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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The Lorentz Factor

In the previous section we review the basic concepts necessary for calculating bounce-average quantities, finishing with the derivation of the bounce-average angular drift speed, which is dependent on several quantities, not least in this context being the Lorentz factor $\gamma$. In order to calculate how far an electron drifts in a given time, we need to know how $\gamma$ changes with time due to synchrotron losses, which is derived in this section. We also demonstrate the validity of our assumption that the timescale of synchrotron losses is very much greater than the bounce time.

For the application under consideration here, the opening angle of the wedge of enhanced number density of mildly relativistic electrons in our model of V471 Tau, the important timescale is the time taken for a group of mildly relativistic electrons to lose its energy and thermalise. In the case of V471 Tau only synchrotron losses are taken into consideration. In this case Coulomb collisions are unimportant until the number density of thermal electrons is so high that Razin supression reduces the gyrosynchrotron emission to unobservable levels. The other potentially important loss mechanism is precipitation onto the K2 dwarf surface of those electrons with equatorial pitch angle sufficiently small that their mirror points are inside the star. It can be shown (Nicholls and Storey 1998; Kundu et al 1987) that scattering is efficient in replenishing those pitch angles that lead to precipitation, and so it is reasonable to assume an isotropic distribution throughout the magnetosphere. However, scattering is not expected to take place uniformly throughout the magnetosphere and the loss due to precipitation is reduced by a factor related to the volume of the magnetosphere in which scattering occurs. In this paper loss through precipitation is assumed to be negligible, so the opening angle calculated below will be an upper limit.

Kardashev (1962) showed that for synchrotron radiation the energy of the particle decreases as

$${d E\over dt} \propto - B^2 \sin^2\!\alpha E^2,$$ (22)

or in terms of the Lorentz factor (Petrosian 1985)

$${d \gamma\over dt} -{2 e^4\over 3 (4 \pi \epsilon_0) c^3 m_e^3} B^2 \sin^2\!\alpha \gamma^2 \beta^2$$ =

$$-a B^2\sin^2\!\alpha (\gamma^2-1),$$ = (23)

where $a = (2 e^4)/(3 (4 \pi \epsilon_0) c^3 m_e^3)$. Integrating equation (23), yields

$$\gamma(t) = {(\gamma_0+1) +(\gamma_0-1)\exp(-2 a B^2 \sin^2\... ...ver(\gamma_0+1) -(\gamma_0-1)\exp(-2 a B^2 \sin^2\!\alpha t)},$$ (24)

where $\gamma_0$ is the initial Lorentz factor of the electron.

The radiation timescale is the time taken for the electron to reach Lorentz factor $\gamma_f$, and is given by

$$t_f = {1\over 2 a B^2 \sin^2\!\alpha }\ln\left({(\gamma_0-1)(\gamma_f+1)\over(\gamma_0+1)(\gamma_f-1)}\right).$$ (25)

In a uniform field $B^2 \sin^2\!\alpha$ is constant, and for $\sin^2\!\alpha$ close to zero, emission is suppressed as the electron is travelling almost parallel to the magnetic field line and therefore in an almost straight line. However, in a dipole field $B^2 \sin^2\!\alpha$ are changing constantly. If the rate of change of $B^2 \sin^2\!\alpha$ is very much greater than the radiation timescale, that is if the electron is losing energy sufficiently slowly that it undergoes many bounces before its Lorentz factor changes appreciably, then it is a reasonable approximation to average $B^2 \sin^2\!\alpha$ over a bounce period and use the average in equation (28).

For the physical parameters applicable to V471 Tau, see Tables 1 and 2, the bounce period for electrons on field lines that reach the white dwarf is of the order of seconds to a few tens of seconds for Lorentz factors greater than 3 and for the full range of pitch angles. Using the model magnetic field strength at the radius of the white dwarf (Nicholls and Storey 1998), equation (25) yields a lower limit to t_f of 8 x 10⁸s for an initial Lorentz factor of 3, and a final Lorentz factor of 1.1. For initial Lorentz factors higher than 3, t_f will be even longer. So, for V471 Tau it is a reasonable approximation to use the average of $B^2 \sin^2\!\alpha$ over a bounce period.

