Eur. Phys. J. C (2016) 76:612DOI 10.1140/epjc/s10052-016-4464-3
Regular Article - Theoretical Physics
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Web End = Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics
V. I. Denisov1, I. P. Denisova2, A. B. Pimenov1, V. A. Sokolov1,a
1 Physics Department, Moscow State University, Moscow 119991, Russia
2 Moscow Aviation Institute (National Research University), Volokolamskoe Highway 4, Moscow 125993, Russia
Received: 5 September 2016 / Accepted: 23 October 2016 / Published online: 8 November 2016 The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract In this paper we investigate the corrections of vacuum nonlinear electrodynamics on rapidly rotating pulsar radiation and spin-down in the perturbative QED approach (post-Maxwellian approximation). An analytical expression for the pulsars radiation intensity has been obtained and analyzed.
1 Introduction
Vacuum nonlinear electrodynamics effects is an object that piques a great interest in contemporary physics [14]. 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) [57], Helmholtz International Beamline for extreme elds (HIBEF) [8]. It opens up new possibilities in fundamental physics tests [9 11] with an extremal electromagnetic eld intensities and particle accelerations that have never been obtained before.
At the same time the investigation of the effects of vacuum nonlinear electrodynamics in astrophysics gives us an additional opportunity to carry out versatile research using the natural extreme regimes of strong electromagnetic and gravitational elds with intensities unavailable yet in laboratory conditions. Compact astrophysical objects with a strong eld, such as pulsars and magnetars, are best suited for vacuum nonlinear electrodynamics research. Nowadays, there are many effects of vacuum nonlinear electrodynamics predicted in the pulsars neighborhood. For example vacuum electronpositron pair production [12] and photon splitting [13], photon frequency doubling [14], light by light scattering, and vacuum birefringence [15], transient radiation ray bending [16,17] and normal waves delay [18]. Some of the predicted effects are indirectly conrmed by astrophysical observations. For instance the evidence of the absence of the
a e-mail: mailto:[email protected]
Web End [email protected]
high-eld (surface elds more than Bp > 1013G) radio loud pulsars can be explained by pair-production suppression due to photon splitting [19].
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 Sect. 2, we present vacuum nonlinear electrodynamics models and discuss their main physical properties and predictions. In Sect. 3 pulsar radiation in post-Maxwellian approximation is calculated. Section 4 is devoted to an analysis of the pulsars spin-down vacuum nonlinear electrodynamics inuence. In the last section we summarize our results.
2 Vacuum nonlinear electrodynamics theoretical models
Modern theoretical models of nonlinear vacuum electrodynamics suppose that the electromagnetic eld Lagrange function density L = L(I(2), I(4)) depends on both independent
invariants I(2) = Fik Fki and I(4) = Fik Fkl Flm Fmi of the
electromagnetic eld tensor Fik. The specic relationship between the Lagrange function and the invariants depends on the choice of the theoretical model. Nowadays the most promising models are BornInfeld and HeisenbergEuler electrodynamics.
BornInfeld electrodynamics is a phenomenological theory originating from the requirement of self-energy niteness for a point-like electrical charge [20]. In subsequent studies, the attempts of quantization were performed [21,22] and also it was revealed that BornInfeld theory describes the dynamics of electromagnetic elds on D-branes in string theory [2325]. As the main features of BornInfeld electro-dynamics one can note the absence of birefringence (however, there are modications of the BornInfeld theory [26] with the vacuum birefringence predictions) and dichroism for
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electromagnetic waves propagating in external electromagnetic eld [27]. Furthermore, this theory has a distinctive feature the value of electric eld 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 development [28].
Lagrangian function in BornInfeld electrodynamics has the following form:
L =
1 4a2
1
a2
2 I(2)
a4
4 I(4) +
a4
8 I2(2)
1
,
(1)
where a is a characteristic constant of theory, the inverse value of which has a meaning of maximum electric eld for the point-like charge. For this constant only the following estimation is known: a2 < 1.2 1032 G2.
The other nonlinear vacuum electrodynamics the HeisenbergEuler model [15,29] was derived in quantum eld theory and describes one-loop radiative corrections caused by vacuum polarization in a strong electromagnetic eld. Unlike the BornInfeld electrodynamics, this theoretical model possesses vacuum birefringent properties in a strong eld.
