlinear electrodynamics. The behavior of the electric field respect
to $\rho$ is plotted in Fig. 3. From this figure, we observe that in the
presence of the dilaton field, the behaviour of the electric field is similar
to the electric field of END and BID black holes in the previous sections.

(a) $\beta=0.5$

(b) $\alpha=0.6$

Figure 3: The behavior of $E(\rho)$ for LND black holes in (A)dS space versus
$\rho$ for $q=b=1$

##  VI Discussion

Now, we want to study the casual structure and physical properties of the
obtained solutions. Let us remember that the metric function of asymptotically
(Ad)S black holes in EMd gravity has the following form SheAdS

$\displaystyle f(r)_{\rm
EMd}=\left(1-\frac{c}{r}\right)\left(1-{\frac{b}{r}}\right)^{1-\gamma}-\frac{\Lambda
r^{2}}{3}\left(1-\frac{b}{r}\right)^{\gamma},$ (69)

where the two constants $c$ and $b$ are satisfied $q^{2}(\alpha^{2}+1)=bc$.
Expanding the obtained solutions in Eqs. (30), (50) and (66) for large
$\beta$, we find

$\displaystyle f(r)_{\rm BID}$ $\displaystyle=$ $\displaystyle f(r)_{\rm
END}=f(r)_{\rm
EMd}+\frac{q^{4}(\alpha^{2}+1)}{4\,br^{5}\beta^{2}}\left(1-{\frac{b}{r}}\right)^{1-3\gamma}+O\left(\frac{1}{\beta^{4}}\right)$
(70) $\displaystyle f(r)_{\rm LND}=f(r)_{\rm
EMd}+\frac{q^{4}(\alpha^{2}+1)}{8\,br^{5}\beta^{2}}\left(1-{\frac{b}{r}}\right)^{1-3\gamma}+O\left(\frac{1}{\beta^{4}}\right)$

It is obvious that for $r\rightarrow\infty$, the dominant term in all
solutions is $-{\Lambda r^{2}}/{3}$, which guarantees that the asymptotic
behavior of the metric function is (A)dS for ($\Lambda<0$) $\Lambda>0$. The
expansion of these solutions in the absence of dilaton field ($\alpha=0$) and
for $r\rightarrow\infty$ can be written as

$\displaystyle f(r)_{\rm BID}=f(r)_{\rm END}=-\frac{\Lambda
r^{2}}{3}+\left(1-{\frac{b}{r}}\right)\Bigg{\\{}1-\frac{c}{r}+\frac{q^{4}}{4br^{5}\beta^{2}}-\frac{q^{6}}{8br^{9}\beta^{4}}\Bigg{\\}}+O\left(\frac{1}{\beta^{6}}\right),$
$\displaystyle f(r)_{\rm LND}=-\frac{\Lambda
r^{2}}{3}+\left(1-{\frac{b}{r}}\right)\Bigg{\\{}1-\frac{c}{r}+\frac{q^{4}}{8br^{5}\beta^{2}}-\frac{q^{6}}{24br^{9}\beta^{4}}\Bigg{\\}}+O\left(\frac{1}{\beta^{6}}\right).$
(71)

Now, we are going to investigate the geometric nature of the solutions. The
existence of singularity and horizon ensure that solutions are intrinsically
black holes. To cover all the spacetime, we prefer to continue with the new
coordinate $\rho$. One can compute the Ricci and Kretschmann scalars as

$\displaystyle Ricci$ $\displaystyle=$
$\displaystyle-\left(1+\frac{b^{2}}{\rho^{2}}\right)f^{\prime\prime}+\Bigg{[}-4\rho(b^{2}+\rho^{2})R^{\prime}+(b^{2}-4\rho^{2})R\Bigg{]}\frac{f^{\prime}}{R\rho^{3}}+\frac{2}{R^{2}(b^{2}+\rho^{2})}$
$\displaystyle+\Bigg{[}-4\rho(b^{2}+\rho^{2})R^{\prime\prime}R-2\rho(b^{2}+\rho^{2}){R^{\prime}}^{2}+4(b^{2}-3\rho^{2})R^{\prime}R-\frac{2\rho^{3}R^{2}}{b^{2}+\rho^{2}}\Bigg{]}\frac{f}{R^{2}\rho^{3}}$
$\displaystyle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{(b^{2}+\rho^{2})^{2}}{\rho^{4}}{f^{\prime\prime}}^{2}+\Bigg{[}4(b^{2}+\rho^{2})^{2}{R^{\prime}}^{2}+8\rho(b^{2}+\rho^{2})RR^{\prime}+\frac{(b^{4}+4\rho^{4})R^{2}}{\rho^{2}}\Bigg{]}\frac{{f^{\prime}}^{2}}{\rho^{4}R^{2}}$
(72)
$\displaystyle+\Bigg{\\{}-\frac{2b^{2}(b^{2}+\rho^{2})f^{\prime\prime}}{\rho^{5}}+8\Big{[}\rho(b^{2}+\rho^{2})R^{\prime\prime}-(b^{2}-2\rho^{2})R^{\prime}\Big{]}\
\Big{(}(b^{2}+\rho^{2})R^{\prime}+\rho
R\Big{)}\frac{f}{\rho^{5}R^{2}}\Bigg{\\}}f^{\prime}$
$\displaystyle+\Bigg{\\{}\frac{4\rho(b^{2}+\rho^{2})^{2}{R^{\
\prime}}^{4}}{R}+16\rho^{3}(b^{2}+\rho^{2}){R^{\prime}}^{3}+\frac{16\rho^{5}R^{2}R^{\prime}}{(b^{2}+\rho^{2})}+\frac{4\rho^{6}R^{3}}{(b^{2}+\rho^{2})^{2}}$
$\displaystyle+8\Bigg{[}\rho^{2}(b^{2}+\rho^{2})^{2}{R^{\prime\prime}}^{2}-2\rho(b^{2}+\rho^{2})(b^{2}-2\rho^{2})R^{\prime}R^{\prime\prime}+(b^{4}-4b^{2}\rho^{2}+7\rho^{4}){R^{\prime}}^{2}\Bigg{]}R\Bigg{\\}}\frac{f^{2}}{\rho^{6}R^{3}}$
$\displaystyle+\left(-8(b^{2}+\rho^{2}){R^{\prime}}^{2}-16\rho
RR^{\prime}-\frac{8\rho^{2}R^{2}}{b^{2}+\rho^{2}}\right)\frac{f}{(b^{2}+\rho^{2})\rho^{2}R^{4}}+\frac{4}{(b^{2}+\rho^{2})^{2}R^{4}}$

where the prime indicates the derivative with respect to $\rho$. Also,
$R=R(\rho)$ and $f=f(\rho)$ where may find them as

