ON PLASMA ROTATION AND DRIFTING SUBPULSES IN PULSARS: USING ALIGNED PULSAR B0826−34 AS A VOLTMETER

Let us do a more careful derivation of the formula for the drift speed of plasma above the accelerating gap in the pulsar polar cap than that presented in Ruderman & Sutherland (1975). In the reference frame corotating with the NS, Faraday's law for electric and magnetic field as measured in the corotating frame has the same form as in the laboratory frame (Schiff 1939), namely,

$$\begin{equation} \nabla \times \mathbf {E} = \frac{1}{c}\frac{\partial {\mathbf {B}}}{\partial {t}}\,. \end{equation} \tag{ 1 }$$

The electric field in Equation (1) represents the non-corotational component of the electric field—the component that forces plasma to move in the rotating frame. In the corotating frame the temporal variations of the magnetic field in the polar cap can be caused only by fluctuations of currents in the magnetosphere. Even if fluctuations of the electric current are of the order of the Goldreich–Julian current ρ_GJc, the resulting variation of the magnetic field is δB/B ≃ (R_NS/R_LC)^3/2 ≃ 6 × 10⁻⁶P^−3/2; here R_NS is the NS radius and R_LC ≡ c/Ω is the light cylinder radius, Ω_NS is the angular velocity of NS rotation, and P is pulsar period in seconds. Hence, $\nabla \times \mathbf {E}=0$ with high accuracy and circulation of the non-corotational electric field along a closed path is zero (even for an inclined rotator).

Let us consider the circulation of the electric field along a fiducial path abcd—see Figure 1. In the pulsar polar cap the top (ab) and bottom (cd) parts are infinitesimally small and are just below the NS surface (cd) and above the accelerating gap in the region of dense plasma with screened electric field (ab); the lateral sides (bc and da) follow magnetic field lines. Note that we consider a path that is fully in the open magnetic field line region, in contrast to the work of Ruderman & Sutherland (1975), where one lateral path was along the field line along the polar-cap boundary (see their Figure 4). To clarify our derivation, Figure 1 shows our unit vectors along positive directions for the perpendicular component of the electric field, e_⊥, and the drift velocity, e_D. The circulation of the electric field along this path is

$$\begin{eqnarray} \oint {}E\,dl & = & \displaystyle - E_\perp \delta {}r + \int _b^cE_{||}\,dz + \int _d^aE_{||}\,dz \nonumber \\ & = & \displaystyle - E_\perp dr + V_{{\rm bc}} - V_{{\rm ad}}= 0\,, \end{eqnarray} \tag{ 2 }$$

where E_⊥ and E_|| are components of the electric field, perpendicular and parallel to the magnetic field, respectively, while V_bc, V_ad are potential drops in the accelerating gap, along magnetic field lines. E_⊥ = 0 just below the NS surface, which we assume to be a perfect conductor. Taking the limit δr → 0, we get for the non-corotational component of the electric field

$$\begin{equation} E_\perp = -\frac{dV}{dr}\,. \end{equation} \tag{ 3 }$$

The drift velocity of the plasma relative to the NS is then

$$\begin{equation} v_{\rm D} = c\frac{\mathbf {E}\times {}\mathbf {B}}{B^2} = \frac{c}{B} \frac{dV}{dr}\,. \end{equation} \tag{ 4 }$$

The drift velocity of plasma relative to the NS thus depends on the variation of the electric potential in the acceleration zone. For aligned pulsars this result is well-known textbook material (e.g., Beskin 2010), and for inclined rotators the correct general expression for the plasma drift velocity was explicitly mentioned in Fung et al. (2006); as we shall detail below, this result partially supersedes the Ruderman & Sutherland family of models, but it has not yet struck root. Here we want to draw special attention to it, as it has profound consequences for the interpretation of pulsar drift phenomena.

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The reasoning leading up to this conclusion, i.e., the variation of the accelerating potential causes plasma drift, can be put differently and perhaps more insightfully: the electric field is the gradient of the electrostatic potential, $\mathbf {E}=-\nabla {}V$; in the co-rotating frame the potential is the accelerating potential and the perpendicular electric field E_⊥ is the field that causes drift relative to the NS; hence, E_⊥ = −(∇V)_⊥ = −∂V/∂r. If, for example, the potential drop is the same along all polar-cap magnetic field lines, there is no relative rotation of plasma, however large the potential drop is. In that case, though, there is a region very close to the polar-cap boundary where the potential drop plummets to zero at the last closed magnetic field line. Plasma in that region will have a very large drift velocity.

