High-energy Photon Opacity in the Twisted Magnetospheres of Magnetars

The optical depth for a photon emitted at any locale r _E in the magnetosphere (in the observer's coordinate frame; OF) in direction $\,{\hat{{\boldsymbol{k}}}}_{{\rm{E}}}\,$ is given by

$$\begin{eqnarray}&&\tau (l)={\int }_{0}^{l}{ \mathcal R }\left({\varepsilon }_{\infty },{\hat{{\boldsymbol{k}}}}_{{\rm{E}}},{{\boldsymbol{r}}}_{{\rm{E}}};{{\rm{\Theta }}}_{\mathrm{kB}},{\boldsymbol{B}}\right)\,{ds},\end{eqnarray} \tag{ 1 }$$

where $\,{ \mathcal R }\,$ is the attenuation coefficient (in units of cm⁻¹) and l is the cumulative proper length of the photon trajectory. Here εいぷしろん_∞ m_e c² is the observed photon energy at infinity. In considering general relativistic photon propagation, the optical depth is integrated over the geodesic path length s, sampling different local inertial frames (LIFs) with altitude. Thus, B ≡ B _GR is the LIF frame magnetic field, specified in Equation (15) below, and Θしーた_kB is the angle between the field and the photon momentum in the LIF, which satisfies Equation (3). The attenuation length L is defined (Baring & Harding 2001; Story & Baring 2014; Hu et al. 2019) to be the proper distance over which optical depth τたう(L) = 1. The principal quantity of interest in our opacity considerations is the escape energy εいぷしろん_esc, which is defined as the OF photon energy at which the attenuation length L becomes infinite:

$$\begin{eqnarray}&&1={\int }_{0}^{\infty }{ \mathcal R }\left({\varepsilon }_{\mathrm{esc}},{\hat{{\boldsymbol{k}}}}_{{\rm{E}}},{{\boldsymbol{r}}}_{{\rm{E}}};{{\rm{\Theta }}}_{\mathrm{kB}},{\boldsymbol{B}}\right)\,{ds}.\end{eqnarray} \tag{ 2 }$$

Since the attenuation coefficients increase rapidly with photon energy for both photon splitting and pair creation, the optical depth satisfies τたう < 1 for photons with energies εいぷしろん smaller than the escape energy εいぷしろん_esc . Thus, the escape energy εいぷしろん_esc represents the photon energy below which the magnetosphere is transparent. It is a strong function of the emission locale (at which s = 0) and the magnetic field morphology, features that are central to the considerations of the paper.

The geometry of the attenuation process in the twisted magnetosphere of a rotating neutron star is depicted in Figure 1, with the Fermi Gamma-Ray Space Telescope image being from its mission web page. ⁷ The inclination angle between the stellar rotation axis and the magnetic axis is denoted as αあるふぁ , and the colatitude of the emission locale relative to the rotation axis is denoted as θしーた_E . We consider photon propagation in the curved spacetime described by the Schwarzschild metric, fixing the stellar radius at R_NS = 10⁶ cm and stellar mass at M_NS = 1.44 M_⊙ throughout. For the schematic in Figure 1, the photon path starts with emission parallel to B , the focal case of Section 4.1. The optical depth in Equation (1) is then integrated numerically over the geodesic path length s, and at each point along the path, $\,{ \mathcal R }\,$ is specified in the LIF. The trajectories of photons are integrated numerically in our calculation. These protocols are detailed in Section 5.1 of Hu et al. (2019). The opacity is very sensitive to the angle between the external magnetic field and the momentum vector of the propagating photon, which is given by

$$\begin{eqnarray}&&\cos {{\rm{\Theta }}}_{\mathrm{kB}}=\displaystyle \frac{\hat{{\boldsymbol{k}}}\cdot {\boldsymbol{B}}}{| {\boldsymbol{B}}| }\quad \mathrm{or}\quad \sin {{\rm{\Theta }}}_{\mathrm{kB}}=\displaystyle \frac{| \hat{{\boldsymbol{k}}}\times {\boldsymbol{B}}| }{| {\boldsymbol{B}}| }.\end{eqnarray} \tag{ 3 }$$

Although our formalism and associated computations are presented in curved spacetime, flat spacetime results can be easily obtained by specializing to M_NS → 0 and are used as a check of the results (see below).

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In all results in this paper, the modulation of viewing geometry with stellar rotation phase will not be considered, as it was in Section 4.3 of Hu et al. (2019), where the pertinent behavior was detailed sufficiently. The primary interest here is in how twisted field morphology influences escape energies and the volume of magnetospheric opacity. Accordingly, it suffices to consider aligned rotators with αあるふぁ = 0 in the ensuing exposition, so that θしーた_E represents the magnetic colatitude of emission at position vector r _E .

The attenuation of photons is calculated for two linear polarization modes, namely ∥ (ordinary) mode and ⊥ (extraordinary) mode. Here ⊥ and ∥ refer to the states where the photon electric vector is locally perpendicular and parallel to the plane containing the photon momentum vector k and the external field B , respectively. These two linear modes approximately represent the polarization eigenmodes of soft X-rays when vacuum polarization dominates the dielectric tensor of the magnetosphere of a magnetar. As the photon propagates in the magnetosphere, its electric field vector evolves adiabatically following the change of the magnetic field direction, with the birefringent QED vacuum ensuring that the polarization state (i.e., ⊥ or ∥ ) remains unchanged during propagation (see Heyl & Shaviv 2000). This adiabatic polarization evolution persists out to the polarization-limiting radius, which is mostly beyond the escape altitudes for X-rays and gamma rays that are highlighted in the various figures below. This feature simplifies the polarization transport considerably.

The opacity is computed for the two key processes that can be prolific in strong-field QED, namely photon splitting and pair creation. Here we provide a summary of the essentials of the rates for these processes, and detailed discussions of the physics for them can be found in Harding et al. (1997), Baring & Harding (2001), and Harding & Lai (2006).

