3 Discussion and conclusions

We argue in this paper that the presence of a strong non-dipolar magnetic field on the neutron star surface can help to understand the recently discovered radio pulsars with dipolar magnetic field above the photon splitting threshold, as well as to understand the long-standing problems of vacuum gap formation and drifting subpulse phenomenon. We model the actual surface magnetic field as a superposition of the global star-centered (large-scale) dipole and local crust-anchored (small-scale) dipoles $\vec{B}_{\rm s}=\vec{B}_{\rm d}+\sum_i\vec{B}_{{\rm m}i}\approx \vec{B}_{\rm d}+\vec{B}_{\rm MO}$, where $\vec{B}_{\rm MO}$ is the local dipole nearest to the polar cap centre (Fig. 1). Such a model is quite general, as it describes the magnetic field structure even if the star-centered fossil dipole field is negliglible at the star surface. In such a case the global surface dipole field $\vec{B}_{\rm d}$ (inferred from P and $\dot{P}$ measurements) is a superposition of all crust-anchored dipoles calculated at a far distance and projected down to the polar cap surface according to the dipolar law.

We propose a model for radio-loud HBPs with high inferred dipolar magnetic field $B_{\rm d}>10^{13}$ G, even exceeding the critical value $B_{\rm cr}\sim 4\times 10^{13}$ G. Given the difficulty that in a strong magnetic field the magnetic pair creation process is largely suppressed, the puzzling issue remains how these HBPs produce their ${\rm e}^-{\rm e}^+$ pair plasma necessary for the generation of the observable radio emission. Zhang & Harding (2000a) proposed a "lengthened version'' of the stationary SCLF model of inner accelerator (e.g. Arons & Sharleman 1979), in which the pair formation front occurs at altitudes r high enough above the polar cap that $B_{\rm d}\sim B_{\rm cr}(R/r)^3$ degrades below $B_{\rm cr}$, thus evading the photon splitting threshold. Our VG model is an alternative to the lengthened SCLF model, with pair creation occurring right at the polar cap surface, even if the magnetic field exceeds $B_{\rm cr}$. We have assumed that the open surface magnetic field lines result in an actual pulsar from superposition of the star-centered global dipole moment and a crust-anchored local dipole moment. We argued that if the polarities of these two components are opposite, and their values are comparable, then the actual value of the surface magnetic field $B_{\rm s}$ can be lower than the critical field $B_{\rm cr}$, even if the global dipole field $B_{\rm d}$ exceeds the critical value. Thus, the creation of electron-positron plasma is possible at least over a part of the polar cap and these high magnetic field neutron stars can be radio-loud (HBPs). In fact, one should expect that in HBPs, in which by definition $B_{\rm d}\ga B_{\rm cr}\sim B_{\rm q}=4.4\times 10^{13}$ G, the ratio $B_{\rm m}/B_{\rm d}$ should be of the order of unity, since $\vec{B}_{\rm s}= \vec{B}_{\rm d}+ \vec{B}_{\rm m}$ and $10^{13}~{\rm G}\la B_{\rm s}\la B_{\rm cr}\sim 4\times 10^{13}$ G.

Within our simple model of a non-dipolar surface magnetic field $\vec{B}_{\rm s}$ one should expect that both cases $\vec{m}\cdot\vec{d}>0$ and $\vec{m}\cdot \vec{d}<0$ will occur with approximately equal probability. However, from the viewpoint of observable radio emission only the latter case is interesting in HBPs with $B_{\rm d}\ga B_{\rm cr}$. In fact, when $\vec{m}\cdot\vec{d}>0$, then the surface magnetic field $B_{\rm s}>B_{\rm d}\gg B_{\rm cr}$ (Fig. 4) and the photon splitting level is highly exceeded. For $\vec{m}\cdot \vec{d}<0$ we have two possibilities: (i) if $m/d\la (\Delta R/R)^3$ thus $B_{\rm m}\la B_{\rm d}$ at the pole $(r=R, \theta=0)$ then the polar cap (locus of the open field lines) is circular (Fig. 2); (ii) if $m/d>(\Delta R/R)^3$ thus $B_{\rm m}>B_{\rm d}$ then part of the circumpolar field lines are closed and the actual polar cap has the shape of ring (Fig. 3). In both cases (i) and (ii), the actual surface magnetic field $B_{\rm s}$ at the polar cap (or at least part of it) can be lower than $B_{\rm cr}$, even if $B_{\rm d}$ exceeds $B_{\rm cr}$. The values of $B_{\rm m}$ and $B_{\rm d}\ga B_{\rm cr}$ should be comparable to make reduction of The strong surface field $B_{\rm s}$ below $B_{\rm cr}$ possible. In our illustrative examples presented in Figs. 2 and 3 (corresponding to the same pulsar with P=1 s and $B_{\rm p}=2{\rm d}R^{-3}=6.4\times 10^{19}(P\cdot\dot{P})^{1/2}$ G) we used ratios $B_{\rm m}/B_{\rm d}$ ranging from 0.5 to 1.6. These values could be slightly different, say by a factor of a few, thus we can say that the ratio $B_{\rm m}/B_{\rm d}$ should be of the order of unity. If $B_{\rm m}/B_{\rm d}\gg 1$, then the reduction of the surface dipole field is not effective (see example presented and discussed in Fig. 5). On the other hand, the case with $B_{\rm m}\ll B_{\rm d}$ is not interesting, as it represents a weak surface magnetic field anomaly. Thus, among a putative population of neutron stars with $B_{\rm d}\ga B_{\rm cr}$, only those with a ratio $B_{\rm m}/B_{\rm d}=(m/d)(R/\Delta R)^3$ of the order of unity, and with magnetic moment $\vec{m}$ and $\vec{d}$ (Fig. 1) antiparallel at the polar cap surface, that is $\vec{m}\cdot \vec{d}<0$, can be detected as HBPs. Other neutron stars from this population of high magnetic dipole field objects should be radio-quiet. This probably explains why there are so few HBPs detected.

