Stephen W. Hodson

A Numerical Laboratory

In a series of talks in 1946, John von Neumann envisioned the use of high‐speed computers to generate solutions to nonlinear problems, particularly in fluid dynamics. He pointed out that scientists were conducting expensive and difficult experiments to observe physical behavior even when the underlying principles and governing equations were known. “The purpose of the experiment is not to verify a proposed theory but to replace a computation from an unquestioned theory by direct measurements,” he wrote. “Thus wind tunnels are used at present, at least in large part, as computing devices of the so‐called analogy type to integrate the nonlinear partial differential equations of fluid dynamics.”

The discovery of nonradial instability strips for hot, evolved stars

We have performed radial and nonradial, linear, nonadiabatic pulsation analyses of model stellar envelopes in the effective temperature range from 8 x 10/sup 4/ K to 1.5 x 10/sup 5/ K. These models have total masses of 0.6 M/sub sun/ and radii chosen so that they line up along a pre-white dwarf cooling curve computed by Schoenberner. We use three different interior compositions: (1) 100% carbon: (2) 50% /sup 12/C and 50% /sup 16/O (by mass); and (3) 10% /sup 12/C and 90% /sup 16/O (by mass). We find nonradial and radial instability strips for each composition caused by the partial ionization of carbon or oxygen or both. Our objective is to find the cause of the observed pulsations of PG 1159-035, a hot, evolved star with T/sub e/> or =10/sup 5/ K. While this star may be too hot to lie within any of our new instability strips, the closest agreement comes from models with significant amounts of /sup 16/O near the stellar surface. If correct, this result implies that helium burning in envolved stars produces much more /sup 16/O than heretofore believed.

Modal selection in pulsating stars

Initial-value and periodic nonlinear pulsation integrations are carried out for a series of double-mode Cepheid models (P/sub 1//P/sub 0/approx.0.7) in the vicinity of the resonance ..omega../sub 1/+..omega../sub 0/=..omega../sub 3/. None of the models tested shows persistent double-mode behavior. A new, semiquantitative description of modal selection, based upon the iterative theory of Simon, is introduced to analyze the nonlinear results. In the simplest version of this description, modal section categories emerge which are identical to those of Stellingwerf. Our hydrodynamic results also agree at least partially with Stellingwerfs in that we find a region in the red where the models approach (though never reach) simultaneous instability of both limit cycles. Analysis of resonant effects on modal selection in our calculations leads to the conclusion that models lying between the resonances ..omega../sub 1/+..omega../sub 0/=..omega../sub 3/ and P/sub 2//P/sub 0/=0.5 may yet be viable candidates for double-mode pulsation.

Revised masses for the double-mode and bump Cepheids. [Rotation, helium- or metal-rich-layer]

We consider Population I Cepheids with two pulsation modes and those with a bump in the light and velocity curves. Model envelopes for these Cepheids have been altered in several ways in an attempt in remove the discrepancy between masses predicted from the evolution theory mass-luminosity relation an th masses predicted from pulsation theory. One of these ways, the inclusion of rotation, does not change the period ratio of the first overtone mode to the fundamental mode enough to resolve this mass discrepancy. Another way, the inclusion of a helium- (or metal-) rich layer mixed by convection and pulsation between the stellar surface and 70,000 K decreases this period ratio appreciably. The ratio of the second overtone period to the fundamental period is also reduced with this structure. The masses of the double-mode Cepheids U TrA and V367 Sct and the bump Cepheids U Sgr and ..beta.. Dor are found to be much closer to the masses derived from stellar evolution theory.

Very High Resolution Simulations of Compressible, Turbulent Flows

The steadily increasing power of supercomputing systems is enabling very high resolution simulations of compressible, turbulent flows in the high Reynolds number limit, which is of interest in astrophysics as well as in several other fluid dynamical applications. This paper discusses two such simulations, using grids of up to 8 billion cells. In each type of flow, convergence in a statistical sense is observed as the mesh is refined. The behavior of the convergent sequences indicates how a subgrid-scale model of turbulence could improve the treatment of these flows by high-resolution Euler schemes like PPM. The best resolved case, a simulation of a Richtmyer-Meshkov mixing layer in a shock tube experiment, also points the way toward such a subgrid-scale model. Analysis of the results of that simulation indicates a proportionality relationship between the energy transfer rate from large to small motions and the determinant of the deviatoric symmetric strain as well as the divergence of the velocity for the large-scale field.

Theoretical radial pulsation analyses of DA white dwarfs

A linear, nonadiabatic, radial pulsation analysis of DA white dwarfs in the range of effective temperatures: 7 X 10/sup 3/

Theoretical period ratios for RR Lyrae variables and AQ Leonis

The ratio of the overtone to fundamental periods has been calculated for models of RR Lyrae variables by using the linear nonadiabatic theory. Both homogeneous envelope and helium-enriched surface-layer models have been considered in order to predict the observed period ratio for the only known Population II double-mode RR Lyrae variable AQ Leonis. It is shown that the period ratio for homogeneous models in the radial pulsation instability strip depends only on the fundamental mode period at 0.58 M/sub sun/ and 0.65 M/sub sun/ and does not depend on the luminosity or surface effective temperature. The AQ Leo period of 0/sup d/.55 and the period ratio of 0.746 can be explained with a homogeneous composition model with a mass of 0.65 M/sub sun/. No surface helium enrichment is necessary for this variable even though such enrichment seems to be necessary for the more luminous double- and triple-mode Cepheids. It appears that the double-mode behavior of AQ Leo can be explained by mode switching in its blueward or redward horizontal branch evolution. This mode switching is rapid compared with its lifetime as an RR Lyrae variable.

Linear theory period ratios for surface helium enhanced double-mode Cepheids. [Periods between 1 and 7 days]

Linear nonadiabatic theory period ratios for models of double-mode Cepheids with their two periods between 1 and 7 days have been computed, assuming differing amounts and depths of surface helium enhancement. Evolution theory masses and luminosities are found to be consistent with the observed periods. All models give Pi/sub 1//Pi/sub 0/approx. =0.70 as observed for the 11 known variables, contrary to previous theoretical conclusions. The composition structure that best fits the period ratios has the helium mass fraction in the outer 10/sup -3/ of the stellar mass (T< or =250,000 K) as 0.65, similar to a previous model for the triple-mode pulsator AC And. This enrichment can be established by a Cepheid wind and downward inverted ..mu.. gradient instability mixing in the lifetime of these low-mass classical Cepheids.

Shock-bubble interactions: Generation and evolution of vorticity in two-dimensional supersonic flows

We present high resolution simulations with the PPM code of shock-bubble interactions coresponding to the recent Haas-Sturtevant experiments. VCR cinemas show and diagnostic interpretations quantify the variety of shock-vortex interactions

Pulsations of the R Coronae Borealis stars

The radial pulsations of very luminous, low-mass models (L/M ∼ 104, solar units), which are possible representatives of the R CrB stars, have been examined. These pulsations are extremely nonadiabatic. We find that there are in some cases at least one extra (“strange”) mode which makes interpretation difficult. The blue instability edges are also peculiar, in that there is an abrupt excursion of the blue edge to the blue for L/M sufficiently large. The range of periods of the model encompasses observed periods of the Cepheid-like pulsations of actual R CrB stars.