MULTI-WAVELENGTH EMISSION REGION OF γ-RAY EMITTING PULSARS

We start with the Vela pulsar, which has been well studied to test the validity of our simple model. TCS08 considered this source, but they used geometric parameters that are slightly different from ours.

Pulse profiles have been detected in optical to γ-ray bands. The pulse profile in the γ-ray band observed by Fermi (Abdo et al. 2010b) shows a prominent double-peak structure and bridge emission between the two peaks. The first and second peaks are located at the phases ϕ ∼ 0.13 and ϕ ∼ 0.56, respectively, and the separation is Δϕ = 0.43. We show the intensity map for outward emission as a function of the altitude of the emission region and rotation phase in the upper panel of Figure 1(A). The emission altitude producing a peak separation Δϕ = 0.43 is r_ov ∼ 1.05–1.06.

The X-ray data from RXTE (Harding et al. 2002) also show a double-peak structure but the second peak broadens toward early phase. The calculated intensity map is shown for outward and inward emissions in the upper and lower panels of Figure 1(A). The main double peaks are located at the same phases as those in the γ-ray band, so they are interpreted as being formed by outward emission. The broad component before the second peak at ϕ ∼ 0.47 is associated with the caustic formed by the inward emission, as shown in the lower panel. We attempted a fit without the inward emission, but found that the inward emission is needed in order to reproduce the observed X-ray pulse profile. The necessity of inward emission was discussed in TCS08. Thus, the peak positions of γ-ray and X-ray pulses can be explained with the same value of r_ov ∼ 1.05–1.06. The contour map, however, shows a minor peak at ϕ ∼ 0.8 formed by the outward emission. The peak was not observed in the γ-ray band.

We compare our model with the UV data of Romani et al. (2005) and the optical data of Gouiffes (1998). The pulse profiles in both bands are very similar, that is, they have a double-peak structure at the same phases. The peak phases, however, differ from those of the γ-ray and X-ray bands. The first peak of the UV/optical bands shifts to a later phase ϕ ∼ 0.27 and the second peak shifts to an earlier phase ϕ ∼ 0.46, so that the peak separation becomes smaller. It can be seen from Figure 1(A) that such a double-peak structure corresponds to r_ov ∼ 0.65–0.80 for outward emission. The corresponding inward emission cannot be detected since its observable range is r_ov ⩾ 0.9, as shown in the lower panel. The choice of outward emission with r_ov ∼ 0.65–0.80 is also supported by the fact that the second peak at ϕ ∼ 0.46 in the UV/optical ranges is sharper than the first one at ϕ ∼ 0.27 because of their different dependence on r_ov. Thus, we have reproduced the pulse profiles of optical to γ-ray bands by the caustics model without any detailed assumptions about emissivity. From the fitting model, we found three conditions for the emission region. (1) The UV/optical emission region is located at an altitude above the γ-ray and X-ray emission region of r_ov ∼ 1.05–1.06. (2) There is a separation of altitude between the X-ray and optically dominant emission regions. (3) The UV/optical emission range, Δr_ov ∼ 0.15, is broader than that of γ-ray/X-ray emission regions, Δr_ov ∼ 0.02.

The pulsar PSR J0659+1414 has also been observed in the γ-ray to optical bands. We use the γ-ray data from Fermi (Weltevrede et al. 2010), X-ray data from XMM-Newton (De Luca et al. 2005), UV data from Shibanov et al. (2005), and optical data from Kern et al. (2003). The X-ray data are a combination of thermal (blackbody) and non-thermal (power-law) emissions and are consistent with a cooling middle-aged neutron star (e.g., Becker & Trümper 1997). At soft X-rays, the pulse fraction is low and the pulsations are sinusoidal, as is typical for thermal emissions from the surface of a neutron star with non-uniform temperature distribution. At higher energies (>1.5 keV), where the non-thermal component dominates, the pulsed fraction increases and the profile becomes single peaked. We therefore consider the pulse profiles of hard X-rays (>1.5 keV) only.

