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An Analytic Approximation to the Bounce-Average Drift Angle for Gyrosynchrotron-Emitting Electrons in the Magnetosphere of V471 Tauri
Jennifer Nicholls, Michelle C. Storey, PASA, 16 (2), in press.

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Application to V471 Tau

In evaluating

$\langle\phi\rangle$ for a situation such as the opening angle of the wedge of enhanced number density of mildly relativistic electrons in the model for the radio emission from V471 Tau, we have to determine reasonable values for $\gamma_f$ and $\gamma_0$ , and r₀, and our determination of these is discussed in this section.

$\gamma_f$ is chosen so that the population of electrons accelerated in the interaction region of the magnetospheres of the white dwarf and the K2 dwarf has radiated sufficient energy that it is indistinguishable from the ambient population of gyrosynchrotron electrons. This defines the far edge of the wedge, i.e. the edge furthest from the longitude of the white dwarf. Numerical models indicate that the enhanced mildly-relativistic electron number density is much greater than the ambient mildly-relativistic electron number density (Nicholls & Storey, 1998). Hence, at the far edge of the wedge most of the electrons have thermalised, so we set

$\gamma_f = 1.1$ . Gyrosynchrotron emission is by mildly relativistic electrons with Lorentz factors of a few to of order 10, which places an upper limit on $\gamma_0$ .

If the electrons were confined to a flux tube of small cross sectional area, which extended to the white dwarf radius, then the choice of r₀ would be obvious. However, numerical modelling (Nicholls and Storey 1998) indicates that such a model is not consistent with the data, whereas models with a uniform distribution of mildly relativistic electrons throughout the wedge of enhanced electron density is much more consistent with the data, implying that radial diffusion is efficient. Hence, the mildly-relativistic electrons accelerated at r = d_wd rapidly diffuse to fill the entire wedge from the radius of the white dwarf to the surface of the K2 star. It might be thought that the region of enhanced electron density would be narrow near the K2 surface, as electrons near the surface of the star have a slow drift speed and a short lifetime, with the region getting wider and wider as the equatorial radius of the electrons increases resulting in a faster initial drift speed and a longer drift time. However, some types of radial diffusion lead to an increase in energy of the electron (those mechanisms that do not violate the first or second adiabatic invariant, where the first is defined in equation (7) and the second is defined to be $\oint p_\Vert ds$ ). If this type of radial diffusion is important then the electrons that diffuse inwards will drift further in $\phi$ than ones that diffuse due to mechanisms that are energy conserving or energy losing. Further, the same processes that lead to radial diffusion near the white dwarf would be expected to operate in other regions of the magnetosphere of the K2 star, and so the inner regions of the wedge of enhanced density would be replenished by those electrons at the initial radius of r₀ = d_wd, and this would give rise to a region of enhanced number density that is independent of equatorial radius of the electrons, out to the radius of the white dwarf. Hence, we use r₀ = d_wd when evaluating the drift angle. To calculate the exact shape of the enhanced-density region is not necessary for a comparison with our numerical modelling and is beyond the scope of this paper.

A more exact calculation of the pitch angle average between

$\theta_{m\star}$ to $\pi/2$ , where

$\theta_{m\star}$ is the mirror point on the surface of the K2 dwarf for r₀ = d_wd, does not significantly change our results.

Next Section: Results and Conclusion
Title/Abstract Page: An Analytic Approximation to
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Contents Page: Volume 16, Number 2

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