∎

<EMAIL_ADDRESS>
11institutetext: Physics Department, Moscow State University, Moscow 119991,
Russia 22institutetext: Moscow Aviation Institute (National Research
University) Volokolamskoe Highway 4, Moscow 125993, Russia

# Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics

V.I.Denisovaddr1  I.P.Denisovaaddr2  A.B.Pimenovaddr1  V.A.Soklolove1,addr1

(Received: date / Accepted: date)

###### Abstract

In this paper we investigate vacuum nonlinear electrodynamics corrections on
rapidly rotating pulsar radiation and spin-down in the perturbative QED
approach (post-Maxwellian approximation). An analytical expression for the
pulsar’s radiation intensity has been obtained and analyzed.

††journal: Eur. Phys. J. C

##  1 Introduction

Vacuum nonlinear electrodynamics effects is an object that piques a great
interest in contemporary physics a1 ; a2 ; a3 ; a4 . First of all it is
related to the emerging opportunities of experimental research in terrestrial
conditions using extreme laser facilities like Extreme Light Infrastructure
(ELI) a5 ; a6 ; a7 , Helmholtz International Beamline for Extreme Fields
(HIBEF) a8 . It opens up new possibilities in fundamental physics tests a9 ;
a10 ; a11  with an extremal electromagnetic field intensities and particle
accelerations that have never been obtained before.

At the same time investigation of vacuum nonlinear electrodynamics effects in
astrophysics gives us an additional opportunity to carry out a versatile
research using natural extreme regimes of strong electromagnetic and
gravitational fields with intensities unavailable yet in laboratory
conditions. Compact astrophysical objects with a strong field, such as pulsars
and magnetars are best suited for vacuum nonlinear electrodynamics researches.
Nowadays, there are many vacuum nonlinear electrodynamics effects predicted in
the pulsar’s neighborhood. For example vacuum electron-positron pairs
production a12  and photon splitting a13 , photon frequency doubling a14 ,
light by light scattering and vacuum birefringence a15 , transient radiation
ray bending a16 ; a17  and normal waves delay a18 . Some of predicted effects
are indirectly confirmed by astrophysical observations. For instance the
evidence of the absence of the high-field (surface fields more then
$B_{p}>10^{13}G$) radio loud pulsars can be explained by pair-production
suppression due to photon splitting a19 .

In this paper we calculate vacuum nonlinear electrodynamics corrections to
electromagnetic radiation of rapidly rotating pulsar and analyze pulsar spin-
down under these corrections.

This paper is organized as follows. In Sec. 2, we present vacuum nonlinear
electrodynamics models and discuss their main physical properties and
predictions. In Sec. 3 pulsar radiation in post-Maxwellian approximation is
calculated. Sec. 4 is devoted to analysis of pulsar spin-down under vacuum
nonlinear electrodynamics influence. In the last section we summarize our
results.

##  2 Vacuum nonlinear electrodynamics theoretical models

Modern theoretical models of nonlinear vacuum electrodynamics suppose that
electromagnetic field Lagrange function density $L=L(I_{(2)},I_{(4)})$ depends
on both independent invariants $I_{(2)}=F_{ik}F^{ki}$ and
$I_{(4)}=F_{ik}F^{kl}F_{lm}F^{mi}$ of the electromagnetic field tensor
$F_{ik}$. The specific relationship between Lagrange function and the
invariants depends on theoretical model choice. Nowadays the most promising
models are Born-Infeld and Heisenberg-Euler electrodynamics.

Born-Infeld electrodynamics is a phenomenological theory originating from the
requirement of self-energy finiteness for pointlike electrical charge a21 . In
subsequent studies, the attempts of quantization were performed a22 ; a23  and
also it was revealed that Born-Infeld theory describes dynamics of
electromagnetic fields on D-branes in string theory a24 ; a25 ; a26 . As the
main features of Born-Infeld electrodynamics one can note the absence of
birefringence (however there are modifications of the Born-Infeld theory a27
with the vacuum birefringence predictions) and dichroism for electromagnetic
waves propagating in external electromagnetic field a28 . Furthermore, this
theory has a distinctive feature – the value of electric field depends on the
direction of approach to the point-like charge. This property was noted by the
authors of the theory and also eliminated by them in the subsequent model
developmenta29 .

Lagrangian function in Born-Infeld electrodynamics has the following form:

$L=-{1\over 4\pi a^{2}}\Big{\\{}\sqrt{[1-{a^{2}\over 2}I_{(2)}-{a^{4}\over
4}I_{(4)}+{a^{4}\over 8}I_{(2)}^{2}]}-1\Big{\\}},$ (1)

where $a$ is a characteristic constant of theory, the inverse value of which
has a meaning of maximum electric field for the point-like charge. For this
constant only the following estimation is known: $a^{2}<1.2\cdot 10^{-32}$
G-2.

