Orthogonal pulsars as a key test for pulsar evolution
E.M. Novoselov, V.S. Beskin, A.K. Galishnikova, M.M. Rashkovetskyi, A.V. Biryukov
MMNRAS , 000–000 (0000) Preprint 12 May 2020 Compiled using MNRAS L A TEX style file v3.0
Orthogonal pulsars as a key test for pulsar evolution
E. M. Novoselov , V. S. Beskin , (cid:63) , A. K. Galishnikova , , M. M. Rashkovetskyi andA. V. Biryukov , Columbia University, Department of Physics, New York City, 116th str. Broadway, New York, 10027, United States Moscow Institute of Physics and Technology, Dolgoprudny, Institutsky per. 9, Moscow Region, 141700, Russia P.N.Lebedev Physical Institute, Leninsky prosp. 53, Moscow, 119991, Russia Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA The Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel Sternberg Astronomy Institute, Moscow State University, Universitetsky prosp. 13, Moscow, 119234, Russia Kazan Federal University, Institute of Physics, Kremlyovskaya str. 18, Kazan, 420008, Russia
Accepted, Received
ABSTRACT
At present, there is no direct information about evolution of inclination angle χ be-tween magnetic and rotational axes in radio pulsars. As to theoretical models of pulsarevolution, they predict both the alignment, i.e. evolution of inclination angle χ to ◦ ,and its counter-alignment, i.e. evolution to ◦ . In this paper, we demonstrate that thestatistics of interpulse pulsars can give us the key test to solve the alignment/counter-alignment problem as the number of orthogonal interpulse pulsars ( χ ≈ ◦ ) drasti-cally depends on the evolution trajectory. Key words: stars: neutron — pulsars: general.
Fifty years of intensive researching of radio pulsars have notleaded to complete understanding of the nature of many keyprocesses in the pulsar magnetosphere (Manchester & Tay-lor 1977; Lyne & Graham-Smith 2012; Lorimer & Kramer2012). We still know neither the mechanism of coherent ra-dio emission nor the real structure of electric currents thatare responsible for braking of neutron stars. In this article,we try to show how the existence of orthogonal radio pul-sars itself may help us to clarify several important questionsfaced by the theory.Remind that one of the most important unsolved ques-tions is the problem of the evolution of the inclination angle χ between magnetic and rotational axes, i.e. between mag-netic moment m and the angular velocity Ω (see, e.g., Be-skin 2018). Nowadays there are the theories predicting boththe alignment, i.e., evolution of inclination angle χ to ◦ , andits counter-alignment, i.e., evolution to ◦ . In what followswe call the first group as MHD model, as the alignmentevolution is mainly based on the results of numerical simu-lations within ideal magneto-hydrodynamics (Tchekhovskoyet al. 2013), and the second group as BGI model as thecounter-alignment scenario was first proposed by Beskinet al. (1984).Unfortunately, at present it is impossible to determinethe evolution of inclination angle χ of individual pulsars (cid:63) Contact e-mail: [email protected] directly from observations. As to different indirect meth-ods, they give controversial results (Rankin 1990; Tauris& Manchester 1998; Faucher-Giguère & Kaspi 2006; Wel-tevrede & Johnston 2008; Young et al. 2010; Gullón et al.2014). In particular, it was found both directly (i.e., by theanalysis of the χ distribution) and indirectly (i.e., from theanalysis of the observed pulse width) that statistically theinclination angle χ decreases with period P as the dynam-ical age τ D = P/ ˙ P increases. At first glance, these resultsdefinitely speak in favor of alignment mechanism. However,as was demonstrated by Beskin et al. (1993), the averageinclination angle of pulsar population, (cid:104) χ (cid:105) ( τ D ) can decrease even if inclination angles of individual pulsars increase withtime due to dependence of so-called “death line” on the in-clination angle χ . Moreover, recently, by analyzing 45 yearsof observational data for the Crab pulsar, Lyne et al. (2013)found that the separation between the main pulse and in-terpulse increases at the rate of . ◦ per century implyingsimilar growth of χ (see, however, Arzamasskiy et al. 2015;Zanazzi & Lai 2015).Thus, until now it was not possible to formulate a testthat would allow to distinguish between these two modelsof evolution, as both models reproduce well enough the realdistribution of pulsars on the P – ˙ P diagram. For this rea-son, so far the mechanism for pulsar braking has remainedunknown. Further we will show that statistics of orthogonalinterpulse pulsars (with inclination angles close to ◦ ) maygive us a key test to solve this problem as the number of c (cid:13) a r X i v : . [ a s t r o - ph . H E ] M a y E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov
Table 1.
The number of axisymmetrical (SP) and orthogonal(DP) interpulse pulsars. Lower limit corresponds to the number ofcertainly defined pulsars, when different catalogues are in agree-ment with each other; the upper limit corresponds to the highestnumber that can be obtained (see Appendix A). When determin-ing the percentage, the ATNF catalogue (Manchester et al. 2005)is used. P [s] > N SP ÷
10 2 ÷ ÷ . ÷ . . ÷ . . ÷ . N DP ÷
26 3 ÷ ÷ . ÷ . . ÷ . . ÷ . interpulse pulsars directly depends on the inclination angleevolution.As is well-known (Manchester & Taylor 1977; Lyne &Graham-Smith 2012; Malov 1990), the interpulse appears ifwe observe either two opposite poles (a double-pole or DPpulsar) or the same pole twice (a single-pole or SP pulsar);in the latter case, the two peaks correspond to the doubleintersection of the same hollow-cone directivity pattern. Forthe DP case, the inclination angle χ is close to ◦ , while fora SP pulsar, this angle is close to ◦ . Since the procedure fordetermining of the inclination angle (which is based on theanalysis of the polarization properties of mean profiles) con-tains a number of uncertainties, different catalogues (see,e.g., Maciesiak & Gil 2011; Malov & Nikitina 2013) givedifferent number of orthogonal pulsars (see Table 1 and Ap-pendix A). But in any way we can be sure that there areabout dozens of them, therefore several certain claims canbe done.The present paper is organized as follows. In Section 2the necessary data on the properties of orthogonal pulsarsare given. Further, in Section 3.1 we start with defining theregions within the polar cap where the secondary plasmais generated for nearly orthogonal pulsars. Assuming, asis usually done, that the intensity of the radio emission isproportional to the density of the outgoing plasma (in theBGI model, the radio luminosity is a fraction of − ofthe plasma energy flux) this allows us not only to clarifythe shape of the directivity pattern for interpulse pulsars,but also define the parameters of the “death-line” and, inparticular, its dependence on the magnetic field on the neu-tron star magnetic pole B (Section 3.2). Then, accordingto this information, we make an observational constraint onone of the key model parameters (Section 3.3). In Section4 we define the visibility function that connect the “true”distribution by physical parameters to the observed one fororthogonal interpulse pulsars. Then in Section 5 we use thekinetic equation approach to obtain the distributions theo-retically. Finally, in Section 6 we determine the number ofexpecting orthogonal interpulse pulsars for different modelsand show that such an analysis provides a test to distin-guish the alignment and counter-alignment evolution sce-nario. The Section 6.1 is a recap of previous results, while in6.2 we make a crucially important correction to BGI modeland provide its main implications. Then in Section 6.3 weconduct a Monte-Carlo simulation to consider the correctedevolution law more thoroughly and compare its results to observations. Finally, in Appendix we discuss the ATNF cat-alogue limitations and give possible ways that would allowus to overcome the incompleteness of this catalog.It should be emphasized that such a consideration, whentwo-dimensional distribution of the density of the outflowingplasma over the surface of the polar cap in the vicinity ofthe death line is studied, is