2 Modelling the surface magnetic field

Figure 1: Superposition of the star-centered global magnetic dipole "d'' and crust-anchored local dipole "m'' placed at $\vec{r}_{\rm s}=(r_{\rm s}\sim R,\theta=\theta_{\rm r})$ and inclined to the z-axis by an angle $\theta _{\rm m}$. The actual surface magnetic field at radius vector $\vec{r}=(r,\theta)$ is $\vec{B}_{\rm s}= \vec{B}_{\rm d}+ \vec{B}_{\rm m}$, where ${B}_{\rm d}=2{d}/r^3$, ${B}_{\rm m}=2{m}/\vert \vec{r}-\vec{r}_{\rm s}\vert^3$, r is the radius (altitude) and $\theta$ is the polar angle (magnetic colatitude). R is the radius of the neutron star and L is the crust thickness.

We model the actual surface magnetic field by superposition of the star-centered global dipole $\vec{d}$ and a crust-anchored dipole moment $\vec{m}$, whose influence results in small-scale deviations of the surface magnetic field from the global dipole, as presented in Fig. 1. The technical details of the calculations of the resultant surface magnetic field are presented in the Appendix. Here we discuss the main results and their consequences. For simplicity, in this paper we mostly consider an axially symmetric case in which both $\vec{d}$ and $\vec{m}$ are directed along the z-axis (parallel or antiparallel), thus $\theta_{\rm r}=\theta_{\rm m}=\phi_{\rm r}=\phi_{\rm m}=0$ (see Fig. 1). Also, for convenience "$\vec{m}$'' is expressed in units of $\vec{d}$. We use normalized units in which d=P=R=1 and $r_{\rm s}=0.95$. (see caption of Fig. 2 for the normalization convention). All calculations are carried out in three-dimensions, although, for clarity of graphic presentation, in the figures we present only two-dimensional cuts of the open field line regions.

2.1 High-magnetic-field pulsars

Figure 2: Structure of the surface magnetic field for a superposition of the global star-centered dipolar moment $\vec{d}$ and crust-anchored dipole moment $\vec{m}=-10^{-4}~\vec{d}$ (Fig. 1). The open dipolar field lines (solid) and the actual surface open field lines (dashed) are shown in the upper panel. The horizontal axis is labelled as azimuthal angle $\theta$ (magnetic colatitude), which measures the polar cap radius. For purely dipolar field lines the polar cap radius $r_{\rm d}\approx R\cdot\sin\theta_{\rm d}$, which for pulsar period P=1 s is about 0.014 radians (thus $r_{\rm d}\approx 1.4\times 10^4$ cm). The actual polar cap is broader, with the last open lines emanating at the polar angles $\theta _{\rm s}\approx 0.023$(thus the actual polar cap radius $r_{\rm s}=2.3\times 10^4~{\rm cm}=1.65r_{\rm d}$). The actual open surface field lines (solid) reconnect with dipolar field lines (dashed) at distances $z=(r/R)\cdot\cos\theta\approx 1.2$, where r is the radius and $\theta <0.025$ radians. In the lower panel the surface values (r=R) of both dipolar field (dashed horizontal) and the actual field (solid line) are shown. The radial components $B_{\rm r}^{\rm d}=2d\cos\theta /R^3\approx 2$ and $B_{\rm r}^{\rm s}=(\vec{B}_{\rm d}+ \vec{B}_{\rm m})\cdot \vec{r}/r$, and total values $B_{\rm s}=\sqrt {(B_{\rm r}^{\rm s})^2+(B_\theta ^{\rm s})^2}$ are presented (where d=R=1is assumed for convenience).

