Appendix a Radial Velocity

Home , 14 Herculis, Chi Herculis, HD 179949 b, HD 217107, HD 217107 c, HD 73256

UNIVERSIDAD DE CHILE FACULTAD DE CIENCIAS FÍSICAS Y MATEMÁTICAS DEPARTAMENTO DE ASTRONOMÍA

A HIGH RESOLUTION SPECTROSCOPIC SEARCH FOR THE THERMAL EMISSION OF THE EXTRASOLAR PLANET HD 217107 b

TESIS PARA OPTAR AL GRADO DE MAGÍSTER EN CIENCIAS, MENCIÓN ASTRONOMÍA

PATRICIO ERNESTO CUBILLOS VALLEJOS

PROFESOR GUÍA: PATRICIO ROJO RÜBKE

MIEMBROS DE LA COMISIÓN: María Teresa Ruiz González Diego Mardones Pérez John R. Barnes

SANTIAGO DE CHILE MAYO 2011 Resumen

En este trabajo hemos retomado y afinado un método de correlación para buscar directamente, en alta resolución, el espectro de planetas extrasolares sin tránsito. Nuestro objetivo principal es caracterizar las propiedades físicas de estos objetos, específicamente la inclinación de su órbita, su masa y la proporción de los flujos entre el planeta y su estrella.

Esta técnica se vale del efecto Doppler causado por el movimiento orbital del planeta y la estrella en torno al centro de masa del sistema. Para observaciones lo suﬁcientemente extensas, el espectro del planeta se va a desplazar con respecto al de la estrella lo suﬁciente para que sea detectable en observaciones espectroscópicas de alta resolución. Alineando y sumando los espectros de cada noche construimos un modelo del espectro estelar. Este es substraído a cada espectro, dejando un espectro residual compuesto por la emisión del planeta inmerso en ruido.

Dada su baja intensidad, el espectro planetario no es directamente discernible del ruido. Por lo tanto, buscamos la emisión planetaria a través de una función de correlación entre nuestros espectros residuales y modelos de la emisión termal de la atmósfera del planeta. Evaluando para distintos valores de la inclinación de la órbita del modelo, obtenemos una curva de correlación. El valor de esta curva debe ser máximo cuando la inclinación coincida con la inclinación del sistema. Para calcular el valor de la proporción de los ﬂujos entre el planeta y su estrella, recreamos observaciones inyectando espectros sintéticos del planeta con parámetros dados de inclinación y proporción de ﬂujos. Luego, mediante un test de χ2 entre las curvas de correlación, estimamos los parámetros que mejor se ajustan a nuestro resultado.

Presentamos resultados en el sistema planetario HD 217107, observado con el espectrógrafo de alta res- olución Phoenix, en una longitud de onda de 2.14 µm. Como resulatado, no logramos detectar el planeta con los datos disponibles, aunque determinamos límites superiores para su emisión termal, siendo menor a 5 10−3 veces la emisión de su estrella, con 3–σ de certeza. × Además, exploramos el escenario ideal de observación para proyectos futuros, y describimos una estrategia óptima de observación y selección de candidatos que maximice las probabilidades de detección. Finalmente, simulando observaciones realistas para Phoenix, generamos datos sintéticos de observaciones de otros candidatos para demostrar las ventajas de usar nuestra estrategia de observación. Calculamos límites de detectabilidad para este instrumento en los planetas simulados. Nuestra conclusion es que si nos aproxi- mamos al límite de ruido de fotones, si es posible detectar planetas extrasolares con este método. Summary

We have revisited and tuned a correlation method to directly search for the high-resolution signature of non-transiting extrasolar planets. The main objective of this work is to characterize the physical properties of non transiting extrasolar planets, aiming to obtain the inclination of the orbit, the mass of the planet, and the planet-to-star ﬂux ratio.

The technique is based in the out of phase Doppler-shift effect caused by the wobble of the star and planet around the center of mass of the system. For long enough observing runs, the spectral signals will shift with respect to each other, and thus will be detectable with high resolution spectroscopic observations. By aligning and adding the spectra of each night we construct a stellar template, which we subtract to the data, leaving a residual spectrum consisting of the planetary signal embedded in noise.

The planetary spectrum is not readily detectable due to its much fainter signal. Therefore, we search for the planet calculating the correlation between the residual data and thermal emission models of the planet’s atmosphere, assigning different values to the inclination of the orbit of the models, expecting a peak in the correlation when we match the real value of the inclination. To asses the value of the planet-to-star flux ratio, we reproduced the observations using synthetic spectra, injecting a scaled and shifted planetary spectrum according to given flux ratios and inclinations. Then, we determine the best fitting parameters through a χ2 minimization between the data and the synthetic results.

We present the results of this technique on the planetary system HD 217107, observed with the high resolution spectrograph Phoenix, at 2.14 µm. We could not detect the planet with our current data, but we present an upper limit to its thermal emission determined with a Monte Carlo Bootstrap method. With a conﬁdence level of 3–σ we constrain the HD 217107 planet-to-star ﬂux ratio to be no more than 5 10−3. × Furthermore, we explore the ideal observing scenario for future projects, and outline an optimized observational and selection strategy to increase future probabilities of success by considering the best conditions to observe and the best candidates using this method.

Finally, using realistic data sets for the Phoenix instrument we carried out simulations on other planet candidates to demonstrate the improvements achieved when we use our optimal observing strategy. Detectability limit of the method using this instrument and the simulated planets are given. We conclude that with our same number of observations, it is possible to detect extrasolar planets with planet-to-star ﬂux ratios of the order of 10−4 if we approach to the photon noise limit. Agradecimientos

Primero quiero agradecer a Patricio Rojo, mi profesor guía, por toda su ayuda y compromiso con este trabajo. Tanto su ayuda, guía, y experiencia, como constante apoyo anímico y paciencia fueron fundamentales para llevar a cabo este trabajo, y de gran ayuda para mi futuro desarrollo como investigador.

A mi familia le agradezco toda la conﬁanza depositada en mi, dandome libre oportunidad de tomar mis propias decisiones, sin cuestionar, al momento de deﬁnir mi futuro. Gracias a ellos es que he tenido la oportunidad de llegar hasta donde estoy.

También quiero agraceder especialmente a mi polola, Elisa Carrillo, quien estuvo constantemente apoyan- dome y acompañandome hacia el ﬁnal del proceso, gracias por ser tan espectacular conmigo, por saber que decir para darme ánimos y poder terminar esta tesis. No podría estar mas feliz de tenerte a mi lado.

Merecen mención también los profesores de la Universidad de Chile, tanto de Licenciatura como de Magíster, gracias por su disposición. De ellos adquirido un enorme conocimiento tanto en astronomía como en las ciencias relacionadas. El nivel en dominio de la materia y capacidad de enseñar de la mayoría de ellos ha sido de lo mas alto, siendo un ejemplo a seguir.

Finalmente le agradezco a mis compañeros y amigos que he encotrado en mi recorrido a lo largo de estos años como estudiante de la Universidad de Chile, haciendo que los buenos momentos hayan sido realmente espectaculares y dando apoyo en los momentos mas críticos. Fue un agrado realmente compartir aquellos años de Licenciatura con Luis Gutiérrez, Eduardo Godoi, Ricardo Ordenes, entre muchos otros más. Tam- bién a todos mis compañeros en Cerro Calán, Cinthya Herrera, Sergio Hoyer, Maria Fernanda Durán, Matias Jones, Viviana Guzmán, Felipe Murgas, Andrés Guzmán, Matias Vidal, entre tantos otros, han sido la mejor compañía tanto como para pasar un buen momento como de ayuda en mis cursos y trabajo. Contents

1 Introduction 1

2 The Planetary System HD 217107 3

2.1 BackgroundInformation ...... 3

2.2 Radial Velocity ...... 4

2.3 Flux Estimate ...... 7

3 Observations and Data Reduction 11

3.1 TheData...... 11

3.2 Reduction...... 12

3.3 Wavelength calibration ...... 14

4 Data Analysis and Results 17

4.1 Stellar Light Removal ...... 17

4.2 Planet’s Atmospheric Model ...... 20

4.3 Correlation ...... 23

4.4 DataResults...... 25

4.5 Planet-to-Star Flux Ratio Fitting ...... 25

4.6 False Alarm Probability ...... 29

4.7 Data Results Discussion ...... 29

5 The Optimal Acquisition of Data 30

5.1 TargetSelection...... 30

5.2 Optimal Observing Strategy ...... 31

I 5.3 HD 179949 b Simulation ...... 33

5.4 Tau Boo b and HD 73256 b Simulations ...... 38

6 Discussion and Conclusions 42

Appendices 44

A Radial velocity 45

B Error Propagation 47

B.1 Equilibrium Temperature ...... 47

B.2 DataReduction ...... 47

C Observing Log of HD 179949 50

II List of Tables

2.1 HD217107Parameters ...... 3

3.1 Gemini South Telescope Characteristics ...... 11

3.2 Observinglog...... 12

3.3 InfraredLinesCatalog ...... 14

4.1 Theoretical Atmospheric Models...... 22

5.1 Favorable targets for Gemini South ...... 31

5.2 Parameters ...... 33

C.1 ObservationLog ...... 50

III List of Figures

2.1 Detection Methods...... 4

2.2 HD 217107’s radial velocity ...... 6

2.3 HD 217107’s Spectral irradiance ...... 9

2.4 HD 217107 planet to star ﬂux ratios as black bodies ...... 10

3.1 Rawframes ...... 13

3.2 Wavelength Calibration Lamp ...... 14

3.3 Telluric Wavelength Calibration ...... 15

3.4 Telluric wavelength calibration ﬁt ...... 16

4.1 30 km s−1 PlanetaryBlurring ...... 18

4.2 3 km s−1 PlanetaryBlurring...... 19

4.3 Planet Velocity Spans ...... 20

4.4 Theoretical blurring ...... 21

4.5 Planet-Star Separation ...... 23

4.6 Model at inﬁnite resolution...... 24

4.7 Model at instrumental resolution...... 24

4.8 HD 217107 b Correlation Results ...... 26

4.9 Synthetic correlations curves ...... 28

4.10 Synthetic correlations comparison...... 28

5.1 HD 179949 Observability window...... 32

5.2 HD 179949 radial velocity curve...... 34

5.3 SpectralIrradiances...... 35

5.4 Planet-to-star ﬂux ratios as black bodies...... 36

IV 5.5 HD 179949 b’s search simulation...... 37

5.6 Tau Boo Observability window...... 38

5.7 Tau Boo b’s search simulation...... 39

5.8 HD 73256 Observability window...... 40

5.9 HD 73256’s planet search simulation ...... 41

V Chapter 1

Introduction

After several years of search, the discovery of the first extrasolar giant planet in a close orbit (Hot-Jupiters) around the main sequence star 51 Pegasi [Mayor and Queloz, 1995] marked the beginning of a new research field in planetary sciences. Since then, the study of extrasolar planets has became one of the most developing branches in astronomy. As has constantly happened along the history, this discovery brings more questions than answers. 51 Pegasi b, a Jupiter-Mass planet located at only 0.05 AU from his host star defied the existing formation theories [for example, Pollack, 1984], and broke the scheme of planetary systems with less massive rocky planets orbiting near the star and massive gaseous planet in outer orbits as in our own Solar System.

It is thanks to the refinement of the radial velocity measurement method, the most successful technique to discover extrasolar planets today, that the scenario has changed dramatically, knowing now hundreds of other worlds outside the solar System. This technique, measures periodical changes in a star’s radial velocity, as the star and the planet orbit about their common center of mass, the star motion is detectable through the Doppler Effect. Along with other search methods like gravitational microlensing surveys, transit light curve measurements, or pulsar timing monitoring, over 400 extrasolar planets have been discovered so far. Their characterization, then, started to take place, most of these studies are carried out at optical and infra-red wavelengths since it is there where the reflected light and the thermal emission reaches its maximum flux, respectively. The discovery of transiting planets [Charbonneau et al., 2000, Henry et al., 2000] allowed astronomers to constrain new physical parameters such as the radii or the masses of these planets, not measured by the radial velocity method alone. This orbital configuration permits to directly observe the planet, by measuring the dip in a light curve when the planet crosses in front (a transit) or behind the star (a secondary eclipse) blocking part of the light. It is on these systems that, in the last years, the planetary atmospheres characterization has achieved the most exciting advances. Space telescopes have been able to measure secondary eclipses through the use of both spectroscopy and broadband photometry. To mention some examples, it has been possible to identify molecules by fitting theoretical models of the planetary atmospheres, like water absorption in HD 189733b [Tinetti et al., 2007], as the most likely cause of the variation of the planetary radius for the different bands using Spitzer Space Telescope observations [Werner et al., 2004]. Later, for this same planet, Swain et al. [2008] claimed the presence of methane in the atmosphere and confirmed the existence of water using data from Hubble Space Telescope. Another interesting example is the measurement of the variation of the thermal emission with orbital phase as the fraction of the day-side surface we see is changing, for HD 189733b at 8 µm distinguishing also the transit and secondary eclipse [Knutson et al., 2007]. Just recently there have been successful results of measurements of secondary eclipses of transiting extrasolar planets from ground base observations [e.g., Croll et al., 2010].

