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A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
02.12.1; 02.12.3; 02.20.1; 08.01.2; 08.03.3; 08.12.1 Gl 616.2
ASTRONOMY
AND
ASTROPHYSICS
April 17, 2000
The Contribution of the Micro­Turbulent Velocity on the
Modelling of Chromospheric Lines in Late Type Dwarfs ?
D. Jevremovi'c 1;2 , J.G. Doyle 1 , and C.I. Short 3
1 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
email: djc@star.arm.ac.uk jgd@star.arm.ac.uk
2 Belgrade Observatory, Volgina 7, 11160 Belgrade 74, Yugoslavia
3 Department of Physics & Astronomy, University of Georgia, Athens, GA, 30602­2451, USA
receive date, accepted date
Abstract. We discuss the influence of the non­thermal
velocity (micro­turbulence) on the formation of chromo­
spheric lines in the atmospheres of late type dwarfs. A
review of previous work shows a variety of different ap­
proaches to the problem leading to different atmospheric
structures and consequently different computed line pro­
files. In that light, we re­examine the formation of the Hy­
drogen Balmer lines and Na i D lines using twelve different
distributions of the micro­turbulent velocity throughout
the atmosphere. Our results show a wide range of possi­
ble line shapes. Using the analogy with the solar case and
the latest results of the non­thermal component widths as
derived from instruments on­board SOHO we model Hff
and the Na i D lines in an active dMe star Gl 616.2.
Key words: micro­turbulence --- chromospheric lines --
late­type stars
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 atmo­
sphere (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 late­type stars
(Linsky & Wood, 1994; Wood et al. 1996). The excess line
width has been ascribed to micro­turbulence in the at­
mosphere 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 so­
lar 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
Send offprint requests to: D. Jevremovi'c
? Based in part on observations made at Observatoire de
Haute Provence (CNRS), France and SOHO
km s \Gamma1 at a temperature of 10,000K up to 30­35 km s \Gamma1
at the top of the transition region.
With regard to the modelling of chromospheric lines
there have been different approaches and approximations
of the micro­turbulent/non­thermal 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 micro­turbulent
velocities ranging from 1 km s \Gamma1 in the photosphere, to
8.5 km s \Gamma1 at a temperature of 8,000K. For the cooler um­
bral model they varied the micro­turbulent velocity from
zero to 4.1 km s \Gamma1 .
Kelch et al. (1979) in modelling of atmospheres of late
type stars used photospheric non­thermal velocities rang­
ing from 2km s \Gamma1 to 5km s \Gamma1 in the upper layers. Gi­
ampapa et al. (1982) built a set of models for M dwarfs
characterizing the non­thermal motions as isotropic Gaus­
sian micro­turbulence and included the turbulent pressure
in the hydrostatic equilibrium equation. They followed
the treatment of micro­turbulent velocity by Kelch et al.
(1979) with somewhat lower values up to 2km s \Gamma1 . Eriks­
son et al. (1983) produced a model for the late type star fi
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 micro­turbulent velocity
because it influenced some later work on late type dwarfs
(e.g Short & Doyle 1998). They considered two models for
the micro­turbulence. The first of their models equated
the micro­turbulent velocity with the local sound speed.
In the other model they kept the micro­turbulent velocity
at a level of 2 km s \Gamma1 up to the temperature minimum,
then linearly increased it (in logarithm column mass) to
10 km s \Gamma1 and kept this level throughout the remainder of
the atmosphere.
Thatcher et al. (1991) modelled the K2V star ffl Eri­
dani, linearly increasing the micro­turbulent velocity from
1 km s \Gamma1 at the deepest photospheric levels to 5km s \Gamma1 at
300,000K (i.e. the top of their atmosphere). Houdebine
et al. (1995) produced an extensive grid of chromospheric
structures using a value of 1km s \Gamma1 for the photospheric