The original derivation of the expression for v_D in Ruderman & Sutherland (1975) is similar to our derivation here, but instead of expressing v_D in terms of the derivative of the accelerating potential dV/dr, an order-of-magnitude estimate is used: their Equation (31) is equivalent to our Equation (4), but using the entire potential drop along a field line V for dV and half of the polar-cap radius, r_pc/2, as an estimate for dr. Yet, the potential drop in the pair-production zone is expected not to vary much from one field line to another, except at the boundaries of the active zone (e.g., Harding & Muslimov 1998; Hibschman & Arons 2001); the active zone is also expected to be a noticeable part of the polar cap, at least in the space charge limited flow model (Arons & Scharlemann 1979). Hence, the Ruderman & Sutherland (1975) order-of-magnitude prediction for the drift rate is a large overestimate. And indeed, the drift-rate speeds predicted by Ruderman & Sutherland (1975) in their Equation (31) are much larger than those observed—for example, the very slow but steady drift rate observed in PSR B0809+74 (van Leeuwen et al. 2003) is incompatible with this order-of-magnitude estimate. Thus, in the most detailed drifting-subpulse models based on Ruderman & Sutherland (1975), a substantial reduction of the accelerating potential is necessary to reproduce these observed slow drift rates. In Gil & Sendyk (2000) and Gil et al. (2003), this is achieved by assuming highly non-dipolar polar-cap magnetic fields and by fine-tuning the particle flux impact to produce the critical surface temperature.

Furthermore, literal application of Ruderman & Sutherland's (1975) expression for the plasma drift speed (i.e., in the form v_D = cV/r_pcB) has difficulty explaining the drift-rate variations with time or longitude that are observed in several pulsars (such as the bidirectional drift bands in PSR J0815+09; Champion et al. 2005) and is incapable of producing the complete drift direction reversals seen in PSR B0826−34. The explanation for such observed subpulse-drift reversals usually involves aliasing. In the case of B0826−34 this requires the circulation frequency to be fine-tuned to the rotation frequency, such that the aliased subpulses appear to move only very slowly; and indeed Gupta et al. (2004) were able to construct such a model. However, Esamdin et al. (2005) find that the observed reversals influence the intrinsic subpulse brightness, strongly suggesting that the reversal is not just apparent, but actually occurs in the corotating frame. The existence of such real reversals thus suggests a failing of the $\mathbf {E}\times \mathbf {B}$ family of models.

We want to point out that all these apparent discrepancies between model and observation are non-existent. Using the derivation we present here, in Equation (4), no fine-tuning or additional assumptions, such as strong higher-order magnetic fields, are needed to explain the observed subpulse-drift properties. The drift rate depends on the variation of the potential drop from one field line to another. The potential drop can be very large, but if it does not change much across the polar cap, the drift rate will be slow. Depending on whether the potential drop increases or decreases toward the center of the polar cap, the drift rate will be in one or another direction. If the potential-drop derivative at the line of sight varies with time, the observed drift speed changes and can even reverse. In Section 5, we use subpulse-drift data from B0826−34 to show that the variation of the accelerating potential necessary to explain these phenomena is small indeed and that the model is plausible.

We reiterate that we are not solving these problems by introducing extra assumptions on how the potential drop varies. Equation (4) simply is the correct dependence for the drift velocity of the plasma; it is not a fine-tuning or modification of the previously widely used order-of-magnitude expression for v_D. If drifting subpulses are indeed due to plasma motion relative to the NS, all their properties depend on the variation of the potential drop and not on the potential drop itself. In Section 6, we will also argue that all the properties of the accelerating potential necessary for explanation of various properties of drifting subpulses seem to be integral properties of the standard pulsar model.

In pulsars with subpulse drift, Equation (4) can be used to set limits on the local parameters of the pulsar polar cap. There, a measurement of the plasma drift velocity provides the value of the radial derivative of accelerating potential. In practice, however, the quantity that subpulse-drift observations produce is the angular drift velocity of subpulses within the pulsar profile, not the plasma drift velocity in the pulsar coordinate frame that is needed in Equation (4). Below we derive expressions that connect observable parameters to the variation of the accelerating potential.

The momentary angular velocity of plasma relative to a magnetic axis is

$$\begin{equation} \Omega _{\rm D} = \frac{c}{B r} \frac{dV}{dr}\,, \end{equation} \tag{ 5 }$$

where r is the distance between our sight-line path and the magnetic axis. When we derived this expression in Section 2.1, we considered the path segment ab in the region above the accelerating gap where electric field is screened, and so B, r, dr in Equation (5) are also taken in some plane perpendicular to the magnetic axis $\boldsymbol{\mu }$, in the region with screened electric field (see Figure 1).