The other nonlinear vacuum electrodynamics – Heisenberg-Euler model a15 ; a30
was derived in quantum field theory and describes one-loop radiative
corrections caused by vacuum polarization in strong electromagnetic field.
Unlike Born-Infeld electrodynamics, this theoretical model possess vacuum
birefringent properties in strong field.

Effective Lagrangian function for Heisenberg-Euler theory has the following
form:

$\displaystyle L={I_{(2)}\over 16\pi}-{\alpha B_{c}^{2}\over
8\pi^{2}}\int\limits_{0}^{\infty}{e^{-\sigma}d\sigma\over\sigma^{3}}\Big{[}xy\sigma^{2}\hbox{ctg}(x\sigma)\hbox{cth}(y\sigma)$
$\displaystyle+{\sigma^{2}\over 3}(x^{2}-y^{2})-1\Big{]}d\sigma,$ (2)

where $B_{c}=m^{2}c^{3}/e\hbar=4.41\cdot 10^{13}\,G\,$ is the value of
characteristic field in quantum electrodynamics, $e$ and $m$ are the electron
charge and mass, $\alpha=e^{2}/\hbar c$ – fine structure constant and for
brevity we use the notations

$\displaystyle x=-{i\over\sqrt{2}B_{c}}\Big{\\{}\sqrt{{1\over 2}({\bf
B}^{2}-{\bf E}^{2})+i({\bf B\ E})}-$ $\displaystyle-\sqrt{{1\over 2}({\bf
B}^{2}-{\bf E}^{2})-i({\bf B\ E})}\ \Big{\\}},$ (3)

$\displaystyle y={1\over\sqrt{2}B_{c}}\Big{\\{}\sqrt{{1\over 2}({\bf
B}^{2}-{\bf E}^{2})+i({\bf B\ E})}+$ $\displaystyle+\sqrt{{1\over 2}({\bf
B}^{2}-{\bf E}^{2})-i({\bf B\ E})}\ \Big{\\}}.$ (4)

Many attempts to find out the experimental status for each of these theories
were taken for a long time, but nowadays it still remains ambiguous. There are
experimental evidences in favor of each of them. Heisenber-Euler
electrodynamics predictions were experimentally proved in Delbrück light-by-
light scattering a31 , nonlinear Compton scattering a32 , Schwinger pair
production in multiphoton scattering a1 . At the same time the recent
astrophysical observations a33 ; a34  point on the absence of vacuum
birefringence effect which favors the Born-Infeld theory prediction. The
measurements performed for the speed of light in vacuum show that it doest’n
depend on wave polarization with the accuracy $\delta c/c<10^{-28}$. So
clarification of vacuum nonlinear electrodynamics status requires the
expansion of the experimental test list both in terrestrial and astrophysical
conditions. The main hopes on this way are assigned to the experiments with
ultra-high intensity laser facilities a4  and astrophysical experiments with
X-ray polarimetry a35  in pulsars and magnetars neighborhood.

As it follows from Lagrangians (1)-(2) vacuum nonlinear electrodynamics
influence becomes valuable only in strong electromagnetic fields comparable to
$E,B\sim 1/a$ for Born-Infeld theory and $E,B\sim B_{c}$ for Heisenberg-Euler
electrodynamics a36 . In case of relatively weak fields ($E,B<<B_{c}$) the
exact expressions (1) and (2) can be decomposed and written a37  in the form
of unified parametric post-Maxwellian Lagrangian:

$L=\frac{1}{32\pi}\Big{\\{}2I_{(2)}+\xi\Big{[}(\eta_{1}-2\eta_{2})I_{(2)}^{2}+4\eta_{2}I_{(4)}\Big{]}\Big{\\}},$
(5)

where $\xi=1/B_{c}^{2}=0.5\cdot 10^{-27}\ G^{-2}$, and the post-Maxwellian
parameters $\eta_{1}$ and $\eta_{2}$ depend on choice of theoretical model. In
case of Heisenberg-Euler electrodynamics post-Maxwellian parameters $\eta_{1}$
and $\eta_{2}$ are coupled to fine structure constant $\alpha$ a38 :

$\eta_{1}={\alpha\over 45\pi}=5.1\cdot 10^{-5},\quad\eta_{2}={7\alpha\over
180\pi}=9.0\cdot 10^{-5}.$ (6)

For Born-Infeld electrodynamics these parameters are equal to each other and
can be expressed through the field induction $1/a$ typical of this theory a38
:

$\hfil\eta_{1}=\eta_{2}={a^{2}B_{c}^{2}\over 4}<4.9\cdot 10^{-6}.$ (7)