$\displaystyle R(\rho)$ $\displaystyle=$
$\displaystyle\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{\frac{\gamma}{2}}$
(73) $\displaystyle f(\rho)_{BID}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{\gamma}\frac{(b^{2}+\rho^{2})\Lambda}{3}+\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{1-\gamma}$
(74)
$\displaystyle\times\Bigg{\\{}1-\frac{2\beta^{2}(\alpha^{2}+1)}{b}(b^{2}+\rho^{2})^{\frac{3}{2}}\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{2\gamma}\left[\sqrt{1+\eta_{\rho}}-1\right]\Bigg{\\}}$
$\displaystyle f(\rho)_{END}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{\gamma}\frac{(b^{2}+\rho^{2})\Lambda}{3}+\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{1-\gamma}$
(75)
$\displaystyle\times\Bigg{\\{}1-\frac{2q^{2}(\alpha^{2}+1)}{\,b\,\eta_{\rho}\sqrt{b^{2}+\rho^{2}}}\left[1+\sqrt{\eta_{\rho}}\Bigg{(}\sqrt{L_{W}(\eta_{\rho})}-\frac{1}{\sqrt{L_{W}(\eta_{\rho})}}\Bigg{)}\right]\Bigg{\\}}$
$\displaystyle f(\rho)_{LND}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{\gamma}\frac{(b^{2}+\rho^{2})\Lambda}{3}+\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{1-\gamma}$
(76)
$\displaystyle\times\Bigg{\\{}1-\frac{4q^{2}(\alpha^{2}+1)}{b\,\eta_{\rho}\sqrt{b^{2}+\rho^{2}}}\left[\frac{\eta_{\rho}}{1+\sqrt{1+\eta_{\rho}}}+\ln\left(\frac{2}{1+\sqrt{1+\eta_{\rho}}}\right)\right]\Bigg{\\}}$

where

$\eta_{\rho}=\frac{q^{2}}{\beta^{2}(b^{2}+\rho^{2})^{2}}\left(1-\frac{b}{\sqrt{b^{2}+\rho^{2}}}\right)^{-2\gamma}$
(77)

It is a matter of calculation to show that these scalars diverge in the
vicinity of the origin, namely

$\displaystyle\lim\limits_{\rho\rightarrow 0}Ricci$ $\displaystyle=$
$\displaystyle\infty,$ $\displaystyle\lim\limits_{\rho\rightarrow
0}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\infty.$

These confirm that the spacetime associated with the solutions has a
singularity at the origin ($\rho=0$). Also, it can be shown that in large
distances and for various values of $\alpha$, the Ricci and Kretschmann
scalars are $N\Lambda$, where $N$ is a positive number, so it guarantees that
the asymptotical behavior of these solutions is (A)dS for $\Lambda<0$
($\Lambda>0$). Now, we focus on the existence of horizon.

In order to find the location of all horizons one should solve
$f(\rho=\rho_{+})$=0. Due to the complexity of the obtained metric functions,
it is not possible to solve this equation analytically. To have better
understanding on the nature of horizons, we plot $f(\rho)$ versus $\rho$ for
different metric parameters.

(a) $f(\rho)_{{}_{\rm BID}}$ for $q=0.97$

(b) $f(\rho)_{{}_{\rm END}}$ for $q=0.87$

(c) $f(\rho)_{{}_{\rm LND}}$ for $q=0.81$

Figure 4: The behavior of $f(\rho)$ versus $\rho$ for $\beta=0.5$,
$\Lambda=-3$ and $b=1$.

(a) $f(\rho)_{{}_{\rm BID}}$ for $q=1.13$

(b) $f(\rho)_{{}_{\rm END}}$ for $q=1.064$

(c) $f(\rho)_{{}_{\rm LND}}$ for $q=1$

Figure 5: The behavior of $f(\rho)$ versus $\rho$ for $\alpha=1.1$,
$\Lambda=-3$ and $b=1$.

(a) $f(\rho)_{{}_{\rm BID}}$ for $\beta=0.47$

(b) $f(\rho)_{{}_{\rm END}}$ for $\beta=0.33$

(c) $f(\rho)_{{}_{\rm LND}}$ for $\beta=0.29$

Figure 6: The behavior of $f(\rho)$ versus $\rho$ for $\alpha=2$, $\Lambda=-3$
and $b=1$.

According to these figures, the existence of horizons and their numbers are
crucially depend on the values of the metric parameters. By suitable choices
of the different parameters, one can see that since $f(\rho)$ is positive near
the origin as well as large distances, the obtained solution may represent
black holes with two horizons, an extreme black hole or naked singularity
depending on the parameters. To be more precise, the effects of parameters are
plotted in different figures. Fig. 4 shows that by fixing other metric
parameters, increasing of $\alpha$ may increase the number of horizons in all
presented theories. From Fig. 5, one may realize that, when other metric
parameters are fixed, a direct relation between the increase of the nonlinear
parameter $\beta$ and the number of horizon stands for all three metric
functions. Also, by fixing other parameters, the same behavior is seen for
electric charge $q$ in Fig. 6. It is worth noting that the effect of changes
in metric parameters on the number of horizons of BID-(A)dS, END-(A)dS and
LND-(A)dS black holes is quite contrary to the their counterparts in non-(A)dS
spacetime. As mentioned in Shey2 ; ShKa ; ShN  for non-(A)dS dilaton black
holes with nonlinear electrodynamics, by fixing other metric parameters,
decreasing of the electric charge parameter $q$, or nonlinear parameter
$\beta$ or dilaton coupling constant $\alpha$ may increase the number of
horizon, but our solutions treat differently in this regard.

