A Novel Look at the Pulsar Force-Free Magnetosphere 3

Home , Stellar magnetic field

arXiv:1608.07998v1 [astro-ph.HE] 29 Aug 2016 cetdXX eevdYY noiia omZZZ form original in YYY; Received XXX. Accepted MNRAS hlpo ptosy2014 Spitkovsky & Philippov 1999 al. et Contopoulos ⋆ paradigm. magnetospheric existing facil the and of character modification distinct tate essentially have may equation force a of the that with to infinity, monopole. close at free being one near distribution monopolar form current the dipolar poloidal to the star from neutron the smoothly varies field magnetic 2006 (e.g. magnetosphere is simulated (e.g., numerically star and the neutron monopole in the struc- quently, a from magnetospheric far of the represent ture field to ( magnetic believed equation commonly the this of to solution corresponds exact known only 1973 et r eae ytennlna usreuto ( bal- equation pulsar force non-linear the c electromagnetic elec- and by fields related the accelerating the are and of provide the rents distributions aligned to self-consistent screen the are and to ance), axes field plasma magnetic enough tric and is rotation there magnetospheric (whe pulsar model the force-free the ideal affect axisymmetric electron- the should In copious structure. which the plasma contains positron magnetosphere pulsar The INTRODUCTION 1 .A Petrova magnetosphere A. S. force-free pulsar the at look novel A c nttt fRdoAtooy A fUrie Chervonoprap Ukraine, of NAS Astronomy, Radio of Institute -al [email protected] E-mail: 06TeAuthors The 2016 oee,ohreatsltoso h o-ierpulsar non-linear the of solutions exact other However, ; ; cehvkye l 2013 al. et Tchekhovskoy calmn aoe 1973 Wagoner & Scharlemann 000 , 1 – 6 21)Pern 1Spebr21 oplduigMRSL MNRAS using Compiled 2018 September 21 Preprint (2016) ⋆ ; otpuo 2005 Contopoulos ; rla&Jcbo 2014 Jacobson & Gralla ; hn&Blbrdv2014 Beloborodov & Chen h antcfil eie rmterttoa hrceitc ft of cot characteristics the rotational with the consistent from appears derived field magnetic the nrymc oeecety mligr-siaino h tla mag stellar the of re-estimation sc implying plas spatial efficiently, larger the B more much ne of the at much that consumes nebula energy magnetosphere with wind axisymmetric line pulsar the the in configuration, m as geometry, new The directly cylindrical support. observed observational common its than examine construc and equation, rather basis pulsar its the puls on of a solution model of dipolar exact magnetosphere an force-free present axisymmetric stationary The ABSTRACT neutron o words: unification star Key neutron the into betwe insight angle basis. gives random a and with axes along stars rotational neutron the of strength field rbtdi h nevl[0 interval the in tributed n 0 ew ; ihl1991 Michel kmt 1974 Okamoto 3 = . 3 × ; ihl1973 Michel Spitkovsky 10 H lsa usr:gnrl–sas antcfils–stars: – fields magnetic stars: – general pulsars: – plasmas – MHD .Conse- ). − 4 Michel .The ). B/P the ) ur- re i- raSr,4 hro 10,Ukraine 61002, Kharkov 4, Str., orna ) - , ; where , π/ , ] u euti ugsieo nqeata magnetic actual unique a of suggestive is result Our 2]. sfnto ftepla eid npriua,i h magne the if field particular, In magnetic period. stellar pulsar the the of of function re-estimation as substantial a to ao)btas scmae otelse fteclassical the of losses the to ( compared magnetosphere as force-free ro- compared also orthogonal as the but (of only rota- losses tator) not magnetodipolar star common efficiently neutron the more to the much off energy takes tional appear and magnetosphere twisted force-free substantially dipolar the that, Besides 2005 Contopoulos aigwt h eto tr(r sa xeto,o the on exception, an as