Modeling energy distribution of cool dwarfs in the optical and near-infrared

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Modeling energy distribution of

cool dwarfs in the optical and

near-infrared.

by

Maxim Kuznetsov

Thesis submitted in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY IN

ASTROPHYSICS

at the

Instituto Nacional de Astrof´ısica, ´

Optica y

Electr´onica

December 2019

Tonantzintla, Puebla

Under the supervision of:

Dr. Carlos del Burgo D´ıaz

INAOE, Mexico

c

INAOE 2019

The author hereby grants to INAOE permission

to reproduce and to distribute publicly paper and

electronic copies of this thesis document in

whole or in part.

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Declaration

This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgments.

I hereby declare that my thesis entitled:

Modeling energy distribution of cool dwarfs in the optical and near-infrared.

is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other Research Institute or University.

Some parts of this work have already been published in refereed journals as follows:

• Section 2.2, has already partially appeared in G´alvez-Ortiz et al. (2014).

• Chapter 3, Chapter 4, Chapter 7 has already partially appeared in Kuznetsov et al. (2019).

• Chapter 5, Chapter 6, has already fully appeared in Kuznetsov et al. (2019). I further declare that this copy is identical in every respect to the volume examined for the Degree, except that any alterations required by the Examiners have been made.

Date: December 5, 2019 Signed:

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Abstract

M dwarf stars are the dominant stellar component of the Galaxy, and therefore they contain fundamental information about galactic structure and evolution. Furthermore, M dwarfs were recognized as promising targets for the detection of potentially habitable planets.

We present the homogeneous analysis of a sample of Southern early-type M dwarfs in the Solar neighborhood (d<60pc), in an attempt to determine the physical properties of the stars, as well as to constrain and to improve the methods that are used for this purpose.

In order to create a complete sample of known, bright M dwarfs, we have merged the recent datasets of low-mass stars and obtained a list of about 13,000 objects. For the ob-jects from our sample we collected the available high-resolution La Silla/HARPS spec-tra, and mid-resolution VLT/X-shooter optical spectra using the ESO Science Archive Facility. The stars form two subsamples: the ones having HARPS spectra (HARPS sample; N=420), and the ones having X-shooter spectra (X-shooter sample, N=153). We also gathered the observed broadband photometric data in the optical and near-infrared.

For 295 stars from the HARPS sample and 61 stars from the X-shooter sample we obtained the effective temperature (Teff) and metallicity ([Fe/H]) by comparing

ob-served and theoretical broad-band photometry in optical and near-infrared. We also performed the estimation of metallicity based on theJ₋K empirical calibration.

We applied the MCAL technique, based on the empirical calibration of pseudo equivalent widths (pEW), to derive the effective temperature Teff, metallicity [Fe/H]

and activity indexia(Hα)of 420 M stars using HARPS spectra. We found systematic underestimation of our MCAL temperatures of 110 - 160 K in comparison with the literature. We concluded that the MCAL temperatures should be used only in combi-nation with other methods, because of its sensitivity to multiple factors (signal-to-noise ratio of spectra, stellar activity and limits of applicability of the method). Neverthe-less, the MCAL metallicities are in good consistency with the literature with standard deviations around 0.1 dex.

The effective temperature Teff, surface gravity logg, metallicity [Fe/H] and

pro-jected rotational velocity Vrotsini of 153 M0-M6 dwarfs were determined by fitting the observed intermediate-resolution spectra from the VIS arm of VLT/X-shooter with

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a grid of BT-Settl stellar atmosphere models. We used χ2 _{minimisation for the full}

spectrum, and for the parameter-sensitive spectral regions. We slightly improved the standard methodology and developed the multi-region approach to minimise the degen-eration effects. The multi-region approach allows to decrease the uncertainties and to improve the reliability of our results. We estimated the typical uncertainties of the fit with X-shooter spectra by varying region-to-region results by: σTeff ∼50 K,σlogg ∼

0.2, andσ[Fe/H]_∼0.2 dex. The spectroscopic parameters for 10 stars from the sample were obtained for the first time. We have explored systematic differences of our results with other studies and found that the photometric temperatures from GAIA DR2 show overestimations of about 300-400 K and require corrections for cool stars.

We compared our results from different methods to estimate absolute uncertainty in determining the physical properties of M dwarfs. We conclude that despite signifi-cant progress in recent years, the determination of the physical properties of M dwarfs remains challenging. The absolute errors in obtaining physical properties can be esti-mated by comparing results from different methods and with results from literature. We found the typical uncertainties for individual stars of about_∼150 K inTeff, in∼0.2 in

logg, and_∼0.2 in [Fe/H]. The use of a combination of different independent methods is required for more precise and realistic determination of stellar properties.

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Resumen

Las estrellas enanas de tipo M son el componente estelar dominante de la galaxia, y por lo tanto contienen información fundamental acerca de la estructura y evolución galáctica. Además, la enanas M han sido identificadas como objetivos prometedores para la detección de planetas potencialmente habitables.

En este trabajo se presenta un análisis homogéneo de una muestra de estrellas enanas tempranas de tipo M, del hemisferio sur, en el vecindario local (d<60 pc), en un intento por determinar las propiedades f´ısica de estas estrellas, as´ı como de restringir y mejorar los métodos que se utilizan con este propósito.

Para construir una muestra completa de estrellas enanas M conocidas y brillantes, se han unido los conjuntos de datos recientes de estrellas de baja masa, obteniendo una lista de alrededor de 13,000 objetos. Para los objetos de esta muestra, se han conseguido los espectros ópticos disponibles de alta resolución de La Silla/HARPS y de resolución intermedia del VLT/X-shooter, utilizando el Science Archive Facility de la ESO. Estas estrellas forman dos submuestras: las que tienen espectros HARPS (muestra HARPS; N = 420) y las que tienen espectros X-shooter (muestra X-shooter, N = 153). También se han conseguido los datos fotométricos de banda ancha observados en el óptico e infrarrojo cercano.

Para 295 estrellas de la muestra HARPS y 61 de la muestra X-shooter se han obtenido su temperatura efectiva (Teff) y metalicidad ([Fe/H]) comparando la fotometr´ıa de banda

ancha observada con la teórica, en el óptico e infrarrojo cercano. También se realizo la estimación de la metalicidad con base en la calibración emp´ıricaJ₋K.

Se aplico la t´ecnica MCAL, basada en la calibraci´on emp´ırica de pseudo anchos equivalentes (pEW), para derivar la temperatura efectiva Teff, metalicidad [Fe/H] e

´ındice de actividadia(Hα)de 420 estrellas del tipo M utilizando los espectros HARPS. Se encontró una subestimación sistemática, de las temperaturas derivadas por la técnica MCAL, de entre 110 y 160 K, en comparación con la literatura. Se concluye que las temperaturas derivadas por la técnica MCAL se deben utilizar solamente en combi-nación con otros métodos, debido a su sensibilidad a mltiples factores (señal a ruido del espectro, actividad estelar y limites de aplicabilidad del método). Sin embargo, las metalicidades derivadas por la técnica MCAL son consistentes con la literatura con una desviación estándar de alrededor de 0.1 dex.

