A new era in Astronomy: extra-solar planets

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Current techniques

The detection of the rst exoplanets (Wolszczan and Frail, 1992; Mayor and Queloz, 1995; Marcy and Butler, 1996) answered the ancient question about the existence of other worlds, and in addition it shows that planetary systems may be very di erent in comparison to the Solar-system. This enormous advance triggered huge e orts to search for more exoplanets to answer new questions: What is the occurrence of planets? Around which type of stars? How are their populations in terms of mass and size? How do they form and eventually migrate? What are the characteristics of their atmospheres? What is the nature of their composition (super cial and internal structure)? What are the properties of their host star? Can they support life?
The main techniques or methods used to detect these worlds are extreme development of previous techniques applied to the study of stellar binary systems: spectroscopy (Doppler e ect) or astrometry to detect re ex motions, photometry to detect transits or occultations, and high-contrast imaging to directly detect companions. Only visual high-contrast imaging is a direct method, the other techniques are indirect in the sense that we do not directly detect photons from the companion (only detected through the human genius). In addition, there are the timing and microlensing techniques. Each technique has its own pros and cons, here I will discuss the principles of each of them. Only the radial velocity technique will be described with more details, as it is in the focus of this thesis.

De nition of methods

Transits: In some cases the orbital plane of an exoplanet is aligned with the line of sight connecting the host-star and the observer, in this con guration the planet will pass in front of the star blocking part of the stellar light (a transit). In addition, the planet will also pass behind the stellar disc (an occultation, Fig. 1.5). During a transit, the light dimming is proportional to the ratio of the planet and star surfaces, therefore assuming a known stellar radii, it is possible to infer the planet radii. Such an event will be repeated periodically, unveiling the orbital period; from this method it is also possible to derive the orbital semi-major axis and the angle between the sky and orbital planes, however the planetary mass remains unknown (which can be inferred from the combination of this method with RVs). The probability that a given planet shows transits is proportional to the stellar radii and inversely proportional to the semi-major axis and, therefore, this technique is largely biased toward short periods. Specially relevant is that, from this technique, it is possible to characterise the atmosphere of the planet through absorption spectroscopy, if there is any. Indeed, under the presence of an atmosphere the light dimming will be wavelength dependent according to the atmospheric composition. For a detailed description of this method see, e.g., Winn (2010).
Radial velocities: In a system of bodies, each orbits a common centre of mass, where position depends on the body’s masses. The semi-major axis of the orbit of the more massive object (the host star) is smaller than the one of the less massive object (the planet). We can thus infer the presence of a planet by measuring the re ex motion that is induced on the parent star (Fig 1.6). This re ex motion is measured using the Doppler e ect: the spectral lines of the star appear shifted by an amount proportional to the radial velocity. The periodic variation of the RV reveal the presence of the planetary companion and, from such a variation it is possible to infer the orbital period, the minimum mass of the planet (assuming that the stellar mass is known), the eccentricity and the semi-major axis. This technique measures the minimum mass (m sin(i), i being the angle between the orbital and sky planes) because the angle between the orbital and sky planes remains unknown (only one component of the stellar motion is measured), the planet radii is also unknown. At the end of the XXth-century, the radial velocity technique was biased to Jupiter-like planets due to the typical RV precision of 10-20 m s 1. But now with the development of stable and precise spectrographs, Earth-like planets orbiting low-mass stars are accessible. This technique will be reviewed in detail below.
Astrometry: In the same way as described in the radial velocity technique, this method aims to quantify the wobble of the parent star around the common centre of mass; this time, by measuring with an exquisite precision the position of the target star with respect the background stars (Fig 1.6). One of the biggest advantages of this method is that it measures two components (projected in the sky plane) of the stellar motion, allowing to infer all orbital parameters, including the planetary mass (assuming that the stellar mass is known). This method, as the radial velocity method, is biased to massive planets orbiting low-mass stars in close orbits, but also to the closest systems because the astrometric motion is directly proportional to the distance to the system. Unfortunately, today, no detection have been made with this technique. The astrometry method is detailed in, e.g. Quirrenbach (2011).
Microlensing: A foreground star passing close to the line of sight connecting a background star and the observer produces a microlensing event. The e ect is provoked by bending of light due to the gravitational eld of the foreground star. If this star has a companion, a perturbation will appear while the background star is being magni ed (Fig 1.7). Such perturbation will depends on the planetary mass and the planet-star separation. This technique is sensitive to Earth-mass planets at large separation but the event is transient, which is the main disadvantage of the method. The technique is fully described in Gaudi (2011).