coro- (or, different magnetosphere a star at the rotating neutron magnetosphere of the case with the tating tradition on were equation concentrated pulsar analytic ally the Both of literature. studies numerical preceding equa- and pulsar the the in of overlooked solution dipolar tion exact an present we Here h usrwn euaosre ietya uhlarger much at directly e.g., observed (see, nebula scale of spatial wind form the pulsar resembling the geometry, rath cylindrical toroidal common has than magnetosphere force-free dipolar purely support. observational its examine b its and on sis model veloci magnetospheric rotational the and construct current distributions, poloidal unique pulsar the ’complete’ with tion the the that In ( demonstrate equation. equation cannot pulsar we dipole ’truncated’ paper, pure a such a present of that solution known the well veloc be is rotational It the of distribution. form ity grounded physically actual the χ P lw where -law, u otepcla oainlvlct itiuin the distribution, velocity rotational peculiar the to Due stepla eid hnte7odrsatrof scatter 7-order the Then period. pulsar the is kmt 1974 Okamoto χ ) anybcueteewsn daa to as idea no was there because mainly )), sarno uniyuiomydis- uniformly quantity random a is aglsve l 2015 al. et Kargaltsev osalwa xc ioa solu- dipolar exact an allow does ) adn ta.1999 al. et Harding h magnetospheric the t to trrotational star utron ri osdrd We considered. is ar constant ntemgei and magnetic the en eplasobserved pulsars he h geometrical the n dlhstoroidal has odel A T l.I t new its In ale. E o review). a for , tl l v3.0 file style X eoiy(e.g., velocity .Ti leads This ). aoutflow ma ei field, netic ty er a- al s - - - 2 S. A. Petrova tars can be regarded as the axisymmetric force-free rotators, that in the axial region the force-free regime is broken. This their magnetic field strength should be 1010 1011 G, just contrasts with the monopolar case (3), the consequent nu- ∼ − as in the Central Compact Objects (CCOs). Furthermore, merically simulated models (e.g., Contopoulos et al. 1999; with our new magnetic field estimate, an unphysical trend Contopoulos 2005; Spitkovsky 2006; Tchekhovskoy et al. of the stellar magnetic field strength with pulsar period is 2013; Chen & Beloborodov 2014; Philippov & Spitkovsky removed, whereas the residual scatter of the magnetic field 2014; Gralla & Jacobson 2014) and also with the numeri- values can be attributed to the random angle between the cal extensions toward the jet-like structure (Goodwin et al. rotational and magnetic axes of a pulsar. 2004; Lovelace et al. 2006; Takamori et al. 2014), where The plan of the paper is as follows. In Sect. 2, we present A(0) and Ω(0) are finite, though the particle-in-cell sim- an exact dipolar solution of the pulsar equation, and its ulations of the pulsar magnetosphere do arrive at a uniqueness is demonstrated in Appendix A. The dipolar strong accelerating electric field along the magnetic axis force-free model of the pulsar magnetosphere is developed (Yuki & Shibata 2012). in Sect. 3. Observational consequences of our model are ex- amined in Sect. 4. Section 5 contains a brief summary of the results. 