Lyman-$\alpha$ coupling and heating at Cosmic Dawn
  The global 21-cm signal from the cosmic dawn is affected by a variety of
heating and cooling processes. We investigate the impact of heating due to
Lyman-$\alpha$ (Ly~$\alpha$) photons on the global 21-cm signal at cosmic dawn
using an analytical expression of the spectrum around the Ly~$\alpha$ resonance
based on the so-called `wing approximation'. We derive a new expression for the
scattering correction and for the first time give a simple close-form
expression for the cooling due to injected Ly~$\alpha$ photons. We perform a
short parameter study by varying the Ly~$\alpha$ background intensity by four
orders of magnitude and establish that a strong Ly~$\alpha$ background is
necessary, although not sufficient, in order to reproduce the recently detected
stronger-than-expected 21-cm signal by the EDGES Collaboration. We show that
the magnitude of this Ly~$\alpha$ heating is smaller than previously estimated
in the literature by two orders of magnitude or more. As a result, even a
strong Ly~$\alpha$ background is consistent with the EDGES measurement. We also
provide a detailed discussion on different expressions of the Ly~$\alpha$
heating rate used in the literature.