##  VII Closing remarks

Till now exact analytical solutions of Einstein-dilaton gravity in the
presence of nonlinear electrodynamics and in the background of (A)dS spaces
have not been constructed. In this paper, we constructed three classes of
asymptotically AdS dilaton black holes solutions in the presence of nonlinear
electrodynamics. We have considered three type of nonlinear electrodynamics,
namely BID, END and LND field. For this purpose, we utilized a combination of
three Liouville-type dilaton potential. We found the metric functions,
electric field and dilaton field and studied their behaviour for three class
of solutions. We observed that the nonlinearity of the gauge field do not
affect the dilaton field, and for $r\rightarrow\infty$
($\rho\rightarrow\infty$), the dilaton field goes to zero. We proved that our
solution has a singularity at $\rho=0$ ($r=b$), which depending on the value
of the metric function, it can be covered by the one or two horizon. Also, we
realized that the number of horizon increases by increasing nonlinear
parameter $\beta$, electric charge $q$ or dilaton coupling constant $\alpha$.
We studied the behavior of the electric field of all three nonlinear
electrodynamics. Interestingly, we found out that in the presence of the
dilaton field, the electric field has a strange behaviour. Indeed, it vanishes
at $\rho=0$ and increases until reaches a maximum value at $\rho=\rho_{0}$ and
goes to zero smoothly as $r\rightarrow\infty$ ($\rho\rightarrow\infty$). We
observed that the electric field is zero at singularity and increases smoothly
until reaches a maximum value, then it decreases smoothly until goes to zero
as $r\rightarrow\infty$ ($\rho\rightarrow\infty$). The maximum value of the
electric field increases with increasing the nonlinear parameter $\beta$ or
decreasing the dilaton coupling $\alpha$ and is shifted to the origin in the
absence of either dilaton field ($\alpha=0$) or nonlinear gauge field
($\beta\rightarrow\infty$). It seems that the coupling of the nonlinear gauge
field with the dilaton field in the background of (A)dS space leads to such an
unfamiliar behavior. However, the physical reason for this behaviour and its
origin deserves further investigations and we leave them for the future works.
It would be also interesting to carry out the investigation on the
thermodynamics of the obtained solutions. One may also interested in
generalizing these four dimensional static and spherically symmetric solutions
to the rotating and higher dimensional cases with other horizon topology. We
leave these issues for future works.

###### Acknowledgements.

We thank Shiraz University Research Council. This work has been supported
financially by Research Institute for Astronomy and Astrophysics of Maragha,
Iran.

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# Can fallback accretion on magnetar model power the X-ray flares
simultaneously observed with gamma-rays of Gamma-ray bursts?

Wen-Yuan Yu11affiliation: Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China<EMAIL_ADDRESS>, Hou-Jun Lü11affiliation: Guangxi Key
Laboratory for Relativistic Astrophysics, School of Physical Science and
Technology, Guangxi University, Nanning 530004, China<EMAIL_ADDRESS>, Xing
Yang11affiliation: Guangxi Key Laboratory for Relativistic Astrophysics,
School of Physical Science and Technology, Guangxi University, Nanning 530004,
China<EMAIL_ADDRESS>, Lin Lan 22affiliation: CAS Key Laboratory of Space
Astronomy and Technology, National Astronomical Observatories, Chinese Academy
of Sciences, China , Zhe Yang11affiliation: Guangxi Key Laboratory for
Relativistic Astrophysics, School of Physical Science and Technology, Guangxi
University, Nanning 530004, China<EMAIL_ADDRESS>
###### Abstract

The prompt emission, X-ray plateau, and X-ray flares of Gamma-ray bursts (GRB)
are thought to be from internal dissipation, and the magnetar as the central
engine with propeller fallback accretion is proposed to interpret the observed
phenomena of GRBs. In this paper, by systematically searching for X-ray
emission observed by Swift/Xry Telescope, we find that seven robust GRBs
include both X-ray flares and plateau emissions with measured redshift. More
interestingly, the X-ray flares/bumps for those seven GRBs are simultaneously
observed in the gamma-ray band. By adopting the propeller fallback accretion
model to fit the observed data, it is found that the free parameters of two
GRBs (140512A and 180329B) can be constrained very well, while in the other
five cases, more or less, they are not all sufficiently constrained. On the
other hand, this requires that the conversion efficiency of the propeller is
to be two or three times higher than that of the spindown dipole radiation of
the magnetar. If this is the case, it is contradictory to the expectation from
the propeller model: namely, a dirtier ejecta should be less efficient in
producing gamma-ray emissions. Our results hint that at least the magnetar
central engine with propeller fallback accretion model cannot interpret very
well both the GRB X-ray flares simultaneously observed in the gamma-ray band
and the X-ray flares of GRBs with a high Lorentz factor.

###### Subject headings:

Gamma-ray burst: general

##  1\. Introduction

The majority of long-duration gamma-ray bursts (LGRBs) are thought to
originate from the deaths of massive stars, excepting for several specific
events, i.e., GRB 060614 (Gehrels et al., 2006), GRB 191019A (Levan et al.,
2023), GRB 211211A (Rastinejad et al., 2022; Troja et al., 2022; Yang et al.,
2022), and GRB 211227A (Lü et al., 2022), and the $``$collapsar$"$ model has
been widely accepted as the standard paradigm for LGRBs (Woosley, 1993;
MacFadyen & Woosley, 1999). A black hole or a rapidly spinning, strongly
magnetized neutron star (called a millisecond magnetar) as the central engine
of a GRB may be formed after collapse (Usov, 1992; Thompson, 1994; Dai & Lu,
1998; Popham et al., 1999; Wheeler et al., 2000; Zhang & Mészáros, 2001;
Metzger et al., 2008; Lyons et al., 2010; Lei et al., 2013; Lü & Zhang, 2014;
Liu et al., 2017). It successfully launches the relativistic outflow, to break
out of the stellar envelope of the progenitor star, and produces prompt gamma-
ray emission, which passes through the photosphere, internal shocks, or
undergoes magnetic dissipation, becoming transparent (Meszaros & Rees, 1993;
Piran et al., 1993; Rees & Meszaros, 1994; Zhang & Yan, 2011; Zhang, 2018).
The broadband afterglow emission is attributed to synchrotron emission from
the external shock when the outflow is decelerated by a circumburst medium
(Mészáros & Rees, 1997; Sari et al., 1998; Zhang et al., 2016).