3 BASIC FEATURES OF THE DIPOLAR FORCE-FREE MAGNETOSPHERE 3.1 Alfvenic surface 2 PULSAR EQUATION AND ITS EXACT SOLUTIONS The force-free magnetosphere described by Eq. (2) has the peculiar surface at Ωr sin θ = 1, where the differential rota- The presence of an abundant plasma in the pulsar magneto- tion velocity equals the speed of light. For constant Ω(f) it sphere implies that the magnetic and electric field strengths, is known as the light cylinder, whereas in the purely dipolar B and E, differ from those of a vacuum rotating dipole (e.g., −1/3 case (4) it is the torus, r = C1 sin θ, whose large and Michel 1973). In case of an ideal conductivity (E B = 0), −1/3 · small radii both equal to C /2 (see Fig. 1). axial symmetry and stationarity, the fields can be presented 1 The light torus separates the magnetosphere into the in- as ner and outer parts, with the force-free regime holding only f eϕ A(f) in the inner one. Then the poloidal current flowing along the B = ∇ × + eϕ, E = Ω(f) f , (1) r sin θ r sin θ − ∇ force-free magnetic lines should close at the light torus with where (r, θ, ϕ) are the coordinates in the spherical system the consequent energy transmission from the electromag- with the axis along the pulsar axis, the dimensionless func- netic field to the particles (see Sect. 3.2 below). The energy tions f(r, θ) and A(f) are proportional, respectively, to the transmission at distances of order of the light torus size is magnetic flux and poloidal electric current through a circle supported by observations (Beskin 2010; Aharonian et al. of radius ρ r sin θ centered on the magnetic axis at an al- 2012), and, in contrast to the cylindrical geometry, the ≡ titude z r cos θ above the origin, Ω(f) is the dimensionless toroidal one is appropriate for the current circuit closure. ≡ angular frequency characterizing the differential rotation of Leaving a place for an axial jet (at the dipolar magnetic the magnetosphere due to the drop of the electric potential field lines not crossing the light torus), the toroidal configu- V (f) across the magnetic field lines, Ω dV/df. ration of the inner magnetosphere resembles that observed ≡ With the dimensionless electric current and charge den- in the pulsar wind nebulae at a much larger spatial scale sities j and ρe given by the Maxwell’s equations j = (e.g., Kargaltsev et al. 2015). B and ρe = E, the electromagnetic force balance, ∇× ∇ · j B + ρeE = 0, reduces to the pulsar equation (Okamoto × 3.2 Magnetospheric structure beyond the light 1974) torus 2 ∂f dA dΩ 1 Ω2ρ2 ∆f = A + ρ2Ω ( f)2 , (2) Beyond the light torus, the force-free regime is expected to − − ρ ∂ρ − df df ∇ break, in which case the force-free electric field should switch where the functions f, A(f) and Ω(f) are unknown. The off. According to the condition E = 0, this should be ∇× classic monopolar solution has the form (Michel 1973) followed by the rise of the longitudinal electric field component suggestive of particle acceleration and, perhaps, pair f = 1 cos θ, A = f(2 f) , Ω = 1 . (3) − − formation just beyond the light torus. It is easily generalized for an arbitrary Ω(f), in which case Given that the azimuthal current beyond the light torus A = Ωf(2 f) (see, e.g., Petrova 2013). is absent, ( B)ϕ = 0, Eq.(A1) is still valid and the − ∇ × As is verified in Appendix, the pulsar equation (2) also poloidal magnetic field can be presented as a linear com- has a unique exact dipolar solution bination of the multipolar components. The magnetic flux continuity for the field lines crossing the light torus then 2 2 sin θ C1 4C1 yields f = , Ω= 2 ,A = 2 + C2, (4) r f s f sin2 θ sin2 θ f = + α cos θ α cos θ 2 , (5) where C1 and C2 are arbitrary constants. Among all conceiv- r − r able solutions of Eq. (2), it is the one seeming most relevant where α is an arbitrary constant (see Fig. 2 with α = 1). In- to the pulsar case. dependently of α, the magnetic field lines meet the equator At the magnetic axis (f = 0), the exact dipolar solu- at r = 1/f. The quantity re = 1/r∗ (where r∗ is the di- tion (4) exhibits a singularity in both A and Ω, implying mensionless stellar radius) for the field line passing through

