  The optical and near-ultraviolet (NUV) continuum radiation in M dwarf flares
is thought to be the impulsive response of the lower stellar atmosphere to
magnetic energy release and electron acceleration at coronal altitudes. This
radiation is sometimes interpreted as evidence of a thermal photospheric
spectrum with $T \approx 10^4$ K. However, calculations show that standard
solar flare coronal electron beams lose their energy in a thick target of gas
in the upper and middle chromosphere (log$_{10}$ column mass /[g cm$^{-2}$]
$\lesssim -3$). At larger beam injection fluxes, electric fields and
instabilities are expected to further inhibit propagation to low altitudes. We
show that recent numerical solutions of the time-dependent equations governing
the power-law electrons and background coronal plasma (Langmuir and
ion-acoustic) waves from Kontar et al. produce order-of-magnitude larger
heating rates than occur in the deep chromosphere through standard solar flare
electron beam power-law distributions. We demonstrate that the redistribution
of beam energy above $E \gtrsim 100$ keV in this theory results in a local
heating maximum that is similar to a radiative-hydrodynamic model with a large,
low-energy cutoff and a hard power-law index. We use this semi-empirical
forward modeling approach to produce opaque NUV and optical continua at gas
temperatures $T \gtrsim 12,000$ K over the deep chromosphere with log$_{10}$
column mass /[g cm$^{-2}$] of $-1.2$ to $-2.3$. These models explain the color
temperatures and Balmer jump strengths in high-cadence M dwarf flare
observations, and they clarify the relation among atmospheric, radiation, and
optical color temperatures in stellar flares.
