The following is a summary of research being conducted by Professor Werner Däppen. Professor Däppen is working in collaboration with Alan Nayfonov, his postdoctoral research fellow and graduate students Zhigang Gong, Chia-Hsien Lin, and Ladislav Zejda. They are coordinating with several international research teams. The Department is extremely proud of Professor Däppen and his continuing research.
The interest in the physics of stellar matter is not merely motivated by astrophysics. It has turned out that one star - the Sun - is very special in two respects. First, the methods of helioseismology, the art of interpreting the precisely measured solar five-minute oscillation frequencies, allow us to infer conditions in the solar interior very accurately (in particular, sound speed and density). Second, in the solar convection zone, helioseismology presents an opportunity to isolate the question of the equation of state from opacity and nuclear reaction rates, since the stratification is essentially adiabatic and thus determined by thermodynamics. Thus the Sun has become an astrophysical laboratory to study subtle thermodynamic properties of a Coulomb system under conditions that cannot be achieved on Earth.
In stellar models, the equation of state and opacity are, together with nuclear reaction rates, the fundamental material properties. The structure of a star is a result of ( i ) a balance of forces, ( ii ) a balance between the energy loss at the stellar surface and energy generation in the core, and ( iii ) stationary energy transport between the core and the surface. The balance of forces, described as (hydrostatic equilibrium), results in a relation between the pressure gradient and the gravitational acceleration. The force of gravity is determined by the density distribution in the star: thus stellar modeling requires a relation between density and pressure through the specific properties of the matter. The temperature of the stellar interior is determined by the energy balance. In much of the Sun, energy transport takes place through radiation and depends on absorption coefficients, obtained from atomic physics, which determine the opacity of stellar matter. It was also realized early on that the requirements of radiative transport could result in a temperature gradient so steep that a star would become convectively unstable: convection, where hot elements of gas rise and then generally dominate the energy transport. In the Sun this occurs in about the outer 30% of the radius. Due to the efficiency of convective energy transport, it generally requires a temperature gradient only slightly in excess of the adiabatic gradient, a thermodynamic quantity derived from the equation of state. Another consequence of the equation of state, adiabatic sound speed, plays the crucial role in helioseismology.
In the solar convection zone, helioseismology presents an opportunity to isolate the question of the equation of state from opacity and nuclear reaction rates, since the stratification is essentially adiabatic and thus determined by thermodynamics. Accurate analysis of the observations requires use of the full, non-asymptotic behavior of the oscillations. We now have astonishingly accurate results, for instance, sound speed from the solar surface down to the center. The results of these inversions can be used, in a simplifying spirit, as the (data) of helioseismology, disregarding how they were obtained from solar oscillation frequencies. The most important result of the helioseismic equation-of-state analyses was that it is essential to include the leading Coulomb correction to ideal-gas thermodynamics. Under solar conditions, the size of the relative Coulomb pressure correction is largest in the outer part of the convection zone (about -8 percent) and it has another local maximum in the core (about -1 percent).
However, two very recent inversions have had further implications for the equation of state. First, the strong constraints from helioseismology has forced us to include relativistic effects of electrons. This is very surprising and illustrates the fantastic degree of accuracy of helioseismology. Temperature in the solar center is about 107 K, that is, kT is around 1keV. The relativistic effect on sound speed is manifested in the property that the adiabatic exponent is, respectively, 5 / 3, 5 / 3, and 4 / 3 for a classical, a non-relativistic degenerate, and a fully-relativistic gas. Helioseismology has revealed the subtle lowering of the adiabatic exponent from 5 / 3 towards 4 / 3 due to the relativistic part of the electrons. Now, since the kinetic energy of the electrons in the solar center is small (1 keV) compared to their rest energy (511 keV), the lowering brought in by relativistic electrons is merely on the order of 10-3.
Second, there are indications that in the outside 2% of the solar radius, the presence of excited states in hydrogen and helium is revealed. Again, this is a small, subtle effect, because under the prevailing circumstances, the majority of atoms and ions are in the ground state, and when they begin to be excited, normally they are ionized at the same time. Therefore, the correction due to the presence of excited states is small, again on the order of 10-3, and therefore much smaller than the effect of screened Coulomb potentials. It speaks for the power of helioseismology as an astrophysical tool that such small physical effects can be studied.
Future work will concentrate, among other aspects, on the difficulties in interpreting the helioseismic results, in particular due to ( i ) the influence of turbulence on the structure and the acoustic modes, ( ii ) magnetic fields, and ( iii ) effects from the deviation from local thermodynamic equilibrium (LTE) in the solar atmosphere. Removing those uncertainties will enhance the reliability of our findings both in plasma physics (e.g. higher-order non-ideal effects) and solar physics (e.g. helium abundance of the Sun).
Department of Physics & Astronomy / USC Physics & Astronomy Newsletters /