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PHOTON SPLITTING AND PAIR CREATION IN HIGHLY MAGNETIZED PULSARS Matthew G. Baring1 and Alice K. Harding Laboratory for High Energy Astrophysics, Code 661, NASA Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A. [email protected], [email protected] To appear in The Astrophysical Journal, Vol 547, February 1, 2001 issue.

ABSTRACT The absence of radio pulsars with long periods has lead to the popular notion of a high P “death line.” In the standard picture, beyond this boundary, pulsars with low spin rates cannot accelerate particles above the stellar surface to high enough energies to initiate pair cascades, and the pair creation needed for radio emission is strongly suppressed. In this paper we explore the possibility of another pulsar “death line” in the context of polar cap models, corresponding to high magnetic fields B in the upper portion of the period-period derivative diagram, a domain where few radio pulsars are observed. The origin of this 13 high B boundary, which may occur when B becomes comparable to or exceeds Bcr =4.4 10 Gauss, is also due to the suppression of magnetic pair creation, but primarily because of ineffective× competition with magnetic photon splitting. Threshold pair creation also plays a prominent role in the suppression of cascades. We present Monte Carlo calculations of the pair yields in photon splitting/pair cascades which show that, in the absence of scattering effects, pair production is effectively suppressed, but only if all three modes of photon splitting allowed by QED are operating in high fields. This paper describes the probable shape and position of the new “death line,” above which pulsars are expected to be radio quiet, but perhaps still X-ray and gamma-ray bright. The hypothesized existence of radio-quiet sources finds dramatic support in the recent discovery of ultra-strong fields in Soft Gamma-ray Repeaters and Anomalous X-ray Pulsars. Guidelines for moderate to high B pulsar searches at radio wavelengths and also in the soft and hard gamma-ray bands are presented. Subject headings: gamma rays: theory — radiation mechanisms — magnetic fields — stars: neutron — pulsars: general

1. INTRODUCTION some long period pulsars like PSR J2144-3933 can be explained through production of pairs by photons from inverse-Compton Most current theories for the generation of coherent radio scattering (Arons 2000, Zhang, Harding & Muslimov 2000), a emission in pulsar magnetospheres (see summations in Michel 1991; Melrose 1993; Lyutikov et al. 1999) require formation of process which does not require as high a voltage as does curva- ture radiation. an electron-positron pair plasma. Such a pair plasma has been In this paper, we study another area of pulsar phase space, shown to develop via electromagnetic cascades along the open magnetic field lines as a result of particle acceleration to TeV that of very high magnetic fields, where the character of pair cascades is significantly altered by the process of photon split- energies, followed by curvature, synchrotron radiation and one- + ting and other effects such as ground state pair creation and photon pair production in the strong magnetic field, γ e e− , near the neutron star surface (e.g. Sturrock 1971; Ruderman→ & positronium formation. Magnetic photon splitting, γ γγ , a QED process in which a single photon splits into two→ lower- Sutherland 1975, Arons & Scharlemann 1979). Numerical sim- ulations of pair cascades above a pulsar polar cap show that energy photons (Adler 1971, Baring & Harding 1997), operates 103 104 pairs are produced for each accelerated primary elec- efficiently and competes effectively with pulsar pair production − only in magnetic fields above 1013 Gauss (Harding, Baring tron (Daugherty & Harding 1982) in a typical young pulsar. ∼ But if any of the necessary conditions for development of a pair & Gonthier 1997, hereafter HBG97). This region of high mag- netic field strength lies in the upper-right part of the P P˙ cascade, i.e. production of γ-ray photons (requiring particle ac- − celeration to high energy) or production of sufficient pairs, is ab- diagram. We will explore whether photon splitting may be the sent the pulsar should not, according to theory, emit detectable explanation for why there are no classical radio pulsars detected with derived magnetic fields above 1014 Gauss. Our initial radio emission. The existence of an observed “death line” in ∼ the P P˙ distribution of known radio pulsars (see Figure 5 results (Baring & Harding 1998) have suggested that photon below),− to the right of which no pulsars were detected, until the splitting could indeed explain the absence of very high-field ra- recent discovery (Young, Manchester & Johnston 1999) of the dio pulsars. This paper considers in more detail the underlying 8.5 second pulsar PSR J2144-3933, provided some circumstan- assumptions made in that calculation, explores the physics of tial evidence that pairs are required for radio emission. The high-field pair cascades and makes predictions for high-energy slope of this putative boundary fits a line of constant voltage searches for high-field pulsars. Although pulsars in this region 3/2 ˙ 1/2 of phase space may be radio-quiet, those that are to the left of across the open field lines, V P − P , suggesting that radio emission ceases when the particle∝ acceleration is not high the death line are still prolifically accelerating particles to high enough to sustain pair production. The radio emission from energies. Their pulsed emission may therefore be detectable in the X-ray and gamma-ray wavebands, making this relatively-

1Universities Space Research Association 1 2 unexplored region of pulsar phase space extremely interesting. We will therefore explore several possibilities for photon split- There has recently been growing evidence for another class ting mode behavior in our study of high-magnetic field pair cas- of pulsars with ultra-strong magnetic fields. Observations at cades. In Section 2.1, we discuss the physics of pair suppression X-ray energies have yielded detections of both long periods and in high magnetic fields, both by photon splitting and by pair high period derivatives in two types of sources, anomalous X-ray production in low-lying Landau states, as well as the possible pulsars (AXPs) and soft γ-ray repeaters (SGRs), which suggest alternative mechanism of pair suppression by bound-state pair dipole spin-down fields in the range 1014 1015 Gauss. The creation, discussed by Usov & Melrose (1996). In Section 3, we AXPs are a group of seven or eight pulsating− X-ray sources with present results of detailed simulations of splitting/pair cascades periods in the range 6–12 seconds, and are continuously spinning which explore the conditions for suppression of pair creation down (Vasisht & Gotthelf 1997). The SGRs are a type of γ-ray in a pulsar magnetosphere. We compute the boundary in the transient source that undergoes repeated bursts; four (possibly pulsar P P˙ distribution where the escape energies for pho- now five: see Cline et al. 2000 for SGR 1801-23) are currently ton splitting− and magnetic pair production for photons of known to exist. A detailed discussion of these sources and re- polarization are equal. This condition then defines a putative⊥ cent discoveries pertaining to them is presented in Section 4.3. radio-quiescence boundary, posited in Baring & Harding (1998), While conventional radio pulsars tap their rotational energy to above which pulsars may not produce the dense pair plasmas power their emission, if such a new class of ultra-magnetized required for radio emission. For reasonable assumptions about neutron stars or “magnetars” does exist, their long periods in- the location of particle acceleration and the angles of the emit- dicate that their emission cannot be rotationally-powered: they ted photons in high-field pulsars, the computed radio-quiescence may instead be driven by their magnetic energy. With the pos- boundary lies at P˙ above those in the known radio pulsar pop- sible exception of the recent report (Shitov 1999; Shitov, Pu- ulation, potentially explaining the absence of radio pulsars with gachev & Kutuzov 2000) of a low frequency pulsed detection of fields above 1014 Gauss. Our cascade simulations indicate a counterpart (PSR J1907+0919 at 111 MHz) to the soft gamma that such a boundary∼ is viable only if photons of both polariza- repeater SGR 1900+14, which has not been confirmed by pulsa- tions can split; otherwise pair creation is generally postponed a tion searches in Aricebo data at higher frequencies (Lorimer & generation rather than suppressed. In the latter case, significant Xilouris 2000) none of these sources has detectable pulsed radio reductions in pair production can ensue only if the maximum emission. energy of primary photons is below or not much above the pair In several previous papers, we have computed the photon creation escape energy. If pair suppression is rife at these high splitting attenuation lengths of photons emitted parallel, or at P˙ , it suggests the possible existence of a new class of radio-quiet, small angles, to the magnetic field near the surface of strongly ultra-magnetized pulsars that may be related to the emerging magnetized neutron stars and compared them to the attenua- class of magnetars. tion lengths for one-photon pair production (Harding, Baring & Gonthier 1996; 1997). Both processes depend on the photon 2. PHYSICAL EFFECTS INFLUENCING PAIR CREATION energy ω and the angle θkB of photon propagation to the local + field. However, since photon splitting has no threshold, it may The most important competitor to pair creation γ e e− occur before the photon reaches the threshold for pair produc- at high magnetic field strengths as a mechanism for attenuating→ 2 tion at ω sin θkB =2mec . We found that the photon splitting photons in pulsar magnetospheres is magnetic photon splitting attenuation lengths are lower than those for pair production in γ γγ . Hence this process motivated the suggestion (Bar- fields higher than 3 1013 Gauss, for emission in the open field ing→ & Harding 1998) that splitting can potentially suppress pair region near the neutron∼ × star surface. One must therefore incor- creation and inhibit radio emission in highly-magnetized pul- porate photon splitting in cascade models of highly-magnetized sars; accordingly it forms the centerpiece of the discussion of pulsars. This has been done in the case of PSR1509-58 (Hard- this Section. Yet, the creation of pairs in low Landau levels or ing, Baring & Gonthier 1997), which has a field of 3 1013 Gauss the ground state can also suppress subsequent pair creation, so as estimated using Equation (9). We considered both× the case this effect is addressed in Section 2.2. We also include a brief where all linear polarization modes of photon splitting allowed discussion of the role of positronium formation, since the res- by CP invariance operate, and the case where only the idence of pairs in neutral bound states can inhibit a range of mode (our polarization convention is defined near the⊥→kk begin- coherent mechanisms for producing radio emission. The section ning of Section 2) allowed by selection rules (Adler 1971) in the concludes with a discussion of how the combined effects of pho- weakly dispersive limit operates. The latter case, we argued, ton splitting and threshold pair creation change the nature of was most appropriate for pulsars. In both cases, we found that pair cascades as the magnetic field increases. photon splitting attenuation is a plausible explanation for the At this point, it is appropriate to identify conventions adopted very low spectral cutoff between 10 and 30 MeV in PSR1509- throughout this paper. The photon linear polarizations are such 58 (Kuiper et al. 2000). Since the rates of these attenuation that refers to the state with the photon’s electric field vector mechanisms are very sensitive to the polarization states of the parallelk to the plane containing the magnetic field and the pho- incoming photons, it is evident that the polarization properties ton’s momentum vector, while denotes the photon’s electric of the radiated photons are very important in modeling both field vector being normal to this⊥ plane. Furthermore, all photon the output spectrum and pair suppression in high-field pulsars. and electron energies will be written in dimensionless form, be- 2 In ultra-strong magnetic fields, the vacuum dispersion in- ing scaled by mec . Since general relativistic effects will play an creases to the point where the selection rules derived for weak important role in our considerations, we adopt the convention of linear dispersion (appropriate for B Compton scattering. Synchrotron and cur- T = 2 ω 2 ⊥→⊥⊥ 60π λ– Bcr M vature radiation are central to the hard X-ray and gamma-ray   emission of polar cap models (e.g. Daugherty and Harding 1982; in the frame where photons propagate perpendicular to the field, 1996), and are essentially identical in their polarization proper- where ties in the classical picture. For monoenergetic electrons, polar- 4 ization levels = n˙ n˙ / n˙ +˙n of between 50% and − ∞ B ds sBcr /B 100% are achievedP (e.g.| ⊥ − seek| Fig.| ⊥ 6.7 ofk| Bekefi 1966), while for 1 = e− M Bcr 0 s power-law electrons, the degree of polarization is generally in   Z the 60%–70% range (e.g. Bekefi 1966; Rybicki and Lightman 3 s cosh s 3+2s2 s cosh s + + 2 + 3 , 1979), with photons dominating in both cases. At B >Bcr , × −4s 6 sinh s 12 sinh s 2 sinh s (  ) there is some⊥ degree of quantum depolarization, though∼ in re- (2) 4 ality this is generally small since photon angles with respect − ∞ B ds sBcr /B 2 = e− to local fields are always small. The dominance of photons M Bcr s applies also to the products of resonant Compton⊥ scattering,   Z0 which has more recently been considered (e.g. Sturner and Der- 3 cosh s 3 4s2 3s2 + − 2 4 mer 1994) as a primary emission mechanism. In the Thomson × (4s sinh s 4 sinh s − 2 sinh s) limit, the cross-sections for scatterings of polarized photons can be derived from the results of Herold (1979), indicating that and αf is the fine structure constant and λ–=¯h/(mec)isthe photons are produced at 3 times the rate of photons. ⊥ k Compton wavelength of the electron divided by 2π .Atlow fields, 1 and 2 are independent of B, but at high fields M M3 4 possess 1 B− 2 B− dependences. These are the ex- 2.1. Competition with Magnetic Photon Splitting pressionsM used∝ in thisM paper∝ because of their broad applicability 2.1.1. Photon Splitting to the pulsar problem. Deviations from this low energy limit near pair creation threshold are presented in detail by Baier, The relevance of photon splitting γ γγ to neutron star → Mil’shtein, & Shaisultanov (1996) and Baring & Harding (1997), environments was emphasized by Adler (1971), Mitrofanov et and are mentioned below where appropriate. An analytic ap- al. (1986) and Baring (1988). In the context of pulsars, it has proximation to the total rate in the B Bcr limit is presented been discussed by Baring (1993), who proposed it as a mecha- in Baring and Harding (1997), and  analytic forms for + B Bcr nism for suppressing the appearance of e e− annihilation lines, other splitting modes can be found in Baring (2000). and by Harding, Baring and Gonthier (1996, 1997) and Chang, The birefringence of the magnetized vacuum implies an al- Chen and Ho (1996), who focused on its action in attenuating teration of the kinematics of strong field QED processes (Adler gamma-ray pulsar continua. Splitting is a third-order QED pro- 1971), admitting the possibility of non-collinear photon split- cess with a triangular Feynman diagram. Though it is permitted ting. Hence, while the splitting modes , and by energy and momentum conservation, when B = 0 it is for- are forbidden by CP invariance in the⊥→⊥k limit ofk→⊥⊥ zero disper- bidden by a charge conjugation symmetry of QED known as sion,k→kk dispersive effects guarantee a small but non-zero probabil- Furry’s theorem (e.g. Jauch and Rohrlich 1980), which states ity for the channel. Extensive discussions of linear dis- that ring diagrams that have an odd number of vertices with persion in a⊥→⊥k magnetized vacuum are presented by Adler (1971) only external photon lines generate interaction matrix elements and Shabad (1975); considerations of plasma dispersion (e.g. see that are identically zero. This symmetry, which pertains to the the analysis of Bulik 1998) are not relevant to classical gamma- electron/positron propagators, is broken by the presence of an ray pulsars because of the relatively low densities present in external field. The splitting of photons is therefore a purely the magnetosphere, but may be quite pertinent to soft gamma quantum effect, and has appreciable reaction rates only when repeaters. Adler (1971) showed that in the limit of weak lin- the magnetic field is at least a significant fraction of the quan- ear vacuum dispersion (roughly delineated by B sin θkB

