Towards pulsation mode identification in 3-D: theoretical simulations of line profile variations in roAp stars
Time-resolved spectroscopic observations of rapidly oscillating Ap (roAp) stars show a complex picture of propagating magneto-acoustic pulsation waves, with amplitude and phase strongly changing as a function of atmospheric height. We have recently conducted numerical, non-linear MHD simulations to get an insight into the complex atmospheric dynamics of magnetic pulsators. Here we use the resulting time-dependent atmospheric structure and velocity field to predict line profile variations for roAp stars. These calculations use realistic atmospheric structure, account for vertical chemical stratification and treat the line formation in pulsating stellar atmosphere without relying on the simplistic single-layer approximation universally adopted for non-radial pulsators. The new theoretical calculations provide an essential tool for interpreting the puzzling complexity of the spectroscopic pulsations in roAp stars.
Rapidly oscillating Ap (roAp) stars is a group of cool magnetic Ap stars pulsating in high-overtone, non-radial modes with periods around 10 min. Excitation of these pulsations and the physics of their propagation in the stellar envelopes and atmospheres is closely connected to the presence of global magnetic fields of several kG strength (e.g., Balmforth et al. 2001; Saio 2005).
Recent time-resolved spectroscopic observations of roAp stars (Kochukhov & Ryabchikova 2001; Mkrtichian, Hatzes & Kanaan 2003; Ryabchikova et al. 2007) showed a remarkably complex and diverse pulsational variability of spectral lines of different chemical elements. In particular, one often finds a factor of 100 difference in amplitude and phase jumps of up to radian between the lines which should originate in very similar regions of normal stellar atmosphere. This unique behaviour is understood to be a result of vertical chemical stratification (e.g., Kochukhov et al. 2006), combined with a rapid intrinsic height variation of the magneto-acoustic pulsation waves propagating in stellar atmosphere.
The spatial filtering effect of chemical inhomogeneities opens interesting prospects for horizontal and vertical resolution of pulsation modes in roAp stars, as can be done for no other type of pulsating stars except for the Sun. The horizontal mapping of roAp pulsations has already been performed with the help of an extended Doppler imaging technique (Kochukhov 2004). However, the vertical resolution of pulsation modes turns out to be considerably more challenging because one has to abandon the standard single-layer approximation universally adopted in detailed line profile modelling of non-radially pulsating stars (e.g., Briquet & Aerts 2003; Schrijvers et al. 1997).
In an effort to get a better insight into the complex atmospheric dynamics of these stars, we are conducting numerical, non-linear magneto-hydrodynamic (MHD) simulations of pulsational wave propagation (Khomenko & Kochukhov 2009). Here we use the resulting time-dependent atmospheric structure and velocity field to predict line profile variations for roAp stars accounting for vertical chemical stratification and using realistic line formation calculations. This theoretical modelling represents a key step towards understanding the puzzling complexity of the spectroscopic pulsations in roAp stars and eventually resolving 3-D structure of their pulsation modes.
2 MHD simulations of roAp pulsations
Our approach to modelling pulsational wave propagation in roAp stars has been described in detail by Khomenko & Kochukhov (2009). Briefly, we perturb the lower boundary of a hydrostatic LTE model atmosphere obtained with the LLmodels code (Shulyak et al. 2004) and follow the resulting outward propagation and transformation of magneto-acoustic waves through the numerical solution of the ideal MHD in two dimensions using the code described in Khomenko & Collados (2006) and Khomenko el al. (2008). Adiabatic calculations are performed in the plane-parallel approximation for homogeneous magnetic field. A series of such MHD runs for different magnetic field strengths and inclinations represent a global structure of a low-degree roAp pulsation mode.
In our new series of simulations presented here we have employed an initial static model atmosphere with parameters K and , extended down to to ensure the ratio of gas to magnetic pressure at the lower boundary for the field strengths of up to 7 kG. The temperature, density, sound and Alfvén speeds of our initial model are illustrated in Fig. 1. We consider pulsations with a period of 10 min and perform calculations for 10 different co-latitudes of the kG dipolar magnetic field. Pulsational displacement at the lower boundary is assumed to have only a vertical component, which is given by the , spherical harmonic, aligned with the dipolar field axis.