2 D. Jevremovi'c et al.: Micro­turbulence in dwarf stars
micro­turbulence. In the chromosphere they increased the
micro­turbulent velocity to 2.5 km s \Gamma1 at 8,200K and in
the transition region sharply increase it to 4.6 km s \Gamma1 .
Mauas & Falchi (1996) build models for the quiescent and
flare state of the very active star AD Leo (M4.5Ve) using
a very low micro­turbulent velocity of only 2 km s \Gamma1 .
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
micro­turbulent velocity in the photosphere of 2 km s \Gamma1 ,
increasing it to 10km s \Gamma1 at the top of chromosphere and
to 20 km s \Gamma1 at the top of atmosphere.
As is evident, differences in the approximation of the
non­thermal velocities are significant. The most interest­
ing 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 Hff 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 gcm \Gamma2 ). Several of these latter works have
shown the importance of the higher atmosphere (i.e. tran­
sition 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 x2.1 we describe the basic models and some im­
portant details of the calculations. x2.2 describes in detail
the different models of non­thermal velocity used, while in
x3 we discuss our results. In x4 we discuss the results with
reference to Hff and Na i D line observations of an active
dMe star.
2. Construction of models and details of the
calculations
2.1. Basic atmosphere
There exist two schools of thoughts for constructing `semi­
empirical' models of stellar atmospheres. One school fol­
lows a strict semi­empirical approach (VAL, Giampapa
et al. 1982, Mauas & Falchi 1996) which means that they
guess a starting solution for the atmosphere and then com­
pare the output from the radiative transfer calculations
to the observed spectral lines/fluxes. With the advent of
faster computers there has emerged a new school which
basically calculates grids of models (Houdebine & Doyle
1994, Short et al. 1997) by varying some of the important
parameters in the atmosphere. After calculating the out­
put of the models they compare the calculated results with
observation and try to find which model gives the best
agreement. We can call the latter approach `schematic'.
In order to investigate the different levels of the non­
thermal velocities through the model atmosphere we kept
the temperature structure versus column mass identical
for all models. The temperature stratification is achieved
Fig. 1. The temperature dependence of the column mass for
our test model.
using the `standard approach', choosing the position of the
temperature minimum and transition region. Up to the
temperature minimumwe keep the photospheric model as
calculated in radiative equilibrium, from the temperature
minimum to the top of the chromosphere we consider a
linear temperature rise with the logarithm of the column
mass and in the transition region we keep log T versus
log m constant.
The photospheric model used is the `Next Generation'
model of Allard & Hauschildt (1996) with the effective
temperature of T eff = 3800K and log g = 5 and solar
metallicity. It is interesting that some authors use a lower
effective gravity that we find to be of `historical' origin ­
in the eighties the best photospheric models for late type
stars were models of Mould (1976) which were between
log g = 4:75 and 5:75. We choose a photospheric model
which corresponds to a dwarf star of spectral type M0­M1.
From the temperature minimum, situated at log m =
\Gamma2 , to the transition region, which onsets at tempera­
tures around 8,200K (Houdebine & Doyle 1994, Short &
Doyle 1998) and in our test case at the column mass
log m = \Gamma4:5, we keep dT
dlogm constant. From the onset
of the transition region to the top of the atmosphere at
300,000K we keep dlogT
dlogm = \Gamma6:5. The temperature de­
pendence on the column mass is shown in Fig. 1. Each
atmospheric structure has one hundred depth points. This
model is similar to the model used by Short et. al. (1998)
describing a so called zero­activity M dwarf.
2.2. Models for the non­thermal velocities
To test the influence of the non­thermal velocity on the
electron density stratification and the line formation we
use twelve different model distributions. These can be di­
vided into three groups:
-- ffl A constant non­thermal velocity throughout the at­
mosphere for three different levels of v turb = 1; 2 and
5 km s \Gamma1 .

D. Jevremovi'c et al.: Micro­turbulence in dwarf stars 3
Fig. 2. The upper left panel shows models of constant non­thermal velocity. The upper right panel shows the final distribution
of the electron density throughout the atmosphere while the middle panels show the line profiles of H ff, H fi and the lower
panels show the line profiles of H 08 and Na i D.
-- ffl Models where the non­thermal velocity depends on
the sound velocity defined by
c 2
s = \Gamma 1
p
ae
= \Gamma 1 (1 + x) \Lambda kT
mH
(1)
where \Gamma 1 is the generalized adiabatic exponent:
\Gamma 1 = @lnp
@lnae
= 5 + x(1 \Gamma x)[ 5
2 + (ffl H =kT )] 2
3 + x(1 \Gamma x)
\Theta
3
2 + [ 3
2 + (ffl H =kT )] 2 \Lambda
(see e.g. Mihalas and Mihalas 1984, p.52). ffl H is the
ionization energy from the first level of the Hydrogen
atom and x is the degree of ionization.