Magnetic photon splitting is a third-order QED process in which a single photon splits into two lower-energy photons. The CP invariance of QED permits only three splitting channels, namely ⊥ → ∥∥, ∥ → ⊥∥, and ⊥ → ⊥⊥ (see Adler 1971). In the domain of weak vacuum dispersion, ⊥ → ∥∥ is the only CP-permitted channel that satisfies energy-momentum conservation (Adler 1971). This kinematic selection rule may not hold when plasma dispersion competes with the vacuum contribution in lower fields at higher altitudes where the cyclotron resonance can become influential. Thus, in the following presentation we will consider all the CP-permitted splitting channels and focus especially on polarization-averaged results, which differ only modestly from results where only ⊥ → ∥∥ is activated.

In the low-energy limit well below the pair creation threshold, applying to hard X-rays that are of principal interest here, the reaction rates for all the CP-permitted modes can be expressed as (Baring & Harding 2001)

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal R }}_{\perp \to \parallel \parallel }^{\mathrm{sp}} & = & \frac{{\alpha }_{f}^{3}}{60{\pi }^{2}{\unicode{x0019B}}_{{\rm{C}}}}\,{\varepsilon }^{5}\,{B}^{6}\,{{ \mathcal M }}_{1}^{2}\,{\sin }^{6}{{\rm{\Theta }}}_{\mathrm{kB}}=\frac{1}{2}\,{{ \mathcal R }}_{\parallel \to \perp \parallel }^{\mathrm{sp}},\\ {{ \mathcal R }}_{\perp \to \perp \perp }^{\mathrm{sp}} & = & \frac{{\alpha }_{f}^{3}}{60{\pi }^{2}{\unicode{x0019B}}_{{\rm{C}}}}\,{\varepsilon }^{5}\,{B}^{6}\,{{ \mathcal M }}_{2}^{2}\,{\sin }^{6}{{\rm{\Theta }}}_{\mathrm{kB}}.\end{array}\end{eqnarray} \tag{ 4 }$$

Here αあるふぁ _f is the fine-structure constant, ƛ_C is the reduced Compton wavelength of electron, and the field strength B is expressed in the units of the quantum critical field B_cr . The LIF photon energy εいぷしろん is scaled by the rest-mass energy of electrons. For unpolarized photons, the attenuation coefficient can be obtained averaging Equation (4):

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{ave}}^{\mathrm{sp}}=\frac{{\alpha }_{f}^{3}}{120{\pi }^{2}{\unicode{x0019B}}_{{\rm{C}}}}\,{\varepsilon }^{5}\,{B}^{6}\left(3{{ \mathcal M }}_{1}^{2}+{{ \mathcal M }}_{2}^{2}\right)\,{\sin }^{6}{{\rm{\Theta }}}_{\mathrm{kB}},\end{eqnarray} \tag{ 5 }$$

a result detailed in Hu et al. (2019). The reaction rate coefficients $\,{{ \mathcal M }}_{1},{{ \mathcal M }}_{2}\,$ derived from squares of the matrix elements are purely functions of B in this low-energy εいぷしろん ≪ 1 domain, possessing the forms

$$\begin{eqnarray}&&{{ \mathcal M }}_{\sigma }=\displaystyle \frac{1}{{B}^{4}}{\int }_{0}^{\infty }\displaystyle \frac{{ds}}{s}{e}^{-s/B}\,{{\rm{\Lambda }}}_{\sigma }(s),\end{eqnarray} \tag{ 6 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Lambda }}}_{1}(s) & = & \left(-\displaystyle \frac{3}{4s}+\displaystyle \frac{s}{6}\right)\displaystyle \frac{\cosh s}{\sinh s}+\displaystyle \frac{3+2{s}^{2}}{12{\sinh }^{2}s}+\displaystyle \frac{s\cosh s}{2{\sinh }^{3}s},\\ {{\rm{\Lambda }}}_{2}(s) & = & \displaystyle \frac{3}{4s}\displaystyle \frac{\cosh s}{\sinh s}+\displaystyle \frac{3-4{s}^{2}}{4{\sinh }^{2}s}-\displaystyle \frac{3{s}^{2}}{2{\sinh }^{4}s}.\end{array}\end{eqnarray} \tag{ 7 }$$

For low fields B ≪ 1 , $\,{{ \mathcal M }}_{1}\approx 26/315\,$ and $\,{{ \mathcal M }}_{2}\approx 48/315\,$ are independent of B , but at highly supercritical fields B ≫ 1 they possess $\,{{ \mathcal M }}_{1}\propto {B}^{-3}\,$ and $\,{{ \mathcal M }}_{2}\propto {B}^{-4}\,$ dependences.

One-photon pair creation is a first-order QED process that is allowed in the presence of a strong external field because momentum conservation orthogonal to B is then not operable. This conversion process is extremely efficient when B ≳ 0.1 and can only proceed when the photon energy is above the $\,{\varepsilon }_{\perp }\equiv \varepsilon \sin {{\rm{\Theta }}}_{\mathrm{kB}}=2\,$ threshold. The produced electrons occupy excited Landau levels in the external magnetic field. The attenuation coefficient exhibits a sawtooth structure since it diverges at the threshold of each Landau level accessed (see Daugherty & Harding 1983; Baier & Katkov 2007, for detailed calculations). The attenuation coefficient for pair creation can be expressed in the form of

$$\begin{eqnarray}&&{{ \mathcal R }}_{\perp ,\parallel }^{\mathrm{pp}}=\frac{{\alpha }_{f}}{{\unicode{x0019B}}_{{\rm{C}}}}\,B\sin {{\rm{\Theta }}}_{\mathrm{kB}}{{ \mathcal F }}_{\perp ,\parallel }\left({\varepsilon }_{\perp },B\right)\end{eqnarray} \tag{ 8 }$$

for the ⊥, ∥ photon linear polarizations. Since the forms we employ for the $\,{{ \mathcal F }}_{\perp ,\parallel }\,$ functions have been presented in several other papers and are somewhat lengthy, they are listed in Appendix A.

