Collisionless shock waves in space and astrophysical plasmas can accelerate
electrons along the shock layer by an electrostatic potential, and scatter or
reflect electrons back to the upstream region by the amplified magnetic field
or turbulent fluctuations. The notion of the critical pitch angle is
introduced for non-adiabatic electron acceleration by balancing the two timescales under a quasi-perpendicular shock wave geometry in which the upstream
magnetic field is nearly perpendicular to the shock layer normal direction.
An analytic expression of the critical pitch angle is obtained as a function
of the electron velocity parallel to the magnetic field, the ratio of the
electron gyro- to plasma frequency, the cross-shock potential, the width of
the shock transition layer, and the shock angle (which is the angle between
the upstream magnetic field and the shock normal direction). For typical
non-relativistic solar system applications, the critical pitch angle is
predicted to be about 10

Collisionless shocks in space and astrophysical plasmas are unique
in that electrons are efficiently accelerated there.
Different types are possible for the acceleration mechanisms.
For a quasi-parallel shock at which the upstream magnetic field
is nearly aligned with the normal direction of the shock layer,
the electrons can be efficiently trapped by turbulent fluctuations on the both sides
of the shock. Since the incoming flow speed on the upstream side
is higher than that on the downstream side,
the electron trapping leads to a diffusive shock acceleration

Naively speaking, for a smaller pitch angle, the electrons are nearly
field-aligned (with respect to the magnetic field) and should have
sufficient time to stay in the shock transition layer for an efficient
acceleration by the electrostatic potential. For a larger pitch angle, the
electrons are scattered and eventually kicked away from the shock transition
layer, either in the upstream and downstream directions. We define the
critical pitch angle through the balance between two timescales, the
electron acceleration time

One may roughly estimate the acceleration timescale as

We use the equation of motion (in a non-relativistic sense) for the electron
in the parallel or field-aligned component:

The acceleration timescale can explicitly be obtained by regarding
Eq. (

Figure

Critical pitch angle as a function of the dimensionless quantity

The critical shock angle is formulated as a function of the electron velocity
parallel to the magnetic field, the ratio of the electron gyro- to plasma
frequencies, the electrostatic potential, the width of the shock transition
layer, and the shock angle (which is the angle between the upstream magnetic
field and the shock normal direction). From the above simple (and algebraic)
estimate, particularly in Eq. (

A caveat needs to be addressed here.
As we have assumed non-adiabatic motion of electrons,
the spatial gradient of the magnetic or electric field must be
smaller than the electron gyro-radius such that
the magnetic moments of electrons are no longer constant.
The condition of non-adiabatic motion for
the electric field is

Understanding the relationship between the cross-shock potential and the
particle dynamics such as trapping, parallel acceleration, and perpendicular
scattering with respect to the mean magnetic field has various applications
to the collisionless shocks in the solar system and in astrophysical systems.
The method of Liouville mapping provides high-time-resolution electron
velocity distribution functions and is a useful tool to evaluate the
cross-shock potential

H. Comişel thanks Manfred Scholer and Octav Marghitu for stimulating and useful discussions which led to developing and formulating the notion of the critical pitch angle for this article. The work by H. Comişel and U. Motschmann in Braunschweig is supported by an extended program of Collaborative Research Center 963, “Astrophysical Flow, Instabilities, and Turbulence” of the German Science Foundation. The work conducted by H. Comişel in Bucharest is supported by Romanian Ministry for Scientific Research and Innovation, CNCS – UEFISCDI, project number PN-II-RU-TE-2014-4-2420. Edited by: C. Owen Reviewed by: one anonymous referee