Dynamics of exoplanets and exosatellites in binaries

RAMON MARIA ALLER ASTRONOMICAL OBSERVATORY DEPARTMENT OF APPLIED MATHEMATICS Dynamics of exoplanets and exosatellites in binaries. Pedro Pablo Campo Díaz Santiago de Compostela, 2019 Dynamics of exoplanets and exosatellites in binaries. by Pedro Pablo Campo Díaz DOCTORAL DISSERTATION Submitted for the degree of DOUTOR EN MATEMATICAS Universidade de Santiago de Compostela Santiago de Compostela, 2019 Dynamics of exoplanets and exosatellites in binaries. Ado. Pedro Pablo Campo Díaz Memoria para optar ao grado de Doutor realizada no Observatorio Astronómico Ramón María Aller dentro do Programa de Doutoramento de Matemáticas da Universidade de Santiago de Compostela, baixo a dirección do Profesor José Ángel Docobo Durántez. Santiago de Compostela, a 30 de agosto de 2019 Ado. José Ángel Docobo Durántez DECLARACIÓN DO AUTOR DA TESE. Dynamics of exoplanets and exosatellites in binaries. Presento a miña tese, seguindo o procedemento axeitado ao Regulamento, e declaro que: •A tese abarca os resultados da elaboración do meu traballo. •De ser o caso, na tese faise referencia ás colaboracións que tivo este traballo. •A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. •Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. Santiago de Compostela, a 30 de agosto de 2019 Ado. Pedro Pablo Campo Díaz AUTORIZACIÓN DO DIRECTOR DA TESE Dynamics of exoplanets and exosatellites in binaries. Prof. José Ángel Docobo Durántez, Catedrático de Astronomía do Departamento de Matemática Aplicada da Universidade de Santiago de Compostela INFORMA: que a presente tese, correspóndese co traballo realizado por D. Pedro Pablo Campo Díaz, baixo a miña dirección, e autorizo a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como director desta non incorre nas causas de abstención establecidas na Lei 40/2015. Santiago de Compostela, a 30 de agosto de 2019 Ado. Prof. José Ángel Docobo Durántez system of pre-main sequence stars. In the fourth article, we review the scenarios of four-body systems with exoplanets and exosatellites and conduct a dynamical study of a system with two stars, a planet, and a satellite. viii Resumen En esta tesis se incluye el trabajo realizado por Pedro Pablo Campo Díaz bajo la dirección de José Ángel Docobo Durántez, catedrático en Astronomía y director del Observatorio Astronómico Ramón María Aller (OARMA) de la Universidade de Santiago de Compostela. El nucleo de la tesis está compuesto por una compilación de artículos de investigación sobre sistemas binarios, exoplanetas y exosatélites. En primer lugar se incluye una Introducción en la que se hace una revisión de la historia y el estado del arte en estos campos de investigación, incluyéndolos dentro de su contexto científico. La investigación en exoplanetas (y sus satélites) es una extensión natural de la realizada en sistemas estelares múltiples, donde con respecto a la dinámica lo que cambia fundamentalmente es la relación de masas. Incluso a nivel observacional muchas de las técnicas utilizadas para el descubrimiento de los planetas extrasolares provienen del estudio de binarias, salvado el impedimento de la sensibilidad de los detectores que en otras épocas impedía la detección de cuerpos pequeños como planetas. Sigue un capítulo sobre las estrellas dobles, en el que se tratan los tres tipos de binarias existentes, visuales, espectroscópicas y eclipsantes. Se repasan las técnicas de observación usadas en cada tipo y los métodos principales de cálculo de los elementos orbitales. En el caso de las binarias visuales se considera también la paralaje dinámica y su cálculo, además de presentar una implementación original del método de Edwards para la separación de espectros. En la sección sobre binarias espectroscópicas se habla sobre la estancia realizada en el Observatorio de Cambridge, que fue parte de una fructífera colaboración con el profesor Roger F. Griffin. Hay también dos secciones específicas dedicadas a binarias espectro-interferométricas y a sistemas múltiples. El segundo capítulo está dedicado a los exoplanetas y exosatélites, comenzando por su clasificación y las técnicas de observación utilizadas, incluyendo una revisión de las principales misiones pasadas, presentes y futuras, además de la incipiente búsqueda de exosatélites. Después se trata el tema dinámico, haciendo un repaso de trabajos clásicos como los de Harrington, Dvorak y otros, anteriores incluso al descubrimiento del primer exoplaneta, y de trabajos más actuales, algunos de ellos sobre estabilidad en sistemas binarios. Se desarrolla también un trabajo original sobre la posibilidad de descubrir exosatélites mediante velocidades radiales a partir de las perturbaciones que causan en la órbita del planeta. Para ello se hacen simulaciones utilizando el paquete de integración TIDES. Finalmente se habla sobre habitabilidad, que es un concepto exclusivo de este campo. En el último capítulo se incluyen los artículos publicados en revistas internacionales de impacto sobre los temas tratados en la tesis. Hay tres artículos sobre binarias, dos de ellos calculando órbitas muy precisas de binarias espectro-interferométricas, que se enmarcan en la colaboración con el profesor R. F. Griffin, y otro sobre el estudio de binarias con medidas ix concentradas en arcos cortos de observación, en el que además se hace el estudio de un sistema de estrellas pre-secuencia principal. El cuarto artículo revisa los escenarios de sistemas de cuatro cuerpos con exoplanetas y exosatélites, realizando un estudio dinámico de un sistema con dos estrellas, un planeta y un satélite. x CONTENTS Acknowledgements iii Abstract v Introduction 1 I Double stars 9 I.1 Visual binaries ................................ 10 I.1.1 Observation .............................. 10 I.1.2 Orbit calculation ........................... 16 I.1.3 The Baize-Romani algorithm ..................... 21 I.1.4 The Edwards process ......................... 23 I.1.5 Application of the methodology ................... 26 I.2 Spectroscopic binaries ............................ 27 I.2.1 Observation .............................. 30 I.2.2 The Cambridge Observatory ..................... 32 I.2.3 Spectroscopic orbit calculation .................... 33 I.2.4 Spectro-interferometric binaries ................... 36 I.3 Eclipsing binaries ............................... 36 I.3.1 Observation .............................. 36 I.3.2 Morphology ............................. 38 I.3.3 Determination of the parameters ................... 40 I.4 Multiple stellar systems ............................ 41 II Extrasolar planets 45 II.1 Classification ................................. 45 II.2 Observation techniques ............................ 47 II.2.1 Radial velocities ........................... 48 II.2.2 Transits ................................ 49 II.2.3 Imaging ................................ 51 II.2.4 Astrometry .............................. 54 II.2.5 Timing ................................ 56 II.2.6 Pulsar Timing ............................. 56 xi II.2.7 Microlensing ............................. 58 II.3 Dynamics of extrasolar planets ........................ 60 II.3.1 Dynamics of exoplanets in binary sistems .............. 61 II.4 Exosatellites .................................. 63 II.4.1 Equations of motion ......................... 64 II.4.2 Method of integration ........................ 64 II.4.3 Results ................................ 65 II.4.4 First integration ............................ 66 II.4.5 Variations in the inclination ..................... 70 II.4.6 Variations in the eccentricity ..................... 74 II.4.7 Conclusions .............................. 74 II.5 Habitability .................................. 79 III Published articles 85 Bibliography 87 xii LIST OF FIGURES 1 Artistic representation of the planet 51 Pegasi b, also known as Dimidium. Image: ESO/M. Kornmesser/Nick Risinger (skysurvey.org) ......... 2 2 Field of view of the Kepler space telescope. Image: NASA ......... 3 3 Portrait of William Herschel. Image: public domain ............. 5 4 The Doppler effect in binaries. Image: Margaret Murray Hanson ...... 6 5 Portrait of Ramón María Aller ........................ 7 I.1 position angle, θ, and angular separation, ρ, of a binary system. The blue and orange circles represent the primary and the secondary components, respectively. .................................. 11 I.2 Schematic of a filar micrometer and steps for a micrometric measurement of θand ρ. The blue and orange circles represent the primary and the secondary components, respectively. The solid perpedicular lines stand for the crosshair of the micrometer, and the dashed line indicate the mobile thread. ..... 11 I.3 The Rayleigh criterion. The maximum of the PSF of each point source matches the first minimum of the PSF of the other source. Image: Prudyus et al. (2017).................................. 13 I.4 Laser beacon shot by the ESO’s VLT for its adaptive optics system. Image: ESO, Y. Beletsky ............................... 15 I.5 Representation of the relative orbit (solid black line) and the aparent orbit (dashed dark blue line). We can see also represented the inclination (I), the angle of the node (Ω) and the argument of the periastron (ω), as long as the distance (r), the true anomaly (f) and the position angle (θ). The line of the nodes is indicated with a dashed red line and the periastron is in the position marked with a T. ............................... 18 I.6 Radial velocity curve of the double-lined binary HD30090. Included in the article (Docobo et al., 2014b). ........................ 31 I.7 Radial velocity curve of the single-lined binary HD99842. Included in the article (Boffin et al., 2012). .......................... 31 I.8 0.91 m. telescope of the Cambridge Observatory. The spectrometer is located inside the black box in the red structure placed at the right side of the telescope. 33 I.9 Observation of the double-lined binary HD26441 performed in Cambridge. Included in the article (Docobo et al., 2017a). ................ 34 xiii I.10 Light curves of eclipsing binaries of the types EA, EB and EW. Image: Included in the article (Hümmerich, Bernhard, and Srdoc, 2013). ...... 37 I.11 Diagram of the Roche lobes of detached, semi-detached, and over-contact binaries. Image: Original included in the article (Terrel, 2002). ....... 39 I.12 Right handimage: hierarchical quadruplesystemwithtwodouble subsystems (P1+P2and P3+P4). Left hand image: schematics of the hierarchy. ..... 42 I.13 Right hand image: hierarchical quadruple system with a double subsystem (P1+P2), and the third and fourth components at successively larger distances (P3and P4). Left hand image: schematics of the hierarchy. ......... 42 I.14 Schematics of the hierarchy of the sextuple system Castor. ......... 43 II.1 Artistic representation of a hot Jupiter. Image: NASA/Ames/JPL-Caltech. . 47 II.2 Diagram of an echelle spectrograph. Image: HIRES/Keck. ......... 49 II.3 Transit schematic. Image: NASA ....................... 50 II.4 Planet orbiting the star, Fomalhaut (αPsA). Image: NASA, ESA, P. Kalas, J. Graham, E. Chiang, E. Kite (University of California, Berkeley), M. Clampin (NASA Goddard Space Flight Center), M. Fitzgerald (Lawrence Livermore National Laboratory), K. Stapelfeldt and J. Krist (NASA Jet Propulsion Laboratory) .................................... 53 II.5 Outer orbit of the system Gl22. ........................ 54 II.6 Variations in the transits of KOI-872b. Image: included in Nesvorný et al. (2012)..................................... 57 II.7 Variations in the period of rotation of the pulsar PSR B1257+12. Image: A. Wolszczan, D. Frail .............................. 59 II.8 Schematic of the detection of an exoplanet by means of microlensing. Image: NASA, ESA, and A. Feild (STScI) ...................... 59 II.9 Diagram of S, P, and L type orbits in binaries. Image: R. Schwarz. ..... 62 II.10 Case 1. Variations of the orbital elements of the giant planet caused by the second planet ................................. 67 II.11 Case 2. Variations of the orbital elements of the giant planet caused by the satellite .................................... 68 II.12 Comparison between the radial velocities in both scenarios ......... 69 II.13 Radial velocity curves in both scenarios ................... 70 II.14 Case 1. Variations of the orbital elements of the giant planet caused by the second planet with a mutual inclination of 5◦................ 71 II.15 Case 2. Variations of the orbital elements of the giant planet caused by the satellite with a mutual inclination of 5◦.................... 