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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Review of Bounce-Average Quantities

In this section we present a brief review of the quantities and concepts involved in calculating a bounce-average quantity in a dipolar magnetic field, and derive a bounce-average angular drift speed for the mildly relativistic electrons. As we are concerned with gyrosynchrotron emission by electrons we will refer to electrons, but many remarks are equally valid for other charged particles. Similarly, although we use a dipolar field, many of the concepts are applicable to other magnetic field geometries.

Figure 1: Diagram of a dipolar field line showing the co-ordinates r and $\theta$, and the equatorial radius r₀.

In spherical polar co-ordinates, $(r, \theta,\phi)$, see figure (1), the dipolar magnetic field, in SI units, is

$${\mu_0 m_m\over 4\pi} {2 \cos\theta\over r^3},$$ B_r =

$$B_\theta {\mu_0 m_m\over 4\pi} {\sin\theta \over r^3},$$ =

$$B_\phi$$ = 0,

where $\mu_0$ is the permeability of free space and m_m is the dipole magnetic moment. The differential equations of a field line are

$${dr\over B_r} = {r d\theta\over B_\theta},\qquad d\phi = 0,$$ (1)

which can be integrated to give

$$r = r_0 \sin^2\theta;\qquad \phi = \mbox{const},$$ (2)

where r₀ is the equatorial radius of the field line, see figure (1). A field line is totally specified by its equatorial radius and its $\phi$ co-ordinate. The element of field line arc length ds follows from equation (2)

$$ds = (dr^2 + r^2 d\theta^2)^{1\over2} = r_0 \sin\theta\left(1+3\cos^2\theta\right)^{1\over2} d\theta.$$ (3)

Setting $B_0 = (\mu_0 m_m)/( 4\pi r_\star^3)$ to be the strength of the magnetic field on the surface of the star at the equator, with $r_\star$ being the radius of the star, we have

$$B(\theta) B_0 \left({r_\star\over r}\right)^3 \left(1+3\cos^2\theta\right)^{1\over2}$$ =

$$B_0 \left({r_\star\over r_0}\right)^3 {\left(1+3\cos^2\theta\right)^{1\over2}\over\sin^6\! \theta }.$$ = (4)

In a dipolar magnetic field there are three important timescales relating to the motion of charged particles - the cyclotron time, t_c, the bounce time, t_b, and the drift time, t_d, - which we will consider in turn. In general the cyclotron time is very much shorter than the bounce time which in turn is very much shorter than the drift time, t_c<<t_b<<t_d, and the derivations below depend on these relations holding. For more extensive reviews of this material see Roederer (1970) or Schulz and Lanzerotti (1974) and references therein.

The pitch angle, $\alpha$, of an electron is the angle between the velocity vector and the magnetic field. The electron experiences a Lorentz force which causes it to gyrate about the magnetic field line. The period of one gyration is known as the cyclotron period or gyroperiod. In the frame of reference in which an observer sees the electron in a periodic orbit perpendicular to the magnetic field, known as the guiding centre system (GCS), the cyclotron period is

$$\tau_c = {2 \pi m_e \gamma\over e B},$$ (5)

where m_e is the rest mass of the electron and $\gamma$ is its Lorentz factor. The gyroradius, also known as the cyclotron radius and Larmor radius, is the radius of gyration of the electron and is given by

$$\rho_c = {p_\perp\over e B},$$ (6)

where $p_\perp = p \sin\alpha$ is the component of the electron's momentum, p, perpendicular to $\bf B$.

In a field such as a static dipolar field, an electron moving along a field line experiences a change in the strength of the field, which in the GCS appears as a change in field strength with time. If the changes in magnitude of $\bf B$ are very much slower than the cyclotron period, it can be shown that

$${p_\perp^2\over 2 m_e B} = \mbox{const},$$ (7)

This is known as the first adiabatic invariant. In a static dipolar field the field lines are equipotentials and so as long as a particle follows a given field line, its kinetic energy will remain constant. In this case the above equation reduces to

$${\sin^2\!\alpha (s)\over B(s)} = {\sin^2\!\alpha _i \over B_i} = \mbox{const},$$ (8)

where s is the field line arc length, measured from an arbitrary point, labelled i, on the field line. We use the point where the field line crosses the equator as our reference point, and denote it by the subscript 0.

As the electron moves away from the equator the strength of the field increases and hence $\sin^2\!\alpha (s)$ increases until it reaches 1, at which point the electrons speed parallel to the magnetic field is zero. Since the component of the gradient of $\bf B$ parallel to $\bf B$, $\nabla_\Vert B$, is non-zero, there is a component of the Lorentz force acting on the particle that is also parallel to $\bf B$ which drives the electron back the way it came. The point at which the electron is reflected is known as the mirror point. For a dipolar field line the mirror points (with magnetic field strength, B_m) occur symmetrically about the equator, and a particle will bounce between the mirror points. Since $\sin^2\!\alpha = 1$ at the mirror point, from equation (8) we can see that

$$B_m = {B_0\left({r_\star\over r_0}\right)^3\over \sin^2\!\alpha _0} = {B(s)\over \sin^2\!\alpha (s)}.$$ (9)

