Shock Acceleration of Electrons

The non-thermal radiation emitted by the system indicates particle acceleration to relativistic energies which can result from a variety of mechanisms. For example, several other binary systems containing neutron stars feature non-thermal X-ray emission (X-ray binaries) which is attributed to accretion of stellar matter onto the neutron star surface. Other mechanisms include interaction of stellar material with the pulsar magnetosphere, Be star coronal emission, and Bremsstrahlung of heated gas in the stellar outflow. However, Tavani and Arons [1997]

discussed the latter cases and found, that the observational data is not compatible with these scenarios and consider shock acceleration of electrons to be the most likely effect.

In the following, the characteristics of the Be star and pulsar wind interaction and the impli-cations are discussed with respect to the observational data.

Be Star – Pulsar Wind Interaction

Figure 2.14 illustrates the basic properties of the interaction of the pulsar wind with the stellar outflows [see also Melatos et al., 1995, Tavani and Arons, 1997]. At a given distance from the pulsar, the momentum flux of the pulsar wind matches that of the stellar outflow (contact discontinuity) resulting in reverse shocks to be formed – the stellar outflow and the pulsar wind termination shock. The shock surface is bent in the direction of the star or the pulsar, depending on the strength of the wind, forming a bow-shockwith its apex on the line between the pulsar and the star, in contrast to the spherically symmetric shock in the case of the Crab Nebula.

Therefore, the plasma in the down-stream region between the contact and the shock surface expands in the direction away from the apex and a flow component parallel to the shock surface is established, different from the axial symmetric situation in the Crab synchrotron nebula, influencing the energy loss mechanisms of shock-accelerated particles in these regions.

pulsar

Figure 2.14: Schematic view of the interaction between the stellar outflow and the pulsar wind.

Since the density of the stellar outflow is enhanced within the equatorial disk, a

“comet-like” situation arises when the pul-sar approaches periastron and enters the disk, such that the shocks appear close to the pul-sar and the stellar material flows around the pulsar. As the shock geometry changes with the orbital phase, the non-thermal processes, which are expected to occur near the shocks, can produce a variability of non-thermal radi-ation emitted from the interaction region on a timescale of a few weeks in case the disk lies in the orbital plane. In the contrary case of an orbital inclination, the pulsar passes the equatorial disk two times close to periastron, atτ1andτ2, significantly changing the shock geometry on an even shorter timescale of a few days. Figure 2.15 illustrates this scenario with respect to an observer at earth, assum-ing the most likely disk geometry (see also Table 2.1).

Figure 2.15: Sketch of the orbit of PSR B1259−63 with re-spect to the line of sight. The pulsar approaches the equatorial disk prior to periastron (τ₁) while it is “behind” the compan-ion star and turns towards the observer before it crosses the disk the second time (τ₂) after periastron [based on Johnston et al., 1999, Fig. 3]. During the interaction of the disk of the Be star and the pulsar, a more comet-like bow shock is formed.

The two shocks can pro-duce two different populations of shock-accelerated particles, orig-inating from the stellar outflow and the pulsar wind, respectively.

Both shock regions should have different properties such as par-ticle energy spectrum and den-sity, depending on the initial pre-shock attributes of the plasma like the mean kinetic energy of the particles, and the magnetic field strength within the plasma.

Be Star Electrons

Within the above picture of colliding winds, the transient non-thermal radio emission (see Sec. 2.3.1) was explained by Ball et al. [1999] as synchrotron emis-sion of electrons accelerated in

the Be star shock and evolving in the corresponding down-stream region. Therein, two ar-guments are given, showing that the electrons originate from the Be star and not from the pulsar

wind. Firstly, the low energy of the synchrotron photons indicates a parent spectrum of low energy electrons, which would imply, if the electrons originate from the pulsar, that the pulsar wind has a Lorentz factor ofγ≈ 10, five orders of magnitude lower than that assumed to be the case for young pulsars. Since the electrons from the unshocked Be star outflow are expected to be not highly relativistic, they could easily produce the observed synchrotron spectrum in the post-shock region. Secondly, the electron number density expected for the shocked pulsar wind is far from being sufficient in order to explain the observed flux level.

Furthermore, it was shown, that the observed flux is associated with the shock induced by the equatorial disk outflow rather than that from the polar wind, since the ratio of the fluxes observed at periastron and apastron is more than an order of magnitude higher than expected for the interaction with the polar wind only.

The observed light curves around periastron can be qualitatively reproduced with a model, where the pulsar wind interacts at a limited time interval with the disk material, and leaving behind a “synchrotron bubble” which evolves at timescales of weeks by synchrotron emission only, allowing to explain the slow decay of the observed flux. Possible adiabatic losses were neglected because the bubble was considered to expand very slowly. This naturally explains the evolution of the two flux maxima and provides further evidence for a significant disk inclination.

In the model, the disk crossings were assumed to occur at the times indicated by the pulsar eclipse.

It is also important to note, that the maximum flux of the unpulsed emission occurs some days later than the assumed disk crossing times, which is explained in the model by an accumu-lation of accelerated electrons implying a fast acceleration compared to the timescale of energy losses through synchrotron radiation. However, no good explanation is found for the different flux levels of the maxima for different periastron passages (see Fig. 2.10) but it was speculated about inhomogeneities of the stellar outflow causing a varying pressure onto the pulsar wind [Connors et al., 2002]. The time evolution of the synchrotron spectrum allows a rough estimate of the magnetic field strength in the vicinity of the radiating plasma, and it was determined from the 2000 periastron passage data to be≈1.6 G [Connors et al., 2002].