As mentioned above, the formation of the VG inner accelerator requires a very high magnetic field $B_{\rm s}\ga 10^{13}$ G on the surface of the polar cap (Usov & Melrose 1995, 1996; Gil & Mitra 2001). This can be achieved not only in pulsars with high dipolar field $B_{\rm d}\ga 10^{13}$ G. In fact, some of the low-field pulsars with $B_{\rm d}\ll 10^{13}$ G can have a surface field $B_{\rm s}\ga 10^{13}$ G if $B_{\rm m}\gg B_{\rm d}$ (thus $m\gg 1.25\times 10^{-4}$ d). We discuss such normal, low-field pulsars later in this paper. Presently let us consider the HBP with a dipolar surface field at the pole $B_{\rm d}=6.4\times 10^{19}(P\cdot\dot{P})^{1/2}$ G exceeding the photon splitting limit $B_{\rm cr}\sim B_{\rm q}$. If all photon splitting modes operate, such a pulsar should be radio-quiet. Alternatively these pulsars could be radio-loud if the effective surface field is reduced below $B_{\rm cr}$. Such a scenario can be achieved if the polarities of magnetic moments $\vec{d}$ and $\vec{m}$ are opposite, that is $\vec{d}$ and $\vec{m}$ are antiparallel. Figure 2 presents a case with $\vec{m}=-10^{-4}~\vec{d}$ and $\Delta R/R=0.05$. The actual surface values of $B_{\rm s}=\sqrt {(B_{\rm r}^{\rm s})^2+(B_\theta ^{\rm s})^2}$ as well as radial components of $B_{\rm r}^{\rm s}=\vec{B}_{\rm s}\cdot \vec{R}/R$ and $B_{\rm r}^{\rm d}={\vec B}_{\rm d}\cdot{\vec R}/R$ are presented in the lower panel of Fig. 2 (note that all radial components are positive and that the total $B_{\rm s}$ is almost equal to $B_{\rm r}^{\rm s}$ in this case). At the pole (radius r=R and polar angle $\theta=0$) the ratio $B_{\rm m}/B_{\rm p}=(m/d)\cdot(R/\Delta R)^3=0.8$ and thus $B_{\rm s}=B_{\rm p}-B_{\rm m}=B_{\rm d}(1-0.8)=0.2 B_{\rm p}$. As one can see from this figure, all surface field lines between $-\theta_{\rm s}$ and $+\theta_{\rm s}$ are open, but the ratio $B_{\rm s}/B_{\rm d}$ increases towards the polar cap edge, reaching a value of about 0.5 in the region between polar angles $\vert\theta_{\rm d}\vert$ and $\vert\theta_{\rm s}\vert$. The ratio $B_{\rm m}/B_{\rm d}$ is also about 0.5 in this region. Thus, the global dipolar field ( $B_{\rm d}=2$in our units) is effectively reduced by between 2 and 5 times in different parts of the polar cap (defined as the surface area from which the open magnetic field lines emanate). This means that the ratio $B_{\rm m}/B_{\rm d}$ ranges from 0.5 to 0.8 across the polar cap. The actual polar cap is broader than the canonical dipolar polar cap (two dashed vertical lines correspond to the last open dipolar field lines emanating at the polar angles $\theta_{\rm d}=\pm 0.014$ radians for a typical period P=1 s). The ratio of actual to dipolar polar cap radii is $\theta_{\rm s}/\theta_{\rm d}\sim 5/3$ in this case. Thus, using the argument of magnetic flux conservation of the open field lines, one can say that an effective surface magnetic field of the polar cap is about 2.8 times lower than the dipolar surface field measured from the values of P and $\dot{P}$. If the estimated dipole field $B_{\rm p}\approx 10^{14}$ G (like in the case of PSR J1814-1744, Table 1) then the actual surface field at the pole is only $B_{\rm s}\sim 2.5\times 10^{13}$ G, well below the photon splitting death line $B_{\rm cr}=4.4\times 10^{13}$ G. Such a pulsar can be radio-loud without invoking the lengthened SCLF accelerator proposed by ZH00. As shown by Gil & Mitra (2001), in such a strong surface magnetic field the vacuum gap accelerator can form, which implies low altitude coherent radio emission (Melikidze et al. 2000, see Sect. 3 in this paper) at altitudes $r_{\rm em}\sim 50 R$ (for a typical pulsar with P=1 s) in agreement with observational constraints on radio emission altitudes (Cordes 1978, 1992; Kijak & Gil 1997, 1998; Kijak 2001).

Figure 3: As in Fig. 2 but for $\vec{m}=-2\times 10^{-4}~\vec{d}$.

