the search for exoplanets around low-mass stars

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A brief introduction of star-planet formation and evolution

Stars and planets are intrinsically linked throughout their lives. As a consequence, understanding the physical processes driving the star formation is critical to study the birth of planetary systems and their evolution in the early stages of their lives. In this section, we briefly describe the paradigm of planet formation and early evolution around low-mass stars (i.e., of mass smaller than ∼1.5 M⊙ ). After a short introduction about the zoology of evolved low-mass stars, we describe how stars form from giant clouds of gas and dust to the main-sequence. We then detail the current paradigm of planet formation from the accretion disk surrounding newly-formed low-mass stars, before outlining their orbital evolution during the few Myr following their birth. For more information about stellar formation and evolution, the reader is invited to consult the books of Maeder (2009), Bodenheimer (2011) and Beech (2019) from which Sections 1.1.1 and 1.1.2 inspire. For planet formation and evolution, I recommend the recent reviews of Baruteau et al. (2016) and Armitage (2018).

Zoology of low-mass stars through the HR diagram

Following their formation within giant collapsing clouds of gas and dust, stars start burning hy-drogen through thermonuclear fusion in their cores. When nuclear fusion becomes the domi-nant mechanism of energy production (and thus compensates the energy released by gravitational contraction), stars enter the so-called main-sequence (MS) phase where they will spend most of their lives. Low-mass stars remain roughly stable in effective temperature Teff and luminosity1 throughout the MS and occupy a well-defined region of the Hertzsprung-Russel (HR) diagram (or temperature-luminosity diagram; see Figure 1.1). Depending on their effective temperature, MS stars exhibit different prominent spectral lines used as a basis of the so-called Harvard spectral classification, commonly used in the community. Seven spectral types (O, B, A, F, G, K and M, from the hottest to the coolest; see the different stellar spectral types on the X-axis on top of Figure 1.1) are used in the Harvard spectral classification, each spectral type being divided into 10 sub-types. Low-mass stars, on which we focus in this section, belong to spectral types G, K and M (i.e., Teff .6000 K).
The typical inner structure of low-mass stars of mass larger than ∼0.35 M⊙ is illustrated on the Sun in Figure 1.2. For these stars, the energy released by the thermonuclear fusion of hydrogen in
1 In practice, the production of helium from the combustion of hydrogen in stellar cores increases the opacity of the stellar interior which, in turn, induces an increase in stellar luminosity (by a few tens of percents throughout the MS). As a result, the stellar radius slowly increases throughout the MS (by a few percents).
the radiative core is transported outward by radiation through the densest and hottest region of the star: the radiative zone. This region is surrounded by an envelope featuring a steep temperature gradient. In the Sun, for example, the temperature goes from ∼2×106 K on top of the radiative zone to ∼6000 K close to the surface. As a result, the Schwarzschild criterion is no longer fulfilled in the envelope and energy is primarily transported by convection. The radial extent of the convective envelope increases with decreasing mass. Whereas this zone extends no further than ∼0.29 R⊙ for the Sun, stars of mass lower than ∼0.35 M⊙ are entirely convective (Baraffe et al., 1998).
The stellar mass Ms drives the post-MS evolution. The most massive low-mass stars (Ms &0.5 M⊙ ) spend typically a few Gyr on the MS2, until their hydrogen supply is exhausted. At this stage, the temperature Tc within the dense helium stellar core is too low to initiate the thermonuclear fusion of helium (∼108 K). The star evolves on nuclear time scales (e.g., ∼1 Gyr for the Sun) into a red giant: its core contracts and the temperature and density increase therein. On the other hand, as a result of the thermonuclear combustion of hydrogen occuring in the layers surrounding the core, helium keeps accumulating in the stellar core which ends up being degenerated (i.e., the particles cannot get any closer without violating the Pauli exclusion principle). The ignition of the thermonuclear fusion of helium in the degenerated core is extremely intense and brief (a few thousands of seconds for the sun; e.g., Deupree, 1996; Mocák et al., 2010), and referred to as the helium flash in the literature. This runaway process rapidly increases the core temperature until the thermal pressure overcomes the degeneracy of the stellar core which then expands and cools down while keeping burning helium on nuclear time scales. After running out of nuclear fuel, the star is not massive enough to initiate the combustion of carbon in the so-called CNO cycle. As a result, the star expels its outer layers through superwinds and its core contracts and cools down into a white dwarf. Finally, very-low-mass stars, whose lifetime is larger than the age of the Universe, (i.e., Ms .0.5 M⊙ ) have not yet been observed after the MS.