Although, there have been great improvements in characterizing the composition of transiting Hot-Jupiters, they represent less than 20% of the population of the known radial velocity extrasolar planets. For the rest of

1 the non-transiting extrasolar planets, the only way to be able to constrain physical parameters or characterize their atmospheres, is through the direct detection of their light, rather than observing the radial reflex motion induced in the star. The very small flux ratios between the planet and their host stars, makes a direct detection a very challenging goal. From secondary eclipse observations from Spitzer we know that planet-to-star flux ratios can be as high as 2.5 10−3 (measured at wavelengths between 3.6 and 24 µm), at 2.4 µm the expected flux should be less than these× values. From ground based telescopes, several authors have tried to observe the high resolution spectra of the planetary systems, separating the planetary and stellar spectra given their Doppler-shift wobble. In the optical, Collier Cameron et al. [1999] performed a first attempt of the direct detection of the Doppler-shifted signature of starlight reflected from the giant exoplanet orbiting Tau Boötis implementing a least-squares deconvolution technique, he set an upper limit to the albedo and radius, later applied the same technique in the search of υ Andromeda b [Collier Cameron et al., 2002]. In the recent years, Rodler et al. [2008] searched for the visible spectra, first in HD 75289Ab, and later in Tau-Boötis b [Rodler et al., 2010] by means of data synthesis (aided by theoretical models of the reflected spectra), also finding upper limits for their albedos. While in the near-infrared, there is also an extensive list of attempts to detect the thermal emission signature of Hot-Jupiters from ground base telescopes. Wiedemann et al. [2001] searched for methane in the spectrum of Tau-Boötis using a cross correlation method. Lucas and Roche [2002] searched, this time H2O absorption features, using low spectral resolution observation of several stars with planetary companions. Later, using a least-square deconvolution method, Barnes et al. [2007a,b, 2008], constrained the upper limit for the emitted flux of HD 189733 b, HD 75289 b and HD 179949 b. Lastly Barnes et al. [2010] searched for H2O and carbon bearing molecules in the atmosphere of HD 189733 b, finding again upper limits in the planet to star flux ratios. More recently, [Janson et al., 2010] presented the first spectrum of the angularly resolved image of an extrasolar planet. Nonetheless, those resolved systems present a very small fraction of known extrasolar planets.

In this work, we present an effort to constrain new physical parameters of the non-transiting Hot-Jupiter HD 217107 b using high resolution spectra, and models of its atmospheric spectrum in the infrared. With positive results this method could provide new information of non-transiting extrasolar planets and improve the calibration of high-resolution atmospheric models. At the same time, it will also validate a method that could potentially be used to characterize atmospheres of non-transiting planets. Direct detection of extrasolar planets’ emitted or reﬂected spectra, coupled with broadband photometry, would provide complementary information on its characteristics, such as its temperature, chemical composition, and the presence of chemical tracers associated with life while improving conﬁdence in the models.

In Chapter 2 we review the available information of the planetary system HD 217107 and its physical properties. In Chapter 3 we describe the observations, and the reduction and calibration of the data. In Chap- ter 4 we detail the method used to separate the planetary of the stellar spectrum, extract the planetary signal and search the Doppler-shifted signature of the planet, we also describe the theoretical planet atmospheric spectrum, and present the results of our observations of HD 217107. In Chapter 5 we discuss the sensitivity of the method and a strategy of the ideal data acquisition situation, then show simulations to illustrate the improvements that can be achieved. And ﬁnally, in Chapter 6 we summarize the main conclusions of this work. In addition, in Appendix A take an in-depth analysis of the radial velocity method equations, mentioned in Chapter 2. In Appendix B we detail the error propagation formulas used throughout this work. And in Appendix C we complement the information of our simulations with an observing log of the planetary system HD 179949.

2 Chapter 2

The Planetary System HD 217107

2.1 Background Information

HD 217107 (also HR 8734 or HIP 113421) is a main sequence star similar to the Sun in mass, radius, and effective temperature, its spectral type, G8 IV, indicates that it is starting to evolve into the red giant phase. Table 2.1 summarizes the properties of this planetary system.

Table 2.1: HD 217107 Parameters Parameter Value References Star: Spectral type G8 IV W07

Te f f (K) 5 646 26 W07 ± K (mag) 4.536 0.021 C03 ± d (pc) 19.72 0.30 P97 ± Ms (M⊙) 1.02 0.05 S04 ± 0.04 Rs (R⊙) 1.08 0.03 T07 −1 ± Ks (ms ) 140.6 0.7 W07 −1 ± vg (kms ) -14.0 0.6 N04 ± [Fe/H] 0.37 0.05 W07 ± Right ascension (h:m:s) 22:58:15.54 0.000486s P97 ± Declination (deg:m:s) -02:23:43.39 0.005364s P97 ± Planet: P (days) 7.12689 0.00005 W07 ± Tp (JD) 2449998.50 0.04 W07 ± e 0.132 0.005 W07 ± mp sini (M ) 1.33 0.05 W07 Jup ± a (AU) 0.074 0.001 W07 ± ω (deg) 22.7 2.0 W07 ± Notes .— W07: Wittenmyer et al. [2007], C03: Cutri et al. [2003], P97: Perryman and ESA [1997], S04: Santos et al. [2004], T07: Takeda et al. [2007], N04: Nordström et al. [2004].

3 The presence of HD 217107 b was first reported by Fischer et al. [1999] through radial velocity measurements of the star at Lick and Keck observatories, the detection was then confirmed by Naef et al. [2001] using CORALIE data. Later, Fischer et al. [2001] identified a linear trend in the residuals of the radial velocity curve fit, and Vogt et al. [2005], hinted by a unusually large eccentricity also, postulated the existence of a second planetary companion (HD 217107 c) in an external orbit with a period of 8.5 years. The presence of this third object in the system promoted the study of this system in subsequent surveys [Butler et al., 2006, Wittenmyer et al., 2007, Wright et al., 2009], finding a period of about 11.8 years and a minimum mass of 2.6 MJup with 10% of error for HD 217107 c, although a full orbit has not been observed yet, while HD 217107 b’s orbital parameters were more precisely constrained.

2.2 Radial Velocity

Currently, the one method that stands out above all others, in terms of detection of new extrasolar planets, is the radial velocity technique (See ﬁgure 2.1). The basis of the method is relatively simple: As the planet and its host star inﬂict each other a gravitational tug, the spectra of that star is Doppler shifted as it revolves around the center of mass of the system. The high precision achieved in radial velocity measurements has made the plethora of planet detections possible. Nowadays, velocity variations down to 1 m s−1 can be achieve with HARPS (High-Accuracy Radial Velocity Planetary Searcher), the most precise Doppler-measurements instrument [Mayor et al., 2003].

Figure 2.1: Chart of the different detection methods, with 399 extrasolar planets observed (up to February 2010), the radial velocity detection method has allowed the detection of most of them.

The radial velocity curve of the star in a binary system as seen from Earth, can be expressed as (see Appendix A for details):

vs sini = γ + Ks (cos(ν + ω) + ecosω) (2.1)

4 Where γ is the radial velocity of the center of mass of the system, Ks is the radial velocity amplitude of the star, ω is the angle between the line of nodes and the line from the star to the planet at periastron (the argument of the periastron), and ν is the position angle measured from the periastron (the true anomaly).

The time dependence of the true anomaly is given implicitly, through the eccentric anomaly (E), by the set of equations:

ν 1 + e E tan = − tan (2.2) 2 r1 e 2 2π (t − T)= E − esinE (2.3) P

Equation 2.3 is known as the Kepler equation (see for example Murray and Dermott [1999]), where T is the time of periastron passage of the planet. Thus, the orbital parameters P, T, e, and ω can be determined through measurements of equation 2.1.

Then, by determining the value of these orbital parameters, it is possible to numerically solve equations 2.2 and 2.3 and obtain the radial velocity curve of an object for any given time. Figure 2.2 shows the radial velocity curve for HD 217107 due to HD 217107 b, phased over an orbit, and setting the origin in phase (φ = 0) at the time of periastron. The existence of HD 217107 c introduces a long term variation (of 11.7 years of period) in the radial velocity of the star of 37.5 m s−1 of amplitude [Wright et al., 2009], but the planet’s smaller mass (2.6 MJup) and its greater distance from the star (5.32 AU) makes the interaction with HD 217107 b of a secondary order in importance.

From these orbital parameters, we can derive other physical properties of the orbit. From the velocity amplitude of the star, written in terms of the orbital parameters:

2π as sini Ks = (2.4) P √1 − e2

Where as is the semi-major axis of the star’s orbits around the center of mass, and from Kepler’s third law:

2π 2 a3 = G(M + m ) (2.5) P s p µ ¶

with a = as + ap, and making use of the relation as Ms = ap mp; the minimum mass of the planetary companion (mp sini) can be deduced (under the approximation M mp) as: ≫

1/3 P 2/3 2 mp sini Ks Ms √1 − e (2.6) ≈ 2πG µ ¶ From a quick analysis of equation 2.6, it is not surprising why the ﬁrst extrasolar planets discovered are as massive (or more) as Jupiter and in short period orbits, we see that Ks (the observable) is proportional to mp, 2/3 it is (weakly) inversely proportional to the period of the orbit, and it is proportional to Ms .

The radial velocity of the planet, vp sini is given by its reﬂex motion around the center of mass of the system, is proportional to the star’s radial velocity curve and it is shifted half of the phase:

5 150

100

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1 ¡

0 (m s i sin

v 50

100

150

0.0 0.2 0.4 0.6 0.8 1.0 Orbital Phase

Figure 2.2: Radial velocity curve of HD 217107 vs. orbital phase. The crosses mark the observations of Wittenmyer et al. [2007] used to compute this orbital solution. The boxes over the curve indicate the coverage of our observations, the ﬁlled boxes represent the runs utilized in the analysis, while the open boxes represent the discarded runs (details in Chapter 4).

Ms vp(t)sini = − vs(t)sini sini (2.7) mp sini ×

Unfortunately, one of the disadvantages of the radial-velocity technique, is that the sine of the inclination remains unknown unless the orbit of the secondary body is observed, or it is deduced from another method (for example, transit observations), as a consequence, the true mass of the planet cannot be known.

As we have seen in this section, vs(t)sini, is directly measured by the observations, furthermore, the orbital parameters obtained from equation 2.1 lets us determine its value for any desired time, solving the Kepler equation (eq. 2.3). The approximated value of the minimum mass of the planet is also known, and it is given by the equation 2.6.

Stellar masses are computed from evolutionary tracks based on the position of the star in the Hertzsprung- Russell Diagram, by interpolating the theoretical isochrones, using the absolute magnitudes, and the effective temperature (obtained from the spectroscopy). As the search of extrasolar planets is performed only in the vicinity of the Sun, all these planet host stars have a measurable parallax, thus their absolute magnitude can be deduced from measurements of their apparent magnitude. Then, the mass of this star, Ms, is also known.

In conclusion, the radial velocity curve of the planet is a distinctive curve in time, where the only unknown parameter is the inclination of the orbit. Other than transit observations, a direct detection of the radial velocity of the planet seems to be the only way to determine the value of the inclination, which would allow to obtain the more important property, the mass of the planet.

6 2.3 Flux Estimate

By means of simple assumptions we can gain an insight of how luminous are extrasolar planets in comparison to their host stars, this will give an idea of the expected order of magnitude of the planet to star ﬂux ratio as a function of wavelength.

The total luminosity of non transiting extrasolar planets consist of three components: the directly reﬂected stellar light (which does not contribute to the heating of the planet, hence it does not interfere in the energy balance), the thermally re-emitted ﬂux of the absorbed energy from the star, and the intrinsic radiation due to the object’s contraction and cooling.