2.1.2. Attenuation Lengths surface field B0 , the escape energy εesc is uniquely defined As pair creation is a first order QED process, whereas split- and cleanly delineates the energy ranges of source opacity and transparency. The escape energy also depends on the radius of ting is third-order in the fine structure constant α = e2/(¯hc), f emission (see Figure 5). The dependence of on such it is not immediately obvious that γ γγ can ever domi- R0 εesc + → parameters is shown in Figure 1 for photons that are initially nate γ e e− in pulsar environs. Yet it is the propagation → of polarization . Escape energy plots for unpolarized photons of the photons at small angles to the field through the pulsar ⊥ magnetosphere that affords photon splitting an opportunity to are given in Figures 3 and 5 of HBG97. The feature of Fig- ure 1 that is most salient for the considerations of this paper compete effectively with pair creation when B >B , since the cr can be obtained by comparing the plots for splitting and pair pair threshold ω sin θkB = 2 is always crossed∼ from below. An approximate assessment of the relative importance of magnetic production. For low fields, pair production escape energies are + below those for splitting, but the situation is reversed in high photon splitting and pair creation γ e e− was performed by Harding, Baring and Gonthier (1997)→ by computing atten- fields; the escape energies are roughly equal for a narrow band of fields around B0 0.5Bcr . For field strengths B0 >Bcr , uation lengths L for each of these processes. These are the ∼ scalelengths for attenuation in the neutron star magnetosphere, photon splitting can attenuate photons well below pair∼ thresh- and are defined to be path lengths over which the optical depth old at significant colatitudes. Hence splitting can be expected to dominate γ e± in supercritical fields for the attenuation → l of photons of polarization. ⊥ τ(Θ,ε,l)= T (θkB,ω)ds (3) The escape energies in Figure 1 generally decline with Θ and Z0 are monotonically decreasing functions of B0 for the range of fields shown. Consider first the dashed curves in each panel, is unity, i.e. τ(Θ,ε,L) = 1 . In computing such attenuation corresponding to initial propagation of photons along the field. lengths, it is essential to fully include the effects of general rela- The divergences as Θ 0 are due to the divergence of the tivity. The reason for this is that the quantum transition rates field line radius of curvature→ at the poles. The maximum an- for both splitting and pair creation are strong functions of the gle θkB achieved before the field falls off and inhibits attenua- photon energy and angle and the magnitude of the magnetic tion is proportional to the colatitude Θ . For photon splitting, field strength in the local inertial frame, all of which are strongly since the rate in equation (1), and therefore also the inverse of influenced by curved spacetime. Such an analysis is confined to 6 the attenuation length L , is proportional to ω5 sin θ ,itfol- the Schwarzschild metric because the dynamical timescales for kB lows that the escape energy scales as ε Θ 6/5 near the gamma-ray pulsars are considerably shorter than their period esc − poles. At fields B > 0.3B , there is a diminishing∝ depen- (e.g. =015 sec. for PSR1509-58), so that rotation effects in 0 cr P . ∼ the Kerr metric, for example those due to frame-dragging, can dence of B in the attenuation coefficient (e.g. see Adler 1971; be neglected in our photon attenuation analysis. Throughout Baring & Harding 1997 for graphical illustrations) so that a sat- uration of the photon splitting attenuation lengths and escape this paper, we assume a neutron star mass, M =1.4 M and energies arises in highly supercritical fields. For pair produc- radius, R =106 cm. tion, the behaviour of the rate (and therefore 1 )isexponen- Values for L are computed assuming that test photons are /L tial in 1/(ωB sin θkB) (e.g. see Daugherty & Harding 1983), emitted on or above the neutron star surface at some polar co- 1 latitude Θ (usually chosen to be at the polar cap rim) and which then quickly yields a dependence εesc Θ− near the poles for 0 1 . This behavior extends∝ to higher surface propagate outward, initially at a specified angle θ to the B0 < . Bcr kB,0 fields because∼ production then occurs at threshold, which deter- gravitationally-modified dipole magnetic field; a depiction of 1 mines εesc 2/θkB Θ− . The pair production escape energy the geometry is presented in HBG97. Frequently in this pa- ∼ ∝ per, a surface origin of the photons is chosen to provide a con- curves are bounded below by the pair threshold 2/ sin θkB at cise and representative presentation of the attenuation prop- the point of pair creation (not photon emission), and for high Θ erties of the pulsar magnetosphere. Such attenuation lengths approach the pair threshold (1 + 1+2B/Bcr)/ sin θkB (with 5/7 sin θ 1 determined by geometry), blueshifted by the fac- possess power-law behavior at high energies, with L ε− kB ∼ 2 1/2 p for photon splitting, and L 2/ε for pair creation just∝ above tor (1 2GM/Rc )− 1.3 . The field dependence in this − ∼ threshold (HBG97; proportionalities' that hold in both curved saturation energy arises because the creation of pairs by - ⊥ and flat spacetime). Furthermore, they display the property polarized photons cannot leave both the electron and positron that the attenuation length declines with colatitude of emis- in the lowest Landau level (e.g. see Daugherty & Harding 1983); sion, a consequence of the associated increase in curvature of the minimum energy configuration requires one member of the the field lines. At low energies, the attenuation lengths diverge pair to be in the first excited state. and photons escape the magnetosphere without attenuation. Since pulsar cascade high energy emission from curvature ra- The L asymptotes define escape energies (HBG97), be- diation, inverse Compton or synchrotron by relativistic particles low (above)→∞ which spectral transparency (opacity) is effectively with Lorentz factor γe will not beam the photons precisely along guaranteed. The existence of such escape energies is a conse- the magnetic field, but within some angle 1/γe to the field, it 3 ∼ quence of the r− decay of the dipole field: there are always is important to illustrate the effect on the escape energies of a sufficiently low photon energies for which the field declines be- non-zero angle of emission θkB,0 of the photons relative to B. 2 fore the photons have had sufficient time to attenuate in their This is done via the solid curves in Figure 1, where θkB,0 =10− passage through the magnetosphere. (i.e. 0.57◦ )istakentowards the dipole axis; qualitatively and 2 quantitatively similar behavior arises when θkB,0 =10− is 2.1.3. Escape Energies taken away from this axis. Curvature radiation-initiated cas- 7 cades generally have γe 10 (e.g. Daugherty & Harding 1989; For each photon trajectory through the magnetosphere, cor- see also Harding & Muslimov∼ 1998), while inverse-Compton responding to a specific set of values for the colatitude Θ and 5 6 seeded pair cascades yield γe 3 10 –10 (e.g. Sturner θkB,0 of emission at the surface, and for a particular dipole ∼ × 5

Fig. 1.— The escape energy (i.e. where L ) for (a) photon splitting and (b) pair production as a function of magnetic colatitude for photon →∞ emission both along B (dashed curves) and at angle θkB,0 =0.01 radians ( = 0.57◦ ) to the field (solid curves). The escape energies for each process are monotonically decreasing functions of B for the range of parameters shown. The θkB,0 = 0 curves have slopes of -6/5 (splitting) and -1 (pair creation) at small Θ , as discussed by Harding, Baring & Gonthier (1997), and diverge near Θ = 0 , where the field line radius of curvature becomes infinite. For the solid curves, at low magnetic colatitudes Θ < 10θkB,0 , the field curvature is so low that photon attenuation is insensitive to the value of Θ and is well described by the uniform field results in Eqs.∼ (1) and (2). The escape energies are presented for surface photon emission, R0 = R .

