Condensed surfaces of magnetic neutron stars, thermal surface emission, and particle acceleration above pulsar polar caps
Recent calculations indicate that the cohesive energy of condensed matter increases with magnetic field strength and becomes very significant at magnetar-like fields (e.g., 10 keV at G for zero-pressure condensed iron). This implies that for sufficiently strong magnetic fields and/or low temperatures, the neutron star surface may be in a condensed state with little gas or plasma above it. Such surface condensation can significantly affect the thermal emission from isolated neutron stars, and may lead to the formation of a charge-depleted acceleration zone (“vacuum gap”) in the magnetosphere above the stellar polar cap. Using the latest results on the cohesive property of magnetic condensed matter, we quantitatively determine the conditions for surface condensation and vacuum gap formation in magnetic neutron stars. We find that condensation can occur if the thermal energy of the neutron star surface is less than about 8% of its cohesive energy , and that a vacuum gap can form if (i.e., the neutron star’s rotation axis and magnetic moment point in opposite directions) and is less than about 4% of . For example, at G, a condensed Fe surface forms when K and a vacuum gap forms when K. Thus, vacuum gap accelerators may exist for some neutron stars. Motivated by this result, we also study the physics of pair cascades in the (Ruderman-Sutherland type) vacuum gap model for photon emission by accelerating electrons and positrons due to both curvature radiation and resonant/nonresonant inverse Compton scattering. Our calculations of the condition of cascade-induced vacuum breakdown and the related pulsar death line/boundary generalize previous works to the superstrong field regime. We find that inverse Compton scatterings do not produce a sufficient number of high energy photons in the gap (despite the fact that resonantly upscattered photons can immediately produce pairs for G) and thus do not lead to pair cascades for most neutron star parameters (spin and magnetic field). We discuss the implications of our results for the recent observations of neutron star thermal radiation as well as for the detection/non-detection of radio emission from high-B pulsars and magnetars.
keywords:radiation mechanisms: non-thermal – radiation mechanisms: thermal – stars: magnetic fields – stars: neutron – pulsars: general.
Recent observations of neutron stars have provided a wealth of information on these objects, but they have also raised many new questions. For example, with the advent of X-ray telescopes such as Chandra and XMM-Newton, detailed observations of the thermal radiation from the neutron star surface have become possible. These observations show that some nearby isolated neutron stars (e.g., RX J1856.5-3754) appear to have featureless, nearly blackbody spectra (Burwitz et al. 2003; van Kerkwijk & Kaplan 2007). Radiation from a bare condensed surface (where the overlying atmosphere has negligible optical depth) has been invoked to explain this nearly perfect blackbody emission (e.g., Burwitz et al. 2003; Mori & Ruderman 2003; Turolla et al. 2004; van Adelsberg et al. 2005; Perez-Azorin et al. 2006; Ho et al. 2007; but see Ruderman 2003 for an alternative view). However, whether surface condensation actually occurs depends on the cohesive properties of the surface matter (e.g., Lai 2001).
Equally puzzling are the observations of anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) (see Woods & Thompson 2005 for a review). Though these stars are believed to be magnetars, neutron stars with extremely strong magnetic fields ( G), they mostly show no pulsed radio emission (but see Camilo et al. 2006, 2007; Kramer et al. 2007) and their X-ray radiation is too strong to be powered by rotational energy loss. By contrast, several high-B radio pulsars with inferred surface field strengths similar to those of magnetars have been discovered (e.g., Kaspi & McLaughlin 2005; Vranevsevic, Manchester, & Melrose 2007). A deeper understanding of the distinction between pulsars and magnetars requires further investigation of the mechanisms by which pulsars and magnetars radiate and of their magnetospheres where this emission originates. Theoretical models of pulsar and magnetar magnetospheres depend on the cohesive properties of the surface matter in strong magnetic fields (e.g., Ruderman & Sutherland 1975; Arons & Scharlemann 1979; Cheng & Ruderman 1980; Usov & Melrose 1996; Harding & Muslimov 1998; Gil, Melikidze, & Geppert 2003; Muslimov & Harding 2003; Beloborodov & Thompson 2007). For example, depending on how strongly bound the surface matter is, a charge-depleted acceleration zone (“vacuum gap”) above the polar cap of a pulsar may or may not form, and this will affect pulsar radio emission and other high-energy emission processes.
