As the wind from a pulsar flows out from the pulsar’s light cylinder, it is initially “cold”, meaning that the momenta of the particles are not isotropically distributed. There-fore, the particles in the “wind zone” do not radiate except possibly in Gamma-rays through inverse Compton scattering. Eventually, however, the wind reaches the inter-stellar medium. Interaction of this wind with the surrounding medium causes a standing termination shock wave [Rees and Gunn, 1974] which is believed to be the location of acceleration of particles to very high energies (see Figure 3.3).
3.2.1 The Wind Zone
The energy contained in the wind consists of two parts, the Poynting flux, FE×B, and the particle energy flux,Fparticle. Based on these fluxes one can determine the so called magnetization parameter,σ, defined as
σ= FE×B
Fparticle = B2 4πργc2,
where B,ρ and γ are the magnetic field, mass density of particles, and Lorentz factor, in the wind, respectively. Evidence suggests that as the wind flows from the pulsar light cylinder to the pulsar wind shock region, at some point this parameter undergoes a drastic change. Typical values of σ > 104, implying Poynting flux dominated wind, are observed near the light cylinder, while values less than 1, implying particle energy dominated wind, are observed just behind the shock region. One of the best studied examples of this phenomenon is the Crab Nebula.
Kennel and Coroniti [1984] assumed that the entire spin-down luminosity of the pulsar was carried away by relativistic MHD wind with a pure electron-positron plasma. By matching the velocity and pressure of the post shock flow at the boundary of the outer Nebula, they found that σ∞ ≈ 3×10−3. However, models of pair-production in polar cap gaps indicate that only 10−4-10−5 of the polar cap potential is converted into particle kinetic luminosity which would imply σL ≈ 104−105 inside the light cone (i.e. rL = c/Ω ≈ 1.5×108 cm for the Crab [Ruderman and Sutherland, 1975]). Coroniti [1990]
found that this discrepancy could be explained if the Poynting flux, in the form of a magnetically striped wind (see Figure 3.4), is converted into particle kinetic energy through a process of “magnetic reconnection” in the region between the light cone and the standing termination shock of the Nebula.
3.2.2 The Termination Shock
As the cold (i.e. strongly anisotropic) relativistic wind flows out from the light cylinder, it evenutally reaches a pressure equilibrium with the interstellar medium (ISM) where the bulk flow of the wind must stop creating a standing termination shock at a radiusrts
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Figure 3.3: The locations and radiation mechanisms of non-thermal emission from a lep-tonic pulsar wind. Top: within the light cylinder where the magnetospheric pulsed radiation from radio to Gamma-rays is produced. Middle: the wind of cold relativistic plasma between the light cylinder and the shock region with radiation production through the IC mechanism. Bottom: the surrounding synchrotron nebula (plerion) which emits broad-band electromagnetic radi-ation from radio to multi-TeV Gamma-rays through the synchrotron and IC channels. R, O, X and γ stand for radio, optical, X-ray and Gamma-ray emission, respectively. CR, IC and Sy stand for curvature, inverse Comp-ton and synchrotron radiation, respectively. The orientation of the magnetic field lines (B) is also indicated. (Figure taken from Aharonian and Bogovalov [2003].)
3 Pulsars and Pulsar Wind Nebulae
Figure 3.4: Sketch of magnetic topology in a MHD wind. The light cylinder, defined as the distance from the pulsar where a co-rotating observer would be traveling at the speed of light, i.e. rlc =c/Ω, is shown by the dashed vertical lines.
Beyond that radius, the field lines are open and expand outward in a helical structure at the speed of light. Figure taken from Coroniti [1990].
from the pulsar. For a pulsar with spindown luminosity ˙E this radius can be calculated as follows. If the particles are assumed to be moving radially symmetrically at the speed of light, c, then the total amount of energy within the termination shock is given by E˙rtsc and the pressure is thus ˙Ertsc /(4/3πr3ts), and setting that equal to the ISM pressure gives
E˙
4/3πr2tsc =PISM (3.10)
and solving forrts gives
rts =
s E˙
4/3πPISMc (3.11)
For example, for a moderately powerful pulsar of ˙E= 1036erg/s and, since typical values of the pressure of the ISM range from 103 to 104 K cm−3 [Jenkins, 2004], such a pulsar would have a termination shock radius of∼0.8 to 2.5 pc.
Further acceleration of the electrons at the termination shock may be due to a Fermi-type process [Achterberg et al., 2001], where randomization of the particle momenta within the shock region produces a population of electron energies up to & 100 TeV, capable of producing the inverse Compton radiation observed on Earth.
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3.2.3 Pulsars with Supersonic Motion
Many pulsars have been found to be moving at great speeds, from 100 to 2,000 km/s from their place of birth. This is believed to be due to a pulsar “kick”, which may be caused by an asymmetric supernova explosion This kick velocity acquired at birth is believed to be on average 450±90 km/s, exceeding the escape velocity of binary systems, globular clusters and the Galaxy [Lyne and Lorimer, 1994]. If the pulsar is moving at very high velocity through the ISM, this will result in a ram pressure which may deform the termination shock, crushing it in the forward direction and forming a bow shock ahead of the termination shock. This scenario is shown in Figure 3.5 according to simulations carried out by Bucciantini [2002]. It is possible to calculate the velocity of the pulsar if the relative radii of the forward (rFts) and backward (rtsB) shocks are known as
rFts
rtsB =γ1/2M
[van der Swaluw et al., 2003, Bucciantini, 2002] where M is the Mach number, defined asvpsr =M cs. cs, the speed of sound, is given byc2s =γp/ρ, with pthe pressure, ρ the density and γ the adiabatic constant of a monoatomic gas.
Figure 3.5: A simulation of the structure of the shock region of a a supersonic pulsar.
TS is the termination shock, SS is the spherical shock, CD is the contact discontinuity (dotdashed line), BS is the bow shock. The short-dashed lines represent two analytic solutions, the long-dashed line is the analytic solu-tion for the correct wind/environment ram pressure ratio, the dotted lines indicates the sonic surfaces (the subsonic region is in the head). The circle corresponds to the region in which the stellar wind values are fixed. Figure taken from Bucciantini [2002]
PWNe
In 2004 H.E.S.S. discovered its first unidentified, or “dark” source, HESS J1303−631. i.e., a VHE Gamma-ray source without known counterparts at other wavelengths. Since then, over 20 unidentified sources have been discovered by H.E.S.S., nearly all of them extended and lying in the Galactic plane [Aharonian et al., 2008]. One possible explanation for such sources is that they represent evolved pulsar wind nebulae. As the nebula expands the magnetic field density is expected to drop causing a drop in synchrotron brightness. On the other hand, the Gamma-rays produced by inverse Compton scattering are independent of the ambient magnetic field and may continue emitting VHE Gamma-rays for thousands of years, until adiabatic losses and IC cooling take over. This process can produce “dark” pulsar wind nebulae.