The pulse profile in the γ-ray band shows a relatively broad single peak, which lags the radio maximum peak by ϕ ∼0.2 in phase. The peak in the non-thermal X-ray pulse is at ϕ ∼ 0.7–0.8, which is different from the phase of the γ-ray peak. This phase difference cannot be ignored, although the peaks in the γ-ray and X-ray data are rather broad, and hence the difference may be diminished somewhat by including the phase error. We interpret these pulse profiles as being emissions at different phases and in different directions: the peak in the γ-ray band is formed by outward emission, whereas that in the X-ray band is formed by inward emission. The intensity maps are given in the upper panel (outward emission) and the lower panel (inward emission) of Figure 1(B), respectively. From this figure, we see that a peak separation Δϕ = 0.55 between the γ-ray and X-ray data corresponds to an emission altitude of r_ov ∼ 1.13–1.14 by shifting the reference phase by δϕ = 0.06 to an earlier phase. The emission altitude cannot be fixed without the X-ray data: a shift of peak phase is allowed, so r_ov is unknown. This ambiguity is removed by considering multi-wavelength light curves. From the intensity map, we expect another very sharp caustic at ϕ ∼ 0.65, but there is no counterpart in the γ-ray observations (top panel of Figure 4 in Weltevrede et al. 2010). We discuss this missing peak in Section 4.2. In the lower right panel of Figure 4 in Weltevrede et al. (2010), the light curve for inward emission is given. The X-ray data may be a combination of inward and outward emissions, but the X-ray profile observed by De Luca et al. (2005) is similar to that of inward emission only. This means phenomenologically that outward emission of X-rays is weak in this source.

The pulse profiles in the UV (Shibanov et al. 2005) and optical bands (Kern et al. 2003) are almost the same shape and have a clear double-peak structure unlike the single peak in the γ-ray and X-ray bands. The first peak at ϕ ∼ 0.02 is later than the γ-ray peak and the second peak at ϕ ∼ 0.10 is later phase than the X-ray peak. From the upper panel of Figure 1(B), we see that the observed first peak can be reproduced by outward emission at an altitude r_ov < 1.10. Here we have a weak condition because the peak position depends only weakly on r_ov. For the second peak, the inward emission forms caustics for 0.90 < r_ov < 1.04. Thus, the altitude of the emission region in the UV/optical bands is identified as r_ov ∼ 0.90–1.04, where the lower limit is set by a coarse bin of the phase in the observational data.

Kern et al. (2003) have already investigated the multi-wavelength light curve of this pulsar using a similar method to ours, but could not explain the profile using geometrical parameters which are consistent with radio polarization data (Everett & Weisberg 2001). The reason for this is that the lower boundary of the emission region was chosen as the last-open field lines in the vacuum dipole field, that is, r_ov  =  1.0. In our analysis, by allowing r_ov ⩾ 1.0, the phase of peaks can be successfully fitted by using observed geometrical parameters. This suggests that the actual lower boundary of the gap is slightly different from the last-open field lines in a vacuum dipole.

The γ-ray pulse profile observed by Fermi (Abdo et al. 2009a) shows a double-peak structure. The first peak is offset from the radio peak by ϕ ∼ 0.08, and the second is at ϕ ∼ 0.57. The separation is Δϕ = 0.49. The X-ray data from RXTE (Livingstone et al. 2009) are consistent with the results from XMM-Newton (Kuiper et al. 2010). The spectral shape can be fitted by a power law such that most of the emission is non-thermal and the thermal component is constrained by the upper limit (Kuiper et al. 2010). The observed X-ray pulse profile shows two peaks aligned in phase over a wide energy range of ∼0.5–270 keV, and is also very similar to that of the γ-ray band.

We show the intensity maps for outward and inward emissions in the upper and lower panels of Figure 1(C), respectively. As seen in the upper panel, the emission altitude for the double peak with Δϕ = 0.49 is r_ov ∼ 0.97–0.98. A shift of the reference phase δϕ is not necessary in this source. As argued in TCS08, outward emission dominates in the light curve for a young pulsar with a strong non-thermal X-ray component, like the Crab pulsar. PSR J0205+6449 is the youngest pulsar in our sample (characteristic age τ_c ∼ 5 × 10³ yr) and shows rather strong non-thermal radiation in the X-ray band. Thus, it is likely that only outward emission contributes to the observed X-ray light curve in this pulsar.