The electromagnetic field equations for the post-Maxwellian vacuum
electrodynamics with the Lagrangian (5) are equivalent a36  to equations of
Maxwell electrodynamics of continuous media

$\partial_{m}F_{ik}+\partial_{i}F_{km}+\partial_{k}F_{mi}=0,$ (8)
$\frac{\partial Q^{ki}}{\partial x^{i}}=-\frac{4\pi}{c}j^{k},$ (9)

with specific nonlinear constitutive relations a18 :

$Q^{ki}=F^{ki}+\xi\Big{[}(\eta_{1}-2\eta_{2})I_{(2)}F^{ki}+4\eta_{2}F^{ki}_{(3)}\Big{]},$
(10)

where $F^{ki}_{(3)}=F^{kn}F_{nm}F^{mi}$ is the third power of the
electromagnetic field tensor. Tensor $Q^{ik}$ can be separated into two terms
$Q^{ki}=F^{ki}+M^{ki}$, one of which $M^{ki}$ will have a meaning similar to
substance polarization tensor in electrodynamics of continuous media.

Also it should be noted that in post-Maxwellian approximation stress-energy
tensor $T^{ik}$ and Pointing vector ${\vec{S}}$ have a form:

$\displaystyle T^{ik}={1\over
4\pi}\Big{\\{}(1+\xi\eta_{1}I_{(2)})F_{(2)}^{ik}-{g^{ik}\over
8}\Big{[}2I_{(2)}+$
$\displaystyle+\xi(\eta_{1}+2\eta_{2})I_{(2)}^{2}-4\eta_{2}\xi
I_{(4)}\Big{]}\Big{\\}},$ (11) $S^{\mu}=cT^{0\mu}={c\over
4\pi}\Big{[}1+\xi\eta_{1}I_{(2)}\Big{]}F_{(2)}^{0\mu},$ (12)

where $F_{(2)}^{ik}=g^{ni}F_{nm}F^{mk}$ is the second power of the
electromagnetic field tensor, $g^{ik}$ – is the metric tensor and the greek
index takes a value $\mu=1,2,3$.

As it was shown in a39  that post-Maxwellian approximation turns out to be
very convenient for vacuum nonlinear electrodynamics analysis, so we will use
this representation (8)-(12) to calculate radiation of the rapidly rotating
pulsar.

##  3 Rapidly rotating pulsar radiation in post-Maxwellian nonlinear
electrodynamics

Pulsars are the compact objects best suited for vacuum nonlinear
electrodynamics tests in astrophysics. They possess sufficiently strong
magnetic field with the strength varying from $B_{p}\sim 10^{9}G$ up to
$B_{p}\sim 10^{14}G$, so as this values are close to $B_{c}$ the vacuum
nonlinear electrodynamics influence can be manifested. At the same time, the
pulsar’s fast rotation may enhance nonlinear influence on its radiation.

Let us consider a pulsar of radius $R_{s}$, rotating around an axis passing
through its center with the angular velocity $\omega$. We shall suppose that
the rotation is fast enough, so the linear velocity for the points on the
pulsar’s surface is comparable to speed of light $\omega R_{s}/c\sim 1$. We
assume that pulsar’s magnetic dipole moment $\bf{m}$ is inclined to the
rotation axis at the angle $\theta_{0}$, therefore cartesian coordinates of
this vector varies under rotation as ${\bf
m}=\\{m_{x}=m\sin\theta_{0}\cos\omega t,\ m_{y}=m\sin\theta_{0}\sin\omega t,\
m_{z}=m\cos\theta_{0}\\}$.

As the vacuum nonlinear electrodynamics influence in post-Maxwellian
approximation has the character of a small correction to Maxwell theory one
can represent the total electromagnetic field tensor $F^{ki}$ in form

$F^{ki}=F^{ki}_{(0)}+f^{ki},$ (13)

where $F^{ki}_{(0)}$ is the electromagnetic field tensor of the rotating
magnetic dipole ${\bf m}$ in Maxwell electrodynamics and $f^{ik}$ – is the
vacuum nonlinear correction. Substituting (13) in to (10) and retaining only
the terms linear in small value $f^{ik}$ it can be found that:

$Q^{ik}\simeq f^{ik}+F^{ik}_{(0)}+M^{ik}_{(0)},$ (14)

where $M^{ik}_{(0)}=M^{ik}(F_{(0)}^{nj})$ – polarization tensor calculated in
approximation of the Maxwell electrodynamics field $F^{nj}_{(0)}$.
Electromagnetic field equations (8)-(9) with the account of (13)-(14) then
will take a form:

$\displaystyle\partial_{m}F_{ik}^{(0)}+\partial_{i}F_{km}^{(0)}+\partial_{k}F_{mi}^{(0)}+\partial_{m}f_{ik}+\partial_{i}f_{km}+$
$\displaystyle+\partial_{k}f_{mi}=0,$ $\displaystyle\frac{\partial
f^{ki}}{\partial x^{i}}+\frac{\partial F^{ki}_{(0)}}{\partial
x^{i}}+\frac{\partial M^{ki}_{(0)}}{\partial x^{i}}=-\frac{4\pi}{c}j^{k}.$
(15)

The solution of these equation may be obtained by successive approximation
method. In initial approximation we assume that $F_{mi}^{(0)}$ is the solution
of Maxwell electrodynamics equations

$\displaystyle\partial_{m}F_{ik}^{(0)}+\partial_{i}F_{km}^{(0)}+\partial_{k}F_{mi}^{(0)}=0,$
$\displaystyle\frac{\partial F^{ki}_{(0)}}{\partial
x^{i}}=-\frac{4\pi}{c}j^{k},$ (16)

corresponding to rotating magnetic dipole ${\bf m}$, current density for which
$j^{k}$ is represented in right hand side of these equations. In this case,
from (3) follows that vacuum nonlinear electrodynamics corrections $f_{ik}$ my
be obtained as a solution of linearized equations:

$\displaystyle\partial_{m}f_{ik}+\partial_{i}f_{km}+\partial_{k}f_{mi}=0,$
(17) $\displaystyle\frac{\partial f^{ki}}{\partial x^{i}}+\frac{\partial
M^{ki}_{(0)}}{\partial x^{i}}=0.$ (18)

To satisfy homogeneous equation (17) electromagnetic potential $A^{k}$ should
be introduced $f_{ki}=\partial_{k}A_{i}-\partial_{i}A_{k}$. Using this
potential the inhomogeneous equation (18) under the Lorentz gauge will take a
form

$\partial_{n}\partial^{n}A^{k}=\frac{\partial M^{ki}_{(0)}}{\partial x^{i}}.$
(19)

It is more convenient to rewrite the last equation in terms of the
antisymmetric Hertz tensor $\Pi^{ki}$ defined as:

$A^{k}=-\frac{\partial\Pi^{ki}}{\partial x^{i}}.$ (20)

In this case equation (19) will take a simple form

$-\partial_{n}\partial^{n}\Pi^{ki}=\Box\Pi^{ki}=M^{ki}_{(0)},$ (21)

where $\Box=-\partial_{n}\partial^{n}$ – is D’Alembert operator. Six
independent equations in (21) may be expressed in vector form by introducing
Hertz electric ${\bm{\Pi}}$ and magnetic ${\bf Z}$ potentials a40 :

${\bm{\Pi}}^{\alpha}=\Pi^{\alpha 0},\quad{\bf
Z}^{\alpha}=\frac{1}{2}\epsilon^{\alpha\mu\nu}\Pi_{\mu\nu},$ (22)

where $\epsilon^{\alpha\mu\nu}$ – Levi-Civita symbol and all of the indexes
take values $\alpha,\mu,\nu=1,2,3$. In terms of these potentials equations
(21) can be rewritten:

$\Box{\bm{\Pi}}={\bf P}_{0},\quad\Box{\bf Z}={\bf M}_{0},$ (23)

where the source vectors ${\bf P}_{0}$ and ${\bf M}_{0}$ are expressed from
polarization tensor $M^{(0)}_{ik}$ by equalities:

${\bf P}_{0}^{\alpha}=M^{\alpha 0}_{(0)},\quad{\bf
M}_{0}^{\alpha}=\frac{1}{2}\epsilon^{\alpha\mu\nu}M^{(0)}_{\mu\nu}.$ (24)

The explicit components of these vectors may be easily obtained in Minkowski
space-time with the using of (10) and (24):

${\bf P}_{0}=2\xi\\{\eta_{1}({\bf E}_{0}^{2}-{\bf B}_{0}^{2}){\bf
E}_{0}+2\eta_{2}({\bf B}_{0}\ {\bf E}_{0}){\bf B}_{0}\\},$ (25) ${\bf
M}_{0}=2\xi\\{\eta_{1}({\bf E}_{0}^{2}-{\bf B}_{0}^{2}){\bf
B}_{0}-2\eta_{2}({\bf B}_{0}\ {\bf E}_{0}){\bf E}_{0}\\},$ (26)

where ${\bf E}_{0}$ and $\bf{B}_{0}$ are the electromagnetic field components
of the rotating magnetic dipole in Maxwell electrodynamics, the expressions
for which are well described in literature a41  and the field vectors
themselves have the form:

$\displaystyle{\bf B}_{0}({\bf r},t)={3({\bf m}(\tau)\ {\bf r}){\bf
r}-r^{2}{\bf m}(\tau)\over r^{5}}-{{\dot{\bf m}}(\tau)\over cr^{2}}+$
$\displaystyle+{3({\dot{\bf m}}(\tau)\ {\bf r}){\bf r}\over
cr^{4}}+{(\ddot{\bf m}(\tau)\ {\bf r}){\bf r}-r^{2}\ddot{\bf m}(\tau)\over
c^{2}r^{3}},$ (27) ${\bf E}_{0}({\bf r},t)={[{\bf r},\dot{\bf m}(\tau)]\over
cr^{3}}+{[{\bf r},\ddot{\bf m}(\tau)]\over c^{2}r^{2}},$ (28)

where $\tau=t-r/c$ is the retarded time and the dot corresponds to the
derivative of the magnetic dipole moment ${\bf m}(\tau)$ with the respect to
the retarded time $\tau$. Therefore, the right hand side of the equations (23)
can be obtained by using of (25)–(28). The equations (23) themselves are the
unhomogeneous hyperbolic equations the exact solution methods of which are
well developed and described in literature a42 ; a43 ; a44 . Since we are
interested only with the radiative solutions for the pulsar’s field, when
solving equations (23) one should retain only the terms decreasing not faster
than $\sim 1/r$ with the distance to the pulsar. At the same time there is no
restrictions on the rotational velocity so $\omega R_{s}/c\sim 1$. Due to
excessive unwieldiness here we will not represent the whole solutions for the
Hertz potentials ${\bm{\Pi}}$ and ${\bf Z}$, but we will use the results for
them to find the components of the electromagnetic field tensor $f_{ik}$ and
radiation properties such as Pointing vector ${\bf S}$ and tonal intensity
$I$. The Pointing vector components represented by (12) in post-Maxwellian
electrodynamics, can be simplified by the radiative asymptotic condition
$S^{\mu}\sim 1/r^{2}$ which actually means that for radiation description we
can use the Maxwellian expression for this vector:

$S^{\mu}=cT^{0\mu}\sim{c\over 4\pi}F_{(2)}^{0\mu}.$ (29)

Finally, the total intensity can be obtained by integrating of Pointing vector
by the surface with the normal ${\bf n}$ directed to the observer located at
the large distance $r>>R_{s}$ from the pulsar:

$I=\int({\bf S}\ {\bf n})r^{2}d\Omega,$ (30)

where $d\Omega$ is the solid angle.

Solutions of equations (23) with the right hand side (25), (26) lead to the
following expression for the pulsar radiation intensity:

$\displaystyle I$
$\displaystyle=\frac{2\omega^{4}B_{p}^{2}R_{s}^{6}}{3c^{3}}\sin^{2}\theta_{0}\Big{\\{}1+\frac{2}{35Y^{3}}\frac{B_{p}^{2}}{B_{c}^{2}}\Big{(}24Y^{9}\Big{[}\frac{1}{15}\eta_{1}-\eta_{2}\Big{]}$
(31) $\displaystyle\times
Ci(2Y)\sin^{2}\theta_{0}+4Y^{9}\Big{[}\eta_{2}-\frac{311}{45}\eta_{1}\Big{]}Ci(2Y)$
$\displaystyle+Y^{3}\Big{[}\frac{\eta_{1}-15\eta_{2}}{5}(2Y^{4}-3Y^{2})-18\eta_{1}-10\eta_{2}\Big{]}\cos(2Y)\sin^{2}\theta_{0}$
$\displaystyle+\frac{Y}{6}\Big{[}\frac{45\eta_{2}-311\eta_{1}}{15}(2Y^{6}-3Y^{4})+(172\eta_{1}-60\eta_{2})Y^{2}$
$\displaystyle-336\eta_{1}\Big{]}\cos(2Y)+Y^{2}\Big{[}\frac{30\eta_{2}-2\eta_{1}}{5}(2Y^{6}-Y^{4})$
$\displaystyle-\frac{41\eta_{1}+85\eta_{2}}{5}Y^{2}+9\eta_{1}+5\eta_{2}\Big{]}\sin(2Y)\sin^{2}\theta_{0}$
$\displaystyle+\frac{1}{3}\Big{[}\frac{311\eta_{1}-45\eta_{2}}{15}(2Y^{8}-Y^{6}+3Y^{4})$
$\displaystyle+(15\eta_{2}-141\eta_{1})Y^{2}+84\eta_{1}\Big{]}\sin(2Y)\Big{)}\Big{\\}},$

where the following notations are used for brevity: $k=\omega/c$ and
$Y=kR_{s}$, also $B_{p}$ – is the surface magnetic field inductance and
$Ci(x)=\int\limits_{\infty}^{x}\frac{\cos u}{u}du$ – is an integral cosine.

