We compute two-component theoretical chromosphere models for K2 V stars with di †erent levels of magnetic activity. The two components are a nonmagnetic component heated by acoustic waves and a magnetic component heated by longitudinal tube waves. The Ðlling factor for the magnetic component is determined from an observational relationship between the measured magnetic area coverage and the stellar rotation period. We consider stellar rotation periods between 10 and 40 days. We investigate two di †erent geometrical distributions of magnetic Ñux tubes : uniformly distributed tubes, and tubes arranged as a chromospheric network embedded in the nonmagnetic region. The chromosphere models are constructed by performing state-of-the-art calculations for the generation of acoustic and magnetic energy in stellar convection zones, the propagation and dissipation of this energy at the di †erent atmospheric heights, and the formation of speciÐc chromospheric emission lines that are then compared to the observational data. In all these steps, the two-component structure of stellar photospheres and chromospheres is fully taken into account. We Ðnd that heating and chromospheric emission is signiÐ-cantly increased in the magnetic component and is strongest in Ñux tubes that spread the least with height, expected to occur on rapidly rotating stars with high magnetic Ðlling factors. For stars with very slow rotation, we are able to reproduce the basal Ñux limit of chromospheric emission previously identiÐed with nonmagnetic regions. Most importantly, however, we Ðnd that the relationship between the Ca II H]K emission and the stellar rotation rate deduced from our models is consistent with the relationship given by observations.

Context. The heating process that balances the solar chromospheric energy losses has not yet been determined. Conflicting views exist on the source of the energy and the influence of photospheric magnetic fields on chromospheric heating.Aims. We analyze a 1-h time series of cospatial Ca ii H intensity spectra and photospheric polarimetric spectra around 630 nm to derive the signature of the chromospheric heating process in the spectra and to investigate its relation to photospheric magnetic fields. The data were taken in a quiet Sun area on disc center without strong magnetic activity.Methods. We have derived several characteristic quantities of Ca ii H to define the chromospheric atmosphere properties. We study the power of the Fourier transform at different wavelengths and the phase relations between them. We perform local thermodynamic equilibrium (LTE) inversions of the spectropolarimetric data to obtain the photospheric magnetic field, once including the Ca intensity spectra.Results. We find that the emission in the Ca ii H line core at locations without detectable photospheric polarization signal is due to waves that propagate in around 100 s from low forming continuum layers in the line wing up to the line core. The phase differences of intensity oscillations at different wavelengths indicate standing waves for ν < 2 mHz and propagating waves for higher frequencies. The waves steepen into shocks in the chromosphere. On average, shocks are both preceded and followed by intensity reductions. In field-free regions, the profiles show emission about half of the time. The correlation between wavelengths and the decorrelation time is significantly higher in the presence of magnetic fields than for field-free areas. The average Ca ii H profile in the presence of magnetic fields contains emission features symmetric to the line core and an asymmetric contribution, where mainly the blue H 2V emission peak is increased (shock signature).Conclusions. We find that acoustic waves steepening into shocks are responsible for the emission in the Ca ii H line core for locations without photospheric magnetic fields. We suggest using wavelengths in the line wing of Ca ii H, where LTE still applies, to compare theoretical heating models with observations.

Abstract. We describe a method to construct theoretical, time-dependent, two-component and wave heated chromosphere models for late-type dwarfs. The models depend only on four basic stellar parameters: effective temperature, gravity, metallicity and filling factor, which determines the coverage of these stars by surface magnetic fields. They consist of non-magnetic regions heated by acoustic waves and vertically oriented magnetic flux tubes heated by longitudinal tube waves with contributions from transverse tube waves. Acoustic, longitudinal and transverse wave energy spectra and fluxes generated in stellar convection zones are computed and used as input parameters for the theoretical models. The waves are allowed to propagate and heat both components by shock dissipation. We compute the time-dependent energy balance between the dissipated wave energy and the most prominent chromospheric radiative losses as function of height in the stellar atmosphere. For the flux tube covered stars, the emerging radiative fluxes in the Ca II and Mg II lines are computed by using a newly developed multi-ray radiative transfer method.

We investigate the physical reality of acoustic heating in the solar chromosphere. Evidence is provided that contrary to previous claims by Fossum & Carlsson, high-frequency acoustic waves are indeed sufficient to heat the nonmagnetic solar chromosphere. This assessment is based on three different lines of evidence, which are (1) a discussion of the inherent problems of the limited sensitivity of TRACE when assessing the three-dimensional solar chromospheric topology, (2) a study of the acoustic chromospheric wave energy flux, and (3) a new look at the heating and emission of chromospheric basal flux stars such as t Ceti.

We propose a new numerical method to compute one-dimensional time-dependent wave propagation in stellar atmospheres that incorporates the time-dependent treatment of hydrogen ionization together with an evaluation of radiation losses under departures from local thermodynamic equilibrium (NLTE). The method permits us to calculate acoustic waves and longitudinal magnetohydrodynamic (MHD) tube waves. We have tested the method for the solar atmosphere by calculating the propagation of three types of waves, namely, a monochromatic acoustic wave, a stochastic acoustic wave, and a stochastic longitudinal tube wave. It was found that with a time-dependent treatment of the hydrogen ionization (as well as the Mg ionizations) the degree of ionization (H + /H) and the Mg ii/Mg ratio become insensitive to the temperature fluctuations, even in the presence of weak and moderately strong shocks. Only when strong shocks appear do the transition rates become large enough to cause a high correlation between the degree of ionization and the high postshock temperatures. Our calculations show that a mean degree of ionization gets established that increases with height and is very little perturbed by the local temperature fluctuations of the wave. In stochastic calculations, strong shocks appeared periodically (roughly every 3 minutes), which in their postshock regions carried a zone of high or complete ionization. Tests with different numbers of frequency and height points, as well as of the rate of convergence of the Ã-iteration, were performed.