The light curve for γ-rays observed by Fermi (Abdo et al. 2009b) shows an asymmetric single peak at ϕ ∼ 0.49. The tail extends down to ϕ ∼ 0.2. The peak position depends slightly on the energy range above 100 MeV, but the amount of shift is only ∼0.04. The X-ray pulse profile observed by XMM-Newton (Abdo et al. 2009b) shows a double-peak structure. No peak is seen at the γ-ray peak phase. The separation between the first X-ray peak and the γ-ray peak is Δϕ ∼ 0.32, and the separation between the second X-ray peak and the γ-ray peak is Δϕ ∼ 0.14.

The intensity map for outward emission is shown in the upper panel of Figure 1(D). We consider the formation of the γ-ray peak and two X-ray peaks as being due to outward emission only. Such a solution is possible by choosing an emission altitude of r_ov ∼ 1.01–1.02 with a small phase shift δϕ = 0.03. The intensity map for inward emission shows a sudden decrease in the number counts for r_ov < 1.06. The peak becomes broad and hence the contribution of the inward emission is not very important. We show the light curve of inward and outward emissions for r_ov ∼ 1.01–1.02 in Figure 1(D). The outward emission curve is very similar to the observations in the γ-ray and X-ray bands.

In this pulsar, we need to use all the light curves simultaneously in order to determine the range of r_ov. Since emissions with smaller values of r_ov are not seen, if the γ-ray and optical emission regions are separated by Δr_ov > 0.10, similar to the Vela pulsar and PSR J0659+1414, we predict that an optical pulse profile cannot be observed or will be only very weakly detected. This is consistent with the fact that there have been no reports of the detection of a pulse in the lower energy band for this pulsar.

The γ-ray light curve from Fermi (Weltevrede et al. 2010) shows a broad peak at ϕ ∼ 0.2–0.5. This peak may consist of two components, but it is not clear in the current photon statistics. The X-ray pulse profile from ASCA is detected weakly at a marginal level, and shows a broad peak at ϕ ∼ 0.6–0.7 which is different from the γ-ray peak (Roberts et al. 2001; Becker 2009). Recently, Marelli et al. (2011) listed this object as a non-thermal dominated source in X-ray. The pulsed X-ray profile is likely to originate from the non-thermal component.

Since the light curve of this pulsar and its geometrical parameters are similar to those of PSR J0659+1414, we adopt the same interpretation. That is, the γ-ray peak is formed by outward emission, whereas the X-ray peak is formed by inward emission. From the intensity map in Figure 1(E), an emission altitude of r_ov ∼ 1.10–1.11 corresponds to one broad peak at ϕ ∼ 0.2–0.5 by outward emission and another at ϕ ∼ 0.6–0.7 by inward emission. Here a small shift δϕ = 0.10 toward earlier phase is used. Since there are similarities in both the γ-ray and X-ray light curves and the geometrical parameters between this pulsar and PSR J0659+1414, we expect a similar double-peak pulse profile in the optical band, if it is detected.

Observations in the γ-ray band have been obtained by Fermi (Abdo et al. 2009c) and AGILE (Halpern et al. 2008). The observed light curve shows a sharp double-peak structure. The first peak is offset from the radio peak by ϕ ∼ 0.16 and the two peaks are separated by Δϕ ∼ 0.47. The X-ray light curve in Abdo et al. (2009c) shows a relatively sharp peak associated with the first peak in the γ-ray light curve albeit with weak photon statistics. In this paper, we assume that at least the first peak is non-thermal in origin. The possible contribution of non-thermal X-ray emission is also discussed in Hessels et al. (2004) and Van Etten et al. (2008). We expect that this assumption will be tested by phase-resolved spectra from future observations.

As seen in the upper panel of Figure 1(F), the emission altitude is r_ov ∼ 0.97–0.98 for which there is a γ-ray double-peak profile with separation Δϕ = 0.47 and a relative shift δϕ = 0.06 toward later phase. The peak of non-thermal emission at ϕ = 0.15–0.20 in the X-ray light curve is found to be formed by outward emission only. The relatively weak second peak in the X-ray band is consistent with the case of PSR J0205+6449. Thus, the three model parameters for this pulsar and PSR J0205+6449 are very similar, as shown in Table 1.

The γ-ray light curve from Fermi (Abdo et al. 2010c) shows a broad peak at ϕ ∼ 0.25–0.65. This probably consists of two components, but it is not clear. De Luca et al. (2005) extract only a power-law component of the X-ray light curve and their Figure 13 shows two peaks at ϕ ∼ 0.2–0.3 and ϕ ∼ 0.9–1.0, although the data are very coarse. We regard the light curve as being produced by a non-thermal X-ray component.