The cohesive property of the neutron star surface matter plays a key role in these and other neutron star processes and observed phenomena. The cohesive energy refers to the energy required to pull an atom out of the bulk condensed matter at zero pressure. A related (but distinct) quantity is the electron work function, the energy required to pull out an electron. For magnetized neutron star surfaces the cohesive energy and work function can be many times the corresponding terrestrial values, due to the strong magnetic fields threading the matter (e.g., Ruderman 1974; Lai 2001).
In two recent papers (Medin & Lai 2006a, b, hereafter ML06a,b), we carried out detailed, first-principle calculations of the cohesive properties of H, He, C, and Fe surfaces at field strengths between G to G. The main purpose of this paper is to investigate several important astrophysical implications of these results (some preliminary investigations were reported in Medin & Lai 2007). This paper is organized as follows. In Section 2 we briefly summarize the key results (cohesive energy and work function values) of ML06a,b used in this paper. In Section 3 we examine the possible formation of a bare neutron star surface, which directly affects the surface thermal emission. We find that the critical temperature below which a phase transition to the condensed state occurs is approximately given by , where is the cohesive energy of the surface. In Section 4 we consider the conditions for the formation of a polar vacuum gap in pulsars and magnetars. We find that neutron stars with rotation axis and magnetic moment given by are unable to form vacuum gaps (since the electrons which are required to fill the gaps can be easily supplied by the surface), but neutron stars with can form vacuum gaps provided that the surface temperature is less than (and that particle bombardment does not completely destroy the gap; see Section 6). In Section 5 we discuss polar gap radiation mechanisms and the pulsar death line/boundary in the vacuum gap model. We find that when curvature radiation is the dominant radiation mechanism in the gap, a pair cascade is possible for a large range of parameter space (in the – diagram), but when inverse Compton scattering (either resonant or nonresonant) is the dominant radiation mechansim, vacuum breakdown is possible for only a very small range of parameter values. Implications of our results for recent observations are discussed in Section 6. Some technical details (on our treatment of inverse Compton scattering and vacuum gap electrodynamics of oblique rotators) are given in two appendices.
2 Cohesive Properties of Condensed Matter in Strong Magnetic Fields
It is well-known that the properties of matter can be drastically modified by strong magnetic fields. The natural atomic unit for the magnetic field strength, , is set by equating the electron cyclotron energy keV, where , to the characteristic atomic energy eV (where is the Bohr radius):
For , the usual perturbative treatment of the magnetic effects on matter (e.g., Zeeman splitting of atomic energy levels) does not apply. Instead, the Coulomb forces act as a perturbation to the magnetic forces, and the electrons in an atom settle into the ground Landau level. Because of the extreme confinement of the electrons in the transverse direction (perpendicular to the field), the Coulomb force becomes much more effective in binding the electrons along the magnetic field direction. The atom attains a cylindrical structure. Moreover, it is possible for these elongated atoms to form molecular chains by covalent bonding along the field direction. Interactions between the linear chains can then lead to the formation of three-dimensional condensed matter (Ruderman 1974; Ruder et al. 1994; Lai 2001).
The basic properties of magnetized condensed matter can be estimated using the uniform electron gas model (e.g., Kadomtsev 1970). The energy per cell of a zero-pressure condensed matter is given by
and the corresponding condensation density is
where are the charge number and mass number of the ion (see Lai 2001 and references therein for further refinements to the uniform gas model). Although this simple model gives a reasonable estimate of the binding energy for the condensed state, it is not adequate for determining the cohesive property of the condensed matter. The cohesive energy is the (relatively small) difference between the atomic ground-state energy and the zero-pressure condensed matter energy , both increasing rapidly with . Moreover, the electron Fermi energy (including both kinetic energy and Coulomb energy) in the uniform gas model,
may not give a good scaling relation for the electron work function when detailed electron energy levels (bands) in the condensed matter are taken into account.