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Se determino la temperatura efectiva Teff, gravedad superficial logg, metalicidad

[Fe/H] y velocidad rotacional proyectadaVrotsinipara 153 enanas M0 - M6, ajustando los espectros observados de resolución intermedia del brazo VIS del VLT/X-shooter a una red de modelos de atmósferas estelares BT-Settl. Se utilizo la minimización deχ2

para el espectro completo, y para las regiones espectrales sensibles a los parámetros, se mejoro ligeramente la metodolog´ıa estándar desarrollando un enfoque multi-región para minimizar los efectos de degeneración. El enfoque multi-región permite reducir las incertidumbres y mejorar la fiabilidad de los resultados. Se estimaron las incer-tidumbres t´ıpicas del ajuste con los espectros de X-shooter variando los resultados de región a región: σTeff ∼ 50 K, σlogg ∼0.2, y σ[Fe/H] ∼ 0.2 dex. Los parámetros

espectroscópicos para 10 estrellas de la muestra se han obtenido por primera vez. Se han explorado las diferencias sistemáticas de estos resultados con otros estudios y se encontró que las temperaturas fotométricas de GAIA DR2 muestran sobrestimaciones de alrededor de 300 - 400 K y requieren correcciones para las estrellas fr´ıas.

Se han comparado los resultados de los diferentes métodos descritos, para estimar la incertidumbre absoluta en la determinación de las propiedades f´ısicas de las enanas M. Se concluye que a pesar del progreso significativo en los ltimos años, la determinación de las propiedades f´ısicas de las enanas M sigue siendo un desaf´ıo. Los errores absolu-tos en la obtención de propiedades f´ısicas se pueden estimar comparando resultados de los diferentes métodos y con resultados de la literatura. Se encuentran incertidumbres t´ıpicas para estrellas individuales de aproximadamente_∼150 K enTeff,∼0.2 en logg,

y_∼0.2 en [Fe/H]. Se requiere el uso de una combinación de diferentes métodos inde-pendientes para una determinación más precisa y realista de las propiedades estelares.

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Acknowledgments

I would like to thank my advisor, Dr. Carlos del Burgo.

I am very grateful to Dr. Yakov Pavlenko for his invaluable advice

during the research.

I am also very grateful to Dr. Yalia Divakara Mayya, Dra. Elsa

Re-cillas Pishmish, Dr. Roberto Giovanni Terlevich, Dra. Yilen Go ´

mez

Maqueo and Dr. David Hiriart my thesis evaluation committee and

ex-aminers, for undertaking the task of reviewing this manuscript and for

their useful suggestions.

I wish to thank the coordinators Dr. Daniel Rosa Gonz´alez and Dr.

Abraham Luna Castellanos for their help in coordinating the research

process.

I would like to thank the members of my semester evaluation

commit-tee Dr. Emanuele Bertone, Dr. Vahram Chavushyan, Dr. Ivˆanio Puerari

and Dra. M´onica Rodr´ıguez for their comments and suggestions.

I also wish to thank my collaborators Dra. Mar´ıa Cruz G´alvez Ortiz,

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Dr. James Frith, Dr. D. J. Pinfield and Dr. H. R. A. Jones for their

comments and recommendations.

I would like to thank my colleagues from INAOE, especially Dr.

Ri-cardo Ch´avez Murillo, Dr. Leonardo Chavez Velazquez, Dr. Devaraj R.,

Dra. Emmaly Aguilar Perez, Dra. Ana Torres Campos and Dr. Eduardo

Ibarra Medel for their support and hospitality.

I also would like to thank my first teacher Dr. Leushin V.V., who show

me the way.

I would like to thank the the CONACyT (Consejo Nacional de Ciencia

y Tecnolog´ıa) program for PhD studies. Without their scholarship this

thesis would not have been possible. I also acknowledges support by

CONACyT research grant CB-2012-183007 (Mexico).

Last, but no least, I would like to thank my family and friends for their

love and their help. If it was not for their continuous support it would

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Chapter 1 Introduction

1.1 The importance of M dwarf stars

M dwarfs are cold, low-mass stars on the main sequence that have effective tempera-tures in the range of 2300–4000 K and masses below_∼0.5 solar mass. These stars are a dominant stellar component of the Galaxy: they account for about 70% of the stars in the Milky Way and 40% of its stellar mass (Bochanski et al., 2010; L´epine & Gaidos, 2011; Chabrier, 2003). M dwarfs may belong to any population, from old, metal-poor objects in the globular clusters and the galactic halo (Renzini et al., 1996; Cool, Piotto

& King, 1996) to young metal-rich dwarfs in open clusters (Leggett, Harris & Dahn, 1994).

Therefore, our understanding of the Galaxy must be supported with the characterisa-tion of this component. Indeed, low-mass stars have been employed in Galactic studies

as they carry the fundamental information about the structure, chemical and kinematic evolution of the Milky Way (Reid, Sahu & Hawley, 2001).

Spectral type M is defined by the presence of strong absorption bands of the tita-nium oxide molecule (TiO) in the optical spectrum. However, the boundaries between spectral classes K and M are blurred since the TiO bands can also be seen in late K

stars in the far red and near-infrared. (Kirkpatrick, Henry & McCarthy, 1991; Bessell, 1991).

The absorption lines of other molecules, such as H2, H2O, VO, OH, CO, CaOH, SiH, CaH, CrH, FeH, and MgH also appears both in the visual and in the near-infrared

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spectra of M dwarfs.

M class cover a more extended range in mass than the spectral classes F, G, and K together (Boyajian et al., 2012). The massive (M >0.35M⊙) M dwarfs have partially

convective structure, while the lighter objects convert to fully convective (Chabrier & Baraffe, 1997).

M dwarfs evolution is much slower than the Sun-like or brighter stars due to their

lower masses (Tarter et al., 2007). For example, 0.1M⊙ star will burn hydrogen into

helium for 12 trillion (1012_{) years (Laughlin, Bodenheimer & Adams, 1997). Current}

models of stellar structure and evolution typically underestimate the radii and overesti-mate the temperatures of low mass stars at the level of 5-10%. (Morales et al., 2009).

The lower limit for sustained proton-proton nuclear fusion (_∼0.08 M⊙) is at the

lower end of the M spectral sequence (Chabrier & Baraffe, 1997). The border between the hydrogen fusing stars and the brown dwarfs (BDs) depends on the combination of mass and metallicity of an object. The massive brown dwarfs can belong to the late M subtypes and therefore should be distinguished from the hydrogen-burning late M dwarf stars.

In addition to the stellar and Galactic studies, M dwarfs were recognized as promis-ing targets for the detection of potentially habitable planets. A potentially habitable planet implies an Earth-like low-mass rocky planets within the stellar habitable zone, where the temperature of a planet permits stable liquid water on its surface (Tarter et al.,

2007). The habitable zones around low mass stars are located at a smaller orbital ra-dius than Sun-like stars due to their lower temperatures and luminosity. The smaller ratio between the radii and mass of the star to those of the planet lead to higher pho-tometric transit depths and Doppler shifts. This makes the planets more likely to be discovered (Nutzman & Charbonneau, 2008; Quirrenbach et al., 2010; Laughlin, Bo-denheimer & Adams, 2004; Reiners et al., 2018). However, the intrinsic stellar activity

of M dwarfs can produce distortion in photometric and radial velocity data that might be misinterpreted as a planetary signal (Sarkis et al., 2018). The recent discoveries of low-mass planets in the habitable zones around M dwarfs (Udry et al., 2007; Mayor et al., 2009; Vogt et al., 2010; Rivera et al., 2010; Bonfils et al., 2013; Robertson & Mahadevan, 2014; Dittmann et al., 2017; Bonfils et al., 2018; Ment et al., 2019, e.g.) including the detection of a rocky planet orbiting Proxima Centauri (Anglada-Escud´e

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et al., 2016) and a super-Earth orbiting Barnards Star (Ribas et al., 2018), encourage further investigation in this area. These predictions are being tested by several upcom-ing exoplanet missions like CARMENES, GAIA, TESS, JWST and PLATO which all

plan on targeting a selection of M dwarfs.