MNRAS 000, 1–6 (2016) A novel look at the pulsar force-free magnetosphere 3

2 2 2 6 8 −2 the intersection of the light torus with the stellar surface with ξ 3Ω∗/2R∗c . For R∗ = 10 cm we have ξ 10 P ≡ −2 1 2 ∼ presents one more magnetospheric scale. and r 10 P / ; then the pair formation front lies close to ≪ the stellar surface, as in the common polar gap models. The 2 two surfaces intersect at the polar angle θPFF √ξr∗, which ≈ 3.3 Appearance of the closed field line region is always larger than that of the light torus footprint at the stellar surface, θLT r∗, with the difference, θPFF θLT Note that the dipolar solution (4) allows arbitrary choice of ≈ − ≈ 0.225r∗, presumably defining the actual width of the closing the constants C1 and C2, in which case the poloidal current current layer as well as the size of the resultant hot spot at distribution may differ substantially. Hereafter we dwell on the stellar surface. the case C1 = 1, C2 = 4, which is most similar to what − is commonly thought of a stable pulsar magnetosphere (see, e.g., Contopoulos 2005). (The case C2 = 0 corresponds to 3.4.2 Energy transfer in the current-carrying region a corona-like structure resembling that suggested for the magnetars (Thompson et al. 2002) and pulsars (Gruzinov In the regions encompassing the pair formation front (PFF) 2006)). Then the closed field line region lies entirely inside and the light torus (LT), the azimuthal magnetic field com- the light torus and touches it at the equator (see Fig. 3). At ponent can be presented as the boundary of the closed region, f = 1, we have A = 0 and Aσ(µ1) A [1 σ(µ2)] Ω = 1. A rigid corotation with the neutron star, Ω 1, and B 1 = , B 2 = , (9) ≡ ϕ ϕ − the absence of a poloidal current, A 0, are also the typical r sin θ r sin θ ≡ attributes of the force-free closed region itself. The numeri- where σ(µ) stands for the smoothed Heaviside function, cal solution of the pulsar equation (2) in the closed force-free whereas µ1(r, θ) and µ2(r, θ) are identically zero at the PFF region with the boundary condition f = sin2 θ/r = 1 is pre- and LT, respectively. Then the corresponding currents read sented in Fig. 3. dA f eϕ Aδ(µ1) µ1 j1 = ∇ × σ(µ1)+ ∇ eϕ , df r sin θ r sin θ µ1 × |∇ | 3.4 Physical picture of the current-carrying force-free region dA f eϕ Aδ(µ2) µ2 j2 = ∇ × [1 σ(µ2)] ∇ eϕ , At f < 1, the dipolar force-free magnetosphere appears df r sin θ − − r sin θ µ2 × |∇ | substantially twisted. Although the poloidal magnetic field (10) component dominates the azimuthal one even at the light torus, the twist is strong in a sense that the resultant where δ(µ) dσ/dµ is the smoothed Dirac delta-function. ≡ electric charge and poloidal current densities ρe and jp The first and second terms in the above equations corre- exceed the Goldreich-Julian ones, ρGJ and ρGJc (where spond to the volume poloidal current and the closing current ρGJ B/P c is the charge density necessary to screen the sheets, respectively. ≡ induction electric field of a vacuum rotating magnetic dipole The current sheets are not force-free, and the non- (Goldreich & Julian 1969) and c the speed of light): compensated azimuthal force is

2 jp 2 ρe 2 1 + sin θ A µ1,2 = , = . (6) Fϕ1,2 = 2 2 ∇ f , (11) 2 2 2 2 ± r sin θ µ1,2 ×∇ ρGJc f 1 f ρGJ f √1 + 3 cos θ − |∇ | The super-Goldreich-Julianp current density dictated by where the plus sign corresponds to the subscript ’1’ and vice versa. The moment of the force (11), K r Fϕeϕ, acts the dipolar force-free magnetosphere cannot be realized by ≡ × the charges of one sign and therefore cannot be supplied to spin up the PFF and to spin down the LT, adjusting the directly from the neutron star surface. We suppose that at current sheets to their neighbourhood. The corresponding power, K Ω, taken off at the PFF and released at the LT, f < 1 the force-free region matches the vacuum one at the − · surface corresponding to the equality of the electric poten- is tials. It is the surface where the closing current sheet should AΩ µ1,2 w = ∇ , f, eϕ . (12) coexist with the pair formation front supplying the plasma r sin θ µ1,2 ∇ necessary to sustain the force-free poloidal current (Petrova |∇ | 2014). The above equation appears exactly equal to the current losses in the sheets, j E, and also to the Poynting flux 1,2 · (E, B, µ/ µ ) carried in the force-free region through ∇ |∇ | 3.4.1 Geometry of the pair formation front the element of the revolving surface with the unit normal µ/ µ . ∇ |∇ | With the vacuum (Goldreich & Julian 1969) and force-free Taking into account that eϕ µ/ µ is the unit vector ×∇ |∇ | electric field potentials Φ and V in the form tangent to the line µ = 0, eτ dr/dθ, integration of Eq.(12) ≡ 5 2 3 3 over the whole surface is reduced to B∗Ω∗R∗ 3 cos θ 1 B∗Ω∗R∗ R Φ= − , V = (7) 3cR3 2 2c3 2 − − sin θ W = A(f)Ω(f)df (13) (where R rc/Ω∗ is the dimensional coordinate) the equa- ≡ Z tion for the inner boundary of the force-free region reads and yields