The effective Lagrangian function for HeisenbergEuler theory has the following form:
L =
I(2)
16
2I(2) +
(1 22)I2(2) + 42I(4)
B2c
82
e d
3
xy2ctg(x)cth(y)
, (5)
where = 1/B2c = 0.5 1027 G2, and the post-
Maxwellian parameters 1 and 2 depend on the choice of the theoretical model. In the case of HeisenbergEuler electrodynamics the post-Maxwellian parameters 1 and 2 are coupled to the ne structure constant [37]:
1 =
45 = 5.1 105, 2 =
d, (2)
where Bc = m2c3/eh = 4.41 1013 G is the value of the
characteristic eld in quantum electrodynamics, e and m are the electron charge and mass, = e2/hc is the ne structure
constant, and for brevity we use the notations
x =
i 2Bc
1
2(B2 E2) + i(B E)
+
2
3 (x2 y2) 1
7180 = 9.0 105. (6)
For BornInfeld electrodynamics these parameters are equal to each other and they can be expressed through the eld induction 1/a typical of this theory [37]:
1 = 2 =
a2B2c
4 < 4.9 106. (7)
The electromagnetic eld equations for the post-Maxwellian vacuum electrodynamics with the Lagrangian (5) are equivalent [35] to the equations of Maxwell electrodynamics of continuous media,
m Fik + i Fkm + k Fmi = 0, (8)
Qki
xi =
4c jk, (9)
with the specic nonlinear constitutive relations [18]
1
2(B2 E2) i(B E)
, (3)
y =
1 2Bc
1
2(B2 E2) + i(B E)
+
1
2(B2 E2) i(B E)
. (4)
Many attempts to nd the experimental status for each of these theories were made a long time ago, but nowadays it still remains ambiguous. There is experimental evidence in favor of each of them. HeisenbergEuler electrodynamics
predictions were experimentally proved in Delbrck light by light scattering [30], nonlinear Compton scattering [31], Schwinger pair production in multiphoton scattering [1]. At the same time the recent astrophysical observations [32,33] point on the absence of the vacuum birefringence effect which favors the BornInfeld theory prediction. The measurements performed for the speed of light in vacuum show that it does not depend on wave polarization with the accuracy c/c < 1028. So clarication of the vacuum nonlinear electrodynamics status requires the expansion of the experimental test list both in terrestrial and astrophysical conditions. The main hopes as regards this way to proceed are assigned to the experiments with ultra-high intensity laser facilities [4] and astrophysical experiments with X-ray polarimetry [34] in pulsars and magnetars neighborhood.
As follows from the Lagrangians (1)(2) vacuum nonlinear electrodynamics inuence becomes valuable only in strong electromagnetic elds, comparable to E, B 1/a
for BornInfeld theory and E, B Bc for Heisenberg
Euler electrodynamics [35]. In the case of relatively weak elds (E, B << Bc) the exact expressions (1) and (2) can be decomposed and written [36] in the form of a unied parametric post-Maxwellian Lagrangian:
L =
1 32
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Eur. Phys. J. C (2016) 76 :612 Page 3 of 8 612
Qki = Fki +
(1 22)I(2)Fki + 42Fki(3)
, (10)
where Fki(3) = Fkn Fnm Fmi is the third power of the electromagnetic eld tensor. The tensor Qik can be separated into two terms Qki = Fki + Mki, one of which, Mki, will
have a meaning similar to the matter polarization tensor in electrodynamics of continuous media.
Also it should be noted that in a post-Maxwellian approximation the stress-energy tensor T ik and Poynting vector S have the form
T ik =
1 4
(1 + 1I(2))Fik(2)
gik
2I(2)
+ (1 + 22)I2(2) 42 I(4)
8 , (11)
S = cT 0 =
4c jk. (15)
The solution of these equations may be obtained by the successive approximation method. In the initial approximation we assume that F(0)mi is the solution of the Maxwell electro-dynamics equations
m F(0)ik + i F(0)km + k F(0)mi = 0, Fki(0)
xi =
Mki(0)
xi =
F0(2), (12)
where Fik(2) = gni Fnm Fmk is the second power of the electromagnetic eld tensor, gik is the metric tensor; and the Greek index takes the value = 1, 2, 3.
