Teff and log g dependence of velocity fields in M-stars

Abstract

We present an investigation of velocity fields in early to late M-type hydrodynamic stellar atmosphere models. These velocities will be expressed in classical terms of micro- and macro-turbulent velocities for usage in 1D spectral synthesis. The M-star model parameters range between log g of 3.0 - 5.0 and Teff of 2500 K - 4000 K. We characterize the Teff- and log g-dependence of the hydrodynamical velocity fields in these models with a binning method, and for the determination of micro-turbulent velocities, the Curve of Growth method is used. The macro-turbulent velocities are obtained by convolutions with Gaussian profiles. Velocity fields in M-stars strongly depend on log g and Teff. Their velocity amplitudes increase with decreasing log g and increasing Teff. The 3D hydrodynamical and 1D macro-turbulent velocities range from ~100 m/s for cool high gravity models to ~ 800 m/s - 1000 m/s for hot models or models with low log g. The micro-turbulent velocities range in the order of ~100 m/s for cool models, to ~600 m/s for hot or low log g models. Our M-star structure models are calculated with the 3D radiative-hydrodynamics (RHD) code CO5BOLD. The spectral synthesis on these models is performed with the line synthesis code LINFOR3D. Comment: 8 pages, 6 Figures, Proceeding fot the "Recent directions in astrophysical quantitative spectroscopy and radiation hydrodynamics" conference

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arXiv:0908.0820v1 [astro-ph.SR] 6 Aug 2009
Teffand logg dependence of velocity fields in
M-stars
S. Wende∗, A. Reiners∗and H.-G. Ludwig†
∗Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund Platz 1, D-37077
Göttingen, Germany
†GEPI, CIFIST, Observatoire de Paris-Meudon, 5 place Jules Janssen, 92195 Meudon Cedex,
France
Abstract. We presentaninvestigationofvelocityfieldsinearlytolateM-typehydrodynamicstellar
atmosphere models. These velocities will be expressed in classical terms of micro- and macro-
turbulent velocities for usage in 1D spectral synthesis. The M-star model parameters range between
logg of 3.0 – 5.0 and Teffof 2500 K – 4000 K. We characterize the Teff- and logg-dependenceof the
hydrodynamical velocity fields in these models with a binning method, and for the determination
of micro-turbulent velocities, the Curve of Growth method is used. The macro-turbulent velocities
are obtained by convolutions with Gaussian profiles. Velocity fields in M-stars strongly depend on
logg and Teff. Their velocity amplitudes increase with decreasing logg and increasing Teff. The 3D
hydrodynamical and 1D macro-turbulent velocities range from ∼ 100 m/s for cool high gravity
models to ∼ 800 m/s–1000 m/s for hot models or models with low logg. The micro-turbulent
velocities range in the order of ∼ 100 m/s for cool models, to ∼ 600 m/s for hot or low logg
models. Our M-star structure models are calculated with the 3D radiative-hydrodynamics (RHD)
code CO5BOLD. The spectral synthesis on these models is performed with the line synthesis code
LINFOR3D.
Keywords: Radiative transfer - Line: profiles - Stars: atmospheres, low-mass, kinematics
PACS: 95.30.Jx, 95.30.Ky, 95.30.Lz, 95.75.Fg
INTRODUCTION
The measurement of line broadening in cool stars is in general a difficult task. For
example, the investigation of the rotation-activity connection among field M-dwarfs
requires the measurement of rotational line broadening with an accuracy of 1 km/s
[1]. The spectral lines have to be very narrow and well isolated to detect slow rotation.
But in cool stars most of the individual atomic lines become very weak at these low
temperatures and are dominated by pressure broadening. Also the measurement of the
magnetic field strength is dependent on the line width, and detection of Zeeman splitting
becomesmoredifficultatlowtemperaturesduetotheaforementionedreasons. Sinceitis
possibletousethenarrowandwellisolatedFeH moleculelinesincoolstarstodetermine
radial velocities or magnetic field strength [2], it would be very helpful to characterize
the impact of macroscopic velocity fields – primarily driven by convection – on the
lineshapesfromhydrodynamicalmodelatmospheres.Wecalculate3D-CO5BOLDstellar
atmosphere models [3] which serve as input to the line formation program LINFOR3D
[based on 4]. Velocity fields and thermal inhomogeneity are naturally represented in
the hydrodynamical models. The influence on the modeled spectral lines can then be
investigated and translated into effective micro- and macro-turbulent velocities used