There is a small fraction of LGRBs in which the prompt emission shows two or
more sub-bursts of emission in the light curves, with quiescent times of up to
hundreds of seconds, detected by both Fermi/Gamma-ray Burst Monitor (GBM) and
Swift/Burst Alert Telescope (BAT) (Koshut et al., 1995; Lazzati, 2005; Burlon
et al., 2008; Bernardini et al., 2013; Hu et al., 2014; Lan et al., 2018). Lan
et al. (2018) investigated the spectral and temporal properties of two sub-
bursts of emission in the prompt emission, and they did not find any
statistically significant correlation in the duration and peak energy ($E_{\rm
p}$) of the two sub-bursts of emission. This suggests that those two or more
sub-bursts of emission of GRB are likely to be from the same origin.

Thanks to Swift, a multiwavelength GRB mission (Gehrels et al., 2004) has led
to great progress in understanding the nature of this phenomenon (review for
Zhang 2007). Especially, the prompt slewing capability of the onboard X-ray
Telescope (XRT; Burrows et al. 2004) has revealed the discovery of a canonical
X-ray light curve following the prompt emission, which shows successively four
power-law decay segments (i.e., an initial steep decay segment, a shallow
decay segment, a normal decay segment, and a post-jet-break phase), with
superimposed erratic flares (Nousek et al., 2006; O’Brien et al., 2006; Zhang
et al., 2006). Moreover, a small fraction of GRBs show an internal
plateau111An internal plateau with a near flat light curve, before rapidly
falling off with a decay index $\alpha>3$. Throughout the paper, the
convention $F_{\nu}\propto t^{-\alpha}\nu^{-\beta}$ is adopted. in the X-ray
light curve (Liang et al., 2007; Troja et al., 2007; Lyons et al., 2010;
Rowlinson et al., 2010, 2013; Lü & Zhang, 2014).

From the theoretical point of view, both the internal plateau and the X-ray
flares are inconsistent with any external shock model, but must be attributed
to the internal dissipation of a central engine wind. There are two forms of
continuous energy injection into the external forward shock. One involves
invoking a long-lasting central engine such as a spinning-down millisecond
magnetar (Dai & Lu, 1998; Zhang & Mészáros, 2001). The other one involves a
stratification of the ejecta Lorentz factor in an impulsively ejected fireball
(Rees & Mészáros, 1998; Sari & Mészáros, 2000; Uhm et al., 2012). The widely
discussed model for dealing with the energy injection process involves
assuming a millisecond magnetar as the central engine (Zhang & Mészáros, 2001;
Toma et al., 2007; Troja et al., 2007; Lü & Zhang, 2014; Rea et al., 2015;
Stratta et al., 2018; Fraija et al., 2021). Within the scenario of a magnetar
central engine, the energy injection to explain the plateau (or shallow decay)
phase is from the dipole radiation of the magnetar spindown. This has
previously been studied for LGRBs (e.g., Lyons et al. 2010; Dall’Osso et al.
2011; Lü & Zhang 2014), short GRBs (e.g., Rowlinson et al. 2013; Lü et al.
2015, 2017), as well as the extended emission of short GRBs (e.g., Gompertz et
al. 2013). These works assumed a constant rate of spindown and therefore a
constant level of dipole luminosity.

Another component, a flare/bump in the X-ray emission, is also of an important
clue to understanding the physical process in the framework of the magnetar
central engine. This has been discovered in a good fraction of GRBs (Barthelmy
et al., 2005; Chincarini et al., 2007; Falcone et al., 2007; Cusumano et al.,
2010; Margutti et al., 2010). In most cases, flares/bumps are superposed on
either the steep decay phase or the shallow decay phase (Nousek et al., 2006;
Zhang et al., 2006). Peng et al. (2014) found that a good fraction of GRB
X-ray flares, which were observed simultaneously by both BAT and XRT on board
the Swift mission, obeyed the same power-law spectral fit. By studying the
properties of both temporal and spectral behaviors, this suggests that the
X-ray flares are produced by late central engine activities, and may share the
same physical origin as the prompt emission of GRBs (Burrows et al., 2005; Yi
et al., 2016). If this is the case, a small fraction of the GRB X-ray flares
at least can be used to constrain the Lorentz factors, which range from a few
tens to hundreds (Jin et al., 2010; Yi et al., 2015). On the other hand,
several models have also been invoked to interpret the X-ray flares, such as
the tail of prompt emission with a high-latitude curvature effect (Liang et
al., 2006), delayed magnetic dissipation activity as the ejecta decelerates
(Giannios, 2006), and anisotropic emission in the blast wave comoving frame
(Beloborodov et al., 2011). Moreover, the magnetar central engine with the
delayed onset of a propeller regime, which accelerates local material via
magnetocentrifugal slinging, has also been proposed to explain the X-ray
flares in GRBs (Gibson et al., 2017, 2018). This model has also been invoked
to account for supernova explosions (Piro & Ott, 2011) and X-ray plateaus for
short GRBs (Gompertz et al., 2014). However, the main issue of the propeller
mechanism for explaining the prompt emission of GRBs is baryon loading. The
propeller mechanism expels a lot of unaccreted material from the system, and
the outflow must be very dirty and nonrelativistic if the material is expelled
in the funnel region. This means that no matter how large the power is, it
cannot power a GRB itself.

One interesting question is whether the propeller mechanism with fallback
accretion on the millisecond magnetar model can indeed interpret an X-ray
light curve of a GRB that is composed of both plateau emission and an X-ray
flare/bump. In this paper, by systematically searching for X-ray emission
observed by Swift/XRT, we find X-ray light curves of seven GRBs composed of
plateau emission and X-ray flares with measured redshift. The basic propeller
with fallback accretion model of the magnetar central engine is shown in
Section 2. In Section 3, we present the criteria for the sample selection, and
the fitting results for those seven GRBs with the propeller model, by adopting
the method of Markov Chain Monte Carlo (MCMC, Mackay 2003) are reported in
Section 4. Finally, we summarize our discussion and conclusion in Section 5.
Throughout the paper, a concordance cosmology with the parameters
$H_{0}=71~{}\rm km~{}s^{-1}~{}Mpc^{-1}$, $\Omega_{M}=0.30$, and
$\Omega_{\Lambda}=0.70$ is adopted.

