Inverse Compton Emission of Ultra-Relativistic Electrons

The synchrotron origin of the detected X-rays, discussed in the last section, implies electron acceleration to TeV energies. However, this represents only an indirect evidence and it cannot be firmly excluded that the non-thermal emission can be produced by other mechanisms. The detection of the IC component of the accelerated electron population would provide more direct evidence for the acceleration of electrons to TeV energies.

Unshocked Pulsar Wind Electrons

The cold, highly relativistic pulsar wind of the Crab Nebula remains unseen so far at all radiation wavelengths. However, in the case of PSR B1259−63, the companion star provides a very high-density photon field causing the pulsar wind electrons to undergo IC scattering leading to an observable flux of γ-rays [Ball and Kirk, 2000, Ball and Dodd, 2001]. The scattering cross section depends highly on the angle between the electron flow direction and that of the target photons, with its main contribution if they are oriented head on towards each other, and the scattered photons are emitted in the direction of the initial electron directions. Thus, the flux of γ-rays observable on Earth is maximal, if the angle between the line of sight and the line between both stars is minimal θ_p = 90^◦−i = 55^◦ when the pulsar is “behind” the star shortly prior to periastron (compare with Fig. 2.15), in contrast to the case when the pulsar is in “front”

of the star withθ_p= 125^◦post-periastron.

Figure 2.18 (left) shows the expected spectral energy distribution of γ-rays, E²dN/dE = EFE, for a minimal and maximal θp and for two values of the pulsar wind Lorentz factor, γ_wind = 10⁶,10⁷ (solid and dashed line, respectively). The Lorentz factorγ_wind determines the maximum energy a photon can obtain from the up-scattering. In fact, at the considered energies, the IC process takes place in the deep relativistic Klein-Nishina regime, resulting in a line-type spectrum ofγ-rays since the electron transfers nearly all its momentum to the photon in the rest frame of the observer.

Figure 2.18 (right) shows the light curve for an orbital time range of 200 days around perias-tron. The overall shape of the light curve is dominated by the scaling of the target photon density with the separation between pulsar and companion, n_star ∝ (D/R?)⁻², resulting in a maximum

10⁻⁹

Figure 2.18: Spectra and light curve for IC γ-ray emission from the unshocked pulsar wind of PSR B1259−63. Left: Spectral energy distribution EFE ofγ-rays forθ_p = 55^◦,125^◦and wind Lorentz factors ofγ_wind = 10⁶,10⁷[solid and dashed lines Ball and Kirk, 2000]. Right: Integrated energy flux of VHEγ-rays as a function of time with respect to periastron for a wind Lorentz factor ofγ_wind = 10⁶ [Ball and Dodd, 2001]. The solid and dashed lines correspond to a confinement of the pulsar wind by IC losses (drag) or by a termination shock, respectively.

flux at periastron. The dependence on θ_p introduces an asymmetry with respect to periastron since the corresponding modulation of the light curve is symmetrical toθ_p = 55^◦, several days prior to periastron. Additionally, the absolute flux of IC photons depends on the size of the unshocked pulsar wind region. Most likely, the wind is terminated by a shock induced by the dense Be star outflow. However, if the pressure of this outflow onto the pulsar wind is weak, the wind particles may have lost most of its energy due to the IC scattering (IC drag) and the wind is decelerated to subsonic speeds and remains unshocked. These two cases correspond to the dotted and solid line in Fig. 2.18 (right).

Shock-Accelerated Pulsar Wind Electrons

The scenario in which pulsar wind electrons are accelerated by the wind termination shock was intensively discussed in Sec. 2.3.3. Kirk et al. [1999] calculated the resulting IC emission of the accelerated electrons in a – compared to the detailed modeling by Tavani and Arons [1997]

– simplified approach and derived spectra and light curves considering different energy loss mechanisms.

In the model by Kirk et al. [1999], it was assumed that the shock is spherically symmetric, and that the ratio of the energy densities of the magnetic field and the stars photon field is constant which determines the shock position to scale with the orbital separation. The latter assumption implies an isotropic stellar outflow and requires the ram pressures of both winds to have the same radial dependence, which might be the case considering the uncertainties of recent Be star wind models, i.e. for a stellar outflow with a constant velocity profile withn= 2 (see Table 2.1).

The radiation spectra of shock-accelerated electrons were determined for the following pa-rameters:

. the power-law index α^inje of the electron injection spectrum, produced by the acceleration

process,

. the magnetic field strengthB within the radiating plasma in the down-stream region of the shock,

. the pulsar wind Lorentz factor γ₁, and the corresponding upper cutoff of the acceleration spectrum, assumed to beγ2= 100γ1.

The model parameters were tuned such that the computed spectra of synchrotron radiation match theASCAandOSSEX-ray data (see Sec. 2.3.2).

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Figure 2.19: Spectral energy distributions for synchrotron and IC emission of shock-accelerated electrons [based on Kirk et al., 1999, Fig. 2, 4, 6] for different energy loss mechanisms (see text).

