  The radial density of planets increases with depth due to compressibility,
leading to impacts on their convective dynamics. To account for these effects,
including the presence of a quasi-adiabatic temperature profile and entropy
sources due to dissipation, the compressibility is expressed through a
dissipation number, $\mathcal{D}$, proportional to the planet's radius and
gravity. In Earth's mantle, compressibility effects are moderate, but in large
rocky or liquid exoplanets (Super-Earths), the dissipation number can become
very large. This paper explores the properties of compressible convection when
the dissipation number is significant. We start by selecting a simple Murnaghan
equation of state that embodies the fundamental properties of condensed matter
at planetary conditions. Next, we analyze the characteristics of adiabatic
profiles and demonstrate that the ratio between the bottom and top adiabatic
temperatures is relatively small and probably less than 2. We examine the
marginal stability of compressible mantles and reveal that they can undergo
convection with either positive or negative superadiabatic Rayleigh numbers.
Lastly, we delve into simulations of convection performed using the exact
equations of mechanics, neglecting inertia (infinite Prandtl number case), and
examine their consequences for Super-Earths dynamics.