There have been few quantitative studies of infinite chains and zero-pressure condensed matter in strong magnetic fields. Earlier variational calculations (e.g., Flowers et al. 1977; Müller 1984) as well as calculations based on Thomas-Fermi type statistical models (e.g., Abrahams & Shapiro 1991; Fushiki et al. 1992), while useful in establishing scaling relations and providing approximate energies of the atoms and the condensed matter, are not adequate for obtaining reliable energy differences (cohesive energies). Quantitative results for the energies of infinite chains of hydrogen molecules H over a wide range of field strengths () were presented in Lai et al. (1992) (using the Hartree-Fock method with the plane-wave approximation; see also Lai 2001 for some results for He) and in Relovsky & Ruder (1996) (using density functional theory). For heavier elements such as C and Fe, the cohesive energies of one dimensional (1D) chains have only been calculated at a few magnetic field strengths in the range of – G, using Hartree-Fock models (Neuhauser et al., 1987) and density functional theory (Jones, 1985). There were some discrepancies between the results of these works, and some adopted a crude treatment for the band structure (Neuhauser et al., 1987). An approximate calculation of 3D condensed matter based on density functional theory was presented in Jones (1986).
Our calculations of atoms and small molecules (ML06a) and of infinite chains and condensed matter (ML06b) are based on a newly developed density functional theory code. Although the Hartree-Fock method is expected to be highly accurate in the strong field regime, it becomes increasingly impractical for many-electron systems as the magnetic field increases, since more and more Landau orbitals are occupied (even though electrons remain in the ground Landau level) and keeping track of the direct and exchange interactions between electrons in various orbitals becomes computational rather tedious. Compared to previous density-functional theory calculations, we used an improved exchange-correlation function for highly magnetized electron gases, and we calibrated our density-functional code with previous results (when available) based on other methods. Most importantly, in our calculations of 1D condensed matter, we treated the band structure of electrons in different Landau orbitals self-consistently without adopting ad-hoc simplifications. This is important for obtaining reliable results for the condensed matter. Since each Landau orbital has its own energy band, the number of bands that need to be calculated increases with and , making the computation increasingly complex for superstrong magnetic field strengths (e.g., the number of occupied bands for Fe chains at G reaches 155; see Fig. 16 of ML06b). Our density-functional calculations allow us to obtain the energies of atoms and small molecules and the energy of condensed matter using the same method, thus providing reliable cohesive energy and work function values for condensed surfaces of magnetic neutron stars.
In ML06a, we described our calculations for various atoms and molecules in magnetic fields ranging from G to G for H, He, C, and Fe, representative of the most likely neutron star surface compositions. Numerical results of the ground-state energies are given for H (up to ), He (up to ), C (up to ), and Fe (up to ), as well as for various ionized atoms. In ML06b, we described our calculations for infinite chains for H, He, C, and Fe in that same magnetic field range. For relatively low field strengths, chain-chain interactions play an important role in the cohesion of three-dimensional (3D) condensed matter. An approximate calculation of 3D condensed matter is also presented in ML06b. Numerical results of the ground-state and cohesive energies, as well as the electron work function and the zero-pressure condensed matter density, are given in ML06b for H and H(3D), He and He(3D), C and C(3D), and Fe and Fe(3D).
Some numerical results from ML06a,b are provided in graphical form in Figs. 1, 2, 3, and 4 (see ML06a,b for approximate scaling relations for different field ranges based on numerical fits). Figure 1 shows the cohesive energies of condensed matter, , and the molecular energy differences, , for He, Fig. 2 for C, and Fig. 3 for Fe; here is the atomic ground-state energy, is the ground-state energy of the He, C, or Fe molecule, and is the energy per cell of the zero-pressure 3D condensed matter. Some relevant ionization energies for the atoms are also shown. Figure 4 shows the electron work functions for condensed He, C, and Fe as a function of the field strength. We see that the work function increases much more slowly with compared to the simple free electron gas model [see Eq. (4)], and the dependence on is also weak. The results summarized here will be used in Section 3 and Section 4 below.