It is obvious that obtained intensity can be represented in form which
distinguishes Maxwell radiation intensity and vacuum nonlinear electrodynamics
correction. In this representation it is convenient to introduce the
"correction function" $\ \Phi(\theta_{0},Y)$ which is a multiplier before the
scaling factor ${B_{p}^{2}}/{B_{c}^{2}}$ determining how strong the vacuum
nonlinear electrodynamics influence on the pulsar radiation is:

$I=\frac{2\omega^{4}B_{p}^{2}R_{s}^{6}}{3c^{3}}\sin^{2}\theta_{0}\Big{\\{}1+\frac{B_{p}^{2}}{B_{c}^{2}}\
\Phi(\theta_{0},Y)\Big{\\}}.$ (32)

For the most known rapidly rotating pulsars a45  with $Y\sim 1$ the factor
${B_{p}^{2}}/{B_{c}^{2}}\ll 1$ is small which matches the requirements of
post-Maxwellian approximation. At the same time this means that vacuum
nonlinear electrodynamics corrections will be sufficiently suppressed in
comparison with Maxwell electrodynamics radiation. However, this assessment
may be waived for special sources of so called Fast Radio Bursts (FRB’s), six
cases of which have recently been discovered a46 . One of the hypotheses
explaining the nature of FRBs assumes that their source is a rapidly rotating
neutron star with the strong surface magnetic field $B_{p}>B_{c}$ called
blitzar a55 . In this case vacuum nonlinear electrodynamics corrections to
pulsar radiation became significant but at the same time this makes strict
solution (31) unapplicable because it was obtained in low-field limit. So our
further evaluations will be applied to the case of the typical rapidly
rotating pulsar, for instance PSR B1937+21 with $B_{p}\sim 4.2\cdot 10^{8}G\ll
B_{c}$, and maybe for blitzars but with the restriction $B_{p}<B_{c}$. The
main purpose of our analysis will be in identification of new qualitative
features of the pulsar radiation and comparing vacuum nonlinear corrections to
the electromagnetic radiation with the other weak energy loose mechanisms.

Let’s investigate the properties of the correction function
$\Phi(\theta_{0},Y)$. First of all, it should be noted that there is no
radiation when the pulsar dipole moment is coaxial with the rotation axis i.e.
when $\theta_{0}$ is zero. The correction function depends both on the angle
$\theta_{0}$ and the angular velocity through $Y=\omega R_{s}/c$, so the
$\Phi(\theta_{0},Y)$ may be represented as a surface defined in the region
where it’s coordinates take values $0\leq Y<1$ and $0\leq\theta_{0}\leq\pi/2$.
Some isolines – the relations $\theta_{0}(Y)$ at which this surface takes a
constant value $\Phi(\theta_{0},Y)=const$ are represented at the Fig.1, the
numerical values for which were obtained with the $\eta_{1}$ and $\eta_{2}$
from the Heisenberg-Euler theory.

Figure 1: Correction function isolines and the best contrast line

The obtained isolines differ from each other by the absolute value of the
correction function but all of them have pronounced extremum at some point
which lies on the red line. This means that for each fixed angle $\theta_{0}$
between the pulsar dipole moment and rotation axis there is an angular
velocity at which vacuum nonlinear electrodynamics corrections become the most
pronounced. Increasing $Y$ at the constant $\theta_{0}$ up to the value marked
by the red line increases correction of vacuum nonlinear electrodynamics. The
subsequent $Y$ and angular velocity increase becomes ineffective because the
vacuum corrections in this case will be reduced. It should be noted, that
increasing $Y\to 1$ also will enhance total pulsar luminosity which is
$I\propto\omega^{4}\sin^{2}\theta_{0}$ but at the same time, as it was
mentioned, this will decrease vacuum nonlinear electrodynamics correction on
the Maxwell radiation background. For instance, if $\theta_{0}\sim\pi/2$ the
correction will most significantly stand out for the pulsars with $Y\sim 0.5$.
So the correction function $\Phi(\theta_{0},Y)$ plays a role of a contrast.
And the red line in Fig.1 marks the relation between $\theta_{0}$ and $Y$ for
the best contrast.

Another distinctive feature of the pulsar radiation is manifested in
sophisticated, non-polynomial dependence between the radiation intensity (31)
and the angular velocity, which greatly complicates the analysis. Performing
power-law approximation of (31) will allow us to describe vacuum nonlinear
electrodynamics influence on the pulsar spin-down, in traditional terms of
braking-indexes and torque-functions a47 . It also provides a possibility for
comparison of the pulsar spin-down caused by different non-electromagnetic
dissipative factors and the power-low relation between the radiation intensity
and angular velocity, for instance with the quadrupole gravitational
radiation. Let us investigate the features of the pulsar spin-down as a result
of the radiation, with the amendments of vacuum nonlinear electrodynamics.

