# Spindown of magnetars: Quantum Vacuum Friction?

###### Abstract

Magnetars are proposed to be peculiar neutron stars which could power their X-ray radiation by super-strong magnetic fields as high as G. However, no direct evidence for such strong fields is obtained till now, and the recent discovery of low magnetic field magnetars even indicates that some more efficient radiation mechanism than magnetic dipole radiation should be included. In this paper, quantum vacuum friction (QVF) is suggested to be a direct consequence of super-strong surface fields, therefore the magnetar model could then be tested further through the QVF braking. Pulsars’ high surface magnetic field interacting with the quantum vacuum result in a significantly high spindown rate ( ). It is found that QVF dominates the energy loss of pulsars when pulsar’s rotation period and its first derivative satisfy the relationship s, where is the ratio of the surface magnetic field over diploe magnetic field. In the “QVF magnetodipole” joint braking scenario, the spindown behavior of magnetars should be quite different from that in the pure magnetodipole model. We are expecting these results could be tested by magnetar candidates, especially the low magnetic field ones, in the future.

Vol.0 (200x) No.0, 000–000

^{1}

^{1}institutetext: School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,

Beijing 100871, China; gaocy@pku.edu.cn

## 1 Introduction

Kinematic rotation was generally thought to be the only energy source for pulsar emission soon after the discovery of radio pulsars until the discovery of accretion-powered pulsars in X-ray binaries. However, anomalous X-ray pulsars/soft gamma-ray repeaters (AXPs/SGRs, magnetar candidates) have long spin periods (thus low spindown power) and no binary companions, which rules out spin and accretion in binary system as the power sources. The first SGR-giant flare was even observed in 1979 ([Mazets et al. 1979]), and Paczynski (1992) then pointed out that the super-strongmagnetic field may explain the super-Eddington luminosity. AXPs and SGRs are thereafter supposed to be magnetars, peculiar neutron stars with surface/multipole magnetic fields ( G G) as the energy source, while the initially proposed strong dipole fields could not be necessary (e.g., [Tong et al. 2013]). Moreover, the discovery of low magnetic field magnetars ([Zhou & Chen 2014, Rea et al. 2010, Rea et al. 2012, Scholz et al. 2012]) in recent years indicates that some more efficient radiation mechanism than magnetic dipole radiation should be included. Besides failed predictions and challenges in the magnetar model ([Xu 2007], [Tong & Xu 2011]), one of the key points is: can one obtain direct evidence of the surface strong fields? Here we are suggesting quantum vacuum friction (QVF) as a direct consequence of the surface fields, and calculating the spindown of magnetar candidates with the inclustion of the QVF effect.

Magnetodipole radiation could dominate the kinematic energy loss of isolate pulsars (e.g., [Manchester & Taylor 1977, Dai & Lu 1998, Lyubarsky et al. 2001, Morozova & Ahmedov 2008]). The derived braking index ( is the angular velocity of rotation) of a pulsar is expected to be 3 for pure magnetodipole radiation. As a result of observational difficulties, only braking indices of a few rotation-powered pulsars are obtained with some certainty ([Yue et al. 2007, Livingstone et al. 2007], http://www.atnf.csiro.au/research/pulsar/psrcat/). They are PSR J1846-0258 (), PSR B1509-58 (), PSR J1119-6127 (), PSR B0531+21 (the Crab pulsar, )), PSR B0540-69 () and PSR B0833-45 (the Vela pulsar, ). These observed breaking indices are all remarkably smaller than the value of , which may suggest that other spin-down torques do work besides the energy loss via dipole radiation ([Xu & Qiao 2001, Beskin et al. 1984, Ahmedov et al. 2012, Menou & Perna 2001, Contopoulos et al. 2006, Alpar et al. 2001, Chen et al. 2006, Ruderman 2005, Allen et al. 1997, Lin et al. 2004, Tong & Xu 2014, Tong 2015]).