Representative variations of the temperature, density and velocity obtained in our MHD calculations are illustrated in Fig. 2 for an intermediate magnetic co-latitude. One can see an important role of the subphotospheric density inversion, the presence of which leads to substantial variations of thermodynamic quantities. At the same time, velocity amplitude rapidly increases as waves propagate outwards and there is a gradual phase change as well. The phase of pulsations at a given height also noticeably varies across the stellar surface due to different superpositions of the fast and slow magneto-acoustic wave components.
3 Line profile variations
Simulations presented above and by Khomenko & Kochukhov (2009) reveal pulsational wave behaviour broadly consistent with the general depth-dependence of the radial velocity amplitude and phase inferred in recent time-resolved spectroscopic studies of roAp stars. However, the ultimate quantitative comparison with observations requires theoretical calculation of the line profile variation (LPV) using the same spectral lines as studied by observers. Theoretical line profile modelling is also essential for resolving the dispute about the nature of puzzling asymmetric profile variability observed in some rare-earth lines (Kochukhov et al. 2007; Shibahashi et al. 2008).
Here we present the first preliminary results of the spectrum synthesis using our MHD models of pulsating roAp atmospheres. We have chosen to model the Nd iii 5851 Å spectral line as well as several Fe i, ii and Ca i lines. We adopt thermodynamic variables and velocity field from the snapshots of MHD simulations covering 3 pulsation cycles and calculate profile variations of chosen spectral lines at different locations on the stellar surface. The spectrum synthesis calculations are carried out with the help of a modified version of the SynthMag polarized radiative transfer code (Kochukhov 2007).
Line formation is treated accounting for typical vertical chemical inhomogeneities encountered in roAp stars (e.g., Shulyak et al. 2009). These vertical chemical profiles are illustrated in Fig. 1c. For Nd we explore three distributions (a homogeneous one and two different distributions with a large overabundance in the upper layers) to mimic diverse behaviour of different rare-earth ions. For Ca and Fe we adopt stratified distributions with both elements enhanced in the lower atmospheric layers, as found in all cool Ap stars.
Fig. 3 presents profile variations of the Nd iii line at different magnetic co-latitudes. These profiles correspond to the stratified distribution “Nd c” (see Fig. 1c), sampling the outermost atmospheric layers. Calculations show a non-negligible effect of temperature and density variation, which introduces a small asymmetry in the LPV pattern. At the same time, magnetic field has a prominent influence on the pulsational variability, concentrating it towards the magnetic pole.
Fig. 4 compares theoretical LPV of the Nd line computed with three different stratified abundance distributions of this element with the variability predicted for a typical Fe i line. As expected from the simulations results in Fig. 2, we find a dramatic increase of the LPV amplitude towards the outer atmospheric layers. Interestingly, our models predict that the low-amplitude variability of the Fe and Ca lines formed in the lower atmospheric layers is caused mainly by the strong temperature and density fluctuations just below . These results are verified by the spectrum synthesis using constant thermodynamic structure and velocity field from the MHD simulations. Since the variability of temperature and density in the low atmospheric layers is related to the subphotospheric density inversion, pulsational variations of the Fe and Ca lines can be used to probe and explore the structure of deep A-star atmospheric layers, hardly observable otherwise.
We have presented the first calculations of the pulsational LPV in roAp stars based on detailed MHD models and realistic treatment of the spectral line formation in magnetic, chemically stratified atmosphere.
In agreement with observations, our calculations show a large increase of the pulsational amplitude and a gradual delay in phase from Fe and Ca to rare-earth spectral lines.
Even for the simple , pulsation mode considered in our modelling the behaviour of line profiles, as well as pulsational amplitude and phase, depends strongly on the local field strength and inclination.
Pulsational changes of pressure and temperature contribute non-negligibly to the profile variations of rare-earth lines and can dominate variations of Fe and Ca lines.
Acknowledgements.O.K. is a Royal Swedish Academy of Sciences Research Fellow supported by grants from the Knut and Alice Wallenberg Foundation and the Swedish Research Council. E.K. is supported by the Spanish Ministerio de Ciencia e Innovación through projects AYA2007-63881 and AYA2007-66502.
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