Figure 2.19 shows the best fit spectral energy distributions for the resulting synchrotron (X-ray) and IC (γ-ray) radiation compo-nent at an orbital phase 12 days prior to periastron, where the maximum IC flux is expected (see below).

For the case of dominant en-ergy loss by plasma expansion (adiabatic losses, solid lines), the radiating electrons follow an en-ergy spectrum with an index iden-tical to the injection index α_e = α^inj_e . From the measured syn-chrotron spectrum follows an in-jection spectrum withα_e = 2ΓX− 1 ≈ 2.4 = α^inj_e . The ratio of synchrotron and IC energy flux is determined by the magnetic field strength – an increasing magnetic field strength results in stronger synchrotron losses ˙n_e ∝ B² and

therefore reduces the number of electronsne available for inverse Compton emission and vice versa. Since the synchrotron component is fixed by observations, a variation of Bresults in a different flux level of expected ICγ-rays. The chosen values ofB= 0.32 G and 3.2 G were jus-tified by the MHD calculations of Tavani and Arons [1997, see Fig. 2.17] and the estimate from the unpulsed radio emission near periastron (see Sec. 2.3.1). The spectrum of the IC compo-nent slightly deviates from a power law, since the scattering takes place in the transition region between the Thompson and Klein-Nishina regime (see also Sec. 6.2.1), with a corresponding γ-ray photon index of ¹₂(α_e+ 1) < Γγ < α_e +1. The sharp cutoff in the spectra is an artifact resulting from considering only accelerated electrons in a discrete energy range.

If the plasma expansion proceeds on a larger timescale than on which radiative energy losses of the electrons can occur, the initial injection spectrum gets significantly curved. Two cases were considered, dominant synchrotron losses for B = 3.2 G and dominant IC losses for B = 0.32 G, requiring injection spectra withα^inj,sy_e = 1.2 and α^inj,IC_e = 1.4, respectively. Note that such hard injection spectra may be difficult to produce with common shock-acceleration

[days] [days] [days]

τ τ τ

time relative to time relative to time relative to

B=3.2G B=3.2G B=0.32G

dominant adiabatic losses dominant radiative (synchrotron) losses dominant radiative (IC) losses

−40 −40 −40

Figure 2.20: Energy flux of synchrotron and IC radiation of shock-accelerated electrons as a function of time at orbital phases around periastron. The light curves are shown for dominant adiabatic (left) and synchrotron (middle) losses for B=3.2G, and dominant IC losses for B=0.32G (right).

processes.

Within the model, the two loss mechanisms produce quite similar spectra in the VHEγ-ray energy range, making it difficult to distinguish between them. However, the radiative losses strongly depend on the magnetic field strength and the ambient photon field density and are thus expected to strongly vary with the phase of the pulsar orbit, producing a characteristic temporal behaviour of the flux and maybe even a variable spectral shape in case of competing radiative and adiabatic energy losses. Figure 2.20 shows the model light curves – the energy flux for radiation at X- andγ-ray energies as a function of time – for the three cases of dominant adiabatic (left), synchrotron (middle), and IC (right) losses together with the measured X-ray flux (asterisks). The light curves atγ-ray energies show an asymmetry with respect to periastron which is a result of the asymmetry of the IC scattering as discussed for the unshocked wind emission. However, since the shock-accelerated electrons are isotropically distributed and only the target photons have a preferred direction, the effect is less dominant. Furthermore, theγ-ray flux should be noticeably reduced shortly pre-periastron due to the effect of photon-photon pair production by interaction with the soft photons after the IC scattering took place (dotted vs.

solid lines).

For dominant adiabatic losses (Fig. 2.20, left), the shape of the electron spectrum remains the same throughout the orbit and the orbital flux modulation is dominated by the confinement of the pulsar wind by the stellar outflow, which was assumed to be isotropic, resulting in a maximum at periastron for both, the synchrotron and IC emission, in contradiction with the X-ray data. For radiative losses, the electron spectrum index and norm varies with orbital phase, resulting in a minimum of the synchrotron emission at periastron for a given photon energy. The dominance of one of the two competing radiation mechanisms – synchrotron and IC cooling – is determined by the magnetic field strength. For a magnetic field of B = 3.2 G, synchrotron losses dominate, and the main energy is released in form of X-rays, while for B = 0.32 G IC losses are dominant and the maximum energy is contained in VHEγ-rays (see Fig. 2.20, middle and right, respectively). The evolution of the X-ray flux is not correctly reproduced in any of the three scenarios, which could be avoided if one considers adiabatic losses to be dominant for most of the orbit, but being suppressed by radiative losses near periastron, as already suggested

by Tavani and Arons [1997].

In any case, the inverse Compton emission of highly relativistic electrons accelerated within the PSR B1259−63 system results in an unambiguous characteristic shape of the light curve.