Recently, the research of Davies et al. shows that the QVF effect could be a basic electromagnetic phenomenon ([Davies 2005, Lambrecht et al. 1996, Pendry 1997, Feigel 2004, Tiggelen et al. 2006, Manjavacas et al. 2010]). If the quantum vacuum friction exists, the dissipative energy by QVF would certainly be from rotational kinetic energy of pulsar. The loss of rotational kinetic energy of pulsar by QVF may also transform into pulsar’s thermal energy or the energy of pulsar’s radiating photons which might not be isotropic. This is the same argument as in the work of Manjavacas et al. (2010), in which the authors argue that at zero temperature, the friction produced on rotating neutral particles by interaction with the vacuum electromagnetic fields transforms mechanical energy into light emission and produces particle heating. Pulsar may transfer its angular momentum to the vacuum when pulsars rub against quantum vacuum since the angular momentum is conserved. In this case, vacuum may work as an standard medium ([Dupays et al. 2008]). Dupays et al. (2008, 2012) even calculated the energy loss due to pulsars’ interaction with the quantum vacuum by taking account of quantum electrodynamics (QED) effect in high magnetic field. The calculations indicate that when the pulsars’ magnetic field is high, QVF would also play an important role to cause the rotation energy loss of pulsars. Thus, it is necessary to take QVF into the rotation energy loss of pulsars, especially for highly magnetized pulsars on surface, like magnetars.

In this paper we assume that pulsar interacts with quantum vacuum as in the work of Dupays et al. (2008) and consider the difference between the surface/toroidal magnetic field and dipole/poloidal magnetic field. The braking indices for pure QVF radiation and surface magnetic field of magnetars for the “QVF magnetodipole” joint braking model are calculated.

The paper is organized as following. After an introduction, we deduce the relation between the dipole magnetic field and the braking index of magnetars in the second section. The calculated results and analysis are presented in the third section. Finally, conclusions and discussions are presented.

## 2 Spindown and braking index of magnetars

A pulsar has the power of magnetodipole radiation of

(1) |

where

(2) |

is magnetic dipolar moment and is the speed of light in vacuum, is the dipole magnetic field, is the pulsars’ radius, is the inclination angle. For general pulsars, surface magnetic field approximately equal to dipole magnetic field because multipole magnetic field attenuate to little. However, for magnetars there is a surplus of attenuate multipole magnetic field as its extraordinarily strong surface magnetic field. So magnetars’ surface magnetic field include dipole magnetic field and multipole magnetic field. We suppose that the ratio of surface magnetic field and dipole magnetic field

(3) |

is a constant. The pulsar rubs against the quantum vacuum and then loses its rotation kinetic energy ([Dupays et al. 2008]) of

(4) |

where is the coupling constant of electromagnetic interaction, is the QED critical field and is the spin period.

The pulsars’ typical radius cm is adopted. Set inclination angle for the sake of simplicity. Considering the relation (2) between the magnetic moment of pulsars and magnetic field in polar region of pulsars, we can obtain the ratio of the energy loss due to QVF over that due to magnetodipole radiation

(5) |

Assuming the pulsars’ rotation energy loss coming from both magnetodipole radiation and QVF, i.e. , the total energy loss of pulsars are given by

(6) |

From the pulsars’ rotation energy loss , where is the inertia of momentum with typical value gcm, we can obtain a relationship between pulsar’s period and the period derivative with respect to time

(7) |

Using the relation of and , the braking index can be obtained

(8) |

Numerically, the braking index can be written as

(9) |

where

(10) |

with . We can also express the ratio of the energy loss due to QVF over that due to magnetodipole radiation by pulsar’s period() and period derivative() from equation (5) and (7)

(11) |

Numerically, the above equation can be written as

(12) |

## 3 The numerical results

The periods of observed pulsars are distributed mainly in the range from s to s (The ATNF Pulsar Catalogue: http://www.atnf.csiro.au/research/pulsar/psrcat/). Using Eq. (5) we plot the ratio of , as a function of the period in Fig. 1 for and in Fig. 2 for . From Fig.1 we can see that QVF may play an important role when the dipole magnetic field is higher than G for pulsars whose period are between s and 1s. Most of observed pulsar’s magnetic field derived from pure magnetodipole radiation are in the region G, however, if QVF is included in pulsars’ energy loss, the derived magnetic field could be lower. Thus it is necessary to independently measure the magnetic field of pulsars so that we can judge whether QVF has important contribution to pulsars’ rotation energy loss.

