Stochastic and quasi-adiabatic electron heating in quasi-parallel shocks

Using Magnetospheric Multiscale observations at the Earth’s quasi-parallel bow shock we demonstrate that electrons are heated by two different mechanisms: a quasi-adiabatic heating process during magnetic field compression, characterized by the isotropic temperature relation T/B=(T0/B0)(B0/B)α with α = 2/3 when the electron heating function ∣χe∣1 . Both processes are controlled by the value of the stochastic heating function χj=mjqj−1B−2div(E⊥) for particles with mass m j and charge q j in the electric and magnetic fields E and B . Test-particle simulations are used to show that the stochastic electron heating and acceleration in the studied shock are accomplished by waves at frequencies (0.4–5) f ce (electron gyrofrequency) for bulk heating, and waves f>5fce for acceleration of the tail of the distribution function. Stochastic heating can give rise to flat-top electron distribution functions, frequently observed near shocks. It is also shown that obliquely polarized electric fields of electron cyclotron drift and ion acoustic instabilities scatter the electrons into the parallel direction and keep the isotropy of the electron distribution. The results reported in this paper may be relevant to electron heating and acceleration at interplanetary shocks and other astrophysical shocks.