Then, assuming that the total re-emitted power by the planet (Pout) is being balanced by the incident power from the star (Pin) plus the internal production of energy (Pint), we can set the equation:

Pout = Pin + Pint (2.8)

The bond albedo, A, defined as the ratio of the power of the radiation reflected out to space to the power of the total incident radiation on the planet, determines the energy absorbed by the planet as 1 − A times the stellar flux incident on the planet (Fs), integrated over the intersected cross section:

− 2 Pin = (1 A)Fs πRp (2.9)

To estimate this stellar ﬂux, we note that the spectra of stellar objects follow, in a broad approximation, the shape of the Planck function. Also called Black-Body Radiation, this is the radiation of a cavity in thermodynamic equilibrium at a ﬁxed temperature T. Its radiation is isotropic and the power per unit area, per unit solid angle per unit frequency of a black body at temperature T is:

2hc2 Bλ(T)= (2.10) λ5 exp hc/(λkT) − 1

with k the Boltzmann constant, c the speed of© light£ in vacuum,¤ ª and h the Planck constant. If this is integrated over all wavelengths, and over all angles we obtain the black body irradiance or ﬂux density:

4 F = dλ dΩBλ(T)= σT (2.11) Z Z with σ the Stefan-Boltzmann constant. Evaluating at the surface of the star, the temperature is called the effective temperature, this is the Stefan-Boltzmann law. Knowing this, we write the stellar flux at the planet distance in terms of the stellar surface flux using the inverse square law of fluxes:

2 2 a Fs(a) = R Fs(Rs) (2.12) · s · And using the Stefan-Boltzmann law, combining the expressions 2.11 and 2.12 we can rewrite equation 2.9 for the incident power on the planet as:

R 2 P = (1 − A)πR2 σT 4 s (2.13) in p s a µ ¶ 7 Assuming that the energy absorbed by the planet is thermalized before being re-emitted, the planet will adopt a surface effective temperature, Te f f . Accordingly, the thermally emitted power is equal to its surface ﬂux emitted integrated over the emitting area:

2 4 Pout = 4πRp σTe f f (2.14)

The equilibrium temperature of a planet (Teq), is defined as its effective temperature when the planet balances the energy received from its host star with its the thermal radiation emission. For highly irradiated atmospheres, the internal energy in the energy balance equation is negligible in comparison to the strong stellar irradiation, thus, neglecting Pint from equation 2.8 and using equations 2.13 and 2.14 we have for the equilibrium temperature: 1 − A 1/4 R 1/2 T = s T (2.15) eq 4 a s µ ¶ µ ¶ As mentioned at the beginning of this section, the total luminosity of the planet has also a directly reflected light component, for a grey atmosphere, this component should be an attenuated copy of the stellar flux, peaking at optical wavelengths and having little contribution in the infrared. The next figures do not consider the reflected star light on the planet.

Then, for a reference value of the bond albedo of A = 0, we found an equilibrium temperature for HD 217107 b of Teq = 1040 19 K. This value allow us to have a broad idea of the spectral irradiance of the planet compared to that of± the star as a function of wavelength when approximated as a Planck function. The spectral irradiance observed, as seen from Earth, of each one is obtained integrating over the solid angle:

R 2 Fλ(T) = dΩBλ(T)cosθ = πBλ(T) (2.16) d Z µ ¶

Where R is the radius of the object, and d is the distance from the observer to the object. Figure 2.3 shows a plot of the spectral irradiance of HD 217107 and HD 217107 b. From the giant extrasolar planets with measured radius, most of the values lie within one and two Jupiter radii, we expect that the unknown value of HD 217107 b’s radius is most likely within this range. We plot then, the planet ﬂux for three different radii, between one and two Jupiter radii.

Important conclusions can be made from Fig. 2.3, we see that the planetary spectrum peaks in the infrared region of the spectrum between 2 and 3 µm, while the star spectrum peaks at visual wavelengths, near 0.4 µm. Toward shorter wavelengths the ﬂuxes drop exponentially (leftward of 1 µm for the planet, while the star’s ﬂux starts to drop at even shorted wavelengths). On the other side, for∼ wavelengths longer than 10 µm it falls as a power law, proportional to λ−4.

This behavior can be deduced also from equation 2.10. For short wavelengths its called Wien’s approximation, we have λ hc/kT, and the exponential term dominates the emission: ≪ 2hc2 hc Bλ(T)= exp − (2.17) λ5 λkT µ ¶

While in the long wavelengths approximation, known as the Rayleigh-Jeans law, λ hc/kT, we have: ≫ 2ckT Bλ(T)= (2.18) λ4

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10-14

10-1 100 101 102 Wavelength ( ¢m)

Figure 2.3: Log-log plot of the spectral irradiance of HD 217107 (black) and of HD 217107 b (simulated for three radii between 1 to 2 Jupiter radii) as function of wavelength, emitting as black body. The effective temperatures of the star and the planet are 5646 K and 1040 K, respectively. The vertical dashed line marks the waveband of our data around 2.14 µm.

More illustrating is the planet to star ﬂux ratio as a function of wavelength, which is given by:

2 Fλ(T ) Bλ(T = 1040 K) R Flux ratio = planet = p (2.19) Fλ(T ) Bλ(T = 5646 K) R star µ s ¶

Figure 2.4 shows the flux ratio between HD 217107 b and HD 217107 for three different planetary radii. In the top panel, we see that for shorter wavelengths, as the star light dominates the spectrum emission, the flux ratio decreases, while for longer wavelengths the flux ratio increases, and from wavelengths greater than 10 microns remains nearly constant. Although the best flux ratios are found for longer wavelengths, the −4 net flux from the star and the planet fall as Bλ(T) λ .In consequence, the Signal to Noise ratio for longer wavelengths will be significantly lower than in the∝ near infrared range. The bottom panel shows the flux ratio around our observing band (from 2.136 to 2.145 µm). Within a narrow band like this, the flux ratio does not change significantly with wavelength, but for the different radii of the planet the flux ratio goes from 3 10−5 to 1.5 10−4 for sizes from 1 to 2 Jupiter radius, respectively, thus the radius of the planet has an important× effect× in the flux ratio.

A black-body approximation gives an estimate of the order of magnitude of the ﬂux ratio. A better de- termination of the planet’s emergent spectrum involve the consideration of a number of other factors as the atmospheric chemistry or radiative transference (See Section 4.2).

9 10-2

10-3

10-4

10-5 Planet to star flux ratio

-6 10 2.0 RJup

1.5 RJup

1.0 RJup 10-7 100 101 102

Wavelength ( m) ¤

10-2

2.0 RJup

1.5 RJup

1.0 RJup 10-3

10-4 Panet to star flux ratio

10-5

10-6

2.130 2.135 2.140 2.145 2.150 2.155 Wavelength ( ¥m)

Figure 2.4: Top: Planet to star ﬂux ratio of the system HD 217107 emitting as black bodies for the planet radii: 2.0, 1.5, and 1.0 Jupiter radius (red, green, and blue respectively). Bottom: Same as above, but zoomed on the region around 2.14 µm, the two vertical lines enclose the wavelength range selected for our observations.

10 Chapter 3

Observations and Data Reduction

3.1 The Data

We observed the planetary system HD 217107 using Phoenix instrument [Hinkle et al., 2003], a high spectral resolution near-infrared spectrometer. The instrument is mounted on a 8.1-meter diameter altitude-azimuth telescope at Gemini South Observatory, in Cerro Pachón, Chile, at an altitude of 2722 meters. Phoenix is a high spectral resolution echelle spectrometer built by the National Optical Astronomy Observatory (NOAO). An individual spectrum generated by Phoenix is single order and covers a very narrow wavelength range, corresponding to a radial velocity range of 1500 km s−1. The spectrograph is equipped with a InSb Aladdin II array. An argon hollow cathode wavelength calibration source is supplied with the instrument. Table 3.1 summarizes the main characteristics of the telescope and spectrograph.

Table 3.1: Gemini South Telescope Characteristics Phoenix at Gemini South Value Observatory latitude -30:14:26.700 Observatory longitude -70:44:12.096 Primary mirror diameter 8.1 meters Detector 256 1024 InSb Aladdin II × Gain 9.2 e− / ADU Read out noise 40 e− Filter K 4667 Spectral range 2.136 – 2.145 µm Slit length 14 arcseconds Slit width 3 pixels Dispersion 10−5 µm ∼ Spectral resolution 40 000

We observed 11 nights, between August 14 and November 28 of 2007, collecting over 950 frames of our target, which represent 15.4 hours of observation (see Table 3.2). This wavelength band presented a few number of telluric absorption lines, leaving most of the spectrum available for further processing, but these lines are enough to perform a wavelength calibration. The observations were obtained in service mode using

11 the standard ABBA nodding sequences (the telescope is nodded back and forth along the slit, in order to register the sky at the same pixels of the target). After receiving the data from the first runs, we had to tune our observational set-up since the instrument was not fully characterized for use on the Gemini Telescope, lengthening the exposure time to increase the required signal to noise ratio in the frames, without saturation of the instrument. For the first four nights, the exposure time was set to 25 seconds, whereas for the rest of the nights it was set to 80 seconds. We requested as well, dark frames (10 per night), flat-field frames (10 per night), and lamp calibration exposures (1 for each nod position each night).

Table 3.2: Observing log Date Number of Target exposure Orbital Time span Star velocity span UT frames time (min) Phase hours m s−1 14 Aug 108 45 0.775 2.98 15.72 16 Aug 108 45 0.079 2.07 12.29 22 Aug 52 22 0.902 1.37 2.77 26 Aug 108 45 0.465 2.87 3.57 02 Oct 144 192 0.645 6.17 25.51 19 Nov 72 96 0.373 2.50 2.61 23 Nov 72 96 0.930 2.33 2.79 24 Nov 72 96 0.072 2.40 13.90 25 Nov 72 96 0.211 2.38 12.33 26 Nov 72 96 0.350 2.42 3.84 28 Nov 72 96 0.632 2.58 10.23

3.2 Reduction

We implement customized IDL routines for the data reduction, processing each night and slit position as an independent data set in order to minimize any systematic effects that might arise due to different atmospheric conditions or instrumental set up. Figure 3.1 shows an example of the raw frames.

We constructed a master Dark frame image for each night data set from the median of a set of (typically 10) dark frames, which we subtracted from each raw data frame, also built a master Flat-field image for each night in a similar way. The routines, used the flat-field images to identify hot pixels, and mark them as bad pixels if they have a value beyond 3.5 sigma from the median of the values of the 9 pixels in its neighborhood, iterating three times, masking the discarded pixels before the next iteration. Bad pixels were not considered in any further processing stages. We then divided the frames by the master flat-field to correct for the pixel-to-pixel variation in the CCD response. Then, to remove the sky emission we subtract each image from their corresponding opposite A or B image. Appendix B.2 gives a thoroughly description of the reduction.

Finally, we extracted the spectra from the frames using an IDL implementation1 of the Optimal Spectrum Extraction algorithm [Horne, 1986]. The algorithm produces the best attainable signal to noise ratio by applying nonuniform pixel weights in the extraction, assigning lower weights to noisy pixels containing a small fraction of light included in the object spectrum, in this way, avoiding waste of information. The algorithm comprises the following steps (where we have already fulﬁlled the two ﬁrst):

1http://physics.ucf.edu/∼jh/ast/software/optspecextr-0.3.1/doc/index.html

12 Figure 3.1: Raw frames from Phoenix spectrograph from October 2nd. Top: Nod position A. Bottom: Nod position B. As seen in this image, the vertical axis represents the spatial position in sky along which the slit is oriented (14 arc seconds length from top to bottom), while the light is being dispersed along the horizontal axis. During the observations, the star was located at two different nodding spatial positions, thus, for a pixel that captured the star light in a frame, in the next frame will see only the sky emission.

Step 1: Initial image processing (Dark subtraction, ﬂattening and sky subtraction).

Step 2: Initial variance estimation of the object and the sky frames (see Appendix B).

Step 3: Fit sky background outside the horizontal extraction boundaries at each wavelength in the data. Cre- ating at the same time a mask of bad pixels.

Step 4: Extract standard spectrum, Summing each background-subtracted wavelength within the bounds.

Step 5: Construct spatial proﬁle.

Step 6: Revise variance estimates. Initially creates the spatial profile by (reduced - background) / (standard spectrum). Then iteratively fits a function on each column (which includes a sigma rejection scheme) with variance / spectrum2 as weights, the fit is evaluated at all values and returned. All values are then made positive, and each wavelength is normalized to 1.

Step 7: Mask cosmic ray hits.

Step 8: For each wavelength the optimal spectrum is extracted and bad pixels are rejected, iterating until no bad pixels are found. The optimal extraction achieves the highest possible signal to noise ratio by weighting the pixels proportional to the proﬁle divided by the variance.

Step 9: Iterate 3 by itself to find cosmic rays in the background section. Iterate 5 and 6 to find the spatial profile image, but do not use the bad pixel mask found in step 5 for the extraction. Rather, next iterate steps 6, 7, and 8 to mask cosmic rays and optimally extract the spectrum.