Figure 3 presents another case of opposite polarities $\vec{m}=-2\times 10^{-4}~\vec{d}$, with magnitude of $\vec{m}$ being two times stronger than in the previous case (Fig. 2). Again for $\Delta R/R\sim 0.05$, $B_{\rm m}/B_{\rm p}=(m/d)(R/\Delta R)^3= 1.6$ at the pole (r=R and $\theta=0$) and $B_{\rm s}=B_{\rm p}-B_{\rm m}=-0.6B_{\rm p}$. The negative sign of the ratio $B_{\rm s}/B_{\rm d}$ means that the surface magnetic field $\vec{B}_{\rm s}$ is directed opposite to $\vec{B}_{\rm d}$near the pole, that is the circumpolar field lines with polar angles $-\theta_{\rm s1}<\theta<\theta_{\rm s1}$ (where $\theta_{\rm s1}\sim 0.031$) are closed. The last open surface field lines (solid) emanating at polar angles $\theta_{\rm s}=\pm\theta_{\rm s2}$ (where $\theta_{\rm s2}\sim 0.037$) reconnect with the last open dipolar field lines (dashed) at altitudes $z \sim 1.2$ (thus about 2 km above the surface). The actual polar cap, which is the surface through which the open magnetic field lines emanate, has the shape of a ring $(0.031\la \vert\theta_{\rm s}\vert\la 0.037)$ located outside the circle of the dipolar polar cap with angular radius $\theta_{\rm d}=0.014$(or diameter $r_{\rm d}\approx\theta_{\rm d}\cdot R\approx 1.4\times 10^4$ cm). Again, the magnetic flux conservation argument leads to $B_{\rm s}/B_{\rm d}=(\theta_{\rm s_2}^2-\theta_{\rm s_1}^2)/\theta_{\rm d}^2=(0.037^2{-}0.031^2)/0.014^2=0.48$, thus $B_{\rm s}$ is about $0.5B_{\rm d}$ within the ring of the open field lines (note that $\vert B_{\rm m}\vert/B_{\rm d}\sim 1.6$ in this region). The actual values of the surface magnetic field (radial $B_{\rm r}^{\rm s}$ and total $B_{\rm s}=\sqrt {(B_{\rm r}^{\rm s})^2+(B_\theta ^{\rm s})^2}$) are shown as solid lines in the lower panel of Fig. 3, in comparison with radial components of the dipolar field $B_{\rm d}$ (dashed horizontal line). As one can see, $B_{\rm r}^{\rm s}<B_{\rm r}^{\rm d}=2m\cos\theta/R^3$ and $B_{\rm r}^{\rm s}$ is negative for $\vert\theta\vert<0.022$. If the dipolar surface component of the pulsar magnetic field is $B_{\rm d}\sim 7\times 10^{13}$ G (like PSR J1726-3530 given in Table 1), then the actual surface magnetic field $B_{\rm s}\sim 4\times 10^{13}$ G, is below the photon splitting threshold.

2.2 Normal pulsars

Figure 4: As in Fig. 3 but for $\vec{m}=2\times 10^{-4}~\vec{d}$.

Figure 4 presents a case with $\vec{m}=2\times 10^{-4}~\vec{d}$ in which both magnetic moments have the same polarity. Obviously, in such a case the surface magnetic field will be stronger as compared with pure dipole (m=0). The flux conservation argument gives a surface magnetic field $B_{\rm s}/B_{\rm d}=(\theta_{\rm d}/\theta_{\rm s})^2=(0.014/0.008)^2\sim 3$ (the ratio $B_{\rm m}/B_{\rm d}\la 1.6$). Thus the actual surface field is about 3 times stronger than the inferred dipolar field $B_{\rm p}=6.4\times 10^{19}(P\cdot\dot{P})^{1/2}$ G. Such cases of increasing an effective magnetic field can be important in normal pulsars with low dipolar field $B_{\rm d}\ll 10^{13}$ G (Gil & Mitra 2001).

Figure 5: As in Fig. 4 but for $\vec{m}=-4\times 10^{-4}~\vec{d}$.

It is then interesting to examine how different polarities of $\vec{d}$ and $\vec{m}$ would influence normal pulsars with $B_{\rm d}\ll B_{\rm cr}$. If $m/d\sim(\Delta R/R)^3$ thus $B_{\rm m}\sim B_{\rm d}$then of course $B_{\rm s}$ can be slightly lower than $B_{\rm d}$, as in the case of HBPs (Fig. 2). In such a case, however, the VG cannot form. In fact, as argued by Gil & Mitra (2001), the formation of the VG requires that $B_{\rm s}$ is close to 10¹³ G or even above, thus $B_{\rm m}\gg B_{\rm d}$ is required in normal pulsars (see also Gil et al. 2001). Figure 5 illustrates a case of high surface magnetic field with $B_{\rm m}=3.2 B_{\rm p}$, in which the VG can apparently form. As one can see from this figure, the values of $B_{\rm s}$ at the ring-shaped polar cap are close to dipolar values $B_{\rm s}=B_{\rm d}(r=R,\ \vert\theta\vert\sim 0.05)$. One can show that this is a general situation, that is, $B_{\rm s}\sim B_{\rm d}$ no matter by how much $B_{\rm m}$ exceeds $B_{\rm d}$ at the pole. This follows from the fact that the angular location $\theta$ of the polar cap ring increases with increasing ratio $B_{\rm m}/B_{\rm d}\approx (m/d)(\Delta R/R)^3\simeq 8\times 10^3(m/d)$. For example, in the case presented in Fig. 5 $\vec{m}=-4\times 10^{-4}~\vec{d}$ and the last open field lines emanate at polar angles $\theta_{\rm s}\approx\pm 0.055$ radians, or at polar cap radii $R_{\rm p}\sim 6\times 10^4$ cm (for P=1 s). Thus, the narrow polar cap ring is located far from the local dipole $\vec{m}$, whose influence is weak at this distance. The circumpolar field lines between polar angles -0.053 to +0.053 are closed.

