Shock geometry and inverse Compton emission from the wind of a binary pulsar

Lewis Ball, Jennifer Dodd, PASA, 18 (1), in press.
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Subsections

Inverse Compton scattering

Ball & Kirk [2000a; Equation 15] derived an expression for the radiation transfer that determines the intensity of the radiation emitted at the normalised energy

$\epsilon_{\rm out}=E_{\rm out}/m_{\rm e}c^2$. Furthermore, it was shown that absorption of the scattered photons due to pair production on the Be-star photons is negligible. It follows that when the termination of the wind is included, the scattered intensity can be written as a line of sight integral of the scattering due to electrons moving radially out from the pulsar,

$\displaystyle I(\epsilon_{\rm out})=\int_{0}^{r_{\rm T}(\theta)} \left({L_{\rm ... ...} \; \frac{{\rm d}N_{\gamma}} {{\rm d}\epsilon_{\rm out}{\rm d}t}\,{\rm d}s \;,$     (5)

where the fraction of the wind momentum carried by ions is assumed to be negligible. The quantity

$\gamma_{\rm w}(0)$ is the initial Lorentz factor of the pulsar wind and

${\rm d}N_{\gamma}/{\rm d}\epsilon_{\rm out}{\rm d}t$ is the differential rate of emission of inverse Compton scattered photons by a single electron. In the calculations of Ball & Kirk [2000a] the TS was assumed to be very distant from the pulsar, so that the upper limit of the integration was effectively infinity.

The termination of the pulsar wind will necessarily have the largest effect on its inverse Compton emission for small values of $\theta $, since regardless of the value of $\eta $,

$r_{\rm T}(\alpha)$ is a monotonically increasing function. Furthermore, these effects will be most significant if $\eta $ is small, since in such cases the pulsar wind terminates close to the pulsar. The orbit of the PSR B1259-63 system is inclined at $i=35^\circ$ to the plane of the sky, so an observer samples angles $\theta $ between $90-i=55^\circ$ and

$90+i=125^\circ$ over the binary period. The variation of $\theta $ over the orbit is shown in Figure 3. If $\eta \ll 1$ the pulsar wind will terminate very close to the pulsar for small $\theta $, decreasing the observable emission from the unshocked wind most significantly at binary phases which correspond to values of $\theta $ that are close to the minimum, i.e. between around day -200 and periastron.

Figure 3: The variation of the angle $\theta $ between the line of sight to PSR B1259-63 and the line joining the pulsar and its Be-star companion over the binary orbit. Day 0 corresponds to periastron, which next occurs on 2000 October 17.
\begin{figure} \epsfxsize=6 cm \centerline{\epsffile{ballfig3.ps}}\end{figure}

Spectra

Figure 4: Inverse Compton emission spectra for the unterminated (dashed lines) and terminated (solid lines) pulsar wind, for $\eta =0.01$ (left panels) and $\eta =0.1$ (right panels). The top row shows the spectra for

$\theta =55^{\circ }$, and the bottom row is for

$\theta =125^{\circ }$, the extremes of this angle that can be sampled from Earth. All the spectra shown have been calculated for a pulsar wind with an initial Lorentz factor

$\gamma _{\rm w}(0)=10^6$.

\begin{figure} \epsfxsize=12 cm \centerline{\epsffile{ballfig4.ps}}\end{figure}

Figure 4 shows the spectra of inverse Compton emission for two values of $\eta $ for which the Be-star wind dominates the pulsar wind, which therefore terminates quite close to the pulsar for small $\theta $. The quantity plotted is the spectral energy distribution EFE where FE is the energy flux at Earth of inverse Compton scattered photons of energy E. The solid lines show the results when the termination of the pulsar wind is included, and the dashed lines show the spectra calculated by Ball & Kirk [2000a] when the shock termination is very distant from the pulsar. The spectra are shown assuming that the separation of the pulsar and the Be star is that which applies at periastron, and have been calculated for the extreme values of $\theta $ that can be sampled from Earth. Both of these extreme values occur very close to periastron, as can be seen from Figure 3, so the assumption of periastron separation is appropriate. Furthermore, it is assumed that the spectrum of target photons from the Be star can be approximated as a monochromatic distribution at a normalised energy

$\epsilon_0 = 2.7k_{\rm B}T_{\rm eff}/(m c^2)\approx 10^{-5}$ where $T_{\rm eff}$ is the star's effective temperature. Calculations including a more realistic Be-star spectrum have been presented by Ball & Kirk [2000b] and Kirk, Ball & Skjæraasen [2000]. The effects of including a better approximation to the target photon spectrum are generally small, but may warrant closer investigation if the system is detected in hard $\gamma$-rays.