From Fig.2 we can see that QVF may play an important role when pulsars dipole magnetic field for most pulsars’ braking. For millisecond pulsars the derived magnetic field from magnetodipole radiation is already so low (G) that we can neglect the QVF’s contribution to its rotation energy loss, but for magnetars the derived magnetic field from QVF is already so high (G) that we have to consider the QVF’s contribution. We can also express the ratio of the energy loss due to QVF over that due to magnetodipole radiation by pulsar’s period() and period derivative() as shown in Eq. (12). From this equation we can obtain that QVF dominates the energy loss of pulsars when pulsar’s rotation period and its first derivative satisfy the relationship s, where is the ratio of the surface magnetic field over diploe magnetic field. According to above relationship and current observed data for confirmed magnetars (see Table 1) QVF will dominate the rotation energy loss in all of the magnetars’ spindown.

Name | (s) | (s/s) | (G) | (G, ) | (G, ) |

CXOU J010043.1-721134 | 8.020392(9) | 1.88(8) | 3.9 | 5.946 | 5.946 |

4U 0142+61 | 8.68832877(2) | 0.20332(7) | 1.3 | 3.342 | 3.342 |

SGR 0418+5729 | 9.07838827 | ||||

SGR 0501+4516 | 5.76209653 | 0.582(3) | 1.9 | 4.817 | 4.817 |

SGR 0526-66 | 8.0544(2) | 3.8(1) | 5.6 | 7.082 | 7.082 |

1E 1048.1-5937 | 6.4578754(25) | 2.25 | 3.9 | 6.565 | 6.565 |

1E 1547.0-5408 | 2.06983302(4) | 2.318(5) | 2.2 | 8.791 | 8.791 |

PSR J1622-4950 | 4.3261(1) | 1.7(1) | 2.7 | 6.766 | 6.766 |

SGR 1627-41 | 2.594578(6) | 1.9(4) | 2.2 | 7.905 | 7.905 |

CXO J164710.2-455216 | 10.6106563(1) | 0.083(2) | 0.95 | 2.541 | 2.541 |

1RXS J170849.0-400910 | 11.003027(1) | 1.91(4) | 4.6 | 5.516 | 5.516 |

CXOU J171405.7-381031 | 3.825352(4) | 6.40(5) | 5.0 | 9.718 | 9.718 |

SGR J1745-2900 | 3.76363824(13) | 1.385(15) | 2.3 | 6.655 | 6.655 |

SGR 1806-20 | 7.6022(7) | 75(4) | 24 | 15.145 | 15.145 |

XTE J1810-197 | 5.5403537(2) | 0.777(3) | 2.1 | 5.229 | 5.229 |

Swift J1822.3-1606 | 8.43772106(6) | 0.00214(21) | 0.14 | 1.078 | 1.078 |

SGR 1833-0832 | 7.5654091(8) | 0.439(43) | 1.8 | 4.194 | 4.194 |

Swift J1834.9-0846 | 2.4823018(1) | 0.796(12) | 1.4 | 6.430 | 6.430 |

1E 1841-045 | 11.7828977(10) | 3.93(1) | 6.9 | 6.494 | 6.494 |

3XMM J185246.6+003317 | 11.55871346(6) | ||||

SGR 1900+14 | 5.19987(7) | 9.2(4) | 7.0 | 9.856 | 9.856 |

1E 2259+586 | 6.9789484460(39) | 0.048430(8) | 0.59 | 2.466 | 2.466 |

PSR J1846-0258 | 0.32657128834(4) | 0.7107450(2) | 0.49 | 10.379 | 10.379 |

Substituting the observed value of and into Eq. (7), the magnetic field of pulsars can be calculated. We compute the currently confirmed magnetars’ magnetic field and list the results in the last column of Table 1. The fourth column is derived from pure magnetodipole radiation. The calculated results manifest that the derived dipole magnetic field from pure magnetodipole radiation is about () and () times larger than obtained by combining QVF and magnetodipole radiation. And the derived surface magnetic field from pure magnetodipole radiation is about 100 times larger than inferred by combining QVF and magnetodipole radiation for both and .

