This thesis focuses on the study of atmospheres of main sequence cool stars (spectral types G2V, K0V, and M2V, with solar metallicity) by means of three-dimensional time-dependent magneto-hydrodynamic
simulations of their near-surface convection and photospheres. The simulations are computed with the
MANCHA code, which is adapted to simulate other cool stars apart from the Sun.
The opacity is computed and optimized for four spectral classes (F3V, G2V, K0V, and M2V). For that,
monochromatic opacity lookup tables are computed with the SYNSPEC code and used to create opacity
distribution functions, that are compared against the results from the ATLAS code. This comparison serves
to verify the method and assess the importance of some opacity contributors in the different stellar regimes
(such as molecules in the M2V star).
The opacity binning method is needed in the simulations to reduce the number of computations required
by the solution of the radiative transfer equation, which entails too many operations per time step for
nowadays computing power. The method focuses on reducing or completely removing the wavelength
dependence of the opacity, and substitute it by representative values that accurately reproduce the bolometric
radiative transfer quantities. In this work, the opacity binning method is tested using different binning
setups to solve the radiative transfer equation in representative one-dimensional models of the four stellar
atmospheres. Three strategies are studied for the binning of the opacity: optimization of four opacity tau-
bins, increment of the number of tau-bins, and partial preservation of the frequency dependence through the
combined use of tau- and lambda- bins. To assess the degree of suitability of the tau-bins combination, we use a
deviation measure that consists in comparing the values of Q obtained with the binned opacities to those
computed with the corresponding opacity distribution functions (which reproduce with high accuracy the
Q computed with monochromatic opacity).
To optimize the four opacity tau-bins for each of the four stellar models, the bolometric radiative energy
exchange rate Q is computed with all the possible combinations of the tau-bins in a discrete grid of optical
depths. We find that the bins can be successfully optimized to reduce significantly the deviations of Q for
all stars. The exception is the M2V star, for which the deviations are larger and only reduced substantially
with the use of {tau, lambda}-bins.
To study the dependence of Q on the number of tau-bins, Q was computed for increasing number of bins
uniformly distributed in a reference optical depth of each of the four stellar models. In the limit of a large
number of bins, the deviations of Q saturates and the result does not converge to the opacity distribution
function (ODF) solution. Owing to such saturation, the Q rate cannot be improved by increasing the number
of bins to more than about 20. The most effective strategy is to select the optimal location of fewer bins.
The results with a large number of bins do not converge to the ODF solution even when the opacity
is binned with tau- and lambda- bins, because in each bin contributors to the opacity with different variation with
height are being mixed in one opacity value. The use of {tau, lambda}-bins reduces the deviations, but with the cost
of larger computing expenses and an additional layer of complexity when optimizing the distribution of the
bins.
Once the different opacity strategies are tested in one-dimensional stellar models, the quality of the
opacities is studied in three-dimensional simulations. For that, realistic box-in-a-star simulations of the
near-surface convection and photosphere of the G2V, K0V, and M2V spectral types were run with grey
opacity. After reaching the stationary state, one snapshot of each of the three stellar simulations was used
to compute the radiative energy exchange rate with grey opacity, opacity binned in four tau-bins, and opacity
binned in 18 {tau, lambda}-bins. These rates were compared with the ones computed with opacity distribution
functions. Then, stellar simulations were run with grey, 4-bin, and 18-bin opacities to see the impact of the
opacity setup on the mean stratification of the temperature and its gradient after time evolution.
The simulations of main sequence cool stars with the MANCHA code are compared with those in the
literature, showing that they reproduce consistent mean stratifications, aspect of the convection, and velocity
fields. The ranges of pressures scale heights in which the radiative energy exchange takes place varies
with the spectral type, accordingly with the description of the veiled granulation. A sharp surface that
concentrates most of the cooling in the atmosphere is found for the G2V star. In the other extreme, a
smeared surface, where the atmosphere gradually cools, is characteristic of the M2V star.
For the three stars, the radiative energy exchange rates computed with 18 bins are remarkably close to
the ones computed with the distribution functions. The rates computed with four bins are similar to the
rates computed with 18 bins, and present a significant improvement with respect to the rates computed with
the Rossleand opacity, especially above the stellar surface. The Rosseland mean can reproduce the proper
rates in sub-surface layers, but produces large errors for the atmospheric layers of the G2V and K0V stars.
In the case of the M2V star, the Rosseland mean fails even in sub-surface layers, owing to the importance
of the contribution from molecular lines in the opacity, underestimated by the harmonic mean. Similar
conclusions are reached after studying the mean stratification of the temperature and its gradient after time
evolution.
Finally, magnetic fields are seeded with the Biermann battery term in the induction equation and en-
hanced by the action of the small-scale dynamo until saturation in the simulations of the G2V, K0V and
M2V stars with 4 tau-bins. The turbulent magnetic fields are consistent with those described in the literature
for solar and stellar simulations. No significant differences are found in the mean stratification or aspect
of the granulation of any of the three stars comparing the purely hydrodynamic and small-scale dynamo
simulations. The difference in kinetic energy between the hydrodynamic simulations and the small-scale
dynamo simulations is similar to the generated magnetic energy. This suggests a transformation of kinetic
into magnetic energy, consistent with a dynamo action.
No Ohmic diffusion is included in the induction equation and only the diffusion from the numerical
scheme is present in the simulations. For the case of the Sun, it has been confirmed that the diffusion
from the numerical scheme is two orders of magnitude larger than the Ohmic diffusion (Khomenko et al.
2014, 2017). We find that the action of the small-scale dynamo is possible in the main sequence cool stars
studied, with almost equal values of the saturated magnetic field at tau = 1 and differences in the magnetic
field between the spectral types up to tens of Gauss in sub-surface layers.