Rearranging gives

$$\sin^2\!\alpha (s) {\sin^2\!\alpha _0\, B(s)\over B_0\left({r_\star\over r_0}\right)^3}$$ =

$${\sin^2\!\alpha _0 \left(1+3\cos^2\theta\right)^{1\over2}\over \sin^6\! \theta },$$ = (10)

where we have used equation (4). Using this, and denoting the particle speed by v and the component of the speed parallel to $\bf B$ by $v_\Vert = v(1-\sin^2\!\alpha )^{1\over2}$, the bounce period of the electron is given by

$$\tau_b = 2 \int^{s'_m}_{s_m} {ds \over v_\Vert(s)} = {2\over... ...\over B_0\left({r_\star\over r_0}\right)^3}\right)^{1\over2}},$$ (11)

where the integral is along the field line between mirror points s'_m and s_m. A change of variables from s to $\theta$, the symmetrical placing of the mirror points about the equator and using equation (3), yields

$$\tau_b ={4r_0\over v} \int^{\pi/2}_{\theta_m} { \sin\theta ... ...eta\right)^{1\over2}\over \sin^6\! \theta }\right)^{1\over2}}.$$ (12)

The bounce period is much greater than the cyclotron period, $\tau_b\gg\tau_c$. A related quantity is the half-bounce length, $S_b={1\over 2} v \tau_b$, ie the distance along a field line between the two mirror points.

Figure 2: Diagram (exaggerated) showing the drift of an electron due to a component of the gradient of $\bf B$ perpendicular to $\bf B$, $\nabla _\perp {\bf B}$. The stronger field region (bottom of figure) results in a smaller gyroradius than the weaker field region (top of figure), causing the gyroradius to vary periodically, resulting in the electron drifting across field lines.

The third timescale to consider is that of the azimuthal drift of the electrons, which arises because the dipolar field is not uniform in space. The gradient of the field means that during one gyration the electron does not experience the same field at all points on its path, which results in the radius of gyration changing in a periodic fashion. Hence the electron does not follow a strictly circular orbit, resulting in a drift across the field lines (see figure (2)). The curvature of the field also gives rise to a drift related to a centrifugal force, ${\bf F_c} = (m v_\Vert/R_c) {\bf n}$ where R_c is the radius of curvature of the field line and $\bf n$ is the unit vector normal to $\bf B$ along the radius of curvature. Both the curvature and the gradient drifts are in the same direction and always appear together, and if the curvature and gradient of the field are small enough that there is very little change in the magnetic field strength over a gyroperiod, then the drift velocity, ${\bf V}_{CG}$, of an electron, caused by the curvature and gradient of $\bf B$, is given by

$${\bf V}_{CG} = {\gamma m_e v^2\over 2 e B R_c} (2-\sin^2\!\alpha )\, {\bf e}\times{\bf n}$$ (13)

where e is the charge on the electron, and ${\bf e}$ is the unit vector tangent to $\bf B$.

Figure 3: Cartoon showing how the associated speed V_0s is related to the instantaneous drift speed, V_CG.

Equation (13) is the instantaneous drift speed of the electron, and as B, R_c and $\sin^2\!\alpha$ all change with $\theta$ along a field line, then V_CG changes along a field line as well. A more useful quantity can be defined as follows. For an electron passing through a point P there is a related point on the equator, 0, found by tracing down the field line passing through P to the equator, see figure (3). While the electron at P is being displaced by $V_{CG}\delta t$, the associated point 0 is being displaced by $V_{0s}\delta t$. From figure (3) we can see that the associated speed, V_0s, is related to the instantaneous speed $V_{CG}(\theta)$ through $r_0\phi$:

$$V_{0s} \delta t = r_0\phi,$$ (14)

and

$$V_{CG}(\theta) \delta t= r_0 \sin^3\theta \phi,$$ (15)

so

$$r_0\phi = V_{0s} \delta t = {V_{CG}(\theta)\over \sin^3\theta} \delta t.$$ (16)

Hence we can define the bounce-average drift speed, which is the drift speed of the electron's guiding centre at the equator, averaged over all the associated speeds, V_0s, during one bounce,

$$\langle V_0\rangle = {2\over \tau_b} \int^{s'_m}_{s_m}V_{0s}{ds \over v_\Vert(s)}.$$ (17)

Hence the bounce-average angular drift speed is

$$\left\langle \dot{\phi}\right\rangle \langle {V_0\over r_0}\rangle = {2\over r_0 \tau_b} \int^{s'_m}_{s_m} V_{0s}{ds \over v_\Vert(s)}$$ =

$${4\over r_0 v \tau_b} \int^{\pi/2}_{\theta_m}\left({V_{CG}(\theta... ...left(1+3\cos^2\theta\right)^{1\over2}\over \sin^6\! \theta }\right)^{1\over2}},$$ = (18)

where we have used the same change of variables as for equation (12). Upon substitution of $V_{CG}(\theta)$, equation (13), and using equation (4) for $B_0(\theta)$, and

$$R_c = \left({r_0\over 3}\right) {\sin\theta\left(1+3\cos^2\theta\right)^{3\over2}\over(1+\cos^2\theta)}$$ (19)

we obtain

$$\langle \dot{\phi}\rangle = {3 m_e c^2 \gamma \beta^2\over 2 e B_0 r_\star^2} \left({r_0\over r_\star}\right) g(\alpha_0),$$ (20)

where $\beta = v/c$ and c is the speed of light. Here $g(\alpha_0)$ is the ratio of two integrals over $\theta$. Both integrals have an integrable singularity at $\theta_m$ and due to the complexity of their integrands can only be evaluated numerically. A reasonable fit to this ratio, i.e. better than 1% for $6^\circ \le \alpha_0\le 180^\circ$ (and even at $3^\circ$ having only a 1.5% error) is

$$g(\alpha_0) = 0.7 + 0.3 \sin\alpha_0.$$ (21)

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Title/Abstract Page: An Analytic Approximation to
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Contents Page: Volume 16, Number 2