13 3.3 Wavelength calibration

First, we calibrated the wavelength dispersion using the ThAr lamps, identifying the line positions and strengths in a high resolution ThAr line atlas [Hinkle et al., 2001]. Table 3.3 shows the lines and wavelengths from the line catalog in the range of our data that matched the lines in our lamps exposures. Figure 3.2 shows one of the lamp exposures from November 25th with the identiﬁed lines labeled.

Table 3.3: Lines catalog from Hinkle et al. [2001]. Line Wavelength µm Ar I 2.1454598 Ar I blend 2.1452818 Ar II 2.1428609 Unidentiﬁed 2.142620 Ar II 2.1420588 Ar II 2.1398896 Th I 2.1375105 Ar I 2.1373705

600

Ar I blend 2.14528 ¦m 500

400

300 Th I

Flux (counts) 2.13751 ¦m 200 Ar I Ar II Ar II Ar I Unid 100 Ar II

0 0 100 200 300 400 500 600 700 800 900 Pixel

Figure 3.2: Lamp calibration exposure for the observing run of November 25th. Eight of the lines of the data (labeled lines) matched lines in the catalog, we labeled two of them with their corresponding wavelength as reference.

As we had available only one lamp for each night and nod position and (sub pixel) offsets in wavelength are present in the data, this wavelength solution represented only a rough wavelength calibration. To reach the high precision needed for this work, we ﬁne tuned the calibration using a high resolution spectrum of the Sun2 to identify the telluric lines and use them to perform a more precise calibration.

2http://bass2000.obspm.fr/solar_spect.php

14 1.05 Sun spectrum

1.00

0.95 Data spectrum

0.90 Normalized flux

0.85

0.80

2.138 2.140 2.142 2.144 Wavelength ( §m)

Figure 3.3: Reference solar spectrum (top) for the wavelength calibration and an averaged data spectrum (below). We identiﬁed ﬁfteen absorption telluric lines common to both spectra (red marks). The lines cover the whole spectrum leaving few pixel where the solution is extrapolated rather than interpolated.

We identiﬁed the telluric lines as those lines appearing in both the solar spectrum as well as in an average spectra of our data set. Considering that HD 217107 is a star of a spectral type similar to the Sun, both star should share similar stellar absorption lines, nevertheless, given the non-zero proper radial velocity of the target with respect to Earth and given the dispersion relation of the instrument of 10−5, corresponding to a velocity difference of 1.4 km s−1 per pixel, those lines should not be aligned, therefore,∼ avoiding the possibility of a misidentiﬁcation∼ of lines.

For this telluric calibration, first we calculated the relative shifts between the frames in each night and nod position set, we selected the first spectrum as reference, while the rest of the spectra was shifted (using spline interpolations) in intervals of 0.01 pixels for a range of two pixels in each direction, then calculated their correlation with the reference spectrum, and minimized the root-mean-square of that value to find the shift that returned the best match. Next, we co-align all of the spectra in the set, and construct an average spectrum. We used this average spectrum to calibrate the wavelength by identifying the absorption lines common to both spectra (see Figure 3.3). A total of fifteen lines across the spectrum were selected and recorded the values of the center of their absorption in the solar and in the average spectrum.

Finally, we ﬁtted a second order polynomial to retrieve the wavelength solution of the form:

2 wavelength = c0 + c1 p + c2 p (3.1) · · in this formula, the wavelength is expressed in microns, with p the value of the pixel position. Figure 3.4 shows the fit and the corresponding residuals. We performed a third order polynomial fit, but it did not represent an improvement in the residuals, therefore, we selected the quadratic fit.

The most prominent absorption lines in the spectra are produced by the Earth atmosphere, these are known to present a high variability in strength in short amounts of time, making the telluric absorption lines change from frame to frame, thus, in order to avoid systematic errors produced by a incorrect removal of the telluric

15 lines, we decided to mask the pixels at wavelengths dominated by the identiﬁed telluric absorptions, and discard them from all upcoming processing stages. As consequence, from the 1024 initial pixels of each spectrum, on average, 670 pixels remained for the upcoming data analysis step.

0 100 200 300 400 500 600 700 800 900

2.144

2.142

m) ¨

2.140 Wavelength ( Wavelength 2.138

2.136

2e-5 Pixel position m) ¨1e-5

0.0

-1e-5 Residuals ( Residuals -2e-5 0 100 200 300 400 500 600 700 800 900 Pixel position

Figure 3.4: Polynomial fit of the line positions. Top panel: Wavelength of the lines (from the solar spectrum) vs. pixel position (from the average spectrum), the crosses mark the values of the line positions, the solid line is the second order polynomial fit. Bottom panel: Residuals of the fitting, all the points have residuals less than 1 10−5 µm and no pattern is seen in the residuals. The × −5 coefficients of the best fitting polynomial (equation 3.1) are: c0 = 2.145407, c1 = −1.0305 10 , −10 −6 × and c2 = −1.8496 10 , the dispersion relation of the residuals is: RMS = 4.21 10 µm. × ×

16 Chapter 4

Data Analysis and Results

A quick estimation of the photon noise limit in our data vs. the estimated planet to star ﬂux ratio, makes us notice that it is impossible to directly distinguish the planet’s signature from the stellar one in a single spectrum. Because of this, we use all the spectra collected in each run to subtract the stellar component, and search for the planetary Doppler-shift signature through a correlation method between the remaining residual spectra and synthetic models of the thermal emission spectrum of the planet following the idea of Deming et al. [2000] and Wiedemann et al. [2001].

4.1 Stellar Light Removal

The technique is based in the Doppler Effect, this principle tells that an observer in relative motion to or from the source of a front of waves will detect a change in the wavelength of the received signal, the light observed from stars and planets is affected by this phenomenon. For non relativistic velocities (vr c), the change in wavelength for electromagnetic waves can be expressed as: ≪

∆λ λ − λ v = obs rest = r (4.1) λ λrest c

where λobs and λrest are the observed and rest frame wavelength, and vr the radial velocity of the source (as a convention, the velocity is positive if the source is moving away from the observer and negative if the source is moving towards the observer). Then, for a beam of light emitted at a wavelength λem, from Earth it will be observed at a wavelength:

∆v λ = λ 1 + (4.2) obs em c µ ¶ Where ∆v is the relative velocity of the object with respect to Earth, for an object in a binary system, this velocity can be broke down into their different components: the radial velocity of the center of mass of the system (γ), the orbital motion of the object projected along the line of sight (vsini), and the barycentric velocity of the Earth toward the object (v⊕):

∆v = γ + vsini − v⊕ (4.3)

17 To remove the stellar ﬂux from our data, within each night and for each nod position, we align the spectra in a reference system in which the star remains at rest (Doppler-shifting them according to the formula 4.3 for the Star radial velocity, using a spline interpolation). Having the spectra lined up, we construct a stellar template from the average of the set. Then, the stellar templates and the spectra are normalized such their continuum is set to a median of one. Finally, the stellar templates are shifted back to the star’s radial velocity of each spectrum before being subtracted to each frame. Although a combination of the different nights to obtain the stellar template would smear more the planetary component, we avoided this possibility, since it is highly probable that other systematics would be introduced, for example, from the different instrumental set up or different atmospheric conditions.

Since the planet is approximately a thousand times less massive than its host star, the planetary Doppler wobble is the same order of magnitude greater (see equation 2.7), added to the out of phase motion of the planet with respect to its star (as they are orbiting each other), the planetary signature will not be added coherently, and thus will be blurred in the stellar template. Figure 4.1 shows the effects of the averaging on the planetary spectrum if we could see the planetary spectrum alone.

1.0

0.8

0.6

0.4 Normalized planet flux

0.2 first last

smear 30 km s 1 0.0

2.136 2.138 2.140 2.142 2.144 Wavelength ( ©m)

Figure 4.1: Simulated blurring of the planetary spectrum over one run using synthetic spectra of HD 217107 b. The blue curve represents the planetary spectrum as seen in the ﬁrst exposure of the night, the red curve represents the same spectrum by the end of the night, shifted 30 km s−1 in wavelength with respect to its host star. The bottom black curve shows how the planetary spectrum would look after the average of all the spectra comprised between the initial and ﬁnal exposure. The planetary spectrum is clearly blurred loosing its original shape (for an instrumental spectral resolution of 40 000).

As seen in ﬁgure 4.1, the planet spectrum is blurred in the stellar template due to the relative Doppler shifts, the bigger the shifts are, the more blurred the planetary spectrum will be. But, when the velocity span is not enough to produce a signiﬁcant shift, the planetary signature in the stellar template will be similar to the original spectra, then, when we subtract this template to each frame, the planet signature will be subtracted as well, as happened in some of our nights. from equation 2.7, the maximum radial velocity span of the planet (sini = 1.0) given the radial velocity span of the star during the observing run is given by:

18 Ms (∆vp sini)max ∆vs sini (4.4) ≈ mp sini

Plugging in the values from table 2.1, the maximum velocity span for HD 217107 b in a night is:

∆v sini ∆v sini p 0.803 s (4.5) km s−1 ≈ × ms−1 µ ¶max µ ¶

Then, for ﬁve of our runs (see table 3.2) the planetary spectrum will not shift more than a few km s−1. Figure 4.2 shows the blurring of the planetary spectrum for small Doppler shifts.

1.0

0.8

0.6

0.4 Normalized planet flux

0.2 first last

smear 3 km s 1 0.0 2.136 2.138 2.140 2.142 2.144

Wavelength ( m)

Figure 4.2: Same as ﬁgure 4.1 but for a shift of only 3 km s−1 during the night, the averaged spectrum appears slightly blurred due to the small velocity span.

Two factors will determine the amount of blurring in a given night. Clearly the ﬁrst, is the time extent of the run, a longer time span will yield a larger velocity span. A second, less obvious but very important factor, is the timing of the observation, given the particular sinusoidal shape of the radial velocity curve, there are moments when the relative motion between the planet and the star is minimal as seen from Earth (near greater elongation of the orbit), while at the moment inferior or superior conjunction the radial velocity span is maximal. Thus, for an equal time span, the orbital phase of the system when the system is being observed can drastically change the amount of blurring. Figure 4.3 illustrates this point.

Adequate planetary systems observed at the right time, can usually shift more than 20 km s−1 in the ma- jority of the nights, which is enough to blur the signature of the planet (for a 3 hours long observation), while the best systems (in this aspect) can shift up to 40 km s−1 for 3 hours observing windows. Figure 4.4 contrasts the planetary blurring for different velocity spans.

As bottom line, the keyword here is velocity span, which depends on the length of each observation, the timing of that observation, and the selection of an ideal candidate. All this variables must be considered in

v =12.33 m/s 40

t =2.4 h ) 60

1 (m s

i 80 sin s v

100 t =2.5 h

v =3.84 m/s 120

0.20 0.25 0.30 0.35 Orbital Phase

Figure 4.3: Close up, of radial velocity curve of ﬁgure 2.2 on the nights of the 25th (gray) and 26th (white) of November. Although both observing runs span about the same time extent, the radial velocity span for the 25th (12.3 m s−1) is more than four times that of the 26th (3.8 m s−1). the planning of an observation, since, for example, an unfortunate timing in the observation can render a data set useless, regardless of the length of the run. Notice also that the blurring depends on the (undetermined yet) sine of the inclination.

The subtraction of the stellar template, leaves a residual spectrum consisting of the signature of the planet, attenuated in some degree during the averaging process and immersed in the poisson photon noise. This residual spectrum is ready to be correlated with the theoretical planetary atmospheric models.

4.2 Planet’s Atmospheric Model

Since the discovery of the ﬁrst extrasolar giant planet orbiting a star, astrophysicists have quickly treated the problem and tried to understand and predict the effects of intense stellar insulation at such small orbital distances, developing theoretical models of the atmospheres of extrasolar giant planets.

Initial simplified models [Saumon et al., 1996, Guillot et al., 1996] assumed that a fraction of the light of the parent star is reflect as a gray-body (the reflected spectral emission is a copy of received black-body distribution, reduced by a constant factor at all wavelengths) and that other fraction is absorbed by the planet, and re-emitted as a black body emission at the temperature of the planet.