All the spectra in Figure 4 are sharply peaked at an energy close to

$5\times 10^5\;\rm MeV$, somewhat lower than the Thomson limit of

$\gamma_{\rm w}(0)^2 m_{\rm e}c^2 \epsilon_0\approx5\times 10^6\;\rm MeV$ because of the importance of Klein-Nishina effects. In the Klein-Nishina regime, i.e. when

$\gamma\epsilon_0 \,\raisebox{-0.4ex}{$\stackrel{>}{\scriptstyle\sim}$}\,1$, a scattering particle loses a significant fraction of its energy to the scattered photon and the maximum scattered energy is then

$\epsilon_{\rm out}^+\sim \gamma$. The upper limit of the emission spectrum is unaffected by the inclusion of the TS, since the scatterings that produce photons of this energy involve undecelerated electrons (with Lorentz factors

$\sim \gamma_{\rm w}(0)$) and occur very close to the pulsar. As the pulsar wind propagates away from the pulsar the wind electrons lose energy via the inverse Compton scattering process. Lower energy scattered photons are generated farther from the pulsar, and so tend to be suppressed by the termination of the wind.

For small $\theta $ and small $\eta $ the TS is very close to the pulsar where inverse Compton scattering is still very efficient because the target photon density is large and the scatterings are close to head on. The wind is terminated before it has been decelerated to Lorentz factors significantly below

$\gamma_{\rm w}(0)$, so the scattering electrons are essentially monochromatic, as are the target photons in our approximation. This has the effect of dramatically reducing the emission at all energies below the upper cutoff, producing a scattered spectrum that is even more sharply peaked than that resulting from the unterminated wind. Such a case is illustrated in the top left panel for $\eta =0.01$ and

$\theta =55^{\circ }$, for which the scattered spectrum is essentially monochromatic at the upper cutoff.

The TS radius increases rapidly with increasing $\alpha$ (for a given $\eta $), with the rate of increase approaching infinity as $\alpha$ approaches

$180^\circ-\psi$ where $\psi$ is the half-opening angle of the cone and $\eta < 1$ is assumed. At energies near the upper cutoff the scattered spectrum from the terminated wind therefore rapidly approaches that from the unterminated wind as $\theta $ increases. The suppression at lower energies, which are produced by scatterings well away from the pulsar where the wind has been significantly decelerated, persists to larger $\theta $. This effect is dramatically illustrated by a comparison between each top panel of Figure 4 with the corresponding lower panel. For the case where

$\theta =125^{\circ }$ and $\eta =0.01$ the emission at and just below the cutoff energy is essentially unaffected by the inclusion of the TS, because the scatterings that produce photons of these energies occur closer to the pulsar than does the TS.

For larger values of $\eta $ the TS occurs further from the pulsar. It follows that for a given $\theta $, the difference between the inverse Compton emission from the terminated and the unterminated wind decreases as $\eta $ increases. For the case shown in the lower right panel of Figure 4, with $\eta =0.1$ and

$\theta =125^{\circ }$, the TS occurs so far from the pulsar that it has essentially no effect on the scattered emission in the energy range shown, though it does reduce the scattered flux at even lower energies.

Light curves

For $\eta > 1$ the TS wraps around the Be star and the line of sight to the pulsar does not intersect the shock if $\theta > \psi$. Equation (4) implies that for

$\eta \,\raisebox{-0.4ex}{$\stackrel{>}{\scriptstyle\sim}$}\,7$,

$\psi \,\raisebox{-0.4ex}{$\stackrel{<}{\scriptstyle\sim}$}\,55^\circ$ which is the minimum observable value of $\theta $ over the binary orbit of PSR B1259-63. Thus if

$\eta \,\raisebox{-0.4ex}{$\stackrel{>}{\scriptstyle\sim}$}\,7$ the line of sight doesn't intersect the TS at any phase of the binary orbit, the TS has no effect on the observable inverse Compton emission from the freely-expanding wind, and the results of Ball & Kirk [2000a] require no modification.

