Spindown of magnetars: Quantum Vacuum Friction?
aa r X i v : . [ a s t r o - ph . H E ] J un Research in Astronomy and Astrophysics manuscript no.(L A TEX: qvf.tex; printed on July 1, 2015; 0:59)
Spindown of magnetars: Quantum Vacuum Friction?
Xue-Yu Xiong, Chun-Yuan Gao ∗ and Ren-Xin Xu School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University,Beijing 100871, China; [email protected]
Abstract
Magnetars are proposed to be peculiar neutron stars which could power their X-rayradiation by super-strong magnetic fields as high as & G. However, no direct evidencefor such strong fields is obtained till now, and the recent discovery of low magnetic fieldmagnetars even indicates that some more efficient radiation mechanism than magnetic dipoleradiation should be included. In this paper, quantum vacuum friction (QVF) is suggested tobe a direct consequence of super-strong surface fields, therefore the magnetar model couldthen be tested further through the QVF braking. Pulsars’ high surface magnetic field inter-acting with the quantum vacuum result in a significantly high spindown rate ( ˙ P ). It isfound that QVF dominates the energy loss of pulsars when pulsar’s rotation period and itsfirst derivative satisfy the relationship P · ˙ P > . × − ξ − s , where ξ is the ratio ofthe surface magnetic field over diploe magnetic field. In the “QVF + magnetodipole” jointbraking scenario, the spindown behavior of magnetars should be quite different from that inthe pure magnetodipole model. We are expecting these results could be tested by magnetarcandidates, especially the low magnetic field ones, in the future. Key words: pulsars: general — radiation: dynamics — stars: magnetars — stars: neutron
Kinematic rotation was generally thought to be the only energy source for pulsar emission soon after thediscovery 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 pe-riods (thus low spindown power) and no binary companions, which rules out spin and accretion in binarysystem as the power sources. The first SGR-giant flare was even observed in 1979 (Mazets et al. 1979), andPaczynski (1992) then pointed out that the super-strongmagnetic field may explain the super-Eddingtonluminosity. AXPs and SGRs are thereafter supposed to be magnetars, peculiar neutron stars with sur-face/multipole magnetic fields ( G ∼ G) as the energy source, while the initially proposed strongdipole fields could not be necessary (e.g., Tong et al. 2013). Moreover, the discovery of low magnetic fieldmagnetars (Zhou & Chen 2014, Rea et al. 2010, Rea et al. 2012, Scholz et al. 2012) in recent years indi-cates that some more efficient radiation mechanism than magnetic dipole radiation should be included.
X.-Y. Xiong, C.-Y. Gao & R.-X. Xu
Besides failed predictions and challenges in the magnetar model (Xu 2007, Tong & Xu 2011), one of thekey points is: can one obtain direct evidence of the surface strong fields? Here we are suggesting quan-tum vacuum friction (QVF) as a direct consequence of the surface fields, and calculating the spindown ofmagnetar 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 de-rived braking index n = Ω ¨Ω / ˙Ω ( Ω is the angular velocity of rotation) of a pulsar is expected to be3 for pure magnetodipole radiation. As a result of observational difficulties, only braking indices n n = 2 . ± . ), PSRB1509-58 ( n = 2 . ± . ), PSR J1119-6127 ( n = 2 . ± . ), PSR B0531+21 (the Crab pulsar, n = 2 . ± . )), PSR B0540-69 ( n = 2 . ± . ) and PSR B0833-45 (the Vela pulsar, n = 1 . ± . ).These observed breaking indices are all remarkably smaller than the value of n = 3 , which may