The first case to be considered is when the photons are emitted parallel to the magnetic field at the point of emission. For magnetars, this closely matches the expectations of the popular resonant inverse Compton scattering (RICS) models for the production of their persistent hard X-ray emission above 10 keV (Baring & Harding 2007; Fernández & Thompson 2007; Beloborodov 2013; Wadiasingh et al. 2018), discussed below. It is also relevant to curvature radiation emission components from high-field pulsars such as PSR B1509–58 (Harding et al. 1997), if they are generated by primary electrons accelerated in polar cap or slot gap potentials with E · B ≠ 0 in the inner magnetosphere (Daugherty & Harding 1996; Muslimov & Harding 2004).

The opacity for a polarized photon emitted at a specific locale can be obtained using Equation (1) with the attenuation coefficients of photon splitting and pair creation specified by Equations (5) and (8), respectively. Then, the escape energy εいぷしろん_esc is determined by adjusting the photon energy so that the cumulative optical depth in propagating to infinity equals unity, effectively a root-solving algorithm for Equation (2). Thus, εいぷしろん_esc depends on the initial conditions, namely the emission position and direction, and subsequently also on the resultant path of the photon. Figure 3 illustrates the escape energies εいぷしろん_esc of photon splitting (solid) and pair creation (dashed) as functions of the emission colatitudes θしーた_E for photons originating on specific field loops, with their initial photon momentum parallel to the local field, i.e., k ∥ B . Escape energies for combined photon splitting and pair creation with surface polar fields B_p = 100 and B_p = 10 are illustrated in Figures 4 and 5, respectively. The escape energy loci in both of these figures were determined for field loops with one of the footpoint colatitudes located at θしーた _f = 15° and for four different twist parameter p values. On each curve in Figure 3, the black circles and diamonds mark the colatitudes where the emission locales have radii of 10R_NS and 100R_NS, respectively. Decreasing the p value enlarges the field loop so that its maximum altitude is larger (see Table 1), corresponding to much larger radii of field curvature on average.

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Table 1.  $\,{\theta }_{f}\mbox{--}{r}_{\max }\,$ Relations for Different p Values

	$\,{r}_{\max }\,({R}_{\mathrm{NS}})\,$
p	θしーた f = 15°	θしーた f = 30°	θしーた f = 45°	θしーた f = 60°
1.00	10.37	3.01	1.67	1.22
0.75	29.92	5.29	2.22	1.38
0.50	266.5	18.12	4.30	1.84
0.25	2.13 × 105	909.6	42.48	5.69

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The field loop escape energies in Figure 3 are maximized at colatitudes somewhat smaller than 90° and resemble the bell-shaped behavior presented in Figure 10 of Hu et al. (2019; p = 1 case only). This is expected because of the high altitudes of emission at quasi-equatorial colatitudes, where the magnetic fields are relatively weak. Pair creation dominates the attenuation and determines the escape energy at colatitudes remote from the footpoints, and these domains are marked as heavyweight portions of the curves in Figure 4 (top left); the lower average fields along the photon trajectories tend to favor the dominance of pair conversion over photon splitting once the energy threshold is exceeded (e.g., Baring & Harding 2001). While the pair conversion escape energies (for an observer at infinity) in Figure 3 generally exceed the threshold of 2m_e c² for pair creation, there are noticeable portions near the highest emission colatitudes where values slightly lower than 2m_e c² are apparent. These correspond to inward emission cases where the LIF frame photon energy actually exceeds the pair threshold for the inner portion (near periastron) of the trajectory to infinity. For both processes, the escape energy curves are asymmetric about the equator ( θしーた_E = 90°), since photons emitted from upper and lower hemispheres possess different trajectories. Photons emitted outward travel to regions of lower field magnitudes and larger radii of curvature. In contrast, those emitted inward sample fields that are stronger and field lines that are more highly curved (see the sample trajectories in Figure 9), conditions conducive to greater opacity, thereby lowering εいぷしろん_esc.

The key feature of the impact of twists to the magnetospheric geometry is that the escape energy for loop emission increases as p decreases. This is caused by two effects. At small colatitudes, decreasing p enhances the radial $\,{B}_{\hat{r}}\,$ component. Thus, the initial photon momentum k is typically more radial than it is in a dipole field, and Θしーた_kB increases more slowly for lower p owing to the straighter field lines, unless the photon is very far from the stellar surface. For moderate colatitudes, decreasing p raises the emission locale to a higher altitude (see the black circles and diamonds in Figure 4), which also reduces the opacity, as the typical value of ∣ B ∣ along the trajectory is lower. Accordingly, the higher the axisymmetric MHD twist, the more transparent the magnetosphere becomes to splitting and pair creation for photons emitted parallel to B along field lines. As p declines, the opacities should rise with the increase of the field magnitude ∣ B ∣ that is evident in Figure 2. However, this influence is actually dominated by the associated increase in the radii of field-line curvature for higher twists, which reduces the magnetospheric opacity as the field morphology progresses toward that of a split monopole.

Escape energies for photons emitted from the stellar surface are also depicted in Figure 4 as dashed curves. These curves start at θしーた_E = 0° and decrease with the increase of θしーた_E because the field-line curvature radii are smaller at nonpolar colatitudes. All the curves are truncated at θしーた_E = 90°, beyond which the emitted photons propagate inward and therefore never enter the magnetosphere. The escape energy curves for the loop emission touch the surface curves at the footpoint colatitudes θしーた _f , where the field loops are anchored at the surface. The escape energies for surface emitted photons also increase with the decrease of p , observing that the initial photon momentum k is more radially directed when p is small.