72 II.16 Radial velocity curves in both scenarios with a mutual inclination of 5◦. . 73 II.17 Case 1. Variations of the orbital elements of the giant planet caused by the second planet with an orbit of eccentricity 0.5 ................ 75 II.18 Case 2. Short-term variations of the orbital elements of the giant planet caused by the satellite with an orbit of eccentricity 0.5 ............ 76 xiv II.19 Case 2. Mid-term variations of the orbital elements of the giant planet caused by the satellite with an orbit of eccentricity 0.5 ................ 77 II.20 Radial velocity curves in both scenarios with eccentricity 0.5. ........ 78 II.21 Habitability zones depending on the distances to the star and the temperature of the star. The Recent Venus and Early Mars zones are not depicted. Image: Barbara Aulicino ............................... 81 xv LIST OF TABLES I.1 Examples of the application of the methodology ............... 28 I.2 Comparison between the Edwards results and this work ........... 29 xvii Another fruitful technique despite its drawbacks is gravitational microlensing. There are two main programs that use it, the Optical Gravitational Lensing Experiment (OGLE Udalski et al., 1992), a cooperation among the Carnegie Institution of Washington, Princeton University, and the Warsaw University Observatory, which consists of a 1 m. telescope located at the Las Campanas Observatory (Chile) and the Microlensing Observations in Astrophysics (MOA Muraki et al., 1999), a Japan and New Zealand collaboration that operates from the University of Canterbury Mt. John Observatory in New Zealand, with a 1.8 m .telescope. Most of these discoveries were carried out with techniques used in the study of double stars, mainly radial velocities and transits (corresponding to spectroscopic and eclipsing binaries, respectively), as well as imaging and astrometry (visual binaries). From a dynamical point of view, the study of exoplanets in binary and multiple systems is much more interesting. That is why these fields, double and multiple stars as well as extrasolar planets, are closely related. Double stars, along with the work fields related to their study, constitute a fundamental subject of research in Astronomy. In the XVII century, it was discovered that the star, Mizar, in the Ursa Major constellation, had a close companion apart from the pair that it forms with Alcor which is visible with the naked eye. Mizar was observed as a double star by G. B. Riccioli around 1650 (Riccioli, 1651) although some of the letters that are preserved in the National Library in Florence indicate that it had already been seen as a double by B. Castelli and G. Galilei in 1617 (Fedele, 1949). During the XVII and XVIII centuries, other pairs of stars were observed for the first time, such as α-Centauri, α-Geminorum (Castor), or γ-Virginis. The increasing number led J. Mitchell to postulate that they weren’t random alignments but physical systems (Heintz, 1978). However, it was W. Herschel who started to systematically follow pairs of close stars at the end of the XVIII century. His goal was to observe the annual paralactic movement in order to measure the parallax and, therefore, their distance to us. He was not able to achieve his objective but this practice led him to discover the orbital motion of double stars (Herschel, 1803). The distance to a star was determined for the first time a few decades later by F. W. Bessel who, in 1838, estimated the parallax of the star, 61-Cygni, to be 0”3136 (which is equivalent to a distance of 3.19 parsecs), by comparing its position with another six stars (Heintz, 1978). Curiously, this was a binary star from the list of Herschel. Also around that time, J. Goodricke proposed two models to explain the variability of the star, Algol in Perseus which was already known since antiquity (Goodricke, 1783). He suggested that this phenomenon might be caused either by spots similar to those on the photosphere of the Sun or by the transit of a giant planet. However, these explanations were not accepted because of the size that the planet or the spots should have in order to produce the observed variations in brightness. The astronomers at that time noticed another possibility that there was another star in the system which, when eclipsing (and being eclipsed by) the other component, diminished the measured brightness of the system. At the beginning of the XIX century, F. G. W. Struve began to measure double stars by using a filar micrometer (Struve, 1837,1852), a variation of the device invented by W. 4 Figure 3: Portrait of William Herschel. Image: public domain Gascoigne around 1639 (Towenley, 1666). The pioneering works by W. Herschel and F. G. W. Struve were initially continued by their respective sons, J. Herschel and O. Struve (Heintz, 1978). However, F. Savary was the first person to calculate the orbit of a binary star, ξ-Ursae Majoris (Savary, 1827) a few years ahead of J. Herschel (Herschel, 1833). Bessel attributed the variations observed in the proper motions of Sirius and Procyon to the presence of companion stars (Bessel, 1844) which was the first discovery of components of a system by means of astrometric techniques (without direct observation). Sirius B was first observed by A. G. Clark in 1862 whereas Procyon B was seen by J. M. Schaeberle in 1896 (Heintz, 1978) The discovery of the Doppler-Fizeau effect halfway through the XIX century cleared the way for the third technique for the study of double stars (Heintz, 1978), along with direct observation and the study of eclipses. Thanks to this well known physical effect, it is possible to determine the radial component of the orbital velocity of a star in a binary system through the displacement of the absorption lines in its spectrum. After a first attempt (Huggins, 1868), E. C. Pickering was able to observe (Pickering, 1890) the periodic duplicity of the spectral lines in Mizar due to the orbital motion. Many other discoveries followed, including the confirmation of the binarity of Algol (Vogel, 1890). This type of binary is called spectroscopic and they are, in turn, divided in two subtypes. If the spectral lines of both components can be seen in the spectrum, they are called double-lined spectroscopic 5 Figure 4: The Doppler effect in binaries. Image: Margaret Murray Hanson binaries (SB2) whereas, if only the lines of the brightest component appear, they are known as single-lined spectroscopic binaries (SB1). Regarding the visual observations, in 1906, S. W. Burnham published his catalog of double stars (BDS) with more than 13000 entries that were based mainly on observations carried out at the Lick and Yerkes Observatories (Burnham, 1906). This catalog was incorporated into the Catalog of Double Stars by R. S. Aitken (ADS) in 1926, with more than 17000 stars, along with measurements by Doolittle and by himself (Aitken, 1926). During the XX century, a large number of observers significantly increased the number of known binaries and micrometer measurements, with thousands of observations. Some of them were, for example, P. Baize, G. van Biesbroeck, W. H. van den Bos, P. Couteau, W. D. Heintz, R. T. A. Innes, G. P. Kuiper, P. Muller, R. A. Rossiter, G. A. Starikova, R. H. Wilson, and C. E. Worley, and many others (Docobo, 2016). Around that time, the work on binaries began in Spain. J. Comas Solá, among his multiple activities, performed measurements of a large number of double stars even before becoming the Director of the Fabra Observatory. He first worked with R. Patxot and later he worked alone, and he discovered a new pair, SOL1 (Comas Solá, 1898,1899,1900,1902). However, R. M. Aller was the true pioneer in this field in Spain. In addition to the large number of micrometer measurements carried out both at Lalín and at Santiago de Compostela (Aller, 1930,1934,1936), he calculated the first orbits in our country (Aller, 1935,1939). In 1887, A. A. Michelson and E. W. Morley published their famous experiment about the speed of light (Michelson and Morley, 1887). As early as 1900, J. M. Barr proposed the use of the Michelson interferometer for the study of the multiple star, Capella (Barr, 1900), although said study was delayed until 1920 (Michelson, 1920b,a). These works initiated the use of interferometric techniques for the study of visual double stars. J. A. Anderson published the interferometric measurements of several pairs (Anderson, 1920a,b), followed several years 6 Figure 5: Portrait of Ramón María Aller later by P. W. Merril (Merrill, 1922). The next milestone is attibuted to F. W. Finsen who, in 1951, developed an eyepiece interferometer (Finsen, 1951) that he used to conduct a large number of observations of close pairs at the Johannesbourg Observatory. However, interferometry reached its maturity in 1970 due to the development of the so-called “speckle interferometry” by A. Labeyrie (Labeyrie, 1970). This technique permits the attenuation of the effect of the atmosphere in order to reach the diffraction limit of the telescope, thereby allowing the observation of very close pairs. Many authors have used this technique in the last decades, attaining an unprecedented precision in the measurement of the relative positions. Some of them are H. A. McAlister, Y. Balega, W. I. Hartkopf, A. Tokovinin, E. P. Horch, B. D. Mason, J. L. Prieur, M. Scardia, G. Weigelt, as well as the research group of OARMA directed by J. A. Docobo. A Ph.D. dissertation elaborated by J. Gómez and directed by Professor Docobo was recently presented, containing a description of the work carried out at the OARMA in this field (Gómez, 2019). The study of systems that can be detected by means of different techniques is especially interesting becausetheyprovidemoreastrophysical information. Thestudy ofbinariesthatare both spectroscopic and eclipsing is customary, as these techniques favor short period systems. However, our group focuses on the study of visual binaries, therefore it is our preference to research spectro-interferometric systems, with both visual (obtained from interferometric measurements) and spectroscopic orbits that are calculated from their radial velocities. As we will see in the chapter corresponding to double stars, this type of system permits the obtention of the individual masses of the components as well as the orbital parallax which 7 is an independent test for the data obtained by astrometric missions such as Hipparcos or Gaia. From the beginning of the interferometric observations, our interest in these systems was established and many authors started to obtain combined orbits (Balega, Bonneau, and Foy, 1984; Balega and Ryadchenko, 1984; Bonneau et al., 1986; McAlister, 1976,1977, 1978). R. F. Griffin of the Cambridge Observatory contributed largely to this effort. He has published more than 260 articles about spectroscopic orbits and many of those systems have also been observed by means of interferometry. In the last years, our team at OARMA has published several works in this field (Docobo et al., 2014b,2017a,2018a,b), two of them in collaboration with Professor Griffin that have been included in this dissertation. We have also published a methodology to calculate the tridimensional orbit of a spectroscopic binary by using one visual observation (preferentially a high resolution measurement) and the parallax (Docobo et al., 2014a). The Ph.D. dissertation by A. Abushattal, also conducted under the supervision of J. A. Docobo at OARMA (Abushattal, 2017), collected different techniques to work with spectroscopic binaries, some original methodologies included, as well as a compilation of the works by R. F. Griffin and OARMA until that time. 8 Chapter I Double stars As it was established in the Introduction of this Memory, the study of binaries and, in general, of double stars (today the term “binary” is reserved preferentially for the closest cases), has a great significance for obtaining the fundamental physical parameters of the stars and thus being able to formulate precise models of their behavior and evolution. Double stars are classified according to the technique with which their binarity is detected and further study is conducted. There are three types, visual, if it is possible to observe both components by means of optical instrumentation (e.g. speckle interferometry); spectroscopic, if we can measure the radial velocity caused by the Doppler-Fizeau effect when the components of the system revolve around the orbit; and eclipsing, if our line of sight is contained (or close to) the orbital plane, so that the components undergo mutual periodic eclipses. We can also consider astrometric binaries as a particular case of the visual. They are systems in which it is only possible to observe the brighter star and its movement with respect to other neighboring stars due to the difference in magnitude. This classification is not exclusive and there may be systems that belong to two or even three of the types. These stars are the most interesting because they provide more astrophysical information. At OARMA, the main field of research is visual double stars, following the school that Ramón María Aller established in the USC (Aller, 1943,1957; Docobo, 2011,2016). However, due to the advances in the observational devices, we work with more and more spectro-interferometric stars. In this chapter we review the techniques that are used for the study of visual double stars and, more succinctly, of spectroscopic and eclipsing binaries. We discuss the algorithm proposed by P. Baize and L. Romani (Baize and Romani, 1946; Heintz, 1978) for the determination of the dynamical parallaxes and we focus on the Edwards algorithm which allows us to obtain the individual spectra of a double star from the combined spectrum and the difference in magnitude between the components. We will present a novel formulation of the algorithm that we designed and that generalizes the original process by using the bolometric corrections. The final section is a summary of the state of the art in multiple star research. 