Direct imaging: This technique is to spatially resolves the planet in a high contrast image, where the main di culty is to get rid of the stellar brightness. The stars radiation is several orders of magnitude greater than the planet radiation (as an example, Jupiter emits a few 10 9 of the solar luminosity), under this scenario, this method masks the star light using an adapted coronograph and perform a correct treatment of residuals (Fig 1.7). Some advantages of this method is that it is possible to directly derive physical and chemical properties of the planet (like its e ective temperature or atmosphere composition) and also unveil the presence of debris disks; however, it is currently limited to young and nearby systems.
For further details see, e.g., Traub and Oppenheimer (2011).
Timing: This technique reveals the presence of planets by measuring perturbations in periodic events, like variations in pulsars timing, in pulsating stars, or the central time of transits. The mass of the companion dynamically a ects the periodic event, hence, allowing us to quantify the planetary mass.
In the case of pulsars timing, the huge precision in measuring the timing gives access to planets with the I had brie y described the detection techniques, which are complementary in the search for exoplanets. This is shown in gure 1.8, where we visualise the diversity of known planets. At the date, more than 1500 exoplanets are known (transits: 1061; RV: 439; timing: 61; microlensing: 16; imaging: 8)1 and there are about 3000 transiting candidates from the Kepler mission. Great technological challenges were { and are { addressed to reach this quantity.
In the search for transiting exoplanets, the expected 1% light dimming of Jupiter-sized planets transiting Sun-like stars is accessible from the ground (where it is worth the use of reference stars). Since the rst discovery made by radial velocities, several ground-based projects took place to constantly scan the sky in the search for the appreciate light dimming { e.g. TrES, Alonso et al. (2004); WASP, Pollacco et al. (2006); or Mearth (searching for planets transiting low-mass stars, Nutzman and Charbonneau, 2008). However, a great advantage is the use of space-based observatories because the lack of variations introduced by Earth’s atmosphere. Indeed, golden times arrived with space missions like COROT (with a photometric precision of 0.01% which allows to detect transiting super-Earths, Barge et al., 2006) or the Kepler mission to search for transiting Earth-sized planets (achieving a 0.001% precision, Borucki et al., 2009). With its xed eld-of-view, Kepler con rmed planets exceeds 1000 and its candidates are around three times this number. The eld is very promising with future spacecrafts dedicated to search for transits (TESS, PLATO) or even the non-dedicated mission JWST. This is also the case for searches from the ground, with even more dedicated surveys like NGTS or ExTrA, where the latter will be specially dedicated to search for planets transiting cool-stars in the nIR (previous and current surveys operates in the optical).
The radial velocity semi-amplitude ranges from 50 m s 1 for a hot-Jupiter orbiting a Sun-like star to 10 cm s 1 for a Earth-like planets orbiting the same type of stars in a 1 yr period. In section 1.3.3 I review some of the characteristics of dedicated spectrographs that pushed the RV precision to lower values, here I highlight some instruments. CORAVEL was a spectrophotometer installed at the Haute-Provence Observatory (OHP), it achieved a RV precision of 200 m s 1. Such precision was improved with the stabilised and high-resolution spectrograph ELODIE (OHP, Baranne et al., 1996, and its copy CORALIE on the Euler telescope at La Silla) with an instrumental velocity error of 13 m s 1 (low enough to detect the signal of 51Pegb, Mayor and Queloz, 1995). Based on the experience and success of ELODIE, a second generation of super-stable velocimeter see the light with the High Accuracy Radial velocity Planet Searcher (HARPS ) mounted in a 3.6 telescope (ESO). HARPS reaches the m s 1 precision, or even the sub-m s 1 when bright stars data are acquired with simultaneous reference lamps. The instrumental stability is 1 m s 1 over several years (Mayor et al., 2003), giving access to super-Earth planets orbiting low-mass stars. Even more, the near future is very promising with the next generation instrument ESPRESSO (Pepe et al., 2010), that will be installed in a 8m-class telescope (VLT, ESO). ESPRESSO is an ultra-stable, high-resolution spectrograph conceived to break the 10 cm s 1 precision, giving access the to domain of Earth-like planets orbiting Sun-like stars. Cool-stars are also in the target of future instruments designed for the hunt of extrasolar planets, as is the case of SPIRou, a nIR high-precision velocimeter and spectropolarimeter to be mounted in the CFHT (Delfosse et al., 2013). The exoplanet hunt is part of the science case for the upcoming generation of extreme large telescopes, as an example, CODEX@E-ELT (Pasquini et al., 2008) was designed to be an ultra-stable spectrograph that envisages the 2 cm s 1 precision.
The combination of adaptive-optics and coronographs allows the direct imaging of exoplanets in the nIR. NACO is a camera mounted in a 8m-class telescope (VLT-ESO) used for the rst imaged exoplanet P icb (Lagrange et al., 2009); this event encourages even more the design of specialised instruments with extreme adaptive-optics like SPHERE (Beuzit et al., 2008) or GPI (Macintosh et al., 2008), which will allows the direct detection of Jupiter-like planets. The era of Extreme large telescopes, of course, envisages the integration of this kind of instruments, as is the case of METIS@E-ELT. Finally, astro-engineering is not exempt to address the technique of astrometry; GAIA (ESA), with its micro-arcsecond precision, will surely gives excellent news for this technique.
Figure 1.8: The planet mass as a function its separation to the parent star. Each detection technique lls di erent parts of the diagram, showing that each of them favoured the detection of planets of di erent mass regimes and con guration. As a reference, Earth- to super-Earth mass planets ll the regions between 0.003 and 0.03 Jupiter masses.