Over the last few years, the algorithms have evolved to a great degree of complexity, and now integrate and consider a number of different parameters and their interactions, such as, the intense irradiation, the atmospheric structure, temperature proﬁles, chemistry, the planet’s cooling and contraction history, dynam-

20 1.0

0.8

0.6

0.4

0.2 Normalized planet flux

0.0 1 smear 3 km s 1 smear 12 km s

0.2 1 smear 22 km s 1 smear 30 km s

2.136 2.138 2.140 2.142 2.144 Wavelength ( m)

Figure 4.4: Planetary spectrum blurring for different radial velocity spans (see legend). We can compare how effectively the planetary spectrum is lost in the stellar average, for very low radial velocity spans ( 3kms−1) as in some of of our observing runs, for low spans ( 12 km s−1) as in the best of our observing∼ runs, for typical velocity spans of good targets ( 22∼ km s−1) and for good runs of good targets ( 30 km s−1). Not only lines start to blend, but also,∼ the depth of the absorptions is more reduced∼ for larger shifts.

21 ics, and stability (e.g.: Fortney et al. [2008], Burrows et al. [2008], Madhusudhan and Seager [2009]). For example, the atmospheric models of Fortney et al. [2008] account for:

An algorithm of the radiative transfer accounting, both, the incident radiation from the parent star and • the thermal radiation from the planet’s atmosphere and interior.

A complex chemistry of elemental abundance and the calculation of chemical equilibrium composi- • tions, considering the sequestering of elements into condensates, and their removal from the gas phase (“rain-out”).

The use of a large and constantly updated opacity database including also the opacity of clouds, such • as Fe-metal and Mg-silicates.

Calculations on the atmospheres of extrasolar planets reveal a large sensitivity to the amount of irradiation, addressing two classes of day-side atmospheres. The most warm planets present temperature inversions (hot stratospheres), appear bright in the mid-infrared secondary eclipse, and feature molecular bands in emission rather than absorption, they will have large day/night temperature contrasts and negligible phase shifts between orbital phase and thermal emission light curves because radiative timescales are much shorter than possible dynamical timescales. On the other side, those that are cooler, absorb incident ﬂux deeper in the atmosphere, where atmospheric dynamics will more readily redistribute absorbed energy, leading to cooler day sides, warmer night sides, and larger phase shifts in thermal emission light curves.

For the high-resolution synthetic planetary spectra of HD 217107 b, we used customized theoretical thermal emission models of its atmosphere [model described in Fortney et al., 2005, 2006]. Since the orbit of HD 217107 b has a non negligible eccentricity, it is expected a variation in its temperature with orbital phase, we address this using three models adjusted to the different orbital distances. Table 4.1 presents the parameters of the three models available, while Figure 4.5 shows the orbital distance of HD 217107 b along one orbit, indicating the bounds for which we used each one of the models. These models are all solar metallicity, with gravity g =20 ms−2, cloud-free, and use the molecular abundances that are appropriate for chemical equilibrium. At these effective temperatures, the main absorbing molecules are H2O, CH4, CO, and CO2. The chemistry is described in detail in Lodders and Fegley [2002] and Visscher et al. [2006].

Table 4.1: Theoretical Atmospheric Models. Model Semi-major axis Temperature A.U. K 0.066 1095.0 HD 217107b 0.073 1029.0 0.082 981.0

The models originally have an inﬁnite spectral resolution (See Figure 4.6), therefore, to emulate the models as observed through the telescope, it is necessary to lower their resolution to the instrumental spectral resolution. We empirically characterized the instrument through the analysis of the emission lines in the calibration lamps, measuring the full width at half maximum (FWHM) of an isolated line, from which we determined the spectral resolution (R) through the equation:

λ λ R = = (4.6) ∆λ FWHM

22 0.085

T = 981 K

0.080

0.075 T =1029 K

0.070 Separation (AU) T =1095 K 0.065

0.060

0.0 0.2 0.4 0.6 0.8 1.0 Orbital Phase

Figure 4.5: Orbital distance of HD 217107 b as a function of the orbital phase. The horizontal dashed lines delimit the selected ranges for each one of the models, labeling on the left the model used, the gray and white boxes, as in Figure 2.2 denote the used and discarded observing runs.

We found a value of R 40000 corresponding to a velocity resolution ∆v = 7.49 km s−1(∆λ = 5.35 µm in wavelength). Figure 4.7≈ shows a model after being convolved with a Gaussian function with a FWHM representing the planetary spectrum as seen in our data.

4.3 Correlation

An important statistical test that can establish the “sameness” of two data sets is the correlation function. The returned value of this function lies within −1 and 1. A positive value of this function indicates that the two data sets have some degree of correlation, that is, they are similar, while a negative one, denotes anti-correlation. A value near zero tells that the two sets are uncorrelated.

We apply this test, between our residual spectra and the theoretical models, to determine if, for a certain inclination, the model spectra match our observations. In the ﬁnal result we sum up the correlations from all our residual spectra, and divide it by the number of spectra (to keep the ranges within the bounds −1 and 1).

The models before being correlated with the data, must be Doppler-shifted to mimic the radial velocity of the planet at the time of the observation of each frame, To do this, we use the ephemeris from the spectrum to obtain the radial velocity of the star, using equation 2.1. As discussed in section 2.2, the radial velocity of the planet is parametrized by the inclination of the orbit, therefore, assuming a value of sini, we derive the velocity through equation 2.7. Then, to ﬁnd the radial velocity with respect to Earth we use equation 4.3, we plug the that result into equation 4.2, and ﬁnally apply the corresponding shift to the wavelengths of the model.

We calculate the correlation degree as a function of the inclination, C(i), according to the correlation

23 5

4 )

1 Hz

1 s

2 3 erg cm 7 2 10

Flux ( Flux 1

2.136 2.138 2.140 2.142 2.144 Wavelength ( m)

Figure 4.6: Planetary atmospheric model at inﬁnite resolution. The main absorbing molecules are H2O, CH4, CO, and CO2. The equilibrium temperature is T = 1029 K.

4 )

1 Hz

1 s

2 3 erg cm 7 2 10

Flux ( Flux 1

0 2.136 2.138 2.140 2.142 2.144

Wavelength ( m)

Figure 4.7: Planetary atmospheric model at an instrumental resolution corresponding to R = 40000.

24 function [also known as Pearson’s correlation coefﬁcient, Press et al., 1992]:

N Nk 1 rk j − r¯k τk j(i) − τ¯k(i) C(i)= j=1 · (4.7) N k=1 NPk ¡− ¯ 2 ¢ ¡ Nk −¢¯ 2 X j=1 rk j rk j=1 τk j(i) τk(i) r nP ¡ ¢ onP ¡ ¢ o ¯ Where we have used the notation fk j for the value of the function at the pixel j of the spectrum k, fk is the mean value of the function in the spectrum k:

Nk ¯ 1 fk = fk j (4.8) Nk j=1 X In equation 4.7, “r” refers to the residual spectrum, while “τ(i)” refers to the planetary model spectrum (shifted according to sini), with Nk the number of pixels in spectrum k, and N the total number of observed spectra.

We produce then, a curve of the correlation degree vs. the inclination of the orbit, evaluated in the range: 0 < i < π/2. If the two data sets are uncorrelated, and the number of data points is large (as in our case), then the correlation degree is expected to be zero. Then, as consequence of the random nature of the Poisson noise, the correlation of the residual spectra with the models should be close to zero correlation, except when the adopted i matches that of the planetary system. if the planet’s signature is strong enough in comparison with the noise, an appreciable peak in the correlation would represent a successful detection of the planetary signature and would indicate the value of the inclination. By constraining the inclination with this method, the mass of the planet would be immediately found.

4.4 Data Results

As discussed in section 4.1 we excluded those nights with a star’s velocity span less than 10 m s−1 (See column 6 of Table 3.2) since they do not represent any signiﬁcant improvement in the results, as the shift of the planet is less than the instrumental resolution.

Figure 4.8 Top shows the correlation curve of our observations as a function of sini (as explained in section 4.3). The degree of correlation found is close to zero at all inclinations, and we do not distinguish any distinctive positive peak that indicated an atmospheres with absorption features resembling those of the models.

4.5 Planet-to-Star Flux Ratio Fitting

Unfortunately, the results from section 4.3 indicate only the inclination of the orbit, since the value of the degree of correlation does not tell if the observed correlation is statistically signiﬁcant per se. On account of this, in this section, we describe a further analysis of the data to assess the statistical signiﬁcance of the value of the correlation degree reached, and also, the factors that make a correlation degree increase or decrease.

If the planet-to-star ﬂux ratio (Fp/Fs) is low (leaving the planet’s spectrum deeply buried into the noise in the residual spectra) there are little chances to detect the planet, as the correlation will be small at all

25 0.006 0.004 0.002 0.000

0.002

0.004

0.006 Correlation 0.0 0.2 0.4 0.6 0.8 1.0 sin i 10-2 1.0

0.9

0.8

0.7 -3

10 ) 2

0.6

0.5

0.4 -4

10 exp( 0.3

0.2 Planet-to-Star flux ratio 0.1

10-5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 sin i

Figure 4.8: Top: Correlation results for our data set of HD 217107 b, as a function of sin i. The correlation remains flat along all inclinations without any distinctive peak, the maximum value is reached at sini = 0.71. Bottom: Goodness of fit χ2-map, the horizontal and vertical axes refer to the fitting parameters sini and Fp/Fs of the synthetic planetary spectrum added. The gray scale denotes the goodness of fit, as a function of chi-square relative to the best fit , ranging from black for 2 the best fit (at χmin) to white for the poorest fit. Additionally, with bootstrap procedures, we determined the solid lines that mark the 1, 2, 3, & 4–σ (from bottom to top) confidence levels of the false alarm probability. The white cross marks the best fit, located at sini = 0.84 and −3 Fp/Fs = 3.6 10 , unfortunately below the 3–σ confidence level. ×

26 inclinations. On the other side, if Fp/Fs is large, we should expect a noticeable correlation at the inclination of the planet. It is reasonable, then, to state that the planet-to-star ﬂux ratio is the main physical parameter bounded to the correlation degree. Based on this, we will search for the best parameters in the parameter space: [Fp/Fs , sini].

We, thus, recreate and process our observations using synthetic spectra, with known parameters for the planet, to obtain “synthetic” correlation curves, which we can compare the data results, and determine which are the most probable parameters that gave rise to our result, and also estimate what conﬁdence can we put into them. We construct the set of synthetic correlations according to the following scheme:

Step 1: We rearrange the order of the data set with random permutations within each night, but keeping the original order of the dates of the observations. As a consequence, any real planet signature would disappear, but the noise level of the data will remain.

Step 2: Using the atmospheric models of the planet, we inject a synthetic spectrum in the scrambled data set, Doppler-shifted and with a relative ﬂux according to speciﬁc values of sini and Fp/Fs, respectively. For simplicity, we adopt a constant Fp/Fs along the orbit. Step 3: We process these synthetic data through the same routines used in our original data (sections 4.1 −5 −2 through 4.3). We then iterate for a grid of values in the ranges: 0 i π/2 and 10 Fp/Fs 10 , ≤ ≤ ≤ ≤ obtaining a set of synthetic correlations for sini and Fp/Fs.

There are two effects that affect the intensity of the emitted flux over time. Firstly, there is expected a phase variation in the planet’s flux due to the changing distance between the planet and the star along the orbit, expecting a greater emission near the periastron of the orbit. Secondly, there should be a change in the observed planet’s brightness as we observe a greater or smaller portion of the day side surface. This second phase variation, should not be of such a great importance as in the more extreme irradiated planets (pM class planets, according to the nomenclature of Fortney et al. [2008]). For HD 217107 b (a pL class planet) it is expected a lower day/night side temperature contrast since they should present a more important redistribution of energy. None of these effects are simulated in this analysis, a refinement of the method would improve the accuracy of this technique in the future.

Once we have such models, we search for the best fit parameters through a χ2 minimization between the data correlation curve and the synthetic correlation curves, generating a goodness-of-fit map. We plot, in a 2 2 gray scale, χ relative to the best fit (χmin) through the formula:

f (i, flux ratio) = exp −α χ2 − χ2 exp −α ∆χ2 (4.9) · i,fr min ≡ · i,fr © ¡ ¢ª ¡ ¢ This function takes values within the range: 0 < f 1, at the position of the best fitting parameters has the value 1, and decreases as the fit get poorer. The scale≤ goes then, from white for a value of 0 and it gets darker as the value approaches to 1. The parameter α, it is just a plotting device that enables a good contrast in the plots, for consistency we adopted the same value for all the plots in this work.

Bottom panel of Figure 4.8 presents a goodness of ﬁt map of the data results of HD 217107 b with the synthetic correlations parametrized by the sine of the inclination and planet-to-star ﬂux ratio of the injected planet’s spectrum (section 4.5).

Figure 4.9 shows an example of the synthetic correlations, for an injected planet with sini = 0.82. As suspected, greater values of Fp/Fs enhance the peak of the correlation degree. Figure 4.10 shows slices of Figure 4.9 for three values of Fp/Fs.