actually carried out for the firsttime. Until now, the efficiency of particle production hasbeen discussed either in the one-dimensional case (see, e.g.,Shibata 1997; Beloborodov 2008; Timokhin 2010; Timokhin& Arons 2013; Timokhin & Harding 2015) or, more recently,far from the death line (Bai & Spitkovsky 2010; Lockhartet al. 2019). In any case, in these works the strong distortionof the radiation pattern of orthogonal pulsars was not dis-cussed in detail. As a result, it is still often assumed that fororthogonal pulsars the directivity pattern has standard hol-low cone structure (see e.g. Johnston & Kramer 2019). It isclear that this question becomes the key one in the analysisof the visibility function of orthogonal pulsars. To begin with, let us discuss two examples that show whatkind of information we can gain from the very existenceof orthogonal interpulse pulsars. For simplicity, we put theinclination angle χ = 90 ◦ in this section.At first, remember that morphological and polarizationproperties of both pulses in observable interpulse pulsars aresimilar to ordinary ones (see, e.g., Keith et al. 2010). Thus,the process of secondary plasma generation passes with thesame pace as it is for ordinary pulsars. On the other hand, asis shown in Figure 1, main pulse and interpulse can belongto areas with different signs of Goldreich & Julian (1969)charge density (the one that is required to screen the longi-tudinal electric field) ρ GJ = − ΩB πc , (1)i.e., to different signs of the normal vector of the electricfield in areas of plasma particles acceleration. Hence, one canconclude that the injection of particles from the star surfacedoes not play important role in pair creation mechanismas the work function and moreover masses of electrons andprotons differs significantly.In other words, the very existence of orthogonal in-terpulse pulsars supports Ruderman & Sutherland (1975)theory, according to which the ejection of electrons fromthe star surface was assumed to be inefficient, in oppositionto Arons (1982) model, in which it was supposed that ejec-tion is free. Remind that latest simulations of particle gen-eration in polar regions of neutron stars (Timokhin 2010;Timokhin & Arons 2013; Timokhin & Harding 2015) haveactually proved that the injection from the surface does playa small role.We know that to support the high efficiency of pair cre-ation in the polar region of neutron star (which is necessaryto detect a neutron star as a radio pulsar), the potentialdrop near the star surface is to be similar to one for or-dinary pulsars. For Ruderman-Sutherland-type models thepotential drop ψ in the pair creation region can be evaluated MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution W > > > > > > > Figure 1.
Geometry of an orthogonal radio pulsar with inclina-tion angle χ = 90 ◦ . Observer located over the equatorial planedetects two pulses corresponding to two areas with different signsof Goldreich-Julian charge density ρ GJ (1). as ψ ≈ πρ GJ H , (2)where H is the inner gap height. This implies that potentialdrop ψ (2) is proportional to Goldreich-Julian charge den-sity ρ GJ ∝ B cos θ b , where θ b is the angle between angularvelocity Ω and magnetic field B .On the other hand, for orthogonal interpulse pulsars cos θ b near the polar caps is rather small: cos θ b ∼ (cid:18) Ω Rc (cid:19) / . (3)For this reason, for interpulse pulsars their magnetic field B should be much larger than for ordinary pulsars to maintainthe same efficiency of pair production. As even for fast pul-sars with period P about . − . seconds (as was shownin Table 1, most interpulse pulsars belong to this range) thecorresponding factor (Ω R/c ) − / = 20 – , observed inter-pulse pulsars must have ten times stronger magnetic fieldthan for ordinary pulsars. Hence, to analyze their statisticalproperties, the magnetic field distribution should be takeninto account.To sum up, one can conclude that detailed study ofproperties of orthogonal interpulse pulsars (that has notbeen done yet) should allow the significant advance in un-derstanding the key processes in the neutron star magneto-sphere. Moreover, as we show in what follows, a test thatmay answer the question about the evolution of the inclina-tion angle can be formulated. Before formulating the test itself, we need to find the shapeof the directivity pattern of orthogonal pulsars, which mustbe associated with the region of plasma generation in theirpolar cap. It is an essential question, as we need to findthe range of the inclination angles where both poles canbe observable. The main difference with the pulsars havingmoderate inclination angles χ is that for almost orthogonalpulsars the particle creation is to be depressed not only nearthe magnetic pole where the curvature radius R c of mag-netic field lines is very large, but also near the line wherethe Goldreich-Julian charge density (1) vanishes. Indeed, ac-cording to (2), in this case, the potential drop ψ is too smallto create pairs. Hence, the shape of the directivity patternthat highly depends on the charge density is to differ signif-icantly from the standard hollow cone. To estimate the height of the inner gap H (a domainwith nonzero longitudinal electric field where the accelera-tion of primary particles occurs) we use the relations pre-sented recently by Timokhin & Harding (2015). As was men-tioned above, these expressions actually do not differ fromones obtained by Ruderman & Sutherland (1975). Howeverwe will take the dependence of charge density on the angle θ b between magnetic field and rotational axis into consid-eration. It is easy to do if one change Ω to Ω | cos θ b | . As aresult, we obtain H RS = 1 . × | cos θ b | − / R / , P / B − / cm . (4)Here and in the following similar expressions the magneticfield B is expressed in G, pulsar period P in seconds,and curvature radius R c , in cm.Below we also use the results from the same work formultiplicity of particle generation λ = n e /n GJ : λ = 5 . × R − / , P − / B / , B < , (5) λ = 1 . × R − / , P − / B − / , B > . (6)As a result, we may write down the outflowing plasma num-ber density as n e = λ Ω B | cos θ b | πce . (7)Introducing now the polar coordinates ( θ m , φ m ) on the polarcap of the neutron star, we obtain for dipole magnetic fieldin the limit Ω R/c (cid:28) θ b = χ − θ m sin φ m . (8)Accordingly, for dipole magnetic field the curvature radius R c at the star surface looks like R c = 43 Rθ m . (9)Thus, using relations (4)–(9) one can find the distribution ofthe the number density n e of the outflowing plasma withinopen field line region.It is important to remember that equation (2) was ob-tained on the assumption that the height of the acceleratingarea H is less that its width. Usually, the polar cap radius R cap ≈ (Ω R/c ) / R is used as a characteristic size. And theequality H RS = R cap is taken as a condition of the "death-line" on P − ˙ P diagram (Manchester & Taylor 1977; Lyne &Graham-Smith 2012). But for orthogonal pulsars, we mustclarify this criterion. Below, as a condition for the applica-bility of the one-dimensional approximation, we put H RS < R min (10)where R min is the distance towards nearest characteristicpoint or lines: magnetic pole, edge of the polar cap or theline, where ρ GJ = 0 . In more detail, we put λ = λ RS (5)–(6)for H RS < R min , and use linear function λ = K R min match-ing it to the boundary H RS = R min so that λ vanishes atthe characteristic lines.In Figure 2 we show the areas of plasma generationregion for different periods P , magnetic fields B , and incli-nation angles χ . Level lines correspond to number density n e of outflowing plasma; critical number density n cr determin-ing the shape of the directivity pattern will be found in thenext Subsection. Here we make an assumption that the polar MNRAS , 000–000 (0000)
E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov a P = 0.1 b P = 0.2 c P = 0.3 n e / n cr (a) Different periods P ( χ = 85 ◦ , B = 2 ). a B = 5 b B = 3 c B = 2 n e / n cr (b) Different magnetic fields B ( χ = 85 ◦ , P = 0 . ). a= 80 o b= 85 o c= 89 o n e / n cr (c) Different inclination angles χ ( P = 0 . , B = 2 ). Figure 2.