Our path abcd has lateral sides that follow magnetic field lines; hence, if Φ is the magnetic flux limited by some closed path passing through our fiducial point (e.g., the path A or A_* in Figure 1), then Br = ∂Φ/∂r∂φ, where φ is the azimuthal angle. For any magnetic field that is axisymmetric relative to some magnetic axis, the plasma drift relative to the NS will be a rotation around this magnetic axis. The magnetic flux can be measured as the flux through a circle of radius r, and Equation (5) takes the form

$$\begin{equation} \Omega _{\rm D} = 2 \pi {}c\,\frac{dV}{d\Phi }\,. \end{equation} \tag{ 6 }$$

It will be convenient to normalize the potential drop along a magnetic field line to the potential drop across magnetic field lines at the NS surface between the rotation axis and the boundary of the polar cap in an aligned pulsar with the same period:

$$\begin{eqnarray} \Delta {V}_{\rm vac}& = & \displaystyle - \int _0^{r_\mathrm{pc}} E\,dr = \frac{1}{2}\frac{\Omega_{\rm NS} }{c} B r_\mathrm{pc}^2 = \frac{\Omega _{\rm NS}\Phi _\mathrm{pc}}{2\pi {}c}\nonumber \\ & \simeq & \displaystyle 6.6\times 10^{12}R_{{\rm NS},6}^3 B_{12} P^{-2} \; \mbox{V}\,, \end{eqnarray} \tag{ 7 }$$

where r_pc is the polar-cap radius, Φ_pc is the magnetic flux through the polar cap, B₁₂ is the magnetic field strength in units of 10¹² G, and R_{NS, 6} is the NS radius in 10⁶ cm. In doing this numerical estimate, we assumed that the polar cap is small enough that the variation of the magnetic field over it is negligible. ΔV_vac is a good estimate for the available potential drop—if there were vacuum in the open magnetic field line zone, the potential drop along magnetic field lines in the polar cap would reach a value comparable to ΔV_vac at the height ∼r_pc from the NS surface and then vary very slowly with the altitude, approaching its maximum value ΔV_vac from below.

Normalizing the magnetic flux to Φ_pc, we get from Equation (6)

$$\begin{equation} \frac{\Omega _{\rm D}}{\Omega _{\rm NS}} = \frac{d\tilde{V}}{d\tilde{\Phi }}, \end{equation} \tag{ 8 }$$

where $\tilde{V}\equiv {}V/\Delta {V}_{\rm vac}$ and $\tilde{\Phi }\equiv {}\Phi /\Phi _\mathrm{pc}$. The normalized magnetic flux is $\tilde{\Phi }=(r/r_\mathrm{pc})^2= (\theta /\theta _\mathrm{pc})^2$, where θ is the colatitude measured from the magnetic axis. Let us denote the colatitude to the colatitude of the polar-cap boundary as

$$\begin{equation} \xi \equiv \frac{\theta }{\theta _\mathrm{pc}}; \end{equation} \tag{ 9 }$$

then in terms of ξ, Equation (8) takes the form

$$\begin{equation} \frac{\Omega _{\rm D}}{\Omega _{\rm NS}} = \frac{1}{2\xi }\frac{d\tilde{V}}{d\xi }. \end{equation} \tag{ 10 }$$

After one rotation the difference in phase between a fiducial point at the NS surface and an emitting column will be

$$\begin{equation} \delta {\phi }_1 = \Omega _{\rm D} P = \frac{\omega _{\rm D}\pi }{180^\circ }\,, \end{equation} \tag{ 11 }$$

where ω_D is the drift velocity in units of degrees/period. Using Equation (10), we can then write the derivative of the accelerating potential as (hereafter we use V instead of $\tilde{V}$)

$$\begin{equation} \frac{dV}{d\xi } = \xi \frac{\omega _{\rm D}}{180^\circ }. \end{equation} \tag{ 12 }$$

Thus, if we measure the angular velocity at one colatitude, we can now calculate the derivative of the accelerating potential at that point. If we could measure it at two different colatitudes, we can set limits on the accelerating potential, assuming a gradual variation of the potential across the polar cap. For such a study our line of sight must cross two separate drift bands at different colatitudes. An ideal case would be a bright, slightly inclined rotator where the line of sight crosses two regions at different colatitudes, each for a significant fraction of period such that any drifting subpulses are clearly visible; B0826−34 is such a pulsar. It has two separate drift bands that are thought to represent emission from different colatitudes. This pulsar provides us a unique opportunity to put direct observational constraints on the physical conditions in the polar-cap accelerating zone—the heart of the pulsar.