The radial velocity method

This thesis focuses on the detection of exoplanets by the radial velocity method. The principles of the method were brie y described above, and the general solution of the two-body problem is addressed in, e.g., Murray and Correia (2011) and concretely for radial velocities in, e.g., Lovis and Fischer (2011). Here I will directly discuss how orbital parameters and the (minimum) planetary mass is derived from the RV measurements.

Orbital parameters

In the two-body problem, each object orbits the common centre of mass and it is natural to locate the origin of the reference frame in this point. Considering a star with mass M? and a planet with mass mp, each body follow its own elliptical orbit whit the focus located in the centre of mass (Fig. 1.10), they shared the same period P and eccentricity e, their periastron (the orbital point of least distance to the centre of mass) di ers by an angle of , and their semi-major axis scale as where a is the semi-major axis of the relative orbit (maximum separation between the star and the planet, a = a? + ap). Figure 1.10 shows the con guration of the system, where the z^ axis points toward the observer (the line of sight); we are interested in the star velocity vector projected into the line of sight. Firstly we de ne:
Figure 1.10: Representation in 3-dimensions of a Keplerian orbit. Each body orbits the centre of mass (CM, red cross) following the black ellipse (star), and the grey one (planet); for clarity, only the stellar periastron is represented (p). The z-axis is from the centre of mass toward the observer (the observer line of sight).
The bracketed term in equation (1.8) gives the shape of the signal (see gure 1.11) and K the strength of the signal. We note that assuming the stellar mass M? and deriving the eccentricity from the shape of the curve, we derive the planetary minimum mass mp sin(i); combining the radial velocity method with the transits method allows to infer the true mass.
Generally speaking and from equation (1.7), we note that K / mp M? 2=3 P 1=3, which means that the radial velocity method is more sensitive to massive planets, lower stellar mass plays a considerable role and it has some sensitivity to shorter periods. In terms of detectability, a RV signal will be detectable depending on the amplitude of the signal K, the overall RV uncertainty vr and the number of RV measurements N; assuming a circular (or almost circular) orbit, the shape of the signal will have a low impact on its detectability (see Fig. 1.11). With this in mind, a RV signal-to-noise ratio can be expressed In equation (1.9) one assumes an uniform sampling of RV measurements along several periods. An interesting exercise is to estimate the semi-amplitude of Solar-system and others outstanding planets detected by transits. I suppose a detection5 with S=Nvr = 20, without any additional source of noise (stellar, instrumental, etc). As we see from table 1.1, true Mars-like planets are beyond the capabilities of the RV method even if an accuracy of 10 cm s 1 is reached. A reasonable upper limit of data is around 300, with this, it is evident how important is to achieve precision of cm s 1 to access to Earth-like planets.
Figure 1.11: The e ect of the eccentricity and the longitude of periastron on the shape of the radial velocity signal. The semi-amplitude K, the period P and the periastron passage t0 are constants.
Table 1.1: Expected RVs semi-amplitude K and required number of observations (N) for a S=Nvr = 20 detection with di erent RV uncertainties vr .