The formation of low-mass stars

In the current paradigm of star formation, low-mass stars form from the gravitational collapse of denser regions of giant clouds of gas and dust in the interstellar medium. They then go through a protostellar phase where they accrete most of their mass from their surrounding en-vironment, and through a pre-main-sequence (PMS) phase where they contract until reaching the main-sequence. Through the different stages of their formation, young stellar objects (YSOs; i.e., Protostars and PMS stars and their circumstellar environment) exhibit peculiar spectral energy distributions (SEDs) enclosing information on their structure and on the various processes driving their evolution. These SEDs are used to classify YSOs into four classes (0, I, II and III) reflecting their evolutionary stage between their formation and the MS.
YSOs are often observed in gravitationally-bound clusters or OB associations immersed in regions of dust and gas called star-forming regions (SFRs). The closest SFRs such as the ρ Ophiuchi cloud complex (located at ∼130 pc away from the Sun Wilking et al., 2008), the Taurus molecular cloud (at ∼140 pc; Torres et al., 2009), or the Orion nebula (at ∼400 pc; Kuhn et al., 2019), are excellent laboratories to provide observational constraints on the processes driving star (and planet) formation, e.g., by measuring the stellar initial mass function (see the review of Offner et al., 2014), or by probing the surrounding environment of their YSOs with high-contrast imagers and interferometers (e.g., O’Dell & Wong, 1996; ALMA Partnership et al., 2015). In this section, I briefly describe the main phases of the star formation from molecular clouds to the MS. An overview of the evolutionary stages described in this section is given in the cartoon shown in Figure 1.3.
Figure 1.3 – Illustration of the main stages of the formation of solar-like stars (source: Greene, 2001).

Molecular clouds

Star formation arises within gravitationally bound molecular clouds of width ∼10 pc in the in-terstellar medium. These quiescent and cold (T ∼ 10 K) regions are mostly composed of gaseous molecular Hydrogen, H2, with traces of dust and heavier gases (e.g., CO and NH3, used to probe the cloud structure; see the reviews of van Dishoeck et al., 1993; Langer et al., 2000). Giant molecular clouds are sculpted by filaments featuring column densities of remarkably constant inner widths of ∼0.1 pc (e.g., André et al., 2010; Arzoumanian et al., 2011, 2013; André et al., 2014; Arzoumanian et al., 2018) which accrete the surrounding material into ∼1-pc wide denser inner regions, called pre-stellar cores, where star formation arises (Larson, 1969).
Pre-stellar cores are balanced between gravitational energy, which favours their collapse, and thermal, turbulent, rotational, and magnetic energies, that tend to prevent collapse. Under the effects of ambipolar diffusion of the magnetic field (Nakamura & Li, 2005), turbulence-induced shocks within the clouds (Mac Low & Klessen, 2004; Crutcher, 2012), and/or external phenomena (e.g., exploding supernovae, cloud-cloud collision or galactic density waves; see Preibisch et al., 2002; Bodenheimer, 2011), pre-stellar cores initiate their gravitational collapse. The latter oc-curs in two successive phases (Larson, 1969; Masunaga et al., 1998; Masunaga & Inutsuka, 2000). During the so-called first collapse, the pre-stellar core contracts isothermally until the thermal pres-sure in the central region compensates the gravitational forces (corresponding to a core density of ρ ∼ 10−10 cm−3; Maeder, 2009). As this stage, the temperature in the central region is large enough (i.e., &2000 K) to dissociate molecular hydrogen. This process releases enough energy for the pre-stellar core to undergo a phase of adiabatic collapse, which continues until the center density reaches ∼1020 cm−3, when the protostar is said to be born (Tomida et al., 2013). Since the second collapse only occurs in the central region of the first core, the protostar is surrounded by the remnants of the first core. If the pre-stellar core features a very low angular momentum, the surrounding envelope is rapidly accreted onto the protostar (Masunaga & Inutsuka, 2000). Otherwise, the first core remnants evolve into a rotationally-supported circumstellar disk after the protostar formation (Machida & Matsumoto, 2011; Machida & Basu, 2019). In what follows, we only consider rotating protostars hosting a circumstellar disk, where planet formation is thought to arise.

