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We perform the analytic study of the the buoyancy-drag equation with a time-dependent acceleration $\gamma(t)$ by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of $\gamma(t)$ define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of $\gamma(t)$ including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exhibit two r\'egimes for the asymptotic (large time $t$) solution of the buoyancy-drag equation. %\textbf{ It is shown that they describe a mixing zone driven by the Rayleigh--Taylor instability and the Richtmyer--Meshkov instability, respectively., Comment: 15 journal pages, no figure, to appear, Journal of nonlinear mathematical physics

The spectacular recent development of modern high-energy density laboratory facilities which concentrate more and more energy in millimetric volumes allows the astrophysical community to reproduce and to explore, in millimeter-scale targets and during very short times, astrophysical phenomena where radiation and matter are strongly coupled. The astrophysical relevance of these experiments can be checked from the similarity properties and especially scaling laws establishment, which constitutes the keystone of laboratory astrophysics. From the radiating optically thin regime to the so-called optically thick radiative pressure regime, we present in this paper, for the first time, a complete analysis of the main radiating regimes that we encountered in laboratory astrophysics with the same formalism based on the Lie-group theory. The use of the Lie group method appears as systematic which allows to construct easily and orderly the scaling laws of a given problem. This powerful tool permits to unify the recent major advances on scaling laws and to identify new similarity concepts that we discuss in this paper and which opens important applications for the present and the future laboratory astrophysics experiments. All these results enable to demonstrate theoretically that astrophysical phenomena in such radiating regimes can be explored experimentally thanks to powerful facilities. Consequently the results presented here are a fundamental tool for the high-energy density laboratory astrophysics community in order to quantify the astrophysics relevance and justify laser experiments. Moreover, relying on the Lie-group theory, this paper constitutes the starting point of any analysis of the self-similar dynamics of radiating fluids., Comment: Astrophys. J. accepted

This work provides analytical solutions describing the post-shock structure of radiative shocks growing in astrophysics and in laboratory. The equations including a cooling function $\Lambda \propto \rho^{\epsilon} P^{\zeta} x^{\theta}$ are solved for any values of the exponents $\epsilon$, $\zeta$ and $\theta$. This modeling is appropriate to astrophysics as the observed radiative shocks arise in optically thin media. In contrast, in laboratory, radiative shocks performed using high-power lasers present a radiative precursor because the plasma is more or less optically thick. We study the post-shock region in the laboratory case and compare with astrophysical shock structure. In addition, we attempt to use the same equations to describe the radiative precursor, but the cooling function is slightly modified. In future experiments we will probe the PSR using X-ray diagnostics. These new experimental results will allow to validate our astrophysical numerical codes.

In this paper, radiating fluids scaling laws are studied. We focus on optically thin and optically thick regimes which are relevant for both astrophysics and laboratory experiments. By using homothetic Lie groups, we obtain the scaling laws, the similarity properties and the number of free parameters which allow to rescale experiments in the two astrophyscial situations., Comment: accepted paper

In this paper we study specific classes of radiating shocks which are widely spread in astrophysical environments. We present more general solutions of their structure and proceed to the analytical determination of physical quantities.

Radiative shocks (also called supercritical shocks) are high Mach number shock waves that photoionize the medium ahead of the shock front and give rise to a radiative precursor. They are generated in the laboratory using high-energy or high-power lasers and are frequently present in a wide range of astronomical objects. Their modelisation in one dimension has been the subject of numerous studies, but generalization to three dimensions is not straightforward. We calculate analyticaly the absorption of radiation in a grey uniform cylinder and show how it decreases with chi R, the product of the opacity chi and of the cylinder radius R. Simple formulas, whose validity range increases when chi R diminishes, are derived for the radiation field on the axis of symmetry. Numerical calculations in three dimensions of the radiative energy density, flux and pressure created by a stationary shock wave show how the radiation decreases whith R. Finally, the bidimensional structures of both the precursor and the radiation field are calculated with time-dependent radiation hydrodynamics numerical simulations and the influence of two-dimensional effects on the electron density, the temperature, the shock velocity and the shock geometry are exhibited. These simulations show how the radiative precursor shortens, cools and slows down when R is decreased., Comment: Reference of the published article added

The Rayleigh-Taylor instability plays an important role in the dynamics of several astronomical objects, in particular, in supernovae (SN) evolution. In this paper we develop an analytical approach to study the stability analysis of spherical expansion of the SN ejecta by using a special transformation in the co-moving coordinate frame. We first study a non-stationary spherical expansion of a gas shell under the pressure of a central source. Then we analyze its stability with respect to a no radial, non spherically symmetric perturbation of the of the shell. We consider the case where the polytropic constant of the SN shell is $\gamma=5/3$ and we examine the evolution of a arbitrary shell perturbation. The dispersion relation is derived. The growth rate of the perturbation is found and its temporal and spatial evolution is discussed. The stability domain depends on the ejecta shell thickness, its acceleration, and the perturbation wavelength., Comment: 16 pages