As was shown in [38], post-Maxwellian approximation turns out to be very convenient for vacuum nonlinear electro-dynamics analysis, so we will use this representation (8)(12) to calculate the 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 sufciently strong magnetic elds with the strength varying from Bp 109G up to Bp 1014G; as these values are
close to Bc the vacuum nonlinear electrodynamics inuence can be manifested. At the same time, the pulsars fast rotation may enhance the nonlinear inuence on its radiation.
Let us consider a pulsar of radius Rs, rotating around an axis passing through its center with the angular velocity . We shall suppose that the rotation is fast enough, so the linear velocity for the points on the pulsars surface is comparable to the speed of light Rs/c 1. We assume that the pulsars
magnetic dipole moment m is inclined to the rotation axis at the angle 0, therefore the cartesian coordinates of this vector vary under rotation as m = {mx = m sin 0 cos t, my =
m sin 0 sin t, mz = m cos 0}.
As the vacuum nonlinear electrodynamics inuence in post-Maxwellian approximation has the character of a small correction to Maxwell theory one can represent the total electromagnetic eld tensor Fki in the form
Fki = Fki(0) + f ki, (13)
where Fki(0) is the electromagnetic eld tensor of the rotating magnetic dipole m in Maxwell electrodynamics and f ik is
the vacuum nonlinear correction. Substituting (13) into (10) and retaining only the terms linear in a small value f ik it can
be found that
Qik f ik + Fik(0) + Mik(0), (14)
where Mik(0) = Mik(Fnj(0)) is the polarization tensor calcu
lated in the approximation of the Maxwell electrodynamics eld Fnj(0). The electromagnetic eld equations (8)(9) with account of (13)(14) then will take the form
m F(0)ik + i F(0)km + k F(0)mi + m fik + i fkm + k fmi = 0, f ki
xi +
Fki(0)
xi +
c 4
1 + 1I(2)
4c jk, (16)
corresponding to rotating magnetic dipole m and a current density which by jk is represented in the right hand side of these equations. In this case, from (15) it follows that the vacuum nonlinear electrodynamics corrections fik may be obtained as a solution of linearized equations:
m fik + i fkm + k fmi = 0, (17)
f ki
xi +
Mki(0)
xi = 0. (18)
To satisfy the homogeneous equation (17) electromagnetic potential Ak should be introduced fki = k Ai i Ak. Using
this potential the inhomogeneous equation (18) under the Lorentz gauge will take the form
nn Ak =
Mki(0)
xi . (19)
It is more convenient to rewrite the last equation in terms of the antisymmetric Hertz tensor ki dened by
Ak =
ki
xi . (20)
In this case Eq. (19) will take the simple form
nn ki = ki = Mki(0), (21)
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where = nn is the DAlembert operator. Six indepen
dent equations in (21) may be expressed in vector form by introducing the Hertz electric [Pi1] and magnetic Z potentials [39]:
[Pi1] = 0, Z =
1
2 , (22)
where is the Levi-Civita symbol and all of the indices take values , , = 1, 2, 3. In terms of these potentials
Eq. (21) can be rewritten
[Pi1] = P0, Z = M0, (23)
where the source vectors P0 and M0 are expressed from polarization tensor M(0)ik by the equalities
P0 = M0(0), M0 =
nd the components of the electromagnetic eld tensor fik and radiation properties such as the Poynting vector S and the tonal intensity I. The Poynting vector components represented by (12) in post-Maxwellian electrodynamics can be simplied by the radiative asymptotic condition S 1/r2,
which actually means that for the radiation description we can use the Maxwellian expression for this vector:
S = cT 0
c4 F0(2). (29)
Finally, the total intensity can be obtained by integrating of the Poynting vector by the surface with the normal n directed to the observer located at the large distance r >> Rs from the pulsar:
I =
1
2 M(0). (24)
The explicit components of these vectors may easily be obtained in Minkowski space-time with the use of (10) and (24):
P0 = 2{1(E20 B20)E0 + 22(B0 E0)B0}, (25)
M0 = 2{1(E20 B20)B0 22(B0 E0)E0}, (26) where E0 and B0 are the electromagnetic eld components of the rotating magnetic dipole in Maxwell electrodynamics, the expressions for which are well described in the literature [40] and the eld vectors themselves have the form
B0(r, t) =
3(m() r)r r2m()r5 m
() cr2
(S n)r2d , (30)
where d is the solid angle.