3 Condensation of Neutron Star Surfaces in Strong Magnetic Fields
As seen from Figs. 1, 2, and 3, the cohesive energies of condensed matter increase with magnetic field. We therefore expect that for sufficiently strong magnetic fields, there exists a critical temperature below which a first-order phase transition occurs between the condensate and the gaseous vapor. This has been investigated in detail for hydrogen surfaces (see Lai & Salpeter 1997; Lai 2001), but not for other surface compositions. Here we consider the possibilies of such phase transitions of He, C, and Fe surfaces.
A precise calculation of the critical temperature is difficult. We can determine approximately by considering the equilibrium between the condensed phase (labeled “s”) and the gaseous phase (labeled “g”) in the ultrahigh field regime (where phase separation exists). The gaseous phase consists of a mixture of free electrons and bound ions, atoms, and molecules. Phase equilibrium requires the temperature, pressure and the chemical potentials of different species to satisfy the conditions (here we consider Fe as an example; He and C are similar)
where we treat the gaseous phase as an ideal gas. The chemical potential of the condensed phase is given by
where is the energy per cell of the condensate and is the energy per cell at zero-pressure (we will label this simply as ). We have assumed that the vapor pressure is sufficiently small so that the deviation from the zero-pressure state of the condensate is small; this is justified when the saturation vapor pressure is much less than the critical pressure for phase separation, or when the temperature is less than the critical temperature by a factor of a few.
For nondegenerate electrons in a strong magnetic field the number density is related to by
where for and for are the Landau degeneracies, is the electron thermal wavelength, and the last equality applies for . The magnetic field length is . For atomic, ionic, or molecular Fe the number density is given by
with the internal partition function
and . Here, the subscript represents the atomic, ionic, or molecular species whose number density we are calculating (e.g., Fe or Fe) and the sum is over all excited states of that species. Also, is the Fe particle’s thermal wavelength, where is the total mass of the particle ( is the number of “atoms” in the molecule, is the atomic mass number, and ). The vector represents the center-of-mass momentum of the particle. Note that we have assumed here that the Fe particle moves across the field freely; this is a good approximation for large . The internal partition function represents the effect of all excited states of the species on the total density; in this work we will use the approximation that this factor is the same for all species, and we will estimate the magnitude of this factor later in this section.
The equilibrium condition for the process yields the atomic density in the saturated vapor:
where is the cohesive energy of the condensed Fe. The condition for the process yields the molecular density in the vapor:
is the “surface energy” and is the energy per ion in the molecule. The equilibrium condition for the process , where Fe is the th ionized state of Fe, yields the vapor densities for the ions:
and so on. Here, and is the Bohr radius, and represents the ionization energy of the th ionized state of Fe (i.e., the amount of energy required to remove the th electron from the atom when the first electrons have already been removed). The total electron density in the saturated vapor is
The number densities of electrons [Eq. (19)] and ions [e.g., Eqs. (17) and (18)] must be found self-consistently, for all ion species that contribute significantly to the total vapor density. The total mass density in the vapor is calculated from the number densities of all of the species discussed above, using the formula
Figure 5 (for Fe) and Fig. 6 (for C) show the the densities of different atomic/molecular species in the saturated vapor in phase equilibirum with the condensed matter for different temperatures and field strengths. These are computed using the values of , , and presented in ML06a,b and depicted in Figs. 2 and 3. As expected, for sufficiently low temperatures, the total gas density in the vapor is much smaller than the condensation density, and thus phase separation is achieved. The critical temperature , below which phase separation between the condensate and the gaseous vapor occurs, is determined by the condition . We find that for Fe:
and for He:
In terms of the cohesive energy, these results can be approximated by
Note that in our calculations for the iron vapor density at - we have estimated the magnitude of the internal partition function factor ; the modified total density curves are marked on these figures as “”. To estimate we use Eq. (13) with a cutoff to the summation above some energy. For , and we calculate or interpolate the energies for all excited states of atomic Fe with energy below this cutoff, in order to find . The energy cutoff is necessary because the highly excited states become unbound (ionized) due to finite pressure and should not be included in (otherwise would diverge). In principle, the cutoff is determined by requiring the effective size of the excited state to be smaller than the inter-particle space in the gas, which in turn depends on density. In practice, we choose the cutoff such that the highest excited state has a binding energy significantly smaller than the ground-state binding energy (typically 30% of it). As an approximation, we also assume that the internal partitions for Fe molecules and ions have the same as the Fe atom. Despite the crudeness of our calculation of , we see from Fig. 5 that the resulting is only reduced by a few tens of a percent from the value assuming .
