Stable higher order finite-difference schemes for stellar pulsation calculations

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in either 1D or 2D stellar pulsation codes is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grids approach. However these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions. Methods: We propose here a new method, inspired from the staggered grids strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order. Results: This new approach is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh-drift.Comment: 15 pages, 11 figures, accepted for publication in A&

Internal rapid rotation and its implications for stellar structure and pulsations

Massive and intermediate mass stars play a crucial role in astrophysics. Indeed, massive stars are the main producers of heavy elements, explode in supernovae at the end of their short lifetimes, and may be the progenitors of gamma ray bursts. Intermediate mass stars, although not destined to explode in supernovae, display similar phenomena, are much more numerous, and have some of the richest pulsation spectra. A key to understanding these stars is understanding the effects of rapid rotation on their structure and evolution. These effects include centrifugal deformation and gravity darkening which can be observed immediately, and long terms effects such as rotational mixing due to shear turbulence, which prolong stellar lifetime, modify chemical yields, and impact the stellar remnant at the end of their lifetime. In order to understand these effects, a number of models have been and are being developed over the past few years. These models lead to increasingly sophisticated predictions which need to be tested through observations. A particularly promising source of constraints is seismic observations as these may potentially lead to detailed information on their internal structure. However, before extracting such information, a number of theoretical and observational hurdles need to be overcome, not least of which is mode identification. The present proceedings describe recent progress in modelling these stars and show how an improved understanding of their pulsations, namely frequency patterns, mode visibilities, line profile variations, and mode excitation, may help with deciphering seismic observations.Comment: Proceedings for the CoRoT 3/KASC 7 meeting in Toulous

Inequalities on stellar rotational splittings derived from assumptions on the rotation profile

Context: A number of pulsating stars with rotational splittings have been observed thanks to the CoRoT and Kepler missions. This is particularly true of evolved (sub-giant and giant) stars, and has led various groups to investigate their rotation profiles via different methods. Aims: We would like to set up some criteria which will help us to know whether a decreasing rotation profile, or one which satisfies Rayleigh's stability criterion, is compatible with a set of observed rotational splittings for a given reference model. Methods: We derive inequalities on the rotational splittings using a reformulated version of the equation which relates the splittings to the rotation profile and kernels. Results: These inequalities are tested out on some simple examples. The first examples show how they are able to reveal when a rotation profile is increasing somewhere or inconsistent with Rayleigh's criterion in a main sequence star, depending on the profile and the $\ell$ values of the splittings. The next example illustrates how a slight mismatch between an observed evolved star and a reference model can lead to erroneous conclusions about the rotation profile. We also show how frequency differences between the star and the model, which should normally reveal this mismatch, can be masked by frequency corrections for near-surface effects.Comment: 15 pages, 19 figures, accepted for publication in A&

Velocity variations of an Equatorial plume throughout a Jovian year

Features in the equatorial zone of Jupiter show that the equatorial plume reported by Pioneer 10 has existed for an 11-year interval. During this interval the plume has shown an acceleration which can be interpreted as a constant component of 3 x 10 to minus 8th power m/sq cm and a sinusoidal component which anticorrelates with the planetocentric declination of the sun, D sub s, and has an amplitude of -0.96 meters per second per degree change of D sub s. The sinusoidal component has been interpreted in terms of solar heating. Throughout this interval of time the equatorial zone has appeared abnormally dark and has contained many dark projections along the northern edge. When the plume approaches to within 25 to 30 deg of these features they are deflected in the direction of motion of the plume and then dissipate or become obscured as the plume passes. After passage of the plume normal features are again observed

Regular Oscillation Sub-spectrum of Rapidly Rotating Stars

We present an asymptotic theory that describes regular frequency spacings of pressure modes in rapidly rotating stars. We use an asymptotic method based on an approximate solution of the pressure wave equation constructed from a stable periodic solution of the ray limit. The approximate solution has a Gaussian envelope around the stable ray, and its quantization yields the frequency spectrum. We construct semi-analytical formulas for regular frequency spacings and mode spatial distributions of a subclass of pressure modes in rapidly rotating stars. The results of these formulas are in good agreement with numerical data for oscillations in polytropic stellar models. The regular frequency spacings depend explicitly on internal properties of the star, and their computation for different rotation rates gives new insights on the evolution of mode frequencies with rotation.Comment: 14 pages, 10 figure

Using seismic inversions to obtain an internal mixing processes indicator for main-sequence solar-like stars

Determining accurate and precise stellar ages is a major problem in astrophysics. These determinations are either obtained through empirical relations or model-dependent approaches. Currently, seismic modelling is one of the best ways of providing accurate ages. However, current methods are affected by simplifying assumptions concerning mixing processes. In this context, providing new structural indicators which are less model-dependent and more sensitive to such processes is crucial. We build a new indicator for core conditions on the main sequence, which should be more sensitive to structural differences and applicable to older stars than the indicator t presented in a previous paper. We also wish to analyse the importance of the number and type of modes for the inversion, as well as the impact of various constraints and levels of accuracy in the forward modelling process that is used to obtain reference models for the inversion. First, we present a method to obtain new structural kernels and use them to build an indicator of central conditions in stars and test it for various effects including atomic diffusion, various initial helium abundances and metallicities, following the seismic inversion method presented in our previous paper. We then study its accuracy for 7 different pulsation spectra including those of 16CygA and 16CygB and analyse its dependence on the reference model by using different constraints and levels of accuracy for its selection We observe that the inversion of the new indicator using the SOLA method provides a good diagnostic for additional mixing processes in central regions of stars. Its sensitivity allows us to test for diffusive processes and chemical composition mismatch. We also observe that octupole modes can improve the accuracy of the results, as well as modes of low radial order.Comment: Accepted for publication in Astronomy and Astrophysic

Static load versus settlement for geometric shapes on cohesionless soil

Static loading versus settlement for circular plate, cone, and sphere on cohesionless soil - spacecraft landing gear stud