Geometry of plasma generation region for differentparameters. Critical number densities n cr , which determine theshape of the directivity pattern, are shown in bold line. cap has circle shape with its radius R cap = f / ∗ (Ω R/c ) / R ,where the dimensionless polar cap area f ∗ belongs to intervalbetween 1.59 while χ = 0 ◦ and 1.96 while χ = 90 ◦ (Beskinet al. 1983; Gralla et al. 2017). As one can see, while theinclination angle χ goes to ◦ , geometry of plasma gener-ating areas become more and more complicated. For largeperiods P and small magnetic fields B generation of sec-ondary plasma obviously brakes down. Now, we have to determine so-called ”death line”, i.e., theboundary of the area of pulsar parameters that admit effec-
Table 2.
Values of n cr for three different values of A .A 0.5 1.0 2.0 n cr (10 cm − ) tive secondary plasma generation. In what follows we assumethat it is this condition that determines the boundary of thedirectivity pattern of the radio emission.As was shown by Beskin et al. (1993) (see also Arza-masskiy et al. 2017), within Ruderman-Sutherland model,the condition for the efficient secondary plasma generationwhich is necessary for the generation of radio emission, canbe written down as Q BGI < , where Q BGI = A P / B − / cos d − χ. (11)Here A ≈ and d ≈ . . Since the accuracy of determiningthe value of A is actually small, in what follows we assumethat A lies in the range from 0.5 to 2. To correlate thisvalue with our numerical results we suppose that generationof radio emission takes place only for large enough numberdensity of the outflowing plasma n e > n cr .As one can see from Fig. 3, the death line obtainednumerically fits the theoretical θ -dependence with an accu-racy about 10%. We take here magnetic field B = 1 as aclear mean value. Appropriate values n cr are given in Ta-ble 2. Thus, we can use these critical number densities todetermine the transverse structure of effective pair gener-ation region for orthogonal pulsars (and, hence, the shapeof their directivity pattern). In Figure 2 the correspondingcritical densities are shown in bold line. A It is important that the value of A can be constrained fromobservations while considering a subset of pulsars with mea-sured periods P , their derivatives ˙ P and inclination angles χ . Namely, pulsar death line equation Q BGI = 1 (11) can berewritten in the form cos / χ = P · ( A / B − / ) , (12)where B is model-consistent estimate of pulsar magneticfield taken again in G. Therefore, the “death line” ofpulsars is conveniently depicted on the plane of the incli-nation angles and the modified periods P · ( A / B − / ) .The value of magnetic field B explicitly depends on P , ˙ P ,and χ and can be directly obtained from pulsar spin-downlaw. We considered 153 normal pulsars with inclination an-gles χ evaluated by Lyne & Manchester (1988) and Rankin(1993) which magnetic fields were estimated within bothBGI and MHD approaches (see Appendix B). We adoptedfive reasonable values for A = 0 . , . , . , . and . andtested whether these pulsars satisfy the death line condition(12). As one can see from Figure 4, for both models the val-ues A (cid:46) . presume a remarkable gap between the cloudof pulsars and proposed “death line”. On the other hand,the values A > . presume significant number of objectsbeyond the “death line”.Of course, the data used above cannot be regarded as MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution Period , s , o . A = 0.5 Period , s o . A = 1 Period , s , o . A = 2 Figure 3.
Theoretical (11) (dashed curve) and numerical (solidcurve) “death lines" for different coefficient A . Appropriate valuesof n cr are given in Table 2. accurate due to the measurement errors in pulsar inclina-tion angles. And the problem here is that most of observersignore measurement errors for χ since that are affected bysignificant systematic uncertainties. Indeed, estimate of χ strongly depend on the adopted model of pulsar emissiongeometry. Nevertheless, we believe, that estimations madeby Lyne & Manchester (1988) and Rankin (1993) are rele- vant and their systematic errors, being unknown, still sig-nificantly less, than the whole scatter of χ throughout thepulsar subset. Thus, we conclude that A can be constrainedas . – . for real pulsars. In this section we determine the beam visibility function ofinterpulse pulsars V visbeam , that obviously plays the main rolein their statistics. For ordinary pulsars it is determined bythe width of the directivity pattern W r , i.e., by the radia-tion radius r rad , as for dipole magnetic field (Manchester &Taylor 1977; Lyne & Graham-Smith 2012) W = 32 (cid:18) f ∗ Ω Rc (cid:19) / (cid:16) r rad R (cid:17) / . (13)Here and below W corresponds to the pulse width at the50% intensity level usually presented in catalogs (factor / is well-known broadening of the dipole magnetic field). Ac-cordingly, we suppose that the total width W r = 2 W .Remember that observable width of the mean pulse de-pends on the inclination angle χW obs50 = W sin χ , (14)where, according to (13), W can be present as W = W √ P . (15)In what follows W is taken in degrees as it will be moresuitable in further calculations.For ordinary pulsars (i.e., for pulsars with inclinationangles χ < ◦ ) the visibility function is V visbeam = sin χ W r . (16)For orthogonal interpulse pulsars we have to replace W r withthe width of the area δW where both two poles can be ob-servable: V visbeam , = δW . In Fig. 5 two possible realizationsare shown. In the first case, when the inclination angle χ is not so close to ◦ , it is possible to observe interpulsethat correspond to the same sign of Goldreich-Julian chargedensity ρ GJ (1). On the contrary, when χ ≈ ◦ we haveto observe the regions that correspond to different signs of ρ GJ . Both cases can be implemented under certain condi-tions simultaneously. The total visibility function can bedetermined as δW = δW + δW .Certainly, the directivity pattern (and, therefore, thevisibility function δW ) depends on the generation level r rad .Below we consider two cases: r rad = 5 R and r rad = 7 R . Ac-cording to (13) we have W = 3 ◦ for r rad = 5 R , and W = 5 ◦ for r rad = 7 R correspondingly (sf. Rankin 1990; Maciesiak &Gil 2011). Assuming now again that the generation regionof radio emission repeats the shape of the plasma genera-tion domain, we can reconstruct the directivity pattern bytransferring corresponding plasma generation profiles shownin Fig. 2 along dipole magnetic field from the neutron starsurface r = R to the generation level r = r rad .In Fig. 6a–6c we present the visibility functions δW fordifferent pulsar parameters. As one can see, for ordinarymagnetic field B ∼ G the possibility to observe the
MNRAS , 000–000 (0000)
E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov (cid:0) ✁ ✂✄☎✆✝✝✂✞✝ (cid:0) ✁ ✂✄✝✆✝✝✂✞✝ (cid:0) ✁ ✂✄✟✠✡☛ ✆✝✝✂✞✝ (cid:0) ✁ ✂✄✞✆✝✝✂✞✝ (cid:0) ✁ ☞✆✝✝✂✞✝ ✌✍✎(cid:0)✏✑✒✏✓ ✔✏✕✖✗✒✏✓✘✙ ✚✂ ✂✙✆✝ ✂✙✝ ✂✙✞✝ ☞ ☞✙✆✝ (a) BGI model. (cid:0) ✁ ✂✄☎✆✝✝✂✞✝ (cid:0) ✁ ✂✄✝✆✝✝✂✞✝ (cid:0) ✁ ✂✄✟✠✡☛ ✆✝✝✂✞✝ (cid:0) ✁ ✂✄✞✆✝✝✂✞✝ (cid:0) ✁ ☞✆✝✝✂✞✝ ✌✍✎(cid:0)✏✑✒✏✓ ✔✏✕✖✗✒✏✓✘✙ ✚✂ ✂✙✆✝ ✂✙✝ ✂✙✞✝ ☞ ☞✙✆✝ (b) MHD model.