We note that our calculation of the saturated vapor density is very uncertain around , since Eqs. (14) – (18) are derived for while the critical temperature of the saturated vapor density is found by setting . However, since the vapor density decreases rapidly as decreases, when the temperature is below (for example), the vapor density becomes much less than the condensation density and phase transition is unavoidable. When the temperature drops below a fraction of , the vapor density becomes so low that the optical depth of the vapor is negligible and the outermost layer of the neutron star then consists of condensed matter. The radiative properties of such condensed phase surfaces have been studied using a simplified treatment of the condensed matter (see van Adelsberg et al. 2005 and references therein).
4 Polar Vacuum Gap Acclerators in Pulsars and Magnetars
A rotating, magnetized neutron star is surrounded by a magnetosphere filled with plasma. The plasma is assumed to be an excellent conductor, such that the charged particles move to screen out any electric field parallel to the local magnetic field. The corresponding charge density is given by (Goldreich & Julian 1969)
where is the rotation rate of the neutron star.
The Goldreich-Julian density assumes that charged particles are always available. This may not be satisfied everywhere in the magnetosphere. In particular, charged particles traveling outward along the open field lines originating from the polar cap region of the neutron star will escape beyond the light cylinder. To maintain the required magnetosphere charge density these particles have to be replenished by the stellar surface. If the surface temperature and cohesive strength are such that the required particles are tightly bound to the stellar surface, those regions of the polar cap through which the charged particles are escaping will not be replenished. A vacuum gap will then develop just above the polar cap (e.g., Ruderman & Sutherland 1975; Cheng & Ruderman 1980; Usov & Melrose 1996; Zhang, Harding, & Muslimov 2000; Gil, Melikidze, & Geppert 2003). In this vacuum gap zone the parallel electric field is no longer screened and particles are accelerated across the gap until vacuum breakdown (via pair cascade) shorts out the gap. Such an acceleration region can have an important effect on neutron star emission processes. We note that in the absence of a vacuum gap, a polar gap acceleration zone based on space-charge-limited flow may still develop (e.g., Arons & Scharlemann 1979; Harding & Muslimov 1998; Muslimov & Harding 2003).
In this section we determine the conditions required for the vacuum gap to exist using our results summarized in Section 2. The cohesive energy and electron work function of the condensed neutron star surface are obviously the key factors. We examine the physics of particle emission from condensed surface in more detail than considered previously.
4.1 Particle Emission From Condensed Neutron Star Surfaces
We assume that the NS surface is in the condensed state, i.e., the surface temperature is less than the critical temperature for phase separation (see Section 3). (If , the surface will be in gaseous phase and a vacuum gap will not form.) We shall see that in order for the surface not to emit too large a flux of charges to the magnetosphere (a necessary condition for the vacuum gap to exist), an even lower surface temperature will be required.
4.1.1 Electron Emission
For neutron stars with , where is the magnetic field at the polar cap, the Goldreich-Julian charge density is negative at the polar cap, thus surface electron emission (often called thermionic emission in solid state physics; Ashcroft & Mermin 1976) is relevant. Let be the number flux of electrons emitted from the neutron star surface. The emitted electrons are accelerated to relativistic speed quickly, and thus the steady-state charge density is . For the vacuum gap to exist, we require . (If , the charges will be rearranged so that the charge density equals .)