9 Chapter I. Double stars I.1 Visual binaries I.1.1 Observation The first step in the study of visual double (Docobo, 2002,2016) stars is observation in order to be able to calculate their orbits. The orbit, along with the parallax (π), permits us to determine the physical parameters of the stars, mainly the sum of the masses of the components or, in some instances, even the individual masses. Other physical data can be obtained from the observations, for example, the difference in magnitude between the components, ∆m. During an observation, we measure the position of the faintest component (secondary) with respect to the brightest component (primary) and, if they have the same magnitude, the star with the largest right ascension is chosen as the primary. We will see the projection of both components on the plane perpendicular to our line of sight and, as we are observing objects at huge distances, that projection can be considered to be cylindrical. In each observation, we obtain a triplet of data (θ, ρ;t), where trepresents the time of the observation, usually given in the Besselian epoch (B1950.0), although the International Astronomical Union (IAU) Commission G1, which is in charge of the study of double and multiple stars, has stated the necessity of changing to the use of the Julian epoch (J2000.0) in accordance with the recommendations of the IAU General Assembly celebrated in Grenoble in 1976 (Aoki et al., 1983). θstands for the angle between the North direction and the radio vector that joins the primary and the secondary components. This angle is measured in degrees and follows the N-E-S-W path (see Figure I.1). Finally, ρindicates the angular separation between both stars and it is measured in arcseconds although, when observing closer systems, it is necessary to use miliarcseconds. Several procedures have been used throughout history in order to obtain the measurements (θ, ρ). The main instrument before speckle interferometry was the filar micrometer. Prior to the introduction of the micrometer by F. G. W. Struve, W. Herschel, and others already performed observations of double stars, although with lower precision. The simplest design of a micrometer (de Villiers, 1999) consists of a part that is placed between the eyepiece and the telescope, with two thin and resistant filaments (they were formerly made of threads from spider nests, today of synthetic fibers) forming a crosshair and another thread parallel to one of the other. The observation begins by fixing the position of the primary star in the center of the crosshair, and then the micrometer is rotated until the secondary is placed on the fixed thread perpendicular to the mobile one. This movement measures the position angle with respect to the North direction by means of a graduated circle in the micrometer. Next, the mobile thread is displaced onto the secondary star by means of a graduated screw which permits the measurement of the angular separation (see Figure I.2). Usually, the telescope is moved to center the secondary in the crosshair and the mobile thread is placed over the primary star. The mean of both measurements is taken as the final value in order to avoid index errors. 10 I.1. Visual binaries FigureI.1: positionangle, θ, andangular separation, ρ, ofa binary system. The blue and orange circles represent the primary and the secondary components, respectively. Figure I.2: Schematic of a filar micrometer and steps for a micrometric measurement of θand ρ. The blue and orange circles represent the primary and the secondary components, respectively. The solid perpedicular lines stand for the crosshair of the micrometer, and the dashed line indicate the mobile thread. 11 Chapter I. Double stars Despite their simplicity, these devices have two downsides that led to the adoption of the modern digital methods. On one hand, this technique is heavily dependent on the experience of the observer and it requires a lot of practice to master the positioning of the threads. On the other hand, it is largely affected by the atmospheric conditions and a bad seeing can cause large errors in a measurement. Many observations were conducted using photographic plates (see for example GuntzelLingner, 1962; Güntzel-Lingner, 1962; Hertzsprung, 1917,1919,1920,1940,1942b; Hertzsprung and Albada, 1958; Thiele, 1903,1907) although the development of photography could not surpass the micrometer as the preferred tool in the double star studies, at least not before the arrival of digital photography. This technique was limited by observer errors, measurement errors, and errors inherent to the image, in addition to the seeing conditions (Heintz, 1978; Hertzsprung, 1942a). All of these errors made it very difficult to achieve angular separations lower than 100 . This excluded the systems with shorter periods which had a wider arc of observations and permitted to¡he determine¡ation of more precise orbits. The introduction of digital photography overcame these difficulties, mainly with the inclusion of CCD (Coupled Charged Device) chips, and the use of the micrometers has been gradually abandoned. The development of high resolution techniques in the last decades caused a qualitative leap in the research of visual double stars. The purpose of these techniques is to eliminate (or at least largely reduce) the effect of Earth’s atmosphere in order to attain the diffraction limit of the telescope which is given by the formula: ρ=1.22 λ D,(I.1) where ρindicates the minimum separation attainable by a telescope with Daperture, while observing in the λwavelength. The wavefront that reaches the telescope can be considered to be plane due to the large distance towards the astronomical sources. When observing a point source of light with the telescope in absence of atmospherical turbulence, the object does not appear as a point in the plane of the image; the different parts of the objective emit wavefronts that produce an interference pattern called Point Spread Function (PSF) when they overlap in the focus. For a circular or annular aperture, as is the case of most telescopes, the shape of the PSF becomes a bright central disc with concentric rings around it and it is called the Airy function (or Airy disc). The center of the disc corresponds to the maximum of this Airy function, whereas the minima appear in the separation between the disc and the first ring, and between consecutive rings, due to the destructive interference between the different wavefronts. The Rayleigh criterion points to the distance between the maximum and the first minimum and, in the case of two point sources with that separation, the maxima of their PSF would overlap with the first minimum of the other PSF, and the images can be separated (see Figure I.3). 12 I.1. Visual binaries Figure I.3: The Rayleigh criterion. The maximum of the PSF of each point source matches the first minimum of the PSF of the other source. Image: Prudyus et al. (2017) This ideal situation is not found in practice. The wavefront is distorted when passing through the atmosphere and the PSF is not the Airy disc but the instant image of the stars is a variable pattern of speckles. For a more prolonged exposition, the PSF obtained is a fuzzy disc that is called the “seeing disc”. The diameter of the seeing disc increases with the turbulence and, in any case, it is always larger than the diameter of the Airy disc, thus the real resolution never attains the theoretical limit of the telescope. The atmospheric conditions may vary rapidly, sometimes in minutes, and these limitations are not easy to overcome. Several techniques have been developed to minimize the effect of the atmospheric turbulence. The first of them is speckle interferometry. This procedure consists of taking a large number of images with short expositions (on the order of miliseconds) and combining them by means of Fourier analysis after correcting the possible offsets among them (using a method called shift-and-add). The images obtained in this way are practically instantaneous and the speckle pattern becomes visible. The fundamental hypothesis of the work of Labeyrie is that the minimum size of the speckles matches the size of the theoretical Airy disc of the star. If we call of the object O(α, β), and the intensity distribution of the image I(α, β), we can write the following equation: I(α, β)=O(α, β)⊗ |p(α, β)|2,(I.2) where p(α, β)is theFouriertransform oftheperturbed pupilofthe telescope, P(x,y), therefore the equation I.2 shows how the telescope deforms the real image of the object, O, to obtain the captured image, I, under certain seeing conditions. If we apply Fourier transforms to both sides of the equation, and square them, we obtain: 13 Chapter I. Double stars the satellites Hipparcos or Gaia, for example), the adjustment between the common orbital elements obtained by means of other techniques, such as radial velocities, etc. Besides, as for every orbit of the family, we can determine the constant of the areas. It is enough to compare those results with the value of that constant calculated from all the observations. Using the constant of the areas as a control, it would be like working with the Thiele-Innes-van den Bos method. The basis of this method consists of establishing a mapping from the interval (0,2π) into the set of elliptical Keplerian orbits for which their apparent orbits pass through the base points. Obviously, this base points are chosen to match high quality observations. If E3and E1are the eccentric anomalies corresponding to the epochs, t3and t1, respectively, an orbit is obtained for each value V=E3−E1, although there may be values of Vwithout a periodic solution (Docobo, 1985,2012). In this way, if the three points are separated by less than a period, we have that V∈ (0,2π), unlike with the constant of the areas that, when V→2π, it turns out that c→+∞. Anyway, cis univocally determined by Vas long as the three points are in the same revolution. Also, each orbit obtained for a value of Vwill yield different ephemerides out of the three base points. In essence, the methods of Thiele-Innes-van den Bos and Cid are particular cases of the Docobo Method. It is a very versatile method that has been used to calculate hundreds of double star orbits, both by the OARMA personnel and by other researchers. The author of this Memory has developed an implementation of the method using Matlab, derived from the original program in Fortran, that includes the following features: •The Docobo Method. •The calculation of ephemerides from the orbital elements and the evaluation of the rms error with respect to a set of observations. •The precession correction of the observations. •The calculation of the dynamical parallax from the orbital elements, the difference in magnitude between the components, and the spectral types by means of the BaizeRomani algorithm (Baize and Romani, 1946) with the calibrations given by Docobo and Andrade (Docobo and Andrade, 2013). •The calculation of the sum of the masses of the components from the dynamical and trigonometric (Hipparcos or Gaia) parallaxes and the individual masses from the dynamical parallax and the difference in magnitude. •The graphical representation of the orbit and the observations. •The improvement of the orbit by means of a least squares minimization