##  2\. The magnetar propeller with fallback accretion model

Previously, the propeller with fallback accretion model of a magnetar central
engine was invoked to explain the short GRB with extended emission (Gompertz
et al., 2014; Gibson et al., 2017; Lan et al., 2020), X-ray plateau emission
and X-ray flares in long-duration GRBs (Gibson et al., 2018), as well as some
stripped-envelope Supernovae (Lin et al., 2021). In this section, we will
briefly introduce the model of magnetar propeller with fallback accretion.

The propeller regime is basically defined by the relationship between the
Alfvén radius ($r_{\rm m}$) and the co-rotation radius ($r_{\rm c}$).
Generally speaking, when $r_{\rm c}<r_{\rm m}$, the materials already within
$r_{\rm c}$ accrete to the surface of the magnetar, while the materials within
the range of $r_{\rm c}$ and $r_{\rm m}$ are propelled away at $r_{\rm m}$. If
the power of propeller is not strong enough for materials, it cannot reach the
potential well. Then, the materials will return to the disc without any
emission to be detected (called “propeller regime” Illarionov & Sunyaev 1975).
Based on the paper of Gibson et al. (2017), $r_{\rm m}$ and $r_{\rm c}$ can be
defined as follows,

$r_{\mathrm{m}}=\mu^{4/7}(GM)^{-1/7}\left(\dfrac{3M_{\mathrm{d}}}{t_{\nu}}\right)^{-2/7},$
(1) $r_{\mathrm{c}}=(GM/\omega^{2})^{1/3},$ (2)

where $G$ is the gravitational constant, $\mu=B_{\rm p}R^{3}$ is the magnetic
dipole moment, $B_{\rm p}$ is the surface magnetic field, $M$ and $R$ are the
mass and radius of magnetar, respectively. $t_{\nu}=R_{\rm d}/\alpha c_{\rm
s}$ is the viscous time-scale, $\alpha$ and $c_{\rm s}$ are the viscosity
prescription and the speed of sound in the disc, respectively. $\omega$ is the
angular frequency of the magnetar, $M_{\rm d}$ and $R_{\rm d}$ are the mass
and radius of disc, respectively. In our calculations, we adopt $\alpha=0.1$
and $c_{\rm s}=10^{7}\mathrm{cm}\cdot\mathrm{s}^{-1}$ (Gompertz et al., 2014).

At the Alfvén radius $r_{\rm m}$, the effect of materials from the accretion
is smaller than that of magnetic field of magnetar. While, at the co-rotation
radius $r_{\rm c}$, the rotation rate of matter in the disc is the same as
that of the stellar surface. Since $r_{\rm m}$ and $r_{\rm c}$ depend on the
disc mass and the frequency of the magnetar, respectively. The evolution of
disc mass and frequency with time can be expressed as follows (Gibson et al.,
2017),

$\dot{M}_{\mathrm{d}}=\dot{M}_{\mathrm{fb}}-\dot{M}_{\mathrm{prop}}-\dot{M}_{\mathrm{acc}},$
(3) $\dot{\omega}=\frac{N_{\mathrm{acc}}+N_{\mathrm{dip}}}{I}.$ (4)

Here, $\dot{M}_{\mathrm{fb}}$ is the fallback mass changed with time,
$\dot{M}_{\mathrm{prop}}$ and $\dot{M}_{\mathrm{acc}}$ are the mass lost via
the propeller mechanism and the accretion to the magnetar, respectively.
$N_{\mathrm{acc}}$ and $N_{\mathrm{dip}}$ are the accretion and dipole torques
acting on the magnetar, respectively. $I=0.35MR^{2}$ is the magnetar moment of
inertia.

Based on the description of Gibson et al. (2017), $\dot{M}_{\mathrm{fb}}$,
$\dot{M}_{\mathrm{prop}}$, and $\dot{M}_{\mathrm{acc}}$ can be defined as
follows,

$\dot{M}_{\rm{fb}}=\frac{M_{\rm{fb}}}{t_{\rm{fb}}}\left(\dfrac{t+t_{\rm{fb}}}{t_{\rm{fb}}}\right)^{-5/3},$
(5) $\dot{M}_{\rm{prop}}=\eta\left(\dfrac{M_{\rm{d}}}{t_{\nu}}\right),$ (6)
$\dot{M}_{\rm{acc}}=(1-\eta)\left(\dfrac{M_{\rm{d}}}{t_{\nu}}\right),$ (7)

where $M_{\rm{fb}}=\delta M_{\rm d,0}$ is the available fallback mass, $M_{\rm
d,0}$ is the initial mass of disc, and $\delta$ is the ratio between fallback
mass and initial disc mass. $t_{\rm{fb}}=\epsilon t_{\nu}$ is the fallback
time-scale, and $\epsilon$ is the ratio between fallback time-scale and
viscous time-scale. $\eta=\dfrac{1}{2}(1+\tanh[n(\Omega-1)])$ is the
efficiency of the propeller mechanism, and it related to the ”fastness
parameter”
$\Omega=\omega/(GM/r_{\mathrm{m}}^{3})^{1/2}=(r_{\mathrm{m}}/r_{\mathrm{c}})^{3/2}$.
In this work, we adopt $n=1$ to do the calculations.