sug-gest 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 electromag-netic 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 cer-tainly be from rotational kinetic energy of pulsar. The loss of rotational kinetic energy of pulsar by QVFmay also transform into pulsar’s thermal energy or the energy of pulsar’s radiating photons which mightnot be isotropic. This is the same argument as in the work of Manjavacas et al. (2010), in which the au-thors argue that at zero temperature, the friction produced on rotating neutral particles by interaction withthe vacuum electromagnetic fields transforms mechanical energy into light emission and produces parti-cle heating. Pulsar may transfer its angular momentum to the vacuum when pulsars rub against quantumvacuum 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’ interactionwith the quantum vacuum by taking account of quantum electrodynamics (QED) effect in high magneticfield. The calculations indicate that when the pulsars’ magnetic field is high, QVF would also play an im-portant role to cause the rotation energy loss of pulsars. Thus, it is necessary to take QVF into the rotationenergy 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 magneticfield. 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 dipolemagnetic field and the braking index of magnetars in the second section. The calculated results and analysisare presented in the third section. Finally, conclusions and discussions are presented. pindown of magnetars: Quantum Vacuum Friction? 3 A pulsar has the power of magnetodipole radiation of ˙ E dip = − c − µ Ω , (1)where µ = 12 B dip R sin θ (2)is magnetic dipolar moment and c is the speed of light in vacuum, B dip is the dipole magnetic field, R is thepulsars’ radius, θ is the inclination angle. For general pulsars, surface magnetic field approximately equalto dipole magnetic field because multipole magnetic field attenuate to little. However, for magnetars thereis a surplus of attenuate multipole magnetic field as its extraordinarily strong surface magnetic field. Somagnetars’ surface magnetic field B surf include dipole magnetic field B dip and multipole magnetic field.We suppose that the ratio of surface magnetic field and dipole magnetic field ξ = B surf B dip (3)is a constant. The pulsar rubs against the quantum vacuum and then loses its rotation kinetic energy(Dupays et al. 2008) of ˙ E qvf ≃ − α π R sin θcB c B P , (4)where α = e / ~ c ≃ / is the coupling constant of electromagnetic interaction, B c = 4 . × G isthe QED critical field and P = 2 π/ Ω is the spin period.The pulsars’ typical radius R = 10 cm is adopted. Set inclination angle θ = 90 ◦ for the sake ofsimplicity. Considering the relation (2) between the magnetic moment of pulsars and magnetic field in polarregion of pulsars, we can obtain the ratio of the energy loss due to QVF over that due to magnetodipoleradiation ˙ E qvf ˙ E dip = 7 . × − B P ξ . (5)Assuming the pulsars’ rotation energy loss coming from both magnetodipole radiation and QVF, i.e. ˙ E = ˙ E dip + ˙ E qvf , the total energy loss of pulsars are given by ˙ E ≃ − µ Ω c − α π
16 sin θcB c B R P ξ . (6)From the pulsars’ rotation energy loss ˙ E = I Ω ˙Ω , where I is the inertia of momentum with typical value I = 10 g · cm , we can obtain a relationship between pulsar’s period and the period derivative with respectto time ˙ P = 2 π R sin θ c I B P + 3 αR sin θ πB c Ic B
P ξ . (7)Using the relation of Ω and P , the braking index can be obtained n = 1˙ P π R sin θIc B P + 3 α π R sin θB c Ic B
P ξ ! . (8)Numerically, the braking index can be written as n = 7 .