Figure 4.
Observational test for the death line equation (12) using 153 pulsars with known inclination angles χ for different models.The “death line” is shown by red solid line, while pulsar positions within these plots depend on the adopted value of A . As one can see,the values A ≈ . − . seem optimal for matching the border of pulsar cloud for both models. MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution Figure 5.
Two cases when it is possible to observe interpulse fororthogonal pulsars. interpulse is limited to very small periods P (cid:54) . s only.Observation of the interpulse for pulsars with periods of P of the order of 0.5 s becomes possible, as was noted above,only for large enough magnetic fields B ∼ G, which isvery rare.In particular, for the case when the main pulse andinterpulse correspond to different signs of the charge den-sity ρ GJ , the “death line” does not depend on the width ofthe directivity pattern. It should be so, since at χ = 90 ◦ the possibility to observe radiation from one pole meansthat the second one will be also registered. In other words,the “death line” is associated only with the cessation of sec-ondary plasma generation, and not with the observerâĂŹsexit from the directivity pattern. For this realization one canobtain numerically for the maximum period P numcr (see alsoFig. 6a–6c) P numcr ≈ . A − . B . . (17)This estimate can be easily obtained analytically from rela-tion (11) if, according to (3), we put cos χ = (Ω R/c ) / . Itgives P cr ≈ . A − / B / . (18)As to the case when the main pulse and interpulse cor-respond to the same sign of the charge density ρ GJ , theirdeath line is to depend on the width of the directivity pat-tern. In this case, the critical period can be represented withgood accuracy as P cr ( W = 3 ◦ ) ≈ . B / , (19) P cr ( W = 5 ◦ ) ≈ . B / . (20)Thus, we come to important conclusion that the condi-tion of the possibility to observe interpulses π/ − χ < W r (21)which was usually used is to be corrected. As δW (cid:28) W r ,only a small part of this region corresponds to the possi-bility to observe the interpulse. As a result, the number ofinterpulse pulsars turns out to be much less than it has beenestimated so far. Recently we have already analyzed the pulsar distributionon the base of the kinetic equation describing the evolutionof neutron stars (Arzamasskiy et al. 2017). As so-calleddynamic age of the interpulse pulsars τ D = P/ P is usually Period , s , o . W = o Period , s , o . W = o (a) A = 0 . . Period , s , o . W = o Period , s , o . W = o (b) A = 1 . Period , s , o W = o Period , s , o . . W = o (c) A = 2 . Figure 6.
Interpulse visibility function δW for two different win-dow width W (corresponding to two different generation level r rad ) and different A for magnetic field B = 1 . . Dotted linecorresponds to the width of the total directivity pattern W . small due to their small periods P < . s (see Table 1),it was assumed that the magnetic field for ordinary pulsarscan be considered constant.Therefore the kinetic equation describing the distribu-tion of pulsars N ( P, χ, B ) by period P , inclination angle χ and magnetic field B should be determined from the equa-tion ∂∂P ( ˙ P N ) + ∂∂χ ( ˙ χN ) = Q, (22)where the source Q depends actually on two unknown func-tion, i.e., on initial periods P and inclination angle χ (as tomagnetic field distribution, it can be evaluated from observa-tions). As to the values ˙ P and ˙ χ , they should be determinedby the specific model of pulsar braking. In particular, forBGI model we have (Beskin et al. 1993) ˙ P ≈ πf ∗ B R Ω I r c i cos χ, (23) ˙ χ ≈ − f ∗ B R Ω I r c i sin χ. (24) MNRAS000
Interpulse visibility function δW for two different win-dow width W (corresponding to two different generation level r rad ) and different A for magnetic field B = 1 . . Dotted linecorresponds to the width of the total directivity pattern W . small due to their small periods P < . s (see Table 1),it was assumed that the magnetic field for ordinary pulsarscan be considered constant.Therefore the kinetic equation describing the distribu-tion of pulsars N ( P, χ, B ) by period P , inclination angle χ and magnetic field B should be determined from the equa-tion ∂∂P ( ˙ P N ) + ∂∂χ ( ˙ χN ) = Q, (22)where the source Q depends actually on two unknown func-tion, i.e., on initial periods P and inclination angle χ (as tomagnetic field distribution, it can be evaluated from observa-tions). As to the values ˙ P and ˙ χ , they should be determinedby the specific model of pulsar braking. In particular, forBGI model we have (Beskin et al. 1993) ˙ P ≈ πf ∗ B R Ω I r c i cos χ, (23) ˙ χ ≈ − f ∗ B R Ω I r c i sin χ. (24) MNRAS000 , 000–000 (0000)
E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov
Here again f ∗ ∼ is dimensionless polar cap area, I r is themoment of inertia of a star, and i = I/I GJ is dimension-less electric current circulating in the magnetosphere. Theserelations can be rewritten as ˙ P − = B P Q
BGI cos χ, (25) ˙ χ = 10 − Q BGI B P sin χ cos χ, (26)where ˙ P − = 10 ˙ P , and Q BGI is just the pa-rameter (11) entered above. Accordingly, MHD modelgives (Tchekhovskoy et al. 2016) ˙ P ≈ π B R Ω I r c (1 + sin χ ) , (27) ˙ χ ≈ − B R Ω I r c sin χ cos χ. (28)Remind that the observable distribution function N obs ( P, χ, B ) should be connected with N ( P, χ, B ) by re-lation N obs ( P, χ, B ) = V visbeam V vislum N ( P, χ, B ) . (29)Here V vislum is the luminosity visibility function (we cannotobserve far dim objects), and, as before, V visbeam is the beamvisibility function depending on the width of the directiv-ity pattern. As to luminosity visibility function V vislum , thestandard evaluation is V vislum ≈ P − (30)(see Taylor & Manchester 1977; Gullón et al. 2014; Arza-masskiy et al. 2017 for more detail). Below we use this eval-uation for MHD model.On the other hand, for BGI model we need to correct V vislum function, as it depends on inclination angle χ , when χ → ◦ . Remember that according to BGI model (Beskinet al. 1993) pulsars radio luminosity L rad reach only − ofthe particle energy flux W part near the surface of the neutronstar. On the other hand, for fast pulsars ( Q BGI < ) we have W part = Q W tot . (31)As a result, for space-homogeneous distribution of pulsarsthe luminosity visibility function V vislum ∝ L − can be pre-sented as V vislum = P − / B / cos d − χ. (32)Finally, for interpulse pulsars the beam visibility function V visbeam (16) is to be changed with the visibility width δW .The convenience of the kinetic approach is that, inthe presence of integrals of motion (Beskin et al. 1993;Tchekhovskoy et al. 2016) I BGI = P sin χ , (33) I MHD = P sin χ cos χ , (34)equation (22) can be integrated. Moreover, due to very sim-ple observable distribution N obs ( P ) ∝ P / in the domain .
033 s < P < . s (see Arzamasskiy et al. 2017 for moredetail) just overlapping almost all interpulse pulsars this in-tegration can be produced analytically. As a result, it was found that N MHD ( P, χ ) = K MHD ( π/ − χ − sin χ cos χ )cos χ P , (35) N BGI ( P, χ ) = K BGI ( χ − sin χ cos χ )sin χ cos d − χ P , (36)where again d ≈ . , and the coefficients K MHD , K
BGI areto be determined from the normalization to the entire num-ber of pulsars in the range .