Within the lengthened SCLF model there is an upper limit around $B_{\rm d}=2\times10^{14}$ G for radio-loud HBPs (ZH00, ZH01, Zhang 2001). As ZH00 argued, detecting a pulsar above this limit would strongly imply that only one mode of photon splitting occurs. Without the alternative model of HBPs proposed in this paper, such a detection would really be of great importance for the fundamental physics of the photon splitting phenomenon. In our VG-based model there is no natural upper limit for the radio-loud HBPs. However, it is known that due to the magnetic pressure the neutron star surface would tend to "crack'', which should occur at magnetic field strengths approaching 10¹⁵ G (Thompson & Duncan 1995). It is unclear how the radio emission would be affected by such a cracking process.

Figure 6: As in Fig. 2 but for $\vec{m}=-1.25\times 10^{-4}~\vec{d}$. See also text for explanation.

To illustrate the above argument, let us consider Fig. 6 which presents yet another case of opposite polarities $\vec{m}=-1.25\times 10^{-4}~\vec{d}$. With $\Delta R/R\sim 0.05$ this gives $B_{\rm m}/B_{\rm p}=1.0$ and $B_{\rm s}=B_{\rm p}-B_{\rm m}=0$ at the pole (r=R, $\theta=0$). The dashed horizontal line at B=0.2 in the lower panel corresponds to the surface magnetic field $B_{\rm s}$ which is 10 times weaker than the global dipole component $B_{\rm d}=2$ (not shown in the figure). Thus if, for example, $B_{\rm p}=4\times10^{14}$ G (well above the lengthened SCLF limit $B_{\rm d}=2\times10^{14}$ G; such a pulsar was not observed so far), then the actual surface field $B_{\rm s}$ is well below $B_{\rm cr}\sim 4\times 10^{13}$ G, at least in the inner part of the polar cap between $\pm$ $\theta_{\rm v}=0.0035$ rad. This "pair-forming effective'' polar cap is about 2.5 times smaller than the canonical polar cap with radius $\theta_{\rm d}=0.014$ rad, and about 7 times smaller than the entire polar cap with radius $\theta_{\rm s}=0.027$ rad. Near the last open field lines at polar angles $0.027\ga\vert\theta\vert\ga 0.014$the actual surface magnetic field $B_{\rm s}$ is only about 2 times lower than $B_{\rm d}$, while in a narrow circumpolar area with $\vert\theta\vert<\theta_{\rm v}$ the surface field region $B_{\rm s}$ can even be more than 10 times weaker than $B_{\rm d}$. Thus, within our model one can expect a radio-loud HBP with $B_{\rm d}$ even exceeding $4\times10^{14}$ G. However, their radio beams should be much narrower than those expected in normal pulsars, at least a few to several times less than $(r_{\rm em}/R)^{1/2}P^{-1/2}$ degrees (where ${r_{\rm em}}$ is the radio emission altitude; Kijak & Gil 1997, 1998). This would make such sources difficult to detect.

Figure 7: As in Fig. 6 but with the local dipole shifted off center by $\Delta \theta =0.005$ radians. See also text for explanation.