The observed light curve may be regarded either as a broad peak consisting of weak peaks and a relatively bright bridge emission or as a result of the range of the emission region widening toward lower altitudes. In the latter interpretation the fitted r_ov is only a lower limit. We thus focus on the width of the γ-ray peak. From the upper panel of Figure 1(G), the emission altitude is r_ov ∼ 0.93–0.94. Even if we assume a double-peak structure with the first peak at ϕ ∼ 0.31 and the second peak at ϕ ∼ 0.59 following Abdo et al. (2010c), we have r_ov ∼0.90–0.91, which is very similar to the value obtained above. Thus we have r_ov ∼0.90–0.95 in either case. The phase shift is δϕ = 0.10 in this pulsar. The first peak in the X-ray light curve is formed by outward emission, but the second one cannot be produced for the same altitude. This may be a drawback to our model, but the present X-ray data are coarse and a much more precise non-thermal X-ray light curve is needed.

In the previous section, we have shown that the peak phases of seven pulsars emitting γ-rays and X-rays can be successfully fitted using the TCS08 outer gap model, in which both γ-rays and X-rays originate from the same magnetic field line characterized by an altitude r_ov. A parameter r_ov > 1 is needed in the light curve fitting for some sources. Moreover, the inclusion of inward emission for X-rays causes a variety of pulse profiles in both bands. The parameter r_ov could not be determined solely using γ-ray data for a single γ-ray peak pulsar. But, by considering the X-ray light curve, the parameter is uniquely determined for PSRs J0659+1414, J2229+6114, and J1420−6048.

It is worthwhile to explore the general dependence of the altitude r_ov on other characteristics, if any, although there may not be enough data for a proper statistical analysis. In Figure 3, r_ov is plotted as a function of inclination angle α, spin-down luminosity L_SD, characteristic age τ_c, and surface dipole magnetic field B_s. We found that there is a significant correlation between r_ov and the inclination angle α only; the relations of r_ov with the other parameters are very weak. This correlation suggests that the deviation from a vacuum rotating dipole field is large for small inclination angle. It is very interesting to compare this result with that in a force-free magnetosphere. Bai & Spitkovsky (2010b) proposed that the separatrix layer at an altitude of 0.90–0.95 times the height of the last-open field line is relevant to emissions in a three-dimensional inclined force-free magnetosphere. This altitude, which is not exactly symmetric with respect to the magnetic azimuthal angle ϕ_m, but which can be approximated by the value at ϕ_m = 0, is plotted in Figure 3 as purple downward and blue upward triangles. Two linear fitting lines are also shown. The altitude r_ov decreases with the inclination angle α in both our model and in the separatrix layer model of a force-free magnetosphere. However, the emission region in the separatrix layer model extends even outside the light cylinder, whereas ours is well localized around null points. Accounting for this difference may be important for further improvement of the model of the emission region based on a force-free magnetosphere.

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The thickness of the gap region, w, is not known, but it is sometimes assumed to decrease with the spin-down luminosity L_SD (Watters et al. 2009; Romani & Watters 2010). We have w = 1 − r_ov if the lower boundary of the gap is fixed as the last-open field line in the vacuum dipole field. This assumption is tested in the lower left panel of Figure 3 in which the relation (1 − r_ov) ≈ (L_SD/10³³erg s⁻¹)^−1/2 is plotted as a light green curve. (The curve is not fitted to the data points.) This suggests that the assumption of maximum altitude, r_ov = 1.0, is not a good one. This discovery affects the expected number of γ-ray pulsars in the observation. For geometrical reasons, the pulsed emission by caustics is limited to a certain range between inclination and viewing angles.

Romani & Watters (2010) showed the range of observable pulsars with r_ov =0.95, 0.90, and 0.70 for the outer gap model in their Figure 16. We recalculate it and show the result in Figure 4. The observable range of viewing angle ξ is below the curves. Our finding in Figure 3 is that r_ov is a function of the inclination angle, which is similar to that of the separatrix layer model. We also show the observable range according to the empirical relation obtained in Figure 3 as a black solid line, for which the altitude is chosen as 0.925 times the height of the last-open field line in a force-free magnetosphere. The figure shows that sources with low inclination and viewing angles become observable. For example, a pulsar with inclination angle α = 30° can be detected for ξ > 60° for r_ov = 0.95, but not for ξ > 30°. Thus the expected number approximately doubles for sources with low inclination and viewing angles.

