  In observations of flare-heated electrons in the solar corona, a longstanding
problem is the unexplained prolonged lifetime of the electrons compared to
their transit time across the source. This suggests confinement. Recent
particle-in-cell (PIC) simulations, which explored the transport of
pre-accelerated hot electrons through ambient cold plasma, showed that the
formation of a highly localized electrostatic potential drop, in the form of a
double layer (DL), significantly inhibited the transport of hot electrons (T.C.
Li, J.F. Drake, and M. Swisdak, 2012, ApJ, 757, 20). The effectiveness of
confinement by a DL is linked to the strength of the DL as defined by its
potential drop. In this work, we investigate the scaling of the DL strength
with the hot electron temperature by PIC simulations, and find a linear
scaling. We demonstrate that the strength is limited by the formation of
parallel shocks. Based on this, we analytically determine the maximum DL
strength, and find also a linear scaling with the hot electron temperature. The
DL strength obtained from the analytic calculation is comparable to that from
the simulations. At the maximum strength, the DL is capable of confining a
significant fraction of hot electrons in the source.