Thus, we conclude that the actual pulsar surface magnetic field $B_{\rm s}$ can significantly differ (say by an order of magnitude) from the inferred dipolar field $B_{\rm d}$ only in the case when the polarities of the global $\vec{d}$ and local $\vec{m}$ dipole (Fig. 1) are the same, as illustrated in Fig. 4. If this is the case, then $B_{\rm s}$ can largely exceed $B_{\rm d}$, which seems to be important from the viewpoint of vacuum gap formation, which requires $B_{\rm s}\ga 10^{13}$ G (see Gil & Mitra 2001). Therefore, in normal VG-driven radio pulsars the polar cap should be circular, or at least filled - if the axial symmetry does not hold. A ring-shaped polar cap can occur only in normal pulsars with $B_{\rm d}\la B_{\rm cr}$ and in radio-loud HBPs with $B_{\rm d}\ga B_{\rm cr}$.

In the accompanying paper Gil et al. (2002) explored consequences of the vacuum gap model interpretation for drifting subpulses observed in PSR B0943+10, in which 20 sparks move circumferentially around the perimeter of the polar cap, each completing one circulation in 37 pulsar periods (Deshpande & Rankin 1999, 2001). Gil et al. (2002) considered both the curvature radiation (CR) and resonant inverse Compton radiation (ICS) seed photons as sources of electron-positron pairs and determined the parameter space for the surface magnetic field structure in each case. For the CR-VG the surface magnetic field strength $B_{\rm s}>2\times 10^{13}$ G and the radius of curvature of surface field lines $0.6\times 10^{5}~{\rm cm}<{\cal R}<1.2\times 10^{5}~{\rm cm}$, while for the resonant ICS-VG $B_{\rm s}>2\times 10^{13}$  G and $10^{6}~{\rm cm}<{\cal R}<3\times 10^{6}~{\rm cm}$ (of course, in both caseS $B_{\rm s}<B_{\rm q}\sim 4.4\times 10^{13}$ G). The CR-VG with such a curved surface magnetic field does not seem likely (although it cannot be excluded), while the ICS-VG gap supported by the magnetic field structure determined by the parameter space specified above guarantees a system of 20 sparks circulating around the perimeter of the polar cap by means of the $\vec{E}\times \vec{B}$ drift in about 37 pulsar periods.

Further, Gil et al. (2002) modelled the magnetic field structure determined by the ICS-VG parameter space (specified above), using the numerical formalism developed in this paper. Since $B_{\rm p}=6.4\times 10^{19}(P\cdot \dot{P})^{1/2}~{\rm G}=4\times 10^{12}$ G in this case, to obtain $\vec{B}_{\rm s}\sim (2\div 3)\times 10^{13}$ G one needs $B_{\rm m}\gg B_{\rm d}$ and the same polarity of both components. Following the symmetry suggested by the observed patterns of drifting subpulses in PSR B0943+10, the local dipole axis was placed at the polar cap center. A number of model solutions corresponding to $r_{\rm s}\sim 0.97$ and $m\sim(1\div 2)\times 10^{-4}$ d and satisfying the ICS-VG parameter space, was then obtained. As a result of this specific modelling Gil et al. (2002) obtained a number of interesting and important conclusions: (i) The conditions for the formation of the ICS-VG are satisfied only at the peripheral ring-like region of the polar cap, which can just accommodate a system of 20 sparks performing $\vec{E}\times \vec{B}$ drift. (ii) The surface magnetic field lines within the actual gap are converging, which stabilizes the $\vec{E}\times \vec{B}$ drifting sparks by preventing them from rushing towards the pole, as opposed to the case of a diverging dipolar field (e.g. Fillipenko & Radhakrishnan 1982). (iii) No model solutions with $B_{\rm s}\sim (3\div 4)\times 10^{13}$ G and ${\cal R}\sim(0.6\div 1.2)\times 10^5$ cm, could be obtained which corresponding to the CR-VG parameter space, which in turn favors the ICS-VG in PSR B0943+10.