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in classical analyses based on 1D hydrostatic model atmospheres. The comparison
with 1D-models provides some insight when it is necessary to apply 3D-models in the
spectral analysis of cool stars.
In the first part of the paper we describe our hydrodynamical model atmosphere code
and give an overview of the models, in the second part we investigate the velocity fields
in the models and their dependence on logg and Teff. We will express the influence of
velocity fields on line shapes in terms of micro- and macro-turbulent velocities.
METHODS AND MODELS
CO5BOLD is the abbreviation for “COnservative COde for the COmputation of COm-
pressible COnvection in a BOx of L Dimensions with L=2,3” [5]. It can be used to
model solar and stellar surface convection. In solar-like stars, a CO5BOLD model repre-
sents the 3D flow geometry and its temporal evolution in a small (relative to the star’s
radius) Cartesian domain at the stellar surface (“box in a star” set-up). The spatial size of
the domain is chosen to be sufficient to include the dominant convective scales, i.e. the
computational box is large enough to include a number of granular cells at any instant in
time. A CO5BOLD model provides a statistical realization of the convective flow. In this
investigation we usually average over five flow fields taken at different instances in time
(“snapshots”) to improve the statistics.
CO5BOLD solves the coupled non-linear time-dependent equations of compressible
hydrodynamics coupled to the radiative transfer equation in an external gravitational
field in 3 spatial dimensions. As set of independent quantities are chosen the mass
density ρ, the three spatial velocities vx, vy, and vz, and the internal energy εi. With these
quantities, the 3D hydrodynamics equations, including source terms due to gravity, are
the mass conservation equation
∂ρ
∂t+∂ρvx
∂ρx+∂ρvy
∂ρy+∂ρvz
∂ρz= 0,
(1)
the momentum equation
∂
∂t


ρvx
ρvy
ρvz

+∂
∂x


ρvxvx+P
ρvyvx
ρvzvx

+∂
∂y


ρvxvy
ρvyvy+P
ρvzvy

+∂
∂z


ρvxvz
ρvyvz
ρvzvz+P

=


ρgx
ρgy
ρgz

, (2)
and the energy equation which includes the radiative heating term Qrad
∂ρεik
∂t
+∂(ρεik+P)vx
∂x
+∂(ρεik+P)vy
∂y
+∂(ρεik+P)vz
∂z
= ρ(gxvx+gyvy+gzvz)+Qrad. (3)
εikdenotes the sum of internal and kinetic energy. The gas pressure P is related to den-
sity ρ and internal energy εivia a (tabulated) equation of state P = P(ρ,εi). For the
local models used here the gravity field is given by the constant vector? g =


0
0
−g

.