To calculate the electron emission flux from the condensed surface, we assume that these electrons behave like a free electron gas in a metal, where the energy barrier they must overcome is the work function of the metal. In a strong magnetic field, the electron flux is given by
where , is the potential energy of the electrons in the metal, is the electron kinetic energy, and
is the Fermi-Dirac distribution function with the electron chemical potential (excluding potential energy). Integrating this expression gives
where is the work function of the condensed matter and the second equality assumes . The steady-state charge density supplied by the surface is then
where and is the spin period in units of 1 s. For a typical set of pulsar parameters (e.g., and ) , but can range from 23 for millisecond pulsars to 35 for some magnetars. Note that the requirement is automatically satified here when is less than . The electron work function was calculated in ML06b and is depicted in Fig. 4.
4.1.2 Ion Emission
For neutron stars with , the Goldreich-Juliam charge above the polar cap is positive, so we are interested in ion emission from the surface. Unlike the electrons, which form a relatively free-moving gas within the condensed matter, the ions are bound to their lattice sites.111The freezing condition is easily satisfied for condensed matter of heavy elements (see van Adelsberg et al. 2005). To escape from the surface, the ions must satisfy three conditions. First, they must be located on the surface of the lattice. Ions below the surface will encounter too much resistance in trying to move through another ion’s cell. Second, they must have enough energy to escape as unbound ions. This binding energy that must be overcome will be labeled . Third, they must be thermally activated. The energy in the lattice is mostly transferred by conduction, so the ions must wait until they are bumped by atoms below to gain enough energy to escape.
Consider the emission of ions with charge from the neutron star surface (e.g., Fe would have ). The rate of collisions between any two ions in the lattice is approximately equal to the lattice vibration frequency , which can be estimated from
where is the ion plasma (angular) frequency and is the ion cyclotron frequency (). Not all collisions will lead to ejection of ions from the surface, since an energy barrier must be overcome. Thus each surface ion has an effective emission rate of order
The energy barrier for ejecting ions of charge is equivalent to the energy required to release a neutral atom from the surface and ionize it, minus the energy gained by returning the electron to the surface (e.g., Tsong 1990). Thus
where is the cohesive energy, is the th ionization energy of the atom (so that is the energy required to remove electrons from the atom), and is the electron work function. The surface density of ions is , where is the mean spacing between ions in the solid. Thus the emission flux of -ions is
The steady-state -ion number density supplied by the surface is then
where . For a typical set of pulsar parameters (e.g., and ) , but can be as large as 33 for magnetars with and .
4.1.3 Effect of Electric Field on Charge Emission
The discussion in Sections 4.1.1 and 4.1.2 includes only thermal emission of charged particles from the condensed surface. A strong electric field, of order , may be present. Since this electric field is much less than the characteristic field inside the condensed matter (where is the mean particle separation), this field cannot directly rip charges off the surface. Nevertheless, the electric field may enhance the thermal emission of charge particles. We now estimate the magnitude of this effect.
In the presence of a vacuum gap, the electric field at the stellar surface points outward () for stars with and inward () for stars with . A charge moved to some small height above the surface gains a potential energy given by , where the first term is due to the interaction between the charge and the perfectly conducting metal surface, and the second term is due to the external field.222In the vacuum gap, the electric field is not exactly uniform, but since the maximum is attained at a rather small height compared to the gap thickness, this nonuniformity is unimportant for our consideration here. The potential reaches a maximum value
at the height . Thus, compared to the case, the energy barrier for particle emission is now reduced by the amount .
and the steady-state charge density due to ion surface emission (for stars) is
For , we have eV. This is typically much smaller than either or .