Solutions of Eq. (23) with the right hand side (25), (26) lead to the following expression for the pulsar radiation intensity:
I =
24B2p R6s
3c3 sin2 0
1 +
2
35Y 3
B2p
B2c
24Y 9 1
151 2
Ci(2Y ) sin2 0 + 4Y 9
2
311 Ci(2Y )
+Y 3 1 1525 (2Y 4 3Y 2)
181 102
cos(2Y ) sin2 0
+
+
3( m() r)r
Y
6
cr4 +
( m() r)r r2 m()c2r3 , (27)
E0(r, t) = [
452 3111
15 (2Y 6 3Y 4) + (1721 602)Y 2
3361 cos(2Y ) + Y 2
302 21
5 (2Y 6 Y 4)
c2r2 , (28)
where = tr/c is the retarded time and the dot corresponds
to the derivative of the magnetic dipole moment m() with respect to the retarded time . Therefore, the right hand side of Eq. (23) can be obtained by using of (25)(28). Equations (23) themselves are the inhomogeneous hyperbolic equations the exact solution methods of which are well developed and described in the literature [4143]. Since we are interested only in the radiative solutions for the pulsars eld, when solving Eq. (23) one should retain only the terms decreasing not faster than 1/r with the distance to the pulsar.
At the same time there are no restrictions on the rotational velocity so Rs/c 1. Due to the excessive unwieldiness
here we will not represent the whole solutions for the Hertz potentials [Pi1] and Z, but we will use the results for them to
r, m()]
cr3 + [
r, m()]
411 + 852
5 Y 2 + 91 + 52
sin(2Y ) sin2 0
+
3111 452
15 (2Y 8 Y 6 + 3Y 4)
+(152 1411)Y 2 + 841
1 3
sin(2Y ) , (31)
where the following notations are used for brevity: k = /c
and Y = k Rs, also Bp is the surface magnetic eld inductance
and Ci(x) =
x cos uu du is an integral cosine.
It is obvious that the intensity obtained can be represented in a form which distinguishes the Maxwell radiation intensity and the vacuum nonlinear electrodynamics correction. In this representation it is convenient to introduce the correction function (0, Y ), which is a multiplier before the scaling factor B2p/B2c determining how strong the vacuum nonlinear
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Eur. Phys. J. C (2016) 76 :612 Page 5 of 8 612
electrodynamics inuence on the pulsar radiation is,
I =
24B2p R6s
3c3 sin2 0
1 +
1.6
1.4
B2p
B2c
(0, Y )
. (32)
For most known rapidly rotating pulsars [44] with Y 1
the factor B2p/B2c 1 is small, which matches the require
ments of the post-Maxwellian approximation. At the same time this means that the vacuum nonlinear electrodynamics corrections will be sufciently suppressed in comparison with the Maxwell electrodynamics radiation. However, this assessment may be waived for special sources of so-called fast radio bursts (FRBs), six cases of which have recently been discovered [46]. One of the hypotheses explaining the nature of FRBs assumes that their source is a rapidly rotating neutron star with the strong surface magnetic eld Bp > Bc called blitzar [45]. In this case vacuum nonlinear electrodynamics corrections to the pulsar radiation became signicant but at the same time this makes a strict solution (31) inapplicable because it was obtained in the low-eld limit. So our further evaluations will be applied to the case of the typical rapidly rotating pulsar, for instance PSR B1937+21 with Bp 4.2 108G Bc, and maybe for blitzars but with the
restriction Bp < Bc. The main purpose of our analysis will be the identication of new qualitative features of the pulsar radiation and comparing vacuum nonlinear corrections to the electromagnetic radiation with the other weak energy loss mechanisms.