The protostellar phase

At the beginning of the protostellar phase, the system is composed of a rotating pre-stellar core surrounded by a circumstellar disk bathed in uncollapsed residuals of the primordial envelope. The protostar lies in the class-0 stage, characterized by a SED dominated by a cold black body emis-sion (mostly in the far-infrared and submillimetric domains). The mass of the envelope remains significantly larger than that of the protostar, as most of the envelope material has enough angular momentum not to fall in the central core. The orbital energy of this material dissipates through collisons until the infalling material reaches a minimum-energy configuration while conserving an-gular momentum, forming a coplanar circumstellar disk which extends up to distances &100 au on time-scales of ∼104 yr (e.g., Masunaga & Inutsuka, 2000; Machida & Matsumoto, 2011). The grad-ual increase in the disk density triggers episodic gravitational instabilities, leading to time-variable accretion of the disk material onto the protostar accompanied by high-velocity collimated bipolar jets (Machida & Basu, 2019). The protostar accretes mass from its circumstellar disk at a large rate of ∼10−6 M⊙ /yr until the envelope is depleted. When the protostar is more massive than the circumstellar envelope, it enters class I protostar (see panel c of Figure 1.3), characterized by weaker accretion rate (∼10−7 M⊙ /yr), and less powerful jets and outflows. In this stage, the SED features a nearly black body continuum at mid-infrared wavelengths from the protostar thermal emission, with a significant excess of continuum at far-infrared and submillimetric wavelengths. This phase lasts for roughly 105 yr, until the protostar reaches a core temperature of ∼106 K and starts burning deuterium in its center (Larson, 2003; Evans et al., 2009). In this process, the released energy is high enough for the convection to overcome radiation in the proto-stellar interior (Stahler & Palla, 2005): a star is born.

Pre-main sequence phase

Low-mass PMS stars (see panels d and e in Figure 1.3) quickly consume their deuterium supply and the released nuclear energy is not sufficient to balance the gravitational energy. As a consequence, the stars contract and their luminosity decreases at roughly constant temperature in the so-called Hayashi tracks of the HR diagram (see Hayashi, 1961, and Figure 1.4). After having spent a few Myr on the Hayashi tracks, PMS stars of mass larger than ∼0.5 M⊙ develop a radiative core and undergo a phase of temperature increase at roughly constant luminosity, along the so-called Henyey tracks of the HR diagram (Henyey et al., 1955). Less massive stars keep following the Hayashi track up to the main sequence and remain fully-convective. Due to stellar contraction, the central temperature of the star keeps increasing during both Hayashi and Henyey tracks until the hydrogen combustion threshold (i.e., 107 K) is reached. As a result of their degenerated core preventing the central temperature from rising above this threshold, PMS stars less massive than 0.08 M⊙ never initiate the thermonuclear fusion of hydrogen and are referred to as brown dwarfs.
When it enters the PMS phase (class II SED, panel d of Figure 1.3), the system is composed of a central star surrounded by a keplerian disk and the primary envelope is now mostly depleted. The SED of the system is composed of the continuum emission from the PMS star (resembling that of the future MS star) along with an infrared/submillimetric excess induced by the emission of the dust in the disk. The central star, called classical T Tauri star (cTTS), keeps accreting the disk material at rates of ∼10−8 M⊙ /yr (see Hartmann et al., 2016, for a review of accretion processes during the PMS phase). CTTSs feature intense and complex magnetic fields of several kilogauss (e.g., Johns-Krull et al., 1999a; Johns-Krull, 2007), whose surface topology has only recently started to be constrained (see Donati et al., 2010, 2012, and the results of the MAPP science program). These magnetic fields play a central role in the evolution of cTTSs. First, the magnetic field of the star creates a cavity (called magnetospheric gap) of typically 5-10 stellar radii, where gas elements, mostly ionized, are no longer dominated by thermal pressure and fall onto the star. Due to a magnetic coupling between the disk and the star at the edge of the magnetospheric cavity, the disk material is funneled along the field lines onto the star (Bouvier et al., 2007). Under the effect of hydrodynamic turbulence in the disk and/or magnetized winds ejecting the particles from the surface of the disk (see the reviews of Armitage, 2015; Hartmann et al., 2016, and the references therein), the disk material drifts towards its inner edge, where it is accreted onto the central star. Finally, the magnetic field plays a crucial role in the evolution of the stellar rotation rate. Classical TTSs appear to rotate much slower than one would expect from their contraction (Rebull et al., 2004). In the standard disk-locking paradigm, the star-disk magnetic coupling leads the star to co-rotate with the inner edge of the disk (Ghosh & Lamb, 1979). In practice, disk-locking is a complex adaptive process which depends on the magnetic field and on the various sources of angular momentum gain/loss (see Bouvier et al., 2014, for a review).
On time scales of about 1-10 Myr (Bell et al., 2013; Richert et al., 2018), the circumstellar disk material is progressively depleted by accretion onto the star, photoevaporation and planet formation (Armitage, 2011). Once the inner disk material is exhausted, the star becomes a weak-line T Tauri star (wTTS, class III pre-stellar object). Its SED resembles that of the central star, except that a weak excess at far-infrared and submillimetric wavelengths can still betray the presence of a debris disk around the star (Hughes et al., 2018). Released from the disk-braking, wTTSs spin up under the effect of their contraction. The shrinking of low-mass stars lasts throughout both the weak-line T Tauri phase (10-15 Myr, Martin, 1997) and the post T Tauri phase3, until the star reaches the main sequence. The more massive the PMS star, the shorter the contraction time scale. For example, the contraction phase lasts for ∼20 Myr for a solar-mass star, whereas it takes more than 100 Myr for a star with Ms = 0.5 M⊙ . Around the zero-age main-sequence (ZAMS), the contraction progressively slows down and the star starts losing angular momentum through magnetized stellar winds (Weber & Davis, 1967; Kawaler, 1988; Réville et al., 2015; Gallet & Bouvier, 2015), which lasts throughout the MS phase.

Planet formation around low-mass stars

Structure and properties of protoplanetary disks

Planets form in the accretion disk (called protoplanetary disk) of gas and dust surrounding PMS stars, whose structure and main properties are illustrated in Fig. 1.5. At this stage, the star has almost reached its final mass, and the disk is typically 100 times less massive than the star, although measurements of the disk mass are often plagued with large errors (based on the disk’s supposedly optically thin integrated intensity at radio wavelengths and assuming a gas-to-dust ratio of 0.01, a highly uncertain value based on interstellar medium studies; see Williams & Cieza, 2011; Williams
& Best, 2014). In the radial direction, the disk surface density decreases roughly as r−1, where r is the radial distance to the star, up to the outer edge of the disk, at r ∼ 100 au (e.g., Andrews et al., 2010). Perpendicularly to the disc mid-plane, the pressure scale height evolves slightly faster than linearly with r, explaining the observed « bowl »-shape vertical structure of edge-on disks (e.g., see the high-resolution observations of the HH 30 disk with the Hubble Space Telescope in Cotera et al., 2001), and reaches a aspect ratio (i.e., the ratio between of the height of the disk to its radius) of 0.1 at the outer edge of the disk. The latter can be roughly described as an optically thick cold interior, whose temperature structure is mostly controlled by stellar irradiation (T ∝ r−1/2, Kenyon
& Hartmann (1987)), sandwiched between an optically thin warmer molecular layer, heated by the stellar UV emission, and a hot atmosphere heated up by X-ray emission triggered by the accretion of disk material in the vicinity of the star (Armitage, 2015). Observationally, protoplanetary disks are mostly probed from the dust thermal emission (i.e., submillimetric and radio excesses) and narrow gas emission lines (mainly CO isotopologs, see Carmona, 2010).

The different steps of planet formation

There is currently no consensus on how planets form within protoplanetary disks. In this paragraph, we briefly describe the planet formation from the so-called core accretion scenario, which currently stands as a paradigm for planet formation, even though some crucial steps of the process remain poorly understood mainly due to very few direct observations of forming planets.
3 Post T Tauri stars (pTTSs) are PMS stars featuring intermediate properties between T Tauri stars, associated to the molecular clouds where they formed, and MS stars (Mamajek et al., 2002). Similarly to T Tauri stars, pTTSs are located above the MS in the HR diagram and are thereby relatively well separated from their more evolved counterparts. Distinguishing between wTTSs and pTTSs is much harder (Herbig, 1978). Weak-line T Tauri stars of mass lower than the Sun exhibit a surface abundance of lithium which is carried down on time scales of a few Myr by convection to layers of high temperature where it is destroyed. As a consequence, pTTSs exhibit much weaker lithium absorption signatures than wTTSs (Martin, 1997). Note however that this criterion is questioned by the relatively high dispersion of the observed lithium abundances on both pTTSs and wTTSs, making it difficult to determine a threshold to unambiguously disentangle between the two types of stars (Jensen, 2001).
Figure 1.5 – Illustration of the structure of a protoplanetary disk viewed from the side. In the bottom panel, we show a cartoon of a cTTS star accreting its circumstellar disk material (source: Hartmann et al., 2016). The red solid lines indicate the magnetic field lines. In the top panel, we show an illustration of physical processes controlling the thermal and emission properties of protoplanetary disk (source: Armitage, 2015).
Under the effect of aerodynamic forces from the disk gas, µm-sized dust grains and progressively grow everywhere in the disk until they turn into pebbles of 1 to 10 mm. The growth of pebbles beyond the centimeter level remains unclear. Given their sizes, pebbles are highly sensitive to the aerodynamic drag exerted by the disk gas and tend to drift inwards at high velocity (up to 50 m s−1 for a m-sized body, resulting in a drift time scale as short as a few hundred years at a few au in a typical disk model). Moreover, their collision with other pebbles results either in the fragmentation of the system (at high relative velocity) or to a simple bouncing (see the results of laboratory collisions in Güttler et al., 2010; Weidling et al., 2012). To explain the growth of pebbles to km-sized planetesimals, we need to account for the drag induced by the dust on the gas (a.k.a. dust feedback) in the so-called streaming instability (Youdin & Goodman, 2005). The dust feedback allows to slow the radial drift of clumps of pebbles, resulting in increasingly long dust filaments that gravitationally collapse into planetesimals of 10-100 km when the filament density is larger than the Roche limit (Johansen et al., 2014). Note however that the growth of pebbles is still actively debated in the current literature (see Section 2 of Baruteau et al., 2016).
At the planetesimal stage, gravity dominates over surface forces and the growth of planetesimals occurs through two-body collisions. From masses of ∼10−8 to ∼10−2 M⊕ , the size R of a planetary embryo increases like R2: it is the runaway growth. After this stage, the remaining accretable planetesimals by the massive body (a.k.a. the oligarch), are sparser with more dispersed orbits and, consequently, the growth rate no longer depends on R: this is the oligarchic growth. Each oligarch accretes all the available mass along or close to its orbit and becomes a planet core. Note that this process is matter of debate. For example, the time scale to build up a sufficiently massive core to explain the formation of Jupiter or Saturn exceeds the typical lifetime of a protoplanetary disk (∼3 Myr; Haisch et al., 2001). Inward migration induced by oligarch-disk interaction could overcome this challenge (Alibert et al., 2005; Ida & Lin, 2008). Another increasingly popular explanation is that planet cores do not arise by collisions of km-sized planetesimals, but rather by the accretion of cm-sized pebbles, which turns out to be a particularly efficient process to build planet cores of several Earth masses (e.g., Johansen & Lacerda, 2010; Lambrechts & Johansen, 2012; Bitsch et al., 2019). If the planet core is massive enough (mass typically larger than ∼10 M⊕ ), it accretes all the disk gas available nearby through a runaway process until no more supply of gas is left, and becomes a gas giant (Pollack et al., 1996).

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Evolution of planetary systeme

During and after their formation, planets undergo interactions with the remaining protoplanetary disk, the host star, and other planets in the system which affect their orbital parameters. The most striking demonstration of these mechanisms is the existence of hot Jupiters (i.e., Jupiter-mass planets orbiting at .0.1 au from their host star such as 51 Pegasi b Mayor & Queloz, 1995). In situ formation of hot Jupiters (hJs) is unlikely because of the lack of gas supply in the vicinity of the star (e.g., Coleman et al., 2017). The current paradigm is that hJs form at a few au from their host star and are massive enough to open a gap along their orbit (see Crida et al., 2006). Under the effect of the outer disk gas exerting a stronger torque than its inner disk counterpart, the planet rapidly migrates inwards (in the co-called type II migration) until reaching the magnetospheric cavity at the inner edge of the disk on time scales of ∼104-105 yr after the planet has reaches its final mass (see Baruteau et al., 2014, and the references therein). At this distance, hJs undergo strong tidal forces from the host star which tend to circularize their orbit and synchronize their rotation period with their orbital period. Planets of mass smaller than ∼ 10 M⊕ are not massive enough to open a gap in the disk and undergo a generally slower inward migration (called type-I migration) on time scales of ∼1 Myr, which is stopped by enhanced opacity regions of the disk (e.g., ice or silicate evaporation lines, Bitsch et al., 2014) or by its dissipation. Finally, intermediate mass planets (i.e., Saturn-mass planets) are massive enough to open a partial gap in the disk along their orbit and migrate, through a complex regime (type III migration), either inward or outward depending on the mass of the disk (Baruteau et al., 2014).
Disk migrations are thought to preserve zero obliquities and low eccentricities for the planet orbits. However, observations have shown that while hJs feature indeed relatively low eccentric-ities, about one third of them exhibit large sky-projected obliquities between the orbital plane and the stellar equatorial planet (Winn & Fabrycky, 2015). Gravitational interactions between disk-migrating planets and other massive planets in the system (planet-planet scattering; Rasio & Ford, 1996; Chatterjee et al., 2008; Jurić & Tremaine, 2008) and/or Kozai-Lidov cycles with a nearby inclined stellar companion (Wu & Murray, 2003; Fabrycky & Tremaine, 2007) are invoked to explain the distribution of hJs obliquities. These mechanisms are likely to significantly enhance the eccentricity and inclination of planet orbits. Planets on highly eccentric orbit undergo tidal forces induced by the star every time they pass through their periastron, which contributes to their migration. Close to the star, the eccentricity of their orbit is rapidly damped through tidal interactions with the host star whereas the orbit alignment process takes significantly longer (Daw-son & Johnson, 2018). High-eccentricity migration scenarios have been suggested to explain the formation of all hot Jupiters (Triaud et al., 2010; Albrecht et al., 2012). However, these scenarios, in which planet migration and orbit circularization occur over typically more than ∼100 Myr, fail at explaining the recently unveiled hJs around stars younger than 25 Myr (e.g., Donati et al., 2016; Yu et al., 2017).

Current status of the search for exoplanets from Doppler and transit surveys

Since the detection of the first exoplanet orbiting a solar-like star by Mayor & Queloz (1995), about 4300 exoplanets have been detected and several thousands of planet candidates still remain to be confirmed. These detections, carried out with various techniques yielding complementary planet parameters, have revealed a great variety of planetary systems which has turned the paradigm of planetary formation and evolution upside down. In this section, we take stock of the search for exoplanets around low-mass stars, focusing in particular on the detection methods based on Doppler spectroscopy and transit photometry that we use in this thesis. We redirect the interested reader towards the book of Perryman (2018) for a comprehensive view of the various techniques for planet detection and characterization.

A brief overview of exoplanet detection techniques

The various methods to unveil and characterize exoplanets can be divided in two main types: those that directly measure the light emitted by the planet, and indirect techniques, based on the effects induced by the planet on its host star.
High-contrast imaging allows to directly detect the light emitted by a substellar object4 (planet or brown dwarf). This measurement is challenging due to the very low star-planet angular sepa-rations and the huge star-planet contrasts (e.g., a Sun-Jupiter system located at 10 pc features an angular separation of 0.5 arcsec and a star-to-planet contrast of ∼109). State-of-the-art imagers, such as SPHERE at the VLT (Beuzit et al., 2008) or GPI at the Gemini-South telescope (Mac-intosh et al., 2014), are thus equipped with adaptive optics, to improve spatial resolution, and coronographic masks, to get rid of the stellar light (see Bowler, 2016, for a review). As things stand, direct imaging is mostly limited to the detection of nearby massive planets (i.e., of a few Mjup) orbiting at large distance from their host star (typically further than a few au) and present-ing a favourable star-to-planet contrast ratio (e.g., young systems with hot contracting planets). Since the first direct detection of a planet by Chauvin et al. (2004), about fifty massive planets were unveiled with direct imaging (e.g., Kalas et al., 2008; Marois et al., 2010; Lagrange et al., 2010, for the most notorious detections). By coupling high-contrast imaging with high-resolution spectroscopy, one has the potential to directly probe the atmosphere of exoplanets. This gave rise to ambitious projects aimed at characterizing the atmosphere of nearby Earth-sized planets with next-generation giant telescopes like the ELTs (Snellen et al., 2013, 2015; Lovis et al., 2017).
On the other hand, indirect methods complement the landscape of directly imaged exoplanets. By measuring the radial velocity wobbles induced by a planet on its host star, Doppler spectroscopy (or velocimetry, see Section 1.2.2) enables to access the planet mass, orbital period, and eccentricity. In a complementary way, transit photometry, which measures the light dimming when the planet
4 A substellar object (i.e., of mass lower than ∼ 0.08 M⊙ ≈ 80 Mjup ) is referred to as a planet if its mass lies below ∼13 Mjup , and as a brown dwarf otherwise (Burrows et al., 1997). passes between the star and the observer, yields the planet radius as well as its orbital period and inclination. Velocimetric and spectroscopic observations of planetary transits enable to (i) access the mutual inclination between the stellar rotation axis and the planet orbital plane (through the Rossiter-MacLaughlin effect; Fabrycky & Winn, 2009) and (ii) probe the structure and composi-tion of the planet atmosphere (Madhusudhan, 2019). Other indirect exoplanet detection methods such as microgravitational lensing (Tsapras, 2018), astrometry (Perryman, 2018), interferometry (Gravity Collaboration et al., 2019) or chronometry (e.g., the detection of planets around pul-sars; Wolszczan & Frail, 1992) contribute to the versatility of the current exoplanet landscape, but are not discussed in detail in this manuscript which focuses on Doppler spectroscopy and transit photometry.