4.2 Conditions for Gap Formation
No vacuum gap will form if the electrons or ions are able to fill the magnetosphere region above the polar cap with the required Goldreich-Julian density; i.e., the vacuum gap will cease to exist when or . From Eqs. (39) and (38) we can see that no polar gap will form if
for a negative polar magnetosphere (), and
For neutron stars in general, the electron work function is much less than keV (see Fig. 4), so electrons can easily escape from the condensed surface. No gap forms for a negative polar magnetosphere under neutron star surface conditions. (This is contrary to the conclusions of Usov & Melrose 1996 and Gil et al. 2003.) The ion binding energy [given by Eq. (33)], on the other hand, can be larger than keV under certain neutron star surface conditions (see Figs. 1, 2, and 3). Ions can tightly bind to the condensed surface and a polar gap can form under these conditions. Figure 7 shows the critical temperature (determined by ) below which a vacuum gap can form for the Fe, C, and He surfaces.
5 Vacuum Gap Acclerators: Pair Cascades and the Pulsar Death Line/Boundary
Pair cascading in the magnetosphere of a pulsar is an essential ingredient for its radio emission (e.g., Melrose 2004). The pair cascade involves: (a) acceleration of primary particles by an electric field parallel to the magnetic field; (b) gamma ray emission by the accelerated particles moving along the magnetic field lines (either by curvature radiation or inverse Compton upscattering of surface photons); (c) photon decay into pairs as the angle between the photon and the field line becomes sufficiently large. To initiate the cascade an acceleration region is required; the characteristics of this particle accelerator determine whether pulsar emission can operate or not (the so-called “pulsar death line”; e.g., Ruderman & Sutherland 1975; Arons 2000; Zhang et al. 2000; Hibschman & Arons 2001). Depending on the boundary condition at the neutron star surface, there are two types of polar gap accelerators: If charged particles are strongly bound to the neutron star surface by cohesive forces, a vacuum gap develops directly above the surface, with height much less than the stellar radius (Ruderman & Sutherland, 1975); if charged particles can be freely extracted from the surface, a more extended space-charge-limited-flow (SCLF) type accelerator develops due to field line curvatures (Arons & Scharlemann, 1979) and the relativistic frame dragging effect (e.g., Muslimov & Tsygan 1992). Because the cohesive strength of matter at G was thought to be negligible (based on the result of Neuhauser et al. 1987), most theoretical works in recent years have focused on the SCLF models (e.g., Arons 2000; Muslimov & Harding 2003, 2004).
Our results in Section 4 show that for sufficiently strong magnetic fields and/or low surface temperatures, a vacuum gap accelerator can form. Such a vacuum gap may be particularly relevant for the so-called high-B radio pulsars, which have inferred magnetic fields similar to those of magnetars (e.g., Kaspi & McLaughlin 2005; Burgay et al. 2006). In this section we discuss the conditions under which a vacuum gap will be an effective generator of pulsar emission. As discussed in Section 4, since electrons are weakly bound to the condensed stellar surface, such a vacuum gap is possible only for pulsars with (as suggested in the original Ruderman-Sutherland model).
Our analysis is similar to the original Ruderman-Sutherland model, except that we extend our discussion of the cascade physics to the magnetar field regime, which introduces some corrections to previous works (e.g., Ruderman & Sutherland 1975; Usov & Melrose 1996). We also consider photon emission due to inverse Compton scattering, in addition to curvature radiation, in the cascade (cf. Zhang et al. 1997, 2000; Hibschman & Arons 2001).
5.1 Acceleration Potential
When the temperature drops below the critical value given in Section 4, the charge density above the polar cap decreases quickly below , and a vacuum gap results. In the vacuum region just above the surface (), the parallel electric field satisfies the equation . The height of the gap is determined by vacuum breakdown due to pair cascade, which shorts out the electric field above the gap (i.e., for ). Thus the electric field in the gap is
where is the actual magnetic field at the pole, and differs from the dipole field by a factor . The potential drop across the gap is then
With this potential drop, the electrons and positrons can be accelerated to a gamma factor
where (with G the QED field), , and is the spin period in units of 1 s. The voltage drop across the gap can be no larger than the voltage drop across the polar cap region , where is the radius of the polar cap through which a net postive current flows:
Thus the gap height is limited from above by
where we have adopted km.