The resonant upscattering of the surface thermal emission by relativistic e⁺/e⁻ is very efficient in the magnetospheres of magnetars, because the resonance at the cyclotron frequency ( εいぷしろん = B ) increases the cross section by around 2–3 orders of magnitude above the classical Thomson value when B ≲ 1 (e.g., Gonthier et al. 2000). The velocity of the ultrarelativistic e⁺/e⁻ (Baring & Harding 2007; Wadiasingh et al. 2018) is essentially parallel to the magnetic field direction owing to rapid cyclotron cooling perpendicular to the field, modulo small drift velocity components due to the slow magnetar rotation. Thus, the momentum of the upscattered photon is nearly parallel to the magnetic field as a consequence of Doppler beaming. This also applies to other emission mechanisms, including curvature radiation and synchrotron emission for γがんま-ray pulsars. The scattered e⁺/e⁻ are assumed to stay in the ground Landau states, which is relevant to situations where the scattering samples energy near or below the fundamental cyclotron resonance. This is generally valid owing to rapid cooling of e⁺/e⁻ near the cyclotron resonance, which can normally prevent e⁺/e⁻ from encountering higher cyclotron harmonics; see Baring et al. (2011) for a discussion.

For this case of scatterings being dominated by those sampling the cyclotron frequency, the maximum scattered photon energy $\,{\varepsilon }_{\max }\,$ is realized for a head-on collision with a photon scattering angle of 180° in the electron rest frame, which leads to (Baring et al. 2011)

$$\begin{eqnarray}&&{\varepsilon }_{\max }={\gamma }_{e}(1+{\beta }_{e})\left(\displaystyle \frac{B}{1+2B}\right).\end{eqnarray} \tag{ 17 }$$

Here γがんま_e is the Lorentz factor of the e⁺/e⁻ , βべーた_e is the ratio of lepton speed to c , and B is the field strength at the scattering locale. This form was employed in Hu et al. (2019) in Figure 8 for the dipole case. For fixed lepton velocity, the maximum scattered photon energy $\,{\varepsilon }_{\max }\,$ solely depends on the field strength and obviously saturates at γがんま_e (1 + βべーた_e )/2 when B ≫ 1 .

The gravitational-redshifted maximum scattered photon energy $\,{\varepsilon }_{\max }/(\sqrt{1-{\rm{\Psi }}})\approx 1.32{\varepsilon }_{\max }\,$ (for Ψぷさい = 0.425 ) along the p = 1 field loop is displayed as a sparsely dashed green curve in the top left panel of Figure 4. This is specifically for a fixed γがんま_e = 10 , a value that naturally emerges as a result of rampant RICS cooling (Baring et al. 2011; Wadiasingh et al. 2018) in magnetar-strength fields. The $\,{\varepsilon }_{\max }\,$ trace is symmetric about the magnetic equator ( θしーた_E = 90° ), and it intersects the escape energy curve at θしーた_E ≈ 50° and 100° , which are labeled by green triangles. At low/high colatitudes outside these two intersections, the maximum photon energy $\,{\varepsilon }_{\max }\,$ is larger than the escape energy εいぷしろん_esc; therefore, photon splitting will attenuate the highest energies of hard X-ray photons produced by RICS. The $\,{\varepsilon }_{\max }\,$ curves for field loops with smaller p values share similar behavior and so are not displayed in the figures. The $\,{\varepsilon }_{\max }\,$ values decline along the trace as the colatitude moves away from the poles toward the equator because the emission locales are then farther from the surface and the field strength is weaker. Intersections of loci for $\,{\varepsilon }_{\max }\,$ and εいぷしろん_esc for loops with different p values are labeled in the figures with triangles (γがんま_e = 10) and squares (γがんま_e = 100) in the three panels that depict escape energies. The separation of the two intersection points increases with a decrease of p , so that when p is small, photon splitting is more likely to attenuate photons at energies below $\,{\varepsilon }_{\max }\,$ near the field-line footpoints.

The bottom left panel of Figure 4 illustrates the ratio of the ⊥ mode (⊥ → ∥ + ∥ , ⊥ → e⁺ e⁻) escape energy to the polarization-averaged escape energy. The ratios increase with emission colatitude θしーた_E at small θしーた_E and sharply drop to around unity when pair creation dominates the opacities (the heavyweight portion demarcates pair conversion dominance for εいぷしろん_esc,⊥) at quasi-equatorial colatitudes. For the quasi-polar domains where photon splitting determines the escape energy, the ratio can be intuitively estimated as $\,{\varepsilon }_{\mathrm{esc},\perp }/{\varepsilon }_{\mathrm{esc},\mathrm{ave}}\,\approx {\left[2{{ \mathcal M }}_{1}^{2}/(3{{ \mathcal M }}_{1}^{2}+{{ \mathcal M }}_{2}^{2})\right]}^{-1/5}$. This approaches the value (338/1083)^−1/5 ≈ 1.26 in the subcritical field domain, which is in agreement with the ratio cusps apparent in Figure 4. For pair creation considerations, the escape energy ratio can be estimated assuming $\,3\exp [-8/(3B{\varepsilon }_{\mathrm{esc},\mathrm{ave}})]/2\approx \exp [-8/(3B{\varepsilon }_{\mathrm{esc},\perp })]\,$ in the subcritical field domain. This estimate is obtained using the asymptotic expressions for the polarization-dependent and polarization-averaged pair creation rates in Erber (1966), which can also be deduced by taking the εいぷしろん_⊥ ≫ 2 limit of the expressions in Equation (A3) in Appendix A. This relation yields $\,{\varepsilon }_{\mathrm{esc},\perp }/{\varepsilon }_{\mathrm{esc},\mathrm{ave}}\approx 1+3B{\varepsilon }_{\mathrm{esc}}\mathrm{ln}(3/2)/8\approx 1+0.152B{\varepsilon }_{\mathrm{esc}}\,$, which is very close to unity for high-altitude emission near the equator. Since these ratios are not vastly different from unity, it is sufficient to employ just polarization-averaged opacity determinations for deriving the representative character of opacity in the ensuing exposition.