03 s < P < . s N tot = (cid:90) . . d P (cid:90) π/ dχ V vis ( P, χ ) N ( P, χ ) . (37)In turn, the number of orthogonal interpulse pulsars in thesame range can be determined as N ort = (cid:90) . . d P (cid:90) π/ dχ V visbeam , ( P, χ ) V vislum ( P, χ ) N ( P, χ ) . (38)Note that the question of normalization constant N tot ,i.e., the total number of isolated pulsars with .
03 s
03 s < P < . s there are f orth = 2 . ÷ . % of orthogonal rotators amongthe overall population of active pulsars . Such a wide cred-ible interval is due to relatively small number of actuallyobserved orthogonal pulsars and uncertainties in the proce-dure to decide if given pulsar is an orthogonal one or not.The analysis produced by Arzamasskiy et al. (2017)has shown that the number of observable interpulse pulsarswith χ ≈ ◦ can be explained within both BGI (counter-alignment) and MHD (alignment) models due to consider-able uncertainties in the initial pulsar distribution Q ( P, χ ) .In turn, this approach gave us the possibility to evaluatebirth functions Q P ( P ) and Q χ ( χ ) on pulsar initial periodsand inclination angles. As was found, for BGI model in thedomain .
03 s < P < . they look like Q BGI P ( P ) = P, Q
BGI χ ( χ ) = 2 π . (39)Accordingly, for MHD model in the domain .
03 s < P < . we have Q MHD P ( P ) = 1 , Q MHD χ ( χ ) = sin χ. (40)As to interpulse pulsars with χ ≈ ◦ , analysis was per-formed only for MHD model, which also gave the reasonablenumber of orthogonal interpulse pulsars.However, this analysis did not include into considerationthe real visibility function for interpulse pulsars V visbeam ≈ δW which, as was shown above, is to diminish drastically the In Table 1 we use the normalization for the total number ofpulsars in the corresponding range of pulsar periods P .MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution predicted number of orthogonal interpulse pulsars. Never-theless, below we utilize distribution functions (42)–(51) ob-tained earlier, since the necessary additional corrections tothis study refer not so much to the equation itself as to thevisibility function V visbeam and features of the behavior of itssolution when χ → ◦ .Below we assume, as was done by Arzamasskiy et al.(2017), that pulsar birth function Q can be presented asa product Q P ( P ) Q χ ( χ ) Q B ( B ) . Indeed, the effects of thedistribution over the magnetic field were not taken intoaccount, since integrals of motion (34)–(33) does not de-pend on magnetic field. This implies that magnetic field canonly change the rapidity of the individual pulsar motionalong their evolutionary path. Therefore, we can now takeinto account the distribution on the magnetic field simplyby multiplying the previously obtained distribution func-tions N ( P, χ ) by Q B ( B ) /B k , where according to (22)–(28) k BGI = 10 / and k MHD = 2 . On the other hand, as wasshown by Beskin et al. (1993), with high accuracy one canput Q B ( B ) = (cid:18) BB n (cid:19) a (cid:18) BB n (cid:19) − − a − b , (41)where B = 10 G , a = 2 , and b = 0 . .As a result, the pulsar distribution function in BGImodel N BGI ( P, χ, B ) can be written down as N BGI ( P, χ, B ) = K BGI Q B ( B ) B / ( χ − sin χ cos χ )sin χ cos d − χ P , (42)where K BGI = N tot W I I I , (43)and I = (cid:90) π/ ( χ − sin χ cos χ )sin χ d χ = 1 , (44) I = (cid:90) . . P / d P ≈ . , (45) I = (cid:90) ∞ Q B ( B ) B / d B ≈ . . (46)In (44) we put d = 0 . . Accordingly, distribution functionin MHD model N MHD ( P, χ, B ) looks like N MHD ( P, χ, B ) = K MHD Q B ( B ) B ( π/ − χ − sin χ cos χ )cos χ P , (47)where K MHD = N tot W I I I , (48)and now I = (cid:90) π/ ( π/ − χ − sin χ cos χ ) sin χ cos χ d χ = π , (49) I = (cid:90) . . P / d P ≈ . , (50) I = (cid:90) ∞ Q B ( B ) B d B ≈ . . (51) Table 3.
Comparison of the fraction of approximately orthogonalpulsars N ort , in percents for different parameters W and AW ◦ ◦ A MHD 0 . . . . . . . . . . . . . . . . Now we can return to our main goal, i.e., to formulatinga test that may clarify the direction of the inclination an-gle evolution. As was already mentioned above, the centralidea is connected with the amount of orthogonal interpulsepulsars, as their number should depend substantially on thesign of the derivative ˙ χ . For this reason, for orthogonal pul-sars the predictions of MHD and BGI model are to differsignificantly.Remember that according to Arzamasskiy et al. (2017)within MHD model the distribution function N MHD ( P, χ ) (47) reproduces good enough the number of both alignedand orthogonal interpulse pulsars. In particular, the num-ber of orthogonal pulsars in the range .
03 s < P < . is ÷ , in good agreement with observations (see Table 1).However, in this paper, it was supposed that the visibilityfunction of orthogonal interpulse pulsars V visbeam , is deter-mined by the condition π/ − χ < W r / . In Figures (6a)–(6c) it corresponds to the complete filling of the area abovethe dashed line. As was shown in Sect. 4, this assumptionsignificantly overestimates the number of orthogonal inter-pulse pulsars.On the other hand, now the precise accounting forthe plasma generation region within magnetic poles, i.e.,more accurate determination of the directivity pattern, al-lows us to specify the number of interpulse pulsars and,hence, to clarify the direction of the inclination angle evo-lution. In Table 3 we present the number of interpulse pul-sars N ort , (38) within the domain .
