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Astron. Astrophys. 358, 575-582 (2000)

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1. Introduction

Line widths of upper chromospheric and in particular transition region lines are broadened far in excess of their thermal widths. Over the last two decades there have been numerous measurements of line widths in the solar atmosphere (see references in Doyle et al. 1999) and with the advent of IUE and HST there exists measurements of high temperature transition region lines in many late-type stars (Linsky & Wood, 1994; Wood et al. 1996). The excess line width has been ascribed to micro-turbulence in the atmosphere on scale lengths of the photon mean free path in the atmosphere. With the launch of SOHO in 1995, the measurements of spectral line widths in different solar features, coronal holes, `quiet' and active regions has been published (Teriaca et al. 1999, Doyle et al. 1999 and reference therein). The measured velocities range from 10 km s-1 at a temperature of 10,000 K up to 30-35 km s-1 at the top of the transition region.

With regard to the modelling of chromospheric lines there have been different approaches and approximations of the micro-turbulent/non-thermal velocity. The most important solar models are the `so called' VAL models (Vernazza et al. 1976, 1981, Maltby et al. 1986). In their reference model, Maltby et al. (1986) used micro-turbulent velocities ranging from 1 km s-1 in the photosphere, to 8.5 km s-1 at a temperature of 8,000 K. For the cooler umbral model they varied the micro-turbulent velocity from zero to 4.1 km s-1.

Kelch et al. (1979) in modelling of atmospheres of late type stars used photospheric non-thermal velocities ranging from 2 km s-1 to 5 km s-1 in the upper layers. Giampapa et al. (1982) built a set of models for M dwarfs characterizing the non-thermal motions as isotropic Gaussian micro-turbulence and included the turbulent pressure in the hydrostatic equilibrium equation. They followed the treatment of micro-turbulent velocity by Kelch et al. (1979) with somewhat lower values up to 2 km s-1. Eriksson et al. (1983) produced a model for the late type star [FORMULA] Ceti (G9.5 III). Although the effective gravity in this star is much lower (log g=2.9) than in M dwarfs it is important to mention their treatment of the micro-turbulent velocity because it influenced some later work on late type dwarfs (e.g Short & Doyle 1998). They considered two models for the micro-turbulence. The first of their models equated the micro-turbulent velocity with the local sound speed. In the other model they kept the micro-turbulent velocity at a level of 2 km s-1 up to the temperature minimum, then linearly increased it (in logarithm column mass) to 10 km s-1 and kept this level throughout the remainder of the atmosphere.

Thatcher et al. (1991) modelled the K2V star [FORMULA] Eridani, linearly increasing the micro-turbulent velocity from 1 km s-1 at the deepest photospheric levels to 5 km s-1 at 300,000 K (i.e. the top of their atmosphere). Houdebine et al. (1995) produced an extensive grid of chromospheric structures using a value of 1 km s-1 for the photospheric micro-turbulence. In the chromosphere they increased the micro-turbulent velocity to 2.5 km s-1 at 8,200 K and in the transition region sharply increase it to 4.6 km s-1. Mauas & Falchi (1996) build models for the quiescent and flare state of the very active star AD Leo (M4.5Ve) using a very low micro-turbulent velocity of only 2 km s-1.

Short & Doyle (1998) built another grid of models and introduced the treatment of Eriksson et al. (1983) into the modelling of late type dwarfs. They use a constant micro-turbulent velocity in the photosphere of 2 km s-1, increasing it to 10 km s-1 at the top of chromosphere and to 20 km s-1 at the top of atmosphere.

As is evident, differences in the approximation of the non-thermal velocities are significant. The most interesting point is that only Giampapa et al. (1982), Short & Doyle (1998) and Houdebine et al. (1995) were able to produce absorption in the case of the H[FORMULA] line although Giampapa at al. (1982) considered the transition region at an unrealistic height, namely at column mass of log m = -12 (with m in g cm-2). Several of these latter works have shown the importance of the higher atmosphere (i.e. transition region) on the modelling of some chromospheric lines, however the question of what influence the micro-turbulence has on the formation of these lines has not been addressed properly.

The material in the following sections is organized as follows: in Sect. 2.1 we describe the basic models and some important details of the calculations. Sect. 2.2 describes in detail the different models of non-thermal velocity used, while in Sect. 3 we discuss our results. In Sect. 4 we discuss the results with reference to H[FORMULA] and Na I D line observations of an active dMe star.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000