Figure 5 is a B_p = 10 analog of the escape energy results in Figure 4. This value is close to the surface field strengths of most magnetars. Since the attenuation of both photon splitting and pair creation is positively correlated with the field strength, the escape energies are generally higher than those in Figure 4. So the domination of pair creation (weighted curves) covers a larger portion of the escape energy curves. Yet the general shapes of the escape energy curves are quite similar to those in Figure 4, a consequence of the employment of an identical selection of field morphologies.

Modest or large emission angles Θしーた_e to the local field are expected for magnetar bursts and flares, where radiation comes from highly optically thick "fireballs" along magnetic flux tubes populated by quasi-thermalized electron–positron pair plasma. To accommodate this situation and provide a contrast to results from Section 4.1, we explore the escape energy for photons emitted initially perpendicular to the local B field, which might be the primary direction for radiation to escape from the fireball in the closed-field region. Here we consider four different azimuthal directions for such Θしーた_e = 90° cases, namely primarily outward emission (${\boldsymbol{k}}\propto {\boldsymbol{B}}\times \hat{\phi }$ so that ${\boldsymbol{k}}\cdot \hat{r}\gt 0$) and inward emission (${\boldsymbol{k}}\propto \hat{\phi }\times {\boldsymbol{B}}$ with ${\boldsymbol{k}}\cdot \hat{r}\lt 0$), both in the r–θしーた planes, and two sideways emission cases ($\pm {\boldsymbol{k}}\propto ({\boldsymbol{B}}\times \hat{\phi })\times {\boldsymbol{B}}$) that are essentially orthogonal to active magnetic flux tubes.

To benchmark the ensuing depictions against the results displayed in Figures 4 and 5, the ratios of escape energies of outward emission cases to parallel emission cases are displayed in Figure 6 for four different p values and a fixed θしーた _f = 15° . Note the logarithmic representation of the ratios, employed because they span around five decades in range. The weighted portion of the curves marks the outward emission locales from where photon attenuation is dominated by pair creation instead of photon splitting. The ratio is smaller than unity for emission colatitude θしーた_E < 90° and larger than unity for θしーた_E > 90° . For θしーた_E < 90° , at the emission local Θしーた_kB ≡ Θしーた_e = 90° for outward-moving photons, and the escape energy is markedly reduced relative to the situation where emission is locally parallel to B . For emission colatitudes θしーた_E > 90° in the other hemisphere, parallel emitted photons travel close to the stellar surface, where the field strength is strong. Yet the Θしーた_e = 90° photons still propagate outward into weaker fields, with trajectories that are symmetric about the magnetic equator, i.e., under the interchange θしーた_E → 180° − θしーた_E . The combination of these influences leads the escape energy ratio to generally be an increasing function of θしーた_E . The deviation of the escape energy ratio from unity at small and large colatitudes is enhanced for small p , primarily because the radial component of the photon momentum vector is enhanced with larger twists for the case of photons emitted parallel to B .

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Figure 7 illustrates the escape energies for photons emitted perpendicular to field loops that are characterized by footpoint colatitude θしーた _f = 15° and four different p values. In the top two panels, the solid curves represent the outward emission case where photons possess initial momenta ${\boldsymbol{k}}\propto {\boldsymbol{B}}\times \hat{\phi }$ and propagate away from the star. The escape energies for the inward case (${\boldsymbol{k}}\propto -{\boldsymbol{B}}\times \hat{\phi }$) are displayed in the top left panel as dashed curves. For both emitting directions, the photon trajectories lie in "meridional" (r, θしーた)-planes so that the escape energy curves are symmetric about the magnetic equator. The outward emission escape energy curves resemble the bell shapes in the parallel emitting case, with ratios of the two as depicted in Figure 6. For the inward case, the photon escape energy is strongly diminished near the equator, generating a double-peaked shape. This reduction is because the trajectories of these photons pass close to the stellar surface; the gaps in the inward curves demarcate the locales where emitted photons actually hit the star. These colatitudes of stellar shadowing of the radiation get reduced as the twist increases since the equatorial portions of the loops move to higher altitudes, for which the star becomes more remote. The cusps in the p = 1 outward emission curves near θしーた_E = 72° and 108° are caused by the fact that the pair creation attenuation coefficient is not a strictly increasing function of photon energy between two distinctive Landau levels. Thus, the escape energy exhibits two small steps that correspond to the first and second Landau levels (see Appendix A).

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Escape energy curves for photons emitted in one of the sideways directions (${\boldsymbol{k}}\propto ({\boldsymbol{B}}\times \hat{\phi })\times {\boldsymbol{B}}$) are displayed as dashed curves in the top right panel of Figure 7. The curves are now not symmetric about the equator, and the values of the escape energy are intermediate between those of the outward and inward emission cases. The escape energies for another side direction ($-{\boldsymbol{k}}\propto ({\boldsymbol{B}}\times \hat{\phi })\times {\boldsymbol{B}}$) are reflection symmetric about the equator to the depicted ones, and thus they are not explicitly displayed.

To complete the suite of information, the escape energy ratios εいぷしろん_esc,in/εいぷしろん_esc,out and εいぷしろん_esc,side/εいぷしろん_esc,out are presented in the bottom panels of Figure 7. This row augments the top panel information wherein it is difficult to discern escape energy curves that differ by 10% or less. The εいぷしろん_esc,in/εいぷしろん_esc,out ratios are close to unity for large or small emission colatitudes and drop to around zero near the equator. As the twist increases, the emission locale raises and the ratio drops arise at colatitudes somewhat closer to the equator. Therefore, the differences between outward and inward emission cases decrease with higher field twists. The εいぷしろん_esc,side/εいぷしろん_esc,out ratios also decline near the equator. The equatorial bump in the p = 1 case is caused by the onset of pair creation in the outward emission case. Curve structure bracketing the bump, with analogous variations exhibited for other p values (albeit less pronounced), captures the discontinuities of the pair conversion rates near threshold; see Appendix A. The shapes of the εいぷしろん_esc,side/εいぷしろん_esc,out ratios are generally complicated since photons emitted sideways do not move in fixed (r, θしーた)- planes. Yet the ratios are generally close to unity (20%) for most emission colatitudes.

