Context : Studies of solar and stellar convection often employ simple polytropic setups using the diffusion approximation instead of solving the proper radiative transfer equation .
This allows one to control separately the polytropic index of the hydrostatic reference solution , the temperature contrast between top and bottom , and the Rayleigh and Péclet numbers .

Aims : Here we extend such studies by including radiative transfer in the gray approximation using a Kramers-like opacity with freely adjustable coefficients .
We study the properties of such models and compare them with results from the diffusion approximation .

Methods : We use the Pencil Code , which is a high-order finite difference code where radiation is treated using the method of long characteristics .
The source function is given by the Planck function .
The opacity is written as \kappa = \kappa _ { 0 } \rho ^ { a } T ^ { b } , where a = 1 in most cases , b is varied from -3.5 to +5 , and \kappa _ { 0 } is varied by four orders of magnitude .
We adopt a perfect monatomic gas .
We consider sets of one-dimensional models and perform a comparison with the diffusion approximation in one- and two-dimensional models .

Results : Except for the case where b = 5 , we find one-dimensional hydrostatic equilibria with a nearly polytropic stratification and a polytropic index close to n = ( 3 - b ) / ( 1 + a ) , covering both convectively stable ( n > 3 / 2 ) and unstable ( n < 3 / 2 ) cases .
For b = 3 and a = -1 , the value of n is undefined a priori and the actual value of n depends then on the depth of the domain .
For large values of \kappa _ { 0 } , the thermal adjustment time becomes long , the Péclet and Rayleigh numbers become large , and the temperature contrast increases and is thus no longer an independent input parameter , unless the Stefan–Boltzmann constant is considered adjustable .

Conclusions : Proper radiative transfer with Kramers-like opacities provides a useful tool for studying stratified layers with a radiative surface in ways that are more physical than what is possible with polytropic models using the diffusion approximation .