Abstract
Context.Recent numerical and theoretical considerations have shown that low-degree acoustic modes in rapidly rotating stars follow an asymptotic formula. In parallel, recent studies have revealed the presence of regular pulsation frequency patterns in rapidly rotatingδScuti stars that seem to match theoretical expectations.Aims.In this context, a key question is whether strong gradients or discontinuities can adversely affect the asymptotic frequency pattern to the point of hindering its identification. Other important questions are how rotational splittings are affected by the 2D rotation profiles expected from baroclinic effects and whether it is possible to probe the rotation profile using these splittings.Methods.In order to address these questions, we numerically calculate stellar pulsation modes in continuous and discontinuous rapidly rotating models produced by the 2D Evolution STEllaire en Rotation (ESTER) code. This code self-consistently calculates the rotation profile based on baroclinic effects and uses a spectral multi-domain approach, thus making it possible to introduce discontinuities at the domain interfaces without loss of numerical accuracy. The pulsation calculations are carried out using an adiabatic version of the Two-dimensional Oscillation Program (TOP) code. The variational principle is then used to confirm the high numerical accuracy of the pulsation frequencies and to derive an integral formula for the generalised rotational splittings. Acoustic glitch theory, combined with ray dynamics, is applied to the discontinuous models in order to interpret their pulsation spectra.Results.Our results show that the generalised rotational splittings are very well approximated by the integral formula, except for modes involved in avoided crossings. This potentially allows the application of inverse theory for probing the rotation profile. We also show that glitch theory applied along the island mode orbit can correctly predict the periodicity of the glitch frequency pattern produced by the discontinuity or Γ1dip related to the He II ionisation zone in some of the models. Furthermore, the asymptotic frequency pattern remains sufficiently well preserved to potentially allow its detection in observed stars.
Highlights
Much effort has gone into producing realistic models of rapidly rotating stars
This includes the pioneering works by Roxburgh et al (1965), Ostriker & Mark (1968), and Jackson (1970) and continues on in the present with various 1D codes (e.g. Palacios et al 2003; Eggenberger et al 2008; Marques et al 2013) as well as 2D codes such as the one from the Evolution STEllaire en Rotation (ESTER) project (Rieutord & Espinosa Lara 2009; Espinosa Lara & Rieutord 2013; Rieutord et al 2016)
Reese et al (2017) studied theoretical pulsation spectra with realistic mode visibilities in rapidly rotating 1.8 and 2 M stellar models based on the self-consistent field (SCF) method (Jackson et al 2005; MacGregor et al 2007)
Summary
Much effort has gone into producing realistic models of rapidly rotating stars. This includes the pioneering works by Roxburgh et al (1965), Ostriker & Mark (1968), and Jackson (1970) and continues on in the present with various 1D codes (e.g. Palacios et al 2003; Eggenberger et al 2008; Marques et al 2013) as well as 2D codes such as the one from the Evolution STEllaire en Rotation (ESTER) project (Rieutord & Espinosa Lara 2009; Espinosa Lara & Rieutord 2013; Rieutord et al 2016). A number of recent studies have shown that the pulsation frequencies of low-degree acoustic modes of rapidly rotating stars follow an asymptotic formula Such a formula was first explored on an empirical basis (Lignières et al 2006; Reese et al 2008, 2009) before being justified using ray dynamics (Lignières & Georgeot 2008, 2009; Pasek et al 2011, 2012). Reese et al (2017) studied theoretical pulsation spectra with realistic mode visibilities in rapidly rotating 1.8 and 2 M stellar models based on the self-consistent field (SCF) method (Jackson et al 2005; MacGregor et al 2007) They showed that it may be possible, depending on the configuration, to detect the rotating counterpart to the large frequency separation, or half its value, as well as frequency spacings corresponding to multiples of the rotation rate. The subsections describe the set of pulsation equations, the interface conditions that apply between different domains, the boundary conditions, and the numerical approach
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