In summary, the main message contained in the results depicted in Figures 6 and 7 is that emitting photons at large angles to the local field alters the magnetospheric opacity substantially and that the escape energy is most sensitive to the azimuthal direction of emission around B for equatorial locales. Both these properties emerge naturally from the angle and field dependence of the pair creation and photon splitting rates.

The final focus of our results section is on the regions very close to the magnetic poles. This is primarily motivated by the phenomena of magnetar giant flares, yet it may also be germane (Younes et al. 2021) to the simultaneous detection of an FRB (Bochenek et al. 2020) with a hard X-ray one (FRB-X; Mereghetti et al. 2020; Ridnaia et al. 2021) from the magnetar SGR 1935+2154 on 2020 April 28 (UTC).

Transient giant flares possess enormous luminosities 10⁴⁴–10⁴⁷ erg s⁻¹ at hard X-ray energies and constitute the hardest emission signal known for magnetars. Their initial spikes, generally lasting less than around 0.2 s, are observed to extend up to the MeV-band energies that permit pair creation to possibly be active. Specifically, for the Galactic magnetar giant flares, Hurley et al. (1999a) identified emission up to around 2 MeV for the 1998 August 27 event from SGR 1900+14, while Hurley et al. (2005) reported signals up to around 1 MeV for the 2004 December 27 giant flare from SGR 1806–20. Going beyond the Milky Way, the 2020 April 15 giant flare from a magnetar in NGC 253 was far enough away that only the initial spike was observed, and it was not subject to instrumental saturation influences. This event thus supplied unprecedented time-resolved spectroscopy. Fermi-GBM observations of the transient detected photons up to around 3 MeV (Roberts et al. 2021), and the interpretation was that the initial spike constituted soft gamma-ray emission from relativistic plasma outflow from the polar (perhaps open field line) regions of a rotating magnetar. This picture motivates the focal investigation of this subsection.

For the polar surface field B_p = 100 case, Figure 8 displays the escape energies for photons emitted parallel to the local magnetic field directions from field loops anchored at polar regions, specifically with θしーた _f = 1° and 5° . The left panel partners with three of the panels in Figure 4, while the right panel is to be compared with the solid curves in the top panels of Figure 7. In both cases, the escape energies increase with the emission colatitude θしーた_E and realize power laws when pair production is the dominant mode of photon attenuation, corresponding to emission altitudes larger than around 10R_NS . This is a domain where the GR influences are small, so that a flat spacetime asymptotic analysis can be used to describe the analytic character of the escape energy. For the emission parallel to B , the case in the left panel, this adapts the protocol developed by Story & Baring (2014). The power-law behavior can be well estimated by an analytic approximation:

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{esc}}\approx A(p)\frac{{h}^{2+p/2}}{{B}_{p}{\theta }_{f}}\approx A(p)\frac{{\theta }_{{\rm{E}}}^{1+4/p}}{{B}_{p}{\theta }_{f}^{2+4/p}},\end{eqnarray} \tag{ 18 }$$

where the poloidal field-line relation in Equation (B2) is used in the second approximation, and

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}A(p) & = & \frac{{2}^{5}{\left(p+3\right)}^{p+3}}{3p{\left(p+2\right)}^{p+3}}\\ & & \times \,{\left\{\mathrm{log}\left[\frac{{3}^{2}p{\left(p+2\right)}^{3p/2+5/2}}{{2}^{8}{\left(p+3\right)}^{3p/2+4}}\frac{{\alpha }_{f}{R}_{\mathrm{NS}}}{{\unicode{x0019B}}_{{\rm{C}}}}\right]\right\}}^{-1}.\end{array}\end{eqnarray} \tag{ 19 }$$

Here h = r/R_NS is the altitude of the emission locale scaled by the stellar radius R_NS . This relation is an extension of Equation (20) of Story & Baring (2014) to the twisted field configuration. It is valid for arbitrary twist parameter 0 < p ≤ 1 with θしーた_E ≪ 1. For a specific p , one can infer that ${\varepsilon }_{\mathrm{esc}}\propto {h}^{2+p/2}\propto {\theta }_{{\rm{E}}}^{1+4/p}$, with a power-law index that is independent of the field strength B_p and the footpoint colatitude θしーた _f of the field loops. The derivation of Equation (18) is detailed in Appendix B. Equation (18) is plotted for p = 0.5 and θしーた _f = 1° as the black dotted curve in Figure 8 to compare with the numerical results. Although not depicted in the figure, Equation (18) well approximates (with error ≲30% for θしーた_E ≲ 20° and h > 10) the power-law relationships for all the other p cases. Observe that there is a slight downward curvature at high colatitudes beyond the θしーた_E ≪ 1 domain.

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The right panel of Figure 8 illustrates the escape energies for photons emitted perpendicular (outward directed with ${\boldsymbol{k}}\propto {\boldsymbol{B}}\times \hat{\phi }$) to the field loops for the same polar footpoint colatitudes. Note that the escape energies for other perpendicular emission cases (inward directed, sideways) are almost identical to the outward case. The values of the escape energies are smaller than their counterparts for the parallel emission case, since photons immediately move across field lines, thereby manifesting a different power-law behavior with ${\varepsilon }_{\mathrm{esc}}\propto {\theta }_{{\rm{E}}}^{2+4/p}$. This index can be quickly obtained by inserting Θしーた_kB = πぱい/2 , B ∝ B_p /h^p+2 , and Equation (B2) into B ωおめが_⊥ ≈ constant; similar behavior is realized for other fixed values of Θしーた_kB .