Obtaining the parameters of low-mass stars is also essential for our understanding of a broad range of topics including star and planet formation, the initial mass function, circumstellar disks, structure of cold atmospheres, and brown dwarfs.

1.2 The determination of atmospheric parameters of M

dwarfs

The accurate determinations of the fundamental stellar parameters (effective

temper-ature Teff, surface gravity logg and metallicity [Fe/H]) of cool dwarfs is essential to

the galactic and exoplanetary science. However, this problem still does not have an unambiguous solution due to the difficulties in obtaining high-quality observations and problems in the modeling of low-mass stars.

M dwarfs are intrinsically faint objects, and the majority of their flux is emitted in the near-infrared wavelengths (Reiners et al., 2010; Kirkpatrick et al., 1999). That made them difficult to observe until the development of infrared-sensitive CCDs in the late-1980’s (Laughlin, Bodenheimer & Adams, 1997). As photographic plates were re-placed by more efficient CCDs, highly accurate observations of faint cool stars became possible. The deep sky surveys, such as the 2MASS (Skrutskie et al., 2006),

DE-NIS (Delfosse et al., 1997) and SDSS (York et al., 2000) has stimulated the new stage in stellar astronomy. However, there are still limitations in obtaining high-resolution spectra with high signal-to-noise ratios of the low-mass stars, and there are difficulties in getting homogeneous samples concerning their age and metallicity (Rajpurohit et al., 2018b).

Regarding the modeling of the atmospheres of cool stars, the combination of the low temperatures and high pressures leads to the fact that the main opacity sources within low-mass star atmospheres are due to molecules such as TiO, CO, VO, MgH, CaH, FeH, H2, and water (Allard & Hauschildt, 1995). This creates difficulties in the

cal-culating of the stellar atmospheres and processing of the observed spectra. The proper

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representation of molecular bands requires information about strengths and transitions of millions of their lines (Bean et al., 2006). The strong absorption bands also prevent to identify continuum level and unique atomic features in stellar spectrum (Rajpurohit

et al., 2013).

The convective structure, chromospheric activity, flares, magnetic spots and, for the latest subtypes, the dust clouds, make modeling of M dwarfs even more challenging (see Tsuji, Ohnaka & Aoki, 1996; Allard et al., 2003a; Allard, Homeier & Freytag,

2012a; Rajpurohit et al., 2012).

Despite the difficulties, different techniques have been developed to determine the physical properties of M dwarfs. The effective temperature can be obtained from the direct interferometric measurements of stellar radii (S´egransan et al., 2003; Boyajian

et al., 2012). However, this method can only be applied to a few of the brightest M dwarfs.

Many studies that are dedicated to determining the parameters of M dwarfs are based on different photometric empirical calibrations. For example, Casagrande, Flynn & Bessell (2008) adopted the flux ratio in different bands in visual and infrared to estimate

the effective temperatures.

As for metallicity, binary stars with an FGK-type primary and an M dwarf secondary were used as reference objects by Bonfils et al. (2005). It is assumed that the low-mass secondary and the higher-mass, FGK primary formed in the same stellar nursery and

share a common origin so their metallicities should be similar. Using this assumption, the photometric properties of the low-mass components were used to develop photo-metric calibrations that could be applied to M dwarfs that were not a part of a binary system. The work Bonfils et al. (2005) have formed the basis for similar photometric calibrations (Johnson & Apps, 2009; Schlaufman & Laughlin, 2010; Neves et al., 2012; Hejazi, De Robertis & Dawson, 2015).

The various combinations of spectroscopic indices in the visible and in the near-infrared were also used to develop empirical calibrations for Teff and [Fe/H] For

ex-ample, (Rojas-Ayala et al., 2012) use the infrared H2O-K2 index and the equivalent width of the NaI and CaI lines. The calibrations in H band were performed by (Terrien et al., 2012) and (Mann, Gaidos & Ansdell, 2013) focus on the optical and J band

spec-tra. The relative strength of the TiO and the CaH molecular bands is often used in the

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visible spectrum (L´epine, Rich & Shara, 2007; Dhital et al., 2012; L´epine et al., 2013).

Other techniques, based on using pseudo-equivalent widths from high-resolution optical spectra forTeff and [Fe/H] diagnostics have been presented in two recent studies

by (Neves et al., 2014; Maldonado et al., 2015). These promising methods allow the use of high-resolution spectra from spectrographs such as HARPS or HARPS-N, to obtainTeff and [Fe/H] with typical uncertainties in the order of 70-100 K, and 0.1 dex

respectively. However, these methods can only be used for early M dwarfs which do

not show high levels of chromospheric activity.

The empirical methods do not provide an understanding of the physics of the stellar atmosphere and can be used only for limited purposes. The proper characterization of M dwarfs requires the comparison of synthetic spectra with observations. It allows us

to obtain the complete set of fundamental parameters of a star (Teff, loggand [Fe/H]) as

well as some dynamic characteristics such as the projected rotational velocity (Vrotsini) and properties of the stellar atmosphere (activity, micro-turbulent velocity etc). In spite of difficulties, the models of atmospheres of low-mass stars have greatly improved over the last decade.

Independent scientific groups developed a variety of new models of the cold stars such as the BT-Settl model of Allard, Homeier & Freytag (2012a), the PHOENIX-ACES of Husser et al. (2013) and MARCS model of Gustafsson et al. (2008). The new grids of synthetic spectra were widely used to derive the stellar properties of M dwarfs by comparing them to the medium and high-resolution spectra in the optical and infrared regions ¨Onehag et al. (2012); Rajpurohit et al. (2013); del Burgo et al. (2013);

G´alvez-Ortiz et al. (2014); Lindgren, Heiter & Seifahrt (2016); Passegger, Wende-von Berg & Reiners (2016); Bayo et al. (2017); Pavlenko et al. (2017); Passegger et al. (2018); Rajpurohit et al. (2018a,b)

A number of studies show that the effective temperatures obtained by different

meth-ods deviates significantly from one another (del Burgo et al., 2013; Bayo et al., 2014; Maldonado et al., 2015). Concerning metallicity, Lindgren, Heiter & Seifahrt (2016) demonstrate that [Fe/H] values derived by different calibrations can differ by about 0.6 dex for individual stars.

The results of the existing methods are often not stable and do not always consistent

with each other. Thus, despite significant progress, accurate establishing the parameters

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of cold stars remains challenging.

1.3 Aims of this Work

The main goal of this work is to determine the fundamental stellar parameters of a sam-ple of Southern M dwarfs in the solar neighborhood using a set of independent photo-metric and spectroscopic methods. We aimed to improve the techniques, to undertake a comparative analysis of the results, and to estimate the uncertainties, reliability and limitations of the methods.

Having this objective in mind, we chose to use the high-resolution HARPS spectra, and the intermediate-resolution X-shooter spectra form the public ESO Science Archive Facility. The ESO archive contains the result of years of observations of M dwarfs. Most of these data were used in an attempt to search for extrasolar planets (HARPS

spectra), to characterise individual stars, or for other purposes are not involving the deriving of stellar parameters as a primary objective (X-shooter spectra). Thus, using the HARPS and X-shooter spectra allow providing unique study aimed at determination of the stellar properties of the sample of hundreds of M dwarfs, based on the uniform set of methods.

