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Soft X-ray heating as a mechanism of optical continuum generation in solar-type star superflares

Superflares on the solar type stars observed by \textit{Kepler} demonstrate the contrast in the optical continuum of the order 0.1--1 per cent. The mechanism of formation of this radiation is not firmly established. We consider a model where the stellar atmosphere is irradiated by the soft X rays emitted from the flaring loop filled with the hot plasma. This radiation heats a large area beneath the loop. Subsequent cooling due to {\sc h$^{-}$} and hydrogen free-bound emission can contribute to the observed enhanced continuum. We solve the equations of radiative transfer, statistical equilibrium, ionization balance and radiative equilibrium in the model atmosphere illuminated by the soft X rays, compute the temperature and the electron density in the atmosphere and find the emergent radiation. We found that a flare loop of the length $\sim10^{10}$~cm and plasma density $10^{12}$~cm$^{-3}$ at the temperature $T = 20$ MK can provide the contrast in the \textit{Kepler} bandpass of 0.1 and 0.8 per cent if the heated region covers $\sim$~1 and 10 per cent of the visible stellar surface respectively. The required emission measure is of the order $10^{55}$~cm$^{-3}$.