31 + f ( B dip , P, ξ ) , (9) X.-Y. Xiong, C.-Y. Gao & R.-X. Xu where f ( B dip , P, ξ ) = 18 . B , P ξ (10)with B dip , = 10 − B dip . We can also express the ratio of the energy loss due to QVF over that due tomagnetodipole radiation by pulsar’s period( P ) and period derivative( ˙ P ) from equation (5) and (7) ˙ E qvf ˙ E dip = 7 . × − − c π R B c + r ( 87689 c π R B c ) + 87683 R πIcB c ξ P ˙ P ! . (11)Numerically, the above equation can be written as ˙ E qvf ˙ E dip = −
12 + r
14 + 3 . × ξ P ˙ P . (12)
The periods of observed pulsars are distributed mainly in the range from . s to ˙ E qvf / ˙ E dip ,as a function of the period P in Fig. 1 for ξ = 10 and in Fig. 2 for ξ = 100 . From Fig.1 we can see thatQVF may play an important role when the dipole magnetic field is higher than ∼ G for pulsars whoseperiod are between . s and 1s. Most of observed pulsar’s magnetic field derived from pure magnetodipoleradiation are in the region − G, however, if QVF is included in pulsars’ energy loss, the derivedmagnetic field could be lower. Thus it is necessary to independently measure the magnetic field of pulsarsso 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 B dip > for most pulsars’ braking. For millisecond pulsars the derived magnetic field from magnetodipoleradiation is already so low ( B dip < G) that we can neglect the QVF’s contribution to its rotationenergy loss, but for magnetars the derived magnetic field from QVF is already so high ( B dip > G)that we have to consider the QVF’s contribution. We can also express the ratio of the energy loss due toQVF over that due to magnetodipole radiation by pulsar’s period( P ) and period derivative( ˙ P ) as shown inEq. (12). From this equation we can obtain that QVF dominates the energy loss of pulsars when pulsar’srotation period and its first derivative satisfy the relationship P · ˙ P > . × − ξ − s , where ξ is theratio of the surface magnetic field over diploe magnetic field. According to above relationship and currentobserved data for confirmed magnetars (see Table 1) QVF will dominate the rotation energy loss in all ofthe magnetars’ spindown.Substituting the observed value of ˙ P and P 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 B infdip of Table 1. The fourth column B dip is derived from pure magnetodipole radiation. The calculated resultsmanifest that the derived dipole magnetic field B dip from pure magnetodipole radiation is about ( ξ =10 ) and ( ξ = 100 ) times larger than B infdip obtained by combining QVF and magnetodipole radiation.And the derived surface magnetic field B surf from pure magnetodipole radiation is about 100 times largerthan B infdip inferred by combining QVF and magnetodipole radiation for both ξ = 10 and ξ = 100 .If ˙ E = ˙ E QVF , from Eq. (4) we can obtain ˙Ω ∝ Ω , therefore braking index n = 1 for pulsar’s spindownby 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 pindown of magnetars: Quantum Vacuum Friction? 5 B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L E (cid:144) E = - E qvf (cid:144) E dip = E qvf (cid:144) E dip = P H s L E qvf E dip Fig. 1
The ratio of a pulsar’s energy loss rate from QVF over that from magnetodipole radiation,as a function of period, where ξ = 10 . B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L B d i p = H G L E qvf (cid:144) E dip = E qvf (cid:144) E dip = P H s L E qvf E dip Fig. 2
The ratio of a pulsar’s energy loss rate from QVF over that from magnetodipole radiation,as a function of period, where ξ = 100 . ξ , so magnetars have bigger f ( B dip , P, ξ ) function value (see Eq. (10)) which result in QVF dominatingmagnetars’ braking and its braking indices being about . However, for some low magnetic field millisecodpulsar, minor f ( B dip , P, ξ ) function value lead to magnetodipole radiation becoming main energy lossway in its spindown and its braking index is about . Considering pulsar’s spindown by both QVF and X.-Y. Xiong, C.-Y. Gao & R.-X. Xu
Table 1