In the course of our work, two samples were generated: the list of low-mass stars that have the HARPS spectra available (HARPS sample, 420 objects) and the list of low-mass stars for which we have the X-shooter spectra available (X-shooter sample, 153 objects). Most objects in the HARPS sample are early M dwarfs. There are only two stars later than M4. Most stars in the X-Shooter sample are belong to subclasses

M3, M4 and M5. There are 22 M0 to M2 stars and only 5 M6 and M7 stars. The sample contains mostly early to mid-M dwarfs, and there is a lack of the late M stars.

To obtain the fundamental stellar parameters, we use three independent techniques:

• Photometry: comparison of the observed broad-band photometry in Johnson (B, V), Gaia G, and 2MASS (J, H, Ks) bands, with synthetic colors computed for the models BT-Settl Allard, Homeier & Freytag (2012a) (SED method);

• High-resolution spectroscopy: the MCAL technique Neves et al. (2014), based on the empirical correlation of pseudo-equivalent widths (pEW) of high-resolution HARPS spectra with the physical properties;

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• Intermediate-resolution spectroscopy: fitting synthetic spectra BT-Settl Allard, Homeier & Freytag (2012a) to the medium-resolution flux-calibrated X-shooter spectra.

The comparison of the results derived by the independent methods provides a unique opportunity to estimate the absolute uncertainties in the determination of the stellar properties, to understand the reliability and the limitations of the methods and to im-prove them. Furthermore, this study will help to understand the physical processes

which occur in cool atmospheres.

1.4 Structure of this Work

Through the second chapter, I will be presenting the theoretical background of cal-culation of synthetic spectra and a detailed description of the PHOENIX models for low-mass stars.

The third chapter describes the sample of known, bright M dwarfs, the observed visual spectra from La Silla/HARPS and VLT/X-shooter as well as broad-band pho-tometry.

The fourth chapter explores the determination of the fundamental physical properties the effective temperatureTeff and metallicity [Fe/H] of stars from both the HARPS and

the X-shooter samples by comparing the observed photometry with model predictions. I also present the estimation of metallicity based on theJ₋K empirical calibration.

In the fifth chapter, I perform the MCAL technique, based on the empirical cali-bration of pseudo equivalent widths (pEW), to derive the effective temperature Teff,

metallicity [Fe/H] and activity indexia(Hα)of 420 M stars using HARPS spectra. In the sixth chapter, I fitted the observed intermediate-resolution spectra from the VIS arm of VLT/X-shooter to a grid of BT-Settl stellar atmosphere models to obtain the effective temperature Teff, surface gravity logg, metallicity [Fe/H] and projected

rotational velocity Vrotsiniof 153 M0-M6 dwarfs. I improved the standard χ2 min-imisation methodology decreasing the uncertainties and improving the reliability of the results. The spectroscopic parameters for ten stars from the sample were obtained for the first time.

In the seventh chapter I show the comparison of results coming from different

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ods, including photometric methods the MCAL and the comparison of synthetic and observed spectra. I have estimated also the uncertainties for individual stars by com-paring results from different methods and with results from the literature.

Finally, in the eight chapter, I summarize the general conclusion of this work.

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Chapter 2 Model Atmospheres

2.1 Introduction

The electromagnetic spectrum of a star contains information about its fundamental pa-rameters. In particular, about the kinematics and the chemistry. The comparison of observed spectra with stellar atmosphere models allow us to infer such parameters. The accuracy and precision depend on the achieved resolving power and the quality of the models, but also on the methodology itself.

The observation techniques were improved significantly over the last decades. The last generation of high-resolution spectrographs such as as HARPS (Mayor et al., 2003), HARPS-N (Cosentino et al., 2012), and CARMENES (Quirrenbach et al., 2010) can provide high-resolution, high signal-to-noise ratio spectra of bright M dwarfs. Future instruments such as HPF (Mahadevan et al., 2012) and SPIRou (Cersullo et al., 2017)

will further provide more high-resolution spectra of good quality.

The modelling of the atmospheres of low-mass stars has also evolved to using a line-by-line opacity in spherical symmetry (Hauschildt, Baron & Allard, 1997) and to 3D radiation transfer (Seelmann, Hauschildt & Baron, 2010). Thus far, the calculations of theoretical stellar spectra have been improved to the point where there is a good

agreement with observations (Passegger et al., 2018).

Some factors should be discussed at the stage of choosing the model of the atmo-sphere to analyze a specific type of stars since these factors will affect the quality of the reproduction of stellar spectra.

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The first step needs to be done before starting the numerical calculations is to ana-lyze the combination of approximations that will be applied. For example, is it possible to use the plane-parallel layers or it is necessary to apply a spherical-symmetric model

of atmosphere like for giant stars. It is important to note that using 1D models leads to exclusion of some important effects associated with the small scale differential move-ments that might affect the profiles of spectral lines as well as the energy distribution in the whole spectrum. Another factor is the energy transfer mechanism. It could be radiative, convective or combination of both. The microturbulent velocity is also one of the important factors. This parameter varies with the depth in the real stellar

atmo-sphere. However, it is considered as a constant in the most models. It is also important to mention the effects associated with the chromospheric activity since the emission lines appears in the spectra of many low-mass stars.

The second factor, that is of particular importance for the cool stars, is the method of

description of molecular absorption in the spectrum. The molecular lines are dominated in the spectra of late-type dwarfs, however, the molecular line lists, that amount to millions of lines, are often not sufficiently accurate.

The third factor that needs to be discussed is the local thermodynamic equilibrium

(LTE) approximation. The LTE assumption is allowing the use of the Boltzmann and the Saha equation to determine the populations of energetic levels, which substantially simplify the calculation process. The LTE is proven to be useful in the reproduction of the the modelling of the atmospheres of cool stars (Allard, Homeier & Freytag, 2012b). However, it is essential to note that the LTE does not correspond to a real physical con-dition in stellar atmospheres and which might generate discrepancy between theoretical

spectra and observations.

The modelling of stellar spectra and comparison of theoretical spectra with observed data is of great use for studying of physical properties of stars such as atmospheric stricture, chemical composition, a field of velocity, stellar evolution etc.

The theoretical background of calculation of synthetic spectra is presented in section 2.2. A detailed explanation of the PHOENIX models for low-mass stars is given in section 2.3. A comprehensive description of the BT-Settl/AGSS09 grid of synthetic spectra is presented in section 2.4.

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2.2 Calculation of synthetic spectra

The calculation of synthetic spectra is based on a model of stellar atmosphere and lists of atomic and molecular lines in a given spectral region. The total absorption coefficient α(λk, τ), where τ is the optical depth, is calculated for every wavelength

λk. The contribution of all nearby lines, the absorption in the continuum, and a profile of every individual line should be taken into account. The aim of the first step of the calculations is to determine the two-dimensional array of the absorption coefficients

α(λk, τ). The second step is aimed to obtain the array of the optical depthsτλk. As soon as the α(λk, τ) and τλk are established the radiation flux on a given wavelength

Fλk can be calculated. The output flux of a given line can be computed by the formula :

F = 2

Z ∞

0

S(τ)E2(τ)dτ (2.1)

whereS(τ)in the source function andE2(τ)is the second integral exponential

func-tion.

The approximation of local thermodynamic equilibrium defines the source function as the Plank function:

Bλ(T) =

2hc2

λ5

1

exp(_λkThc )−1 (2.2)

The flux in the continuum is derived by applying the same procedure, by using optical depth in the continuous spectrumτc instead ofτλ.

The functionFλk(λ)is a synthetic spectrum, that can be compared with an observed data. The comparison of synthetic and observed spectra is used to highlight many astrophysical problems, including the determination of physical properties in the atmo-spheres of stars.