of the sum of the rms in θand ρ. 20 I.1. Visual binaries •The calculation of the radial velocity ephemerides and their rms with respect to a set of observations. •The representation of the radial velocity curves. This implementation of the Docobo Method has been used by Professor Docobo and the author of this Memory in the articles corresponding to the calculation of binary star orbits included in this dissertation as well as other articles published in recent years and orbits published in the Information Circular of the IAU Commission G1 (Binary and Multiple Star Systems, http://www.usc.es/astro/circularing.html) with the author code, Docobo-Campo. The determination of the orbits permits the calculation of the sum of the masses of the components in units of solar mass (M) from Kepler’s Third Law, as long as we know the parallax (trigonometric or dynamical), by using the following expression: M1+M2=¯a3(u.a.) P2=a(00) π(00)31 P2(I.10) in which the orbital period, P, must be given in years. I.1.3 The Baize-Romani algorithm This algorithm which was presented by P. Baize and L Romani in 1946 (Baize and Romani, 1946; Heintz, 1978) permits the calculation of the parallax of a binary system from its orbital elements (concretely, the semimajor axis and the period), the spectral types of the components, the difference in magnitude (∆m), and the total magnitude of the system. On one hand, it is based on Kepler’s Third Law and, on the other, on a mass-luminosity relationship (MLR) of the stars as follows: L∝ Mk,(I.11) for a value of k. If we define h3=(M1+M2)π3, where πis the parallax, given in arcseconds, and take it into the equation I.10, we obtain for the semimajor axis, a, in arcseconds, and the period, P, in years: log(h)=log(a)− 2 3log(P)(I.12) The luminosity of a star, L, can be written in terms of the absolute bolometric magnitude, L=100.4(MBol −M0)(in units of solar luminosity, L), with MBol the bolometric magnitude of the star, and M0the reference bolometric magnitude. In the case of main-sequence stars, M0 is usually chosen as the bolometric magnitude of the Sun. In order to obtain the bolometric 21 Chapter I. Double stars magnitude, we need to measure the luminosity along the entire electromagnetic spectrum but it can be calculated from the visual magnitude by means of the equation, MBol =MV+BC, where BC is called the bolometric correction, and it is determined empirically. There are multiple calibrations that list these corrections, for example, those included in Bessell, Castelli, and Plez (1998), Flower (1996), Gray (2005), and Straizys and Kuriliene (1981). In principle, any calibration can be used as long as we are careful to correct them according to the zero-point (Torres, 2010). We can reformulate the equation I.11 in terms of the bolometric magnitudes: MBol =M0−5 2k log(M) (I.13) And taking into account the known relationship between the absolute magnitude, M, and the apparent magnitude, m: M=m+5+5log(π),(I.14) which is valid both for the visual and bolometric magnitudes and, if we define: A=2.5[log(1+10−0.4∆m+0.4(BC2−BC1))(I.15) −k log(1+10(−0.4∆m+0.4(BC2−BC1))/k)(I.16) −log((1+10−0.4∆m+0.4(BC2−BC1))/(1+100.4∆m))] (I.17) B=m1−2.5log(1+100.4∆m)(I.18) C=BC1+5−M0(I.19) D=7.5k log(h)(I.20) we can combine the equations I.12,I.13 and I.14 in order to obtain the dynamical parallax: log(πdyn)=A+B+C+D 7.5k−5.(I.21) This parallax is quite accurate for main-sequence stars because their MLR is well determined, and it is reasonably accurate for subgiant stars whereas, for giant and supergiant stars, it is not applicable. With this parallax, we can calculate the sum of the masses of the components using Kepler’s Third Law, as well as the individual masses by means of the relationship: 22 I.1. Visual binaries log(M1+M2)=3log(h)−3log(π)(I.22) k log(M1 M2)=0.4∆M(I.23) The algorithm is based on two parameters, kand M0, which are obtained empirically, and they depend on the chosen calibration. In our implementation of this algorithm that we use with Docobo’s method, we take the values provided by J. A. Docobo and M. Andrade (Docobo and Andrade, 2013) that are based on the calibrations of V. Straizys and G. Kuriliene (Straizys and Kuriliene, 1981). The adopted values are k=4.23 and M0=MB=4.74 for main-sequence stars and k=3.64 and M0=3.88 for subgiants. I.1.4 The Edwards process As we have seen in the previous section, the knowledge of the individual spectra permits us to obtain information about the astrophysical parameters of the components of the system. In the case of single stars, the spectrum can be measured directly by using a spectrometer but, when we have a binary system, if the separation between the components is lower than the resolution of the spectrograph, we will not be able to get both spectra. We will obtain a combined spectrum in which features of the spectra of both components appear unless the difference in magnitude between the components is high. There are several methods to separate the spectra of the binary components. If we can determine, or at least estimate, the spectrum of one of the components (for example, during a total eclipse in the case of eclipsing binaries), we can substract it from the combined to obtain the one from the other star (Griffin and Griffin, 1986). In other cases, the spectral features of each component can be easily identified in the combined spectrum, and the process of separation is essentially straightforward (Ferluga et al., 1997; Pilachowski and Sowell, 1992). However, if the spectral lines of the components overlap, this is not possible and we need to resort to more sophisticated techniques. One option is the use of tomographic algorithms for image reconstruction (Bagnuolo and Gies, 1991). Also, K. P. Simon and E. Sturm (Simon and Sturm, 1994) developed a method based on obtaining several combined spectra from different phases (non-eclipsing) that are later transformed in an over-determined, rank-deficient system which can be solved. This method also permits the determination of the elements of the spectroscopic orbit. Finally, P. Hadrava designed a method based on the Fourier transform to determine both the spectra and the spectroscopic orbit (Hadrava, 1995). This procedures are widely used in the study of spectroscopic and eclipsing binaries. Many interferometric visual binaries are also spectroscopic or eclipsing because they are close systems, therefore their spectra are determined. However, for wider systems with lower radial 23 Chapter I. Double stars velocities, the spectra overlap completely and even the most sophisticated techniques are unable to separate them. Thus, we only know the combined spectrum in many visual systems that have not been observed with high resolution. In 1976, T. W. Edwards (Edwards, 1976) proposed a method to obtain the individual spectra of binaries from the combined and the difference in magnitude by using a calibration of the luminosities of the MK system. The article was based on a previous work by J. W. Christy and R. L. Walker (Christy and Walker, 1969) and it consists of performing an interpolation of the spectral types by means of the following expressions: S(1)+xS(2)=(1+x)S(1+2),and (I.24) M[S(1)]− M[S(2)] =∆m=−2.5log(x).(I.25) where S(1),S(2), and S(1+2)represent the spectral types of the primary, the secondary, and the combined, respectively. M[S]is the calibration of the luminosity and ∆mis the difference in magnitude. However, there is the question of how to interpret the quantity that represents the spectral type, S(i), in order to be able to interpolate it. A first approximation is to use a calibration of the absolute magnitude, MV, of the combined spectrum and to obtain the individual magnitudes from the equation: MV1+x MV2=(1+x)MV,(I.26) in which MVi stands for the absolute visual magnitude of the component iand xis calculated from ∆mand the second equation of I.24. Although this approximation yields satisfactory results when it is applied to real systems, its physical interpretation is more problematic. Another approximation is based on the work by W. I. Beavers and D. B. Cook (Beavers and Cook, 1980). In order to follow this work, it is necessary to clarify several concepts beforehand. The thermal radiation produced by a star is similar to that of a black body with the same surface temperature. The brightness per surface unit of a black body in thermodynamical equilibrium at temperature T, for a wavelength, λ, can be calculated from Planck’s Law: Bλ(T)=2hc λ5 n2 λ exp(−hc λkT )−1,(I.27) where hand kare the Planck and Boltzmann constants, respectively, nλrepresents the refraction index of the medium for that wavelength, and cis the speed of light in the free 24 I.1. Visual binaries space. If we integrate the brightness with respect to the wavelength, we obtain the StefanBoltzmann Law which, applied to a semi-sphere yield the luminosity, L, of a star, i.e., the total emission of energy per unit of time: L=4πσR2T4.(I.28) Rrepresents the radius of the star, Tstands for the surface temperature, and σis the StefanBoltzmann constant. For a certain wavelength, λ, the spectral flux density, fλ, is defined as the amount of energy incoming from a star (with Rradius and Ttemperature) that crosses a unit of surface at a distance (D) and it can be calculated from the brightness by means of the following expression: fλ(T)=πR2 D2Bλ(T).(I.29) If we integrate fλwith respect to the wavelength, it yields the flux, which is related to the luminosity and the distance through the equation: F=πR2L D2.(I.30) There are also relationships between both the spectral flux density in the visual band and the visual magnitude, and between the flux and the bolometric magnitude: mV=−2.5log(fV),(I.31) mBol =−2.5log(F).(I.32) Finally, the normalized flux for a given wavelength λis defined as: Fλ=fλ F.(I.33) Back to the work of Beavers and Cook, the basis of it is that, given a binary system and a constant k≥0, the monochromatic normalized flux of the system, Fλ, can be obtained from the individual fluxes (F1,λ,F2,λ) as follows: Fλ=F1,λ +k F2,λ 1+k.(I.34) 25 Chapter I. Double stars The similarity with the Edwards formulation is evident and if we consider k=F2 F1, the equation I.34 can be written in the following way: fλ F = f1,λ F1+kf2,λ F2 1+k.(I.35) which is more physically sound than the equation I.26 but it requires the knowledge of the flux. However, if we combine the I.35 and I.31 equations, we obtain: (1+k)10−0.4mV 10−0.4mBol =10−0.4m1,V 10−0.4m1,Bol +k10−0.4m2,V 10−0.4m2,Bol .(I.36) and if we consider the relationship between the apparent and absolute magnitudes given in the equation I.14, it yields: (1+k)10−0.4MV 10−0.4MBol =10−0.4M1,V 10−0.4M1,Bol +k10−0.4M2,V 10−0.4M2,Bol .(I.37) Finally, if we take into account the relationship between the visual and bolometric magnitudes, BC :=MBol −MV, the equation I.37 is equivalent to: (1+k)100.4BC =100.4BC1+k100.4BC2.