03 s < P < . for W = 3 ◦ and W = 5 ◦ obtained through the visibility func-tion V visbeam , = δW and distribution functions (42) and (47).We see that for MHD model the number of orthog-onal interpulse pulsars is always lower than 1% Which isbarely consistent with f orth mentioned above. On the otherhand, BGI model predicts larger fraction . ÷ . % for A ∼ . ÷ . . This probably may indicate that BGI ap-proach is better consistent with the observations. Neverthe-less, no choice between two models can be made at thisstage. Here we come to another key subject of our consideration.The point is that the results presented above in Table 3do not allow us to determine the total number of orthogo-nal interpulse pulsars N ort within BGI model. Indeed, theoriginal version of BGI model determines good enough onlyone component of the braking torque which is parallel tothe magnetic moment. This torque resulting from symmetric MNRAS , 000–000 (0000) E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov (north-south even within polar cap) part of the longitudinalcurrents i in (23)–(24) vanishes for orthogonal rotator. Aswithin BGI model individual pulsars evolve to 90 degrees,clarification of the braking law for orthogonal rotator is ofparticular importance.On the other hand, as was shown recently by Beskinet al. (2017), the braking torque perpendicular to mag-netic moment depends on fine details of the distributionof electric currents on the surface of the polar cap, whichcould not be determined analytically. This has become pos-sible only in recent years, based on the results of numeri-cal modeling (Spitkovsky 2006; Kalapotharakos et al. 2012;Tchekhovskoy et al. 2013; Philippov et al. 2014). As to MHDmodel, it does not require correction at all, since both theevolution equations (27)–(28) (and, hence, the integral ofmotion I MHD (34)) stay true for any inclination angles.As shown in Fig. 7, the exact following to invariant I BGI (33) leads to unlimited accumulation of pulsars in the region χ = 90 ◦ . This results from neglecting the second term C inthe exact equation of evolution, that in general form can bewritten down as (Beskin et al. 2017) ˙ P − = B P ( Q BGI cos χ + C ) , (52) ˙ χ = 10 − Q BGI B P sin χ cos χ. (53)Here again B = B / , and ˙ P − = ˙ P / − . As toextra small factor C (cid:28) , it just describes the evolution oforthogonal rotators along the line χ = 90 ◦ .As was already stressed, the value of C within theframework of the BGI model cannot be determined withthe necessary accuracy. In the original work of Beskin et al.(1983), in which only the action of volume magnetosphericcurrents was taken into account, it was shown that C canbe estimated as (Ω R/c ) i A where i A = j (cid:107) /j GJ is the ratioof the longitudinal electric current to the Goldreich-Juliancurrent; recall that in BGI model i A = 1 . However, as wasshown later (Beskin et al. 2017), this estimate did not allowus to explain the pulsar braking ˙Ω (27) for χ = 90 ◦ withinMHD model, because in this model i A ≈ (Ω R/c ) − / whichis too small to get the desired result C ≈ correspondingto MHD model.To resolve this contradiction it was assumed that inorthogonal case the energy losses can be connected with ad-ditional currents that circulate in magnetosphere along theseparatrix separating the areas of open and closed magneticfield lines (see Beskin et al. 2017 for mode detail). Suppos-ing now that the additional separatrix current is propor-tional to volume current circulating in the pulsar magneto-sphere, i.e. C = Ki A , we obtain from the MHD model that K ≈ (Ω R/c ) / . Assuming now that relation C = Ki A canbe also used for the BGI model, we finally obtain C ∼ (cid:18) Ω Rc (cid:19) / . (54)Since this quantity is not determined with sufficient accu-racy, we assume in what follows that C = ε P − / , (55)where the value of ε belongs to the range between . and . .As is also shown in Fig. 7, for nonzero C the pulsars are P , s , Figure 7.
Evolution of individual pulsars for BGI model for A =0 . and ε = 0 . . The green dashed trajectories correspond toinvariant I BGI (33). Blue solid lines are more realistic trajectoriesdefined by (52)–(53) and were used in Monte-Carlo simulation. to evolve along the axis χ = 90 ◦ gradually increasing theirperiod P until they cross the “death line”. For such pulsars,we can determine the distribution function N ( P, B ) , whichsatisfies the kinetic equation dd P ( ˙ P N ) = ( N BGI ˙ χ ) χ → ◦ . (56)Here, according to (52), ˙ P ≈ − C B
P , (57)and the source in the r.h.s. is determined by the pulsar flowto the region χ = 90 ◦ according to (42) and (53).On the other hand, it is clear that the analytical consid-eration carried out in Sections 6.1–6.2 (corresponding trajec-tories are shown by green dashed curves in Fig. 7) does notallow us to reproduce exactly the evolution trajectory of in-dividual pulsars. Indeed, as shown in Fig. 7, sequential con-sideration of more accurate evolution equations (52)–(53)leads to trajectories presented by blue solid curves whichsignificantly deviate from the trajectories corresponding toconservation of invariant I BGI (33) just in the area whereorthogonal interpulse pulsars should be observed. More rig-orous analysis based on Monte-Carlo simulation is presentedin the next Subsection. Here we carry out a qualitative con-sideration based on the kinetic equation method.For this reason, as a zero-order estimation we assumethat the trajectory reach the boundary χ = 90 ◦ not with aperiod P , but with a period P + P , where P ≈ . s (seeFig. 7). As a result, solution of the kinetic equation (56) N ( P, B ) = 1˙ P (cid:90) P ( N BGI ˙ χ ) χ → ◦ ,P (cid:48) → P (cid:48) + P d P (cid:48) (58)on the r.h.s. of which we made a replacement P → P + P looks now like N ( P, B ) = 7 πA K BGI ε Q B ( B ) B [( P − P )( P + 3 P )] P / . (59)Here we neglect the power 1/14. Accordingly, observable dis-tribution function N obs90 ( P ) of such pulsars can be found as MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution Table 4.
Comparison of observable (Table A1) and BGI predicted(60) distributions of orthogonal interpulse pulsars χ = 90 ◦ by theperiod P for ε = 0 . and A = 1 P [s] ÷ ÷
10 7 ÷ ÷ ÷ BGI 0 N obs90 ( P ) = (cid:90) ∞ N ( P, B ) V visbeam , V vislum , d B. (60)Here the beam visibility function V visbeam , is to be equal to δW , and for luminosity visibility function V vislum (32) we haveto replace cos χ with characteristic value (Ω R/c ) / , so that V vislum , ≈ . P − / B / . (61)Finally, it is necessary to stress that according to (57)magnetic field for orthogonal pulsars χ = 90 ◦ is to be esti-mated as B ≈ ε − / P / ˙ P / − , i.e., as B ≈ ε − / . P / ˙ P / − . (62)As to magnetic field distribution function N obs90 ( B ) , we ob-tain N obs90 ( B ) = (cid:90) ∞ N ( P, B ) V visbeam , V vislum , d P. (63)In Table 4 we present the comparison of the observable(Table A1) and predicted (60) distributions of orthogonalinterpulse pulsars with χ = 90 ◦ by period P for ε = 0 . and A = 1 . As we see, the observable distributions is in goodagreement with the prediction of BGI model. But, as wasalready stressed, it was a pretty rough evaluation. Analyticaltrajectories of the pulsars on P − χ diagram (see Fig. 7),differ from real ones, so there we needed to make a shiftof the period, to reduce the artificial amount of orthogonalpulsars with the period less than . s. Nevertheless, sucha good correlation can be treated as a credible test, that isdone for reasonable values of parameters. To verify the analytical results presented above, we study theevolution of radio pulsars in the framework of Monte-Carloapproach as well. To reconcile the results of the Monte-Carlosimulations with results obtained within kinetic equationmethod, we certainly have to use the same birth functions(39) and (40) of pulsars on periods Q P ( P ) and inclinationangles Q χ ( χ ) (Arzamasskiy et al. 2017). It is these birthfunctions that lead to good agreement between the BGI andMHD predictions of the number of aligned interpulse pulsars N SP (see Table 1) and observations. Accordingly, the evolu-tion of individual pulsars for BGI model is to be determinedby Eqns. (52)–(53), and by Eqns. (27)–(28) for MHD one.For simplicity, we take into account only three physi-cal parameters of pulsars: period P , inclination angle χ andmagnetic field B . At the start of simulation a big amountof pulsars is generated according to some initial parameterdistribution. After that at each step period P and inclina-tion angle χ of all existing pulsars are evolved using one of Table 5.
Visible orthogonal interpulse pulsars fractions (in per-cents) with .
033 s < P < . obtained by Monte-Carlo simula-tion. Those values that lie within the interval . ÷ . % obtainedfrom observations are underlined. W ◦ ◦ A ε = 0 .