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2.2.1 Profile of absorption line in a stellar spectrum

The absorption coefficient in a line for one atom is:

kλ =k0 ∗H(a, u)∗(1−e

−hν

kT ) (2.3)

The expression in the brackets associated with the stimulated emission andk0is the

absorption coefficient in the center of a line:

k0 =

√

πe2

mν0v

f (2.4)

wheremis mass of the electron,eis the charge of the electron,ν0 is the absorption

frequency of a stationary atom, v is the mean velocity of the chaotic movement of atoms (thermal and turbulent),f is the oscillator strength (the probability of absorption or emission of a photon in transitions between energy levels) for a given line.

H(a, u) is the Voigt function, that describes a line profile. The Voigt function is the convolution of two mechanisms of broadening: a Lorentzian profile and a Gaussian

profile, associated with the Doppler broadening. It can be written as:

H(a, u) = a

π

Z +∞

−∞

e−y2

a2_{+ (}_u₋_y₎2dy (2.5)

whereu = ∆λ/∆λD, and ∆λ is the distance from the distance from the center of the line to a given point.

∆λD =

λ c

s

2RT µ +v

2

turb (2.6)

whereµis the atomic weight of a given element, R is the universal gas constant and

v2

turb is the microturbulent velocity.

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The parameterathat is included in the formula 2.5 is calculated by the formula:

a= λ

2

c γ

4π∆λD

(2.7)

whereγ is the attenuation constant.

The algorithms for the approximate calculation of the Voigt function were developed in Kielkopf (1973); Kurucz (1979). It is shown that the algorithm from Kurucz (1979)

is suitable for the fast calculations. This approach is based on the representation of the Voigt functionH(a, u)in the form of Taylor-Harris series (Harris, 1948) and quadrature formulas.

The attenuation constantγ depends on the optical depthτ. The attenuation constant

can be represented as the sum of three terms for an atomic lines:

γ(τ) =γr+γst+γvdw (2.8)

whereγris the attenuation constant associated with radiation, γst is the attenuation constant associated with the quadratic Stark effect andγvdw is the attenuation constant associated with the van der Waals forces (collisions with neutral particles). These con-stants might be obtained experimentally or calculated.

The approximate ”classic” formula can obtain the attenuation constant for radiation

γr.

γ(r)_≈0.22/λ2 (2.9)

whereλ2 _in _cm_{. The lifetime of the corresponding atomic levels should be}

estab-lished to derive the value ofγ(r)more precise. The several studies demonstrate that the ”classic” formula provides reliable results for stellar lines (Groth, 1961; Carbon et al.,

1982).

Since theγstandγvdw depend on the physical conditions, these parameters are func-tions of the depth of atmosphere. The approximate formula is used to calculate the attenuation constant associated with the quadratic Stark effect:

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γ(st) = (13.595Z_{ef f}2 /(χion−χup))3/2∗10−8 (2.10) where χion and χup are the ionization potential and the excitation potential of the higher level of the atom respectively.

The approximate formula for the van der Waals effect is:

γ(vdw) =γ(vdw)(H)(n(H) + γ(vdw)(He)

γ(vdw)(H) n(He) +

γ(vdw)(H2)

γ(vdw)(H)n(H2)) (2.11)

or

γ(vdw) = γ(vdw)(H)(n(H) + 0.42n(He) + 0.85n(H2)) (2.12)

where

γ(vdw)(H) = 17.0(C₆2/5v_i3/5ni) (2.13)

and

C6 ≈0.3∗10−30(

k2

(χion−χl−hν)2 −

k2

(χion−χl)2

) (2.14)

hereχion is the ionization potential, χlis the excitation potential of the lower level andhνis the photon energy in the line (all values are in eV). For a neutral atomsk = 1, for a singly ionized atomsk= 2.

It should be noted that the contribution ofγ(vdw)in the attenuation constantγ(τ)

is significant in the cold main sequence stars (F-G-K-M etc.). The value ofγ(vdw)is negligible in comparison withγr+γst for atmospheres of giant stars.

The uncertainty of the attenuation constant can affect accuracy of obtaining of chem-ical composition. The value ofγ(τ)is often calculated by using the approximate

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mulas, thus the question, how the uncertainty inγ(τ)affects the derived chemical com-position is of great importance.

The method described above was designed for the lines of metals. Here, we do not consider the influence of the magnetic field on the spectral lines (the Zeeman effect). The hydrogen lines broadening that are carried out by the specific physical processes are beyond this work.

The concentration of atoms at the lower levelnlshould be multiplied onkλto derive the absorption coefficient in a line . The determination of populations of energy levels in general case is a difficult task. However, in the approximation of local thermody-namic equilibrium, the Boltzmann equation and the Saha ionization equation are used to calculatenl.

ni

nl

= bi

bl

gi

gl

exp(₋Ei−El

kT ) (2.15)

ne

ni

nl

= bi

bl

2(2πmkT)3/2

h3 exp(−

Ei−El

kT ) (2.16)

whereni andnlare the numbers of atoms on the energetic levelsiandl,Ei andEl are excitation energies of the energetic levels iandl, gi, gl are the statistic weights,k is the Boltzmann constant,neis the electron concentration,bi,blare the coefficients of deviations of the population of level from LTE. (bi= 1 in LTE case).

The dissociative equilibrium equation for molecules besides atoms and ions is:

nk₁₂₃+_..m = n1n2n3..nm

nk e

ε (2.17)

whereεis the dissociation constant:

ε = U(n

k

123..m(2πM123..mkT /h2)3/2[2(2πmekT /h2)3/2]k

U1(2πM1kT /h2)3/2...Um(2πMmkT /h2)3/2 ∗

exp(₋[E(nk₁₂₃+_..m)₋E1−E2..Em]/kT) (2.18)

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herenk+

123..m is the concentration of a molecule that consist ofm atoms in degree of ionizationk,Ui is the partition function of the componenti,Miis the mass of the atom

i,meis the mass of electron,M123..mis mass of molecule,Eiis the bond energy of the componenti,E(nk₁₂₃+_..m)is the bond energy of the molecule,E(nk₁₂₃+_..m)₋E1−E2..Em is the bond-dissociation energy of the molecule.

Thus, we can derive the radiation fluxes emitted from the atmosphere of a star in the line or in the continuum by using the calculations for the concentration of atoms and

molecules and monochromatic absorption coefficient.

2.2.2 Calculation of star atmosphere models

The model or the structure of stellar atmosphere is a set of parameters that describes the physical state of matter (temperature, pressure, electron density etc.) on a given grid of

the depths. The developing the models of stellar atmospheres is not a trivial. Therefore there are some simplifications are usually used. The modelling of the atmospheres of cold stars and brown dwarfs (spectral classes are M-L-T) require the absorption of radi-ation by molecules taken into account. The conventional simplificradi-ations in calculradi-ations of stellar atmosphere of dwarf-stars are:

•The energy transfer in a stellar atmosphere is carried out by radiation and convec-tion simultaneously. The flux of energy do not change with the depth since the sources of energy are deep under the atmosphere and there are no sources of energy in the atmosphere.

• The thickness of the atmosphere is small when compared to the radius of a star. The atmosphere can be represented as a complex of a plane-parallel layers.

•The atmosphere is in thermodynamic equilibrium (the gas pressure is balanced by the force of gravity).

ρ(d2r/dt2) =₋ρg+dP/dr= 0 (2.19)

where ρ is the density, g is the acceleration of gravity (that is constant due to the approximation of the thin atmosphere, g = GM/R2, M and R are stellar mass and radius respectively), P is the gas pressure.

