  The linear stability threshold of the Rayleigh-Benard configuration is
analyzed with compressible effects taken into account. It is assumed that the
fluid obeys a Newtonian rheology and Fourier's law of thermal transport with
constant, uniform (dynamic) viscosity and thermal conductivity in a uniform
gravity field. Top and bottom boundaries are maintained at different constant
temperatures and we consider here boundary conditions of zero tangential stress
and impermeable walls. Under these conditions, and with the Boussinesq
approximation, Rayleigh (1916) first obtained analytically the critical value
27pi^4/4 for a dimensionless parameter, now known as the Rayleigh number, at
the onset of convection. This manuscript describes the changes of the critical
Rayleigh number due to the compressibility of the fluid, measured by the
dimensionless dissipation parameter D and due to a finite temperature
difference between the hot and cold boundaries, measured by a dimensionless
temperature gradient a. Different equations of state are examined: ideal gas
equation, Murnaghan's model and a generic equation of state, which can
represent any possible equation of state. We also consider two variations of
this stability analysis. In a so-called quasi-Boussinesq model, we consider
that density perturbations are solely due to temperature perturbations. In a
so-called quasi-anelastic liquid approximation, we consider that entropy
perturbations are solely due to temperature perturbations. In addition to the
numerical Chebyshev-based stability analysis, an analytical approximation is
obtained when temperature fluctuations are written as a combination of only two
modes. This analytical expression allows us to show that the superadiabatic
critical Rayleigh number quadratic departure in a and D from 27pi^4/4 involves
the expansion of density up to the degree three in terms of pressure and
temperature.