Let us investigate the properties of the correction function (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 0 is zero. The correction function depends both on the angle 0 and the angular velocity through Y = Rs/c, so (0, Y ) may be represented as a
surface dened in the region where its coordinates take values 0 Y < 1 and 0 0 /2. Some isolines the
relations 0(Y ) at which this surface takes a constant value, (0, Y ) = const are represented in the Fig. 1, the numer
ical values for which were obtained with the 1 and 2 from the HeisenbergEuler theory.
The obtained isolines differ from each other by the absolute value of the correction function but all of them have a pronounced extremum at some point which lies on the red line. This means that for each xed angle 0 between the pulsar dipole moment and the rotation axis there is an angular velocity at which the vacuum nonlinear electrodynamics corrections become the most pronounced. Increasing Y at constant 0 up to the value marked by the red line increases the correction of vacuum nonlinear electrodynamics. The subsequent Y and angular velocity increase become ineffective because the vacuum corrections in this case will be reduced. It should be noted that increasing Y 1 also will enhance
0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.2
1
0, rad
0.8
0.6
0.4
0.2
Y
Fig. 1 Correction function isolines and the best contrast line
the total pulsar luminosity, which is I 4 sin2 0, but at the
same time, as mentioned, this will decrease the vacuum nonlinear electrodynamics correction on the Maxwell radiation background. For instance, if 0 /2 the correction will
most signicantly stand out for the pulsars with Y 0.5. So
the correction function (0, Y ) plays the role of a contrast. And the red line in Fig. 1 marks the relation between 0 and Y for the best contrast.
Another distinctive feature of the pulsar radiation is manifested in a sophisticated, non-polynomial dependence between the radiation intensity (31) and the angular velocity, which greatly complicates the analysis. Performing a power-law approximation of (31) will allow us to describe the vacuum nonlinear electrodynamics inuence on the pulsar spin-down, in traditional terms of braking-indices and torque-functions [47]. It also provides a possibility for comparison of the pulsar spin-down caused by different non-electromagnetic dissipative factors with the power-law 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 [47] can be expressed by the derivative of the angular velocity as
=
I
J =
2B2p R3s
3J sin2 0
Y 3 +
B2p
B2c
, (33)
where J is the pulsars inertia momentum and the dot means the time derivative. For a description in terms of torque func-
Y 3
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Table 1 Expansion coefcients
0 7 104 6 104 5 104 4 104 3 104 /2 1.7 4.0 2.4 0.3 0.2
/3 1.5 3.8 2.7 0.5 0.3
/6 1.1 3.6 3.3 0.9 0.5
tions, the right hand side of Eq. (33) should be represented in a polynomial form of the angular velocity,
Y 3 (0, Y ) =
N
n
Let us compare the 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 the two most relevant scenarios quadrupole mass radiation [50] and the radiation caused by Rossby waves [51], called r-modes.
Quadrupole gravitational radiation may originate 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 by
= KQ 5 =
32
5
G J2c5 5, (38)
where G is a gravitational constant and is the pulsar ellipticity, which is in accordance with modern representations < 104 [47].
Another reason for gravitational wave emission by an isolated pulsar are the oscillations modes induced by the pulsar rotation. Gravitational radiation is caused by the instability of such oscillations. As was shown by Owen et al. [52] for young rapidly rotating pulsars, spin-down caused by r-modes can be expressed in the form
= KR 7 =
217 F2GM2 R6s2sat
3752 Jc7 7, (39)
where M and Rs are the pulsar mass and its radius, the r-mode oscillations saturation amplitude 107 sat 105 was
dened by [53], and the dimensionless constant F as has been shown in [54] is to be strictly bounded within 1/(20)
F 3/(28).
So the torque function for the quadrupole gravitational radiation KQ can be compared with the nonlinear electrodynamics torque K5 and the r-mode radiation torque KR can be compared with the torque K7. For this comparison we suppose the pulsar with the typical radius Rs = 30 km, mass
M = 2M and inertia momentum J = 1045 g cm2. Also we
assume that the dipole moment inclination is 0 = /2 and
the post-Maxwellian parameters correspond to Heisenberg Euler theory (the choice of BornInfeld parameters in rst estimation gives a similar order).