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•The atmosphere is homogeneous in every plane-parallel layer. This approximation reduces modeling to one-dimensional case.

•The abundance of chemical elements do not change in the atmosphere with depth. This assumption is based on the high efficiency of mixing of matter by convection in the stellar photosphere.

Opacities in stellar atmospheres

The theoretical energy distribution in the stellar spectrum depends heavily on the opac-ity in the continuum and the lines. The absorption coefficient in any given frequency is the sum of absorption of all possible transition (bound-bound and bound-free and

free-free transitions) of all chemical elements and their chemical compounds that can absorb photons on a given frequency. The LTE assumption simplified the calculations significantly because the absorption coefficients in the LTE case are functions of the density (or the number of particles) and the temperature only. The variables associated with the chemical composition are the parameters of the calculations. In the atmo-spheres of the late-type stars that are having the solar chemical composition negative

ions of atoms and molecules provide a significant contribution to opacity. For example, in the Sun-like stars, the primary source of bound-free absorption is theH− _{ion. The}

free-free absorption of H−

2 became important in M type stars. The Rayleigh

scatter-ing provides substantive contributions in the opacity of G and K stars. The opacities associated with the bound-bound transitions (the spectral lines) are essential in stars of all spectral types. The neutral and singly ionised metals of the group of iron play an

essential role in Sun-like stars. The molecular bands appear in cool stars and dominate the spectra of M-L-T types in the optical and infrared.

The bound-bound absorption affects the structure of the stellar atmosphere and the energy distribution in the stellar spectrum. This is the so-called ’line blanketing’ effect.

The relationship between the absorption coefficientkλand the temperatureT owing to the impact of the ’line blanketing’ effect in the layers where the continuous spectrum and the spectral lines are formed. The fraction of the energy absorbed in the lines on a given depth is reflected in the underlying layers. The density of the radiative energy and the temperature are increasing due to this process. This is the backworming effect Lyubchik & Pavlenko (2000). The distributions of the gas pressurePgand the electronic

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pressurePeare changing together withkλ andT in the models. These effects affect the resulting synthetic fluxesFλ.

The line blanketing effect and the numerous molecular lines are of great importance in the late-type stars. The molecular bands are sensitive to temperature. Thus even small differences in the temperature distribution change the intensity of the molecular lines dramatically.

The chemical composition

The chemical composition is one of the fundamental parameters of stars together with

the effective temperature Teff and gravity logg. There are variables depends of the

chemical composition in the calculations of the continuum absorption coefficient kλ and the electronic pressure Pe. The blanketing effect, that is responsible for addition opacity by multiple spectral lines, is also depends on the chemical composition. The parameter[M/H] =logN(X)−logN⊙(X)is used for characterisation of metallicity,

wherelogN⊙(X)is abundance of an elementX in the solar atmosphere.

The variations of the abundances of chemical elements affect the structure of the stellar atmosphere in the following circumstances:

• The element contributes significantly to the gas pressure due to high abundance.;

• The opacity associated with the free-free and the bound-free transitions of a given element contributes significantly in the full absorption coefficientkλ(τ);

• The given element is one of the primary providers of the free electrons, so the electronic pressurePeis affected by the abundance of the element;

• The atomic lines of an element or the lines of a molecule, having the element, provide substantial contribution in the blanketing effect.

The variations of chemical composition play important role in the atmospheres of cool dwarfs. The structures of the atmospheres and the spectra are strongly depend on relative abundances of C, N and O (Pavlenko & Yakovina, 1994). In general, the chemical composition affects the spectra in the complex way: the spectral regions have

different sensitivity to the abundances of different elements. On the other hand, only the methods of modelling of stellar atmospheres allow to provide a detailed study of the stellar chemical composition.

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Convection

The energy transfer in stellar atmospheres is being carried out through two mechanism: the radiative transfer and the convection. The full radiative flux can be derived as:

πF =πFrad+πFconv =σRTef f4 (2.20)

whereσRis the StefanBoltzmann constant.

According to numerical calculations, the radiative transfer is dominated in the atmo-sphere of early spectral types stars. The convection became more and more important starting with F type stars and colder (Mihalas, 1970). In M dwarfs the radiative zone is limited optically thin layer (Allard & Hauschildt, 1995). The atmospheric structure and therefore spectra of M dwarfs arise from the effects of convection (Baraffe et al.,

1995; Freytag et al., 2010).

The standard mixing length theory (Allard & Hauschildt, 1995) is the most common approach to model convective energy transfer in stars. However, it can be considered only as first approximation.

The method named ’penetrative convection’ was proposed in Zahn (1991), Kurucz (1992). The method is based on the assumption that the centre of the convective element stops at the boundary of the convective zone, and on the distance of one radius of the convective element there is a convective flow above the boundary of the convective zone The flux is found by calculating the convective element of energy transfer in the

approximation of the theory of the length of mixing, and then smoothed to the length of the lengths of the active element.

There are number of models of convection were developed (Chan et al., 1991; Gross-man, 1996; Kim et al., 1996). However, they are very complex and were not applied to

models of M dwarfs. It should be notest that several studies have shown that the the-oretical spectra are not very affected by the specific of convection mechanism (Brett, 1995; Baraffe et al., 1997). This fact makes cold stars poor test objects for testing of models of convection in stellar atmospheres, but, it also makes it possible to sim-plify calculations by applying simple convection mechanisms, like the standard mixing length approximation.

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Line broadening mechanisms

The contribution of atomic lines in the spectra of cool dwarfs are less critical than for hotter stars because the molecular absorption bands are dominated in M dwarfs and the low temperatures of photospheres lead to an increase of a number of atoms in higher excitation and ionisation levels.

As a result, there are only a few the strongest resonance and subordinate lines can be identified and used for diagnostics of the photospheric parameters. This is NaI lines at 5889,5896, 8183, 8195 and 10746, 10749,10835 ˚A; KI lines at 6911, 6939, 7665, 7699, 9950, 9954, and 10480,10482,10487 ˚A, RbI lines at 7950 ˚A and Ba I at 7911, 7913 ˚A to name a few. Despite the difficulties of working with atomic lines, the proper modelling of atmospheres of cool stars requires for a complete account of the opacity

in the hotter layers of the inner atmosphere. The profiles of atomic lines are sensitive to the fundamental physical properties of stars, making them essential to define the gravity, metallicity and rotation velocity.

The natural broadening is several orders of magnitude weaker than pressure and

Stark broadening so it can be neglected in the weakly ionized atmospheres at Teff<

3000 K. It is because the temperature of a gas is not high enough to provide a necessary amount of ionisation, and the electron and proton densities are considerably smaller than the densities of the neutral atoms and molecular species. The broadening of KI and NaI resonance lines is dominated by the pressure effects, but the theory of these mechanism is not particularly advanced. The pressure broadening is calculated by using

two methods: the collisional approximation (van der Waals theory) and the quasi-static theory (Allard et al., 2003b; Burrows & Volobuyev, 2003).

The total thermal plus micro-turbulent line widths are always much smaller than the line width associated with the van der Waals broadening:

γvdW = 17C62/5v3/5Np (2.21)

The formula describes the interaction between two different, unpolarized neutral particles within the impact or static approximation. HereγvdW is the full-width half-maximum damping constant of the resulting Lorentz profile,v is the relative velocity

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between perturber and absorber, and Np is the number density of perturbers. The in-teraction constantC6 can be determined for the atomic hydrogen. However, there is no

direct method for slow molecular hydrogen perturbers that dominate in the atmospheres

of low-mass stars (Guillot et al .,1994).