2 2 1 r∗ 1+ 1 r∗ 4 2 2 2ξr = sin θ(3 cos θ 1) (8) W = −2 log − . (14) − p r∗ − pr MNRAS 000, 1–6 (2016) 4 S. A. Petrova

As the dimensionless stellar radius is small, r∗ 1, the lat- (P > 1 s), though it is still within the limits expected for −2 ≪ ter is simplified to W r∗ . For the dimensional quantities, the neutron stars. ≈ the surface integral of the power transmitted is Interestingly, if the magnetars with their customary 14 15 2 2 4 magnetic fields B 10 10 G and periods P = 5 12 B∗ Ω∗R∗ ∼ − − W , (15) s can be regarded as force-free aligned rotators, their actual ≈ 4c fields should be 1010 1011 G, just as those known for ∼ − where B∗ is the magnetic field strength at the stellar surface, the CCOs (Mereghetti et al. 2015; Kaspi & Kramer 2016). Ω∗ the angular rotation frequency of the star, R∗ the stellar Because of the dramatic difference in the magnetic field radius and both hemispheres are taken into account. strength, CCOs used to be regarded as ’anti-magnetars’ The assumed coexistence of the polar gap with the (Halpern et al. 2007) or magnetars with the magnetic field closing current sheet seems to have microphysical grounds buried under the crust (Vigano & Pons 2012). Furthermore, (Petrova 2014). The particles flowing across the poloidal it is the CCOs that do not exhibit any signature of the mag- magnetic field lines emit synchrotron photons, which are be- netospheric activity, being the most probable candidate for lieved to be an important ingredient of the pair production classical magnetodipolar losses, in which case the existing scenario. The primary particles are accelerated in the longi- estimates of their magnetic field should remain true. tudinal electric field resulting from the switching on of the Note that the magnetar magnetic fields in the range force-free field on condition that E = 0. Thus, it is 1014 1015 G are generally believed to be directly inferred ∇ × − the current losses that should ultimately provide the par- from the X-ray observations of cyclotron features (see, e.g., ticle distribution sustaining the force-free regime at higher Tiengo et al. 2013). It should be kept in mind, however, that altitudes. such estimates correspond to an assumption of proton lines. In the differentially rotating force-free region, the en- In case of electron lines, the magnetic field estimates should − ergy is transmitted along the magnetic field lines via the be less by a factor of 5 10 4, i.e. 1011 G, well in line × ∼ Poynting flux and is ultimately deposited to the particles with our result. at the light torus. The power transmitted is generally be- Our radical re-estimation of the magnetar magnetic lieved to be supplied by the current closing at the stel- field rules out the magnetic field energy as a dominant lar surface and exerting a decelerating moment on the star energy source of these objects. The magnetic field decay (Beskin et al. 1983). In our scheme, however, most of the (Thompson & Duncan 1996) is usually introduced because closing current sheet neighbours with the vacuum region, the magnetar rotational energy loss rate is too low to account though a small segment does lie at the stellar surface. Ap- for the persistent X-ray luminosities observed, whereas their parently, it is the vacuum electromagnetic field that feeds giant flares require the energy budget far in excess of the to- the force-free region with energy and also adjusts the stellar tal stellar rotational energy. Note, however, that the X-ray rotation via interaction with its interior field. luminosities of CCOs, with their low magnetic fields, also exceed their rotational energy loss rate. Perhaps, both types of neutron stars may be powered by the stellar thermal en- 4 APPLICATIONS TO PULSARS ergy, which appears high enough to account for the magnetar energetics (Heyl 2005). Leaving aside the details of energy transformation, we con- A plethora of magnetar observational manifestations, front the power (15) transmitted in the force-free region with including transient emission and drastic changes in the ro- the loss rate of the stellar rotational energy, IΩ∗Ω˙ ∗ (where tational characteristics (see, e.g., Mereghetti et al. 2015, for I is the neutron star’s moment of inertia and Ω˙ ∗ the tem- a recent review), hint at an unstable adjustment of the dif- poral derivative of the stellar rotation frequency Ω∗). The ferentially rotating force-free region with its neighbourhood. magnetodipolar losses of the neutron star, Indeed, the neutron star deceleration implies an increase of 2 4 6 2 the light torus, and this process does not necessarily hold 2B∗ Ω∗R∗ sin χ Wdip = (16) monotonically. 3c3 Figures 4,a and 4,b show the magnetic field strength (where χ is the angle between the magnetic and rotation derived for the cases of the vacuum orthogonal and aligned axes of a pulsar) are absent not only in the aligned rotator force-free rotators as functions of the pulsar period for the case (χ = 0), since the radiation of a frequency Ω∗ should ∼ pulsars from the ATNF Pulsar Catalogue (Manchester et al. be formed over a region of a size c/Ω∗ actually occupied ∼ 2005). The non-physical trend of B with P present in the for- with the force-free plasma rather than vacuum (Beskin et al. mer case is completely excluded in the latter one. In Fig. 4,b, 1983). the pulsars are expected to move with age toward longer pe- Comparison of Eqs. (15) and (16) proves that the riods at a constant B/P , approaching ultimately the death aligned force-free magnetosphere can consume the stellar ro- line. Take note of the sources with transient (RRATs) and tational energy much more efficiently than the vacuum or- quenched (INSs) radio emission near the expected death line. thogonal rotator (χ = π/2), with the surface magnetic field The large and apparently random scatter of the points strength being re-estimated as in Fig.4,b is most probably attributable to the unaccounted 0 −4 non-zero angle χ in real pulsars. The distribution of the nor- Bnew 3.3 10 B/P , (17) ≈ × mal radio pulsars in B/P appears accurately symmetric (in where B = 3.2 1019(P P˙ )1/2 G is the customary value contrast to the distribution in B) and well fits the cot χ-law × based on magnetodipolar losses of the orthogonal rotator with χ being the random quantity uniformly distributed in 45 2 6 and it is taken that I = 10 g cm and R∗ = 10 cm. Thus, the range [0, π/2] (see Fig.5 and Appendix B for details). 0 Bnew appears much less, especially for long-period pulsars Then the energy loss rate (15) of an aligned force-free rota-