##  4 Pulsar spin-down

The observed spin-down rate a47  can be expressed by derivative of the angular
velocity as

$\dot{\omega}=-\frac{I}{J\omega}=-\frac{2B_{p}^{2}R_{s}^{3}}{3J}\sin^{2}\theta_{0}\Big{\\{}\,Y^{3}+\frac{B_{p}^{2}}{B_{c}^{2}}\,Y^{3}\,\Phi\Big{\\}},$
(33)

where $J$ is the pulsar’s inertia momentum and the dot means the time
derivative. For description in terms of torque functions, the right hand side
of equation (33) should be represented in polynomial form of angular velocity

$Y^{3}\Phi(\theta_{0},Y)=\sum\limits_{n}^{N}\alpha_{n}(\theta_{0})\,Y^{n}=\sum\limits_{n}^{N}\alpha_{n}(\theta_{0})\,\Big{(}\frac{\omega
R_{s}}{c}\Big{)}^{n},$ (34)

where $\alpha_{n}$ are decomposition coefficients and the number of the terms
$N$ should be selected sufficient to ensure the required accuracy of the
decomposition. We will take the number of terms in the expansion (34) equal to
$N=8$. This choice ensures the accuracy of power-law approximation for the
pulsars with the $Y>0.6$ better then 0.1%. It should be noted, that the series
does not converge at $Y\sim 1$ but it’s replacement by the partial sum with
the specially selected number of terms allows to accomplish the polynomial
approximation which provides a good match with the exact expression near
$Y\sim 1$, but leads to significant errors when $Y\ll 1$. In this case the
expansion coefficients (with the $\eta_{1}$ and $\eta_{2}$ from the
Heisenberg-Euler theory) for the terms providing the largest contribution are
represented in the Table 1. The coefficients not listed in the table are small
and can be neglected in further consideration.

Table 1: Expansion coefficients $\theta_{0}$ | $\alpha_{7}\cdot 10^{4}$ | $\alpha_{6}\cdot 10^{4}$ | $\alpha_{5}\cdot 10^{4}$ | $\alpha_{4}\cdot 10^{4}$ | $\alpha_{3}\cdot 10^{4}$
---|---|---|---|---|---
$\pi/2$ | $1.7$ | $-4.0$ | $2.4$ | $-0.3$ | $-0.2$
$\pi/3$ | $1.5$ | $-3.8$ | $2.7$ | $-0.5$ | $-0.3$
$\pi/6$ | $1.1$ | $-3.6$ | $3.3$ | $-0.9$ | $-0.5$

For quantitative analysis, we will take the inclination angle is equal to
$\theta_{0}=\pi/2$. This choice is justified because it provides the greatest
total intensity of the pulsar radiation and in our comparison of nonlinear
electrodynamics spin-down with the other non-electrodynamics dissipative
factors, it gives the upper limit of nonlinear electrodynamics influence.

After the expansion, the right hand side of the spin-down equation (33) will
take the form:

$\dot{\omega}=K_{M}+\sum\limits_{n}K_{n}\omega^{n},$ (35)

where $K_{M}$ corresponds to the torque function of the dipole magnetic
radiation in Maxwell electrodynamics a48 ; a49 :

$K_{M}=-\frac{2B_{p}^{2}R_{s}^{6}}{3Jc^{3}}\sin^{2}\theta_{0},$ (36)

and $K_{n}$ are the torques originating form nonlinear vacuum electrodynamics:

$K_{n}=\alpha_{n}(\theta_{0})K_{M}\Big{(}\frac{B_{p}}{B_{c}}\Big{)}^{2}\Big{(}\frac{R_{s}}{c}\Big{)}^{n-3}.$
(37)

Let us compare pulsar spin-down caused by nonlinear vacuum electrodynamics and
dissipation caused by gravitational waves radiation. Among several possible
ways of gravitational radiation by an isolated pulsar we will choose two most
relevant scenario – quadrupole mass radiation a50  and the radiation caused by
Rossby waves a51 , called r-modes.

Quadrupole gravitational radiation can be originated by the strain caused by
the pulsar rotation, which is especially likely for rapidly rotating pulsars.
The spin-down under this kind of radiation can be represented as:

$\dot{\omega}=K_{Q}\;\omega^{5}=-{32\over 5}{GJ\varepsilon^{2}\over
c^{5}}\;\omega^{5},$ (38)

where $G$ is a gravitational constant and $\varepsilon$ is the pulsar
elipticity, which is in accordance with modern representations
$\varepsilon<10^{-4}$ a47 .