For the dipole torque, we adopt the classical solution as given by (Shapiro &
Teukolsky, 1983) and (Piro & Ott, 2011),

$N_{\text{dip}}=-\dfrac{\mu^{2}\omega^{3}}{6c^{3}}.$ (8)

Moreover, $N_{\mathrm{acc}}$ has two forms that depend on the relationship
between $r_{\rm m}$ and $R$. It reads as

$\displaystyle
N_{\mathrm{acc}}=(GMr_{\mathrm{m}})^{1/2}(\dot{M}_{\mathrm{acc}}-\dot{M}_{\mathrm{prop}}),~{}r_{\rm
m}>R$ (9) $\displaystyle
N_{\mathrm{acc}}=(GMR)^{1/2}(\dot{M}_{\mathrm{acc}}-\dot{M}_{\mathrm{prop}}),~{}r_{\rm
m}<R$ (10)

Then, do the integration of equations 3 and 4, one can estimate the components
of propelled luminosity and dipole luminosity,

$L_{\text{prop}}=\eta_{\text{prop}}\left[-\omega
N_{\text{acc}}-\left(\eta\dfrac{GMM_{\text{d}}}{r_{\text{m}}t_{\nu}}\right)\right],$
(11) $L_{\mathrm{dip}}=\eta_{\mathrm{dip}}\dfrac{\mu^{2}\omega^{4}}{6c^{3}}.$
(12)

Here, $\eta_{\text{prop}}$ and $\eta_{\text{dip}}$ are the conversion
efficiencies of propeller and dipole energy-luminosity, respectively. The
total luminosity is therefore defined as follows,

$L_{\mathrm{tot}}=(1/f_{\mathrm{B}})(L_{\mathrm{dip}}+L_{\mathrm{prop}}),$
(13)

where $1/f_{\rm B}$ is the fraction of the stellar sphere that is emitting,
and it is related to the half-opening angle of the jet ($\theta_{j}$). One has
$f_{\mathrm{B}}=1-\operatorname{cos}\left(\theta_{\mathrm{j}}\right)$ (Rhoads,
1999; Sari et al., 1999).

By given some typical parameters of magnetar propeller with fallback accretion
model, such as $R=10^{6}~{}\rm cm$, $M=1.4~{}M_{\odot}$, $B_{\rm
p}=10^{15}~{}\rm G$, $P_{0}=10^{-3}~{}\rm s$, $M_{\rm d,0}=10^{-3}M_{\odot}$,
$R_{\rm d}=10^{7}~{}\rm cm$, $\epsilon=0.1$, $\delta=10^{-4}$, $\eta_{\rm
prop}=0.8$, $\eta_{\rm dip}=0.1$, and $1/f_{B}=10^{1.7}$, one can plot the
picture of luminosity as function of time for both propeller and dipole
radiations (see Figure 1).

##  3\. Sample Selection and Data Reduction

The XRT data is downloaded from the Swift data archive and the UK Swift
Science Data Center222http://www.swift.ac.uk/burst_analyser/ (Evans et al.,
2007, 2009). Our entire sample includes more than 1718 GRBs observed by
Swift/XRT between January 2005 and June 2023. The magnetar signature typically
exhibits a shallow decay phase (or plateau) followed by a normal decay in
X-ray emission when it is spinning down, and the delayed onset of a propeller
regime that accelerates local material via magneto-centrifugal slinging can
produce the X-ray flares. We only focus on the long-duration GRBs with both
flare/bump and plateau emissions observed in X-ray afterglow, and 154 GRBs are
too faint to be detected in the X-ray band, or do not have enough photons to
extract a reasonable X-ray light curve. Then, we select the GRBs whose X-ray
emission exhibits the feature of plateau followed by a normal decay, and adopt
a smooth broken power-law function to fit (Liang et al., 2007),

$\displaystyle
F_{1}(t)=F_{01}\left[\left(\frac{t}{t_{p}}\right)^{\omega\alpha_{1}}+\left(\frac{t}{t_{p}}\right)^{\omega\alpha_{2}}\right]^{-1/\omega},$
(14)

where fixed $\omega=3$ represents the sharpness of the peak and $t_{p}$,
$\alpha_{1}$, and $\alpha_{2}$ are the peak time, and decay slope of plateau
and normal decay phase, respectively.

Within the above scenario, three criteria are adopted for our sample
selection. (i) We focus on those long-duration GRBs that show such a
transition from shallow decay to normal decay in the X-ray light curves, but
require that the decay slope of the normal decay segment following the plateau
phase should be in the range of $-1$ to $-2$ (Lü et al., 2018); (ii) It must
include the X-ray flare/bump in the X-ray light curve; (iii) In order to
estimate the intrinsic luminosity of the plateau emission, the redshift needs
to be measured. By adopting the criteria for our sample selection, only seven
robust cases identified to satisfy the above three criteria. The X-ray light
curves are shown in Figure 3 (also see Table 1 for a summarized).

More interestingly, it is found that the X-ray flare/bump for those seven GRBs
is simultaneously observed by Swift/BAT in $\gamma-$ray band by searching in
GCN Circulars Archive case by case. So we downloaded the Swift/BAT data of the
seven GRBs from the NASA Archive, and extracted the light curves with the
standard Swift scientific tools. Based on the methods from Hu et al. (2014)
for Swift/BAT, we employed the Bayesian Block (BBs; Scargle 1998) algorithm to
search for possible signals and identify the light curves. In order to search
for a low-significance signal before and after the duration, the extracted BAT
light curves usually cover 300 s before the BAT triggers and after duration of
the GRBs. More details, please refer to the paper on data analysis with the
Bayesian Block algorithm (Hu et al., 2014). Phenomenally, we find that the
prompt emission in $\gamma-$ray band of the seven GRBs are composed of two
sub-bursts with the quiescent time. The light curves of the two sub-bursts
exhibit different behavior for our sample, e.g., a softer sub-burst prior to
(3 out of 7) or following (4 out of 7) the stronger sub-burst of prompt
emission. The light curves of both prompt emission and X-ray flares for those
seven GRBs are shown in Figure 2.

##  4\. Fitting results with magnetar propeller-fallback-accretion model

One motivation is to test whether magnetar propeller-fallback-accretion model
can be invoked to interpret the GRBs that we selected sample above. In order
to find out the best fit parameters of magnetar propeller-fallback-accretion
model, we adopt the MCMC simulation package (Foreman-Mackey et al., 2013) to
fit the X-ray flare/bump and X-ray plateau in the afterglow. There are nine
free parameters (i.e., $B_{\rm p}$, $P_{\rm 0}$, $M_{\rm d,0}$, $R_{\rm d}$,
$\epsilon$, $\delta$, $\eta_{\rm dip}$, $\eta_{\rm prop}$, and $1/f_{\rm B}$)
by invoking the propeller and dipole radiations of magnetar. The MCMC is one
of the very popular methods in the field of high energy astronomy, and it is
very effective in constraining the free parameters which are in degeneracy
between each other. More details can refer to Gibson et al. (2017, 2018).