27 Figure 4.9: Example of synthetic correlation curves for an injected planetary spectrum with sini = 0.82, and different given Fp/Fs (y–axis). The correlation curves then are plotted as a function of inclination (z−axis vs. x–axis), for example, the highlighted black curve. The correlation curve peaks at the inclination of the injected planet and takes values near zero far from that inclination.

0.030 3 F /F =2.8 10

0.025 p s 3

Fp /Fs =1.0 10

0.020 4

Fp /Fs =3.6 10 0.015

0.010

Correlation 0.005

0.000

0.005 0.0 0.2 0.4 0.6 0.8 1.0 sin i

Figure 4.10: Synthetic correlation curves from Figure 4.9, for three different Fp/Fs of the injected planetary spectrum (see legend), as expected the correlation degree increases with larger Fp/Fs.

28 As corollary, the stronger the planetary signal is compared to the noise, the greater the correlation degree will be, and the easier to detect the planet. Thus, our ability to detect the planetary spectrum in the photon noise can be quantified by the planet-to-star flux ratio and the stellar flux, according to the expression (fluxes in number of photons):

(Fp/Fs) Fs Fp/Noise = · (Fp/Fs) √Fs (4.10) Fs + Fp ≈ · | | | | p 4.6 False Alarm Probability

In addition, to assess the conﬁdence of the results, we calculate false alarm probability limits for this map, using a bootstrap procedure. Following the idea of Collier Cameron et al. [2002], we determine the frequency with which the correlation degree exceeds a given value as a result of noise in the absence of a planet signal.

The procedure consist in repeating steps 1 and 3 of section 4.5 a big number of trials (in this work we usually iterated between 3 000 and 5 000 times), recording after each trial the correlation curve. This set of correlation curves represent the correlation found in the absence of a planetary signal, and, being created from the data themselves, define an empirical probability distribution that include both the photon statistics and the systematics errors. Then, at each inclination of the correlation curve, we stack the values of the correlation degree and sort them in increasing order. Finding the value of the correlation degree at the 65, 90, 99 and 99.9 percentiles of the trials, we define the 1, 2, 3 and 4–σ confidence limits, they represent the signal strengths at which spurious detections occur with 35, 10, 1, and 0.1 per cent false alarm probability respectively, for each value of the inclination (See Figure 4.8, bottom panel).

For example, for correlation values above the correlation indicated by the 3–σ curve we would be 99% conﬁdent that the measurement is not a spurious detection arising from systematic effects. This procedure, then, allow us to assess the probability of obtaining a certain correlation degree without the presence of planetary emission, and give a more accurate picture of the noise and systematic effects of our data set.

4.7 Data Results Discussion

−3 Regarding the results of HD 217107 b, the best fit occurs at sini = 0.838 and Fp/Fs = 3.6 10 . This value disagrees with the maximum value of the correlation curve (Figure 4.8 Top), besides,× the relative improvement in χ2 against the surrounding parameters is negligible, and furthermore, the bootstrap results indicate that this values is below the 3–σ confident limit of the signal not being a false positive. Also, this Fp/Fs is much larger than the expected value (from Section 2), where the predicted flux ratio ranges from 2 10−5 and 1 10−4. The natural conclusion is that we can not state the detection of HD 217107 b. This feature,× two orders× of magnitude above the expected flux ratio, should be result of a systematic errors during the processing of the data. The 3–σ confidence limit allows us to establish an upper limit in the flux ratio of the planet at 4–5 10−3 for inclinations greater than sini = 0.6. ×

29 Chapter 5

The Optimal Acquisition of Data

Although our current data does not allow us to claim the direct detection of HD 217107 b, during the analysis we have developed a strategy to maximize the chances of success in a future search. In this chapter we will investigate the limits of the direct search of extrasolar planets with this method, and study how can we develop it. We basically identiﬁed two keystones, that will let us improve this technique:

1.– Select the most appropriated candidates.

2.– Follow an optimal observing strategy.

To test our approach in a real situation, we simulated observations of planetary systems as observed by Phoenix spectrometer at Gemini South, for the time constrained by a proposal for the first semester of a year (From the beginning of February to end of July). We start with a list of extrasolar planets from The Extrasolar Planets Encyclopaedia1, limiting to the confirmed non-transiting extrasolar planets. Even though other (potentially better suited) planetary systems could arise, when we consider other observatories (for example, high resolution infrared spectrometers like CRIRES at VLT or NIRSPEC at Mauna Kea), or if we consider the observable extrasolar planets during the whole year, the main goal of these simulations is to demonstrate that the detection of extrasolar planets with our routines is possible in a real case. Thus, we follow the constraints set by these conditions. For this same reason, and acknowledging that other extrasolar planets should present different spectral features due to their own physical properties, we will disregard these specific spectral features, and rather focus on the broad relative features in the planet and host star spectra, as are the planet-to-star flux ratios. The consideration of the appropriate spectral features like specific absorption or emission lines and their strengths is then let to the theoretical atmospheric modelers.

5.1 Target Selection

The ﬁrst constraint for the selection is that the orbit of the targets must be observable from Gemini South, limiting the declination of the candidates to values up to 50 deg from the latitude of the observatory. If the difference is greater, the observable time span of a night± would be too short (less that 3 hours) due to the larger air mass. Also, the longitude of the observatory limits the right ascension of the targets,∼ since they must be located near opposition of the orbit of the Earth around the Sun during the semester.

1http://exoplanet.eu

30 As indicated by equation 4.10 we prefer candidates with larger star’s apparent brightness for a better signal to noise ratio. In the case of Infrared observations, we look at the K-band magnitudes. Another consideration is to set a lower limit cutoff in the planet’s radial velocity span in a observing run, we set it at 5 km s−1 if the orbits had inclinations of 30◦, this limit corresponds to the FWHM of the instrument spectral resolution, thus any planet orbiting with an inclination larger than 30◦ should have a Doppler shift detectable.

In terms of physical parameters, this criteria translates into smaller semi-major axes, involving higher radial velocity spans (enabling a greater Doppler shift between the planet and the star spectra during the runs), and at the same time, this favors higher planet-to-star ﬂux ratios.

Table 5.1 lists three of the better suited selected planetary systems (HD 217107 listed for comparison), all of the targets have smaller orbits than HD 217107 b, thus having a larger maximum velocity amplitude (column 4).

Table 5.1: Favorable targets for Gemini South

Target a K band Kp AU mag km s−1 HD 179949 0.0443 0.0026 4.936 0.018 158.23 ± ± Tau Boo 0.0481 0.0028 3.507 0.348 150.62 ± ± HD 73256 0.0371 0.0021 6.264 0.022 158.17 ± ± HD 217107 0.073 0.001 4.536 0.021 112.28 ± ±

5.2 Optimal Observing Strategy

To simulate the observations, we recreated the same instrumental settings of our data, that is, a similar amount of total time (distributed in 9 runs of 3–hours length each), same spectral range and resolving power. But, as discussed in section 4.1, carefully selecting the observing schedule. We proceeded as follow:

Step 1: For each one of the available nights in the period, knowing the coordinates of the target and of the observatory, we calculated the air mass as a function of time, therefore we restrict ourselves to the time when the air mass remains under 1.5, constraining the time span available during the night. Step 2: Using the information from the orbital solution, within each nightly time span, we select the 3–hour range that gives the maximum radial velocity span, be it at the beginning of the night or at the end. Step 3: We record then, the radial velocity spans for each night, and choose those nights with the biggest span.

Figure 5.1 shows the observability windows for HD 179949, indicating for each day the UT range when the planet is observable under 1.5 air mass (diagonal black lines), and the most favorable observing runs (vertical black lines). This target has very favorable conditions, since it spends more than 6 hours under 1.5 air mass, and more than one hundred days in the observable range from Gemini South, allowing several nights available to observe with a large radial velocity span.

Table C.1 shows the information collected from Figure 5.1, for each of the observing runs, it presents the time at the beginning of the observation, the phase coverage, the radial velocity span and average radial velocity of the star, and also the maximum radial velocity span reachable for the planet (for a edge on orbit), and maximum average radial velocity of the planet.

31 Figure 5.1: Daily observability windows of the extrasolar system HD 179949 for the ﬁrst period of 2009 (x–axis) as a function of Universal Time (y–axis). The color scale reﬂects the air mass of the system as observed from Gemini South Observatory. The purple stripes mark the twilight (the time between dawn and sunrise, and between sunset and dusk). The diagonal black lines delimit the air mass range set for this system (less than 1.5), and each of the vertical black lines mark the nightly observing runs (3–hours) when the radial∼ velocity span is maximal.

32 To create the synthetic spectra, we used the solar spectrum (from section 3.3) to simulate the stellar component, while for the planetary component we used the atmospheric models of HD 217107 b. After selecting the dates of observation, the spectra are Doppler-shifted according to the target’s orbital parameters. For the planetary spectrum, we choose a planet-to-star ﬂux ratio and inclination to calculate the appropriate radial velocity before adding it to the stellar spectra, then to the composed spectra we add Poisson noise, corresponding to the K band magnitude of the target. Then, after synthesizing the data, we proceed to apply our analysis routines (Chapter 4) just as we did with our real data.

5.3 HD 179949 b Simulation

In this section we present simulations of an observing campaign on a target with the physical parameters of the planetary system HD 179949 (See Table 5.2), for which we will contrast the results when our observing strategy is applied and when it is not, and highlight the aspects in which our idea improves the results. We used the values from this table to compute the orbital solution of the star radial velocity (See Figure 5.2 Top panel).

Table 5.2: Parameters Parameter HD 179949a Tau Booa HD 73256b References Star: a b Te f f (K) 6168 44 6387 44 5636 50 V05, S04 ± ± ± d (pc) 27.05 0.59 15.60 0.17 36.5 1.0 B05 ± ± ± a b Ms (M⊙) 1.14 0.08 1.33 0.11 1.05 0.05 V05, S04 ± ± ± a b Rs (R⊙) 1.19 0.03 1.42 0.02 0.89 V07, U03 −1 ± ± Ks (ms ) 112.6 1.8 461.1 7.6 269.0 8.0 B05 −1 ± ± ± a b vg (kms ) -24.4 0.5 -15.9 0.5 29.5 0.2 V05, N04 ± ± ± Right Ascension (h:m:s) 19:15:33.23 13:47:15.74 08:36:23.02 P97 Declination (deg:m:s) -24:10:45.67 17:27:24.86 -30:02:15.45 P97 Planet:

Tp (JD) 2451002.36 0.44 2446957.81 0.54 2452500.18 0.28 B05 ± ± ± e 0.022 0.015 0.023 0.015 0.029 0.020 B05 ± ± ± a (AU) 0.0443 0.0026 0.0481 0.0028 0.0371 0.0021 B05 ± ± ± P (days) 3.092514 0.000032 3.312463 0.000014 2.54858 0.00016 B05 ± ± ± mp sini (M ) 0.916 0.076 4.13 0.34 1.87 0.27 B05 Jup ± ± ± ω (deg) 192 188 337 46 B05 ± Notes .— V05: Valenti and Fischer [2005], S04: Santos et al. [2004], B06: Butler et al. [2006], U03: Udry et al. [2003], N04: Nordström et al. [2004], P97: Perryman and ESA [1997].

