Australia Telescope National Facility

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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Introduction

The eclipsing binary system, V471 Tauri, is an interesting system for several reasons, not least because it is the only known pre-cataclysmic system to exhibit non-thermal radio emission. V471 Tau comprises a white dwarf and a K2 dwarf only slightly distorted by the Roche potential. Values of the system parameters are given in table (1). As well as emission due to flares (Crain et al. 1986), the quiescent radio emission has been shown to vary in intensity, in phase with the optical light curve (Patterson, Caillaut, & Skillman 1993; Lim, White & Cully 1996).

**Table 1:** System parameters for V471 Tau
Parameter	Value	Parameter	Value
Orbital period	12.51 hours	inclination angle	$80^\circ$
distance from Earth	45 pc	presumed age	0.6 Gyr
R_K2	0.8 $R_\odot$	R_WD	0.01 $R_\odot$
combined mass	1.4 $M_\odot$	Rotation period of WD	9.25 min
distance between stars (d_wd)	$3.1\,R_\odot = 3.9\,R_{K2}$

In another paper (Nicholls & Storey 1998) we calculated the gyrosynchrotron intensity and circular polarization from a three-dimensional model of the system's magnetic field, comprising a dipole field region around the K2 dwarf, and a magnetized stellar wind beyond the closed field lines (based on a model for RSCVn binary systems (Storey 1996)), with a region of enhancement of mildly-relativistic electron number density between the two stars. We assumed that the enhancement in density of mildly relativistic electrons between the two stars was due to the acceleration of electrons in the region where the magnetic fields of the two stars interact. The accelerated electrons are trapped in the region of the dipolar magnetic field of the K2 star and experience curvature/gradient drift, resulting in a drift in azimuthal angle from the white dwarf phase. We explored several models for the variation of electron number density in the region of enhanced number density, including different models for the decrease in number density with azimuthal angle, $\phi$ , caused by gyrosynchrotron radiation losses. We showed that a wedge of enhanced mildly relativistic electron density that precedes the white dwarf, and in which the number density falls as a power law, provides the best fit (and a good fit) to the observed variation in intensity.

The numerical modelling indicated that the enhancement in number density of relativistic electrons fills a significant fraction of the magnetosphere around the K2 star, with the angular extent of the enhancement lying between about $90^\circ$ and $200^\circ$ . However, in the numerical model we assume that such an enhancement forms, and the angle through which the mildly-relativistic electrons drift is determined by the best fit to the data. In this paper we derive an analytic approximation to the angle through which gyrosynchrotron-emitting electrons will drift to before thermalising, and show that, given the assumptions made during the derivation, electrons neither drift so far that any azimuthal structure is smeared out, nor radiate their energy before a wedge of enhanced density has time to form, but that the gyroshynchrotron-emitting electrons are expected to form a wedge of enhanced density with an opening angle that is consistent with the best-fit numerical models.

To accurately calculate the angle through which the mildly relativistic electrons drift before thermalising, it is necessary to consider the non-uniformity of the magnetic field in calculating both the radiation timescale and the variation in drift velocity as the electrons lose energy. An order of magnitude estimate ignores these significant variations, and hence overestimates the angle through which the electrons drift.

The steps we take are

to derive the bounce-averaged angular drift speed of the electrons, which depends on the Lorentz factor, $\gamma$ ,
to derive an expression for the change of $\gamma$ with time and hence the radiation timescale for the electrons
to consider the effect of the dipole field on both $\gamma(t)$ and the radiation timescale
to integrate the bounce-averaged drift speed over time
and to average over pitch angle

which leads to our final expression for the drift angle. Throughout the paper we use the frame of reference corotating with, and centered on, the K dwarf. The drifts due to the curvature and gradient of the magnetic field that we consider are drifts relative to the corotating plasma (Schulz & Lanzerotti 1974, pp4-6).

In section 2 we present a short review of concepts and quantities needed to calculate bounce-average quantities in a dipolar magnetic field. In section 3 we discuss how the Lorentz factors of the relativistic electrons change with time due to synchrotron losses in a dipolar magnetic field, and in section 4 we calculate the extent of the angular drift of relativistic electrons in the time taken for them to radiate all their energy and thermalise. The discussion section, section 5, uses this analytic expression to calculate the extent of the drift for V471 Tau.

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