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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Shock geometry

The winds of the pulsar and its companion star will generally be separated by a pair of termination shocks separated by a contact discontinuity. The shock structures enclose the star with the weaker wind. Giuliani [1982] developed a general description of axisymmetric flows that can be applied to the interaction of two such winds. Girard & Willson [1987; hereafter GW87] followed this description and derived the equations that describe the shape and position of the boundary separating two stellar winds subject to the following assumptions:

1.

sep=0pt

2.

The shape of the shock is not affected by the orbital motion of the two stars. This implies azimuthal symmetry about the line joining the two stars.

3.

The shock region has negligible thickness.

4.

Both winds are emitted radially with no angular dependence.

5.

The winds are non-relativistic.

The resulting set of three ordinary differential equations (ODEs) can be integrated directly to find the position of the boundary, which can be loosely referred to as the `shock', given the assumption of negligible thickness. The winds are characterised by their mass loss rate $\dot M$ and (constant) radial velocity u and these enter the equations that describe the shock geometry only in terms of the ratios

$$m={\dot{M_1}\over \dot{M_2}} \qquad {\rm and} \qquad w={u_1\over u_2}$$ (1)

where the subscripts refer to the different stars. GW87 argued that the results obtained depended only on the quantity $\eta = m w$, although m and w enter the ODEs separately. In an alternative formulation presented by Huang & Weigert [1982] the quantities m and w only enter the ODEs as their product $\eta$.

Assumption (4) listed above is almost certainly not valid for a pulsar wind, which is likely to be highly relativistic. Modelling of the nebula produced by the wind of the Crab pulsar implies that the wind has a bulk Lorentz factor of $\gamma\sim10^6$ at radii well beyond the light cylinder [Kennel & Coroniti 1984]. Melatos, Johnston & Melrose [1995] considered the interaction of a relativistic wind of PSR B1259-63 comprised primarily of electrons and positrons, with a wind from SS2883 dominated by much more massive ions. They showed that when the ions are dominant in both number and energy density, as is expected in PSR B1259-63/SS2883, the dependence of the geometry only on the product of m and w is lost. Nevertheless, for a given value of $\eta$ the opening angles calculated by Melatos, Johnston & Melrose [1995] are within a factor of $\sim 2$ of the results of GW87 for a wide range of values of w/m.

In the following we approximate the shock position in the PSR B1259-63/SS2883 system using the formulation of GW87. This minimises the number of poorly known wind parameters that affect the results. The shock position is represented by its radial distance from the pulsar $r_{\rm T}(\alpha)$ where $\alpha$ is the angle between the lines joining the pulsar to the shock and the pulsar to the Be star.

Figure 1: Shock positions calculated from the formulation of GW87. The pulsar is at (0,0) and the Be star is at (1,0). The values of $\eta$ are as labelled.

Figure 1 shows the position of the termination shock (TS) obtained from numerical solution of the ODEs of GW87 for six different values of $\eta$. The quantity $\eta$ is the ratio of the rates at which the stars are transferring momentum to their winds. When star 1 is taken to be the pulsar and star 2 is the Be star it follows that

$$\eta=\frac{L_p/c}{\dot M v}$$ (2)

where $\dot M$ and v are the mass loss rate and speed of the Be-star wind, and $L_{\rm p}$ is the spin-down luminosity of the pulsar. When $\eta < 1$ the Be-star wind dominates the pulsar wind, and the TS wraps around the pulsar. As $\eta$ increases the apex of the TS moves further from the pulsar, the opening angle increases, and for $\eta > 1$ the pulsar wind is dominant and the TS wraps around the Be star.

The standoff distance from the pulsar to the apex of the TS is

$$r_{\rm T}(0) = \frac{\sqrt{\eta}}{1+\sqrt{\eta}}\; D$$ (3)

where D is the stellar separation. At large distances from the stars the momenta of the winds are almost parallel and the shock tends asymptotically to a cone characterised by a half-opening angle $\psi$ and distance to the apex $\rho_{\rm c}$. A useful empirical approximation for $\psi$ as a function of $\eta$ is [Eichler & Usov 1993]:

$$\psi=2.1\left (1-\frac{\bar{\eta}^{\frac{2}{5}}}{4}\right ) \bar{\eta}^{\frac{1}{3}}$$ (4)

where $\bar{\eta}=\min(\eta,\eta^{-1})$. Figure 2 shows the geometry of the termination shock and illustrates the parameters used to describe it. We use $\theta$ to denote the value of $\alpha$ corresponding to the line of sight from Earth to the pulsar; i.e. $\theta$ is the angle between the line of sight and the line from the pulsar to the Be star.

Figure 2: Geometry of the termination shock.

The Be star companion of PSR B1259-63 possesses an excretion disk which the pulsar passes through near periastron [Johnston et al. 1996, 1999; Ball et al. 1999]. This disk will dominate the termination of the pulsar wind for 50 days or so near periastron, greatly complicating the physics of the pulsar wind system at these epochs. Its effects on unpulsed X-ray emission close to periastron have been considered by Tavani & Arons [1997]. However its presence is irrelevant for the majority of the 1237 day binary orbit and it is neglected here. The results presented below can therefore not be taken to be accurate at epochs close to periastron.

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