Another reason for gravitational waves emission by isolated pulsar are the
oscillations modes induced by the pulsar rotation. Gravitational radiation is
caused by instability of such oscillations. As it was shown by Owen et al. a52
for young rapidly rotation pulsars spin-down caused by r-modes can be
expressed in form:

$\dot{\omega}=K_{R}\;\omega^{7}=-\frac{2^{17}\pi
F^{2}GM^{2}R_{s}^{6}\beta_{sat}^{2}}{3^{7}5^{2}Jc^{7}}\;\omega^{7},$ (39)

where $M$ and $R_{s}$ are the pulsar mass and it’s radius, the r-mode
oscillations saturation amplitude $10^{-7}\leq\beta_{sat}\leq 10^{-5}$ was
defined by a53 , and dimensionless constant $F$ as it has been shown in a54
is to be strictly bounded within $1/(20\pi)\leq F\leq 3/(28\pi)$.

So the torque function for the quadrupole gravitational radiation $K_{Q}$ can
be compared with nonlinear electrodynamics torque $K_{5}$ and the r-modes
radiation torque $K_{R}$ can be compared with the torque $K_{7}$. For this
comparison we suppose the pulsar with the typical radius $R_{s}=30\ km$, mass
$M=2M_{\odot}$ and inertia momentum $J=10^{45}\ g\cdot cm^{2}$. Also we
assume, that dipole moment inclination is $\theta_{0}=\pi/2$ and post-
Maxwellian parameters correspond to Heisenberg-Euler theory (choice of Born-
Infeld parameters in first estimation gives the similar order).

For the pulsar with the surface magnetic field $B_{p}\sim 10^{11}G$, which
elipticity reaches the maximum value $\varepsilon\sim 10^{-4}$, r-mode
saturation amplitude $\beta_{sat}\sim 10^{-6}$, and $F=1/(20\pi)$ the
following estimation takes place $K_{5}/K_{Q}\sim 1.3\cdot 10^{-11}$ and
$K_{7}/K_{R}\sim 2.6\cdot 10^{-7}$. So the quadrupole and r-mode gravitational
radiation torque will significantly exceed nonlinear electrodynamics torque
coupled with the terms $\sim\omega^{5}$ and $\sim\omega^{7}$ in spin-down
equation. For another parameter set the opposite case takes place. If the
pulsar distortion and elipticity is two orders of magnitude lower (
$\varepsilon\sim 10^{-6}$), and the pulsar field is stronger $B_{p}\sim
10^{13}G$ then $K_{5}/K_{Q}\sim 12.6$ and $K_{7}/K_{R}\sim 25.6$. However, it
should be noted that the rapidly rotating pulsars with such a strong field
have not been observed yet. Nevertheless, the theoretical models assuming the
blitzars as the sources of Fast Radio Burst a55  do not eliminate the
possibility of such a strong electromagnetic fields for the rapidly rotating
pulsar. Therefore obtained ratio between the torques seems very exotic but
still can not be completely discarded.

##  5 Conclusion

In this work, we have studied vacuum nonlinear electrodynamics influence on
rapidly rotating pulsar radiation in parameterized post-Maxwellian
electrodynamics. In assumption of flat space-time the analytical description
of radiation intensity (31) was obtained. Despite on the fact that the
expression for the intensity is quite complicating for analysis some new
features of pulsar’s radiation have been obtained. For instance, it was shown
that for the rapidly rotating pulsar vacuum nonlinear electrodynamics
corrections observation is optimal only for certain relations between the
inclination angle $\theta_{0}$ of the magnetic dipole moment to the rotation
axis and the angular velocity $\omega$. Such relation plays a role of the
contrast for nonlinear corrections on total pulsar radiation background. It
follows that enhancing of vacuum nonlinear electrodynamics influence on pulsar
radiation requires not only increasing magnetic field but also needs
compliance of conditions marked on Fig.1 to ensure the best possible contrast
for the nonlinear corrections.

The obtained radiation intensity was used to estimate pulsar spin-down. In
this framework, for description in terms of the torque functions the power-low
expansion of the intensity (31) was carried out (35)-(37) with the
decomposition coefficients listed in Table 1. This provided an opportunity to
compare nonlinear electrodynamics torque with the the weak mechanisms of the
energy dissipation, for instance with gravitational waves radiation. For such
comparison the most realistic scenarios of gravitational radiation by isolated
pulsar were selected – quadrupole gravitational radiation and r-modes
radiation. The quantitative comparison has shown that for the common rapidly
rotating pulsar, gravitational radiation torques significantly exceed
nonlinear electrodynamics torques coupled with the terms of same $\omega$
powers in spin-down equation. This result can be explained by low surface
magnetic field $B_{s}<10^{11}G$ specific for most of rapidly rotating pulsar’s
population. Implementation of similar estimates for the compact object
possessing stronger magnetic field (hypothetical blitzar) $B_{s}\sim 10^{13}G$
shows the possibility of the opposite case when vacuum nonlinear
electrodynamics torques exceed gravitational torque and play more significant
role in spin-down equation under certain conditions.

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