MNRAS 000, 1–6 (2016) A novel look at the pulsar force-free magnetosphere 5 tor can be empirically extended to the arbitrary inclination Contopoulos I., 2005, A&A, 442, 579 χ as Contopoulos I., Kazanas D., Fendt C., 1999, ApJ, 511, 351 4 2 0 2 2 Goldreich P., Julian W. H., 1969, ApJ, 157, 869 R∗Ω∗(B ) cos χ W new , (18) Goodwin S. P., Mestel J., Mestel L., Wright G. A. E., 2004, MN- ≈ 4c 1+ a2 sin2 χ RAS, 349, 213 Gralla S. E., Jacobson T., 2014, MNRAS, 445, 2500 where a 1 is a constant and B0 is the same for all ≫ new Gruzinov A., 2006, arXiv0604364 normal pulsars. At χ 0 we arrive at the magnetar case, → Halpern J. P., Gotthelf E. V., Camilo F., Seward F. D., 2007, whereas at χ π/2 the force-free losses cease, giving place → ApJ, 665, 1304 to the magnetodipolar ones. With the maximum of the his- Harding A. K., Contopoulos I., Kazanas D., 1999, ApJ, 525, L125 togram in Fig.5 at log B/P = 12.25, the normal pulsar mag- Heyl J. S., 2005. Magnetars. 22nd Texas Symposium on Relativis- netic field matches that of the CCOs and the re-estimated tic Astrophysics, 130 magnetar field on condition that a 10 102. Kargaltsev O., Cerutti B., Lyubarsky Y., Striani E., 2015, Space ∼ − Sci. Rev. 191, 391 Kaspi V. M., Kramer M., 2016, ArXiv e-prints. arXiv:1602.07738 Lovelace R. V. E., Turner L., Romanova M. M., 2006, ApJ, 652, 5 CONCLUSIONS 1494 Manchester R. N., Hobbs G. B., Teoh A., Hobbs M., 2005, ApJ, We have found the exact dipolar solution of the pulsar equa- 129, 1993 tion and presented the magnetospheric model on its ba- Mereghetti S., Pons J. A., Melatos A., 2015, Space Sci. Rev., 191, sis. With its general toroidal structure, the magnetosphere 315 leaves a place for an axial jet, in striking contrast to the usual Michel F. C., 1973, ApJ, 180, L133 cylindric force-free models and in line with the jet+torus Michel F. C., 1991, Theory of neutron star magnetospheres. structure of the PWN observed. Chicago, IL, University of Chicago Press, 533 p. The energy budget of the axisymmetric purely dipo- Okamoto I., 1974, MNRAS, 167, 457 lar force-free magnetosphere appears much larger than that Petrova S. A., 2013, ApJ, 764, 129 of the previous force-free models, implying substantial re- Petrova S. A., 2014, Astronomische Nachrichten, 335, 246 estimation of the neutron star magnetic field based on Philippov A. A., Spitkovsky A., 2014, ApJ, 785, L33 the pulsar rotational characteristics observed. In particu- Scharlemann E. T., Wagoner R. V., 1973, ApJ, 182, 951 Spitkovsky A., 2006, ApJ, 648, L51 lar, if the magnetars can be regarded as the axisymmet- Takamori Y., Okawa H., Takamoto M., Suwa Y., 2014, PASJ 66, ric force-free rotators, their actual magnetic fields should be 25 1010 1011 G, just like those of the CCOs, which are be- ∼ − Tchekhovskoy A., Spitkovsky A., Li J.G., 2013, MNRAS, 435, L1 lieved to lose rotational energy via magnetodipolar radiation Thompson, C., Duncan R. C., 1996, ApJ, 473, 322 and therefore to be characterized by the classical magnetic Thompson C., Lyutikov M., Kulkarni S. R., 2002, ApJ, 574, 332 field estimates. Tiengo A., Esposito P., Mereghetti S., Turolla R., Nobili L., The new estimate of the neutron star magnetic field Gastaldello F., Gotz D., Israel G. L., Rea N., Stella L., Zane strength appears to have different dependence on the pul- S., Bignami G. F., 2013, Nature, 500, Is. 7462, 312. sar period. Consequently, an unphysical trend of the stel- Vigano D., Pons J. A., 2012, MNRAS, 425, 2487. lar magnetic field with period is removed. Furthermore, the Yuki S., Shibata S., 2012, PASJ, 64, 43 residual scatter of the magnetic field values derived from the pulsar rotational characteristics may well be attributed to the random angle between the rotational and magnetic axes of pulsars. Then the magnetic field of neutron stars, APPENDIX A: EXACT DIPOLAR SOLUTION similarly to their other basic parameters, such as the mass OF THE PULSAR EQUATION and radius, may be approximately the same for the whole 2 neutron star population (perhaps, except for the millisecond For a pure dipole, f = sin θ/r, we have pulsars with their peculiar evolution). As a result, unifica- ∂2f 1 ∂f ∂2f tion of the neutron star properties (Kaspi & Kramer 2016) + =0 (A1) ∂ρ2 − ρ ∂ρ ∂z2 may be realized primarily on the geometrical basis. and therefore the pulsar equation (2) can be presented in the form