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CO5BOLD uses the convention that the vertical axis points upwards. The radiative heat-
ing term Qradis obtained from the solution of the non-local frequency-dependent radia-
tive transfer equation. The frequency dependence of the radiation field is captured by
considering a small number of representative wavelength bands [“opacity binning”, see
3, 6]. The resulting 3D radiative-hydrodynamic (RHD) models treat convection from
basic physical principles and avoid approximations like mixing-length theory. In the
following,the three dimensional data cubes of the CO5BOLD-models will be called “3D-
models”, and the spectral lines computed from three dimensional atmosphere models
“3D-lines”.
We will characterize the velocity fields in the 3D RHD models and analyze their
influence of FeH lines. For the 1D spectral synthesis the 3D velocity fields will be
expressed in terms of micro- and macro-turbulent velocity. For this investigation, we
choose a set of CO5BOLD-models with Teff= 2500 K – 4000 K and logg = 3.0 – 5.0
[cgs]. Table 1 gives the model parameters. The opacities used in all models are obtained
TABLE 1.
Overview of different model quantities. We simulated main sequence stars and
varied logg slightly for models with changing Teff. The models with almost same Teffwere
started at the same Teffof 3300 K, but they converge at slightly lower or higher Teff.
Model code
d3t33g30mm00w1
d3t33g35mm00w1
d3t33g40mm00w1
d3t33g50mm00w1
d3t40g45mm00n01
d3t38g49mm00w1
d3t35g50mm00w1
d3t28g50mm00w1
d3t25g50mm00w1
Dim.Size(x,y,z) [km]
85000 x 85000 x 58350
28000 x 28000 x 11500
7750 x 7750 x 1850
600 x 600 x 260
4700 x 4700 x 1150
1900 x 1900 x 420
1070 x 1070 x 290
370 x 370 x 270
240 x 240 x 170
Opacities
PHOENIX
PHOENIX
PHOENIX
PHOENIX
PHOENIX
PHOENIX
PHOENIX
PHOENIX
PHOENIX
Teff[K]
3240
3270
3315
3275
4000
3820
3380
2800
2575
logg [cgs]
3
3
3
3
3
3
3
3
3
3.0
3.5
4.0
5.0
4.5
4.9
5.0
5.0
5.0
from the PHOENIX stellar atmosphere package [7] assuming an abundance mixture
according Asplund et al. [8]. The opacity tables were computed after Ferguson et al.
[9] and Freytag et al. [10].
In order to compare 1D model atmospheres with the hydrodynamical 3D-models, we
average the3D-models oversurfaces of equal optical depth. We obtain so-called <3D>-
models which have the same mean thermal profile as in the 3D-models but without
cold and hot regions which stem from the convective granulation pattern. In the <3D>-
modelsthe hydrodynamicvelocity field is not considered. The line-broadening is treated
in the classical way by adding isotropic Gaussian micro- and macro-turbulence. We will
call the spectral lines of these models <3D>-lines.
HYDRODYNAMICAL VELOCITY FIELDS
Spectral lines are broadened by velocity fields where the wavelength of absorption
or emission of a particle is shifted due to its motion in the gas. Here we are mostly
concerned with the macroscopic, hydrodynamic motions but have in mind that the
thermal motions are also constituting an important contribution. If we envision each
voxel in the RHD model cube to form its own spectral line, the whole line consists of

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a (weighted) sum of single lines. The velocity distribution might be represented by a
histogram of the velocities of the voxels. We try to describe the velocity fields in that
sense instead of using the rms-velocities. Since a velocity vector is assigned to each
voxel, we apply a binning method and count all vertical velocities to plot them in a
histogramwithabinsizeof25m/s.Wetookthestandarddeviationσ ofafittedGaussian
normal distributionas a measure for the velocity dispersion in the models (see Fig.1). At
this stage of the investigation time we only concentrate on the vertical direction which
is appropriate if we assume that the major part of the line broadening stems from the
vertical motions. The velocities of the horizontal directions are of roughly the same
order. As a measure for the statistical variability in the 3D RHD model, we average
-1000 -5000 500
velocity in z direction [m/s]
0.00
0.05
0.10
0.15
normalized number of points
z-velocity histogram
gauss fit
FWHM
FIGURE 1.
points is plotted against the vertical velocity in m/s (solid line). The Gaussian (dashed line) fits the
velocity distribution and determined an FWHM value (dashed-dotted line), which is related to σ with
FWHM = 2√2ln2·σ. The underlying model is located at Teff= 2800 K and logg = 5 [cgs].
over five different temporal windows and also compute the standard deviation. The
determined velocities are plotted against Teffand logg to investigate their dependence
(Fig. 2). The velocities increase strongly with decreasing logg and with increasing
Teff. They are as low as a few hundred m/s in the coolest high-gravity models. We
can describe the dependence of the velocity dispersions on Teffwith a second order
polynomial, and in logg with a linear function (The fits for the velocity dispersion are
overplotted in Fig.2).
In order to investigate in which region of the model atmospheres the velocity fields are
generated and what the physical meaning of the σ is, we bin the 3D model velocities
at constant geometrical height and fit a Gaussian to determine the velocity dispersion
for each layer (Fig. 3). Typically, we find maxima of the velocity dispersion around
logτ =
−1. The maximum velocities in the convection zone of each model (Teff
and logg) are plotted in Fig. 2 too. In order to measure the velocities in the region
where the lines originate, we compute the mean of the velocities weighted with the
contribution function of the line depression [11] σweighted=∑−6
bottom panel of Fig. 3. These velocities are also plotted in Fig. 2. In most cases they lie
between thevelocitydispersionand themaximumvelocityin theconvectionzone. Some
models show a strong increase of the vertical velocity in higher layers (Fig. 3). This is
related to the presence of waves which are excited by the stochastic fluid motions [3,
and references therein]. However, it will not affect the spectral lines, because they are
Histogram of the velocity distribution in vertical direction. The normalized number of
τ=2στ·CFτ
∑−6
τ=2CFτ
showed in the

