3 Rotation-driven asymmetry

Figure 1: Top view of orthogonally rotating pulsar with the spin period P=1.5 ms; the sense of rotation is counter-clockwise (indicated); the cartesian coordinates are in units of the light cylinder $R_{\rm lc} = cP/2\pi$. The magnetic field lines are approximated with the static-like dipole of the magnetic moment $\vec \mu$ (indicated). Two photon trajectories (starting from two opposite points on the polar cap) in the inertial observer frame (OF) are marked with two long dotted lines. These are the prototypes of the leading peak (on the left) and the trailing peak (on the right). In the corotating frame (CF) these trajectories follow the long dashed lines. Short segments of the dotted-line trajectories (OF) have been transformed via rotation to several actual photon locations in the magnetosphere, and thus they indicate the local propagation directions $\hat \eta$ of the photons in OF. Solid arrows at these locations point along the directions $\hat \eta^{\rm FP}$ of local free propagation in OF. Note much larger angles between $\hat \eta^{\rm FP}$ and $\hat \eta$ for the photons of the leading peak than for the photons of the trailing peak.

In this section we present in detail the mechanisms which lead to rotation-induced asymmetry in the (otherwise axially symmetric) hollow-cone gamma-ray beam. For the sake of better demonstration of the effects we consider an exaggerated case: propagation of a photon in the equatorial plane of a fast orthogonal rotator. Figure 1 shows the case for the rotation period P=1.5 ms: photons of the same energy in the corotating frame (CF) are emitted from two opposite points on the outer rim of the polar cap tangentially to the magnetic field lines in the CF. To follow their straight-line propagation in the inertial observer frame (OF) three effects have to be taken into account: 1) In the OF the photons propagate at an aberrated direction (dotted lines in Fig. 1) and differ in energy. 2) The rotation-induced electric field $\vec E = -\vec\beta\times\vec B$, which is present in the OF, modifies the rate of the magnetic photon absorption in a different way for photons forming the LP than for photons forming the TP. 3) The electromagnetic field in the OF is time-dependent: because of the rotation the photons propagate through different parts of the magnetosphere.

We find that the second effect - due to rotationally induced $\vec E$ - plays a dominant role in generating the asymmetry in the magnetic absorption rate R between photons of the LP and the TP. An importance of a weak electric field $\vec E \perp \vec B$for the rate R was for the first time recognized by Daugherty & Lerche (1975) who presented also its quantitative treatment. A consequence of its presence is that the rate R does not vanish along the direction of local $\vec B$ any more, and becomes non-axisymmetric with respect to it as well. Instead, the rate R vanishes along two new directions which lie in the plane perpendicular to $\vec E$ and deviate from $\vec B$ by angle $\sim$E/B in such a way that in a local coordinate frame with $\hat z \parallel \vec B$, $\hat y \parallel \vec E$, and $\hat x \parallel \vec E\times\vec B$ the "free propagation" direction $\hat \eta^{\rm FP} = [\eta^{\rm FP}_x, \eta^{\rm FP}_y, \eta^{\rm FP}_z]$has two solutions: $[E/B, 0, \pm(1 - E^2/B^2)^{1/2}]$. For definiteness, hereafter we will consider photons which propagate outwards within the regions above the northern magnetic pole (i.e. with propagation vectors $\hat \eta$ satisfying $\hat \eta \cdot \vec B > 0$) which corresponds to the case of $\eta^{\rm FP}_z = +(1 - E^2/B^2)^{1/2}$. Figure 2 shows the mean free path $\lambda_{\rm mfp} = c/R$ for the magnetic photon absorption for different directions in the plane perpendicular to $\vec E$. The rate R was calculated by performing Lorentz transformation to a frame in which $\vec E^\prime = 0$ and then applying a purely magnetic formula. The formula of Erber (1966) with a modification of Daugherty & Harding (1983) correcting its near-threshold inaccuracy is used throughout the paper:

 

$$R^\prime(\varepsilon^\prime, \vec B^\prime, \sin\theta_B^\prime)=... ...ime B^\prime \exp{[-c_2 f/ (\varepsilon^\prime \sin\theta_B^\prime B^\prime)]},$$ (1)

where $\theta_B^\prime = \angle(\hat \eta^\prime, \vec B^\prime)$, $\varepsilon^\prime$ is the photon energy; c₁ and c₂ are constant quantities, while $f = f(\varepsilon^\prime, \vec B^\prime)$ is the near-threshold correction of Daugherty & Harding (1983). Six lines in Fig. 2 correspond to six values of the ratio E/B: 0.01, 0.1, 0.2, 0.5, 0.9, and 0.999. Note that the free propagation direction deviates from the local $\vec B$by angle $\theta^{\rm FP}$ which increases with increasing contribution of the electric field: $\theta^{\rm FP}_z \equiv \arccos(\eta^{\rm FP}_z) = \arcsin(E/B) \simeq E/B$. Moreover, R increases ( $\lambda _{\rm mfp}$ decreases) monotonically with increasing angle $\angle (\hat \eta, \hat \eta^{\rm FP})$. As long as $E\ll B$, magnetic photon absorption rate Rremains approximately symmetric around the free propagation direction $\hat \eta^{\rm FP}$. The directions of $\hat \eta^{\rm FP}$ at various points within the magnetosphere are shown in Fig. 1 in the OF (with solid arrows). At the points of photon emission the direction $\hat \eta^{\rm FP}$overlaps with the photon direction $\hat \eta$ since this is where the angle $\theta_B$ between the photon direction and the magnetic field line equals $\theta_B \simeq E/B$ and consequently R = 0 (Harding et al. 1978; Zheng et al. 1998). Initially, therefore, the rate Ris the same for photons emitted from two opposite sites of the polar cap rim which in this picture give rise to the leading peak and the trailing peak. But as the photons propagate outward their $\hat \eta$ begin to deviate from local free propagation directions $\hat \eta^{\rm FP}$ opening thus a possibility for magnetic photon absorption and electron-positron pair creation. Three effects are responsible for it to happen: (1) curvature of the magnetic field lines, (2) increase of E/B with altitude, and (3) slippage of magnetic field lines under the photon's path. For a static-like dipole (assumed in Sect. 2) the curvature of lines is symmetric with respect to the magnetic axis. Consequently, symmetry is expected between the rates of magnetic absorption for LP photons and TP photons. Any asymmetry between the LP and the TP may occur via the effects (2) and (3) only. Whenever the curvature of magnetic field lines happens to be relatively large (which is the case at the polar cap rims of pulsars with $P\sim 0.1$ s) the effect (1) significantly dominates over (2) and (3), which means that the resulting asymmetry is quite subtle. However, for a relatively small curvature of magnetic field lines (and appreciable local corotation velocities $\beta$) the resulting asymmetry becomes pronounced [note that these "favourable" conditions are fulfilled within outer-gap accelerators].

Figure 2: Directional dependence of the mean free path $\lambda _{\rm mfp}$ for magnetic absorption of a photon emitted at the center of the reference frame (with the coordinates [0,0]) is shown for six values of E/B (denoted with six different line types). The electric field vector $\vec E$ is pointing at the right angle to the page, and toward the reader; the $\vec E\times\vec B$ vector is parallel to the horizontal axis, and pointing to the left. The mean free path $\lambda _{\rm mfp}$ is symmetric with respect to the horizontal axis.

Figure 3a: Directional and spectral gamma-ray characteristics calculated for the Vela pulsar with the angle $\alpha$ between the spin axis and the magnetic axis set $\alpha = 7.6^\circ$ (nearly aligned rotator). Eight rows are shown (the continuation in Fig. 3b), with three panels each. Left column shows the outgoing photons of energy $\varepsilon > \varepsilon _{\rm limit}$ which are mapped onto the parameter space $\zeta _{\rm obs}$ vs. $\phi$, where $\zeta _{\rm obs}$ is the viewing angle (between the spin axis and the l.o.s) and $\phi$ denotes the phase of rotation. Middle column shows the double-peak pulse profile formed with these photons when $\zeta _{\rm obs}= 7.6^\circ$ is chosen (yielding the peak-to-peak separation equal 0.42). Right column shows the phase-averaged energy spectrum (the flux level $\varepsilon F_{\varepsilon }$ in arbitrary units) for $\zeta _{\rm obs}= 7.6^\circ$ i.e. the same for all rows. Dotted vertical line indicates the part of the spectrum ( $\varepsilon > \varepsilon _{\rm limit}$) which contributes to the corresponding pulse profile on the left. The eight rows correspond to 8 consequtive values of $\varepsilon _{\rm limit}$: 1, 10, 102, 103, $4\times 10^3$, $6\times 10^3$, $8\times 10^3$, and 104 MeV (these values are displayed in the panels of the right column).

Figure 3b: Continued.

We find that the effect (2) - the presence of electric field - is crucial for generating the asymmetry. The way in which this effect operates can be most clearly assessed by inspecting Fig. 2. Let us consider a photon propagation vector $\hat \eta$ anchored at the origin of frame in the figure. For photons emitted close to the star, both in the LP and the TP, $\hat \eta$ initially points along the $\hat \eta^{\rm FP}$ direction which differs only slightly from the direction of $\vec B$ in Fig. 2 since $E\ll B$. As the LP photon moves away from emission point, its propagation vector rotates clockwise in Fig. 2 which reflects the fact that $\vec B$ diverges from $\hat \eta$ due to the magnetic field line curvature. At the same time, however, the directional pattern of $\lambda _{\rm mfp}$rotates counterclockwise due to increase in E/B which additionally enhances the absorption rate. In the case of the TP photon, however, both its propagation vector $\hat \eta$ and the directional pattern of $\lambda _{\rm mfp}$ rotate in the same direction (counterclockwise) with respect to $\vec B$ in Fig. 2 so that the absorption rate is weakened. Thus, for photons within the LP the effects (1) and (2) cummulate, whereas for the TP they effectively tend to cancel out each other, and the expected asymmetry between the peaks is due to stronger absorption suffered by photons within the LP than by photons of the same energy within the TP. Accordingly, the high energy cutoff in the LP spectrum will occur at a slightly lower energy than the cutoff in the TP spectrum. The slippage (3) affects this picture in the way which depends on both the rotation period and the photon position within the magnetosphere, but an overall picture remains unchanged. For most rotation periods (P > a few ms), the slippage reduces the asymmetry only marginally. For the fastest rotators ( $P \sim 1.5$ ms) it enhances the asymmetry by making photons of the TP to propagate along the free propagation direction (or, equivalently, along the magnetic field lines in the CF). The latter case is presented in Fig. 1 where the photon propagation direction $\hat \eta$ as seen in the OF (dotted lines) and the local free propagation direction $\hat \eta^{\rm FP}$ in the OF (solid arrows) are shown for several positions along the photon trajectory in the CF. Strong absorption within the LP is anticipated, whereas within the TP $\hat \eta$ nearly coincides with $\hat \eta^{\rm FP}$.

Figure 4: Close-up view of the pulse profile for $\varepsilon > 1$ MeV from Fig. 3a. The curvature-radiation wings ( $\vert\phi \vert > 0.3$) accompanying both peaks can be now acknowledged. Their shape is modelled analytically (shown with thick solid lines).

Another way to understand this asymmetry is to follow photon trajectories in a reference frame (with primed quantities) where the condition

$$\vec E^\prime = 0$$ (2)

is fulfilled. In such a frame, an asymmetry in $R^\prime$ (cf. Eq. (1)) for the LP and TP photons arises from transformation properties of $\theta_B^\prime$ (aberration), $\varepsilon^\prime$ (red- or blue-shift), and $B^\prime$. One of the reference frames satisfying the condition (2) is a reference frame of local $\vec E\times\vec B$ drift. Denoting dimensionless drift velocity $\vec \beta_{\scriptscriptstyle D} = \vec E\times\vec B/B^2$, the transformations read:

 

$$\varepsilon^\prime = \varepsilon~\gamma_{\scriptscriptstyle D}(1 ... ...\ \ \ \ \ \ \ \ \ \vec B^\prime = \frac{\vec B}{\gamma_{\scriptscriptstyle D}},$$

$$\sin\theta_B^\prime= \frac{[(\eta_x - \beta_{\scriptscriptstyle D... ...\scriptscriptstyle D}^2)]^{1/2}}{(1 - \eta_x\beta_{\scriptscriptstyle D})}\cdot$$ (3)

In the equatorial plane of orthogonal rotator $\eta_y = 0$so that $\eta_x = \mp\sin\theta_B$, where the signs "minus" and "plus" correspond to the leading and the trailing peak, respectively. The transformation of propagation angle reduces then to $\sin\theta_B^\prime = \vert\eta_x - \beta_{\scriptscriptstyle D}\vert/(1 - \eta_x\beta_{\scriptscriptstyle D})$. The Taylor expansion of $\sin\theta_B^\prime$, $\varepsilon^\prime$, and $B^\prime$ around $\beta_{\scriptscriptstyle D} = 0$ reveals that a difference between the magnetic absorption rates in the locally drifting frame and in the OF results primarily from the aberration of photon direction, whereas the changes in $\varepsilon$ and $\vec B$ are second order effects. Obviously, the aberration is asymmetric for the LP and the TP ( $\eta_x < 0$ and $\eta_x > 0$, respectively). Figure 1 presents this "aberration effect" in the rigidly corotating frame, where $\vec E^\prime = 0$ is assumed. Photon trajectories in this frame (dashed curves) indicate clearly that photons of the leading peak encounter larger $B^\prime_\perp$ than photons of the trailing peak of the same energy.