The escape energy curves also realize a horizontal saturation (black dashed line) at small emission colatitudes θしーた_E close to the stellar surface. The saturated escape energy is independent of the twist parameter p and can be estimated in a superstrong field where ${{ \mathcal M }}_{1}\approx 1/(6{B}^{3})$ dominates the reaction rates ${{ \mathcal M }}_{\sigma }$ (see Equations (6) and (7)) and the photon splitting coefficient is independent of the field strength. In this case, Equation (5) yields

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{esc}}\,\approx \,\sqrt{1-{{\rm{\Psi }}}_{{\rm{E}}}}{\left(\frac{1440{\pi }^{2}{\unicode{x0019B}}_{{\rm{C}}}}{{\alpha }_{f}^{3}{R}_{\mathrm{NS}}{\sin }^{6}{{\rm{\Theta }}}_{{\rm{e}}}}\right)}^{1/5}.\end{eqnarray} \tag{ 20 }$$

Here we assume that the stellar radius R_NS is the typical length scale for the attenuation. For emission locales near the surface, Equation (20) gives a saturated escape energy εいぷしろん_esc ≈ 26 keV when Θしーた_e = πぱい/2 and Ψぷさい_E = 0.425 , which is in agreement with the values near the footpoint colatitudes for both θしーた_f = 1° and 5° in the right panel of Figure 8. For Θしーた_e < πぱい/3 , the typical attenuation locales are well removed from the surface, and then Equation (20) accurately expresses the saturation energy when Ψぷさい_E → 0 and ${\varepsilon }_{\mathrm{esc}}\approx 36\,\mathrm{keV}\,{(\sin {{\rm{\Theta }}}_{e})}^{-6/5}$.

A key feature of the polar region emission is that the escape energy for parallel emitted photons increases with the decrease of the field loop footpoint colatitude θしーた _f , for fixed field strength B_p and twist parameter p . Therefore, in the polar region, the escape energy can be very large (>10–100 MeV) even for photons emitted near the surface, since the field lines are only mildly curved. Adding twists to the field configuration will further enlarge the escape energy because the radius of the curvature is increased. According to Figure 8, this is more pronounced for the case of emission parallel to B , and much less so for emission orthogonal to the field. This rise in εいぷしろん_esc as θしーた _f becomes small applies to both processes, so that the competition between pair creation and photon attenuation in polar regions appears to be only modestly dependent on the twist parameter p .

The 3 MeV energy markers on both panels of Figure 8 signify the approximate maximum energy observed from the initial spike of the magnetar giant flare in NGC 253 (Roberts et al. 2021) by Fermi-GBM in 2020. It is clear from the left panel of Figure 8 that photon transparency in the inner magnetosphere for such a signal is guaranteed right down to the surface if the pertinent field-line footpoint colatitude is somewhat smaller than 5° and the emission is along the field. In striking contrast, if the emission is perpendicular to the field and outward directed (right panel), then photon splitting and even pair creation would be rife, precluding the visibility of such a signal if generated at altitudes of 30R_NS or less. The εいぷしろん_esc curves are very similar for inward-directed, Θしーた_e = πぱい/2 emission (not displayed). Moreover, our computed saturation escape energies for photon splitting are 295 keV for Θしーた_e = 10° and 674 keV for Θしーた_e = 5° for all axisymmetric twist scenarios; see Equation (20). While Doppler boosting will likely beam the hardest giant flare emission along the local field, substantial angles Θしーた_e to the field will be germane to emission at energies below 1 MeV. Accordingly, the time-dependent spectra presented in Roberts et al. (2021) likely constrain the inferred quasi-polar emission altitude to at least 10R_NS for highly twisted (p = 0.25) configurations and more than 30R_NS for dipole morphology. Refined emission geometry diagnostics for this NGC 253 transient will be deferred to future work.

In this subsection, an exploration of how the twisted field structure affects the opaque volumes for photon splitting and pair creation in the magnetosphere is presented. In contrast to Section 4, here photons are emitted from a large variety of fixed locales instead of from individual field loops. The twists change the morphology of the magnetic field around the star, thereby altering the transparency of the magnetosphere.

The left panel of Figure 9 displays the opaque regions of polarization-averaged photon splitting for photons with energy εいぷしろん = 200 keV emitted in the twisted magnetospheres with p =1.0, 0.75, 0.5, and 0.25. The colored contours depict the boundaries of the opaque regions in the planar meridional section that contains the center of the star. Thus, at the boundary, the photon splitting escape energy is 200 keV, and inside it εいぷしろん_esc is lower; these zones of opacity are signified in two cases by shading colored to pair with that of the corresponding boundary contours. Given the axisymmetric field construction, the volumes of opacity are formed by rotating these planar sections about the magnetic axis. Field loops are plotted as projections onto the planar meridional section for p = 0.5 on the left-hand side (light-blue curves) and for p = 1.0 on the right-hand side (light-green curves), respectively. The maximum radii ${r}_{\max }$ for the field loops are fixed at 2R_NS, 5R_NS, 10R_NS, 20R_NS, 50R_NS, and 200R_NS; accordingly, the field loops on the left-hand side (p = 0.5) appear flattened by the twists that generate field components out of the meridional plane and move the footpoints closer to the equator.

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Contours on the left-hand side of the panel present the opaque regions for photons emitted perpendicular to the local field directions specifically with their trajectories lying in the meridional plane (${\boldsymbol{k}}\propto {\boldsymbol{B}}\times \hat{\phi }$); therefore, the contours are symmetric about the x-axis. For a fixed photon energy, the opaque volume increases with the increase of the magnetospheric twist, character that is not that easily discerned from the figures in Section 4; there photons are emitted from field loops with fixed θしーた _f (i.e., see Figure 7), and the coupling of escape energies to emission altitudes is highlighted. At the equator in Figure 9, emitted photons are directed away from the star, experiencing weaker field strengths along their trajectories compared with photons from other emission colatitudes. Therefore, the overall opacity is smaller and the contours slightly shrink near the equator. This is more obvious for magnetospheres with large twists, e.g., p = 0.25, where the magnetic field direction changes more rapidly near the equator.