ACKNOWLEDGEMENTS 2 2 2 F1(f)ρ ∆f + F2(f)ρ ( f) = F3(f). (A2) ∇ The work have used the ATNF Pulsar Catalogue available We have to find out whether the set of functions F1 2 3 sat- at http://atnf.csiro.au/people/pulsar/psrcat , , isfying Eq.(A2) exists and whether it is unique. An arbitrary F (ρ,z) is a function of f given that the Poisson bracket, REFERENCES ∂F ∂f ∂F ∂f [F, f] (A3) Aharonian F. A., Bogovalov S. V., Khangulyan D., 2012, Nature, ≡ ∂ρ ∂z − ∂z ∂ρ 482, 507 Beskin V. S., Gurevich A. V., Istomin I. N., 1983, ZhETP, 85, is zero. Applying the Poisson bracket to both sides of 401 Eq.(A2) yields Beskin V. S., 2010, Physics Uspekhi, 53, 1199 2 2 2 Chen A.Y., Beloborodov A. M., 2014, arXiv1406.7834 F1(f) ρ ∆f, f + F2(f) ρ ( f) , f = 0 , (A4) ∇ MNRAS 000, 1–6 (2016) 6 S. A. Petrova which may be further reduced to This paper has been typeset from a TEX/LATEX file prepared by the author. ρ2∆f, f , f = 0. (A5) [ρ2( f)2, f] " ∇ # For the purely dipolar flux function, Eq.(A5) is fulfilled identically, and hence the set of functions F1,2,3(f) satisfying Eq.(A2) really exists. Using Eqs.(A2) and (A4) and keeping in mind the actual form of the pulsar equation (2), we arrive at the equations f dΩ dΩ dA Ω2 = Ω , 2f 2Ω = A (A6) − 2 df df df with a unique solution