Doppler spectroscopy and the cross-correlation function

I described how to detect exoplanets by the radial velocities and derive the orbital parameters (Sect. 1.3.1). But how we measure the stellar radial velocity? The answer comes from stellar absorption lines, which are shifted according to the relative velocity between the star and the observer. First, it is convenient to de ne the Solar-system barycenter as the standard reference frame (IAU). Once in the standard reference frame and neglecting relativistic e ects, the radial Doppler e ect gives the wavelength shift, which is written as = 0 1 + vcr (1.10).
where is the wavelength measured by the observer, 0 is the wavelength at the source and c is the speed of light.
A to be measured in a resolution element of 0.055 A (for a high instrumental resolution
of R = 100000). So, how to achieve precision of the order of m s 1? The Doppler information is
contained in each stellar spectral line, so instead of measuring the position of each line independently, ones can increase the signal-to-noise by accumulating the Doppler information of thousands absorption lines in a one unique line, which is the idea behind cross correlating; therefore, one wants a wide spectral domain (cross-dispersed echelle spectrographs). Initially, physical masks with holes tracking the position of most favourable lines were proposed (Fellgett, 1955) and used (Gri n, 1967; Baranne et al., 1979), where the physical mask is cross-correlated with the observed spectra. Then, the cross-correlation was improved in accuracy and spectral type adaptability by using a numerical mask (Queloz, 1995) and bre-fed spectrographs (SOPHIE, Baranne et al., 1996). The numerical mask (binary template M) has box-shaped emission lines strategically placed in most prominent lines, and it is shifted in velocity space to search the minimum of the correlation. This writes as where C is the cross-correlation function (CCF), S the stellar spectrum and we look for the value of which minimises C (the procedure is detailed in, e.g. Queloz, 1995; Baranne et al., 1996, see gure 1.12 showing a sketch of the procedure). The { approximately Gaussian { shape of the CCF represents the overall shape of spectral lines, hence, its symmetry will depends on stellar lines symmetry; and its full-width at half-maximum (F W HM) is representative of the stellar rotation. The stellar radial velocity corresponds to the centre of a Gaussian t performed to the CCF, while the error is estimated from Monte-Carlo simulations (see Fig. 1.13). Bouchy et al. (2001) gives a procedure to estimate the RV error directly from spectrum (although is not suitable for low signal-to-noise spectra). Bouchy et al. (2001) showed that the RV { photon { precision depends on the spectrum signal-to-noise ratio and the F W HM and depth of stellar lines, that is, narrow and deep lines contain more Doppler information than spread and weak ones. Accordingly, Pepe et al. (2002) proposed a weighted CCF, where the weight of prominent lines is greater; in addition, they found an improvement in RVs when rejecting stellar lines close by 30 [km s 1] (annual Earth’s motion) to telluric lines.