According to Figures 5.3 and 5.4, Top right panels, from the equilibrium temperature of HD 179949 b the planet to star ﬂux ratio should be of the order of 10−4. But, as the purpose of this section is to show the improvement of our observing strategy in contrast with a regular observation, we adopted a larger value of Fp/Fs. The system was then simulated with an inclination, sini = 0.77 and a planet-to-stat ﬂux ratio of −3 Fp/Fs = 3 10 . ×

33 150

100

) 50

0 (m s i sin s

v 50

100

150

0.0 0.2 0.4 0.6 0.8 1.0 HD 179949's Orbital Phase

600

400

200 )

1 "

0 (m s i sin s

!200

!400 !600 0.0 0.2 0.4 0.6 0.8 1.0 Tau Boo's Orbital Phase

300

200

100 )

1 $

0 (ms i sin s

v 100 # #200

300 # 0.0 0.2 0.4 0.6 0.8 1.0 HD 73256's Orbital Phase

Figure 5.2: Radial velocity curve of the extrasolar planet host stars HD 179949 (Top panel), Tau Boo (Center panel), and HD 73256 (Bottom panel) vs. orbital phase. Calculated from the orbital parameters of Table 5.2. The phase origins are set at the orbits’ periapsis. 34 10-2 10-2 HD 217107 HD 179949 10-3 10-3 2.0 RJup planet 2.0 RJup planet 1.5 R planet 1.5 R planet 10-4 Jup 10-4 Jup

1.0 RJup planet 1.0 RJup planet 10-5 10-5 ) )

1 1 ( &10-6 10-6 m m

2 2

& ( 10-7 10-7

10-8 10-8 Flux (W Flux m (W Flux m 10-9 10-9

10-10 10-10

10-11 10-11

10-12 10-12

10-1 100 101 102 10-1 100 101 102

' Wavelength (%m) Wavelength ( m)

10-2 10-2 Tau Boo HD 73256 10-3 10-3 2.0 RJup planet 2.0 RJup planet 1.5 R planet 1.5 R planet 10-4 Jup 10-4 Jup

1.0 RJup planet 1.0 RJup planet 10-5 10-5 ) )

1 1 , *10-6 10-6 m m

2 2

* , 10-7 10-7

10-8 10-8 Flux (W Flux m (WFlux m 10-9 10-9

10-10 10-10

10-11 10-11

10-12 10-12 10-1 100 101 102 10-1 100 101 102

Wavelength ( m) Wavelength ( m) ) +

Figure 5.3: Log-log plot of the black body spectral irradiance of the systems HD 217107 (Top Left panel), HD 179949 (Top Right panel), Tau Boo (Bottom Left panel), and HD 73256 (Bottom Right panel) as function of wavelength. The planets’ spectra are in black color and their respective planets, simulated for three radii between 1 to 2 Jupiter radii in colors (see legend). The planets’ effective temperatures were calculated using the equation 2.15, ﬁnding Teq = 1040 19 K for ± HD 217107 b, Teq = 1541 55 K for HD 179949 b, Teq = 1673 51 K. for Tau Boo b, and Teq = 1331 42 K for HD 73256± b. ± ±

35 10-2 10-2

10-3 10-3

10-4 10-4 Flux ratios Flux ratios

10-5 10-5 2.0 RJup 2.0 RJup

1.5 RJup 1.5 RJup

1.0 RJup 1.0 RJup 10-6 10-6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Wavelength ( m) Wavelength (.m) -

10-2 10-2

10-3 10-3

10-4 10-4 Flux ratios Flux ratios

10-5 10-5 2.0 RJup 2.0 RJup

1.5 RJup 1.5 RJup

1.0 RJup 1.0 RJup 10-6 10-6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Wavelength (/m) Wavelength ( m) 0

Figure 5.4: Semi-log plot of the planet-to-star ﬂux ratio of the systems HD 217107 (Top Left panel), HD 179949 (Top Right panel), Tau Boo (Bottom Left panel), and HD 73256 (Bottom Right panel) as function of wavelength, emitting as black bodies for the planet radii: 2.0, 1.5, and 1.0 Jupiter radius of the planet (red, green, and blue respectively).

36 We present, two simulations for this system, ﬁrst following our observing strategy, using the coordinates and radial velocity of the target we calculated the observability windows already shown in Figure 5.1 and Table C.1. For a good target as this one, there are several favorable nights when the planet can cover a large radial velocity span. We selected 9 of them, to simulate the observation (nights marked with an “o” in row (9) of Table C.1). In the second simulation, we simulated the observations without taking care of selecting the nights with larger radial velocity span (nights marked with an “x” in row (9) of Table C.1). Figure 5.5 show the results of these simulations.

0.025 0.020 0.015 0.010 0.005

0.000 10.005 Correlation 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 sin i sin i 10-2 1.0

0.9

0.8

0.7 -3

10 ) 2

0.6 5

4 3

0.5 2

0.4 -4

10 exp( 0.3

0.2 Planet-to-Star flux ratio 0.1

10-5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 sin i sin i

Figure 5.5: Same as Figure 4.8 but for synthetic spectra, based on the planetary system HD 179949. The −3 injected planet was simulated with: sin i = 0.77 and Fp/Fs = 3 10 . Left panel: Results using our observing strategy. Right panel: Results without our observin× g strategy. The routine −3 successfully recovered the signal at sini = 0.78 and Fp/Fs = 2.8 10 in both cases, although, × when using our strategy, the correlation degree is stronger, and the χ2 peak is better deﬁned in comparison with the right panel.

In both cases the correlation curves (Top panels) mark the inclination of the synthetic orbit with a noticeable increment in the correlation degree near sini = 0.77. While the χ2-maps (Bottom panels) effectively −3 indicate the best ﬁt at sini = 0.78 and Fp/Fs = 2.8 10 . We identify the differences between these two simulations: ×

1.– The correlation degree in the Top-left panel is greater than in the Top-right. This can be understood, given the larger radial velocity spans, the planetary spectrum is blurred in a greater extent in the stellar template (section 4.1), and consequently, less reduced when the template is subtracted, the planetary spectrum signal is thus stronger in the residual spectrum, which will increase the correlation degree.

2.– A consequence of having greater correlation degree is that all the confidence levels are in general lower, since it is less probable to reach such correlation degree by chance in a no-planet spectrum. −4 The 3–σ confident limit lies at Fp/Fs = 3 10 when using our observing strategy (for sini > 0.5), in × −4 the other case, is lies close to Fp/Fs = 4 10 . Therefore, in general we can reach and discern lower planet-to-star flux ratios. ×

3.– Lastly, the peak in the χ2-map is much better determined in the left panel.

37 These ﬁgures evidence that when we adopt this observing strategy there is a number of improvement in the results that make a detection more probable.

As a general remark, we note, from these and the other χ2-maps, a sensitivity bias for this method, favoring the detection of more inclined orbiting systems, thus, leaving a inclination-sensitivity dependence at each planet-to-star ﬂux ratios.

5.4 Tau Boo b and HD 73256 b Simulations

In this section we present two simulations reproducing as close as possible to the physical conditions of the planetary systems Tau Boo and HD 73256, implementing our observing strategy, we examine the chances of a successful direct detection.

The orbital conﬁguration of the system Tau Boo allows a planetary radial velocity span similar to the one of HD 179949 b (See Tables 5.1 and 5.2), the star radial velocity curve is shown in Figure 5.2, Center panel. The calculation of the Equilibrium temperature of the planet (Teq = 1673 51 K), sets a range of −4 −4 ± Fp/Fs between 2 10 and 7 10 (See Figures 5.3 and 5.4 Bottom left panels). Figure 5.6 shows the observability window× of this planet,× as it is close to the cutoff limit in declination, the air mass is greater than in the other simulations.

Figure 5.6: Similar to Figure 5.1, daily observability windows of the extrasolar system Tau Boo for the ﬁrst period of 2009 vs. Universal Time.

38 Figure 5.7 shows the simulation of Tau Boo, the planetary parameters set are: sini = 0.82 and Fp/Fs = 4 10−4. The correlation curve (Top panel) peaks near the injected orbital inclination, and our routines × −4 return the best fit parameters (Bottom panel): sini = 0.79 and Fp/Fs = 3.6 10 , underestimating by a × few percent the values. Nevertheless, the χ2-map shows an improvement in the region near the injected −4 inclination and flux ratio, and the bootstrap results set the 3–σ confidence limit around Fp/Fs 1.5 10 (for × sini > 0.5), making this detection 99% of not being a spurious signal.

0.008 0.006 0.004 0.002 0.000

0.002 6

Correlation 60.004 0.0 0.2 0.4 0.6 0.8 1.0 sin i 10-2 1.0

0.9

0.8

0.7 -3

10 ) 2

0.6 :

9 8

0.5 7

0.4 -4

10 exp( 0.3

0.2 Planet-to-Star flux ratio 0.1

10-5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 sin i

Figure 5.7: Same as Figure 4.8 but for synthetic spectra of the planetary system Tau Boo, using our observing −4 strategy. The injected planet was simulated with: sin i = 0.82 and Fp/Fs = 4 10 . The best ﬁt −4 × parameters found are: sini = 0.79 and Fp/Fs = 3.6 10 . ×

39 We repeated the analysis, this time for the planetary system HD 73256. This planet has one of the largest maximum radial velocities (See Tables 5.1 and 5.2), it has also the greatest planet-to-star Flux ratio of our simulated candidates: between 2 10−4 and 8 10−4 (See Figures 5.3 and 5.4 Bottom Right panels), but the star has a larger K band magnitude× than the other× stars in Table 5.1, which will produce a lower signal to −4 noise ratio. The planet was added with parameters: sini = 0.69 and Fp/Fs = 7 10 . Figure 5.8 shows the observability windows of this system. ×

Figure 5.8: Similar to Figure 5.1, daily observability windows of the extrasolar system HD 73256 for the ﬁrst period of 2009 vs. Universal Time.

Figure 5.9 shows the results of this simulation, the Top panel shows a faint peak near sini = 0.67, but it −4 is not conclusive. The best fit (Bottom panel) is found at: sini = 0.63 and Fp/Fs = 7.8 10 . In this case, × the contrast in the χ2-map between the peak and the surrounding is slighter than in the other cases, probably −4 due to the lower signal to noise of this system. With a 3–σ confident limit around Fp/Fs 2 10 , again, the detection has 99% confidence. ×

40 0.004 0.002

0.000 ;0.002

0.004 ; Correlation 0.0 0.2 0.4 0.6 0.8 1.0 sin i 10-2 1.0

0.9

0.8

0.7 -3

10 ) 2

0.6 ?

> =

0.5 <

0.4 -4

10 exp( 0.3

0.2 Planet-to-Star flux ratio 0.1

10-5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 sin i

Figure 5.9: Same as Figure 4.8 but for synthetic spectra of the planetary system HD 73256, using our ob- −4 serving strategy. The injected planet was simulated with: sin i = 0.69 and Fp/Fs = 7 10 . The −4 × best ﬁt parameters found are: sini = 0.63 and Fp/Fs = 7.8 10 . ×

41 Chapter 6

Discussion and Conclusions

In this chapter we present the main conclusions of this work:

HD 217107: • We have revisited a correlation method to directly search for the high-resolution signature of the non- transiting extrasolar planets, we modified it, considering that analysing each night on their own is important to minimize systematic effects, and applied it to HD 217107 b in the near infrared spectral range. The correlation curve (as a function of the inclination of the orbit) did not present any distinctive peak, and their values were close to zero for all inclinations, the maximum correlation degree was located at sini = 0.71. Furthermore, using the original data, we created synthetic expected spectra for known planet’s inclinations and flux ratios, to fit the sine of the inclination and planet-to-star flux ratio through a χ2 square minimization routine. We found as best fitting parameters: sini = 0.84 and −3 Fp/Fs = 3.6 10 , but this is below our 3-σ confidence limit of false alarm probability. Besides, as can be observed× in Figure 4.8 (bottom panel), there is difficulty in determining the best fitting parameters for the planet, as the relative improvement in χ2 varies negligibly in the parameter space. As a consequence of the faint features in the results and that the peak in the correlation (Figure 4.8, top panel) disagrees with the most probable value of sini (Figure 4.8, bottom panel), we could not claim a detection of HD 217107 b with our current data. According to the sensitivity of our method and our data, we can constrain the planet emission, rejecting the flux ratio of HD 217107 b to its host star to be over 5 10−3 (3–σ confidence, from our bootstrap method). We attribute these results to two main reasons: First,× since the instrument was not well characterized at the time, the data was not as sensitive as expected, and we had to tune the service-mode observations after the first few observing windows. Second, the nonexistence of a proper observing strategy at the time of the observations made us exclude five of our nights given the poor velocity span.

Optimal Observational Strategy: • Although we were unsuccessful in detecting HD 217107 b, we set the outlines of future campaigns, by carefully defining a candidate selection criteria and adopting an optimized observational strategy that enhances the capability of this method. In terms of the ideal candidate, there are two fundamental considerations to take into account: How much radial velocity span can a planet have during an observing run, and how well can we distinguish the planetary spectrum after the stellar component removal. Thus, the systems that are best suited for this technique are those in very close orbits, allowing the planets to have high orbital velocity and higher planet-to-star flux ratios; and also those with high K-band fluxes, for better signal to noise

42 ratios. But a good candidate is not enough to ensure favorable conditions, since a non optimal choice of the observation dates can make our attempts futile. Then we propose an observing strategy where for each night of the period of observations, we calculated the radial velocity evolution in time, and specifically selected the nights where the velocity span of the planet is maximum. To explore the capabilities of the method and the improvement of our strategy, we simulated observations of other planetary systems as observed by Phoenix spectrograph, with the same amount of hours and an appropriate schedule of observations. Under these conditions, we recreated observations of synthetic data based on the extrasolar planet HD 179949 b, contrasting the use of our observing strategy with a regular observation schedule, we recovered the planetary companion signature with parameters −3 sini = 0.77 and Fp/Fs = 3 10 showing that the improvement is reflected in a greater correlation × degree peak, better determined best-fit parameters in the χ2-map, and deeper false alarm probability limits when using our observing schedule. Finally, we performed simulations for two other planetary systems following the most reasonable physical properties, successfully determining the inclination of the orbit of Tau Boo b and determining their best fit parameters.