For the pulsar with the surface magnetic eld Bp
1011G, for which ellipticity reaches the maximum value 104, the r-mode saturation amplitude sat 106
and F = 1/(20), and we have the following estimation:
K5/KQ 1.3 1011 and K7/KR 2.6 107. So the
quadrupole and r-mode gravitational radiation torque will signicantly exceed the nonlinear electrodynamics torque coupled with the terms 5 and 7 in the spin-down equa
tion. For another parameter set the opposite case takes place. If the pulsar distortion and ellipticity is two orders of magnitude lower ( 106), and the pulsar eld is stronger
n(0) Y n =
N
n
n(0) Rs
c
n
, (34)
where n are decomposition coefcients and the number of the terms N should be selected sufcient 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 a power-law approximation for the pulsars with Y > 0.6 better than 0.1%. It should be noted that the series does not converge at Y 1 but its
replacement by the partial sum with the specially selected number of terms allows one to accomplish the polynomial approximation, which provides a good match with the exact expression near Y 1 but leads to signicant errors when
Y 1. In this case the expansion coefcients (with the 1
and 2 from the HeisenbergEuler theory) for the terms providing the largest contribution are represented in Table 1. The coefcients not listed in the table are small and can be neglected in further consideration.
For quantitative analysis, we will take the inclination angle equal to 0 = /2. This choice is justied because it provides
the greatest total intensity of the pulsar radiation and in our comparison of nonlinear electrodynamics spin-down with the other non-electrodynamical dissipative factors, it gives the upper limit of the nonlinear electrodynamics inuence.
After the expansion, the right hand side of the spin-down equation (33) will take the form
= KM +
n
Knn, (35)
where KM corresponds to the torque function of the dipole magnetic radiation in Maxwell electrodynamics [48,49]:
KM =
2B2p R6s
3Jc3 sin2 0, (36)
and Kn are the torques originating from the nonlinear vacuum electrodynamics:
Kn = n(0)KM
Bp
Bc
2 Rs c
n3. (37)
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Eur. Phys. J. C (2016) 76 :612 Page 7 of 8 612
Bp 1013G than K5/KQ 12.6 and K7/KR 25.6.
However, it should be noted that rapidly rotating pulsars with such a strong eld have not been observed yet. Nevertheless, the theoretical models assuming the blitzars as the sources of fast radio bursts [45] do not eliminate the possibility of such strong electromagnetic elds for the rapidly rotating pulsar.Therefore the obtained ratio between the torques seems very exotic but still cannot be completely discarded.
5 Conclusion
In this work, we have studied the vacuum nonlinear electro-dynamics inuence on rapidly rotating pulsar radiation in parameterized post-Maxwellian electrodynamics. Under the assumption of a at space-time the analytical description of radiation intensity (31) was obtained. Despite the fact that the expression for the intensity is quite complicated for an analysis some new features of pulsars 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 0 of the magnetic dipole moment to the rotation axis and the angular velocity . Such a relation plays the role of a contrast for nonlinear corrections on the total pulsar radiation background. It follows that enhancing of the vacuum nonlinear electrodynamics inuence on pulsar radiation requires not only an increasing magnetic eld, but one 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 the pulsar spin-down. In this framework, for a description in terms of the torque functions the power-low expansion of the intensity (31) was carried out (35)(37) with the decomposition coefcients listed in Table 1. This provided an opportunity to compare the nonlinear electrodynamics torque with the weak mechanisms of the energy dissipation, for instance with gravitational wave radiation. For such a comparison the most realistic scenarios of gravitational radiation by isolated pulsars were selected quadrupole gravitational radiation and r-mode radiation. The quantitative comparison has shown that, for the common rapidly rotating pulsar, gravitational radiation torques signicantly exceed the nonlinear electrodynamics torques coupled with the terms of the same powers in the spin-down equation. This result can be explained by the low surface magnetic eld Bs < 1011G specic for most of the rapidly rotating pulsars population. The implementation of similar estimates for the compact object possessing a stronger magnetic eld (hypothetical blitzar)Bs 1013G shows the possibility of the opposite case when
the vacuum nonlinear electrodynamics torques exceed the
gravitational torque and play a more signicant role in the spin-down equation under certain conditions.
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The European Physical Journal C is a copyright of Springer, 2016.
Abstract
In this paper we investigate the corrections of vacuum nonlinear electrodynamics 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.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer