Equation of state and Rankine-Hugoniot shock relations for realistic dynamical processes in stellar atmospheres

Abstract

The complexity of many of the processes that take place in the solar atmosphere and interior has led to the development of large, often multidimensional, numerical models to understand in detail the underlying physics. The study of phenomena from jets, surges and spicules, all the way to coronal mass ejections (CME´s) or solar flares, of structures such as coronal arcs or of processes such as convection or wave propagation, requires combining aspects of hydrodynamics, electromagnetism, plasma physics and radiative transport. The development of numerical codes allows to solve in detail the equations of those fields together. One of the basic aspects of these numerical codes is the equation of state (EOS) that they implement. The EOS can be very simple, or very complex, depending on the degree of realism needed to obtain relevant results. A code at the forefront of the study of dynamical processes in the solar atmosphere is the Bifrost code, developed at the University of Oslo. The fundamental EOS implemented in Bifrost deals with an ideal gas with ionization/recombination and molecular formation/dissociation processes; it is realistic and complex, and contains detailed microphysics. In this Graduation Thesis, we study the Bifrost EOS from a double perspective: on the one hand, we carry out a detailed characterization of the EOS, covering a large variety of aspects of the thermodynamics of partially ionized gases; on the other hand, we study the shock transitions, in particular the Hugoniot curves, allowed by the EOS. Concerning the first aspect, the motivation to carry out the characterization is that the Bifrost EOS is not well documented in the literature, which may constitute a problem for users of the code. Concerning the second, one cannot find publications that describe the general behavior of shocks calculated with this EOS. Given the ionization/recombination and molecular formation/dissociation processes taking place in the solar atmosphere and included in the EOS, this study can be important to understand the properties of shocks in the numerical models. In the first part, the characterization of the EOS is carried out by calculating thermodynamic quantities on the basis of the tables of temperature and pressure as a function of density and internal energy obtained from the EOS. For all the numerical calculations throughout this work original Python programs have been developed independently of Bifrost’s own program suite. To begin with, quantities that do not require advanced numerical methods, such as atomic mass per particle or specific heat at constant volume, are determined. Next, we calculate more advanced quantities, such as the entropy distribution, ionization or dissociation coefficients, the Chandrasekhar adiabatic coefficients, the adiabatic gradient, the specific heat ratio γ, and the speed of sound, most of which require high-order integrations or interpolations in one or two dimensions. The results are compared to those obtained for the equation of state for the simplest ideal gases, i.e., those with no changes in chemical composition, showing the importance of taking into account the ionization and molecular formation processes. In the case of entropy, the integration methods used are discussed, and the results compared with the adiabatic curves facilitated by Dr H Ludwig obtained in the context of the COBOLD code, thus verifying the validity of our results. Additionally, a detailed analytical expression for the internal energy of the gas is developed that includes all the ionization levels for Hydrogen and Helium and the formation of H2 molecules. The obtained formula is tested by making calculations in regions where the chemical composition does not vary, obtaining an excellent fit to the general curves obtained from the Bifrost EOS table in those regions. In this way, we now have at our disposal detailed information about the ionization or molecular formation processes for H and He that take place in the different ranges of density and internal energy (or pressure and temperature) in the solar atmosphere. In the second part of the work, the jumps of pressure, temperature and density across a shock, the corresponding increase in entropy and the incoming Mach numbers allowed by the Bifrost EOS are studied in detail. For this, the Rankine-Hugoniot jump relations are used and the corresponding Hugoniot curves are obtained, comparing the results to the well known ones for simple ideal gases. First, analytical expressions are derived for the jumps allowed when the component of the internal energy associated with ionization and molecular dissociation processes is uniform in the local thermodynamical domains where the pre-shock and post-shock states are located. This leads to unexpected results when that component has a different value before and after the shock transition. Also, an analytical expression for the derivative of post-shock pressure with respect to post-shock density along the Hugoniot curve is calculated in the general case. Then, a program is created to calculate Hugoniot and Mach number curves numerically for the general case. To illustrate the results, curves starting at pre-schock states in five regions of interest are calculated. Those curves have entry states located in regions of simple ideal gas, but are such that, along their path, cross bands of ionization or molecular dissociation. The crossing gives rise to striking consequences: the density jump can become much larger in those shocks than in the standard simple ideal gas ones; duplication in the admissible pressure and temperature jumps for a given density jump is also obtained. The temperature jump for a given pressure jump is much reduced compared to the simplest ideal gas case when the postshock state is in one of those bands, the reason being that the incoming energy in the shock may be used to ionize (or cause molecular dissociation in) the gas to a larger extent than in increasing its temperature. Summarizing, we have derived a large number of thermodynamical and shock-related properties of the gas described by the EOS of use in the Bifrost code exclusively on the basis of the temperature and pressure maps as functions of density and internal energy. The results illuminate the behavior of the gas in various regimes of interest for the calculations in the upper solar interior, photosphere, chromosphere, transition region and corona. Particularly interesting results are obtained concerning the properties of shocks. We expect that the current results can be of use to the community using the Bifrost code in future.