4 D. Jevremovi'c et al.: Micro­turbulence in dwarf stars
Fig. 3. Same as Fig. 2 except for the models of non­thermal velocity as a fraction of the sound speed. Solid line (0:1cs ), dotted
line (0:2cs ), dashed line (0:5cs) and dot­dash line (0:7cs) and long dashed line (cs ).
We consider five levels of the non­thermal velocity in
the atmosphere namely 0:1 c s , 0:2 c s , 0:5 c s , 0:7c s and
c s and we limit the maximum non­thermal velocity to
30km s \Gamma1 .
-- ffl Models where we keep the non­thermal velocity con­
stant (1 km s \Gamma1 ) up to the temperature minimum, then
linearly increasing to (2, 3, 5 and 10 km s \Gamma1 ) at the top
of the chromosphere and then to (5, 10, 20, 30km s \Gamma1 )
at the top of the atmosphere.
For the electron density calculation it is important to take
into account turbulent pressure, therefore the hydrostatic
equilibrium equation is written as:
g \Theta m = p gas + p e + p rad + 1
2 aev 2
turb (2)

D. Jevremovi'c et al.: Micro­turbulence in dwarf stars 5
Fig. 4. Same as the Fig. 2 for the third type of non­thermal velocity distribution.
where g is the gravity acceleration, m the column mass,
p gas , p e and p rad are the gas, electron and radiative pres­
sure and 1
2 aev 2
turb is the turbulent pressure.
2.3. Details of calculation
We use the radiative transfer code MULTI (Carlsson 1986,
1992) to solve simultaneously the equations of radiative
transfer, and the statistical and hydrostatic equilibrium
equations for Hydrogen. Our Hydrogen model atom con­
sists of 15 bound levels plus continuum. The oscillator
strengths were from Green et al. (1957), the transition
probabilities from Reader et al. (1980) and the values for
the Stark broadening from Sutton (1978). Our Sodium
atom has 10 bound levels plus two continuum levels with
the atomic data taken from Bashkin & Stoner (1982).

6 D. Jevremovi'c et al.: Micro­turbulence in dwarf stars
Fig. 5. The non­thermal velocity as determined by Teriaca
et al. (1999) for an active region (squares) plus a `quiet' sun
region (circles). The lines corresponds to 0:5cs and cs .
We use extensively the radiative­collisional switching tech­
nique of Hummer & Voels (1991) to achieve convergence
of the population levels. The population of the levels in
the Hydrogen and Sodium atom are iterated up to the
level when changes are less than 1%.
In this calculation we have used opacities from the Up­
psala opacity package. Andretta et al. (1997) and Short
et al. (1997) have shown that it is necessary to include a
better treatment of background opacities. At the moment
we are working on the inclusion of a depth dependant
micro­turbulent velocity in PHOENIX. Thus to have a
self­consistent treatment, we presently use only Uppsala
opacities.
3. Results
In Figs. 2­4 we present the basic results of our calcula­
tions. The upper panels show the modelled non­thermal
velocity distribution and the calculated electron density.
The reminding panels show line profiles for four lines; H ff,
H fi, H 08 and Na—i D.
3.1. Constant velocity
From Fig. 2 we can see that there is a lowering of the elec­
tron density with increasing non­thermal velocity. Also,
with the increase of non­thermal velocity, the H ff line
shows a deeper self­reversal, while H fi and H 08 change
from emission to absorption.
In the Sodium doublet there are changes in the shape
of the line wings with the lines becoming narrower with
increasing non­thermal velocity. For turbulent velocities
equal to or greater than 5 km s \Gamma1 , the line wings change
dramatically.
3.2. Non­thermal velocity proportional to sound velocity
We can again see that the electron density is changed with
changes in the level of the non­thermal velocity. By chang­
ing the non­thermal velocity from 0:1c s to c s we manage to
change the H ff profile from emission with the self­reversal
to pure absorption. H fi and H 08 both have a very small
emission component even with a non­thermal velocity of
0:1c s and both lines again change their character similar
to H ff.
The Sodium doublet again shows changes in the line
shape. We attribute these changes to changes in the elec­
tron density in the photosphere. With an increase in the
non­thermal velocity, the central emission in the lines
weakens and in the case when the non­thermal velocity
equal 0:7c s , almost completely disappears. There is a large
increase in the intensity of the Na i wings with increasing
turbulent velocity in the photosphere.
3.3. Linearly increasing non­thermal velocity in the upper
chromosphere
In the chromosphere/transition region, changes in the
electron density are in agreement with Eq. (2). Again there
are dramatic changes in the shape of the Balmer lines. H ff
shows the most prominent changes ­ from emission in the
case of a low non­thermal velocity to deep absorption with
an increase in the level of the non­thermal velocity.
The wings of the Sodium doublet stay almost un­
changed, which is easily explained by the constant non­
thermal velocity in the photosphere. Again the emission
core of the D line changes from being very prominent to
almost non­existence with increasing non­thermal velocity
in the chromosphere.
4. Comparison with the observations ­ the case
of CR Dra
On May 21 1997, we obtained high resolution spectra of
the active dM0e star Gl 616.2 (CR Dra) using the 1.9m