Limitations and di culties of the radial velocities method Instrumental

Of course, one of the rst limitations in precise RVs is the need of high signal-to-noise spectra, which is limited by the star’s luminosity and its distance. Therefore, e cient spectrographs and large telescopes are needed to analyse faint objects. Also required are high-resolution (R > 50000) spectrographs with a wide spectral coverage, because the needs to sample stellar lines with enough accuracy and to accumulate the Doppler information from a vast number of lines.
A key step in the radial velocity technique was the ne wavelength calibration of spectra, allowing precision from tens of m s 1 to few m s 1 with the introduction of the gas-cell or ThAr lamps. Iodine-gas cell located at the spectrograph entrance slit superimpose catalogued absorption lines (mostly between 5000-6200 A) to the stellar spectrum, improving RV precision around 2 m s 1. However, caution is needed because changes in the internal pressure of the gas cell may introduce spurious variations of the wavelength calibration; in addition, there is a loss of light when using gas cells and the domain of iodine lines makes almost impossible to use it for M dwarfs. Another method for wavelength calibration is the use of ThAr lamps, which have a similar RV precision than iodine cells. These lamps emit lines in a wider spectral domain (from the optical to nIR) than I2, however, their lines are a ected by a drift when the lamp ages and its internal pressure change. Such drift a ects Th and Ar lines in a di erent way, allowing a correction for a stable RV precision (Lovis and Pepe, 2007). Another approach to push the limits of wavelength calibration is the use of laser frequency comb that generates emission lines arranged in a regular way, enabling a wavelength calibration of the order of 0.01 m s 1 (Murphy et al., 2007). Fabry-Perot etalon are also used for the wavelength calibration (e.g. HARPS; Wildi et al., 2011, demonstrated that 10 cm s 1 was achieved during a night and a 1 m s 1 stability was reached over 60 days), and will certainly be combined with the laser frequency comb (to overcome the high-frequency peaks that spectrographs can not resolve at R 100000).
A deep understanding and monitoring of the instrument pro le is required. To illustrate the di cul-ties, a pixel of the HARPS CCD along the dispersion direction of the spectrograph, represents about 1 km s 1. To ensure RV precision of 1 m s 1 it is thus necessary to monitor the spectrum position at a level of 1/1000th pixel, i.e., in the case of HARPS at a level of 15 nm on the detector (for a 10 m pixel size). By the use of bres, the spectrograph can be placed in a vacuum vessel with a ne temperature control (stable environment) avoiding thermal or mechanical e ects. Additionally, bre-fed instruments overcomes the problem of illumination variations of slit spectrographs. However, variation of the light injection is observed as a function of the centering of the star in the ber entrance.

Table of contents :

1 Introduction 
1.1 Where are we?
1.2 A new era in Astronomy: extra-solar planets
1.2.1 First ideas…then the detection
1.2.2 Current techniques
1.3 The radial velocity method
1.3.1 Orbital parameters
1.3.2 Doppler spectroscopy and the cross-correlation function
1.3.3 Limitations and diculties of the radial velocities method
1.3.4 Radial velocities surveys
2 Search for planets around M dwarfs 
2.1 Basic properties of M dwarfs
2.1.1 M dwarfs activity
2.1.2 Extra-solar planets around M dwarfs
2.2 This thesis
3 The R0H K-index in M dwarfs 
3.1 Dynamo processes
3.2 Stellar activity diagnostics
3.3 Stellar activity and planets detection
3.4 Calibrating the R0HK-index (paper)
4 Optimise radial velocity extraction 
4.1 2-minimisation method
4.1.1 Read data
4.1.2 Building the stellar template
4.1.3 Calculating radial velocities: 2-minimisation
4.1.4 Fraction of rejected wavelength region
4.1.5 Performances
4.2 Applications
4.2.1 Telluric correction
4.2.2 A complete analysis of the radial velocities of GJ 3293, GJ 3341 and GJ 3543
5 Conclusions and future prospects

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