For $\eta < 1$ the TS wraps around the pulsar and is intersected by the line of sight to the pulsar if

$\theta < 180^\circ -\psi$. It follows from Equation (4) that if

$\eta \,\raisebox{-0.4ex}{$\stackrel{<}{\scriptstyle\sim}$}\,0.14$,

$180^\circ - \psi \,\raisebox{-0.4ex}{$\stackrel{>}{\scriptstyle\sim}$}\,125^\circ$ which is the maximum observable value of $\theta $, and thus the line of sight to the pulsar intersects the TS at all binary phases.

Figure 5: Light curves showing the integrated energy flux at the Earth from PSR B1259-63 over the whole orbital period (left panel), and over 200 days centred on periastron (right panel). The initial Lorentz factor of the pulsar wind is

$\gamma _{\rm w}(0)=10^6$. The dashed curves show the emission from an unterminated wind as calculated by Ball & Kirk [2000a]. The solid curves show the emission when the wind is terminated by a shock whose position is determined by the parameter $\eta =0.1$.
\begin{figure} \epsfxsize=12 cm \centerline{\epsffile{ballfig5.ps}}\end{figure}

Figure 5 shows the orbital variation of the integrated flux density

$\int F_E \,{\rm d}E$, expected from PSR B1259-63 for

$\gamma _{\rm w}(0)=10^6$. The dashed curve shows the emission from the unterminated wind as calculated by Ball & Kirk [2000a]. The solid line shows the emission when the wind is terminated by a shock at the position shown in Figure 1 for $\eta =0.1$. The TS is wrapped around the pulsar and is intersected by the line of sight to the pulsar at all binary phases. However, between days +50 and +400 the intersection point is sufficiently distant from the pulsar that it is beyond those radii where the majority of the inverse Compton scattering occurs. The termination of the wind therefore has little effect in reducing the inverse Compton emission in this period, and the light curve from the terminated wind is indistinguishable from that of an unconfined wind.

The effect of the termination of the wind is asymmetric because of the asymmetry in the dependence of the line-of-sight angle $\theta $ about periastron. The angle $\theta $ decreases very rapidly towards its minimum value just prior to periastron, which implies that the radius at which the line of sight to the pulsar intersects the TS does the same. The increase from the minimum value of $\theta $ to the maximum then occurs very rapidly, starting just a few days before periastron. The termination of the wind therefore decreases the emission from the freely expanding portion of the pulsar wind most effectively prior to periastron, and thus has the effect of decreasing the characteristic asymmetry in the $\gamma$-ray light curve about periastron. For $\eta =0.1$ Figure 5 shows that the effect of the TS is to reduce the maximum integrated inverse Compton flux density by a factor of $\sim4$, to make the time at which the maximum occurs a few days later, and to reduce the asymmetry about periastron measured by the ratio of the integrated fluxes at days $\pm50$, from $\sim 3$ to $\sim 1.6$.

Figure 6: Light curves showing the integrated energy flux at the Earth from PSR B1259-63 when the wind is terminated by a shock whose position is determined by the parameter $\eta =0.01$ (solid curve). The initial Lorentz factor of the wind is

$\gamma _{\rm w}(0)=10^6$. The dashed curves show the emission from an unterminated wind as calculated by Ball & Kirk [2000a] (and are identical to those in Figure 5).
\begin{figure} \epsfxsize=12 cm \centerline{\epsffile{ballfig6.ps}}\end{figure}

Figure 6 shows the light curve for a shock-terminated wind with $\eta =0.01$. In this case the shock is so tightly wrapped around the pulsar that its intersection with the line of sight to the pulsar always occurs at a radius where inverse Compton scattering is still effective. The emission from the terminated wind is thus substantially less than that from an unconfined wind throughout the binary orbit. The observable flux in this case is reduced by a factor of between 1.5 and 3 over roughly 80% of the orbital period. The maximum integrated hard $\gamma$-ray flux density is a factor of $\sim 10$ lower than that from an unterminated wind and occurs $\sim 6$ days later, and the $\pm50$ day asymmetry is actually reversed to $\sim0.95$.

Next Section: Discussion and conclusions
Title/Abstract Page: Shock geometry and inverse
Previous Section: Shock geometry
Contents Page: Volume 18, Number 1

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