  The cosmic dawn 21-cm signal is enabled by Ly~$\alpha$ photons through a
process called the Wouthuysen-Field effect. An accurate model of the signal in
this epoch hinges on the accuracy of the computation of the Ly~$\alpha$
coupling, which requires one to calculate the specific intensity of UV
radiation from sources such as the first stars. Most traditional calculations
of the Ly~$\alpha$ coupling assume a delta-function scattering cross-section,
as the resonant nature of the Ly~$\alpha$ scattering makes an accurate
radiative transfer solution computationally expensive. Attempts to improve upon
this traditional approach using numerical radiative transfer have recently
emerged. However, the radiative transfer computation in these treatments
suffers from assumptions such as a uniform density of intergalactic gas, zero
gas temperature, and absence of gas bulk motion, or numerical approximations
such as core skipping. We investigate the role played by these approximations
in setting the value of the Ly~$\alpha$ coupling and the 21-cm signal at cosmic
dawn. We present results of Monte Carlo radiative transfer simulations, without
core skipping, and show that neglecting gas temperature in the radiative
transfer significantly underestimates the scattering rate and hence the
Ly~$\alpha$ coupling and the 21-cm signal. We also discuss the effect of these
processes on the 21-cm power spectrum from the cosmic dawn. This work points
the way towards higher-accuracy models to enable better inferences from future
measurements.