Doppler spectroscopy

General principle

Doppler spectroscopy is an indirect method to unveil exoplanets from the wobbles that they induce on their host star’s radial velocity (RV; i.e., the line-of-sight projection of the velocity vector of the star). Due to the reciprocal star-planet gravitational interaction, the star revolves around the barycenter of the star-planet system. If the orbital plane is not orthogonal to the line-of-sight, the variations of stellar RV along the orbit induce periodic Doppler shifts of the stellar lines that are in principle detectable from the Earth.
Using Newton’s and Kepler’s laws for planetary motion, one can express the RV signature Vp induced by a planet on its host star as a function of the time t and the system parameters such that (see the demonstration in Section 2.1 of Perryman, 2018):
where ω is the argument of the periapsis of the stellar orbit, e its eccentricity, ν its true anomaly (which also depends on the orbital period Porb, phase and eccentricity of the planet), and Ks, the semi-amplitude of the planet-induced signal which depends on the masses of the star, Ms, and of the planet, Mp, such that ip referring to the inclination of the orbital plane of the star-planet system. As a consequence, by monitoring the RV variations of a given star, one can access the orbital properties (especially Porb and e) as well as the minimum mass Mp sin ip of the planet, provided that its orbital cycle is well-covered by the observations. In practice, the measured RV is the combination of the RV signals induced by the planets in the system and by stellar activity (see Section 1.4), making the estimation of the planet parameters tricky.
Figure 1.6 – Illustration of the RV measurement principle. Top right: optical view of the SPIRou spec-trograph (Donati et al., 2018). The input light beam coming from the telescope is indicated by the red arrow. Top left: portion of a nIR echelle spectrum of the M dwarf AD Leo collected with SPIRou (first light, credit: SPIRou team). Each column of three vertical bars corresponds to one SPIRou order. In each column, the first two vertical bars contain the science spectrum (measured in two polarization states), whereas the last bar contains the Fabry-Perot calibration spectrum. Bottom right: illustration of the cross-correlation procedure to compute the average line profile from the observed spectra (credit: Melo 2001 Ph.D. Thesis). Bottom right: RV signature induced by 51 Pegasi b on its host star (source: Mayor & Queloz, 1995).

High-precision RV measurement

Precise velocimetric measurements are carried out using échelle spectrographs. The general princi-ple is to spread the incoming light beam into a wide range of diffraction orders covering the spectral domain of the instrument using an échelle grating (e.g., 72 orders ranging from 380 to 690 nm for HARPS; Mayor et al., 2003). These orders are then dispersed again in the direction orthogonal to the direction of the échelle grating diffraction using a cross-dispersion system (typically a set of prisms or a grating; see the optical view of SPIRou in the upper right panel of Figure 1.6). All the cross-dispersed orders are then simultaneously recorded on a detector (generally a CCD or a H4RG for optical and nIR domains, respectively).
The thousands of stellar lines recorded by the detector are affected in a systematic way by the Doppler shifts induced by the planet on the star. Using a list of N stellar lines covering the entire spectral domain of the instrument, one can compute an average line profile, I, using the cross-correlation method (Baranne et al., 1996) or, equivalently, Least-Squares Deconvolution (see Donati et al., 1997, and the description in Section 3.2.1). This process highly increases the signal-to-noise ratio (S/N) of the average line profile5, and so does the precision on the RV measurement. The RV information lies essentially where the slopes of the average spectral line are steepest, or, equivalently, where |∂I/∂λ| is large (Pepe & Lovis, 2008). Hence the need for the spectrograph to have a large resolving power R = λ/Δλ, where Δλ is the full-width at half maximum (FWHM) of the instrumental profile at wavelength λ, so that the stellar lines are resolved. This condition is typically fulfilled for R &50 000 – 100 000 for a slowly-rotating solar-type star observed in the optical domain (Bouchy et al., 2001).

Table of contents :