Table 2: Model parameters for V471 Tau (from Nicholls and Storey 1998)

Parameter	dipole magnetosphere	wind	enhanced region
surface magnetic field at equator (tesla)	$3.5\times10^{-3}$	$1.0\times10^{-3}$	$3.5\times10^{-3}$
electron density at stellar surface (m-3)*	$4.0\times10^{8}$	$4.0\times10^{8}$	$6.5\times10^{10}$
power law dependence of electron density on radius*	0.0	-2.0	0.0
power law dependence of electron density on energy*	-2.0	-2.0	-2.0
extent in stellar radii	3.9	13.0	3.9
*These parameters refer to the mildly-relativistic electrons.

The bounce-average of $B^2 \sin^2\!\alpha$ is

$$\langle B^2 \sin^2\!\alpha \rangle = {2\over S_b} \int^{\pi/... ...eta\right)^{1\over2}\over \sin^6\! \theta }\right)^{1\over2}}.$$ (26)

Again this yields a ratio of integrals that can only be done numerically due to the complexity of the integrands. The fit,

$$\langle B^2 \sin^2\!\alpha \rangle = B^2_0\left({r_\star\ove... ...6 \left(0.913 +\left({0.75\over \alpha_0}\right)^{3.3}\right),$$ (27)

to the ratio of integrals in equation (26) agrees to better than 10% for all values of $\alpha_0$. Hence the time dependence of $\gamma$ is

$$\gamma(t) = {(\gamma_0+1) +(\gamma_0-1)\exp(-2 a \langle B^2... ...-(\gamma_0-1)\exp(-2 a \langle B^2 \sin^2\!\alpha \rangle t)},$$ (28)

with $\langle B^2 \sin^2\!\alpha \rangle$ given by equation (27).

Substituting equation (27) into the equation for the radiation timescale, equation (25), we get

$$t_f = {1\over a B^2_0\left({r_\star\over r_0}\right)^6 \left... ...(\gamma_0-1)(\gamma_f+1)\over(\gamma_0+1)(\gamma_f-1)}\right).$$ (29)

An inspection of this equation shows that as $\alpha_0 \rightarrow 0,\ t_f \rightarrow 0$, and hence $t_f\le10^9$s, a total reversal from the uniform field case stated above, which can be understood as follows. In a uniform magnetic field the magnetic field strength, and hence the pitch angle of the electron, is unchanging, so emission is most efficient for an electron with initial pitch angle of $\pi/2$, and suppressed for initial pitch angles of close to 0. However, in a dipole field the magnetic field experienced by the electron changes constantly with an associated change in pitch angle. As $\sin^2\!\alpha _0 \rightarrow 0,\ \theta_m\rightarrow 0$, and so the electrons with very small equatorial pitch angles reach very high magnetic field strengths before they mirror. In other words, the electrons with smallest equatorial pitch angles spend the greatest amount of their bounce period in regions where the magnetic field is very large, and with their instantaneous pitch angles close to $\pi/2$ where their radiation is most efficient. Hence they will lose energy extremely rapidly. However, electrons with equatorial pitch angle of close to $\pi/2$ are trapped in low field regions and hence take longer to radiate their energy.

In practise the electrons with small enough pitch angle will precipitate onto the stellar surface, and the average radiation time for the remaining electrons will remain large. For electrons with $\alpha_0 = \pi/2$, and hence trapped on the equator with unchanging conditions, the radiation time is the same as for electrons in a uniform magnetic field of the same strength and with pitch angle of $\pi/2$, as expected. For electrons that mirror just above the stellar surface on a field line with r₀ = d_wd the radiation time is $2\times10^6$s, about 5 orders of magnitude greater than the bounce time. Hence the assumption of the bounce time being very much shorter than the radiation time, used to derive the expression for t_f and $\gamma(t)$, is valid for our parameters of V471 Tau.

Next Section: Calculation of the average
Title/Abstract Page: An Analytic Approximation to
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Contents Page: Volume 16, Number 2