1995; see also Harding & Muslimov 1998), both yielding angles instead of the pair creation escape energies always being mono- θkB,0 very much smaller than the example in the figure. The tonically decreasing functions of B0 , they experience an inver- situation is similar for synchrotron radiation from first genera- sion at small colatitudes and increase with B0 . Suchaneffect 3 4 tion secondary electrons, where γe 10 10 ; such Lorentz was not present in the polarization-averaged escape energies pre- factors result from pairs created by primary∼ − photons (curvature sented in HBG97 because the state could always access the k or resonant Compton) in the 1 GeV–10 GeV range. However, as true pair threshold, i.e. 2/ sin θkB,0 . Note also, that while split- the cascade proceeds, higher generations of pairs achieve lower ting escape energies always drop when θkB,0 is increased from 2 Lorentz factors, resulting in a cumulative power-law spectrum zero to 10− , opposite behaviour is seen in high fields for pair 2 4 between γe 10 and γe 10 (Daugherty & Harding 1982). creation at moderate colatitudes, due to subtleties concerning This behaviour∼ applies to pulsars∼ with low to moderate magnetic the sudden onset of pair creation (HBG97). fields, B0 < 0.2Bcr , a domain that is well-studied in the litera- It is important to emphasize that general relativistic effects ture. For more∼ highly-magnetized pulsars, the restriction of pair are crucial to these calculations. While the dependence of the creation to a single state, discussed at length in Section 2.2, will escape energy on emission colatitude and surface polar field inhibit the creation of second and higher generation pairs, im- strength is similar in curved and flat spacetime, the introduction plying that the dominant synchrotron signal will sample angles of the Schwarzschild metric results in decreases of the escape en- 2 to the field much smaller than θkB,0 =10− . Hence, in summa- ergies for both process by factors of between 2 and 4, as is illus- 4 3 tion, we expect that θkB,0 < 10− 10− will be representative trated in Figure 4 of HBG97. Moreover, these decreases differ − 2 for the considerations of this∼ paper. The θkB,0 =10− choice for splitting and pair creation, making it imperative to carefully in Figure 1 aids clarity of illustration. account for propagation in curved spacetime when comparing 2 The general behaviour of the θkB,0 =10− curves in Figure 1 attenuation characteristics for the two processes. The largest can be simply understood. For the most part, the escape energy effects of the incorporation of general relativity are due to the is insensitive to the emission angle for Θ > 10θkB,0 . For small increase of the surface dipole field strength by roughly a fac- angles, the escape energy decreases and the∼ curves flatten below tor of 1.4 above the dipole spin-down estimate of B0 ,andthe the θkB,0 = 0 curves, converging as Θ 0 to an energy that is correction for the gravitational redshift of the photon, which 6/5 → increases the photon energy by roughly a factor 1.2–1.3 in the proportional to (B0 sin θkB,0)− when B0 Bcr . This con- vergence is a consequence of the field being almost uniform and local inertial frame at the neutron star surface compared to the tilted at about angle θ to the photon path for trajectories energy measured by the observer in flat spacetime at infinity. kB,0 The influence of these modification factors is amplified by the that originate near the pole; this behaviour at low colatitudes + sensitivity of the rates for γ γγ and γ e e− to B and ε was first noted, in the case of pair creation in flat spacetime, by → → Chang, Chen and Ho (1996). These “saturation” energies fol- so that 20%-30% deviations from flat spacetime parameters map low from the dependence of the photon splitting rate on B and over to the factors of 2–4 in the escape energies. Near the polar cap, the curvature of the photon trajectory in a Schwarzschild Θ , and have a declining sensitivity to B0 when B0 > 0.3Bcr . In Fig. 1b, the same effect is seen for pair creation,∼ but this metric does not affect the escape energies, to first order, since it time the “saturation” is at the redshifted threshold energy for is generally compensated for by relativistic modifications to the curvature of the magnetic field (e.g. see Gonthier & Harding γ e± , which varies as (1 + 1+2B/Bcr)/ sin θkB 0 . Hence, → , p 6

Fig. 2.— The solutions to Eq. (4) that define (a) the polar surface field B0 and (b) the escape energy εesc (for either photon splittings or pair + ⊥→kk creation e e− ) as functions of the colatitude Θ of the point of photon emission, which is assumed to take place on the stellar surface. Solutions are ⊥→ 4/15 1 displayed for the four values of θkB,0 , as labelled, with the θkB,0 = 0 case generating the approximate dependences B0 Θ− and εesc Θ− .All ∝ 4 ∝ θkB,0 = 0 cases produce B and εesc that are independent of Θ for sufficiently small colatitudes, namely roughly when Θ < 10 θkB,0 . 6 ∼