D. Jevremovi'c et al.: Micro­turbulence in dwarf stars 7
Fig. 6. The comparison between the observed and calculated fluxes in Hff (left panel) and Na i D lines (right panel) for CR
Dra. The top panels show the case of log m0=\Gamma2:0, middle panel for log m0=\Gamma1:5 and bottom panel log m0=\Gamma1:0. The position
of the transition region is as follows: log TRm=\Gamma4:8 (solid line), \Gamma4:6 (dotted line), \Gamma4:4 (dashed line), \Gamma4:2 (dot­dashed line),
\Gamma4:0 (dot­dot­dashed line) and \Gamma3:8 (long dashed line).
telescope at the Observatoire de Haute Provence (OHP)
and the fibre fed spectrograph ELODIE (Baranne et al.
1994). This spectrograph provides a spectral resolution of
–
\Delta– = 45000 throughout the visible part of the spectrum
(between 3900 and 6700 š A). The OHP is situated at a
relatively low altitude, and seeing is very rarely better
than 2 arc sec (which is the diameter of the fibres).
We show in Fig. 5 the latest solar results based on mea­
surements of both a `quiet' Sun region and an active re­
gion, superimposed with curves for 0:5c s and c s . These re­

8 D. Jevremovi'c et al.: Micro­turbulence in dwarf stars
sults would suggest that perhaps 0:7c s from 10 4 K to 10 5 K
is an appropriate approximation while from 100,000K to
300,000K it is constant at ¸30km s \Gamma1 . For the lower pho­
tospheric region, we have used the results as derived in
x2.2 which implies a level of ¸ 5 km s \Gamma1 for the photo­
spheric layers. As shown in Fig. 2, changes are only ap­
parent in the line profiles for values of greater than or
equal to 5km s \Gamma1 .
To compare our modelling results to the observations,
we build a grid of models with the non­thermal velocity
equal 0:7c s . In this grid of models we change the posi­
tion of the transition region from log m TR= \Gamma4:8 to \Gamma3:8
and the position of the temperature minimum from log
m 0 =\Gamma2:5 to \Gamma1:0. The comparison for Hff and Na i D lines
are given in Fig. 6. We have normalized the fluxes to the
continuum level at 5845 š A and 6540 š A for Na i D and Hff
lines respectively. One can see that the Sodium doublet is
not fitted perfectly, especially in the wings. The main rea­
son for this discrepancy is in the treatment of background
opacities (see Andretta et al. 1997). Additional opacities
would lower the continuum level and consequently increase
the relative fluxes.
The Sodium fits however suggest that the position of
the temperature minimum is around log m 0 =\Gamma1. Using
that value we estimate from the Hff fits, that the transi­
tion region is at log mTR= \Gamma4:3. These values have to be
confirmed with a better treatment of molecular opacities.
From the various fits one can see that the chosen value
of 0.7 c s , which influences the width of the emission cores,
agrees quite well with the observed ones for both Hff and
Na i D.
5. Conclusions
Some earlier modelling, e.g. Houdebine et al. (1995) sug­
gested that the level of micro­turbulence would not have
an influence on the Hydrogen line profiles. This is clearly
not the case as shown in the previous sets of figures. Of
course one could argue that small scale motions with ve­
locities close or even higher than the sound velocity are
physically doubtful because of the generation of shocks.
We think that in the light of new solar observations
(SOHO) and the richness of small scale motions, we have
to examine even these high levels of non­thermal veloci­
ties.
From the modelling it is clear that there is a lowering
of the electron density with increasing turbulent veloc­
ity. Also, with increasing turbulent velocity, H ff shows a
deeper self­reversal, while H fi changes from emission to
absorption. In Na i, there are changes in the shape of the
line wings with the lines becoming narrower with increas­
ing turbulent velocity, particularly for values greater than
5 km s \Gamma1 in the photosphere.
Fairly good results were achieved for fitting the ob­
served profiles of CR Dra. From our modelling we conclude
that the model with transition region at log m TR=\Gamma4:3
and a temperature minimum at log m 0 =\Gamma1:0 gives the
best results for H ff, where Na i D is used to estimate the
position of the temperature minimum.
Acknowledgements. Research at Armagh Observatory is grant
aided by DENI while support for software and hardware is
largely provided by the STARLINK Project which is funded
by the UK PPARC. This work was supported by a grant
(PPA/G/S/1997/00298) from the UK PPARC. SUMER is part
of SOHO, the Solar and Heliospheric Observatory of ESA and
NASA.
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