  One of the most important objectives of solar physics is the physical
understanding of the solar atmosphere, the structure of which is also described
in terms of the density (N) and temperature (T) distributions of the
atmospheric matter. Several multi-frequency analyses show that the
characteristics of these distributions are still debated, especially for the
outer coronal emission.
  We aim to constrain the T and N distributions of the solar atmosphere through
observations in the centimetric radio domain. We employ single-dish
observations from two of the INAF radio telescopes at the K-band frequencies
(18 - 26 GHz). We investigate the origin of the significant brightness
temperature ($T_B$) level that we detected up to the upper corona ($\sim 800$
Mm of altitude with respect to the photospheric solar surface).
  To probe the physical origin of the atmospheric emission and to constrain
instrumental biases, we reproduced the solar signal by convolving specific 2D
antenna beam models. The analysis of the solar atmosphere is performed by
adopting a physical model that assumes the thermal bremsstrahlung as the
emission mechanism, with specific T and N distributions. The modelled $T_B$
profiles are compared with those observed by averaging solar maps obtained
during the minimum of solar activity (2018 - 2020).
  The T and N distributions are compatible (within $25\%$ of uncertainty) with
the model up to $\sim 60$ Mm and $\sim 100$ Mm of altitude, respectively. The
analysis of the role of the antenna beam pattern on our solar maps proves the
physical nature of the atmospheric emission in our images up to the coronal
tails seen in our $T_B$ profiles. The challenging analysis of the coronal radio
emission at higher altitudes, together with the data from satellite instruments
will require further multi-frequency measurements.