1994). of fields of interest to the focus of this paper. The variation of The most concise designation of when the creation of pairs B0 with Θ actually increases very slowly at smaller colatitudes might be suppressed by photon splitting is when the escape en- due to the diminishing dependence of the photon splitting rate ergies of the two processes are set equal: for the mode on field strength in highly supercritical ⊥→kk + fields. The non-zero θkB,0 solutions for B0 are monotonically e e− εesc⊥→kk(B0, Θ,R0)=εesc⊥→ (B0, Θ,R0) , (4) decreasing functions of θkB,0 , and eventually become indepen- 4 dent of colatitude when Θ < 10 θkB,0 , as is expected from the where R0 is the radius of the point of emission. Note that defining a similar equality for the polarization mode, or a horizontal branches of the escape∼ energy plots in Figure 2. Since polarization-averaged alternative, leadsk to only very modest the escape energy solutions trace the behaviour of the pair cre- ation εesc , small increases with θkB,0 are exhibited at moderate changes in the ensuing results. Baring & Harding (1998) con- 4 tended that Eq. (4) is representative of the delineation between to high colatitudes, even though for Θ < 10 θkB,0 , the solutions 1 pulsar parameter regimes where pairs are created in profusion scale as (θkB,0)− , as expected. ∼ and when they are present only in paucity (i.e. at higher B0 ). The purpose of the extensive investigation of this paper is to 2.2. Threshold Pair Production determine under what conditions such a demarcation is truly As discussed in the previous sections, pair production by pho- representative of a suppression of pair creation in pulsars; such tons initially emitted at small angles to the field occurs very an exploration appears in Section 3.2 below. The solutions of near threshold in magnetic fields exceeding 0.1Bcr . In this case, this equality define curves in the (B0, Θ) plane, and therefore there are only a small number of kinematically available electron specify functional dependences of the dipole spin-down mag- and positron Landau states. In very high fields, the accessible netic field B0 and the escape energy (for either process) on the number of excited pair states diminishes, and the pairs created colatitude Θ of emission. These functions are displayed in Fig- by most photons in the primary particle spectrum will occupy ure 2 for different values of the initial angle θkB,0 of the photons the ground state (for photon polarization ) or the first excited relative to the field. These solutions define the basis for our con- state (for photon polarization ). Pairs ink the ground state can- siderations of the suppression of pair creation in the context of not produce synchrotron photons.⊥ Even pairs in the first excited radio pulsars in Section 3 below. 2 state radiate photons of energy ω 1+2B/Bcr + p 1for To understand the behaviour of the curves, first consider the ' z − electron momentum pz parallel to the field, which will gener- θkB,0 = 0 case. The escape energies plotted in Fig. 1 have the ate very few pairs (Harding & Daughertyp 1983), and certainly α β approximate dependence ε B− Θ ,with α 3/4(for esc 0 − none if p2 < 4(1 + 1+2B/B ) for photons of polariza- 4 )and =6∝5 for splitting, and ≈ =1 for z cr Bcr Bcr α Although pairs in the ground state could be excited to higher solutions to Eq. (4) must∼ yield escape energies approximately≈ 1 Landau levels through inverse-Compton scattering of soft X-ray proportional to Θ− . This can be immediately folded into the photons (e.g. Zhang & Harding 2000), we leave discussion of 3/4 6/5 photon splitting proportionality εesc B0− Θ− to yield the potential importance of this effect to the end of this section. ∝4/15 the magnetic field dependence B0 Θ− in Figure 2, an ap- First, we examine the effect of threshold pair production. ∝ proximation that is applicable for Bcr 0.1B ), some photons in the ceeded) as a function of sin . At higher photon energies, 0 cr ω θkB spectrum create pairs in the ground state (0,1) or (1,0) for N (ω sin θ ,B) is computed directly from equation (6) for states kB polarization. By B =0.5B , virtually the entire spectrum of⊥ the particular values of sin that are sampled at the point 0 cr ω θkB photons will create pairs in state (0,1) or (1,0). The “noisiness” of pair production. at the minimum of the B0 =0.1 curve in Figure 3 is a real dis- creteness effect, due to the fact that the field at the pair creation point is increasing with photon energy and the ordering of the (j,k) states depends on the local field strength. From these re- 4 sults, we can conclude that there will be a decrease in pair yield 10 in pulsar polar cap cascades at magnetic fields B0 > 0.1Bcr . Bo/Bcr = .05 For B0 > 0.2Bcr , there will be strong suppression of cascade 3 0.1 pair yields. Note that this condition relates to surface polar 10 fields. The site for pair creation is generally proximate to the pole only for photon energies exceeding around 1 GeV in Fig- 0.15 ure 3; otherwise the local field drops below B0 as the escape 2 10 energy is approached. As mentioned above, inverse-Compton scattering of pairs in 0.2 the ground state could excite them to higher Landau states, 1 10 thereby increasing the number of synchrotron photons radiated and subsequently the pair yields. In the case of highly rela- Number of Pair States Pair of Number 0.5 tivistic particles moving along the magnetic field, the incident 0 photons have angles to the field near zero in the particle rest 10 1 2 3 4 5 6 frame. For such photon incidence angles, the scattering cross 10 10 10 10 10 10 section in a magnetic field is strongly suppressed below the cy- 2 Photon Energy (mec ) clotron resonance, ωB = B/Bcr . Above the resonance, the par- ticle may be excited to a higher Landau state. Although the scattering cross section does not have resonances at photon en- Fig. 3.— The number of kinematically-accessible electron-positron pair ergies above the cyclotron fundamental for incidence along the states Nstates , at the point of pair creation, of a photon having polarization field, excitation through continuum scattering at these energies state , emitted from the neutron star surface, parallel to the magnetic ⊥ is still possible, and becomes more probable with increasing field field at colatitude Θ = 5◦ . Nstates is displayed as a function of the photon energy in the observer’s frame at infinity. The curves are labeled with val- strength (Daugherty & Harding 1986; Gonthier et al. 2000). If ues of the dipole spin-down magnetic field strength B0 . Single-state pair the incident soft photons have a quasi-isotropic blackbody spec- creation becomes rife when B0 > 0.15Bcr . The vertical asymptotes to the trum at temperature T and their average energy is ωs =3kT , left mark the respective escape∼ energies. then scattering above the resonance in the particle rest frame Figure 3 shows the number of available pair states at the 2 3 1 will occur when γ > (B/Bcr) mec /3kT =2 10 (B/Bcr) T6− , point of pair production as a function of the lab-frame energy a condition applicable∼ for all fields provided× that the particles of a photon emitted parallel to the magnetic field at colatitude 8 are fairly relativistic. The mean-free path of these particles to approach and exceed the critical field. To more realistically mea- scattering will then be sure pair suppression in high magnetic fields, one must examine not only the propagation of single high-energy photons through 1 4 1 3 the neutron star magnetosphere (i.e., the escape energies) but λs =7.5 10 η− T6− cm, (7) ' ησT nγ × also the transport of the higher generations of photons and par- 6 ticles, which are produced by the primary photons that do not where T6 T/10 Kand η is a suppression factor for the ≡ escape. Even though the first generation of photons split in- scattering cross section σ = ησT in the Klein-Nishina regime. stead of producing pairs, the second generation of photons may Such mean free paths are short enough to suggest that exci- create pairs. It is therefore necessary to investigate pair yields tation through scattering is potentially important. However, of the pair/photon cascades which are initiated by the primary these photons scattering above the resonance in the electron rest photons. In Section 3.2, we will present results of numerical frame will not form the bulk of the produced pairs, because the simulations of such pair cascades. In this section, we will qual- resonant cross section (and thus the scattering rate) is several itatively discuss how we expect the nature of these cascades to orders of magnitude larger than for non-resonant cross section. change as the magnetic field increases. 2.3. Positronium Formation Another mode of pair creation exists, namely the formation of pairs in a bound state, i.e. positronium. This has been pro- /2:),(/'&$6&$'(6 +,*+),(/'&$6&$'(6 posed as an effective competitor to the production of free pairs B < 0.1 B B > 0.1 B (Shabad & Usov 1985, 1986; Herold, Ruder & Wunner 1985; cr cr Usov & Shabad 1985; Usov & Melrose 1995) because the bind- ing energy lowers the threshold slightly ( 1 %) below the value 3 Splitting Modes  ⊥ −> II II ⊥ −> II II for production of free pairs. Positronium formation fundamen- No Splitting Splitting Only ⊥ −> ⊥ ⊥ tally alters the magnetosphere: it amounts to the suppression of II −> ⊥ II electron or positron currents that can screen the induced electric fields in the rotator. Furthermore, depending on how long the 0 1 0 ⊥ pairs remain bound in their passage along open field lines to the 0 1 light cylinder, such a presence of neutral positronium will help 0 ⊥ 000 0 to suppress any collective plasma modes that might spawn radio 0001 00II II emission. Hence positronium formation is a viable alternative II, ⊥ II ⊥ II ⊥ II ⊥ means of inhibiting the radio signal from pulsars. The relevance of positronium formation to pulsars is poten- tially great if the bound state is stable for considerable times. Fig. 4.— The nature of pair cascades in increasing magnetic fields. Positronium is subject to destruction by three main mecha- Wavey lines represent photons and straight lines represent electrons and nisms: free decay, electric field ionization, and photo-ionization. positrons. Polarization modes of photons ( - parallel or - perpendicu- Using the spontaneous two-photon decay rate (in the positron- lar) are noted where relevant. k ⊥ 14 1 ium rest frame) of 3.5 10 (B/Bcr)sec− computed by Wun- Figure 4 illustrates the behavior of low and high field pair cas- ner & Herold (1979), Bhatia,× Chopra & Panchapakesan (1987) cades. Pair cascades in magnetic fields well below Bcr have been obtained positronium decay lengths considerably shorter (by extensively studied in connection with models of pulsar high en- 3 two orders of magnitude for bulk Lorentz factor γp =10) ergy emission (e.g. Daugherty & Harding 1982, 1996). In fields than the stellar radius for 0 2 ; the length is a strongly B < 0.1Bcr ,both and mode photons above escape energy B . Bcr ⊥ k declining function of B . Hence∼ it appears that positronium is will∼ produce pairs well above threshold, so that the resulting unlikely to be long-lived in the magnetosphere except for a nar- electrons and positrons will initially occupy high Landau states. row range of fields, or for sufficiently high positronium Lorentz Each pair member will make multiple Landau state transitions factors, i.e. from the attenuation of photons above 10 GeV. Ap- before reaching the ground state, radiating many synchrotron proximate estimates of when field ionization becomes important photons. Many of the synchrotron photons will pair produce, were obtained by Herold, Ruder & Wunner (1985) and Usov & initiating more synchrotron emission sequences. Such a process 2/3 can go on for a number of generations, provided that the energy Melrose (1996): for pulsar periods P < 0.5(10B0/Bcr) sec, ionization due to E should convert positronium∼ into free pairs. of the primary photon is high enough, resulting in the produc- This corresponds tok a sizeable portion of the phase space of in- tion of many pairs for each primary photon. terest for the considerations of this paper. In addition, photo- When B > 0.1Bcr , the primary photons create pairs at pair ionization can destroy positronium, forming free pairs. Herold, threshold, as∼ discussed in Section 2.2. This has a profound effect on the pair cascades, since the mode photons produce pairs Ruder & Wunner (1985) claimed that this was highly likely due k to collisions with thermal radiation emanating from the stel- with both members in the ground state, cutting off all further 5 generations. Photons of mode, which have a higher thresh- lar surface for temperatures greater than around 10 K. Usov ⊥ & Melrose (1995) used the calculations of Bhatia, Chopra & old, produce pairs with one member in the first excited state. Panchapakesan (1992) to argue that photo-ionization timescales The single Landau state transition made by this particle will 5 result in primarily mode photons which may create a sec- exceed dynamical ones for surface temperatures 10 K. This ⊥ discrepancy is yet to be resolved. ∼ ond generation pair with one member in the first excited state. Higher generations of pairs are therefore possible, depending on 2.4. Pair Cascades in High Magnetic Fields the energy of the primary photon, but the total pair yield is severely diminished because there is no multiplication of pairs We have discussed several of the physical mechanisms capa- with each generation, as in the low-field cascades. ble of suppressing pair production when the local magnetic fields 9

At fields B > 0.5Bcr , the photon splitting attenuation 3.1. Radio Quiescence Boundaries in the P - P˙ Diagram lengths become shorter than pair attenuation lengths. At this ∼ In Section 2.1.3, the condition for the equality of photon split- point, the nature of the pair cascades will depend on which ting and pair creation escape energies established a region in modes of splitting are operating at what field strengths. In (B0, Θ) space where pair suppression would probably occur. fields B

11/15 13 P − P˙ 7.9 10− , (10) ≈ × 1sec   the slope of which depends on the how strongly the rate of pho- ton splitting depends on B in this regime of fields. It should also be noted that while such putative boundaries of radio quiescence have been labelled by specific values of θkB,0 for the purpose of illustration, in fact the value of this initial angle of photon propagation with respect to the field is a weak function of the colatitude. This is due to the coupling between the Lorentz fac- tor of accelerated electrons (and therefore θkB,0 ) and the polar cap size Θ , a connection discussed below in Section 4.2. Also depicted in Figure 5 is a fiducial positioning (adopted by Tay- lor, Manchester & Lyne 1993) of the conventional death line, with P˙ P 3 (i.e. corresponding fixed open field line voltage, adopt),∝ to the right of which there is just one detected radio pulsar (PSR J2144-3933). It is a well-known problem that the computed position of the pulsar “death line”, assuming pairs are produced only by curvature radiation photons, lies at smaller periods than the observed “death line”, for the assumption of the standard polar cap size (e.g. Arons & Scharlemann 1979). However, increasing the polar cap size to less than twice that of the standard brings the computed “death line” into agreement with virtually the entire radio pulsar population, the notable exception being the recently “re-discovered” 8.5 second Parkes pulsar (Young, Manchester & Johnston 1999) that is conspicu- Fig. 5.— The conventional depiction of pulsar phase space, the P - P˙ diagram, with filled circles denoting the locations of 541 members (those ous on the right hand side of Fig. 5. However, pair production by with P>˙ 0 ) of the latest edition (Taylor, Manchester & Lyne 1995; see inverse Compton-scattered photons can explain radio emission also http://pulsar.princeton.edu/) of the Princeton Pulsar Catalogue, from this pulsar (Zhang et al. 2000). and open circles marking 122 pulsars in the recent Parkes Multi-Beam sur- The Princeton catalog by itself clearly rules out boundaries vey [http://www.atnf.csiro.au/~pulsar/psr/pmsurv/pmwww/]. Three of of radio quiescence due to photon splitting that correspond the handful of gamma-ray pulsars, namely the Crab, Vela and PSR 1509- 4 to θkB 0 > 3 10− for surface emission. The fact that the 58 are highlighted, with point styles as indicated in the inset. The dotted , × diagonal lines denote constant field strength, as inferred from Eq. (9). A θkB,0 =0,∼ R0 = R boundary was comfortably located above fiducial positioning of the conventional death line, from P - P˙ diagrams the entire Princeton collection of radio pulsars was an attractive at http://pulsar.princeton.edu/, is depicted as the dashed line on the feature that underpinned the suggestion of Baring & Harding right. The heavy solid curves give a variety of choices for the boundary of (1998) that this particular case marked the approximate loca- radio quiescence for surface emission, depending on the value of θkB,0 ,the initial angle of photons relative to the field. In addition, the heavy dashed tion of the boundary of radio quiescence, where photon split- curve is a similar boundary for θkB,0 = 0 and emission point half a stellar ting could strongly inhibit pair creation. However, this conclu- radius above the surface (i.e. R0 =1.5R ). Each of these curves, defined sion has been challenged by the dramatic increase in the ob- by solutions to Eqs. (4)–(9), purportedly partitions the P - P˙ diagram into regions of radio loud (below the curve) and radio quiet (above) pulsars, served population due to the exciting new Parkes Multi-Beam the latter being where photon splitting suppresses pair creation for the survey (Camilo et al. 2000; D’Amico et al. 2000; Kaspi et 4 ⊥ polarization state. Values θkB,0 < 10− and r > 1.2R can comfortably al. 2000), with three new pulsars (PSRs J1119-6127, 1726-3530 accommodate the current radio pulsar∼ population.∼ and 1814-1744, all with high dispersion measures in the 700–850 ˙ By inverting Eq. (9) to solve for P in terms of B0 and P , range) having been discovered that are obviously at P˙ above the solutions B0(Θ) to Eq. (4) that demarcate the domain of the θkB,0 =0, R0 = R boundary. 11