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FIGURE 2.
and weighted velocity dispersions (see text) for models with different Teff(left) and different logg (right).
The dashed line on the left plot shows a second order polynomial fit for the velocity dispersion and on the
right plot a linear function is sufficient to fit the logg data.
Plotted are velocity dispersions with error bars, maximal velocities in the convection zone,
FIGURE 3.
velocity (dotted) are plotted against the optical depth in logarithmic scale. Bottom panel: In each bottom
panel are the contribution functions (CF) of a FeH-line at a wavelength of 9954.0 Å˙The width (dashed
triple dotted) and depression (dashed) of the line as a function of optical depth on a logarithmic scale.
Upper panel: σ velocities.In each upper panel the σ -velocities (solid) and the mean σ-

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generated in the region between an optical depth of logτ = 1.0 and logτ = −4.0.
FIGURE 4.
(right).
Micro-turbulent velocities as a function of loggf for different Teff(left) and different logg
FIGURE 5.
logg = 4.9 [cgs] (right). The upper panels show the 3D-line (dots) and the <3D>convolved-line (solid
line) which was broadenedby a Gaussian profile. For comparison we plotted a <3D>-line which was not
broadenedby any velocities (dashedline). In the lower panels are the 3D-<3D>convolvedresiduals plotted.
One can see the asymmetry which stems from the line shifts due to convective motions.
FeH lines for models with Teff = 2800 K, logg = 5.0 [cgs] (left) and Teff = 3820 K,
FIGURE 6.
points are fitted by second order polynomials.
Micro- and macro-turbulent velocities as a function of Teff(left) and logg (right). The data

