SPITZER observations of GJ 3470b: a very low-density Neptune-size planet orbiting a metal-rich M dwarf 

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Planet-stellar mass correlation

The evidence of a planet-stellar mass correlation is not as easy to establish as its metallicity counterpart.
The detection biases are harder to correct, as the RV sensitivity changes both with brightness and stellar mass.
Early works on the planet-stellar mass link have shown a weak correlation of giant planets with increasing mass, with hints that higher mass stars, up to 1.5M may have a higher giant planet frequency (e.g. Laws et al. 2003). This hint was reinforced by works on M dwarfs, where giant planets are scarce (Endl et al. 2006; Butler et al. 2006; Bonfils et al. 2007) and more frequent at intermediate mass subgiants and giants (Johnson et al. 2007; Lovis & Mayor 2007). At the same time it seems that there are indications that planets around more massive stars tend to have higher masses (e.g. Lovis & Mayor 2007). Core-accretion models support these results as it is predicted that planet formation efficiency should increase with stellar mass (e.g. Laughlin et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008; Alibert et al. 2011). Another mechanism that may come into play is the ‘compensation effect’ (e.g. disk mass to allow giant planet formation (and vice-versa). Moreover, the existence of both a planetmetallicity and planet-stellar mass relation suggests that the surface density of dust is very important in the planet formation process (e.g. Laughlin et al. 2004; Ida & Lin 2004b; Robinson et al. 2006). More recently, a study of Johnson et al. (2010a) quantifies a mass-metallicity-planet function, where the frequency of giant planets is defined by the functional form fp = C:(M?=M)a:10b:[Fe=H].
C, a, and b have the best fit values of 0.07, 1.0, and 1.2 respectively. They reach the conclusion that this function has a much higher significance than a similar one with only the [Fe/H] dependence.
Quantitatively, this translates into an Bayesian evidence 2400 times higher for fp than for the metallicity only model, according to Kass & Raftery (1995). However, a recent work by Mortier et al. (2013) has tested different functional forms using a HARPS+CORALIE sample of 1798 stars, and did not find any difference between different functional forms (metallicity & mass dependence, only metallicity dependence, only mass dependence, and a combination of different functions, including cutoff or a constant for lower [Fe/H] values) with stellar mass and [Fe/H] or with only metallicity (see Fig. 1.11).
The Bayesian evidence ratio between the different functions sits between 1.0 and 4.12 meaning a very low preference between models. They also predict that it will only be possible to disentangle between the different functional forms with a sample of about 5000 stars.

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 N = gn u(T)10􀀀q(T)cn ; (2.1).
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 N0 = F(T) Pe ; (2.2).

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 models we can calculate the stellar parameters (metallicity, temperature, surface gravity, microturbulence, and others), and also chemical abundances.

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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.
Every spectral line shows a similar behaviour. A plot like Fig. 2.4(a) is called a curve of growth. Fig. 2.4(b) shows the line profile change with the chemical abundance of the absorbing species.

Microturbulence and velocity fields

In the analysis of line profiles we should be aware that photospheric velocity fields (the turbulence) introduce Doppler shifts, that will be reflected in the spectra: the small scale motion (<< mean free path of photons), microturbulence, can affect the radiation transfer and the large scale motions, macroturbulence and rotation, introduce a broad distribution of Doppler shifts that reshapes the line profile, but does not change its EW.
Microturbulence is an ad-hoc broadening parameter and is almost always incorporated into stellar parameter analysis because it adjusts the EW of saturated lines when they are smaller or greater than predicted. Fig. 2.6 shows the effect of microturbulence on the curve of growth. As we can see, changes in microturbulence can change the shape of the curve of growth and consequently the measured abundance.
To obtain the correct value of microturbulence one has to simply change its value in the atmospheric models until all spectral lines of an element, namely iron, give the same abundance, independently of the EWs.
In M dwarfs, microturbulence is not a very important parameter, amounting only to 1-2 km s􀀀1 and is usually computed around these values or simply neglected on stellar modelling (Reid & Hawley 2005).

Table of contents :

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 

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