Although the boundary of radio quiescence for surface emis- eter. In the results presented in this paper, we generally take sion (chosen for illustrative purposes by Baring & Harding 1998) α =1.6 as a spectral index representative of either primary cur- neatly falls between the population in the Princeton catalog and vature radiation with energy losses (giving α =5/3) or resonant the handful of known anomalous X-ray pulsars and soft gamma inverse Compton emission which is relatively hard. Polarization repeaters (see the blow-up of the P – P˙ diagram in Figure 9), is chosen randomly such that the average distribution has 75% most (or perhaps all) of which are radio quiet, clearly the new in the mode and 25% in the mode, reflecting the po- Parkes survey data questions this assumption of surface emis- larization⊥ produced by the radiationk mechanisms discussed at sion. The quiescence boundary rapidly moves up on the P - the beginning of Section 2. The path of each input photon is P˙ diagram for emission at higher altitudes, which might nat- traced through the magnetic field, in curved spacetime, accu- s urally be expected for traditional curvature radiation-initiated mulating the survival probabilities for splitting, Psurv ,andfor p cascades in pulsars with Crab-like surface fields (e.g. around R pair production, Psurv , independently: above the surface for Daugherty & Harding’s 1996 modeling of the Vela light curve). However, for the high B fields sampled 4 Psurv(s)=exp τ(Θ,ε,l) (12) by the θkB,0 < 10− curves, inverse Compton is probably more − relevant to the∼ acceleration region (see Sturner 1995; Harding & n o Muslimov 1998) and low altitude emission is much more prob- where τ(Θ,ε,l) is the optical depth as defined in Eqn. (3). able, as discussed in Section 4.2 below. One might then expect Each photon may split, produce pairs or escape, based on a that the mean radius of emission satisfies r < 1.5R . Accord- combination of the running survival probabilities for splitting and pair production (see HBG97). If a photon splits, the ener- ingly, we have computed a θkB,0 =0, R0 =1∼ .5R boundary and depicted it in Figure 5; it clearly lies above even the highly- gies and polarizations of the final photons are sampled from the magnetized Parkes Multi-Beam survey pulsars. Intuitively, it distribution from the branching ratios given in HBG97. Each might be anticipated that the location of the boundary as a final photon is then followed in the same way as the parent function of emission radius R0 should scale simply by the rela- photon. If a photon produces pairs, the total energy, Landau tionship between field strength and radius for a dipolar struc- state and parallel momentum of the electron and positron are 3 ture, so that B0 r . This appears to be borne out near the determined. Each member of the pair is assumed to have half stellar surface, however∝ the radial dependence of the location of the energy and the same direction of the parent photon (tech- the quiescence boundary weakens at altitudes r R (in spite nically this is only appropriate for high Landau states), except of the reduced effects of general relativity at r>R∼ ), princi- when the pair is produced at threshold (i.e. in the [0,0] state for polarization and [0,1] or [1,0] for polarization), in which pally because of changes in the flaring of field lines sampled by k ⊥ photons in their flight away from the neutron star. The present case we use the full kinematic equations to determine the en- Parkes survey data constrain computed quiescence boundaries ergy and momentum of each pair member. Each member of the to the r > 1.2R range; in the light of the present indetermi- pair occupying an excited state emits a sequence of cyclotron nacy in the∼ actual locale of the acceleration region (addressed or synchrotron photons. The method used to simulate the cy- in Section 4.2), this is not a serious problem. clotron/synchrotron emission is similar to that of Daugherty & Harding (1996). If the particle Landau level is larger than 20, 3.2. Simulations of Photon Splitting/Pair Cascades the high-energy limit of the quantum synchrotron transition rate (Sokolov & Ternov 1968) is used, in which case we assume that Using the equality of photon splitting and pair creation escape the photons are emitted perpendicular to the magnetic field in energies for polarization as the criterion for when splitting ⊥ the particle rest frame (high-energy limit). When the particle starts to strongly suppress the production of pairs is a simple Landau level is smaller than 20, the exact QED cyclotron tran- choice, motivated largely by the predominance of production of sition rate (Harding & Preece 1987) is used, in which case the photons in the relevant emission processes. It is necessary to ⊥ angles of the emitted photons are sampled from a distribution. determine whether and under what conditions this choice is an In both cases, the emitted photon polarizations are sampled appropriate one. from the corresponding polarization distributions. Each emit- To assess the applicability of the radio quiescence boundaries, ted photon is propagated through the magnetic field from its we have studied pair suppression in high magnetic fields by emission point until it splits, produces pairs or escapes. The means of Monte Carlo simulations of photon splitting/pair cas- cyclotron/synchrotron emission sequence continues until each cades in a neutron star magnetosphere. This calculation gener- particle reaches the ground state. The cascade continues until alizes that of HBG97 by including the cyclotron and synchrotron all photons from each branch have escaped. Throughout the radiation from the electron-positron pairs. This improvement is cascade, a running tally is kept of the number of pairs produced necessary in order to compute the pair yields from all gener- and of the number of photon splittings. General relativistic ef- ations of the cascade. Note that we omit resonant Compton fects of a Schwarzschild metric on the photon momentum and scattering from our simulations since, while it is important as dipole magnetic field, are included in the same way as described a radiation process for primary and secondary particles, it will in HBG97. not significantly impact the spatial transfer of photons in normal We have chosen the total number of pairs produced per in- radio pulsars due to the low densities and small angles; it may, jected photon as a quantitative measure of the pair yield of the however, be significant in soft gamma repeaters. We inject pho- cascade initiated by a particular parent photon spectrum. The tons parallel to the local magnetic fields at the neutron star sur- free parameters are then surface magnetic field strength, emis- face with specified magnetic colatitude Θ and four-momentum sion colatitude Θ , spectral index α , and minimum, ε and . The photon energies are sampled from a power-law distribu- min k maximum, ε , energy of the primary photon spectrum. Cas- tion, max α cade simulations have been run for the cases where one mode of N(ε)=N0ε− ,εmin <ε<εmax , (11) splitting operates (the mode), three modes of splitting ⊥→kk where we generally set εmin = 1 , and let εmax be a free param- operate (the , and modes permitted by ⊥→kk ⊥→⊥⊥ k→⊥k 12

QED), and where splitting is turned off completely (“no split- (see Figure 1), the pair yield should asymptote to a constant ting”). Figure 6 shows the cascade pair yield as a function of value at high field strengths. This constant is just the frac- magnetic field strength and maximum primary photon energy; tion of primary photons above pair threshold: when εmax 1, α 1  since the total number of photons and the pair yield depend on this fraction is approximately (sin Θ/2) − , which evaluates to the value of εmin ,wehold εmin = 1 constant for all calcula- 0.15 for a polar cap size of Θ = 5◦ and α =1.6 . Reducing ' tions. εmax clearly diminishes the proportion of photons above pair threshold. In the presence of photon splitting, above 0 5 ,the Θ 0 θ B0 . Bcr = 5 α = 1.6 kB,0 = 0 cascade is limited to only one pair generation in∼ each polariza- 1 10 tion mode, amounting to one pair per mode photon injected k 8 NO Splitting above the pair escape energy plus two pairs per mode photon εmax = 10 ⊥ 1 Splitting Mode injected above the splitting escape energy (see Figure 4). The re- sults clearly indicate that photon splitting in only one mode does 0 3 Splitting Modes 10 not significantly suppress the pair yield. As long as one photon polarization mode does not undergo splitting, there is always 6

γ 10 a channel for pair production. In fact, photon splitting in one mode can even increase the pair yield, as is evident in Figure 6 /N 4 -1 for field strengths between 0.2Bcr and 0.7Bcr ,for εmax 10 . 10 ≥

pair Consequently, the putative radio quiescence boundaries in Fig. 5 N 4 10 are not borders to domains of significant pair suppression if pho- tons of polarization cannot split. While splittings can competek effectively with pair creation, the two produced⊥→kkpho- -2 2 10 10 tons can only create pairs if splitting of photons is forbiddenk by kinematic selection rules. In such ak case, pair creation is prolific unless the second generation photons are around or + k e e− below the escape energy for pair production, εesck→ , i.e. the -3 + 10 e e− -1 0 1 original photon is of energy < 2εesck→ . Hence, in the en- 10 10 10 ⊥ + + e e− ∼e e− B0/Bcr ergy range εesck→ <ε< 2εesck→ , splittings truly do suppress pair creation; otherwise they⊥→kk merely postpone it Fig. 6.— ∼ ∼ The cascade pair yield (number of pairs per injected pho- a generation. By comparing the number of primary photons ton) as a function of surface magnetic field strength, B0 , in units of the + e e− critical field, Bcr , and maximum primary photon energy, εmax , in units in this range with that above εesck→ , it is quickly estimated 2 of mec . Different line types refer to the cases where no splitting, only that the pair yield is enhanced by a factor of the order of 22 α one mode ( ) or three modes of splitting are allowed. The curves for − ⊥→kk 4 6 the “no splitting” case behave similarly for εmax =10 , εmax =10 and when splitting is switched on, i.e. generally a minimal change. 8 2 6 εmax =10 cases, so only the curves for the εmax =10 and εmax =10 This is precisely the behaviour observed in Figure 6 for all but cases are shown for clarity. Switching on only one mode of splitting leads the =102 case. Effectively, the doubling of the number of 4 εmax to only minimal reductions in the pair yield for εmax > 10 , as explained photons by splittings compensates almost exactly for the sup- in the text. Photons are assumed to be emitted from the∼ stellar surface. + e e− k→ The curves for the “no splitting” case, shown only for εmax = pression of pair creation below 2εesc . Therefore, when only 102 and =106 to avoid confusion, are of course not physi- mode photons can split, pair creation is inhibited only when εmax + ⊥ e e− cal but are included to illustrate the effect of one splitting mode εmax < 2εesck→ . on the pair yield. When εmax is below the pair escape energy The∼ situation changes dramatically if three modes of photon 2 4 (cf. Section 2.1.3), as it is for the εmax =10 and εmax =10 splitting (those allowed by QED) are operating. As shown in 2 4 curves at lower field strengths, the pair yield is considerably Figure 6 for the case of εmax =10 and εmax =10 (the other lower than when εmax is well above pair escape energy (as is cases not shown behave similarly to this case), the pair yield 6 8 the case for the εmax =10 and εmax =10 curves), for which drops dramatically above B 0.4Bcr (depending on Θ ) when pair yields at the lower field strengths can be quite large due photons of both polarization∼ modes are allowed to split. Pure to the pair multiplication effect through successive generations, photon splitting cascades can now take place, and typically ex- asdescribedinSection2.4.Aboveafieldstrengthof0.1Bcr , ceed ten photon generations. In fields above Bcr very few pairs 4 in εmax > 10 cases the pair yields drop because the pairs are are created because the photon splitting cascade degrades the produced∼ in low Landau states at or near threshold, suppressing photon energies below the pair escape energy before the cascade synchrotron emission and thus the pair multiplication effect. In reaches a height where pair production can take over. Figure 7 the absence of photon splitting, the pair yield increases slowly shows the effect of varying the primary photon injection co- with increasing B0 > 0.1Bcr , because the number of generations latitude on the cascade pair yield, as a function of magnetic 2 of the cascade can increase. The rapid rise of the εmax =10 field strength for cascades where three modes of splitting are case when B0 < 0.2Bcr is due primarily to the escape energy for allowed. The field strength at which the pair yield begins to + ∼ e e− drop increases with decreasing colatitude, due to the increas- pair production εesc⊥→ being close to εmax so that the opti- cal depth is of the order of unity or less; the pair yield must then ing field line radius of curvature. This effect was seen in the 4/15 radio quiescence boundary, where B0 Θ− was computed necessarily be a strong function of B0 by virtue of the expo- ∝ nential asymptotic form (see Daugherty & Harding 1983) for the in Section 3.1 by equating pair and splitting escape energies. A pair creation rate. Since the pair escape energy saturates when similar boundary may be predicted by equating the polarization- averaged escape energies for pair creation and photon splitting, B0 > 0.5Bcr at approximately the threshold value of 2/ sin Θ ∼ 13 with the result being a virtually identical dependence of B0 This feature is the radio quiescence line due to photon splitting on Θ , but with a constant of proportionality that is a factor as described in Section 3.1 The period dependence of this line of about 1.7 lower than that in Eq. [10]. Such a polarization- is consistent with that inferred from the escape energy calcula- 4/15 2/15 averaged boundary of radio quiescence reproduces the ( B0 ,Θ) tion, i.e. B0 Θ− P − . The radio “death line” is phase space corresponding to the precipitous declines observed also apparent as∝ a decline∝ in pair yield at large periods. in Figure 7. Hence it can be concluded that the simplified cri- terion for pair suppression by splitting is fairly accurate if all three QED-permitted modes of splitting are operating. 4. DISCUSSION 4.1. Radio quiescence? The results of the previous section show that if three modes 6 α = 1.6 θkB,0 = 0 εmax = 10 of splitting operate at high field strengths, then there is nearly 0 10 complete pair suppression at high field strengths and the approx- imate condition for radio quiescence proposed in Section 3.1 is 0 3 Splitting Modes Θ = 8 valid. If only one mode of splitting (i.e. ) operates, then some other effect must act to reduce the⊥→kk pair yield, otherwise, pair creation in cascades remains prolific at near-critical and supercritical fields, and there is no reason to expect a regime 0 5 ˙