The parameters and the inferred magnetic field of magnetars. The magnetars’ dataof period ( P ), the period derivative ( ˙ P ) and dipole magnetic field ( B dip ∼ pulsar/magnetar/main.html). The lastcolumn of table, B infdip , is inferred magnetic field from our model based on both magnetodipoleradiation and QVF. Name P (s) ˙ P ( − s/s) B dip ( G) B infdip ( G, ξ = 10 ) B infdip ( G, ξ = 100 )CXOU J010043.1-721134 8.020392(9) 1.88(8) 3.9 5.946 5.9464U 0142+61 8.68832877(2) 0.20332(7) 1.3 3.342 3.342SGR 0418+5729 9.07838827 < . < . < . < . SGR 0501+4516 5.76209653 0.582(3) 1.9 4.817 4.817SGR 0526-66 8.0544(2) 3.8(1) 5.6 7.082 7.0821E 1048.1-5937 6.4578754(25) 2.25 3.9 6.565 6.5651E 1547.0-5408 2.06983302(4) 2.318(5) 2.2 8.791 8.791PSR J1622-4950 4.3261(1) 1.7(1) 2.7 6.766 6.766SGR 1627-41 2.594578(6) 1.9(4) 2.2 7.905 7.905CXO J164710.2-455216 10.6106563(1) 0.083(2) 0.95 2.541 2.5411RXS J170849.0-400910 11.003027(1) 1.91(4) 4.6 5.516 5.516CXOU J171405.7-381031 3.825352(4) 6.40(5) 5.0 9.718 9.718SGR J1745-2900 3.76363824(13) 1.385(15) 2.3 6.655 6.655SGR 1806-20 7.6022(7) 75(4) 24 15.145 15.145XTE J1810-197 5.5403537(2) 0.777(3) 2.1 5.229 5.229Swift J1822.3-1606 8.43772106(6) 0.00214(21) 0.14 1.078 1.078SGR 1833-0832 7.5654091(8) 0.439(43) 1.8 4.194 4.194Swift J1834.9-0846 2.4823018(1) 0.796(12) 1.4 6.430 6.4301E 1841-045 11.7828977(10) 3.93(1) 6.9 6.494 6.4943XMM J185246.6+003317 11.55871346(6) < . < . < . < . SGR 1900+14 5.19987(7) 9.2(4) 7.0 9.856 9.8561E 2259+586 6.9789484460(39) 0.048430(8) 0.59 2.466 2.466PSR J1846-0258 0.32657128834(4) 0.7107450(2) 0.49 10.379 10.379 magnetodipole radiation, we use Eq. (9) to calculate the braking indices of magnetars. The results showthat all the magnetars’ braking indices are around 1 for both ξ = 10 and ξ = 100 . In the future, the modelcould be tested by comparing the calculated results to observed braking indices. This comparison can alsoprovide further information to understand QVF. We investigate pulsar’s rotation energy loss from QVF and compare it with that from magnetodipole ra-diation in the different magnetic field range and different period range. We find that if the ratio ξ of thesurface magnetic field over dipole magnetic field is fixed to , QVF could play a critical role forpulsars’ braking when B surf · P > (10 ) G · s, while it can be ignored when B surf · P < (10 ) G · s.Magnetars may have high surface magnetic field and long period ( B surf · P ≫ G · s) if the value ofmagnetic field is inferred by pure classical magnetodipole radiation. Therefore it is necessary to considermagnetars’ rotation energy loss by both magnetodipole radiation and QVF. pindown of magnetars: Quantum Vacuum Friction? 7 We consider the difference between the surface magnetic field and dipole magnetic field of pulsars andcompare the energy loss rate of pulsars due to magnetodipole radiation to that due to QVF. The results showthat when a pulsar has a strong magnetic field or a long period ( B surf · P > G · s for ξ = 10 , B surf · P > G · s for ξ = 100 ), comparing to QVF, the energy loss by magnetodipole radiation can be ignored, whilewhen pulsars have weak magnetic field or short period ( B surf · P < G · s for ξ = 10 , B surf · P < G · sfor ξ = 10 ) the QVF can be negligible. We consider that rotation energy loss of magnetars is the sum ofthe energy loss due to QVF and that due to magnetodipole radiation. Based on this joint mechanism ofenergy loss, the surface magnetic field of magnetars and braking indices are calculated. Our work indicatesthat when QVF is included in the process of rotation energy loss, the surface magnetic field of magnetarsis − times lower than that in pure magnetodipole radiation model. In this joint braking modelQVF dominates the energy loss of pulsars when pulsar’s rotation period and its first derivative satisfy therelationship P · ˙ P > . × − ξ − s , where ξ is the ratio of the surface magnetic field over diploemagnetic 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 magneticdipole radiation. Smaller magnetic field can generate a greater rotation energy loss by QVF comparing tomagnetic dipole radiation. This may explain why magnetars which have great X-ray luminosity and lowmagnetic 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 showboth timing and luminosity features for magnetars, and a data-based research would be necessary and in-teresting. Summarily, further observations for magnetars in the future would test our joint braking model aswell as help us understand QVF in reality. Acknowledgements
The authors thank Yue You-ling, Feng Shu-hua, Liu Xiong-wei and Yu Meng for help-ful discussions. This work is supported by the National Natural Science Foundation of China (11225314),XTP XDA04060604, and SinoProbe-09-03 (201311194-03).