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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Calculation of the average drift angle $\langle\phi\rangle$

To calculate how far the electrons drift before they have lost sufficient energy to be indistinguishable from the ambient population of mildly relativistic electrons we integrate $\langle\dot{\phi}\rangle$ with respect to time, from t=0 to t_f and then average over $\alpha_0$.

The integral over time is

$$\int_0^{t_f} dt\langle\dot{\phi}\rangle = \int_0^{t_f} {3 m_... ...B_0 r_\star^2} \left({r_0\over r_\star}\right) g(\alpha_0) dt.$$ (30)

The only time-dependent quantity in equation (30) is $\gamma \beta^2 = \gamma - 1/\gamma$. With $\gamma(t)$ given by equation (28), the integral becomes

$$\int_0^{t_f} dt\langle\dot{\phi}\rangle = {3 m_e c^2 \over 2... ...in^2\!\alpha \rangle} \ln\left({\gamma_0\over\gamma_f}\right).$$ (31)

All the $\alpha_0$ dependence is in the $g(\alpha_0)/(a \langle B^2 \sin^2\!\alpha \rangle)$ term so the average over $\alpha_0$ becomes

$${1\over a B_0^2 \left({r_\star\over r_0}\right)^6} \int_0^{\... ...ight. = {0.49\over a B_0^2 \left({r_\star\over r_0}\right)^6}.$$ (32)

Bringing this all together gives the average angle through which the electrons drift in time t_f:

$$\langle\phi\rangle = 0.48\, {9 (4 \pi\epsilon_0 )m_e^4 c^5\... ...er r_\star}\right)^7 \ln\left({\gamma_0\over \gamma_f}\right).$$ (33)

For a given system the pitch-angle-average drift angle depends only on the equatorial radius, and initial Lorentz factor. An electron with very large $\gamma_0$ drifts a very large distance before losing sufficient energy to thermalise regardless of its equatorial radius, for two reasons: its drift speed is initially very large and it lives for a very long time, which is why $\langle\phi\rangle \rightarrow\infty$ as $\gamma_0 \rightarrow\infty$. Conversely, an electron with a small initial Lorentz factor will not drift far due to a low initial drift speed and short lifetime. For an electron of given Lorentz factor the drift speed is inversely proportional to both the magnetic field strength and the radius of curvature of the field, both of which decrease with equatorial radius. Hence an electron of given Lorentz factor will drift faster the further it is away from the star. As its lifetime also increases with decreasing field strength, an electron far from the star will drift further than one close to the star.

Next Section: Application to V471 Tau
Title/Abstract Page: An Analytic Approximation to
Previous Section: The Lorentz Factor
Contents Page: Volume 16, Number 2