(I.38) The bolometric correction of the combined spectrum (BC) can be obtained from any calibration and now the equation I.24 of the Edwards method is shown in terms of the bolometric correction, taking into account that k=x=10−0.4∆m. In order to determine the spectral types, we will consider the bolometric correction as a function of the absolute magnitude, BC =φ(MV), that can be interpolated from the calibration values and the order of the equation I.38 can be reduced by means of the difference in magnitude: (1+k)100.4BC =100.4φ(M1,V)+k100.4φ(M1,V+∆m)(I.39) and it can be solved numerically to obtain M1,Vand, therefore, M2,V. The spectral types are obtained from the calibration. I.1.5 Application of the methodology We will now apply the methodology to several visual double stars with calculated orbits. In these examples, we have included only main-sequence stars although it could be applied 26 I.2. Spectroscopic binaries to subgiant stars taking into account the indications by Edwards about the assignment of luminosity classes. The absolute magnitude of giant stars is not monotonically increasing and, therefore, it cannot be used for the interpolation of the bolometric correction. We have selected the calibration of Straizys and Kuriliene (1981) and we have taken the spectral types provided by the SIMBAD database, operated by the CDS in Strasbourg, France. In order to determine the differences in magnitude, we have used the values obtained by means of the speckle measurements included in the Fourth Catalog of Interferometric Measurements of Binary Stars of the U.S. Naval Observatory. When there are several values of ∆m, we calculate the arithmetic mean of those close to the Vband. The results are depicted in Table I.1. In the first two columns the WDS designation and the name of the system are included, respectively, whereas the third column shows the difference in magnitude. The combined and the individual spectral types are represented in columns 4, 5, and 6. If two solutions are possible, correponding to two different measurements of the combined spectrum, they appear in consecutive rows. The interpolation was performed with the method of cubic splines, by using the “interpolate” package included in the “scipy” library of Python. The equation I.39 was solved by means of a Newton method. In the Table I.2, we compare the values obtained in the original work by Edwards with those calculated with our implementation. Therefore, we use the values of the spectral types and differences in magnitude included in the article by Edwards. Part of the slight differences between both works may be attributed to the use of different calibrations. The first four columns follow the format of the previous Table and the last four columns show the individual spectral types, first those calculated by Edwards and, finally, ours. I.2 Spectroscopic binaries Spectroscopic binaries are detected thanks to the Doppler shift that occurs in the spectral lines of the components throughout their periodic movement toward and away from the Earth. This shift can be used to calculate the radial component of the orbital velocity of the binaries by means of the following equation: vr c =λobs −λ0 λ0(I.40) where vrrepresent the radial velocity; c, the speed of light in free space; λobs, the observed wavelength of a determined spectral line; and λ0, the wavelength of that spectral line at rest, as measured in the laboratory. This relationship is valid as long as the observer and the source of radiation are point objects moving with repect to each other which, in reality, will not be true and that is why we need to include several terms of correction. First, we have a daily term 27 Chapter I. Double stars Table I.1: Examples of the application of the methodology WDS Name ∆mSp. type Sp. type Sp. type (combined) (A) (B) 01477 −4358 I52 0.62 F6/7V F5V F8V F5V F9V 02514 −2139 DON43 0.2 F3V F2V F4V 03189 −0101 BU1177 0.18 G8V G7.5V G8.5V 04142 −4608 RST2338 1.34 F8/G0V F5V G3V F8V G7V 04506 +1505 CHR20 0.8 G5 G2.5V G8V 06274 −2544 B114 0.28 K0V G9V K1V 07013 −0906 A671 0.6 F5 F4V F8V 12155 −3106 RST1658 1.15 K7Vk K6V M0V 13044 −1316 HU642 0.31 G0 F9V G1V 14243 −3838 RST1785 0.27 G5V G4V G6V 15332 −2429 CHR232 1.76 A7V A6V F6V 16094 −3103 I557 0.8 A7IV A5IV F0IV A6IV A7V 17115 −1630 HU169 0.695 A7V A6V F0V 18434 −5546 B398 1.23 F7V F4V G2V 19264 +4928 YSC134 0.95 K2.5V K1.5V K5V 20081 −3929 RST2134 0.42 G0V F9V G2V 22007 −5002 I1450 0.19 K0V K0V K0.5V 28 I.2. Spectroscopic binaries Table I.2: Comparison between the Edwards results and this work Edwards This work WDS Name ∆mSp. type Sp. type Sp. type Sp. type Sp. type (combined) (A) (B) (A) (B) 00318 +5431 STT12 0.18 B8V B7.5V B8.5V B7.5V B8.5V 02396 −1152 FIN312 0.15 F6V F6V F6V F6V F6.5V 03175 +6540 STT52 0.45 A3V A2V A4V A2.5V A5V 04199 +1631 STT79 1.38 F9V F7V G6V F6V G5V 05017 +2640 A1844 1.39 G2V G0V G8V G0V G9V 06474 +1812 STT156 0.22 A2V A1V A3V A1.5V A2.5V 07175 −4659 I7 0.70 K2V K1V K4V K1V K4V 08394 −3636 I314 1.45 F3IV F2IV F6V F2IV F7V 09210 +3811 STF1338 0.25 F3V F2V F4V F2V F4V 10361 −2641 BU411 0.98 F6V F4V G0V F4V G0V 11047 −0413BC A676 0.10 M0V M0V M0V M0V M0V 12396 −3717 DAW63 0.35 K5V K4V K6V K4.5V K6V 13123 −5955 SEE170 0.41 B8V B8V B9V B7.5 B9V 14463 +0939 STF1879 0.62 G2V G1V G4V G1V G5V 17082 −0105 A1145 2.00 A3V A1V F3V A3V F4V 18570 +3254 BU648 2.20 G0V F9V K1V F8V K1V 20474 +3629 STT413 1.26 B5V B4V B7V B4V B7.5V 29 Chapter I. Double stars I.2.4 Spectro-interferometric binaries In the case that a system has a visual and a spectroscopic orbit, we can obtain more information about the astrophysical parameter of the system. Among the other elements, we know a00 and Ifrom the visual orbit and, if the spectroscopic orbit is double-lined, we have ¯a1sin I and ¯a2sin I, and we can calculate the semimajor axis in units of distance, ¯a=¯a1+¯a2. Therefore, we can determine the orbital parallax because: πorb =a00 ¯a(I.54) In this way, we have a method based solely on observational data to check the parallaxes obtained by the astrometric satellites Hipparcos and Gaia. Moreover, the visual orbit yields the sum of the masses (Equation I.10) and the spectroscopic orbit provides the ratio of the masses (Equation I.51), therefore we can also calculate the individual masses of the components. Even in the case that the spectroscopic orbit is single-lined, if we know the parallax (for example, from Hipparcos or Gaia), the mass function given in Equation I.50, as long as the inclination and the sum of the masses permit us to obtain the individual masses (Docobo et al., 2018b, see for example). I.3 Eclipsing binaries In this type of binaries our line of sight is contained in (or very close to) the orbital plane. Due to this fact, periodic eclipses (total or partial) are observed from Earth when their components pass in front of each other. These eclipses can be accurately studied and we can obtain some of the orbital elements, concretely the period and the inclination, along with other fundamental parameters such as the radii and the luminosities. As the information about these systems is obtained mainly during the eclipses, they are usually close systems with short periods. In this section, we will mainly follow the formulation given in Abad, Docobo, and Elipe (2017) and Kallrath and Milone (2009). I.3.1 Observation The eclipses cause a variation in the light that we receive from the system and the instrument that is used for their study is a photometer which measures the light intensity from a source. When we observe these stars along time, we obtain the so-called light curve (see Figure I.10) which has an approximately constant value except during the eclipses when a drop in 36 I.3. Eclipsing binaries Figure I.10: Light curves of eclipsing binaries of the types EA, EB and EW. Image: Included in the article (Hümmerich, Bernhard, and Srdoc, 2013). the intensity of light is detected. Unless there are perturbations in the system, the eclipses occur at regular intervals, therefore, these variations are periodic and there are two sets of different eclipses unless both stars have the same size and luminosity. When the brightest star is behind the other, we have the main eclipse with a deeper drop in the luminosity and, when the faintest star is covered, the secondary eclipse occurs. However, not all eclipses are equal and we may produce a classification of these stars on the basis of the typology of their light curves. In this way, we have the following eclipsing binaries of the following types: •EA: they are also called Algol-type or β-Persei due to the star that presents this type of curve that is best known. Within this class, the eclipses stretch through a relatively narrow part of a phase (the representation of one period in the light curve). Outside the eclipses, the light curve is almost flat which indicates little interaction between the components and the minima of the eclipses have different depths which indicates a large difference in brightness. •EB: or β-Lyrae, they show a continuous variation of the brightness along the phase because the proximity between the components cause them to adopt an ellipsoidal shape due to the tidal distortion. The minima have different intensities. 37 Chapter I. Double stars •EW: also known as W-Ursa Majoris, their brightness changes continuously which indicates tidal distortion but the depth of their minima is similar. Several physical effects may also affect the morphology of the light curve. First, we have to take into account the known effect of limb darkening. It consists of a higher brightness in the center of the star than at the border, therefore, the minimum of the eclipse is not flat but also varies continuously. Another known phenomenon that has to be taken into account is the reflection effect which causes the situation that half of each star that is directed to the other is heated by the irradiation received from its companion. If the largest star is eclipsed, the increase of the temperature will increase the brightness of the non-eclipsed part of the visible half of the star, thereby reducing the depth of the minimum. Other sources of perturbation may include starspots on the surfaces of the components, light from a third component, or the presence of circumstellar matter in the system. All of these effects must be considered in order to obtain an accurate model of the system. I.3.2 Morphology These systems are generally very close and the approximation of considering the stars as point objects, or even homogeneous spheres, may introduce large errors in the model. That is why it is necessary to consider tidal effects that cause the stars to acquire an ellipsoidal shape. The tidal forces may also circularize the orbit and induce the coplanarity of the equatorial planes and the stars may start to rotate synchronously or even to exchange mass. In many cases, the effects of the radiation pressure are not negligible and must be incorporated. A classification of close binaries can be done according to them filling their respective Roche lobes. If we assume as a first approximation that the gravitational potential can be determined considering the stars as point masses and adding a centrifugal term, this potential is conservative and we can apply the classical study of the restricted three-body problem, obtaining the zero-velocity (or equipotential) surfaces, and the five Lagrangian points of the system are defined. The inner Lagrangian point, L1, determines the largest equipotential surfaces that contain each component alone and they are called Roche lobes. We have four possible scenarios. •Both stars are smaller than their Roche lobes. In this case, they are called detached systems and the components do not exchange mass in a significant way. •One of the stars is bigger than its Roche lobe. These systems are known as semidetached and said component loses matter which is absorbed by the companion or it remains as circumstellar matter. 38 I.3. Eclipsing binaries Figure I.11: Diagram of the Roche lobes of detached, semi-detached, and over-contact binaries. Image: Original included in the article (Terrel, 2002). •Both stars are bigger than their Roche lobes. The components are joined by a bridge of matter that connects them,and they share a common envelope. This is the case of the over-contact binaries. In these systems, the stars have synchronous rotation and the orbit is circularized. •As a particular case of the former, both stars may fill exactly their Roche lobes. They are usually systems in which one of the components absorbs matter from its companion. The star that gives mass fills its lobe and rotates synchronically, whereas the other spins faster due to the absorption of matter and its lobe shrinks due to the increase in the centrifugal force until it coincides with the surface of the star. These kinds of systems are called double-contact binaries and their orbits also tend to circularize. 