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The caustic model considered in Section 3 provides peak positions consistent with observation, but there are also some additional, unseen peaks. These are interpreted as being prohibited by some mechanism. In this section, we consider an improvement to our model that takes into account a very simple distribution for the emissivity. Detectable γ-rays are radiated with large multiplicity by the pair plasma in the gap region. Therefore, the mean free path of a γ-ray photon should be less than the light cylinder radius (Takata & Chang 2007). The pair creation mean free path is given by λ(r) ∼ 5.6P^13/21(B_s/10¹²G)^−2/7r (Tang et al. 2008) for an assumed limiting distance to the null point r_{n, lim}. The position of the null point of inclined pulsars, where the accelerating electric field arises, depends significantly on magnetic azimuthal angle, so the intensity of γ-ray emission also depends on the magnetic azimuthal angle. Therefore, active field lines should be limited in the azimuthal direction. By taking into account the azimuthal extensions with λ(r_{n, lim}) ≲ 0.2–0.7R_LC listed in Table 1, the fits of the resultant light curves, which are shown in Figure 5, become better. In the same figure, we also show the radial distance to the emission points of the observed photons against the rotation phase. Note that, for the Vela pulsar, the minor third peak at ϕ ∼ 0.8 still remains even after the inclusion of the azimuthal extension limit. The corresponding radial distance of emission points is relatively large, so that the photon energy is expected to be soft. The third peak is not observed in the GeV band, but may appear in a much lower energy band. At least, the minor third peak of the X-ray light curve appears to be associated with the same caustic.

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We have also tried to improve the X-ray light curve with some other simple assumptions, but have not had good results. The reason for this is that there are many ways for X-ray emitting particles to be created: via thermal, magnetospheric emissions and magnetic pair creation. Therefore, the three-dimensional effect of the propagation of γ-ray photons and soft X-ray photons is very important. Without it we cannot successfully explain the light curve.

We explored the UV/optical emission region for Vela and PSR J0659+1414. The results are qualitatively similar: the altitude range of UV/optical emission, Δr_ov ∼ 0.15, is broader than that for γ-rays and X-rays, Δr_ov ∼ 0.02, and both emission regions are not continuous and connected, but are widely separated. The separation may come from two competing mechanisms: a decrease of emissivity and an increase of synchrotron intensity in the UV/optical bands with altitude.

As discussed in TCS08, outward emission is generally dominant in UV/optical emission, as shown in their Figure 4. The explanation is the following. The number of pairs created is the main cause of the difference between inward and outward emissions in the UV/optical bands, since the collision angles with magnetospheric X-rays are not different for outgoing and ingoing γ-ray photons in the acceleration region. More outgoing γ-rays are emitted, and hence more outgoing secondary pairs are produced. Thus, the synchrotron emission for outgoing secondary pairs produced by magnetic X-rays is brighter than that for the ingoing secondary pairs. The observed flux depends strongly on the geometrical configuration. The outward emission in the UV/optical bands may not point toward us even though the intrinsic emission is strong. Our results show that the peaks in the Vela pulsar can be explained by outward emission alone, while those in PSR J0659+1414 require both inward and outward emissions. Our result suggests that outward emission is significantly suppressed in PSR J0659+1414, to the level of the intrinsically weak inward emissions. The stronger component is hidden, because the observable altitude range is narrow, as shown in Figure 1(B). This may explain the fact that the UV/optical flux is smaller than the value extrapolated from non-thermal X-rays, as seen in Figure 4 of Mignani et al. (2010), whereas the flux coincides with the extrapolation in the Vela pulsar. This interpretation may be tested in PSR J1057−5226. We also suggest that this difference is the reason why PSR J0659+1414, which has similar geometrical parameters to the Vela pulsar, has an observable optical spectrum (Mignani et al. 2010), and the flux is slightly smaller than the extrapolation from non-thermal X-ray emission. The pulse profile has not yet been determined, but the peaks should appear at a phase 0.3<ϕ < 0.6 and be due to outward emission.