The dotted horizontal line at B=0.05 in Fig. 6 corresponds to $B_{\rm s}=10^{13}$ G for adopted $B_{\rm d}=4\times 10^{14}$ G. This value of the surface magnetic field is believed to be about the lower limit for VG formation (see Gil & Mitra 2001; Gil et al. 2002). Thus, the shadowed area in Fig. 6 represent a narrow hollow cone above which the VG-driven radio emission cannot occur. A similar hollow cone is marked in Fig. 7, which presents a case similar to that illustrated in Fig. 6, except that the local dipole is shifted off center by $\Delta \theta =0.005$ radians (corresponding to about 0.2 of the actual polar cap radius). The dashed horizontal line at B=0.4 corresponds to $B_{\rm s}=4\times 10^{13}$ G and the dotted horizontal line at B=0.1 corresponds to $B_{\rm s}=10^{13}$ G, both calculated for adopted $B_{\rm d}=2\times10^{14}$ G. The polar angles $\theta_{\rm v1}$ and $\theta_{\rm v2}$ correspond to $-\theta_{\rm v}$ and $+\theta_{\rm v}$ in Fig. 6, respectively. Figure 7 demonstrates that the conclusions of our paper do not depend on where the local dipole is placed.

The above arguments strengthen the possibility that some magnetars can also emit observable radio emission (Camilo et al. 2000; Zhang & Harding 2000b). It is therefore interesting to comment on the apparent proximity of HBP PSRJ 1814-1744 (with $B_{\rm p}=1.1\times 10^{14}$ G) and AXP 1E 2259+586 (with $B_{\rm p}=1.2\times 10^{14}$ G) in the $P-\dot{P}$diagram. In both these cases the inferred surface magnetic field well exceeds the critical value $B_{\rm cr}$. Within our model, the former object can be radio-loud if the strong local dipole and the global dipole have opposite polarities. The radio quiescence of AXP 1E 2259+586 can be naturally explained if the local dipole is not able to decrease the inferred dipole magnetic field below the photon-splitting death-line. Thus, either the polarities are the same or they are opposite but the local dipole is not strong enough to reduce the dipole surface field below $B_{\rm cr}$.

Figure 8: Radius of curvature $\rho _{\rm c}$ (in units of R=106 cm) as a function of normalized altitude $z=(r/R)\cdot\cos\theta$ for actual surface magnetic field lines corresponding to four cases presented in Figs. 2-6. For comparison, the radius of curvature of purely dipolar field lines (in pulsar with P=1 s) is shown (line 1).

In Fig. 8 we show the radii of curvature of actual surface field lines compared with those of purely dipolar field (line 1) as a function of normalized altitude $z=(r/R)\cos\theta$above the polar cap. Within the polar gap at z<1.01 (within about 100 meters from the surface) the curvature radii for all cases presented in Figs. 2-6 have values of the order of a few hundred meters (see Urpin et al. 1986), suitable for curvature-radiation-driven magnetic pair production ( $\rho_{\rm c}=1/\Re<10^6$ cm, where $\Re$ is the curvature of the field lines).

All model calculations performed in this paper correspond to the axisymmetric case in which one local dipole is placed at the polar cap center (except the case presented in Fig. 7). In a forthcoming paper we will consider a general, non-axisymmetric case, including more local dipoles, each with different orientation with respect to the global dipole. Although this generalization will give a more realistic picture of the actual surface magnetic field, it will not change our conclusions obtained in this paper.

It should be finally emphasized that although the lengthened SCLF model for HBPs (ZH00) can solve the problem of pair creation in pulsars with surface dipole field exceeding the photon splitting threshold, it does not automatically warrant generation of the coherent radio emission of such HBPs. The problem is that unlike in the non-stationary VG model, where the low altitude radio emission can be generated by means of two-stream instabilities (Asseo & Melikidze 1998; Melikidze et al. 2000), the stationary SCLF inner accelerator is associated with the high-altitude relativistic maser radiation (e.g. Kazbegi et al. 1991, 1992; Kazbegi et al. 1996). This radiation requires relatively low Lorentz factors $\gamma_{\rm p}\sim 5\div 10$ of a dense secondary plasma (e.g. Machabeli & Usov 1989). It is not clear if such a plasma can be produced within the lengthened SCLF accelerator with delayed pair formation taking place in a purely dipolar magnetic field, either by curvature radiation or by inverse Compton scattering (e.g. Zhang & Harding 2000b) processes. Moreover, the relativistic maser coherent radio emission requires a relatively weak magnetic field in the generation region. With the surface dipole field $B_{\rm d}\sim 10^{14}$ G, such a low field may not exist at reasonable altitudes (about 50% of the light cylinder radius $R_{\rm L}=cP/2\pi$) required by the physics of corresponding instabilities (Kazbegi et al. 1991, 1992; Kazbegi et al. 1996). Thus, if one assumes that the radio-loud HBPs are driven by the SCLF lengthened accelerator as proposed by ZH00, they might not be able to generate observable coherent radio emission. This contradiction seems to be a challenge for the lengthened SCLF scenario for HBPs. In our VG-based model the low-altitude $(r\ll R_{\rm L})$ radio emission of HBPs is driven by just the same mechanism as the one most probably operating in typical radio pulsars (e.g. soliton curvature radiation proposed recently by Melikidze et al. 2000). In fact, the HBPs show apparently normal radio emission, with all properties typical for characteristic pulsar radiation (Camilo et al. 2000).