On the right-hand side of the panel, opaque boundary contours are displayed for photons emitted parallel to the poloidal components of the magnetic field. This is a twisted magnetosphere analog of Figure 9 in Hu et al. (2019). The opaque contours are not symmetric about the x-axis since momentum directions are different for photons emitted in the upper and lower hemispheres. The opaque volumes shrink near the north magnetic pole for smaller p . Thus, the vicinity of the north pole becomes more transparent as the twist increases. This is caused by the enhancement of the radial field component when increasing the twists. Therefore, both the optical depth τたう and the $B\sin {{\rm{\Theta }}}_{\mathrm{kB}}$ factor that controls it grow slowly for photons emitted near the north pole. This is in accordance with the surface-emission curves in Figure 4. At colatitudes near the equator, the opaque volume expands with a decrease in p , because the $B\sin {{\rm{\Theta }}}_{\mathrm{kB}}$ factor sampled by the emitted photon receives significant contributions from the toroidal field component. When the emission colatitude θしーた_E increases across the equator, the opaque volume expands abruptly, creating small dips on the contours above the equator. This is because photons emitted in the lower hemisphere are directed toward the star (see the black photon trajectories in Figure 9). The lower hemisphere is more opaque since emitted photons sample stronger field strengths and shorter radii of field curvature near periastron.

The right panel of Figure 9 presents the opaque volumes to photons with observed energy εいぷしろん = 3 MeV caused by polarization-averaged pair creation. The shapes of the contours are very similar to their photon splitting analogs on the left, yet the scales are larger owing to the higher photon energy required to exceed the pair threshold, something that is also apparent in Figures 3 and 5. As on the left, the transparent volume proximate to the north pole is enlarged as the twist increases, implying reduced rates of pair creation at altitudes ≲ 8R_NS . The green and blue shaded areas, bounded by the dashed black curves, represent the opaque regions where emitted photons will be attenuated by splitting or pair conversion before propagating to infinity or striking the surface. The gray shaded area displays the interior shadow region where emitted photons actually reach the surface without attenuation (τたう < 1). While not explicitly depicted, we remark that for photons of fixed energy εいぷしろん > 2m_e c² emitted parallel to B within the meridional plane, the opaque volumes of pair creation are larger than the photon splitting ones when the emission colatitude is not too small, θしーた_E ≳ 60° .

The results displayed are for a choice of B_p = 10 . When the polar magnetic field is increased to B_p = 100 , the opacity volumes for both splitting and pair creation increase somewhat, as expected: a rise in the field strength throughout the magnetosphere increases the rates for both processes at each locale.

The suite of results presented so far evinces two clear trends. First, there is a general increase of the opacity volumes in the inner magnetosphere when the twist is increased. This coupling is driven by the rise in the magnitude of the field, though a notable exception is in nonequatorial zones for outward emission parallel to B . The second key trend is an increase of the escape energies with larger twists in the case of emission parallel to B on specified field loops somewhat near the poles. This behavior is caused by a general straightening of field lines, which then yields predominantly higher altitudes with lower fields where opaque conditions arise. While these trends may seem somewhat contradictory, they are actually encapsulated in the crossing over of opacity boundaries with different p values in the upper hemispheres of the panels in Figure 9. Addressing the impact of twists specifically on pair creation is also germane, since pair populations are believed to underpin the currents that establish the twisted fields on extended ranges of altitudes in the magnetosphere.

The commentary on Figure 8 in Section 4.3 focused on the gamma-ray transparency for magnetar giant flares. Also apparent in this polar zone figure is that εいぷしろん_esc for the onset of pair creation domination of photon splitting increases modestly as the field twist rises, when the emission is parallel to B . However, these pair onset εいぷしろん_esc values do not change appreciably with variations in the twist parameter p for emission orthogonal to the field. This reasonably implies that once sufficient pair creation is established to support significant twists, the configuration can sustain the twist as long as the energy source for a giant flare continues; the twist does not unduly starve itself of pairs. The pair conversion appears likely to occur at altitudes of 20–30 stellar radii, tapping the giant flare radiation supply there, only to cease at somewhat higher altitudes when the flare plasma outflow (likely ultrarelativistic; see Roberts et al. 2021) becomes optically thin to splitting and γがんま → e^± (yet still Thomson optically thick) and the radiation we see emerges.

The situation for persistent magnetospheric signals from magnetars may be quite different. Earlier figures such as Figures 5 and 7 concentrate on larger footpoint colatitudes that are more or less commensurate with the active twist ones in the plasma simulations of Chen & Beloborodov (2017). For these field lines, the photon energies for the onset of pair creation in quasi-equatorial regions are only moderately increased by a strengthening of the twist. These field zones likely address the persistent hard X-ray tail emission (Thomson optically thin) of magnetars, which is known to not extend up to pair threshold (e.g., Kuiper et al. 2006; den Hartog et al. 2008a, 2008b). Specifically, photons are easily generated to energies well in excess of 1 MeV in resonant inverse Compton models (Wadiasingh et al. 2018), tapping the energies of ultrarelativistic pairs accelerated in the magnetosphere. We anticipate that photon splitting strongly attenuates these gamma rays (Wadiasingh et al. 2019), more so at higher fields and for higher twists, and may effectively starve the radiating plasma of pairs; this implies a possible limit to the maximal twist for normal magnetar activity.

Our results therefore highlight the need for a comprehensive inclusion of photon splitting opacity and its impact on pair creation inhibition in order to enhance simulations of dynamic, twisted magnetar magnetospheres, such as those studied in the works of Beloborodov (2009), Parfrey et al. (2013), and Chen & Beloborodov (2017). It also identifies the compelling need for greater observational sensitivity in the 500 keV–10 MeV band to afford more incisive probes of the shape of magnetar soft γがんま-ray spectra (Wadiasingh et al. 2019). This prospect could be fulfilled by a future Compton telescope such as NASA's COSI SMEX mission, ⁸ under development, and AMEGO ⁹ or its more compact AMEGO-X version (Fleischhack & Amego X Team 2021).

5.2.1. Force-free MHD and Pulsar Context for Twists

5.2.2. Magnetar Radio Emission Connections