Using fitSubPulses (van Leeuwen et al. 2002), we next automatically and visually inspected the data for nulls and mode changes. These are robustly separated in a pulse-energy histogram (Janssen & van Leeuwen 2004) and verified manually. In Figure 3 we show several nulls and a transition from strong to weak mode. In two of the four sessions, PT0168_150 and PT0169_152 (Table 1), the pulsar emitted in the weak mode for the entire session. In the two other observations strong-mode sequences are found, one of ∼2000 and one of ∼7000 s duration (i.e., ∼1000 and ∼4000 individual pulses for this 1.85 s period pulsar). In Figure 4, we show the long-term behavior in the strong and weak mode and a transition between the two.

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Within the strong-mode sequences, we next worked toward identifying drift rates and directions. Our aim was to characterize the variations in subpulse drifting over the wide profile. We thus fit for the location of individual subpulses. This is in contrast to the method described by Esamdin et al. (2005) for B0826−34, where a comb-like template with nine main drift tracks at fixed spacings was used to come to an estimate of the drift rate averaged over all drift tracks. Such a method cannot measure drift rates that vary with pulse longitude, as is seen in many pulsars (Weltevrede et al. 2006). For each strong-mode pulse, we thus fitted for the location of individual subpulses: we smoothed the single-pulse profile by a 4 ms window (the approximate average width of the subpulses) and then determined the location of the maximum. The maximums in subsequent single pulses were next grouped together, if they were less than half the average subpulse separation apart in phase. Only tracks with more than 10 subpulses were retained. In this manner, subpulse tracks of more than 100 pulses were robustly identified. Within each of these tracks, the variation in drift speed and direction was next measured by least-squares fitting to subsequent sets of 10 subpulses, as illustrated in the rightmost panel of Figure 5.

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The accelerating potential estimates we produce only make the general assumption that subpulses are produced by emitting features that are stationary relative to the outflowing plasma in the polar-cap region, and their drift is due to the rotation of this plasma, relative to the pulsar. Here we wish to propose a more specific, but still qualitative and schematic, model for drifting subpulses that well fits the de facto standard force-free model of pulsar magnetospheres and that is consistent with the results presented in this paper.

6.1.1. Timescales

6.1.2. Magnitude of Potential Drop Variations

6.1.3. Plasma and Drifting Subpulses

Certain observations could verify or falsify our model.

In this paper, we relied on the geometrical model of B0826−34 proposed by Esamdin et al. (2005). Only with a more accurate geometrical model can we place more robust constraints on the plasma rotation in the polar cap on this pulsar. Polarization measurements can constrain this emission geometry by measuring the polarization angle sweep. Such observations would have to be undertaken with sufficient collecting area, as only single-pulse polarization measurements can identify and correct for the orthogonal mode changes in B0826−34 (Lyne & Manchester 1988).

Using their averaging technique, Esamdin et al. (2005) followed the strong drift tracks and integrated all single pulses that have the same drift track phase. Those average profiles suggest that drift tracks exist in between the two strongly emitting regions. These tracks sample co-latitudes between our two extremes ξ_{1, 2}. Observations with higher sensitivity than those presented here could measure the pulse-to-pulse drift-rate changes of those weak subpulses and provide an acceleration potential estimate over the intermediate range of colatitudes.

If observations at higher frequencies sample lower emission heights (Cordes 1978), where the polar-cap radius r_pc is reduced, then the observer sight line may start to cross r_pc. From that, the extent of the polar cap and the values of ξ_{1, 2} could be derived. In recent observations at 3094 MHz (Serylak 2011) the pulse profile of B0826−34 still spans 360°, however, so any evidence for a decreasing pulse width would have to be sought at even higher frequencies.

After the report of the existence of the weak emission mode in B0826−34 by Esamdin et al. (2005), no evidence for such a mode was found by Bhattacharyya et al. (2008), down to an upper limit reported to be more constraining than the original detection in Esamdin et al. (2005). In addition to the confirmation of the weak mode at 685 and 3094 MHz in (Serylak 2011), we have here detected B0826−34 in weak mode in all data sets from Table 1, including in the data not originally used by Esamdin et al. (2005), such as the data set in the right panel of Figure 4.

In the model we suggest, the weak mode is a magnetospheric state that is different from the strong mode and should have a different plasma drift rate. Thus, the detection of subpulse drift in the weak mode would provide a stringent test of the general model we propose.