Doppler spectroscopy and the cross-correlation function

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A new era in Astronomy: extra-solar planets

First ideas…then the detection

Having in mind our place in the Universe, next natural questions are: how many worlds like our are there? Are we alone? We already found some speculations about these questions in the ancient Greece. Although the philosopher Epicurus (300 BC) said that everything should be demonstrated through observations or logical deduction, he stated that there are in nite worlds like and unlike ours, and those worlds harbour living creatures and other things we see in our world. Fifteen centuries later, around 1200, history shows that the Catholic bishop Albertus Magnus wondered about the existence of other worlds, asserting that this is one of the most noble and exalted question in the study of Nature. Then, the french philosopher Jean Buridan (1300) speculated about the possibility that there are other inhabited worlds; he also rejected Ptolemy epicycles in favour of eccentric orbits. The existence of other stars and worlds were also advocated by Nicola de Cusa around 1400. The controversial ideas – at that time – of Giordano Bruno (1600) also point to the idea that we are not alone; he pondered that if we suppose that the Sun is like other stars, why there would be no more planets around other stars? In Cosmotheoros (1695), Christiaan Huygens speculate on how would be the sky to inhabitants of other worlds. Later, in 1761, in addition to predicting that stars forms galaxies, and these last form galaxy clusters, Johann Heinrich Lambert wrote that each star is like the Sun, surrounded by their own planets.
In a less speculative way and with the aid of better telescopes, in 1844 Friedrich Bessel measured anomalies in the position of Sirius. He was able to conclude that this bright star is pulled by an unseen companion (Bessel, 1844), which was observed almost twenty years later and found to be a white dwarf. This event was the introduction and success of precise astrometry. Claims of unseen companions with planetary nature were announced around 70 Ophiuchi, 61 Cygni and Barnard’s star (GJ 699), but they all were disproved (Jacob, 1855; Strand, 1944; van de Kamp, 1963; Gatewood and Eichhorn, 1973; Moulton, 1899; Heintz, 1978).
Struve (1952) proposed two di erent techniques to indirectly detect short-period Jupiter-like compan-ions orbiting other stars than the Sun; these techniques were high-precision radial-velocity and transits. The radial-velocities technique looks for Doppler wobbles induced by orbiting planets in the spectra of their hosts stars. Transit observations search for regular eclipses caused by planets transiting the stellar disk. In the case of radial velocities (RVs), an important key element to achieve high precision is the ac-curacy of wavelength calibration of spectra. Initially, telluric spectral lines were used as the reference but the required precision was not achieved at this time to detect Jupiter-mass companions (Gri n, 1973). The next key step, achieving an increase of an order-of magnitude in precision, came with Campbell and Walker (1979). They proposed to use a gas-cell (hydrogen uoride) to generate reference absorption lines. A rst success appeared ten years later, with the discovery of a periodic RV variations produced by an unseen companion (for details of the technique, see 1.2.2). Around HD 114762, Latham et al. (1989) estimated that the HD 114762b has a mass in the lower edge of brown dwarfs. It is to note that with a minimum mass of m sin(i)=10.98MJup, HD 114762b could be classi ed as a very massive planet or a brown dwarf following the inclination i of the system. Then, in 1992 it was discovered the two very rst planets outside the Solar-system; Wolszczan and Frail (1992) announced the detection around a pulsar of objects with masses of 2.8M and 3.4M by measuring pulsar-timing variations (these planets are so-called dead-planets, because they orbit dead stars). Despite this astonishing nding, the community was waiting for the detection of planets around main-sequence stars. The wait was not too long, only three years later came the amazing announcement by Mayor and Queloz (1995) of Jupiter-mass planet orbiting the Sun-like star (51Peg). But the surprise was double, because the RV variations have a pe-riodicity of only 4.2 days, challenging the accepted planetary formation models, which were not able at this time to explain the formation of planets so close to their parent star. Since then, many other planets were detected by di erent groups (e.g. Marcy and Butler, 1996). In the other hand, the transit technique waited a little bit more. A regular light dimming was detected in the Sun-like star HD 209458 (Henry et al., 2000; Charbonneau et al., 2000), corresponding to an unseen planetary companion previously de-tected by RV (Mazeh et al., 2000). The planet shadow was also previously seen by the HIPPARCOS satellite (Soderhjelm, 1999). HD 209458b has a periodicity of 3.5 days for a light dimming consistent with a Jupiter-like planet.

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.

Defnition 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.
Figure 1.6: Sketch of radial velocity and position variations due to an unseen planetary companion. The cross represents the centre of mass of the system to which both bodies orbit, while stars symbols represents the background reference-stars. The target star presents a wobble to the observer: in radial velocities this is translated into a regular decrease and increase, while the star position describes a loop pattern projected in the sky plane. The amplitude of the RV signal and how close is the loop pattern depends (in part) on the angle between the orbital and sky planes).
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.

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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.
Figure 1.7: The left panel shows a scheme of the microlensing technique, where under the presence of a foreground planet system acting as a gravitational lens, the transient light curve of the background star shows a perturbation. The right panel illustrates the direct imaging technique consisting on a correct treatment of a stellar image to unveil the presence of young planets in wide orbits.

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 darfs 
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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