Using Magnetospheric Multiscale (MMS) observations at the Earths 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 $T/B=(T_0/B_0)(B_0/B)^{alpha}$ with $alpha=2/3$ when the electron heating function $|chi_e|<1$, and a stochastic heating process when $|chi_e|>1$. Both processes are controlled by the value of the stochastic heating function $chi_j = m_j q_j^{-1} B^{-2}mathrm{div}(mathbf{E}_perp)$ for particles with mass $m_j$ and charge $q_j$ in the electric and magnetic fields $mathbf{E}$ and $mathbf{B}$. Test particle simulations are used to show that the stochastic electron heating and acceleration in the studied shock is accomplished by waves at frequencies (0.4 - 5) $f_{ce}$ (electron gyrofrequency) for bulk heating, and waves $f>5,f_{ce}$ 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 (ECD) 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.
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