If , from Eq. (4) we can obtain , therefore braking index for pulsar’s spindown by pure QVF. Eq. (9) show that pulsar’s braking index is between in the ‘QVF magnetodipole” joint braking scenario. Magnetars have strong surface magnetic field, longer rotation period and bigger , so magnetars have bigger function value (see Eq. (10)) which result in QVF dominating magnetars’ braking and its braking indices being about . However, for some low magnetic field millisecod pulsar, minor function value lead to magnetodipole radiation becoming main energy loss way in its spindown and its braking index is about . Considering pulsar’s spindown by both QVF and magnetodipole radiation, we use Eq. (9) to calculate the braking indices of magnetars. The results show that all the magnetars’ braking indices are around 1 for both and . In the future, the model could be tested by comparing the calculated results to observed braking indices. This comparison can also provide further information to understand QVF.

## 4 Conclusions and Discussions

We investigate pulsar’s rotation energy loss from QVF and compare it with that from magnetodipole radiation in the different magnetic field range and different period range. We find that if the ratio of the surface magnetic field over dipole magnetic field is fixed to , QVF could play a critical role for pulsars’ braking when Gs, while it can be ignored when Gs. Magnetars may have high surface magnetic field and long period (Gs) if the value of magnetic field is inferred by pure classical magnetodipole radiation. Therefore it is necessary to consider magnetars’ rotation energy loss by both magnetodipole radiation and QVF.

We consider the difference between the surface magnetic field and dipole magnetic field of pulsars and compare the energy loss rate of pulsars due to magnetodipole radiation to that due to QVF. The results show that when a pulsar has a strong magnetic field or a long period (Gs for , Gs for ), comparing to QVF, the energy loss by magnetodipole radiation can be ignored, while when pulsars have weak magnetic field or short period (Gs for , Gs for ) the QVF can be negligible. We consider that rotation energy loss of magnetars is the sum of the energy loss due to QVF and that due to magnetodipole radiation. Based on this joint mechanism of energy loss, the surface magnetic field of magnetars and braking indices are calculated. Our work indicates that when QVF is included in the process of rotation energy loss, the surface magnetic field of magnetars is times lower than that in pure magnetodipole radiation model. In this joint braking model QVF dominates the energy loss of pulsars when pulsar’s rotation period and its first derivative satisfy the relationship s, where is the ratio of the surface magnetic field over diploe magnetic field. Also, we obtain the braking index of magenetars is around 1 in the joint braking model. The efficiency of rotation energy losses generated by QVF in magnetars is very high compared to magnetic dipole radiation. Smaller magnetic field can generate a greater rotation energy loss by QVF comparing to magnetic dipole radiation. This may explain why magnetars which have great X-ray luminosity and low magnetic field ([Zhou & Chen 2014, Rea et al. 2010, Rea et al. 2012, Scholz et al. 2012]).

We are expecting the results presented could be tested by X-ray observations of magnetar candidates, especially for the low magnetic field ones. X-ray data accumulated in space advanced facilities could show both timing and luminosity features for magnetars, and a data-based research would be necessary and interesting. Summarily, further observations for magnetars in the future would test our joint braking model as well as help us understand QVF in reality.

###### Acknowledgements.

The authors thank Yue You-ling, Feng Shu-hua, Liu Xiong-wei and Yu Meng for helpful discussions. This work is supported by the National Natural Science Foundation of China (11225314), XTP XDA04060604, and SinoProbe-09-03 (201311194-03).## References

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