Observational Evidence for Stochastic Shock Drift Acceleration of Electrons at the Earth’s Bow Shock

The first-order Fermi acceleration of electrons requires an injection of electrons into a mildly relativistic energy range. However, the mechanism of injection has remained a puzzle both in theory and observation. We present direct evidence for a novel stochastic shock drift acceleration theory for the injection obtained with Magnetospheric Multiscale observations at the Earth’s bow shock. The theoretical model can explain electron acceleration to mildly relativistic energies at high-speed astrophysical shocks, which may provide a solution to the long-standing issue of electron injection.

Figure 1

Overview of the Earth’s bow shock crossing on 2016 December 9 observed by the MMS1. (a) Ion density. (b) Ion bulk velocity. (c) Magnetic field. (d)–(f) Energy-time spectrogram in units of differential energy flux for (d) ions, (e) high energy electrons measured by FEEPS, and (f) low energy electrons measured by FPI, respectively. The vector quantities are expressed in the GSE coordinate.

Figure 2

Evidence for electron diffusion in shock transition layer. (a) Phase space density (PSD) of electrons. (b) First-order pitch-angle anisotropy of electrons. (c) Frequency-time spectrogram of magnetic field fluctuations. Three solid lines in (c) indicate, from top to bottom, $f_{\mathrm{ce}}$, $f_{\mathrm{ce}}/2$, $f_{\mathrm{ce}}/10$ where $f_{\mathrm{ce}}$ denotes the electron cyclotron frequency. The black dashed lines in (a) represent fitting results with an exponential function. The shaded time interval was used for the fitting.

Figure 3

Schematics illustrating the relation between conventional diffusive shock acceleration and stochastic shock drift acceleration models. Particle acceleration via DSA operates in a spatial extent much larger than the thickness of the shock. In contrast, electrons accelerated by SSDA are confined within the shock transition layer with a typical thickness of ion gyroradius. DSA usually assumes that particles are scattered by MHD waves, which is possible for electrons only if they have relativistic energies. SSDA may accelerate subrelativistic electrons because of intense whistler waves in the transition layer, and may potentially provide a seed population for DSA.

Figure 4

Comparison between theory and observation. (a) Electron phase space density averaged over 1 sec (from 10∶29:19 to 10∶29:20 UT). The red dashed line represents a fitting result with a model function. (b) Scattering rate estimated from the particle intensity profiles. (c) Minimum resonance energy. (d) Magnetic field power spectrum averaged over 5 sec (from 10∶19:15 to 10∶29:20 UT). The magenta dash dotted lines in (c),(d) represent the electron cyclotron frequency. The theoretical threshold shown with the black dashed lines in (b) and (d) were below the measurements in the energy range where the power law was observed.