5 Rotational asymmetry as a function of pulsar parameters

The asymmetry effects are marginal for nearly aligned pulsars with periods $P\sim 0.1$ s. For the above-described model of Vela, they are noticeable only because of the high altitude of the accelerator ( $h_0 = 4\ R_{\rm NS}$). However, for highly inclined ( $\alpha \ga 45^\circ$) and fast pulsars ($P \la 10$ ms) the rotational effects result in asymmetry of considerable magnitude. If detected by GLAST, GeV emission from such objects would provide powerful diagnostics of the polar cap model. Below we present the magnitude of such asymmetry predicted for a wide range of parameters for fast pulsars.

As a measure of rotational effects we consider the escape energy $\varepsilon _{\rm esc}$ which is defined as a maximum energy of a photon (and the photon is emitted tangentially to its "parent" magnetic field line; a footpoint of this line has magnetic colatitude $\theta$) for which the magnetosphere is still transparent, i.e. for which the optical depth integrated along the photon trajectory is less than 1. This energy was calculated with our numerical code and the results are shown below. For the sake of comparison with the case of no rotational effects we recall a simplified, yet quite accurate for magnetic fields weaker than $\sim$ $0.1 B_{\rm cr}$, analytic formula which (after e.g. Bulik et al. 2000) reads

$$\varepsilon_{\rm esc}(\theta) = \frac{\theta_{\rm pc}}{\theta} \cdot \varepsilon_{\rm esc}(\theta_{\rm pc}),$$ (5)

where

 

$$\varepsilon_{\rm esc}(\theta_{\rm pc}) \approx 1\ {\rm GeV}\ \lef... ...}}{0.01~{\rm rad}}\right)^{-1} \left(\frac{r_{\rm m}}{R_{\rm NS}}\right)^{5/2}.$$ (6)

Here $r_{\rm m}$ is the radial coordinate of the emission point (with spherical coordinates ($r_{\rm m}$, $\theta_{\rm m}$, $\phi _{\rm m}$) in the right-handed frame with $\hat z$-axis along the dipole axis) and $\theta \ll 1$ rad is the magnetic colatitude of a footpoint of the parent magnetic field line at the neutron star surface. For field lines originating at the polar-cap rim one should take then $\theta = \theta _{\rm pc}$, where $\theta_{\rm pc} = \arcsin ({\sqrt(R_{\rm NS}/R_{\rm lc}})) \approx 1.45\times 10^{-2}/\sqrt{P}$ radians, and $R_{\rm lc} = cP/2\pi$ is the light cylinder. Note that the escape energy of Eq. (5) is symmetrical with respect to the dipole axis and goes to infinity at the magnetic pole ( $\theta = 0$). Such a behaviour is not the case, however, when the rotational effects discussed in Sect. 3 are taken into account.

Figure 6: The ratio of the number of photons in the trailing peak and the leading peak P2/P1 inferred from the EGRET data for Vela is shown in function of photon energy  $\varepsilon$. The results of model calculations performed for three different values of the altitude h0 of the accelerator are indicated with three lines: dashed, dotted, and solid - for $h_0 = 2 R_{\rm NS}$, $3 R_{\rm NS}$, and $4 R_{\rm NS}$, respectively.

Figure 7: The escape energy $\varepsilon _{\rm esc}$ of photons from the polar cap surface of an orthogonal rotator with $B_{\rm pc} = 10^{12}$ G is shown as a function of normalized magnetic colatitude $\theta /\theta _{\rm pc}$ of the emission points. The points are assumed to lay along the cross-section of the polar cap surface with the equatorial plane of rotation, thus location of each point is determined by $\theta /\theta _{\rm pc}$in the range [0, 1], and the magnetic azimuth $\phi _{\rm m}$ equal either to $\pi /2$ (for the leading half of the polar cap) or $-\pi /2$ (for the trailing half). Three solid lines are labelled with the corresponding spin periods P of 0.1 s, 10 ms, and 1.5 ms. Moreover, to illustrate the difference with the case when rotational effects are ignored, each solid line is accompanied by a dashed line drawn according to Eq. (5).

In Figs. 7, 8, and 9 we present the values of $\varepsilon _{\rm esc}$ obtained numerically for fast rotators with $B_{\rm pc} = 1$ TG, and emission points located on the polar cap surface, i.e. $r_{\rm em} = R_{\rm NS}$ was assumed everywhere. Let us begin with the case of orthogonal rotators (i.e. $\alpha = 90^\circ$) - these are shown in Figs. 7 and 8. Here we consider emission points lying along the crossection of polar cap surface with the equatorial plane of rotation (hence, the results would be relevant for observers located at $\zeta_{\rm obs} = 90^\circ$). Figure 7 shows how $\varepsilon _{\rm esc}$ varies with location of the emission point across the polar cap. The location of each point is determined by the normalized magnetic colatitude $\theta /\theta _{\rm pc}$in the range [0, 1] and the magnetic azimuth $\phi _{\rm m}$ equal either to $\pi /2$ (for the leading half of the polar cap) or $-\pi /2$ (for the trailing half). The spin periods of 0.1 s, 10 ms, and 1.5 ms were considered.

In order to quantify the asymmetry in $\varepsilon _{\rm esc}$ between the leading and the trailing parts of the polar cap we introduce the following parameter:

$${\cal R}_{\rm esc}(\theta) = \frac{\varepsilon_{\rm esc}^{\rm tp}} {\varepsilon_{\rm esc}^{\rm lp}},$$ (7)

where

 

$$\varepsilon_{\rm esc}^{\rm lp} \equiv \varepsilon_{\rm esc}(\phi_{\rm m}=\pi/2,\ \theta),$$

$$\varepsilon_{\rm esc}^{\rm tp} \equiv \varepsilon_{\rm esc}(\phi_{\rm m}=-\pi/2,\ \theta).$$ (8)

For example, ${\cal R}_{\rm esc}(\theta_{\rm pc})$gives the asymmetry between two opposite points located on the outer rim of the polar cap. For spin periods around 0.1 s, typical for the known gamma-ray pulsars, ${\cal R}_{\rm esc}(\theta_{\rm pc})$ remains close to unity; e.g. for the case of P = 0.1 s in Fig. 7, ${\cal R}_{\rm esc}(\theta_{\rm pc}) \simeq 1.3$. However, for P smaller than 0.01 s the parameter ${\cal R}_{\rm esc}(\theta_{\rm pc})$ becomes definitely larger: $\sim$2.5 for P = 0.01 s and $\sim$30 for P = 1.5 ms. Thus, for pulsars with $P \la 0.01$ s and large inclination angles $\alpha$we predict a notable difference (in excess of half a decade in photon energy) between the positions of high-energy spectral cutoff in the gamma-ray emission from the leading and the trailing part of the polar cap.

Another interesting implication of fast rotation is that $\varepsilon _{\rm esc}$ has finite values for any $\theta$, including $\theta = 0$ (the magnetic pole), in contrast to the "static" case of Eq. (5). This can be understood in the following way: consider emission points with decreasing colatitude $\theta$ in the leading part of the polar cap. As we approach the dipole axis ( $\theta \rightarrow 0$), $\varepsilon _{\rm esc}$ increases because the decreasing curvature of magnetic field lines leads to smaller angles between $\vec B^\prime$ and the photon propagation direction $\hat \eta^\prime$in the corotating frame CF. However, photon trajectory bends backwards in the CF (see dashed lines in Fig. 1), which implies that also photons emitted at $\theta = 0$along the straight dipolar axis will quickly encounter $B^\prime_\perp \neq 0$, thus being subject to the magnetic absorption. Now entering the trailing part of the polar cap leads to further increase of $\varepsilon _{\rm esc}$. This is because magnetic field lines start to bend in the same direction as the photon trajectory in the CF (in other words - the efficiency of absorption decreases for emission points in the trailing part of the polar cap). Eventually, at some point (we denote it as $\theta_0$) the escape energy reaches a maximum. This is the point where the magnetic field slippage along with the aberration of photon direction ensure small angles between $\vec B^\prime$ and $\hat \eta^\prime$ over large distances in the photon trajectory. Therefore, the faster is the rotation, the larger is the colatitude $\theta_0$ of that point. For example, $\theta_0/\theta_{\rm pc} \simeq 0.15$for P = 0.1 s whereas $\theta_0/\theta_{\rm pc} \simeq 0.42$ for P = 0.01 s (see Fig. 7). For P=1.5 ms this maximum occurs  close to the outer rim (in the trailing part) and therefore huge asymmetry with respect to the leading rim is predicted: ${\cal R}_{\rm esc}(\theta_{\rm pc}) \simeq 30$. With further increase of $\theta$ (towards the trailing rim of the polar cap) local magnetic field lines start to bend stronger than the photon trajectory in the CF, and this is why $\varepsilon _{\rm esc}$ should now decrease. However, this decrease never compensates the asymmetry in $\varepsilon _{\rm esc}$ at $\theta_{\rm pc}$with respect to $\theta = 0$, i.e. one always ends up with ${\cal R}_{\rm esc}(\theta_{\rm pc}) > 1$.

Figure 8: Escape energy at fixed magnetic colatitude $\theta$ as a function of rotation period. The dashed, solid, and dotted lines correspond to $\theta =2.6^\circ$, $8.3^\circ$, and $22^\circ$, respectively. The thin lines correspond to points on the leading side of the dipole axis (with spherical coordinates ( $R_{\rm NS}$, $\pi /2$, $\theta$)). The thick lines correspond to points on the trailing side (with ( $R_{\rm NS}$, $-\pi /2$, $\theta$)). The curves are for emission in the equatorial plane of orthogonal rotator.

Figure 8 presents escape energy $\varepsilon_{\rm esc}^{\rm lp}$ and $\varepsilon_{\rm esc}^{\rm tp}$as a function of spin period P. This energy was calculated for a fixed position $\theta_{\rm fxd}$ of the emission point, in order to highlight its dependence on rotation. We chose three pairs of oppositely located emission points at: $\theta _{\rm fxd} = 2\hbox{$.\!\!^\circ$ }6$, $8\hbox{$.\!\!^\circ$ }3$, and $22^\circ$, which corespond to $\theta_{\rm pc}$ for P = 0.1 s, 10ms, and 1.5 ms, respectively. As in Fig. 7, the emission points were placed at the neutron star surface, in the equatorial plane of the orthogonal rotator. In the case of slow rotation ($P \sim 1$ s), the values of $\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm fxd})$ for the leading point and $\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm fxd})$ for the trailing point are practically identical, and well approximated by Eq. (5). As rotation becomes faster (P around $\sim$0.1 s) $\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm fxd})$ and $\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm fxd})$ start to diverge due to the asymmetric influence of $\vec E$. At even shorter periods, below $\sim$0.03 s, the maxima in $\varepsilon_{\rm esc}^{\rm tp}$ are reached (at the values of P, which can be reproduced by solving Eq. (9) with $\theta_{\rm fxd}$ in place of $\theta_0$) because slippage of field lines starts to be important (see previous paragraph). For increasing $\theta_{\rm fxd}$ ( $2\hbox{$.\!\!^\circ$ }6$, $8\hbox{$.\!\!^\circ$ }3$, and $22^\circ$ in Fig. 8), the asymmetry parameter ${\cal R}_{\rm esc}(\theta _{\rm fxd})$ decreases for slow rotators ($P \ga 0.1$ s), and it increases for fast (millisecond) rotators.

Figure 9: Escape energy for the leading and the trailing emission points at the polar cap rim as a function of rotation period. Six uppermost lines (thick) correspond to the trailing points. Thin lines are for the leading points (the lines for $\alpha = 70^\circ$, $80^\circ$, and $90^\circ$ overlap).

In Fig. 9 we present $\varepsilon_{\rm esc}^{\rm lp}(\theta_{\rm pc})$ and $\varepsilon_{\rm esc}^{\rm tp}(\theta_{\rm pc})$ as a function of spin period Pfor a set of dipole inclinations $\alpha$. Note that unlike in Fig. 8, the emission points are now placed at $\theta = \theta _{\rm pc}$, i.e. at the rim of polar cap corresponding to P. In the case of small inclinations (e.g.  $\alpha = 10^\circ$), the resulting ratio ${\cal R}_{\rm esc}(\theta_{\rm pc})$ remains close to unity even for millisecond periods. The general increase in $\varepsilon _{\rm esc}$ with P increasing, noticeable in Fig. 9, reflects the approach of emission points to the dipole axis, where the curvature of magnetic field lines is small. The trend is well described by $\varepsilon_{\rm esc}\propto \sqrt{P}$ as given by Eq. (6). In the range of spin periods below $\sim$0.1 s and with $\alpha \ga 45^\circ$, the difference between $\varepsilon_{\rm esc}^{\rm lp}$ and $\varepsilon_{\rm esc}^{\rm tp}$ becomes pronounced, especially for highly inclined millisecond pulsars. If detected by GLAST, high-energy emission from such objects would provide an ideal test of the polar cap model.

However, for the asymmetry effects to be detectable, an additional condition (apart from short period and large inclination) must be fulfilled: the viewing geometry must be of the "on-beam" type, i.e. the observer's line of sight must cross the narrow beam of radiation at the high-energy cutoff - where the magnetic absorption operates. If this is the case, the asymmetry in the absorption may be noticeable even in the phase-averaged spectra. As an example, we show in Fig. 10 the phase-averaged spectrum calculated for a millisecond pulsar with P=2.3 ms, $B_{\rm pc} = 10^9$ G, $\alpha = 60^\circ$, and for $\zeta_{\rm obs} = 60^\circ$. For this rotator we obtain $\varepsilon_{\rm esc}^{\rm lp} \simeq 10^5$ MeV and $\varepsilon_{\rm esc}^{\rm tp} \simeq 5\times 10^5$ MeV at the rim of the polar cap. As a result of this rotationally induced asymmetry in magnetic absorption for the leading and the trailing peak the spectrum at its high-energy cutoff assumes a step-like shape: below $\simeq$10⁵ MeV the spectrum consists of photons from both the leading and the trailing peak, whereas between $\sim$10⁵ MeV and $\sim$ $5\times 10^5$ MeV only photons of the trailing peak contribute to the spectrum; at $\varepsilon \simeq \varepsilon_{\rm esc}^{\rm lp}$ the level of the spectrum drops by a factor of $\sim$2. In these particular calculations of the spectrum, we assumed that the density distribution of primary electrons over the polar cap is dominated by an outer-rim component (see the captions to Fig. 10 for details of the distribution).

Figure 10: Theoretical "on-beam" spectrum of high-energy emission from a millisecond pulsar with P = 2.3 ms, $B_{\rm pc} = 10^9$ G, and $\zeta _{\rm obs} = \alpha = 60^\circ$. The density distribution of primary electrons over the polar cap, which was assumed in this calculation, had a maximum at the polar cap rim and was uniform in magnetic azimuth $\phi _{\rm m}$. The magnetic colatitude profile of this distribution had a Gaussian shape centered at $\theta = \theta _{\rm pc}$, with a width $\sigma = 0.05\ \theta _{\rm pc}$. The initial energy of primary electrons was equal to $2\times 10^7$ MeV. Note the step-like decline near $\sim$100 GeV.

If, however, an emission from the inner part of the polar cap were to contribute considerably to the outer-rim emission, the step-like shape shown in Fig. 10 would be smoothed out, because of contribution of many spectra with different values of $\varepsilon _{\rm esc}$. In such a case the polar cap origin of the observed radiation could be easily revealed by noting strong differences between the high-energy spectral cutoffs in different ranges of the rotational phase (i.e. phase-resolved spectra would have to be obtained). The first obvious candidate to check for this effect (e.g. with GLAST) seems to be J0218+4232 - the only gamma-ray pulsar among all millisecond pulsars (Kuiper et al. 2000). However, this pulsar appears to be a candidate for an "off-beam" case (see Dyks & Rudak 2002 for details).

The strength of magnetic field $B_{\rm pc}$ practically does not affect the shapes of curves shown in Figs. 7-9. The magnetic field only acts as a scaling factor: $\varepsilon_{\rm esc} \propto B^{-1}$; cf. Eq. (6).