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Fundamental parameters of M dwarfs
Measuring accurate stellar parameters from the optical spectra of M dwarfs is a very challenging task. As the abundances of diatomic and triatomic molecules (e.g. TiO, VO, H2O, CO) in the photospheric layers increases with spectral subtype, their forest of weak lines eventually erases the spectral continuum and makes a line-by-line spectroscopic analysis difficult for all but the earlier M subtypes (e.g. Gustafsson 1989; Woolf & Wallerstein 2005). As already shown before, Fig. 1.6 shows a good example of a high-resolution spectrum where a G dwarf star (above) and a M dwarf (below) are compared. Despite the spectra of the M dwarf looking just like noise, it has a higher signal-to-noise ratio.
In the following Section I will make a brief outlook regarding the classical derivation of stellar parameters of FGK stars, adding some details regarding M dwarfs. Afterwards, I will discuss the difficulties of the continuum determination for M dwarfs (Sec. 2.2), as well as an explanation regarding the need for spectral synthesis, in Sect 2.3. Finally, the state of the art regarding the determination of M dwarf parameters will be discussed in Sec. 2.4.
Classic spectroscopic analysis
This section follows Gray (2005); Reid & Hawley (2005), and Bonfils (2012) closely, where I will present a brief outlook on the determination of stellar parameters for a FGK dwarf star. Me and my collaborators applied this procedure to calculate the parameters of the FGK primaries in binaries systems with a M dwarf secondary in order to infer the [Fe/H] of both stars, as detailed in Chapter 3. This work culminated in the publication of the first paper of my Ph.D.
If we take a good look at a regular stellar spectra (Fig. 1.6) we can easily identify the presence of many absorption lines. These lines correspond to electronic transitions among the different levels of atoms and molecules (bound-bound transitions). This is especially true for cooler stars of the FGKM end of the HR diagram, where the atoms and molecules of many species are not fully ionised. The elements other than hydrogen and helium, referred as ’metals’ (e.g., C, O, Mg, Si, Fe, Ti, etc), only account for a tiny percentage of the abundance ( 2%). However, most spectral lines have origin in these metal species.
These lines show different shapes and strengths that derive directly from the conditions in the pho-tosphere of the star (temperature, pressure, radiation, magnetic and velocity fields). The most important aspect in the determination of the stellar parameters is the strength of the line absorption, that depends on the number of absorbers that correspond to a certain electronic transition.
Local thermodynamic equilibrium
If we consider that collisions (rather that radiation) dominate the excitation of the atoms (as a good approximation in the case of FGKM stars), then local thermodynamic equilibrium (LTE) will apply and we can express the ratio between the number of atoms in an energy level n and the total number of the atoms of that species as Nn = gn 10 q(T )cn ; (2.1) N u(T ) where Nn is the population of energy level n, N is the total number of atoms, gn is the degeneracy of level n, cn is the excitation energy of the same level, q(T ) = 5040=T , u(T ) = Sgie ci=kT is the partition function, k is the Boltzmann’s constant and T is the temperature. This is one formulation of the well known Boltzmann equation.
Similarly, the ionisation for the collision dominated gas can be calculated using Saha’s Equation,
N1 F(T )
N0 = Pe ;
where
F(T ) = (pme)3=2(2kT )5=2 u1(T )e I=kT : h3 u0(T )
The N1=N0 is the ratio of ions in a given ionisation state to the number of neutral atoms, u1=u0 is the ratio of ionic to neutral partition functions, me is the electron mass, h is the Plank’s constant, Pe is the electron pressure and I is the ionisation potential. The transition from neutral to first ion, and upwards occurs fairly rapidly with Te f f , as shown in Fig. 2.1, for Iron.
Thermodynamic equilibrium is achieved when the temperature, pressure and chemical potential of a system are constant. In LTE, these thermodynamic parameters are varying in space and time but this variation is slow enough for us to assume that, in some neighbourhood about each point, thermodynamic equilibrium exists (hence the ’local’). When the LTE is valid, each point will behave like a black body of temperature T . This is, of course, an approximation, but it is acceptable for the cases when the ratio of collision to radiation induced transitions is large, as it is the case for photospheres of FGKM stars. In the outer photospheric layers, LTE performs poorly due to the proximity of the open space boundary, where the radiation can escape freely. This boundary is responsible for the formation of the absorption lines. We cannot use strong lines calculated by LTE because their cores form in these upper layers.
In the photospheres of M dwarfs, the temperature is lower and surface gravity is higher, leading to higher matter densities. Both low temperature and high density will increase the opacity of the photosphere. This effect is due to the increase of many low-energy transitions from neutral atoms (the now dominant form in the cooler atmosphere) and molecules, and to the increased density of atoms and molecules, that augment the probability of interaction with the radiation. This combination of factors move the atmosphere towards LTE. Therefore LTE is also valid when modelling M dwarf atmospheres.
As we will see in the following sections, LTE will be used to calculate our model atmosphere and help us find the stellar parameters. We must recall that the temperature in the LTE approximation is the same for all physical processes: thermal velocity distributions, ionisation equilibrium, excitation of atomic populations. It’s a simplification over the real problem but it is very practical.
The behaviour of line strength
The strength or equivalent width (EW) of a spectral line depends on the absorption coefficient (the fraction of incident radiant energy absorbed per unit mass or thickness of an absorber) and on the number of absorbers, derived from Eqs. (2.1) and (2.2). This implies that the line strength depends on temperature, electron pressure and the atomic constants. This is valid only as a good approximation for weak lines (i.e., lines with typical EW . 200 mA)˚. Stronger lines may depend on other factors.
From the accurate measurement of the EW of weak lines, and using the correct atmospheric mod-els we can calculate the stellar parameters (metallicity, temperature, surface gravity, microturbulence, and others), and also chemical abundances.
The Measurement of the EW
Equivalent width is a measure of the intensity of a spectral line. It is defined as the width of a rectangle with height between the level of the continuum, normalised to unity, and the reference zero, having a where Ic is the intensity of the continuum and Il is the intensity of the wavelength at each dl. Usually, the EWs are measured by fitting a gaussian function to the spectral line and to the local continuum. We must note that, in some of the following plots, the equivalent width will be represented by W .
The temperature dependence
Temperature is the most important variable in determining the line strength. This results from transition probabilities of the excitation and ionisation process equations (section 2.1.1). We can appreciate the behaviour of the EW of a typical weak line (e.g., Iron) with Te f f depicted by Fig. (2.3). Four different cases are shown:
In cases 1 and 3 an increase in Te f f leads to an increase in the line strength. This also happens in case 4 up to a maximum value of EW. We can observe, for these cases, that the electron pressure term, W, goes against the increase of the strength, reflecting an increased continuous absorption.
The decrease of the line strength with Te f f in case 4 is caused by the increase in the continuum absorption from the negative hydrogen ion, whereas the decrease in case 2 is due to the ionisation of the absorbing species. In any case, we can predict how a certain line will grow or weaken by considering the ratio of the line absorption coefficient with the continuum absorption coefficient. Note that the weakening of the line due to an increase of the continuum absorption also affects the lines in case 1 and 3. However, this effect is weak compared to the excitation one.
The direction and strength of change will depend on the temperature and on the excitation potential (the minimum energy that an electron of a certain atom needs to make a successful transition between the ground state and an excited state) of the line. For stars similar to the Sun, cases 2 and 4 apply because most elements are ionised. Solar lines of neutral species almost always decrease in strength with Te f f , but ionised species have the opposite behaviour. In the case of cooler M dwarfs, where the atmosphere is composed mostly of neutral species, cases 1 and 3 are the ones of interest. However, as shown in Section 2.2, the existence of hundreds of millions of molecular lines makes a EW-based analysis next to impossible.
The abundance dependence
The abundance is also an important factor in the line strength variation. As the abundance increases, line strength also increases, as expected. However, the EW does not change linearly with abundance, as we can see in Fig. 2.4(a).
There are three different regimes. The first one corresponds to the weaker line behaviour, where the doppler core dominates and the EW is proportional to the abundance A. The second phase begins when the central depth approaches the maximum value and the line saturates and grows asymptotically towards a constant value. The third one starts as the optical depth of the line wings becomes significant compared to the absorption of the continuum. We are only interested in the first phase, where the behaviour of the curve is linear.
Table of contents :
1 Introduction
1.1 Since ancient times
1.2 The first attempts
1.3 The first discovery around a main sequence star
1.4 Planets around M dwarfs
1.5 Planetary system formation
1.6 Host star properties
1.6.1 Planet-metallicity correlation
1.6.2 Planet-stellar mass correlation
1.6.3 Evidence from chemical abundances
1.7 The Thesis
2 Fundamental parameters of M dwarfs
2.1 Classic spectroscopic analysis
2.1.1 Local thermodynamic equilibrium
2.1.2 The behaviour of line strength
2.1.3 The temperature dependence
2.1.4 The abundance dependence
2.1.5 The pressure dependence
2.1.6 Microturbulence and velocity fields
2.1.7 Method
2.2 The continuum problem in M dwarfs
2.3 Spectral synthesis
2.4 State of the art
2.4.1 Metallicity
2.4.2 Effective temperature
2.4.3 Mass & radius
2.4.4 Surface gravity & velocity fields
3 A Comparative study of photometric metallicity scales
3.1 Introduction
3.2 Evaluating the photometric calibrations
3.3 The three photometric [Fe/H] calibrations
3.3.1 Bonfils et al. (2005) calibration
3.3.2 Johnson & Apps (2009) calibration
3.3.3 Schlaufman & Laughlin (2010) calibration
3.3.4 Refining the Schlaufman & Laughlin (2010) calibration
3.4 Discussion
3.5 Paper: A comparative study of photometric metallicity scales
4 Planet-metalllicity and planet-stellar mass correlations of the HARPS GTO M dwarf sample
4.1 Introduction
4.2 A new M dwarf metallicity and effective temperature calibration
4.2.1 Calibration sample
4.2.2 Method
4.3 The metallicity-planet correlation
4.3.1 Bayesian approach
4.3.2 Comparison with the California Planet Survey late-K and M-type dwarf sample
4.4 Metallicity-planet relation from the HARPS+CPS joined sample
4.4.1 Bayesian approach for the joined sample
4.5 The stellar mass-planet correlation bias
4.6 Discussion
4.7 Paper: Planet-metalllicity and planet-stellar mass correlations of the HARPS GTO M dwarf sample
5 SPITZER observations of GJ 3470b: a very low-density Neptune-size planet orbiting a metal-rich M dwarf
5.1 Introduction
5.2 Data analysis
5.2.1 Spitzer photometry
5.2.2 Spectroscopic measurements
5.3 Stellar characterisation
5.4 Planetary and orbital parameters
5.5 Exploring the interior composition of GJ3470 b
5.5.1 Summary
5.6 Paper: SPITZER observations of Gj 3470b: a very low-density Neptune-size planet orbiting a metal-rich M dwarf
6 Conclusions and future prospects
References
A Planet detection techniques
A.1 The radial velocity technique
A.2 Transits
A.3 Other methods
B Exoplanet properties
B.1 Mass distribution
B.2 Period distribution
B.3 Mass-period relation
B.4 Eccentricity-period relation
C The Spectrograph
C.1 The basic principles of a spectrograph
C.2 The echelle spectrograph
C.3 The HARPS Spectrograph
D Publications and communications related to this Thesis
