On the understanding of pulsations in the atmosphere of roAp stars: phase diversity and false nodes

Mon. Not. R. Astron. Soc. 414, 2576–2593 (2011) doi:10.1111/j.1365-2966.2011.18573.x On the understanding of pulsations in the atmosphere of roAp stars: phase diversity and false nodes J. C. Sousa1,2 and M. S. Cunha1,2 1 Centro de Astrofı́sica da Universidade do Porto, Portugal 2 Faculdade de Ciências da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Accepted 2011 February 21. Received 2011 February 16; in original form 2010 November 18 ABSTRACT Key words: waves – techniques: radial velocities – stars: chemically peculiar – stars: oscillations. 1 I N T RO D U C T I O N The rapidly oscillating Ap (roAp) stars are found among the coolest subgroup of chemically peculiar Ap stars, which is located in the main-sequence part of the classical instability strip. They pulsate with periods typically within the range from 5 to 21 min (e.g. Kurtz, Elkin & Mathys 2005), have oscillations with amplitudes between 0.5 and 5 km s−1 in velocity and, as in other Ap stars, have strong, large-scale magnetic fields, with typical intensities of a few kG, although in some stars the magnetic field strength can be higher than 20 kG (e.g. Hubrig et al. 2009). Moreover, they have about two solar masses (Kurtz 1990), and temperatures that range from about 6400 to 8100 K (Kochukhov 2009). The present number of known roAp stars is more than 40. Due to their characteristics, with roAp stars we have the unique opportunity  E-mail: to observe the interaction of acoustic modes with strong large-scale, magnetic fields. The roAp stars have been observed photometrically since their discovery by Kurtz (1982), and until recently the knowledge about their acoustic oscillations was essentially based on high-speed, ground-based, photometric observations. However, in the past few years numerous exciting observational results have been published, both as a result of the acquisition of space-based photometry (e.g. Gruberbauer et al. 2008; Bruntt et al. 2009; Balona et al. 2011) and as a result of the analysis of high-resolution spectroscopic data (e.g. Kurtz, Elkin & Mathys 2006a,b; Ryabchikova et al. 2007; Sachkov et al. 2007). Such high-resolution spectroscopic data hold unique information about the structure and dynamics of the peculiar atmospheres of roAp stars, and have revealed a surprising diversity in the pulsation behaviour of different lines in the roAp spectra. The general picture that emerges from the analysis of time-series of high-resolution spectroscopic data of roAp stars is that pulsational variability is seen predominantly in lines of rare-earth ions,  C 2011 The Authors C 2011 RAS Monthly Notices of the Royal Astronomical Society  Studies based on high-resolution spectroscopic data of rapidly oscillating Ap stars show a surprising diversity of pulsation behaviour in the atmospheric layers, pointing, in particular, to the co-existence of running and standing waves. The correct interpretation of these data requires a careful modelling of pulsations in these magnetic stars. In light of this, in this work we present a theoretical analysis of pulsations in roAp stars, taking into account the direct influence of the magnetic field. We derive approximate analytical solutions for the displacement components parallel and perpendicular to the direction of the magnetic field, that are appropriate to the outermost layer. From these, we determine the expression for the theoretical radial velocity for an observer at a general position, and compute the corresponding pulsation amplitude and phase as a function of height in the atmosphere. We show that the integral for the radial velocity has contributions from three different types of wave solutions, namely running waves, evanescent waves and standing waves of nearly constant amplitude. We then consider a number of case studies to illustrate the origin of the different pulsational behaviour that is found in the observations. Concerning pulsation amplitude, we find that it generally increases with atmospheric height. Pulsation phase, however, shows a diversity of behaviours, including phases that are constant, increasing or decreasing with atmospheric height. Finally, we show that there are situations in which the pulsation amplitude goes through a zero, accompanied by a phase jumps of π, and argue that such behaviour does not correspond to a pulsation node in the outermost layers of the star, but rather to a visual effect, resulting from the observers inability to resolve the stellar surface. Understanding roAp stars  C 2011 The Authors, MNRAS 414, 2576–2593 C 2011 RAS Monthly Notices of the Royal Astronomical Society  cuss the physics underlying our model, along with the assumptions made. In Section 4, we derive the expression for the theoretical radial velocity and describe a Toy Model that will be useful for the interpretation of the results. The results of the analysis of six case studies are presented in Section 5, followed by a general discussion in Section 6. 2 PA R A M E T E R S PAC E Given an underlying equilibrium stellar model, the general problem that we set out to study depends on a number of input parameters. Some of these are intrinsic to the star, namely the magnetic field intensity and topology, characterized by the vector B; the unperturbed (i.e. in the absence of a magnetic field) oscillation mode, characterized by its radial order, n, angular degree, l, and the azimuthal order m; the location of the chemical elements whose spectral lines are used in the derivation of the radial velocity, characterized by an atmospheric depth (or region in depth), as well as by longitudinal and latitudinal limits. Moreover, some parameters depend on the position of the observer, such as the inclination angle between the magnetic axis and the direction of the observer δ (at a given phase of rotation); the inclination angle between the latter and the rotation axis, i. In our analysis we will keep some of the above fixed. In particular, we will only consider magnetic fields of a dipolar topology. We will not consider the effect of rotation on the dynamics and, consequently, will assume that the magnetic and pulsation axis are aligned. This is generally a good approximation for roAp stars, except possibly in cases when the magnetic field is rather low (below ≈1 kG) (Bigot & Dziembowski 2002). Moreover, the unperturbed pulsation modes to be considered are only those axisymmetric about the magnetic axis, thus, characterized by m = 0 in a spherical harmonic decomposition about that axis of symmetry. Finally, we will consider only a fixed phase of rotation, so that our problem will have no dependence on the inclination angle i. We note that the relaxation of most of the above conditions is relatively straightforward, although in some cases will lead to significant additional work. The results of our study will be presented in two papers. The present paper will deal with the underlying mathematical analysis and the in-depth i (...truncated)