39 Chapter I. Double stars I.3.3 Determination of the parameters Once we have at least one light curve of the system at our disposal, we can determine some of the orbital and astrophysical parameters of the components. Concretely, we can obtain the period P, the eccentricity, e, the inclination, I, and the argument of the periastron, ω, of the orbit, as well as the ratio between the luminosities, L2 L1, the ratios between the radii and the semimajor axis of the orbit, Ri a,i=1,2, their surface gravities gi,i=1,2, and their rotational parameters. Because they are close systems and their orbital plane must be close to the line of sight, the components will have high radial velocities and the system is optimal for the study as a spectroscopic binary. If the binary is SB2, we know iand a sin i and we can obtain all of the orbital elements and the individual masses, as well as their radii. The calculation is usually performed by means of a least squares minimization, adjusting the model according to the type of binary that we are studying (EA-EB-EW, detached, semidetached, over-contact, double-contact) and the rest of the contributions that need to be taken into account for each particular case. It must be noticed that these parameters may vary with time. We already commented that the orbits can become circular (e= 0) in over-contact and double-contact binaries. Besides, massloss and mass-transfer phenomena also affect the period and, consequently, the semimajor axis. In the case of mass transfer, if it occurs through the inner Lagrange point and in circular orbits when the mass goes from the least to the most massive star, the semimajor axis increases while, in the opposite case, it diminishes (Andrade, 2007; Negu and Tessema, 2015). If one of the components shows strong stellar winds, the mass transfer may happen through the accretion of those winds by the other star, although in that case, the mass absorbed is a small portion of the mass lost. If the system loses mass, the energy and the angular moment are not conserved, and the behavior of the system will be influenced by the mass-loss mechanism and its intensity. M. Andrade in his PhD dissertation (Andrade, 2007) performed an exhaustive study of the scenarios of mass-loss in binaries taking perturbative phenomena into account, focusing on the periastron effect as well as non-spherical shapes and relativistic effects. The periastron effect causes the intensification of mass-loss when the distance between the components decreases. When we have a time-dependent mass-loss (without a periastron effect), the semimajor axis usually undergoes a secular increase and the eccentricity varies periodically. The periastron effect causes the eccentricity to increase secularly, whereas the semimajor axis and the period diminish. If we combine both types of mass-loss, the result will depend on the dominant contribution, according to the initial conditions (Andrade and Docobo, 2003). The secular influx of the non-spherical shape and the relativistic effects (that must be taken into account because we are working with close systems) can be seen in the variation of the argument of the periastron. 40 I.4. Multiple stellar systems I.4 Multiple stellar systems Previously, we have been considering systems with two components but there are others with more stars as is the case of α-Centauri (3), Polaris (α-Ursa Minoris, 3), Capella (α-Aurigae, 4), or Castor (α-Geminorum, 6). According to several different studies, the rate of multiplicity in our Galaxy depends on the spectral type and it is lower for the later spectral types (Duchêne and Kraus, 2013). The most massive stars (O and B types) are found in multiple systems in more than 70% of the cases, half of which contain three or more components (Peter et al., 2012; Sota et al., 2014). Around 68% of intermediate mass stars (A type) are multiples and close to 60% of them would have only two components (De Rosa et al., 2014). The stars with a mass close to that of the Sun (F and G types) belong to multiple systems inbetween 45% and 65% of the cases, with 35% of them in systems with three or more components (Duquennoy, Mayor, and Halbwachs, 1991; Fuhrmann et al., 2017; Raghavan et al., 2010; Tokovinin, 2014). The rate of multiple stars in the case of red dwarfs (M type) is around 25%, 85% of which are binaries (Ward-Duong et al., 2015; Winters et al., 2019). For brown dwarfs, the percentage is lower than 20% (Fontanive et al., 2018). When we work with multiple systems, the orbits are not Keplerian and we would need to solve a general three-body problem. However, when we deal with hierarchical systems, we can use the theory of perturbations to analytically solve the system of equations (Abad and Docobo, 1988; Docobo, 1977; Harrington, 1969). A hierarchical system is one that can be decomposed into subgroups that are smaller each time, with only one or two components in the subgroups of the last decomposition and the distances within a level of decomposition much smaller than the distances in the upper level. The simplest hierarchical case, a triple system, consists of a double subsystem with the third component much farther away. If we increase the number of components, there are more possible cases. For example, for quadruple stars, there are two possible configurations that maintain the hierarchy, two double subsystems with a large separation between them and a double subsystem with a third component much farther away and another one at a farther distance (see Figures I.12 and I.13. The hierarchical systems are not rare and they are probably the most common within the multiple stellar systems because they are usually more stable. In order to analytically solve the equations of the hierarchical three-body problem, the Hamiltonian of the problem is usually expressed in terms of a small constant parameter (or several parameters in the case of higher order problems) in order to eliminate the non-significant terms, therefore simplifying the equations. In hierarchical systems, the ratios of the orbital semimajor axes are commonly used as parameters although any other can be considered as long as they are small enough. A. Abad, in his PhD dissertation(Abad, 1984; Abad and Docobo, 1988), developed a methodology called the step decomposition, to systematically decompose hierarchical systems and formulate the Hamiltonian and he applied it to study hierarchical triple and quadruple (2+2 configuration) stellar systems. 41 Chapter I. Double stars Figure I.12: Right hand image: hierarchical quadruple system with two double subsystems (P1+P2and P3+P4). Left hand image: schematics of the hierarchy. Figure I.13: Right hand image: hierarchical quadruple system with a double subsystem (P1+P2), and the third and fourth components at successively larger distances (P3and P4). Left hand image: schematics of the hierarchy. 42 I.4. Multiple stellar systems Figure I.14: Schematics of the hierarchy of the sextuple system Castor. Throughout time, the dynamics of these systems have been studied in multiple ways focusing, for example, on the stability (Eggleton and Kiseleva, 1995; Georgakarakos, 2008,2013; Grishin et al., 2017; Li, Fu, and Sun, 2010; Martynova, Orlov, and Rubinov, 2009; Milani and Nobili, 1983; Mylläri et al., 2018; Széll, Steves, and Érdi, 2004; Walker, 1983a,b; Walker and Roy, 1981,1983a,b) or on their secular evolution under the effect of different conditions and perturbations such as the existence of coplanarity or not, and the influence of the inclination and the eccentricity, including the Lidov-Kozai effect (Georgakarakos, 2002,2003,2004,2009; Grishin, Perets, and Fragione, 2018; Naoz et al., 2017), relativistic perturbations (Will, 2017), mass-loss and mass-transfer (Michaely and Perets, 2014), rotation and tidal effects (Borkovits, Forgács-Dajka, and Regály, 2004,2007; Correia et al., 2011; Hamers, 2019), or the influence of the evolutionary state of the components (Toonen, Hamers, and Portegies Zwart, 2016). Many Hamiltonian formulations have developed different orders of perturbation, using different sets of coordinates (Breiter and Vokrouhlický, 2015; Ford, Kozinsky, and Rasio, 2000; Hamers and Portegies Zwart, 2016; Krymolowski and Mazeh, 1999; Lei, Circi, and Ortore, 2018; Naoz et al., 2013), and even specific numerical algorithms have been developed (Beust, 2003). Sophisticated mathematical tools, both analytical and numerical, are required for the longterm study of hierarchical systems. However, in order to determine the masses of the stars for a short term study, the system can be decomposed into independent two-body problems because the mutual influence between distant subsystems is very small. If the difference between the scales of the distances of the different subsystems is large enough, we can calculate a Keplerian orbit for each subsystem without further correction as is the case of the star, Castor (Docobo et al., 2016), a hierarchical sextuple system with the schematics shown in Figure I.14. In this system, for example, the distance between the Aa and Ab subcomponents is 0.127 a.u.., between Ba and Bb, 0.0562 a.u., and between the subsystems A (Aa-Ab) and B (Ba-Bb), 105 a.u. 43 Chapter I. Double stars In other cases, corrections may be necessary such as the case of the Gliese 22 system (Docobo et al., 2008a), a hierarchical triple system for which some measurements did not resolve the inner pair and the position of the third component (B) was referred to the light center of the Aa-Ab subsystem, whereas the most modern observations did resolve it and the B component was measured with respect to Aa and it was necessary to homogenize them in order to calculate the relative orbit of B with respect to Aa-Ab. Therearestudies ofindividual visual, spectroscopic, andeclipsingmultiplesystems(Borkovits et al., 2019; Catanzaro et al., 2019; Docobo and Andrade, 2006; Horch et al., 2019; Jha et al., 1997; O’Brien et al., 2011; Shultz et al., 2019; Tokovinin, 2016a,b) and A. Tokovinin (Tokovinin, 2018) has produced a catalog of known hierarchical systems. 44 Chapter II Extrasolar planets As we have seen in the Introduction, the study of extrasolar planets, or exoplanets, is a field that is growing rapidly within Astronomy. Besides the constant discoveries, many studies are being conducted regarding the dynamics of different scenarios involving exoplanets. In this Chapter, we will review this field of research beginning with their classification. We will then review the observation techniques used for the discovery and the study of the extrasolar planets. Next, we will inspect the research concerning the dynamics of planetary systems, focusing on binaries with exoplanets. Later, we will examine the search for exosatellites orbiting around these planets, reviewing the research, and we will present an original work in which we analyze the dynamics of planet-planet and planet-satellite systems in order to obtain an algorithm that permits the detection of exosatellites through the perturbations that they induce in the spectroscopic orbit of their host planet. Finally we will deal with the study of the habitability of exoplanets and exosatellites, meaning the possibility that liquid water exists on the surface of these bodies, with the goal of finding places with conditions similar to those on Earth. II.1 Classification As it happens in the Solar System, extrasolar planets with different masses and sizes are being discovered. Two types of planets were traditionally considered orbiting the Sun, the Earthtype (or rocky) planets, and the Jupiter-type (or gas giant) planets. Most of the extrasolar planets discovered could be assigned to one of those categories, however, it is not so clear in other cases. That is why several schemes of classification have been proposed over time, both formal and informal. A first option is to use the planetary mass as a criterium for classification. Following this philosophy, S. A. Stern and H. F. Levison (Stern and Levison, 2002) proposed an expansive classification for planets, that in the case of the Solar System, would also include some minor bodies. According to their definition, the planets would be bodies with 45 Chapter II. Extrasolar planets where Irepresents the inclination of the orbit and φ∈ [0,1]is the phase of the planet, with a value of 0 when it reflects all of the light in the opposite direction from the Earth and 1 when the hemisphere of the planet that we see is completely illuminated. The function g(α) is defined as follows: g(α)=sin α+(π−α)cos α π(II.4) In addition to the difference in magnitude, another problem arises which is that, on many occasions, the planet as it is observed from the Earth is located inside the seeing disc of the star, obfuscating the signal of the planet. Even near the theoretical difraction limit of the telescope, the presence of the concentric rings of the Airy function may render the observation impossible. High resolution techniques like the ones commented in the section about visual binaries can be helpful, although they do not currently have the detction power of the transit method or the radial velocities. The development of larger telescopes, the European Extremely Large Telescope (E-ELT), for example, will increase the number of discoveries, as will the use of space telescopes like the JWST. However, there are techniques that permit the improvement of detection with the current telescopes. First, we have the use of coronographs that were originally developed for the observation of the solar corona (Lyot, 1939), as their name indicates. The classic coronography (also known as Lyot coronography) consists in incorporating a mask on the telescope axis in the way that it blocks the light from the central object, improving the SNR of the bodies located off-axis. In the case of exoplanets, the coronography alone is not enough to achieve their detection but when it is combined with adaptive optics (Malbet, 1996; Sivaramakrishnan et al., 2001), it has yielded positive results. There are many other designs of coronographs that are based not only on physical masks but also on the use of destructive self-interference by means of phase shifts (Guyon et al., 2006). There are also methods to reduce the noise of the image that can be applied along with the coronography. We have, for example, Angular Differential Imaging (ADI) which is based on taking a large quantity of short-exposure images (Marois et al., 2006). The device to correct for the paralactic angle is turned off in the telescopes that have it in order to improve the quality of the image. For each image, a synthetic PSF of the star is built from the other observations taking into account that they must be aligned to correct the rotation of the field of view. The synthetic PSF is substracted from the image and that permits the elimination of the quasi-static speckles, reducing the noise level in ∼5. Other technique is Simultaneous Spectral Differential Imaging (SSDI) that consists of taking two simultaneous images with two different narrow-band filters centered on close wavelengths (Racine et al., 1999). ADI and SSDI can be combined with other techniques to further improve the SNR, such as LOCI (Lafrenière et al., 2007), TLOCI (Marois et al., 2014), or SOSIE (Marois, Macintosh, and Véran, 2010). 