STEREO measurements of electron acceleration beyond fast Fermi at the bow shock

Solar wind electrons are accelerated and reflected upstream by the terrestrial bow shock into a region known as the electron foreshock. Previously observed electron spectra at low energies are consistent with a fast Fermi mechanism, based on the adiabatic conservation of the magnetic moment ({\mu}) of the accelerated electrons. At higher energies, suprathermal power law tails are observed beyond the level predicted by fast Fermi. The SWEA and STE electron detectors on STEREO enable measurements of foreshock electrons with good energy resolution and sensitivity over the entire foreshock beam. We investigate the electron acceleration mechanism by comparing observed STEREO electron spectra with predictions based on a Liouville mapping of upstream electrons through a shock encounter. The foreshock electron beam extends up to several tens of keV, energies for which the Larmor radii of electrons is tens of km or greater. These radii are comparable to the scale sizes of the shock, and {\mu} conservation no longer applies. We show that the observed enhancement in the foreshock beam beyond fast Fermi levels begins at energies where this assumption breaks down. We also demonstrate, using the Liouville mapping technique, that the strahl plays an important role in the formation of the bump on tail instability. We discuss this in the context of recent observations in the foreshock and in solar wind magnetic holes.