γ of radio quiescence in the upper portion of the P – P diagram. An answer to the question of how many modes of photon split- -1 0 /N 10 3 ting operate in highly dispersive magnetic fields is not presently

available and will require detailed investigation into the behav- pair 0 ior of the photon splitting rate in this regime. N 1 We have discussed several ways in which a suppression of pair creation at high B can be present if only one splitting mode op- erates. First, if the effective maximum energy εmax of primary + 0 e e− 0.3 photons satisfies εmax < 2ε⊥→ , then the pair yield must -2 esc 10 be dramatically reduced∼ whenever such photons are mostly of polarization. However, there is no reason a priori to believe ⊥ that such low values of εmax should arise when B0 >Bcr .For -1 0 10 10 curvature radiation, the spectrum is only dependent∼ on B0 via acceleration properties. For resonant Compton scattering, the B0/Bcr spectral dependence on B0 is more complicated and needs to be Fig. 7.— The cascade pair yield (number of pairs per injected photon) the subject of future investigation. While the maximum electron as a function of surface magnetic field strength, B0 , in units of the crit- energy declines somewhat with B0 (e.g. Harding & Muslimov ical field, Bcr , for different primary photon injection colatitudes, Θ , for 1998; see the discussion just below), the primary photon spec- the case where three photon splitting modes operate. Pair suppression be- trum is flat up to B2/ε (for soft target photon energy ε and comes substantial in near-critical and supercritical fields when pure photon ∼ s s splitting cascades operate. Fluctuation in the curves at the highest field drops steeply thereafter: Dermer 1990; Baring 1994). Hence, strengths is numerical noise due to low counts. Again, surface emission of properties of the primary spectrum are unlikely to uniformly photons is assumed. produce suppression of pair creation at high field strengths. As a more complete representation of the relevant phase A second possibility that is effective at diminishing, but not space, contour plots of the pair yield as a function of magnetic completely suppressing, the number of pairs when εmax + field and pulsar period are shown in Figure 8 for the cases of e e−  εesc⊥→ is ground state pair creation. It inhibits successive one and three splitting modes. The period is derived from the generations of pair cascading and becomes significant in local surface injection colatitude assuming emission at the rim of a fields B > 2 1012 Gauss. This value is below the spin-down polar cap of standard size Θ as given in Eq. (8). In the left fields of∼ the× most magnetic radio pulsars, so that evidently panel of Figure 8, which shows the case for one splitting mode, ground state pair creation cannot be primarily responsible for the most dramatic feature is the sharp fall-off of pair yield at imposing radio quiescence. Yet this physical effect could be a large pulsar periods. This effect is purely a consequence of the contributory factor, noting that the extant pulsar population polar magnetic field geometry: at large periods (small injection is somewhat thinner in the P – P˙ diagram than might be ex- colatitudes) the radius of curvature of the field lines is larger pected at B > 1013 Gauss. A related possibility (as discussed and the pair production attenuation length can exceed a stellar 0 in Section 2.3)∼ is that positronium formation may inhibit radio radius, suppressing pair cascades. This defines the “death line” emission without reductions in pair creation. But even if sta- for radio pulsars. There is a weak dependence of the “death ble positronium formation does occur in pulsar magnetospheres, line” on magnetic field strength for B > 0.1B because the cr it cannot account for observed radio quiescence in pulsars be- pair attenuation length saturates due to threshold pair produc- ∼ cause the predicted onset of positronium formation occurs at tion. With only one splitting mode in operation, the pair yield field strengths 0.1B , in the middle of the radio pulsar pop- has a relatively weak variation with magnetic field. However, cr ulation. Indeed,∼ Usov & Melrose (1996) invoke bound-state pair there is a maximum in pair yield at fields B < 0.1B and short cr creation in their model for gamma-ray emitting radio pulsars. periods, because pairs are produced in high Landau states. ∼ Finally, we have argued that suppression of pair creation by In the right panel of Figure 8, which shows the case where the conversion of polarization photons that are the product of three splitting modes are operating, the most dramatic feature is splittings backk into the state by the resonant Comp- a sharp fall-off of pair yield at magnetic fields around B Bcr . ⊥→kk ⊥ ∼ ton scattering process, before they can pair produce, will not 14

Number of Pairs/Photon Number of Pairs/Photon 10.0 10.0 ε 6 ε 6 1 Splitting Modes max = 10 3 Splitting Modes max = 10

-1.11 -0.89 -0.96

-1.95

-1.04

-1.18 1.0 cr cr -1.11 1.0 -1.46 -1.60

-1.32 -1.39

-1.25

/ B / B / 0 0 -1.32

-1.46 -1.74 -1.46 -1.53

-1.39 -1.25 B B -1.67 -1.18

-1.81

-1.88

-1.81

0.1 -1.88 0.1

-2 -1 0 1 10-2 10-1 100 101 10 10 10 10 P (s) P (s)

