Jennifer Nicholls, Michelle C. Storey, PASA, 16 (2), in press.
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Title/Abstract Page: An Analytic Approximation to
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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.
In spherical polar co-ordinates,
, see figure (1), the dipolar magnetic field, in SI units, is
Br | = | ||
= | |||
= | 0, |
where is the permeability of free space and mm is the dipole magnetic moment. The differential equations of a field line are
(1) |
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 co-ordinate. The element of field line arc length ds follows from equation (2)
Setting
to be the strength of the magnetic field on the surface of the star at the equator, with being the radius of the star, we have
In a dipolar magnetic field there are three important timescales relating to the motion of charged particles - the cyclotron time, tc, the bounce time, tb, and the drift time, td, - 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, tc<<tb<<td, 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, , 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
(6) |
is the component of the electron's momentum, p, perpendicular to .
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 are very much slower than the cyclotron period, it can be shown that
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
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 parallel to , , is non-zero, there is a component of the Lorentz force acting on the particle that is also parallel to 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, Bm) occur symmetrically about the equator, and a particle will bounce between the mirror points. Since
at the mirror point, from equation (8) we can see that
(9) |
where we have used equation (4). Using this, and denoting the particle speed by v and the component of the speed parallel to by
, the bounce period of the electron is given by
(11) |
The bounce period is much greater than the cyclotron period,
. A related quantity is the half-bounce length,
, ie the distance along a field line between the two mirror points.
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,
where Rc is the radius of curvature of the field line and is the unit vector normal to 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, , of an electron, caused by the curvature and gradient of , is given by
Equation (13) is the instantaneous drift speed of the electron, and as B, Rc and
all change with along a field line, then VCG 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
, the associated point 0 is being displaced by
. From figure (3) we can see that the associated speed, V0s, is related to the instantaneous speed
through :
(14) |
(15) |
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, V0s, during one bounce,
Hence the bounce-average angular drift speed is
= | |||
= | (18) |
where we have used the same change of variables as for equation (12). Upon substitution of
, equation (13), and using equation (4) for , and
(19) |
where and c is the speed of light. Here is the ratio of two integrals over . Both integrals have an integrable singularity at 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
(and even at having only a 1.5% error) is
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Title/Abstract Page: An Analytic Approximation to
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