  The effect of melting in planetary mantles plays a key role in their
thermo-chemical evolution. Because of the laterally heterogeneous nature of
melting, 3D numerical simulations are in principle necessary prohibiting us
from exploring wide ranges of conditions. To overcome this issue, we propose a
new analytical framework allowing to estimate the amount and depths of melting
in a 1D analytical model for a simplified convective system. To do so, we
develop an approach, partly based on an extended version of the mixing length
theory, able to estimate the distribution of the hottest temperatures in
natural systems. The approach involves several free parameters that are
calibrated by fitting 3D numerical simulations. We demonstrate that our
algorithm provides the melting profile at steady-state as well as the long-term
evolution in fairly good agreement with the ones obtained in 3D numerical
simulations. We then apply our framework for a wide variety of planetary sizes
and heating rates. We find that an increase in planetary size increases the
depth of melting for small planets but decreases for large planets. This change
in the trend is caused by the pressure dependence of the solidus.