Fig. 8.— Contour plots of cascade pair yield (number of pairs per injected photon) as a function of surface magnetic field strength, B0 , in units of the critical field, Bcr , and pulsar period for the cases where (left panel) only one mode of photon splitting is allowed, and (right panel) where three photon splitting modes are permitted. Contour levels are equally spaced logarithmic (base ten) intervals. be effective in normal pulsar magnetospheres where the particle less there are many fewer pulsars born with high field strengths, densities are low. the smaller average age of high field pulsars to the left of the While these theoretical issues leave the question of radio qui- conventional death lines cannot account for the dearth of radio escence quite open, several observational issues are pertinent to pulsars at these fields. Concomitantly, the expected clustering this discussion. These relate to whether or not one expects to of pulsars near the death line at subcritical fields is not realized; observe radio pulsars with high P˙ , which divides into issues the beaming of radio emission is obviously a contributor to this of intrinsic population densities in the P – P˙ diagram and ob- effect. The reduction of the polar cap size with period should 1 servational selection effects. There are several obvious natural reduce the solid angle ( P − ) of the cone of emission to more ∝ biases influencing pulsar observability. First, due to the expec- or less compensate the age clustering effect. Such a property 2 4 was invoked by Young, Manchester & Johnston (1999) to ar- tation that pulsars with higher spin-down power (i.e. B0 /P ) are probably more luminous, a property observed in both the gue that the 8.5 second Parkes pulsar PSR J2144-3933 was not X-ray (Becker & Tr¨umper, 1997) and gamma-ray (e.g. Zhang isolated in its existence, but rather representative of a large, un- & Harding 2000) bands, one might anticipate that high-field ra- seen population that challenges conventional theories explaining dio pulsars are more luminous than their less-magnetized coun- the death line. Yet the observed thinning of the population near terparts. There is evidence that this trend is present in the this boundary argues that another factor is influential in deter- observed population: when luminosity distributions for Prince- mining the period distribution; a reduction in luminosity due to ton Pulsar Catalog members are binned in ranges of B0 , radio the onset of pair quenching may provide a partial explanation luminosities Lrad increase with spin-down field up to around of this dilution. Notwithstanding, the beaming phenomenon 2 1013 Gauss, beyond which the trend appears to reverse and impacts the P distribution and not P˙ phase space. × Lrad perhaps begins to decline with B0 . While this reversal There are two main observational selection effects, the first is suggestive of an onset of radio-quiescence at high fields, it is being an intrinsically greater sensitivity at longer periods and presently statistically insignificant. The addition of the Parkes lower dispersion measures, basically due to interstellar disper- Multi-Beam population (for which distances have not yet been sion and scintillation effects, radiometer noise, and pulse shape established from the dispersion measures) will improve statis- and Fourier analysis properties (e.g. see Cordes and Chernoff tics, but perhaps still leave the evidence for a luminosity decline 1997 for a discussion of sensitivities). The second true selec- inconclusive. Uncertainty in source distances pervades estimates tion effect, less evident in the literature, is that pulsar surveys of pulsar luminosity, and since there is a correlation between tend to filter out baseline fluctuations on long (i.e. > 5 second) distance out of the galactic plane and pulsar age (and therefore timescales, thereby selecting against supersecond period∼ pulsars period), we caution against over-interpretation of Lrad trends (Cordes, private communication). This bias is partly driven by for high B0 . past needs to focus on the appropriate range of periods of con- The next factor is the age distribution in the P – P˙ diagram. ventional pulsars, and will be counterbalanced by the increasing Clearly, evolution (with or without field decay) guarantees a scientific interest in magnetars, discussed in Section 4.3 below. denser population at long periods, and the speed at which the In summation, in the light of all these observational consider- period evolves is a rapidly increasing function of B0 . The popu- ations, without performing an extensive population statistics lation density would be constant along diagonal lines of constant analysis, it is fairly safe to argue that radio pulsars with P˙ 11 age P/(2P˙ ) for a uniform birth rates vs. field distribution. Un- in excess of 10− are either rare or non-existent. The confir- 15 mation of a possible radio pulsar counterpart to SGR 1900+14 proximity in the P – P˙ diagram. (discussed in Section 4.3), or otherwise, shall play a prominent role in resolving this issue. 4.3. Magnetars 4.2. Particle Acceleration Locales The importance of a boundary for radio quiescence to the Since the strength of the local magnetic field is one of the study of the radio pulsar population is obvious. Yet the absence key influences on the location of the high-field death line, the of radio pulsars in the high- B0 region of phase space has become site of the high energy emission in the pulsar magnetosphere is of even greater importance with the mounting observational ev- critically important. The initial angle at which the high-energy idence at X-ray and γ-ray energies for the existence of neutron photons are emitted relative to the local magnetic field has also stars with such large fields, which a priori are not discriminated been shown in the previous subsection to be of critical impor- against (for a given age) by known radio selection effects (as tance (see Figure 5). Both of these factors are determined by the discussed in Section 4.1). This evidence includes the detection nature of the particle acceleration above the polar cap. Several of spin-down in the growing number of anomalous X-ray pul- types of models have studied pulsar acceleration due to charge sars (AXPs). These sources have been known to exist for over deficits at different locations in the magnetosphere. Polar cap a decade (e.g. see the biographical summary of Mereghetti & models consider the formation of a parallel electric field in the Stella 1995), though the identification of them as being anoma- open field region near the magnetic poles, while outer gap mod- lous was forged slowly with the accumulation of observational els consider acceleration in the outer magnetosphere, near the data. The designation “anomalous,” coined by van Paradijs, null charge surface (see Mestel 1998 for the most recent and Taam & van den Heuvel (1995), was founded in their periods comprehensive review of pulsar electrodynamics). (6–12 seconds) being relatively short compared with typical ac- Polar cap models for pulsar high-energy emission are all based creting X-ray binary systems, combined with their unusually on the idea, dating from the earliest pulsar models of Sturrock steep X-ray spectra and monotonic increases in periods: short ˙ (1971) and Ruderman & Sutherland (1975; hereafter RS75), of period binary X-ray pulsars usually exhibit P<0 , i.e. spin-up. particle acceleration and radiation near the neutron star surface A list of AXPs with measured P and P˙ is given in Table 1, at the magnetic poles. Within this broad class, there is a large an adaptation of that in Gotthelf & Vasisht (1998). Further- variation, with the primary division being whether or not there more, the association of two of these pulsars (1E 1841-045 and is free emission of particles from the neutron star surface. This 1E 2259+586) with supernova remnants (which are typically question hinges on whether the surface temperature T of the much younger than X-ray binaries) has shifted the focus from neutron star (many of which have now been measured in the accretion torques, as championed by Mereghetti & Stella (1995) range T 105 106 K; Becker & Tr¨umper 1997) exceeds the and van Paradijs, Taam & van den Heuvel (1995), to electro- ∼ − ion, Ti and electron, Te , thermal emission temperatures. If magnetic dipole torques (e.g. Vasisht & Gotthelf 1997) as the TTe ,free born a slow rotator. emission of particles of either sign of charge will occur. The flow The electromagnetic dipole interpretation for the spin-down of particles is then limited only by space charge (SCLF case), immediately implies immense supercritical ( B0 > 4.41 13 × and an accelerating potential will develop (Arons & Scharle- 10 Gauss) fields in these sources: inferred values are given in mann 1979; Muslimov & Tsygan 1992) due to an inability of Table 1. The motivation for such a perspective has been dramat- the particle flow all along each open field line to supply the ically enhanced by the recent detection of similar periods and corotation charge (required to short out E ). In space charge- period derivatives of soft gamma repeaters (SGRs), the neutron limited flow models, the accelerating E isk screened at a height star sources with transient outbursts of soft gamma-ray emis- where the particles radiate γ -rays thatk produce pairs. This so- sion possessing quasi-thermal spectra. Duncan & Thompson called pair formation front (PFF) (e.g. Arons 1983, Harding & (1992) postulated that neutron stars with supercritical magne- Muslimov 1998) can occur at high altitudes above the polar cap. tizations, magnetars, were responsible for SGR activity based Zhang & Harding (2000b) propose that, if both photon po- on the observed 8 second periodicity (Mazets et al. 1981) of the larization modes can undergo splitting at high fields, the radio 5th March 1979 outburst from SGR 0525-66, combined with it’s quiescence line will be much lower for pulsars with vacuum gaps strong directional association (Helfand & Long 1979; Cline et al. (“anti-pulsars” in the language of RS75), than for pulsars with 1982) with the supernova remnant N49 in the Large Magellanic SCLF gaps (“pulsars”). In the case of vacuum gaps the high Cloud. No spin-down P˙ was measurable for this source since energy radiation occurs near the neutron star surface, where the data that unambiguously exhibited periodicity spanned a the local fields are high, whereas in the case of SCLF gaps, the single range of around two minutes. Hence the surface field of 14 high energy emission (and subsequent pair creation) may occur B0 6 10 Gauss inferred by Duncan & Thompson (1992) at high altitudes, especially when photon splitting prevents pair for SGR∼ × 0525-66 was purely circumstantial. Nevertheless, their creation and thus a PFF near the surface. The radio quiescence hypothesis was dramatically bolstered by the detection of a su- line for “anti-pulsars” will be the photon splitting death line for persecond periodicity coupled with a high rate of spin-down in surface emission, while the radio quiescence line for “pulsars” the quiescent counterparts of SGR 1806-20 (Kouveliotou, et al. 14 will be located around B0 =2 10 G (for a surface temper- 1998) and of SGR1900+14 (Kouveliotou et al. et al. 1999); ature of 106K), above which thermionic× emission of electrons parameters for these sources are also listed in Table 1. A giant from the surface is no longer possible and “pulsars” must have burst, very similar to that observed from SGR 0525-66, was seen vacuum gaps. This could account for the observed existence of from SGR1900+14, exhibiting the same 5 s period as was de- radio-loud pulsars such as PSR J1814 and radio-quiet pulsars tected in the quiescent emission (Hurley et al. 1999), strength- such as AXP 1E 2259 at the same field strength and in close ening the magnetar identification for SGRs. While the similar-

16

TABLE 1

a

Spin-Down Parameters for Anomalous X-ray Pulsars and Soft Gamma Repeaters

b

_ _

P P  = P=2P B

1

Pulsar SNR Ref. sec sec sec  yrs Gauss

11 3 15

SGR 1806-20 G10.0-0.3 c 7.47 8:3  10 1:4  10 1:6  10

10 2 15

SGR 1900+14 d 5.16 1:1  10 7:8  10 1:5  10

11 3 15

1E 1841-045 Kes 73 e 11.77 4:7  10 4:0  10 1:5  10

11 3 14

1E 1048-5937 f 6.45 2:2  10 4:6  10 7:6  10

12 4 14

4U 0142+615 g 8.69 2:3  10 6:0  10 2:9  10

13 5 14

1E 2259+586 CTB 109 g 6.98 7:3  10 1:5  10 1:4  10

11 3 15

1RXS J170849.0-400910 h 11 2:3  10 8:0  10 1:1  10

_

NOTE.| a Only sources with measured P are included in the Table. b The magnetic eld esti-

19 1=2

_

mates are obtained using the choice for the dip ole spin-down energy loss rate B =6:4  10 P P 

Gauss adopted by Usov & Melrose 1995, derived in, for example, Shapiro & Teukolsky 1983. Ref-

erences for observed p erio ds and p erio d derivatives are c Kouveliotou et al. 1998, d Hurley et al.

_

1999 for the p erio d and Kouveliotou et al. 1999 for P for SGR 1900+14, e Vasisht and Gotthelf

1997, f  Oosterbro ek et al. 1998; here the estimate obtained from their Table 3 using the maximum time baseline is used, g Mereghetti & Stella 1995, h Israel et al. 1999. ity of P and P˙ and the existence of remnant identifications for Harding (2000b, see Section 4.2), while AXPs are radio-quiet both AXPs and SGRs may provide moderately compelling moti- “anti-pulsars”. vation for classifying the two types of sources together, given the Furthermore, the actual magnetic fields of the AXP and SGR vastly different emission properties between AXPs and SGRs, sources are not likely to be those derived from pure dipole spin- their commonality may be confined just to magnetars being the down. If rapid field decay is powering the luminous emission sites for their activity. from these sources (Thompson & Duncan 1996), then it is likely A property mutual to anomalous X-ray pulsars and soft that transient higher multipoles exist near the surface. Since gamma repeaters is that they are radio quiet pulsars (if one higher multipole radiation contributes relatively less to the spin- excepts the unconfirmed detection of SGR 1900+14 by Shitov down torque, multipole fields significantly stronger than those 1999), an intriguing fact given their high magnetic fields and derived assuming a pure dipole could be present at the sur- the focus of this paper. The spin-down parameters for these face without affecting the spin-down. This would effectively sources (see Table 1), are used to derive their positions on the lower the radio quiescence boundary for the AXPs, both due to P – P˙ diagram in Figure 9. The AXPs and SGRs all lie in the increase in field strength and decrease in radius of curva- the extreme upper right of the P – P˙ phase space, above our ture. Measurement of braking indices larger than 3 (the dipole value) in these sources may reveal the presence of higher mul- derived θkB radio quiescence line. However, one of the radio pul- sars recently discovered in the Parkes Multi-beam Survey, PSR tipoles. In fact, a marginally significant detection ofν ¨ in the J1814-1744, lies very close to the AXP CTB 109, such that AXP 1RXS J170849.0-400910 (Kaspi, Chakrabarty & Stein- a single radio quiescence line cannot separate the radio pulsar berger 1999) would give a braking index in excess of 100 ! Field and AXP populations. This proximity coupled with the fact decay alone would not change estimates of the surface field, but that PSR J1814-1744 has not been detected in X-rays (Pivo- would decrease the age of the pulsar relative to the dipole value varoff, Kaspi & Camilo 2000) strongly suggests that a quantity (Colpi, Geppart & Page 1999). On the other hand, the spin-down of some magnetars could other than P and B0 has a profound influence on the prop- erties of highly-magnetized radio pulsars and AXPs. Note that be influenced by powerful particle winds, in this case lowering there are several effects which could cause a fuzziness in both the estimated surface field strength and increasing the charac- the radio quiescence boundary and the positions of the pulsars teristic age (Harding, Contopoulos & Kazanas 1999, Thompson in the P – P˙ diagram. As mentioned in Section 3.1, the effec- et al. 1999) because the wind increases the spin-down rate by tive quiescence boundary for a particular source depends on the distorting the dipole field inside the light cylinder. This is more altitude of emission. The quiescence boundary applicable for likely to be a factor in SGRs, where there is evidence for par- high-B radio pulsars, whose high-energy emission may occur at ticle emission associated with bursts (Frail, Vasisht & Kulkarni 1997; Frail, Kulkarni & Bloom 1999). Wind luminosities in ex- some altitude above the surface (see Section 4.2), may be higher 36 1 cess of 10 erg s− in SGR1806-20 would lower the derived than the quiescence boundary applicable for AXPs, whose emis- ∼ 14 sion is thought to be predominantly of thermal origin from the surface field strength below 10 G. An episodic wind, which hot neutron star surface. Another possibility, pertinent if the is more likely, would decrease the estimated surface field to a radio detection of SGR 1900+14 is confirmed, is that SGRs lesser degree. Possibly for this reason, the dipole surface fields are “pulsars” with active accelerators allowing pair production of the SGRs are higher than those of most of the AXPs and and radio emission at high altitudes, as discussed by Zhang & more than an order of magnitude higher than the most strongly magnetized radio pulsars. 17