The above equations are for an aligned rotator. For an oblique rotator (where the magnetic dipole axis is inclined relative to the rotation axis), the voltage drop across the polar cap region is larger, of order . But as discussed in Appendix A, the acceleration potential across the vacuum gap is still limited from above by .
5.2 Requirements for Gap Breakdown
There are two requirements for the breakdown of a vacuum gap. First, the photons must be able to create electron-positron pairs within the gap, i.e., the mean free path of photon pair-production is less than the gap height:
Second, the electrons and positrons must be accelerated over the gap potential and produce at least several photons within the gap. If on average only one photon is emitted with the required energy for each electron-positron pair, for instance, then the number of charged particles produced in the gap will grow very slowly and the gap will not break down completely. Therefore, we must have
where is the number of photons emitted within the gap by each electron or positron, and is a number of order –.
5.3 Pair Production
The threshold of pair production for a photon with energy is
where is the angle of intersection of the photon and the magnetic field. Suppose a photon is emitted at an angle . After the photon travels a distance , the intersection angle will grow as , where is the local radius of curvature of the polar magnetic field line. Thus the typical intersection angle (for a photon crossing the entire gap) is
For a pure dipole field, the curvature radius is of order cm, but a more complex field topology at the polar cap could reduce to as small as the stellar radius.
In the weak-field regime, when the threshold condition is well-satisfied (so that the pairs are produced in highly excited Landau levels), the mean free path is given by (Erber, 1966)
where is the Bohr radius. The condition implies for typical parameters (Ruderman & Sutherland, 1975). For stronger magnetic fields (), the pairs tend to be produced at lower Landau levels. Using the general expression for the pair production rate (e.g., Daugherty & Harding 1983), one can check that if the threshold condition Eq. (49) is satisfied, the pair-production optical depth across the gap would also be greater than unity [for , the optical depth is unity when , and by , when .] Thus for arbitrary field strengths, the condition leads to the constraint:
5.4 Photon Emission Multiplicity and the Pulsar Death Line/Boundary
There several possible photon emission mechanisms operating in the vacuum breakdown, each leading to a different “death line”, or more precisely, “death boundary”. We consider them separately.
5.4.1 Curvature Radiation (CR)
The characteristic energy of a photon emitted through curvature radiation is eV, where , and we have used [Eq. (44)]. The emission angle is , which is typically much less than (this can be easily checked a posteriori). Equation (52) then reduces to
The rate of energy loss of an electron or positron emitting curvature radiation is , thus the number of photons emitted through curvature radiation by a single electron or positron across the gap is
The condition [Eq. (48)] then gives
This gives a necessary condition for pulsar emission and defines the pulsar “death line”. For all relevant parameter regimes, , and Eq. (56) simply becomes . The critical pulsar spin period is then
where the dipole polar field is , with . For this is the same as the result of Ruderman & Sutherland (1975).
5.4.2 Resonant Inverse Compton Scattering (RICS)
Here the high-energy photons in the cascade are produced by Compton upscatterings of thermal photons from the neutron star surface. Resonant scattering in strong magnetic fields (e.g., Herold 1979) can be thought of as resonant absorption (where the electron makes a transition from the ground Landau level to the first excited level) followed by radiative decay. Resonance occurs when the photon energy in the electron rest frame satisfies . The resonant photon energy (in the “lab” frame) before scattering is , where is the incident angle (the angle between the incident photon momentum and the electron velocity). After absorbing a photon, the electron Lorentz factor drops to , and then radiatively decays isotropically in its rest frame. The characteristic photon energy after resonant scattering is therefore (e.g., Beloborodov & Thompson 2007)
with typical emission angle . The condition [see Eq. (52)] becomes
For this condition is automatically satisfied, i.e., resonant ICS photons pair produce almost immediately upon being upscattered. For , Eq. (59) puts a constraint on the gap height . As we shall see below, most of the scatterings in the gap are done by electrons/positrons with , where (with the surface blackbody temperature) and is the Lorentz factor of a fully-accelerated electron or positron [Eq. (44)]. For , Eq. (59) yields
For we have