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MICRO- AND MACRO-TURBULENT VELOCITIES
In the following we express the 3D hydrodynamical velocities in terms of classical
micro- and macro-turbulent velocities [see e.g. 12, 13]. We try to see how accurately
the 1D broadening resembles the 3D broadening in the regime of cool stars which, as
wesaw,exhibitvelocityfields of relativelysmallamplitudes.A 1D treatment wouldsave
a lot of CPU-time, and little differences between 1D and 3D broadening could favor the
usage of fast 1D atmosphere codes to simulate M-stars for comparisons with observa-
tions, e.g. to determine rotational- or Zeeman-broadening.
Micro-turbulent velocities: In order to determine the micro-turbulent velocities, we use
the curve of growth (CoG) method [13]. We increase the line strength of an FeH and
an FeI line by (artificially) increasing the loggf value of the line which results in an
increasingly saturated lines. In saturated lines, the effective absorbing cross-section is
enhanced due to the influence of the micro-turbulent velocities. From 3D spectral syn-
thesis we obtain a 3D CoG, which can we compare with the <3D> CoG for different
micro-turbulent velocities which are included in the 1D spectral synthesis. We use a
grid of micro-turbulent velocities that range from 0 km/s to 1 km/s in 0.125 km/s steps
and obtain <3D> CoGs for nine different micro-turbulent velocities. We compare the
<3D> CoGs with the 3D CoG and select the velocity of the <3D> CoG that fits the 3D
CoG best. The determined velocities are plotted in Fig.6. Since strong (saturated) lines
tend to be formed in the upper atmosphere, and weak (unsaturated) lines in deeper lay-
ers of the atmosphere, loggf and the height of formation are related. If we fit a micro-
turbulent velocity to each loggf point of the 3D CoG, we obtain a height-dependent
velocity structure (see Fig.4).
Macro-turbulent velocities: For modeling macro-turbulent broadening we used the
radial-tangential profile from Gray [14], as well as a Gaussian profile in the 1D spectral
synthesis. We compared <3D> absorption lines which were broadened with both pro-
files, and the results differ only very little. Hence it is appropriate to use the Gaussian
profile here. It is remarkable that the broadening due to small 3D RHD velocity fields
in M-stars can be approximated with a simple Gaussian velocity distribution. Two ex-
amples of Gaussian broadened < 3D > FeH lines are shown in Fig. 5. To determine the
macro-turbulent velocities of the 3D RHD models, we first compute the < 3D > lines
with a given micro-turbulent velocity and after this we broaden them with a Gaussian
with a given macro-turbulent velocity until they match the 3D profiles. The results are
plotted in Fig.6. The broadened < 3D > FeH lines fit the 3D FeH lines very well. The
difference of the < 3D > and 3D centroid (C =∑F·v
small velocity fields to 30−40 m/s for strong velocity fields in hot M star models, or
models with low logg. The differences stem from the asymmetry of the 3D line profiles,
which are shifted due to convective motions. However, the errors in the normalized flux
profiles are lower than 1% (see Fig. 5). We fit the micro- and macro-turbulent veloc-
ities as a function of Teffand logg, each pair can be fitted with a polynomial second
order. The fits are overplotted in Fig.6. Both micro- and macro-turbulent velocities show
similar dependence on Teffand logg as the RHD velocity fields considered before (i.e.
they increase with increasing Teffand decreasing logg). Especially the macro-turbulent
velocities for changing Teffagree well with the determined RHD velocities.
∑F) range in the order of a few m/s for

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SUMMARY
We investigated the velocity fields in a set of M-star 3D hydrodynamical models. They
range in Teffand logg from Teff= 2500 K – 4000 K and logg = 3.0 – 5.0 [cgs].
We applied a binning method to characterize the velocity structure and to determine
mean velocities. These velocities range from ∼ 100 m/s for the cool models up to
∼ 800−1000 m/s for hot models or models with small logg values.
In order to compare the 3D RHD velocity fields with velocities needed in 1D spectral
synthesis,weexpressedthemintermsofclassicalmicro-and macro-turbulentvelocities.
For this purpose we computed the 3D and <3D> Curve of Growth and determined
the micro-turbulent velocity from them. They range in the order of ∼ 100 m/s for cool
models to ∼ 600 m/s for hot models or models with small logg. The macro-turbulent
velocities were determined through convolution with a Gaussian profile which, what
turns out, is sufficient in the regime of cool stars. The obtained velocities are of the
order of the 3D RHD velocity fields and show similar dependence from Teffand logg.
We saw that 1D spectral synthesisin cool stars with micro- and macro-turbulent velocity
broadening is able to reproduce the 3D spectral line synthesis.
ACKNOWLEDGMENTS
SW would like to acknowledge the support from the DFG Research Training Group
GrK - 1351 “Extrasolar Planets and their hoststars”. AR acknowledges research funding
from the DFG under an Emmy Noether Fellowship (RE 1664/4- 1). HGL acknowledges
financial support from EU contract MEXT-CT-2004-014265 (CIFIST). We thank Derek
Homeier for providing us with the opacity tables.
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