Besides the van der Waals broadening the quasi-static broadening (Allard et al., 2003b; Burrows & Volobuyev, 2003) is important for understanding of late-M and L dwarfs. The use of a collision approximation (van der Waals broadening) allows us to

describe the profiles of metal lines in early M dwarfs. The quasi-static theory is used in the study of L dwarfs, where energy levels of atomic sodium or potassium are immersed in a sea of molecular hydrogen and are subsequently perturbed by the potential field of the diatomic hydrogen. The modelling of atomic-line broadening for objects that are close to the transition between M and L spectral classes is complicated and this issue is beyond this paper aims.

Dust in cool dwarfs atmospheres

Modelling of dust clouds in the atmospheres of late M, L, T and Y spectral types is one

of the most important challenges in calculations of synthetic spectra of cool stars and brown dwarfs. Tsuji, Ohnaka & Aoki (1996), Tsuji et al. (1996), and Fegley & Lodders (1996) identified dust formation for stars having temperatures below 2600 K and show the importance of condensation of this process.

The presence of dust significantly changes structure of stellar atmosphere. The con-densation of oxygen, titanium, iron, vanadium on the dust particles of CaTiO3, TiO2, and VO2 lead to decrease abundance of these elements in the gas phase. The result of the consideration of the dust in late M and L dwarfs is a significant weakening of the molecular absorption bands of TiO, VO, FeH and exposing the CrH and FeH lines that were otherwise hidden by the stronger bands (Allard et al., 2001). The greenhouse

ef-fect caused by the presence of the dust clouds afef-fect the infrared colours of late M and L dwarfs make them extremely red compared to hotter early M stars with no dust (Allard, Homeier & Freytag, 2012a). The Rayleigh scattering on the dust particles provides veiling to the optical energy distribution of dusty late-M and L dwarfs The high layers of the atmosphere, above the dusty clouds, are depleted from metals that condense be-low and cooler because of the reduced molecular opacities (Allard, Homeier & Freytag,

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2012a). We can also expect inhomogeneous vertical and horizontal distribution of the dust clouds, gravitational settling and rains of condensates in the atmospheres of cold stars and brown dwarfs Baraffe et al. (1997).

The dust composition, that was recovered by using equilibrium chemistry, includes a broad range of chemicals likeZrO2, refractory ceramics (CaT iO3,Al2O3), silicates

(M g2SiO4), salts (CsCl,RbCl,N aCl) and ices (H2O,N H3,N H4SH) (Allard et al.,

2001; Lodders & Fegley, 2006).

The first model of atmospheres for M dwarfs that includes the dust formation and the dust opacities was presented in Tsuji, Ohnaka & Aoki (1996). The next genera-tion of models unified the cloud models were developed by Tsuji (Tsuji, 2002; Tsuji, Nakajima & Yanagisawa, 2004). The limiting effects of dust where discussed in work

(Allard et al., 2001) The two approaches were presented to explore the limiting prop-erties of cloud formation: the sedimentation or gravitational settling is assumed to be fully efficient (the AMES-Cond models) and the gravitational settling is assumed inef-ficient and dust forms in equilibrium with the gas phase (the AMES-Dusty models).

Beyond these extreme cases the distribution of density and size dust particles need

to be considered as a function of depth in the atmosphere. The cloud models of Rossow (1978), that was developed for planetary atmospheres was adopted to cold stars and brown dwarfs in Allard et al. (2003a). However, when applied in the theoretical stellar atmospheres BT-Settl with gravitational settling the underestimation of the production of the dust was found.

The comparison of different cloud models and their impact on model atmospheres was done in (Helling et al., 2008a). Later, the Drift-Phoenix models were developed based on the PHOENIX code (Helling et al., 2008b). Unlike to the other models this work focused on the nucleation and growth of grains as they sediment down into the atmosphere. The Drift-Phoenix models well fit the dusty atmospheres of L dwarfs,

however, the L/T transition was not explained (Witte et al., 2011).

It is important to note that the cloud parameters in all these models were modified in attempt to obtain realistic spectral transition. The problems come from the reproduction of the mixing properties and the resulting diffusion mechanism. The 2D radiation hy-drodynamic (hereafter RHD) model based on the PHOENIX gas opacities was applied

to simulate the mixing and diffusion in cool stars atmospheres in Freytag et al. (2010).

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Freytag et al. (2010) clarifies that the description of the underlying mixing processes requires the establishment of an additional free parameter. The processes of transport of the condensible material from the hotter lower layers to the cloud forming layers

can be simulated by the RHD model. The BT-Settl models were updated by this cloud model, that allow reproducing energy distribution of cool dwarfs down to the L-type objects (Allard, Homeier & Freytag, 2012a).

Despite the progress in the modelling of the dust clouds in stellar and substellar

atmospheres, that has been achieved in recent decades, there are still difficulties in the reproduction of observed spectra of cool stars. For example, the transition between spectral types M-L and L-T, the transition between non-dusty early M dwarfs and dusty late M dwarfs are still understood poorly.

Stellar activity of cold stars

The late type stars show high level of the chromospheric activity. The model of ’α

-Ω’ dynamo, or ’solar dynamo’ predicts a correlation of the magnetic activity of stars with rotation. This correlation was confirmed for the main-sequence stars of spectral types from F to early M by numerous studies (see Noyes et al. (1984), Marilli, Catalano

& Trigilio (1986); Rutten (1986); Delfosse et al. (1998); Pizzolato et al. (2003) etc.). Increasing of the rotational velocity tends to rise of emission in chromospheric activity indicators like lines of CaIIH and K and Hα. However, in some cases, emission declines in the most rapid rotators (James et al., 2000).

According to the model of ’α-Ω’ dynamo, the global dynamo is located at the region between the convective zone and the stable bowels of a star. The ’α-effect associ-ated with the twisting of fields by helical convection and the Ω-effect associated with the generation of toroidal fields from poloidal by differential rotation (Moffatt, 1978; Steenbeck & Krause, 1966). The both effects are sensitive to rotation. The α-effect is sensitive to rotation because the convection is sensitive to the rotation rate. TheΩ-effect affected because the faster rotating stars are produces higher contrasts of the internal

angular velocity. In general, our understanding of the dynamo process is qualitatively consistent with the observed data for the Sun-like stars (Brown et al., 2008).

It is important to note that the stellar rotation and activity might be an indicator to estimate the evolution status of a star. The stellar rotation speed decrease with

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ing age, due to losing of angular momentum. This rotation-activity-age correlation was predicted theoretically and then confirmed by observations (Wilson & Skumanich, 1964).

Despite progress in studying of the Sun-like stars, our understanding of the physics of stellar activity of the low-mass is far less clear. In comparison with FGK-dwarfs, a larger fraction of M dwarfs show evidence of magnetic activity like Hα emission (Delfosse et al., 1998) or frequent flare (Hawley & Pettersen, 1991; Davenport et al.,

2014). High activity of cool stars cannot be explained within the paradigm if ’α-Ω’ dy-namo. Besides, the fully convective late-type stars have unstable interiors, so the mech-anism of generation of the magnetic field and the interaction of the magnetic field with the stellar atmosphere of low-mass stars are different from the Sun-like stars (Chabrier & Baraffe, 2000).