Initially, we set the range of those parameters as following, $B_{\rm
p}\sim(10^{-3}-10)\times 10^{15}$ G, $P_{0}\sim(0.5-100)\times 10^{-3}$ s,
$M_{\rm d,0}\sim(10^{-3}-10^{2})M_{\odot}$, $R_{\rm d}\sim(50-2000)\times
10^{5}$ cm, $\epsilon\sim(10^{-1}-10^{3})$, $\delta\sim(10^{-5}-50)$,
$\eta_{\rm dip}\sim(1-100)\%$, $\eta_{\rm prop}\sim(1-100)\%$,
$1/f_{B}\sim(1-600)$. Then, we adopt MCMC method to fit the X-ray light curve
by invoking the propeller-fallback-accretion model and spin-down dipole
radiation of magnetar.

Figure 3 shows the MCMC fitting results by using the propeller-fallback-
accretion model and spin-down dipole radiation of magnetar for the seven GRBs.
We find that the free parameters of GRBs 140512A and 180329B can be
constrained very well, such as $B_{\rm p}$ and $P_{0}$, which are within a
range of several times $10^{15}$ G and tens of milliseconds, respectively. The
derived parameters of $B_{\rm p}$, $P_{0}$, and $M_{\rm d,0}$ fall into the
reasonable range, and they are consistent with the results from Lü & Zhang
(2014) and Gibson et al. (2018). However, the free parameters of the other
five cases are, more or less, not all constrained well enough, especially, the
main free parameters of propeller-fallback-accretion model, such as $B_{\rm
p}$, $P_{0}$, and $M_{\rm d,0}$. The two-dimensional histograms and parameter
constraints of model fit by invoking MCMC method for our sample are shown in
Figure 4, and the values of parameter constraints are reported in Table 1. On
the other hand, based on the fitting results, we find that the required
conversion efficiency of propeller is two or three times (or even large)
higher than that of spin-down dipole radiation of magnetar (see Figure 5 and
Table 1). Based on the results from Lan et al. (2018) for Fermi/GBM and Peng
et al. (2014) for Swift/BAT, it suggests that those two sub-bursts emission of
GRB, together with the observed simultaneous X-ray flare/bump, are likely to
share the same physical origin. However, the higher efficiency for the
propeller phase in our fits is contradictory to the expectation from the
propeller model, namely, a dirtier ejecta should be less efficient in
producing $\gamma-$ray emissions. It means that at least the magnetar
propeller-fallback-accretion model cannot interpret very well to the GRB X-ray
flares that are simultaneously observed in $\gamma-$ray band, even though it
can be invoked to interpret the X-ray plateaus for short GRBs (Gompertz et
al., 2014) and supernova explosion (Piro & Ott, 2011).

##  5\. Conclusion and Discussion

The prompt emission, X-ray plateau, and X-ray flares of GRBs are thought to be
from internal dissipation (Zhang, 2011). The magnetar as the central engine of
GRB has been proposed to interpret the observed X-ray flares and plateau
emission (Gompertz et al., 2014; Gibson et al., 2017). By systematically
searching for X-ray emission observed by Swift/XRT, we find that seven robust
GRBs include both X-ray flares and plateau emissions with redshift measured.
More interestingly, it is found that the X-ray flare/bump for those seven GRBs
is simultaneously observed by Swift/BAT in $\gamma-$ray band. The prompt
emission in $\gamma-$ray band of the seven GRBs is composed of two sub-bursts
with the quiescent time. Especially, the second sub-burst emission in the
prompt emission is observed simultaneously with X-ray flare.

Then, by adopting the MCMC method, we invoke the propeller-fallback-accretion
model and spin-down dipole radiation of magnetar central to fit the X-ray data
of plateau emissions and flares for our sample. It is found that the free
parameters of two GRBs (GRBs 140512A and 180329B) can be constrained very
well, such as $B_{\rm p}$ and $P_{0}$ are range of several times $10^{15}$ G
and tens of milliseconds, respectively. However, the free parameters of the
other five cases, more or less, are not all constrained well enough,
especially, the main free parameters of propeller-fallback-accretion model,
such as $B_{\rm p}$, $P_{0}$, and $M_{\rm d,0}$. On the other hand, it
requires that the conversion efficiency of propeller is two or three times
higher than that of spin-down dipole radiation of magnetars. If this is the
case, the higher efficiency for the propeller phase in our fits is
contradictory to the expectation from the propeller model, namely, a dirtier
ejecta should be less efficient in producing $\gamma-$ray emissions. Our
results suggest that at least the magnetar propeller-fallback-accretion model
cannot interpret very well to the GRB X-ray flares that are simultaneously
observed in $\gamma-$ray.

We believe that the magnetar propeller-fallback-accretion model can interpret
very well the X-ray plateau emissions of some GRBs. However, this model exists
significant challenges and flaws for the baryon loading issue when it is
related to $\gamma-$ray emission and high relativistic outflow. From the
observational point of view, the X-ray flares are observed in a good fraction
of GRB afterglow, and the constrained bulk Lorentz factor (or just upper
limit) of the X-ray flare outflows ranges from a few tens to hundreds (Jin et
al., 2010; Yi et al., 2015). It means that the dirty of baryon loading issue
still exists if we invoke the magnetar propeller-fallback-accretion model to
interpret the X-ray flares with a highly Lorentz factor. Furthermore, several
physical models without magnetar central engine are invoked to interpret the
observed X-ray flare, plateau emission, as well as two sub-bursts $\gamma-$ray
emission, such as late central engine reactivity (Burrows et al., 2005; Fan &
Wei, 2005; Dai et al., 2006), the relativistic jet and cocoon emissions
(Ramirez-Ruiz et al., 2002; Lazzati & Begelman, 2010; Nakar & Piran, 2017),
two steps in the collapse of the progenitor star (Lipunova et al., 2009), the
collapse of a rapidly rotating stellar core leading to fragmentation (King et
al., 2005), fragmentation in the accretion disc (Perna et al., 2006), the
magnetic barrier around the accretor (Proga & Zhang, 2006), the gravitational
lensing (Paynter et al., 2021; Yang et al., 2021; Lin et al., 2022), and jet
precession model (Gao et al., 2023). However, more or less, each model above
cannot fully interpret all properties of observations for two sub-bursts
emission of GRB from light curve, spectrum, as well as the quiescent time.