2 C1 4C1 Ω= ,A = + C2. (A7) f 2 s f 2

APPENDIX B: RESULTS OF STATISTICAL ANALYSIS OF THE B/P DISTRIBUTION FOR THE NORMAL RADIO PULSARS Based on Fig. 5, we hypothesize that the distribution of B/P for the normal radio pulsars observed may be identified with the cot χ-distribution, where χ is the random quantity uniformly distributed in the interval [0, π/2]. According to the Pearson’s criterium, this hypothesis cannot be rejected at a significance level 13%. The Kolmogorov-Smirnov test proves that the two distributions are the same at a 88%-significance level. For the 13-bin histogram of B/P (not shown), the significance levels for the Pearson’s criterium and Kolmogorov- Smirnov test are 5% and 99.5%, respectively.

MNRAS 000, 1–6 (2016) A novel look at the pulsar force-free magnetosphere 7

Figure 1. The ideal force-free magnetosphere of an aligned rotating dipole: A general scheme; the magnetic ﬁeld lines are shown by solid lines and correspond to the levels of f from 0 to 1 with a step 0.1; the pair formation front is shown by the bold solid line, the light torus by the dashed line; the arrows mark the current ﬂow directions.

MNRAS 000, 1–6 (2016) 8 S. A. Petrova

2.5

z 1.5

0.5

0 0 0.5 1 1.5 2 2.5 3 ρ

Figure 2. The ideal force-free magnetosphere of an aligned rotating dipole: The region outside of the light torus; the light torus is shown by a bold dashed line, solid lines correspond to the magnetic ﬁeld lines given by Eq. 5 with a = 1, dashed lines display the ﬁeld lines of a pure dipole, f changes from 0.2 to 1 with a step 0.2.

0.4

0.3

0.2

0.1

z 0

−0.1

−0.2

−0.3

−0.4 0 0.2 0.4 0.6 0.8 1 ρ

Figure 3. The ideal force-free magnetosphere of an aligned rotating dipole: The closed ﬁeld line region; solid lines represent the numerical force-free solution of the pulsar equation 2 for A ≡ 0, Ω ≡ 1 and the boundary condition f = sin2 θ/r = 1, gray lines show the ﬁeld lines of a pure dipole, the step in f is 0.5.

MNRAS 000, 1–6 (2016) A novel look at the pulsar force-free magnetosphere 9

16 10 radio loud radio quiet high−energy 14 10 binary rrat ins axp 12 10 log B 10 10

8 10

6 10 −3 −2 −1 0 1 2 10 10 10 10 10 10 log P

15 10 raido loud raio quiet high−energy 14 10 binary rrat ins axp 13 10

log (B/P) 12 10

11 10

10 10 −3 −2 −1 0 1 2 10 10 10 10 10 10 log P

Figure 4. Stellar magnetic ﬁeld strength based on the pulsar rotational characteristics P and P˙ from the ATNF Pulsar Catalogue (Manchester et al. 2005); a) the case of magnetodipolar losses of an orthogonal vacuum dipole; b) the case of wind losses of an aligned force-free dipole.

MNRAS 000, 1–6 (2016) 10 S. A. Petrova

800

700

600

500

N 400

300

200

100

0 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 log B/P

Figure 5. Histogram of the force-free - based stellar magnetic ﬁeld for the normal radio pulsars; the solid line with asterisks corresponds to the distribution of cot χ, where χ is the random quantity uniformly distributed in the interval [0, π/2] (see Appendix B for the statistical analysis).

MNRAS 000, 1–6 (2016)