Two concepts have been suggested to explain the behavour of low-mass stars. The generation of the magnetic field depends on the volume of the convection zone. This volume decreasing in proportion to the mass once the stars became fully convective in case of M dwarfs (Noyes et al., 1984). In this case the strength of magnetic field is saturated in the lowest mass stars, and no strong rotation-activity connection would be

expected (Rosner, Golub & Vaiana, 1985; Stauffer et al., 1991). The alternative concept suggests the propagation of the acoustic shocks coming from the boundary of radiative and convective zones. The resulting heating of the chromosphere is efficient in the convective low-mass stars and may by substantial (Mathioudakis & Doyle, 1992; Mul-lan & Cheng, 1993). This concept allows correlating the chromospheric activity with the atmospheric parameters such as age, metallicity and rotation velocity because the

depth of the convection area depends on the physical properties of the star. Dobler, Stix & Brandenburg (2006) and Browning (2008) provided simulations of stellar dynamo in fully convective stars and show that the rotation may affect the field strengths. But these numerical experiments do nor provide understanding of physics of magnetic field gen-eration. Despite the difficulties the recent theoretical works achieved a good progress in description of the large-scale and small-scale magnetic fields in fully convective stars

(Yadav et al., 2015).

The reliable estimations of the magnetic activity of low mass stars is not trivial as well. The Hα line is often used as an indicator of stellar activity. However, the Hα

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is absorption line that may have an emission core in active stars. The combination of the absorption and emission within one line lead to conceptual uncertainties (Cram & Mullan, 1979). The emission lines of CaII (the H and K resonance lines at 3968 and

3934 ˚A) do not have this issue, but it is difficult to detect them in spectra of faint cold stars. The radial velocity searches for extra-solar planets cared out on spectrographs like the HARPS and the HARPS-N provide long-time series of high-resolution spectra for large samples of stars. The resent studies of chromospheric activity of low-mass stars are often used the observed data obtained in the course these searches

(Gonz´alez-´

Alvarez et al., 2019; Astudillo-Defru et al., 2017).

Understanding of cool dwarfs is impossible without consideration of their chromo-spheric activity. We need to explain, for example, stars of the same age or the same mass and rotation velocities having totally different levels of activity. Thus this prob-lem is one of the essential objectives for future theoretical and experimental studies.

Lists of absorption lines of atoms and molecules

Besides the model of the atmosphere, the lists of atomic and molecular lines are an essential component of calculation of synthetic spectra. However, despite considerable efforts by developing lists for molecular opacities, there is still a lack of reliable data for many important species.

Obtaining line lists for hot molecules is proving to be a difficult task because of numerous reasons like: the line lists of molecules may consist of billions of lines; in many cases, it is the difficult to determine absolute line strengths; an assigned spectra required to correct temperature dependence to be reproduced; the line lists should cover

a large range of wavelengths.

In last decades substantial effort has been made to create spectroscopic databases of atomic and molecular line lists:. The most important databases are listed below”

• The ”Vienna Atomic Line Data Base” (VALD) (Kupka et al., 2000; Piskunov et al., 1995) is well-used in stellar physic. The VALD is amid to collect an ac-curate and complete list of atomic lines that are important in astrophysics. The database provides data of about_∼600000 lines, and it is one of the largest homo-geneous collections for atomic transitions. However, there is a lack of molecular data in the VALD.

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• The opacity tables of Kurucz (Kurucz, 2011) is widely used in astrophysics. The tables contain line lists for astrophysically important atoms and diatomic molecules. The database is a compilation of public data and numerical

simula-tions provided by the author. The energy levels, wavelengths, gf-s and damp-ing constants that are not available from the literature were calculated by Ku-rucz. The database includes the line lists for diatomic molecules H2, CII, NH, OH, MgH, SiH, CN, C2, CO, SiO, and TiO that were computed by the author. However, the data for the majority of molecules are approximate, and the list of molecules is not complete

• HITRAN (Rothman et al., 2009) and GEISA (Jacquinet-Husson et al., 2011), provide lists of molecular transitions that play important role at temperatures around 300 K. These databases are aimed to model the Earth’s atmosphere but may be applied for other objects of the Solar system and exoplanets. However, using this data for calculation of stellar atmospheres stars are not optimal since

the databases are designed for significantly colder objects;

• JPL (Pickett et al., 1998) and CDMS (M¨uller et al., 2005), are optimised for the cool interstellar medium and not very useful for stellar physics;

• HITEMP (Rothman et al., 1995, 2010) can be applied for stellar temperatures but for five species only;

• RADEN databank at Moscow State University (Hefferlin, 1999) SCAN data base (Jorgensen, 1996) and UGAMOP data base at the University of Georgia (www.physast.uga.edu/ugamop/) contain partial lists of diatomic molecules;

• ExoMol (Tennyson & Yurchenko, 2012; Yadin et al., 2012) is designed to provide line lists of molecules for cool stars and exoplanets. However, this database is in progress status.

We can consider that, although there are a number of databases of molecular line list, none of them is complete and can alone meet requirements of the modelling of stellar atmospheres. Thereby, the atomic and molecular line lists are often combined from different sources to calculate a particular grid of synthetic spectra.

T iOabsorption bands are the main contributors to the opacity of the spectrum below 1µm in cool, oxygen-rich stars (Allard & Hauschildt, 1995). The theoretical studies Jorgensen (1994) and Plez (1998) provided partial line lists for the molecule. A num-ber of detailed experimental spectroscopic studies was carried out as well (Simard &

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Hackett, 1991; Amiot et al., 1995; Kaledin, Mccord & Heaven, 1995; Ram, Bernath & Wallace, 1996). Schwenke (1998) made a compilation of these data and measurements from earlier laboratory spectra (Galehouse, Davis & Brault, 1980; Brandes &

Gale-house, 1985) with results of theoretical studies to create a detailed TiO line list con-taining 37 million lines. The TiO line list from Schwenke (1998) is the most extensive available list for TiO, and it has been proven to fit well with observed spectra (Allard, Hauschildt & Schwenke, 2000). However, the theoretical line list of Plez (1998) is also widely used. Pavlenko et al. (2006) shows that both sources provide similar opacity in visual and near infrared.

The water lines dominates in the infrared for cool dwarfs. There are severalH2Oline

lists that are used in computations of synthetic spectra of dwarfs: HITRAN Rothman et al. (2009), AMES Partridge & Schwenke (1997) BT2 Barber et al. (2006). The AMES (Partridge & Schwenke, 1997) H2O line list was computed using a potential

energy surface from (Murrell & Garten, 1984). The results were fitted with lines from provided in the HITRAN (Rothman et al., 1995).

However, the most comprehensive H2O line list is the BT2 (Barber et al., 2006).

This is theoretical line list was computed using a discrete variable representation

im-plemented in the DVR3D suite of programs (Tennyson et al., 2004) and a potential energy surface obtained by fitting to laboratory spectra (Schwenke & Partridge, 2000). The BT2 is the most the largest and the most accurate line list. It contains more than 500 million transits and the experimental energy levels are within 0.3 cm1 with the BT2 values in 90% cases.

The molecules of MgH, FeH and CrH are also important opacity sources in the spectra of low-mass stars. Moreover, the lines of MgH and CrH and be applied for the deuterium test: using of the deuterium abundance in brown dwarfs to estimate their properties (Pavlenko et al., 2008).

The massive experimental research of MgH was presented in works Weck et al. (2003a), Weck et al. (2003b) and Skory et al. (2003). The complete line list for the

B′2P+

−X2P+

system of24M gH was calculated using this data.

The FeH line list (Dulick et al., 2003) contains information of the F4_∆

i −X4∆i electronic system based on the spectroscopic constants of rotational levels (v = 0, 1, 2)

of the FeH X and F states. The values forv= 3, 4 andJ-values were extrapolated. The

Modeling energy distribution of cool dwarfs in the optical and near-infrared