52 II.2. Observation techniques Figure II.4: Planet orbiting the star, Fomalhaut (αPsA). Image: NASA, ESA, P. Kalas, J. Graham, E. Chiang, E. Kite (University of California, Berkeley), M. Clampin (NASA Goddard Space Flight Center), M. Fitzgerald (Lawrence Livermore National Laboratory), K. Stapelfeldt and J. Krist (NASA Jet Propulsion Laboratory) 53 Chapter II. Extrasolar planets Figure II.5: Outer orbit of the system Gl22. II.2.4 Astrometry We have seen that a subset of the visual binaries is the astrometric pairs. In that type of system, due to the difference in magnitude, we could only observe the light of the primary which underwent variations of its observed position because of its movement around the center of mass. This is the same case of a star with a planet around it but at a much smaller scale. If arepresents the semimajor axis of the relative orbit of the planet around the star, the semimajor axis of the orbit of the star with respect to the baricenter can be calculated from: α= Mp Mp+M∗ a(II.5) For example, if we consider a planet similar to Jupiter that is at a distance of 5 a.u. from a star with the mass of the Sun and at 10 pc. from the observer, the calculated value is α≃5·10−4 arcseconds. Obviously, this technique is more suitable for massive planets around low mass stars as is the case of the Gliese 22 system (Docobo et al., 2008a), a hierarchical triple system of red dwarfs. This system showed, in the outermost orbit, an astrometric oscillation caused by the possible presence of an object of a mass around 16 MJup. Therefore, it would be in the limit between giant planets and brown dwarfs (see image II.5). For less massive planets, or planets closer to the star, we would have to achieve a precision of micro-arcseconds (µas) which is even lower than the limits of Gaia. At this scale, we need to take into account relativistic corrections of the light path caused by the mass of the Sun. 54 II.2. Observation techniques Even phenomena on the surface of the star such as star spots may produce variations in the position of the photocenter of that magnitude. In order to obtain an astrometric measurement, we need to know other data that affect the position of the photocenter at a larger scale, specifically, the proper motion and the parallax of the system. If those are well determined with a large number of observations, the Campbell elements, P,T,e,a,I,Ω, and ωcan be calculated by observing the variation of the equatorial coordinates of the photocenter. Concretely, we can obtain ∆αcos δand ∆δ, with αand δ being the equatorial coordinates of the star. For a given instant, t, it verifies that (Wright and Howard, 2009): ∆α(t)cos δ=[BX +GY]+∆α0cos δ+πΠα,t+µα(t−t0)(II.6) ∆δ(t)=[AX +FY]+∆δ0+πΠδ,t+µδ(t−t0)(II.7) Here, the Thiele-Innes constants A,B,F, and Gare: A=a(cos ωcos Ω−sin ωsin Ωcos i)(II.8) B=a(cos ωsin Ω+sin ωcos Ωcos i)(II.9) F=a(−sin ωcos Ω−cos ωsin Ωcos i)(II.10) G=a(−sin ωsin Ω+cos ωcos Ωcos i),(II.11) whereas Xand Yare calculated from: X=cos E(t)−e(II.12) Y=p1−e2sin E(t)(II.13) where E(t)is the eccentric anomaly in the instant t.πin the equations II.6 and II.7 represents the parallax, whereas Πα,tand Πδ,tstand for the orthogonal displacements in αand δdue to the parallax, respectively. µα(t−t0)and µδ(t−t0)account for the displacement caused by the proper motion and ∆α0cosδand ∆δ0indicate the difference in the coordinates of the star in t0with respect to the nominal coordinates. It has to be noted that the equations, II.6 and II.7, permit the modelling of multiple planet systems by including the terms [BX +GY]and [AX +FY], for each planet. 55 Chapter II. Extrasolar planets II.2.5 Timing We call “timing” the discovery of an exoplanet due to the variation that it produces in other periodical phenomenon such as another exoplanet previously detected by means of transits, an eclipsing binary, or periodical pulsations of the star. Due to the orbit that the star describes with respect to the center of mass of the star-planet system, the path that the light traverses is longer when it is in the farthest part of the orbit. This causes a delay in the observed moment of the periodical phenomenon or an advance when the star is at the nearest point. Numerically, the amplitude of this variation is given by the following expression: τ=1 cMpa sinIM∗(II.14) where crepresents the speed of light in the vacuum, Mpand M∗indicate the masses of the planet and the star, respectively, and aand Istand for the semimajor axis and the inclination of the orbit of the planet relative to the star. Obviously, this time difference will be very small, therefore we need highly accurate values of the perturbed phenomenon in order to discover exoplanets with this technique. II.2.6 Pulsar Timing One type of periodical phenomena in which we can observe the variations due to the presence of exoplanets is the pulsars. These objects are neutron stars, i. e., the remains of the nucleus of a massive star (> 9 M) that exploded as a supernova. Neutron stars have a typical mass of approximately 1.4 Mand the gravitational attraction forces the matter to condense to a few km. of diameter. Due to the gravitational pressure, the matter inside these objects degenerates and the protons and the electrons combine, forming a superfluid of neutrons. The conservation of the angular moment causes the rotation of these objects to speed up when the radius diminishes, reaching periods of miliseconds. Neutron stars develop strong magnetic fields that produce the emission of radiation (γrays, X rays, visible, and radio) from the magnetic poles. Usually, the magnetic and the rotation axes don’t match and consequently, when the neutron star rotates, the beams of radiation sweep an area of the sky. If that area contains the Earth, periodic pulses of radiation will reach our planet and the neutron star is known as a pulsar. These objects were predicted by Baade and Zwicky in 1933 (Baade and Zwicky, 1934) although their prediction did not receive much attention in that time. It was in 1968 when the first radio pulses from one of these objects were detected in the Mullard Radio Astronomy Observatory by Jocelyn Bell and Antony Hewish (Hewish et al., 1968). The rotational period of the pulsars is highly stable although it decays very slowly due to the radiation and the 56 II.2. Observation techniques Figure II.6: Variations in the transits of KOI-872b. Image: included in Nesvorný et al. (2012) 57 Chapter II. Extrasolar planets magnetically accelerated particles that it emits. This decay follows the formula (Burke and Graham-Smith, 2009): d dt 1 2Iω2=2 3M2 ⊥ω4c−3(II.15) Here ωrepresents the angular velocity, Iindicates themoment of inertia, and M⊥stands for the component of the dipolar magnetic moment that is orthogonal to the rotation axis. The diminution of the rotational velocity is very small with tiny occasional increments that are called “glitches”. After a long time, when the period of rotation reaches several seconds, the spin of the magnetic field is no longer able to feed the radio emission and the pulsar is not observable anymore. If the period of ration of the pulsar shows faster periodical changes, these may indicate the presence of other bodies in the system, for example, planets (see image II.7). The regularity of this movement and its short period make these exoplanets easy to detect even if their mass is as low as 0.020 M⊕. This is the case of Draugr (PSR B1257+12 b) which belongs to the system in which the first confirmed exoplanets were discovered (Wolszczan, 1994). The difficulties arise from the relatively small number of detected pulsars as well as the fact that the planetary systems in the original star are likely destroyed during the supernova phase that produces the pulsar, therefore the exoplanets must form after the explosion with the remaining material that surrounds the nucleus. II.2.7 Microlensing The Theory of General Relativity predicts that a massive object curves the space-time around it and, consequently, if there is another luminous object behind it, the path of the light rays emitted by the most distant object will be distorted thereby yielding the phenomenon of the “gravitational lens”. It is called “macrolensing” when we can resolve the distorted image of the farthest object and deformed ghost images or arcs appear around the closest object. In the case of “microlensing”, the image cannot be resolved, and it is detected as an increase in the brightness of the distant object due to the light rays being focused toward the observer. If the closest object is a star, the presence of an exoplanet around it will cause a secondary peak of smaller magnitude in the light curve observed during the microlensing, thus allowing its detection (see image II.8). This technique has two disadvantages. First, the planet can only be detected during the microlensing which is a one-time event and other methods are needed for a follow-up study. Moreover, the probability of a microlensing event is low, therefore the microlensing observation programs are focused on the observation of the regions of the Galaxy toward the core where the concentration of stars is higher. On the other hand, the observation of microlensing 58 II.2. Observation techniques Figure II.7: Variations in the period of rotation of the pulsar PSR B1257+12. Image: A. Wolszczan, D. Frail Figure II.8: Schematic of the detection of an exoplanet by means of microlensing. Image: NASA, ESA, and A. Feild (STScI) 59 Chapter II. Extrasolar planets is less limited by the distance than other techniques and it has been able to detect planets at distances up to 6500 pc. II.3 Dynamics of extrasolar planets From a mathematical point of view, there are also many problems to study. The dynamics of exoplanets, both in single stars and in double and multiple systems, has opened an extraordinary field of research in Astrodynamics. Among others, there are questions about the study of co-orbital motions, resonances and their influence in the variation of the eccentricity, the stability of orbits, and orbital and rotational perturbations. They have attracted the attention of renowned specialists in Celestial Mechanics, some of them before the discovery of the first exoplanet. Examples of these works are: Beaugé, Ferraz-Mello, and Michtchenko (2003), Beaugé, Michtchenko, and Ferraz-Mello (2006), Callegari, Michtchenko, and Ferraz-Mello (2004), Callegari, Ferraz-Mello, and Michtchenko (2006), Dvorak (1980,1984,2006), Dvorak, Froeschle, and Froeschle (1989), Dvorak and Henrard (1988), Dvorak and Süli (2002), Dvorak et al. (2010), Ferraz-Mello, Beaugé, and Michtchenko (2003), Ferraz-Mello and Michtchenko (2002), Ferraz-Mello (2014,2015), Funk et al. (2004,2011), Funk, Dvorak, and Schwarz (2013), Funk et al. (2009), Giuppone et al. (2010), Hagel and Dvorak (1987), Kubala, Black, and Szebehely (1993), Michtchenko, Beaugé, and Ferraz-Mello (2006,2008a,b), Michtchenko, Ferraz-Mello, and Beaugé (2006), Nesvorný et al. (2002), Pilat-Lohinger and Dvorak (2002), Pilat-Lohinger, Funk, and Dvorak (2003), Rabl and Dvorak (1988), Rodríguez et al. (2011), Szebehely (1979,1980), Szebehely, Black, and Kubala (1995), Szebehely and McKenzie (1977,1981), Szebehely (1967), and Tadeu dos Santos et al. (2015), among others. As a first approximation for the study of the dynamics of planetary systems in single or multiple stars, we can suppose that the movement of the star or stars of the system is not affected by the gravitational force of the surrounding planets. In 1984, A. L. Whipple and V. Szebehely (Whipple and Szebehely, 1984) published an article in which they described the restricted n+νbody problem (n bodies, not necessarily point-like, each one of them with a considerable mass, that are called primaries and νinfinitesimal masses which do not affect the primaries) interacting among them by means of arbitrary forces (not only gravity). For n=ν=1and point masses, we have the two-body problem whereas the case with n=2,ν=1 is the classical restricted three-body problem which can occur, for instance, in a binary star with a planet. The values, n=1and ν=2, would yield a star with two planets, and n=2, ν=2create the so-called restricted 2+2-body problem, for which numerous equilibrium solutions have been obtained (Whipple, 1984), both around the colineal Lagrangian point and the triangular ones. In the cases of n=2,ν=1and n=2,ν=2the motion of the primaries is usually considered to be circular. 