Fig. 9.— The upper region of the P - P˙ diagram, with filled circles again denoting the locations of members of the latest edition of the Princeton Pulsar Catalogue, and the open circles marking pulsars in the recent Parkes Multi-Beam survey. The Crab, Vela and PSR 1509-58 gamma-ray pulsars are highlighted as indicated in the lower left inset, together with the positions of five radio-quiet anomalous X-ray pulsars (open square stars) and S GRs 1806-20 and 1900+14 (filled squares) in the upper right of the diagram; their measured P˙ and inferred fields B0 are listed in Table 1. The conven- tional death line (as in Fig. 5) lies at periods longer than all seven of the putative magnetars. The θkB,0 = 0 (midweight solid diagonal curve) and 3 θkB,0 =10− (midweight dashed curve) depictions of the radio quiescence boundary from Fig. 5 are exhibited. The heavy-weight pairs of curves labelled 2 εmax =10MeVand εmax =10 MeV are contours for the escape energy of photon splitting, , with the solid and dashed line styles denoting 3 ⊥→kk θkB,0 =0 and θkB,0 =10− , respectively. They are shown only above the potential radio quiescence boundaries, where splitting dominates pair creation for the polarization, marking contours for the approximate maximum observable energy permitted in highly-magnetized pulsars; to the right of these the magnetosphere⊥ is transparent to 10 MeV and 100 MeV photons, respectively.

4.4. Guides for Soft and Hard Gamma-Ray Pulsation Searches At near-critical and supercritical fields, the attenuation by pair creation and photon splitting in the EGRET band is dra- To close this section of discussion, it is appropriate to mention matic, as is evident from the contours of maximum energy de- the implications of our analysis for gamma-ray pulsar searches, which are, in a sense, independent of the radio quiescence is- picted in Figure 9. Basically, photon transparency at a given energy is achieved to the right of and below (i.e. at longer peri- sue. If the electrons are accelerated to high Lorentz factors in ˙ profusion, then primary photons will extend into the EGRET ods and lower P ) a given contour so that the positioning of the band and beyond. Therefore, short period pulsars with high εesc = 100 MeV splitting escape energy contours indicates that 2 4 it will not be easy for an experiment like GLAST (depending on spin-down power ( B0 /P ) like the Crab pulsar should be luminous and therefore∝ easily detectable by future experiments its final design) to detect many or most of the known magnetars, such as GLAST. At longer periods near the magnetar regime, and the new Parkes Multi-Beam sources PSR J1119-6127 and the collected observational data and the age distribution bias PSR J1726-3530. Should magnetars emit in the gamma-rays, (see Section 4.1) indicate that spin-powered pulsars will be gen- they can only do so generally below the EGRET band. This erally faint unless they are nearby. Yet the observed hard X-ray conclusion holds even if the magnetospheric field structure is luminosities of AXPs are well in excess of their spin-down lu- non-dipolar, since escape energies are pushed to lower values by minosities so that non-rotational energy sources are indicated. greater field curvature. A possible evasion of this constraint is Hence luminosity biases against long period pulsars may not be that the inferred fields are significant overestimates of the true as pronounced for those with alternative power sources such as surface fields, as might occur for wind-aided spin-down in mag- magnetic field decay, at least in the magnetar regime. The work netars (see Section 4.3), so that the escape energy contours may of this paper clearly underlines the fact that photon attenuation possess pathological distortions in the P - P˙ diagram. More- considerations have as profound an influence on gamma-ray ob- over, such a two-dimensional phase space may be insufficient to servability as do issues of power in the primary radiation. account for the magnetospheric and photon attenuation prop- 18 erties. Hence high energy astrophysicists interested in pulsar ergy around a factor of 2-3 lower than the pair production cutoff. searches at high P˙ should extend their period range to include GLAST (depending on sensitivity below 100 MeV) or a future supersecond periods to maximize their potential harvest. While medium energy γ-ray detector would then see 100% polariza- k this range has historically been given low priority (e.g. see the tion of the highest energy photons, i.e. those between εesc⊥→kk + search in Mattox et al. 1996), the excitement generated by AXP e e− and εk→ . On the other hand, the absence of this signature and SGRs in recent years is changing emphases in such search esc would be consistent with, but not necessarily imply, at least two programs. active splitting modes. These distinctive signatures establish a These attenuation properties emphasize that spectroscopy in strong case for performing gamma-ray polarimetry experiments, a variety of gamma-ray pulsars will probably provide the ability an objective that may not be that far in the future for medium- to discriminate between the applicability of polar cap or outer energy Compton gamma-ray telescopes. gap models. In the polar cap picture, the analysis of this paper shows that spectral cutoff energies possess a strong inverse cor- 5. CONCLUSION relation with surface field strength B0 and the polar cap size Θ . For Crab-like and Vela-like pulsars such cutoffs are coupled to We have presented a detailed study of the comparative at- magnetic pair creation, and appear in the EGRET band. How- tenuation of photons by magnetic pair production and photon ever, for highly-magnetized pulsars such as PSR 1509-58, such splitting, of pair cascades and of conditions for pair suppres- properties are ideally explored with a medium energy gamma- sion in very highly magnetized pulsars. While ground-state pair ray experiment, and probe the action of photon splitting. Ex- creation and positronium formation can act at B0 > 0.1Bcr pectations for trends of gamma-ray cutoffs in outer gap models to reduce the number of free pairs, only photon splitting∼ has are not as well studied. Yet, it appears that their trends with the potential capability of dramatically inhibiting the creation of pairs, bound or free. Our quantitative study of cascade pair B0 and pulsar period should differ significantly from those for polar cap models, due to the inherently different physics in- yields at different field strengths confirms the location of the ˙ volved. Cutoffs in the outer gap scenario represent maximum approximate radio quiescence boundary in the P - P diagram energies of acceleration rather than the effects of photon absorp- proposed by Baring & Harding (1998), assuming that copious tion, and so may actually be independent of or increase with pair production is a requirement for pulsar radio emission. How- surface field strength, and decline with pulse period, contrary ever, it also shows that the existence of such a boundary requires to the indications of the polar gap model. Such distinctive pre- the rate of photon splitting to be non-zero for both photon po- dictions can be probed by significant population datasets such larization modes. This is not believed to be true in the low-field, as those to be afforded by the GLAST mission. A related obser- weakly dispersive limit, where kinematic selection rules prohibit splitting of photons of polarization. Whether photon split- vational diagnostic is spawned by the contention (Chen & Rud- k erman 1993) that gamma-ray emission is not expected in outer ting can significantly suppress pairs in highly magnetized radio gap models when the pulsar period exceeds a sizeable fraction pulsars depends on presently unexplored physics. It therefore of a second. Hence detection of gamma-ray pulsations from a has become critically important to study the photon splitting source with a supersecond period would clearly favor the polar process in the high-field, highly dispersive regime. Understand- cap model. ing the behavior of splitting in this regime will not only resolve Our studies also indicate that the gamma-ray spectra of the radio quiescence question, but is crucial to modeling accel- pulsars should produce distinctive polarization signatures for eration and radiation in high-field pulsars and magnetars. the polar cap model. This should enable discrimination from The circumstantial evidence, that the most highly magnetized outer gap scenarios in a future era when gamma-ray polarime- pulsars (AXPs and SGRs) tend to be radio quiet, argues for try is possible. While the predictions of gamma-ray polar- some pair suppression at high field strengths. The radio pulsar izations might be similar for the continuum spectra in each and magnetar populations are no longer cleanly separated in P - of these two competing models, principally because their con- P˙ space, which indicates that another dimension of phase space tinuum emission processes are similar, the pair creation and is necessary to divide these two source populations. Delineation photon splitting mechanisms generate spectral cutoff energies of these two source groups is complicated by the likelihood of that are polarization-dependent (as emphasized by HBG97), alternate spin-down mechanisms and evolution issues for mag- and therefore immediately distinguishable from outer gap model netars. More systematic surveys of the overlap region of P - P˙ turnovers that have no intimate connection to QED processes space at X-ray and γ-ray energies, correlated with radio pulsar in strong fields. In particular, when B0 >Bcr , the dominance surveys are needed. We have given some guidelines for such of splittings will guarantee that the∼ cutoff energy for high-energy searches based on the expected physics of particle photons exceeds⊥→kk that for ones of polarization. In contrast,k acceleration and photon attenuation, in anticipation of the next at lower fields strengths, photon splitting⊥ is removed from the generation of medium and high-energy gamma-ray experiments picture and the escape energy for pair creation by photons such as GLAST. Spectral and polarization observations may be is slightly lower than that for ones, resulting ink a domi- crucial in identifying signatures of photon splitting and even nance of photons near the maximum⊥ observable energy. Be- in resolving the issue of which splitting modes are operating cause the⊥ strong dependence of the spectral high-energy cutoff in ultra-magnetized sources, an attractive possibility for both in fields B0 >Bcr occurs exclusively when only one mode of astrophysicists and physicists. splitting is operative,∼ it may be possible to resolve by observa- tion the question of whether one or several modes of splitting We are grateful to Peter Gonthier and Marty Knecht for help operate in high fields. The splitting and pair production cutoffs with the splitting cascade code development. We also thank are expected around 100 MeV in long period pulsars (slightly Alex Muslimov, Jim Cordes, Eric Gotthelf, David Thompson, higher if the emission region is above the surface). If only the Peter Gonthier and Matthew Bailes for discussions. This work mode operates, the splitting cutoff will occur at an en- was supported by the NASA Astrophysics Theory Program. ⊥→kk 19

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