1 Introduction: the search for exoplanets around low-mass stars 
1.1 A brief introduction of star-planet formation and evolution
1.1.1 Zoology of low-mass stars through the HR diagram
1.1.2 The formation of low-mass stars
1.1.2.1 Molecular clouds
1.1.2.2 The protostellar phase
1.1.2.3 Pre-main sequence phase
1.1.3 Planet formation around low-mass stars
1.1.3.1 Structure and properties of protoplanetary disks
1.1.3.2 The different steps of planet formation
1.1.4 Evolution of planetary systems
1.2 Current status of the search for exoplanets from Doppler and transit surveys
1.2.1 A brief overview of exoplanet detection techniques
1.2.2 Doppler spectroscopy
1.2.2.1 General principle
1.2.2.2 High-precision RV measurement
1.2.2.3 Brief history of the field and perspectives
1.2.3 Transiting planets
1.2.4 The diversity of exoplanetary worlds
1.2.5 The quest for habitable planets
1.3 The search for planets around M dwarfs and low-mass PMS stars
1.3.1 The case of very-low-mass stars
1.3.2 The case of low-mass PMS stars
1.4 Stellar activity and its impact on RV curves
1.4.1 Dynamo processes, stellar activity
1.4.1.1 Activity and rotation of low-mass stars
1.4.1.2 Surface inhomogeneities
1.4.1.3 Magnetic fields
1.4.1.4 Other sources of stellar activity RV signals
1.4.2 Modeling and filtering techniques
1.5 Observing M dwarfs and young stars with nIR velocimeters
1.5.1 SPIRou: un spectropolarimètre infrarouge
1.5.2 The SPIRou legacy survey and science goals
1.5.3 The challenges of nIR spectroscopy
1.6 Overview of the Ph.D. Thesis
2 Simulating nIR RV observations of low-mass stars with transiting planets 
2.1 Motivation and strategy
2.1.1 Motivation
2.1.2 TRAPPIST-1
2.1.3 K2-33
2.1.4 AU Mic
2.2 Method
2.2.1 Generating realistic stellar activity RV signals
2.2.1.1 Modelling the stellar surface with ZDI
2.2.1.2 Generating realistic densely-sampled stellar activity RV curves
2.2.1.3 Measuring the statistical properties of light-curves
2.2.1.4 Application to TRAPPIST-1, K2-33 and AU Mic
2.2.2 Building mock RV time-series
2.2.2.1 Planetary signals
2.2.3 Modeling the mock RV time-series
2.2.3.1 Quantifying the significance of each planet RV signature
2.3 Results and perspectives
2.3.1 Results for TRAPPIST-1
2.3.2 K2-33
2.3.3 Application to AU Microscopii
3 Modelling the magnetic field and activity of low-mass MS and PMS stars 
3.1 Context: magnetic field and activity of low-mass stars
3.1.1 Measuring magnetic fields
3.1.1.1 The Zeeman effect
3.1.1.2 Measuring magnetic fields from unpolarized spectra
3.1.1.3 Measuring polarized Zeeman signatures
3.1.2 Magnetic fields of M dwarfs and low-mass PMS stars
3.1.3 Magnetic interactions between stars and close-in planets
3.1.4 Modeling stellar activity to improve the filtering of the RV jitter
3.2 Spectropolarimetric analysis of low-mass stars
3.2.1 Spectropolarimetric measurements and data reduction
3.2.2 Mapping brightness inhomogeneities at the surface of low-mass stars with Doppler Imaging
3.2.3 Reconstructing large-scale magnetic topologies of low-mass stars with ZDI
3.2.4 Proxies for magnetic activity
3.3 Application to a sample of low-mass stars
3.3.1 AU Microscopii
3.3.2 Proxima Centauri
3.3.3 EPIC 211889233
3.3.4 V471 Tau
4 Measuring the mass of AU Mic b with SPIRou 
4.1 A Neptune-sized close-in planet around the PMS star AU Microscopii
4.2 Unveiling AU Mic b signature from SPIRou RV time-series
4.2.1 RV measurement process
4.2.2 RV modeling
4.2.3 Detection of the planet
4.2.4 Filtering stellar activity RV signal with ancillary indicators
4.3 Unveiling planet signature using ZDI
4.3.1 3D paraboloid fit
4.3.2 Using ZDI brightness map to filter stellar activity RV signals
4.4 Implications and perspectives
5 Probing the atmosphere of transiting planets around low-mass stars with SPIRou
5.1 Characterizing planet atmospheres
5.1.1 Structure and dynamics of exoplanetary atmospheres
5.1.2 Transit spectroscopy
5.1.3 Probing planet atmospheres with high-resolution spectroscopy
5.2 Unveiling planet atmospheres with SPIRou
5.2.1 Observations and description of the target
5.2.2 Pre-processing the sequences of spectra
5.2.2.1 Step 1: Preliminary cleaning
5.2.2.2 Step 2: Normalizing the spectra
5.2.2.3 Step 3: Detrending with airmass
5.2.2.4 Step 4: Outlier removal
5.2.2.5 Step 5: Correcting residuals with principal component analysis
5.2.3 Modeling the transmission spectrum of an exoplanet’s atmosphere
5.2.4 Correlation analysis
5.2.5 Validation on synthetic data
5.2.6 Preliminary results
5.2.7 Next steps for HD 189733 b and perspectives of improvement
5.3 Future prospects
5.3.1 The ATMOSPHERIX observation program
5.3.2 Transmission spectroscopy of AU Microscopii with SPIRou
6 Conclusions and perspectives

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