The influence of small scale magnetic field on the heating of J0250+5854 polar cap

The influence of surface small scale magnetic field on the heating of PSR J0250+5854 polar cap is considered. It is assumed that polar cap is heated only by reverse positrons, accelerated in pulsar diode. It is supposed that pulsar diode is in stationary state with lower plate nearby the star surface (polar cap model), occupies all pulsar tube crosssection and operates in regime of steady space charge limited electron flow. The influence of small scale magnetic field on electric field inside pulsar diode is taken into account. To calculate the electron-positron pairs production rate we take into account only the curvature radiation of primary electrons and its absorption in magnetic field. It is assumed that part of electro-positron pairs may be created in bound state (positronium). And later such positroniums are photoionized by thermal photons from star surface.


Introduction
Radiopulsar J0250+5854 rotates with period P = 23.54 s [1] and is the slowest pulsar among rotation powered pulsars [2]. It is old pulsar with spin down age τ = 13.7 · 10 6 years, P = 2.71 · 10 −14 , its spin down energy loss rateĖ is equal toĖ = 8.2 · 10 28 erg/s, the strength B dip of dipolar magnetic field at pole estimated by pulsar slowdown is B dip = 5.1 · 10 13 G, distance D DM estimated by dispersion measure is D DM = 1.56 kpc [2]. Such pulsars lie beyond conventional pulsar "death line" , see, for example, [3,4], and usually its radio radiation is explained by the presence of small scale surface magnetic field, see, for example, [5,6,7,8]. It is worth to note that the radio radiation of J0250+5854 also may be explained in case of pure dipolar magnetic field if it is take into account that the B dip value is calculated at assumption aligned pulsar χ = 0 • and braking due to magneto-dipolar losses [9], where χ is inclination angle i.e. angle between vector of magnetic dipole momentum m and vector of angular velocity of star rotation Ω, Ω = 2π/P , see fig. 1 and fig. 2. In case of pulsar braking due to current losses and nearby orthogonal pulsars χ ≈ 90 • dipolar magnetic field is substantially larger than B dip value estimated by slow down [10] that moves the pulsar to "life zone" [9]. In this paper we will not consider a such possibility and assume that the B dip value is the right estimation of dipolar magnetic field strength. In case of large surface magnetic field B surf 4.4 · 10 12 G electron-positron pairs may be produced in bound state (positronium) [11]. The influence of this process on pulsar electrodynamics, pair generation and polar cap X-ray luminosity of radio pulsars has been thoroughly considered in many papers, see, for example, [12,11]. In this paper we consider the influence of small scale magnetic field on polar cap heating by reverse positron current with taking into account positronium generation and its photoionization by thermal photons from star surface. Pulsar is considered in inner gap model with free electron emission from neutron star surface. We assume that pulsar diode is in stationary state and take into account only positron generation due to curvature radiation of primary electrons and its absorption in magnetic field.

Model
Let the neutron star have a radius r ns and dipolar magnetic moment m (its field at magnetic pole is B dip = 2m/r 3 ns ). We assume also that a small-scale magnetic field with strength B sc and characteristic scale ℓ presents nearby the polar cap. For simplicity we model small-scale magnetic field by by additional magnetic moment m sc locating in the polar region of the neutron star at depth ℓ [13,14,15]:  where ρ = r − (r ns − ℓ) e z , m = m e z , B sc = m sc /ℓ 3 -small-scale field strength at (dipolar) magnetic pole. For simplicity we suppose that the vector m sc lies parallel to surface (and m sc · m = 0) in the plane containing m and Ω and is directed "along" Ω, see fig. 1. Hence surface small-scale magnetic field is directed "opposite" pulsar rotation velocity Ω, i.e. ψ Ω = 0, see fig. 2. Also we assume that inclination angle χ is equal to χ = 60 • . We consider only the case of inner gap [16] and assume that the inner gap occupies the entire pulsar tube cross section and resides as low as possible. Let us denote the altitudes of inner gap lower plate (cathode) and upper plate (anode) by z lo and is z hi respectively, see fig. 1. In  most cases the inner gap resides exactly on neutron star surface (z lo = 0), see [17] for details. We suppose that the inner gap is stationary and operates in the regime of charge limited steady flow [18]. Hence in the reference frame rotating with the star we can write, see [19] for details: Φ| z=z lo = 0 and Φ| side = 0 where z is the altitude above star surface, Φ is electrostatic potential, Φ| side is its value at pulsar tube boundary, ρ GJ = ΩB 2πcρ GJ is Goldreich-Julian density [20], ρ = ΩB 2πcρ is total charge density,ρ =ρ − +ρ + ,ρ,ρ − ,ρ + are total charge density, electron and positron densities in units ΩB 2πc correspondingly. We assume that inside pulsar diode the particles move along field lines v B with relativistic velocity v ≈ c. So continuity equation div (ρ ± v) = 0 may be rewritten as ( B · ∇)ρ ± = 0 and hence densitiesρ ± are constant along field lines. Also it is worth to note that without frame draggingρ GJ ( x) ≈ − cosχ, whereχ is the angle between field vector B( x) at point x and angular velocity vector Ω.
For simplicity we take into account the generation of electron-positron pairs only by curvature radiation of primary electrons in magnetic field. We also take into account the generation of pairs in bound state (positronium). For simplicity we assume that the probability P b that a pair is created in bound state is defined as follows [11]: where B is magnetic strength at point of pair creation, B low = 0.04 B cr , B high = 0.15 B cr , B cr ≈ 4.41 · 10 13 G [11]. In order to simplify the calculation we assume that the pair generation and its properties do not depend on photon polarization. However, we take into account the photoionization of positronium by thermal photons from hot polar cap. The photoionization rate is estimated by formula [11] where Γ is positronium Lorentz factor, T ns is neutron star surface temperature, θ ns is angular radius of neutron star at point x, W 0 = 6 · 10 5 s −1 [11]. Due to small polar cap size we neglect positronium photoionization by thermal photons from hot polar cap. In this paper we does not take into account photon splitting and positronium decay. In order to crude estimate the effect of these processes we assume that (1− f ) part of positroniums immediately decays after creation and f part of positroniums does not decay at all.
The calculations of reverse positron current are performed in two models based on extreme assumptions about the rate of parallel electric field E || = ( E · B)/B screening: the model of rapid screening [21] according to which the electron-positron plasma screens parallel electric field almost immediately and the model of gradually screening [22,23], which allows the parallel electric field penetrates deep into electron-positron plasma, see details of calculation in [19]. For simplicity we assume that anode altitude z hi is determined by equatioñ whereρ pair is number unbound electron-positron pairs generated at central field line,ρ r + is reverse positron density calculated according rapid screening model. The input of reverse positron heating to polar cap temperature T pc is estimated as where altitude z = 0 corresponds to star surface, σ B is Stefan-Boltzmann constant and all values calculated at the same field line. Polar cap luminosity due to reverse positron heating is estimated as where we integrate over polar cap surface and θ 0 = Ωr ns /c, see [19] for details.