12 9.4 7.3 4.3 14 11 8.3 5.0 ε = 0 . ε = 0 . ε = 1 the theoretical models, but magnetic field B is assumed tobe constant; also new pulsars are added, the addition rateand parameters are determined by the birth functions fromabove. We are interested in finding the static distributionso the simulation ran until the number of existing pulsarsbecame close to constant. The initial distribution is not soimportant for the result as it affects just the convergencespeed. However in the BGI case the most convenient op-tion is to start from theoretical N BGI (42) with addition of N (59) since the corrections are small. For MHD modelthe theoretical consideration remains exact since there is nocorrection, so the corresponding simulations check both thetheory and the numerical method.Then to determine N obs (29) within our Monte-Carlointegration method we calculated the sum of all interpulsevisibility functions V visbeam (16), V vislum (32) for BGI model or(30) for MHD model to find the proper normalization like in(43) and (48). For this, for BGI model we have to take thegeometric visibility function δW from Fig. 6a–6c instead of V visbeam and luminosity visibility function V vislum (61) for exactlyorthogonal pulsars and (32) for all other ones. For MHDmodel, where there is no correction, we use the luminosityvisibility function (30) for orthogonal pulsars as well.In Table 5 we present the fractions of orthogonal inter-pulse pulsars obtained in Monte-Carlo simulation for differ-ent parameters A , W , and ε . We see that for reasonableparameters ( A ≈ and ε ≈ . ) a good agreement withthe BGI model can indeed be achieved. On the other hand,we have to stress the strong dependence of the number oforthogonal interpulse pulsars on these parameters. We alsonoticed that most of visible orthogonal interpulse pulsars inBGI model should have exactly χ = 90 ◦ . Thus, it was shown that the statistical analysis of orthog-onal interpulse pulsars really allows us to formulate a testthat can determine the direction of the inclination angleevolution. Two new important points that we included intoconsideration should be noted. The first one is a significantrefinement of the directivity pattern of orthogonal pulsars.In fact, the region of secondary plasma generation near thedeath line was first determined (cf., e.g., Qiao et al. 2004;Tsygan 2019). The second point is the correction to BGI
MNRAS , 000–000 (0000) E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov model. We used an updated version of the expression forenergy losses for an orthogonal rotator. All the other sug-gestions (such as the visibility function for ordinary pulsarswith inclination angles χ (cid:54) = 90 ◦ , etc.) did not go beyond thestandard assumptions commonly used in statistical analysisof radio pulsars.As a result, it was shown that BGI model gives goodagreement with observational data; on the other hand, theMHD model predicts too few orthogonal interpulse pulsars.It must be emphasized here that MHD model itself does notrequire any correction for orthogonal rotators. As for at-tracting additional opportunities for reconciling the predic-tions of MHD model with observations (as was done by Arza-masskiy et al. 2015 to explain the observed value of the brak-ing index), this work is certainly beyond the scope of thisstudy.Simply, our result can be explained as follows. In thezero approximation, one can assume that the distribu-tion of the pulsar in the inclination angle χ weakly de-pends on this angle. Then using the beam visibility func-tion V visbeam = sin χ W (16) we can estimate (of course, veryroughly) the total number of aligned and orthogonal inter-pulse pulsars as N SP ∼ N tot W , (64) N DP ∼ N tot W . (65)Here W is measured in radians. This evaluation indeedgives the reasonable values N DP ∼ – and N SP ∼ – .But as was stressed in Sect. 2, we can observe orthogonal in-terpulse pulsars only if their magnetic fields are much largerthan those of ordinary radio pulsars. Otherwise, the genera-tion of secondary plasma (and, therefore, the radio emissionitself) becomes impossible due to too small potential drop ψ ∝ ρ GJ (2) over the pulsar polar cap. As a result, the num-ber of orthogonal interpulse pulsars is to be much smallerthan the evaluation (65) (see Table 3). Only within BGImodel which predicts additional class of almost orthogonalpulsars with χ ≈ ◦ , the agreement with observations canbe achieved.In conclusion, it should be noted that in the statisti-cal analysis of the interpulse pulsars we did not take intoaccount the possible correlations associated with the spatialdistribution of radio pulsars in the Galaxy, evolution of mag-netic field, etc. which is devoted quite a lot of works (see,e.g., Arzoumanian et al. 2002; Faucher-Giguère & Kaspi2006).Finally, it should be noted that the issues discussedabove allow us to take a fresh look at many questions aris-ing in the analysis of observations of radio pulsars. For ex-ample, the above formula (62) for estimating the magneticfield for almost orthogonal pulsars within BGI model givesmuch larger values B ≈ P / ˙ P / − in comparison withstandard estimate B ≈ P / ˙ P / − . Accordingly, such pul-sars can be observed at much larger periods (of course, un-less photon splitting and positronium creation can suppressgeneration of a secondary plasma, see e.g. Usov & Melrose1995; Istomin & Sobyanin 2007). In particular, for radio pul-sar PSR J0250+5854 ( P = 23 . s, ˙ P − = 27 , see Tan et al.2018 for more detail) we obtain B ≈ – , which gives Q BGI < even for such enormous pulsar period.Another example connects with the essential difference of the pair creation domain of orthogonal pulsars comparedwith standard hollow-cone structure (see Fig. 2). This cir-cumstance must be borne in mind in the analysis both themean profiles of radio emission and X-ray radiation. Indeed,as was already stressed, up to now one still often assumethat for orthogonal pulsars the directivity pattern of radioemission has standard hollow cone structure (see e.g. John-ston & Kramer 2019). Moreover, the first results obtained forradio pulsar PSR J0030+0451 by NICER observatory (Ri-ley et al. 2019) can be interpreted as if the shape of theheated regions (which is naturally connected with the regionof effective generation of electron-positron plasma) have thecrescent shape just as shown in Fig. 2. By the way, accordingto radio data (Bilous et al. 2019), this pulsar is really closeto orthogonal, since it has an interpulse exactly between twomain pulses. Authors thank L.I.Arzamasskiy, D.P. Barsukov, A.V.Bilous,A.Jessner, A.A.Philippov and P.Weltevrede for useful dis-cussions. This work was partially supported by RussianFoundation for Basic Research (RFBR), grant 17-02-00788.AB acknowledges the support from the Program of develop-ment of M.V. Lomonosov Moscow State University (LeadingScientific School ‘Physics of stars, relativistic objects andgalaxies’)
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APPENDIX A: “ADVANCED” LIST OFORTHOGONAL INTERPULSE PULSARS
In table A1 we present ”advanced” list of orthogonal inter-pulse pulsars given by Maciesiak & Gil (2011) including pul-sar name, their period P , period derivative ˙ P , magnetic field B evaluated by expression (62) as well as IP/MP radiointensity ratio and IP-MP angular separation. In the lastcolumn a plus sign is placed if there is a coincidence withthe classification specified by Malov & Nikitina (2013), anda minus in the opposite case . Two pluses are given if thereis the confirmation in some other publications. Note, that classification made by Malov & Nikitina (2013) wasbased on combination of various highly model-dependent meth-ods. Moreover, even calculation errors were found for some pulsarsafter the paper was published (E. Nikitina, private communica-tion). Therefore, the contradiction mentioned for eight pulsarsabove should not be considered so earnestly.