60 II.3. Dynamics of extrasolar planets II.3.1 Dynamics of exoplanets in binary sistems In a planetary system around a single star, there is ample tolerance for the existence of stable orbits. It suffices that the planets are sufficiently separated in order to avoid that their mutual gravitational attractions cause large perturbations in their quasi-Keplerian trajectories. Of course, they also have to verify Kepler’s Third Law: M∗+MP1 M∗+MP2 = a3 1/P2 1 a3 2/P2 2 (II.16) between each pair of planetary orbits. We use the following formalism: •M∗, the mass of the star. •MPi, the mass of each planet (i=1,2). •ai, the semimajor axis of the planet with mass Mi(i=1,2). •Pithe corresponding orbital period. (i=1,2). When one planet or more coexist in a double star, things are different. We know from the study of the restricted three-body problem (Battin, 1987; Danby, 1988; Szebehely, 1967) that stable orbits can only exist when the infinitesimal body either moves far away from the primaries or close to one of them or it is located in the Lagrangian points, L4 and L5. The extension of the areas of permitted stable motion depend on the masses of the primaries. According to the criterium of R. Dvorak (Dvorak, 1984), The following types of stable orbits arise: •planet-type orbits (P-type): when the exoplanet moves around both components of the binary system. •satellite-type orbits (S-type): this is the case when the planet orbits one of the stars. •libation-type orbits (L-type): if the planet is located in one of the lagrangian points. The first case would correspond to a close binary, for example a spectroscopic binary with a planet moving around the center of mass of the stars (circumbinary planet). The second situation happens when the planet moves close to one of the components of a double star, usually a wide system such as a visual binary. At the moment, the planets detected in binaries are found in P-type orbits in 23 occasions, and in S-type orbits in 715 cases (Wright et al., 2011). The low number of P-type orbits is due mainly to the technical difficulties for their detection by means of both transits and radial velocities. L-type orbits are expected to be very rare because, in order to be stable, the mass ratio between the components of the binary must be: 61 Chapter II. Extrasolar planets Figure II.11: Case 2. Variations of the orbital elements of the giant planet caused by the satellite 68 II.4. Exosatellites Figure II.12: Comparison between the radial velocities in both scenarios 69 Chapter II. Extrasolar planets Figure II.13: Radial velocity curves in both scenarios the second planet and a wide oscillation in Case 2 due to the perturbation of the periastron, allowing us to discriminate between them (Figure II.13). II.4.5 Variations in the inclination Now we change the inclination of the second planet and the satellite from 0◦to 10◦with 1◦ steps to see if it makes any difference. We can see that an oscillation in the inclination of the planet appears in both scenarios, with higher amplitude and frequency in Case 2 as shown in Figures II.14 and II.15. It also causes an oscillation in the angle of the node in the case of the satellite and a slight secular increase in Case 1, at least within the short term integration. These effects increase with the mutual inclination but the effect in the radial velocities is so small that it is not noticeable (Figure II.16). 70 II.4. Exosatellites Figure II.14: Case 1. Variations of the orbital elements of the giant planet caused by the second planet with a mutual inclination of 5◦ 71 Chapter II. Extrasolar planets Figure II.15: Case 2. Variations of the orbital elements of the giant planet caused by the satellite with a mutual inclination of 5◦ 72 II.4. Exosatellites Figure II.16: Radial velocity curves in both scenarios with a mutual inclination of 5◦ 73 Chapter II. Extrasolar planets Higher mutual inclinations might cause stronger variations in the inclination of the orbit of the planet but, for planetary systems with large satellites, coplanarity or small inclinations seem to be more likely scenarios. In addition to this, high inclinations may cause instability due to the Lidov-Kozai effect (Kozai, 1962; Lidov, 1962). II.4.6 Variations in the eccentricity Now we will study the effect of the eccentricity. We give values from 0.0 to 0.5 (with a 0.1 step) for the eccentricity of both the satellite and the second planet. In Case 1, the increase in the eccentricity causes a boost in the oscillation of the semimajor axis of the first planet during the closest encounter with the Neptune-size planet and an attenuation in the most distant part of the orbit (Figures II.17). The other orbital elements seem unaffected. The radial velocity of the planet doesn’t show significant changes as the difference is too small. However, the change in the eccentricity of the satellite in Case 2 produces two measurable effects on the semimajor axis perturbation. The first is an increase in the amplitude of the oscillation. The second generates a long-term perturbation superimposed over the short-term. A slight secular increase in the argument of the periastron appears which causes a boost in the observed oscillation effect as shown in Figures II.18,II.19, and II.20. II.4.7 Conclusions We have shown that it is possible to detect the presence of a satellite through the perturbations in the radial velocity of the host planet. These perturbations appear mainly as a variation in the periastron passage. The presence of a second planet (Case 1) causes a slow secular precession in the periastron passage while the satellite (Case 2) produces a wide oscillation in that passage. This allows us to distinguish between both scenarios provided that we have observations spanning several periods. Regarding the orbital parameters, the main contributor to the perturbation seems to be the eccentricity whereas the mutual inclination has little effect, at least for low values. The inclination and the angle of the node do not change in the coplanar cases, as expected. Whenweconsider non-zeromutual inclination, we obtaina smalloscillation intheinclination. This oscillation is faster and wider in Case 2 but, nevertheless, it is insufficient to cause a measurable effect in the radial velocity. The angle of the node suffers a precession effect and an oscillation in Cases 1 and 2, respectively. A small eccentricity appears in the initially circular orbits. Both the semimajor axis and the eccentricity suffer oscillations due to the perturbation of the third body, again with greater 74 II.4. Exosatellites Figure II.17: Case 1. Variations of the orbital elements of the giant planet caused by the second planet with an orbit of eccentricity 0.5 75 Chapter II. Extrasolar planets Figure II.18: Case 2. Short-term variations of the orbital elements of the giant planet caused by the satellite with an orbit of eccentricity 0.5 76 II.4. Exosatellites Figure II.19: Case 2. Mid-term variations of the orbital elements of the giant planet caused by the satellite with an orbit of eccentricity 0.5 77 Chapter III Published articles This dissertation has been carried out in the modality of the compilation of articles published in research journals according to the normative of the Escola de Doutoramento Internacional of USC and the RD 99/2011. In this chapter, the published articles are presented, three of them in the journal, Monthly Notices of the Royal Astronomical Society (MNRAS) and another in Astronomy Letters (Astron. Lett.), both indexed in the Journal Citation Report (JCR). All were carried out under the supervision of the Director of this dissertation. Those published in MNRAS concern the study of binaries and those in Astron. Lett. Treat exoplanets and exosatellites. MNRAS is a high impact journal published by the Oxford University Press. They are consistently in the first quartile of JCR as it was in the years in which the articles were published, 2014 (position 12 of 60), 2017 (12/66), and 2018 (15/69). Astron. Lett. is a prestigious Russian journal of Astronomy, the international version of Pis’ma v Astronomicheskii Zhurnal, distributed by Springer. It is also indexed in the JCR. In 2014, the year of publication of the article, it was in the third quartile (39/60). The first two articles are related to the visit to Cambridge and the associated scientific collaboration with Professor Roger F. Griffin. They deal with the study of spectro-interferometric binaries in such a way that the spectroscopic orbit as well as the visual orbit are calculated. In the first article, Elliot P. Horch of Southern Connecticut State University, is one of the authors. He contributed the new interferometric measurements obtained with the 3.5 telescope of the WIYN Observatory (Wisconsin-Indiana-Yale-NOAO) that were used for the calculation of the visual orbit. As we have seen throughout this Memory, the study of this type of binaries is of special importance. On one hand, more precise orbit calculation is permitted which is crucial when studying possible existing planetary systems. On the other hand, the orbital parallax can be determined (which yields an independent measurement of the distance and permits the calculation of the luminosity of the stars) and the individual masses of the components. The masses do not only affect the dynamic study of the planetary systems but, along with the luminosity, they influence the evolutionary development of the stars as can be seen in the second article in which is included an evolutionary study of the binary using the PARSEC models (http://stev.oapd.inaf. 85 Chapter III. Published articles it/cgi-bin/cmd). The masses and the evolutionary state determine the effective temperature and, therefore, the habitability zones around the stars. The url adrresses of the articles are https://academic.oup.com/mnras/article/444/4/3641/1027288 and https://academic.oup.com/mnras/article/469/1/1096/3605384. In the third article, a novel methodology for the study of visual systems in which the arc of observation is small which can determine precise orbits and obtain reliable physical parameters. As indicated in the article, this is of special interest in the case of pre-main sequence stars (PMS) for which there are no good calibrations of mass. These systems can present prominent protoplanetary discs. For that reason, their study is fundamental within the frame of the evolution of planetary systems. The article can be found in https: //academic.oup.com/mnras/article/476/2/2792/4840252. The fourth article is a dynamic study of a model of a planetary system within a binary that consists of two stars with a planet orbiting one of them (orbital type S) and a satellite rotating around the planet. In the first place, the possible four body planetary systems are reviewed using the step formulation developed by A. Abad in his dissertation and classifying them according to the hierarchy that they present. Then the selected model is formulated mathematically. To that end, the Hamiltonian formulation of the system of equations is developed and resolved in an analytic manner by means of the theory of perturbations. First, a change in the Jacobi coordinates is made in order to develop the Hamiltonian in series as a function of three small parameters that are then reduced to two in order to accomplish the necessary truncation. Then the biparametric method of Hori is applied in order to resolve the system. The following link https://link.springer.com/article/ 10.1134%2FS1063773714110012 contains a copy of the article. 86 BIBLIOGRAPHY Abad, A., R. Barrio, F. Blesa, and M. Rodríguez (2011). “TIDES tutorial: Integrating ODEs by using the Taylor Series Method”. 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