Results
The dependence of primary electron currentρ − and diode lower plate (cathode) altitude z lo on strength of small scale magnetic field B sc is shown in fig. 3. According to used model altitude z lo does not depend on pair production at all [17]. And because of pulsar tube radius is very small θ 0 r ns ≪ z hi primary electron densityρ − also does not depend on pair production. The dependence of energy of primary electrons and diode upper plate (anode) altitude z hi on strength of small scale magnetic field B sc are shown in fig. 4 and fig. 5 correspondingly. At B sc 0.5B dip altitude z hi decreases with increasing B sc because of increasing total magnetic field strength and, most importantly, due to increasing field line curvature. Later according considered magnetic field model cathode altitude z lo begins to increase. Hence primary electrons are accelerated at larger altitude where field strength and its curvature are less. And consequently pair production becomes less effective and anode altitude z hi grows with B sc at B sc 0.6B dip . And the increasing of the altitude z hi causes an increase in the energy of primary electron. The dependence of reverse positron currentρ + and the polar cap luminosity L pc caused by this current on strength of small scale magnetic field B sc in case of rapid and gradually screening model are shown in figures 6-9. The dependence of total number of produced unbound or photoionized pairsρ pair on strength of small scale magnetic field B sc is shown in fig. 10. It is worth to note that number of pairs ρ pair produced in case of T ns = 3 · 10 5 K and f = 1 is larger than in case of W 0 = +∞.

Discussion
We consider inner gap model with stationary space charge limited flow in J0250+5854 pulsar and show that this pulsar may lye upper than pulsar "death line" in case of two assumption: the presence of surface magnetic field with very small characteristic scale ℓ ≈ 500 m and neutron star surface temperature T ns ∼ (1 − 3) · 10 5 K. Main problem is that the pulsar is very old τ = 13.7 · 10 6 years. Hence it is difficult to explain why field with so small scale has survived and why star is so hot.
It is worth to note that radiopulsar B0950+08 has spin down age τ = 17.5 · 10 6 years and star surface temperature T ns ∼ (1 − 3) · 10 5 K [24]. A such temperature may be related to internal heating mechanisms like rotochemical heating and heating due to vortex friction [25]. We also may speculate that magnetic field decay event with Hall cascade has occur not so long ago in this pulsar [26]. Hence small scale magnetic field may be generated during Hall cascades and accompanying field decay may heat up the star.
In the paper we does not take into account the photon polarization and, consequently, we can not estimate input of photon splitting effect and positronium decay, see, for example, [27,28,29]. Hence, our conclusion that magnetic field with characteristic scale ℓ ≈ 500 m is enough to explain radio radiation of the pulsar may be too optimistic. But we hope that field with ℓ ≈ 300 m would be enough. Also it is worth to note that we take into account only curvature radiation of primary electrons and resonant compton scattering may give a similar quantity of pairs [30].
Our choice of inclination angle χ = 60 • does not motivated by anything. Although we find that in considered field configuration the pulsar lye down "pulsar death" line in case of χ = 0 • and χ = 30 • . But we guess that it is only artifact of used small scale field model.