Table A1. “Advanced” list of orthogonal interpulse pulsarsName P ˙ P B IP/MP Sep.J [s] − G ratio [ ◦ ]0534 + + ++ + − − − + − ++ − ++ − − − − − + − − − + − ++ − − − − − + − + − − ++ − − − ++ − ++ + − + + + + − + − + APPENDIX B: DETERMINATION OFMAGNETIC FIELD
In this Appendix we remember the procedure of model-consistent estimate of pulsar magnetic field within BGIand MHD approaches. As was shown in Sect. 3.3, pulsardeath line equation can be rewritten and tested in the form cos / χ > P · ( A / B − / ) (12). Here B is model-consistent estimate of pulsar magnetic field taken in G.Within BGI model (Beskin et al. 1984, 1993) the de-termination of magnetic field depends substantially on theparameter Q BGI (11). For Q BGI < (far from the real “deathline” where the theory can only be considered as consistent)the spin-down law (25) results in B BGI12 = A − / P − / ˙ P / − cos − / χ, (B1)where P taken in seconds. In particular, for orthogonal rota-tors the evaluation B ≈ ε − / P / ˙ P / − (62) is to be used.On the other hand, for pulsars in the vicinity of the “deathline” (i.e., for pulsars with Q BGI > ) the evaluation gives B BGI12 = P / cos − χ. (B2)As to MHD model, corresponding magnetic field is ex-pected to be consistent with the spin-down law (27) P ˙ P = π B R Ic (1 + sin χ ) . (B3) MNRAS000
In this Appendix we remember the procedure of model-consistent estimate of pulsar magnetic field within BGIand MHD approaches. As was shown in Sect. 3.3, pulsardeath line equation can be rewritten and tested in the form cos / χ > P · ( A / B − / ) (12). Here B is model-consistent estimate of pulsar magnetic field taken in G.Within BGI model (Beskin et al. 1984, 1993) the de-termination of magnetic field depends substantially on theparameter Q BGI (11). For Q BGI < (far from the real “deathline” where the theory can only be considered as consistent)the spin-down law (25) results in B BGI12 = A − / P − / ˙ P / − cos − / χ, (B1)where P taken in seconds. In particular, for orthogonal rota-tors the evaluation B ≈ ε − / P / ˙ P / − (62) is to be used.On the other hand, for pulsars in the vicinity of the “deathline” (i.e., for pulsars with Q BGI > ) the evaluation gives B BGI12 = P / cos − χ. (B2)As to MHD model, corresponding magnetic field is ex-pected to be consistent with the spin-down law (27) P ˙ P = π B R Ic (1 + sin χ ) . (B3) MNRAS000 , 000–000 (0000) E. M. Novoselov, V. S. Beskin, A. K. Galishnikova, M. M. Rashkovetskyi and A. V. Biryukov
In other words, B MHD12 = 1 . P / P / − (1 + sin χ ) − / , (B4)where we assume neutron star radius R = 12 . km andmoment of inertia I r = 1 . · g cm (Spitkovsky 2006;Philippov et al. 2014). For orthogonal pulsars within MHDmodel one can just put sin χ ≈ and get B MHD , = 0 . P / P / − . (B5) APPENDIX C: ATNF CATALOGUELIMITATIONS
As it was already stressed, the amount of interpulse pulsarsgiven in Tables 1, A1 should be considered as a lower esti-mate. Indeed, the ATNF catalog is not homogeneous and,in particular, it contains a large number of weak sources, forwhich the possible interpulse is beyond the sensitivity limit.Below we try to determine the uncertainty of the normaliza-tion constant N tot (37) which is important for our analysis.Recall that successful detection of a pulsar within a sur-vey depends on the pulsar radio luminosity, distance andpulse broadening due to interstellar dispersion and distor-tion. Weak and wide-pulse pulsars are hard to detect at largedistances (or at large dispersion measures). This is the rea-son for the lack of pulsars with low pseudo-luminosities andlarge dispersion measures on the corresponding diagram, seeFigure C1(a). Grey dots on this plot represent all normal iso-lated pulsars with periods . ÷ . s stored in the ATNFdatabase, while red circles are for orthogonal pulsars fromTable A1.One can see that minimal pseudo-luminosity remainsapproximately constant up to DM ≈ pc · cm − for ANTFpulsars. While at larger values of DM it scales as L min ∝ DM . . At the same time, there are lack of high-SNR pulsarsat DM (cid:38) at Figure C1(b). The effective signal-to-noiseratio was estimated here as SNR eff = F . F Crab (cid:114) P − W W , (C1)where P is pulsar period, F . its radio flux at frequency 1.4GHz, F Crab is the Crab pulsar flux at the same frequencyand W is main pulse width at half maximum intensitytaken in seconds (Johnston & Karastergiou 2017).We conclude therefore, that ATNF catalogue can beconsidered as more or less complete for DM (cid:46) ÷ pc cm . There are ÷ isolated normal pulsars in thisinterval with from 8 to 17 orthogonal ones among them cor-respondingly. Assuming Poissonian statistics for both quan-tities we finally estimate the fraction of orthogonal pulsarsas f orth = (4 ± .
5) % for local galactic volume at σ sig-nificance. And since this local pulsar population is merelyindependent, then one can expect the same fraction for allradio loud galactic pulsars. In the frame of our work wecompare model prediction of f orth for both MHD and BGIapproaches with the number obtained above.Note that the above estimate of the fraction of orthog-onal interpulse pulsars f orth is in good agreement with an-other independent estimate which also can be obtained fromATNF catalog. This criterion connects with discarding pul-sars for which ATNF catalogue does not give mean pulsewidth W10 on the 10% intensity level. (cid:0)✁✂ ✄☎✆✝ ✞✞ ✟✠✡☛☞✌✍✎✏✑✒✓✔✏✕✖✕✗✘✙✌✚✛ ✜✢✣✤✜✢✣✥✜✢✣✦✜✢✧✜✢✦✜✢✥✜✢✤✜✢★ (cid:0)✩✂ ✪✫✬ ✭✭ ✮✯ ✰✱✎✲✲✎✚✳✴✵✎✶✷✸ ✜✢✣✤✜✢✣✥✜✢✣✦✜✢✧✜✢✦✜✢✥✜✢✤ (cid:0)✹✂ ✺✻✼✽✾✿✾❀❁❂ ❃❄❂❅❁✻❅❆❇❈❉ ❀✾✻❊❁❂ ❃❄❂❅❁✻❅✑✴✍✳❋●✚✎✖✙✌✚ ✜✢✣✦✜✢✧✜✢✦✜✢✥ ❍■❏❑▲▼❏■◆❖ P▲✁❏◗▼▲❘ ❑✹ ✹P❙✤✜ ✜✢ ✜✢✢ ✜✢✢✢ Figure C1.
To the discussion of the completeness of the ATNFpulsar database. On the plots (a)-(c) above we show normal ra-diopulsars included into ATNF which periods are within . ÷ . seconds interval (gray dots). Red circles represent known pul-sars with interpulses listed in Table A1. The description of theplots are as follows: (a) pulsar pseudo-luminosity L = F . × D ,where F . is the observed flux at the frequency 1.4 Ghz and D is distance typically based on the dispersion measure estima-tion; (b) Effective signal-to-noise ratio (see Eq.C1); (c) Disper-sion measure-based distance to pulsar. All quantities are plot-ted against the dispersion measure. We interpret the lack of low-luminosity and high-SNR pulsars with at large DM (cid:38) pccm − as a result of the incompleteness of the ANTF database.And assume that this catalogue is more or less complete up to DM ∼ − pc cm − . MNRAS , 000–000 (0000) rthogonal pulsars as a key test for pulsar evolution Indeed, measured W10 in ATNF catalog indicates thatthe noise level for a given pulsar is quite low. Hence, one canbelieve that for such pulsars it is possible to detect an inter-pulse with a sufficiently large intensity ratio IP/MP. And re-ally, in Table A1, pulsar PSR J0826+2637 (IP/MP = 0.01)turned out to be the only exception for which ATNF catalogdoes not give a value of W10. For all other pulsars havingIP/MP > N tot (37) in the range0.033 s < P < f orth = (3 . ÷ .
2) % in good agreementwith the previous estimate.
MNRAS000