We acknowledge the use of the public data from the UK Swift Science Data
Centre. This work is supported by the Guangxi Science Foundation (grant Nos.
2023GXNSFDA026007 and 2017GXNSFFA198008), the National Natural Science
Foundation of China (grant No. 11922301), and the Program of Bagui Scholars
Program (LHJ). Lin Lan is supported by the National Postdoctoral Program for
Innovative Talents (grant No. GZB20230765).

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Table 1The values of derived parameters from the MCMC for our sample.

GRB |  $B_{\rm p}$ $(10^{15}\cdot\rm G)$ |  $P_{0}$ $\rm(ms)$ |  $M_{\rm d,0}$ $(10^{-2}\cdot M_{\odot})$ |  $R_{\rm d}$ $\rm(km)$ | $\epsilon$ |  $\delta$ $10^{-4}$ | $\eta_{\rm prop}$ | $\eta_{\rm dip}$ | $1/f_{B}$ | $z$
---|---|---|---|---|---|---|---|---|---|---
140512A | $2.28^{+0.01}_{-0.01}$ | $69.85^{+1.51}_{-0.82}$ | $8.13^{+0.04}_{-0.04}$ | $148.95^{+0.14}_{-0.12}$ | $19.72^{+0.23}_{-0.23}$ | $73.62^{+0.34}_{-0.34}$ | $0.56^{+0.01}_{-0.01}$ | $0.23^{+0.01}_{-0.01}$ | $7.93^{+0.13}_{-0.14}$ | $0.725^{(a)}$
180329B | $6.30^{+0.01}_{-0.01}$ | $29.93^{+0.23}_{-0.24}$ | $0.44^{+0.01}_{-0.01}$ | $116.53^{+0.10}_{-0.10}$ | $0.28^{+0.02}_{-0.02}$ | $2.32^{+0.12}_{-0.11}$ | $0.24^{+0.01}_{-0.01}$ | $0.08^{+0.01}_{-0.01}$ | $29.04^{+1.30}_{-1.05}$ | $1.998^{(b)}$
060526 | $9.99^{+0.01}_{-0.01}$ | $9.02^{+0.57}_{-0.43}$ | $9.98^{+0.01}_{-0.03}$ | $144.84^{+0.15}_{-0.12}$ | $136.14^{+9.06}_{-22.64}$ | $6.08^{+0.67}_{-0.78}$ | $0.74^{+0.15}_{-0.10}$ | $0.12^{+0.02}_{-0.02}$ | $88.51^{+15.00}_{-14.38}$ | $3.21^{(c)}$
061602 | $1.278^{+0.07}_{-0.09}$ | $7.98^{+1.40}_{-1.94}$ | $9.56^{+0.32}_{-0.57}$ | $81.27^{+1.15}_{-0.94}$ | $0.20^{+0.24}_{-0.08}$ | $37.75^{+16.19}_{-15.16}$ | $0.78^{+0.15}_{-0.21}$ | $0.06^{+0.01}_{-0.02}$ | $10.47^{+3.75}_{-1.78}$ | $2.253^{(d)}$
100906A | $4.00^{+0.18}_{-0.26}$ | $9.32^{+0.43}_{-0.83}$ | $9.75^{+0.18}_{-0.33}$ | $90.96^{+0.39}_{-0.82}$ | $161.43^{+15.16}_{-20.50}$ | $29.64^{+4.70}_{-6.68}$ | $0.79^{+0.13}_{-0.14}$ | $0.02^{+0.01}_{-0.01}$ | $16.25^{+3.46}_{-2.45}$ | $1.727^{(e)}$
130609B | $5.75^{+0.10}_{-0.09}$ | $1.44^{+0.03}_{-0.04}$ | $9.84^{+0.11}_{-0.22}$ | $236.09^{+2.89}_{-1.68}$ | $57.94^{+2.31}_{-2.22}$ | $130.01^{+4.87}_{-4.70}$ | $0.55^{+0.15}_{-0.08}$ | $0.28^{+0.08}_{-0.05}$ | $55.46^{+12.45}_{-12.40}$ | $1.3^{(f)}$
180325A | $3.29^{+0.09}_{-0.07}$ | $9.68^{+0.21}_{-0.45}$ | $9.77^{+0.15}_{-0.36}$ | $65.67^{+1.37}_{-0.95}$ | $109.14^{+6.46}_{-6.10}$ | $883.07^{+28.93}_{-35.85}$ | $0.72^{+0.17}_{-0.16}$ | $0.08^{+0.14}_{-0.06}$ | $6.35^{+1.94}_{-1.10}$ | $2.25^{(g)}$

  * References of redshift measured: (a) de Ugarte Postigo et al. (2014); (b) Izzo et al. (2018); (c) Berger & Gladders (2006); (d) Perley et al. (2016); (e) Tanvir et al. (2010); (f) Ruffini et al. (2013); (g) Heintz et al. (2018).

Figure 1.— X-ray light curve generated by propeller model with fallback
accretion (blue dash-dotted line) and dipole radiation of spin-down (green
dashed line) of magnetar central engine. The black solid line is the sum of
propeller and dipole radiation of spin-down.

Figure 2.— BAT lightcurves (black) of prompt emission and X-ray flares (blue)
for our sample. The solid red line is the background of prompt emission.

Figure 3.— X-ray light curves of GRBs for our sample (red points), as well as
the fits by magnetar propeller model with fallback accretion (red dash-dotted
line) and magnetar spin-down (blue dashed line). The black solid line is the
total luminosity of propeller and spin-down.

Figure 4.— 2D histograms and parameter constraints of model fit for our
sample. The 1D histograms show the distribution for each parameter. The dashed
lines indicate the median and $\pm 2\sigma$ uncertainty of the values.

Fig. 4— continued.

Figure 5.— Efficiency correlation between propeller and dipole radiation. The
dotted line corresponds to $\eta_{\rm prop}=\eta_{\rm dip}$.