Pulsar Wind Electrons

The acceleration of electrons originating from the Be star disk cannot easily account for the observed X-ray emission of PSR B1259−63 since the typical acceleration timescale is much larger than the timescale of synchrotron losses. In contrast, the pulsar wind electrons have already got a high kinetic energy before being shock-accelerated, provided that the wind prop-erties are similar to that of the Crab Nebula with a Lorentz factor of the wind ofγ_wind >10⁵, i.e.

resulting in an injection spectrum starting above several GeV. The observed X-rays then result from synchrotron radiation in the magnetic field within the down-stream region of the pulsar wind shock.

In Tavani and Arons [1997], a complete MHD treatment of the pulsar wind termination shock was performed according to the model of Kennel and Coroniti [1984], developed for the Crab Nebula (see Sec. 2.1.4), but adapted to the environment of the system of PSR B1259−63.

The calculations fit best for pulsar wind parametersγwind= 10⁶andσ=0.02 at the distance of the termination shock. The fact that the magnetisation parameterσis much higher than for the Crab pulsar is explained by the much smaller distance of the shock to the pulsar magnetosphere

such that less magnetic energy was converted into kinetic energy.

Figure 2.16: X-ray luminosity (top) and photon in-dex (bottom) as a function of orbital phase together with the best fit solution of a detailed MHD model of the pulsar wind shock by Tavani and Arons [1997].

In order to reproduce the observed X-ray spectra and light curve, Tavani and Arons [1997] computed the shock char-acteristics for different stellar outflow pa-rameters and for both, an inclined and coplanar disk. They fitted the model to theASCAX-ray data obtained around the 1994 periastron. Figure 2.16 shows the X-ray luminosityLX (top) and photon index ΓX(bottom) in the energy range 1–10 keV as a function of the orbital phase ω for the best model fit to the data. The model clearly favours a disk inclination, with a rather big opening angle θdisk = 30^◦ and, more interestingly, a disk orientation an-gle ω_disk = 120^◦ slightly deviating from the estimate deduced from radio observa-tions and predicting a flux minimum to occur slightly later than periastron.

In Figure 2.17 (left, middle), the shock characteristics corresponding to the best fit model are displayed: the magnetic field strength in the down-stream region of the

pulsar wind shock B2and the distance of the shock from the pulsarrs, showing that the interac-tion with the disk outflow strongly confines the pulsar wind, resulting in a local minimum ofr_s and a corresponding maximum of B₂ near the disk crossings atω ≈ 120^◦ and 300^◦. However, the model assumption of a spherical shock geometry does not consider any asymmetric flow effects, as indicated in Fig. 2.14, which might be important.

Figure 2.17 (right) shows the characteristic timescalest = /˙relevant for the acceleration and evolution of the radiating electrons, in particular:

. taccas the acceleration time,

. tsyas the synchrotron radiation timescale in the down-stream magnetic field,

. tIC,Th,tIC,KNas the timescales on which the electrons undergo inverse Compton scattering with thermal photons emitted by the star, either in the non-relativistic Thompson or relativistic Klein-Nishina regime, respectively,

. tad as the timescale on which adiabatic losses proceed by removing the electrons from the region where acceleration or radiation processes can occur.

Clearly, the acceleration process has the shortest timescale, thus producing a power-law injec-tion spectrum with index α^inj_e and Lorentz factors between γ₁ and γ₂ > γ₁ for the relativistic electrons accelerated in the shock and injected into the down-stream region. The lower end of the spectrum is determined by the Lorentz factor of the pulsar wind, γ1 = γwind ≈ 10⁶ while the upper end remains unknown but is constrained by observations to be γ₂ ≤ 10⁴γ₁, since the extrapolated power law synchrotron spectrum (see Fig. 2.13) is violated by the limits on the

B (r ) [G]2 _s

Figure 2.17: Best fit model for the pulsar wind termination shock [Tavani and Arons, 1997]. Shown are the shock distance from the pulsar r_s, the distance between the pulsar and the Be star D (both middle panel), the down-stream magnetic field strength B₂(rs)(left panel), and the acceleration and energy loss timescales for electrons (right panel, see text) as a function of the orbital phaseω.

soft γ-ray emission found by EGRET(20 MeV–10 GeV). The power-law indexαe of the radi-ating electrons, which could be different from the injection index since the population evolves in time, is given byα_e = 2ΓX−1≈ 2.4 (see Sec. 2.1.3). According to Fig. 2.17, adiabatic ex-pansion seems to be the dominant energy loss mechanism over a wide range of the pulsar orbit, which would indicate a spectrum withα^inj_e = α_e. Since the magnetic field strength is enhanced near the disk crossings, synchrotron losses become comparable to the adiabatic losses for the first disk crossing, shortly prior to periastron. The timescale which applies for inverse Compton losses of the electrons lies somewhere in betweent_IC,Th and t_IC,KN because the scattering takes place in the transition region of both regimes, highly depending on the energy of the electrons (see also Sec. 6.2.1). The maximum of IC scattering near periastron is a direct consequence of the small distance between the pulsar and its companion, where the density of thermal pho-tons originating from the star reaches its maximum. The local minimum of the X-ray flux at periastron can be explained if the corresponding IC timescale becomes shorter than that of the adiabatic losses and especially synchrotron losses which result in the X-ray radiation. Addi-tionally, the dominant IC losses lead to a softer electron and therefore softer X-ray emission spectrum, because the scattering cross section increases with the electron energy, consistent with actual observations (see Fig. 2.16).

It is important to note that the radiated photons resulting from IC losses would have a mini-mum energyEIC ∼_e^min= γ1me ≈5.11×10¹¹eV= 0.511 TeV because in the scattering process the electrons transfers their momentum almost completely to the soft photon. Thus, the IC radiation would be VHEγ-rays, with a flux maximum at periastron.