General conclusions: • We also notice the existence of a sensitivity bias for this method, tending to favor the detection of higher inclination systems leaving an inclination-sensitivity dependence at a given planet-to-star flux ratio. This is expected as greater inclinations implies less massive planets, therefore, they appear more smeared during the average to create the star template spectra. Targets that have an important day/night side brightness contrast, would bias even more a detection for an edge on configuration, since a bigger portion of the day side surface would be observable from Earth. Observations at times of larger radial velocity spans (preferred by our optimal observing strategy) coincides with inferior and superior conjunctions, i.e. when the day side of the planet is maximum and minimum, respectively. Clearly, observations near superior conjunction would capture the largest amount of light possible from the planet and at the same time cover the largest radial velocity span for a determined time extent. We conclude that high resolution instruments as Phoenix at Gemini South Observatory, are capable of detecting extrasolar planets with this method. Our simulations, show that if we perform a careful treatment of the systematics, approaching to the photon noise limit, if we count with appropriate theoretical models for the thermal emission of these objects, and if we follow a smart scheme in the data acquisition, Doppler-shifted signals of extrasolar planets with flux ratios, with respect to their stars, as deep as 104 can be recovered. A refinement of this technique would involve the optimization of the total exposure time, evaluating the optimal distribution of time designated to the length of an observing run vs. the number of nights of observation. The use of others instruments, like CRIRES at VLT Observatory, which has a larger wavelength coverage, and thus, a larger effective sensitivity, would also decrease the total time of observation. While adding a phase-dependent function of the planet’s brightness, to account for the changing observed portion of the day/night side of the planet, as well as a phase function, accounting for different amounts of irradiation for an eccentric orbit, would increase the accurateness of the parameters fitted to the signal to be detected.

43 Appendices

44 Appendix A

Radial velocity

The orbit of a binary system is deﬁned by the so-called orbital parameters (see for example Batten [1973]): the orbital period, P; the inclination of the orbital plane, i; the position angle of the line of nodes, Ω; the argument of the periastron, ω; the semi-major axis of the orbit, a; the eccentricity, e; and the time of passage through the periastron, T.

In a Cartesian system of coordinates, xyz, with the orbital planet in the xy plane and the origin ﬁxed in the center of mass of the system. Let the x axis point toward the periastron of the reduced mass orbit, in this coordinate system the position and the velocity of the reduced mass is given by:

~r = r (cosν xˆ + sinν yˆ) (A.1) ˙ ~r = (r˙ cosν − rν˙ sinν) xˆ + (r˙ sinν + rν˙ cosν) yˆ (A.2)

Where ν is the true anomaly and r is the distance between the masses. In terms of the orbital parameters, the magnitude of the distance and velocity can be expressed as:

1 − e2 r = a (A.3) 1 + ecosν rν˙ esinν r˙ = (A.4) 1 + ecosν

And making use of the relation r2ν˙ = na2√1 − e2 with n the mean motion, we have:

na r˙ = esinν (A.5) √1 − e2 na rν˙ = (1 + ecosν) (A.6) √1 − e2

Then the velocity of the system can be written as:

˙ na ~r = (−sinνxˆ + (e + cosν)yˆ) (A.7) √1 − e2

45 To describe this orbit from an arbitrary system of coordinate, as seen this from an external observer, XYZ, where the Z axis goes along the line of sight. According to the deﬁnition of ω, i, and Ω the is given by a succession of three rotations:

- A rotation around the z axis by an angle ω

- A rotation around the x axis by an angle i, giving the inclination between the xy and XY planes.

- A rotation around the z axis by an angle Ω.

These rotations are represented by the rotation matrices:

cos ω −sin ω 0 1 0 0 cos Ω −sin Ω 0 − Ω Ω P1 =  sin ω cos ω 0  P2 =  0 cos i sin i  P3 =  sin cos 0  (A.8) 0 0 1 0 sin i cos i 0 0 1             Thus, the velocity vector in the XYZ system, is given by:

Vx r˙x r˙x cosω − r˙y sinω cosΩ − r˙x sinω + r˙y cosω sinΩcosi ˙ ˙ − ˙ Ω + ˙ + ˙ Ω  Vy  = P3P2P1  ry  =  ¡rx cosω ry sinω¢sin ¡rx sinω ry cosω ¢cos cosi  (A.9) V 0 r˙ sinω + r˙ cosω sinΩ  z     ¡ x¢ ¡y ¢        ¡ ¢ Plugging in the value of equation A.7, we have:

Vx −sin(ν + ω)cosΩ − cos(ν + ω)sinΩcosi − e(cosΩsinω + sinΩcosω cosi) na Vy = sin(ν + ω)sinΩ + cos(ν + ω)cosΩcosi − e(sinΩsinω − cosΩcosω cosi)   √1 − e2   V cos(ν + ω)sini + ecosω sini  z       (A.10)

We recognize now, from the Z axis, the amplitude of the velocity, K = nasini/√1 − e2, and the radial velocity curve:

V = K(cos(ν + ω) + ecosω) (A.11)

Lastly, adding the velocity of the center of mass, γ, we recover equation 2.1.

46 Appendix B

Error Propagation

In this chapter we will review the error formulas for the calculations done in this work, for this purpose, we make use of the Error Propagation Equation. For a quantity x that is a function of one or more measured 2 2 variables ui (each with variances σu,i), the variance σx for x is given by [Bevington and Robinson, 2003]:

∂x 2 σ2 = σ2 (B.1) x u ∂u u " # X µ ¶

B.1 Equilibrium Temperature

In section 2.3 we calculated the equilibrium temperature of a extrasolar planet (Teq) as a function of the star’s radius (Rs) the semi-major axis of the orbit (a), and the star’s temperature (Ts) through the equation:

1 − A 1/4 R 1/2 T = s T eq 4 a s µ ¶ µ ¶

We then calculated the variance of the equilibrium temperature (σTp), applying the error propagation formula:

σ 2 σ 2 σ 2 σ = T Ts + a + Rs (B.2) Tp eq T 2a 2R sµ s ¶ µ ¶ µ s ¶

with σTs the error in the star’s temperature, σa the error in the semi-major axis of the orbit, and σRs the error in the star’s radius.

B.2 Data Reduction

The probability distribution of photons detected in an observation is known to be Poisson. For which, it is known that for a signal level of N photons in a given pixel, the associated 1 sigma error is given by √N photons [Howell, 2006]. For our raw frames, in each pixel, we have registered the number of counts ADU,

47 which are related to the number of photons as ADU = N/G, where G is the gain of the instrument, then following equation B.1, the variance of a pixel is:

1 2 N σ2 =(√ADU G)2 = | | (B.3) ADU · · G G µ ¶ Considering the error introduced by the read out noise RON (in electrons) that behaves as shot noise, it must be added squared to the variance, we have for the variance of a pixel in the raw data:

N RON 2 σ2 = | | + (B.4) raw G G µ ¶

2 Accordingly, the same formula applies to the variance of the pixels in the raw Dark images (σd) and of the 2 raw Flat-ﬁeld images (σ f ). Next we propagate the errors, during the data reduction (see section 3.2). The master Dark frame is constructed from the median of a set of dark images, that is, at each pixels position, the value of the master dark frame (D) is the value from the set of dark images di that has equally number of values that are greater and lower than. To estimate the error in the master dark{ frame,} we compute the standard deviation of the mean, not the median. The standard deviation of the median is complicated and can only be estimated statistically using a bootstrap method, and it is more computationally expensive. Then, if N is the number of dark images in the set, the formula of the mean is:

1 N D = d (B.5) N i i=1 X And, thus, the corresponding variance of a pixel of the master dark frame is:

1 N σ2 = σ2 (B.6) D N2 d,i i=1 X

for the pixels in the master Flat-field frame (F0), constructed from the median of a set of flat-field images, equation B.6 also holds,

1 N σ2 = σ2 (B.7) F0 N2 f ,i i=1 X

Then, we must subtract the master dark frame (F1 = F0 − D) and the variance changes as:

2 2 + 2 σF1 = σF0 σD (B.8)

And lastly, the ﬂat is normalized to a median value of one:

F = F1/med(F1) (B.9)

48 the ﬁnal variance of the pixels in the master Flat-ﬁeld frame is:

σ 2 σ2 = F1 (B.10) F med(F ) µ 1 ¶ Then, to correct the raw images (R), we subtract the dark frame (D) and divide biy the ﬂat-ﬁeld frame (F):

R − D C = (B.11) F

Applying the error propagation formula for the sum and then for the division, the variance of a dark-ﬂat corrected image is:

1 R − D 2 σ2 = σ2 + σ2 + σ2 (B.12) C F2 raw D F F ( µ ¶ )

Now, the last step is the subtraction of the sky (S), which is just the dark-ﬂat corrected image in the opposed nodding position. The pixels in the ﬁnal corrected sky subtracted image (I = C −S) have then a variance given by:

2 2 + 2 σI = σC σS (B.13)

49 Appendix C

Observing Log of HD 179949

Columns description of table C.1

(1): Calendar day of the year 2009.

(2): Julian date at the start of the target observing run.

(3): Orbital phase at the start of the observing run (the origin, φ = 0, is set at the time of periastron).

(4): Orbital phase at the end of the observing run.

(5): Radial velocity span of the star during the observing run.

(6): Average radial velocity of the star during the observing run.

(7): Maximum radial velocity span of the planet during the observing run (for sini = 1.0).

(8): Average radial velocity of the planet during the observing run (for sini = 1.0).

(9): Selected nights for simulation implemented our observing strategy (o) and for the other simulation (x).

Table C.1: Observation log for HD 179949.

Calendar JD start phase phase ∆vs v¯s ∆vp v¯p selected day −2450000 start end m s−1 ms−1 km s−1 km s−1 night (1) (2) (3) (4) (5) (6) (7) (8) (9) 24 Apr 4945.78 0.15 0.19 28.22 -29.46 -40.63 42.42 o 25 4946.78 0.48 0.52 -4.86 107.57 6.99 -154.86 26 4947.77 0.79 0.83 -24.41 -62.61 35.14 90.14 27 4948.78 0.12 0.16 26.57 -49.46 -38.25 71.21 o 28 4949.76 0.44 0.48 1.21 109.23 -1.74 -157.26 29 4950.76 0.76 0.80 -26.87 -41.94 38.69 60.38 30 4951.79 0.09 0.13 23.89 -67.78 -34.40 97.58 o 1 May 4952.75 0.41 0.45 6.55 106.10 -9.43 -152.76 2 4953.75 0.73 0.77 -28.13 -19.87 40.49 28.61 o 3 4954.79 0.06 0.10 20.23 -83.96 -29.12 120.88 Continued on next page

50 Table C.1 – continued from previous page

Calendar JD start phase phase ∆vs v¯s ∆vp v¯p selected day −2450000 start end m s−1 ms−1 km s−1 km s−1 night (1) (2) (3) (4) (5) (6) (7) (8) (9) 4 May 4955.74 0.37 0.41 11.71 98.74 -16.86 -142.15 5 4956.79 0.71 0.75 -28.31 -6.72 40.75 9.68 6 4957.79 0.03 0.07 15.77 -97.03 -22.71 139.69 x 7 4958.74 0.34 0.38 16.51 87.35 -23.76 -125.76 x 8 4959.79 0.68 0.72 -27.82 13.66 40.05 -19.66 9 4960.79 0.00 0.04 10.61 -106.71 -15.28 153.63 x 10 4961.73 0.31 0.35 20.75 72.32 -29.88 -104.12 o 11 4962.79 0.65 0.69 -26.42 33.57 38.03 -48.33 12 4963.79 0.97 0.01 5.08 -112.40 -7.31 161.82 13 4964.72 0.28 0.32 24.16 54.84 -34.79 -78.95 o 14 4965.79 0.62 0.66 -24.22 51.96 34.87 -74.81 15 May 4966.71 0.92 0.96 -5.43 -112.16 7.82 161.47 16 4967.72 0.24 0.28 26.78 34.60 -38.55 -49.82 o 17 4968.79 0.59 0.63 -21.31 68.50 30.68 -98.62 18 4969.71 0.89 0.93 -11.44 -105.36 16.47 151.68 19 4970.71 0.21 0.25 28.38 12.36 -40.86 -17.79 20 4971.79 0.56 0.60 -17.75 82.84 25.55 -119.27 21 4972.70 0.86 0.90 -16.80 -94.03 24.19 135.37 22 4973.72 0.18 0.23 28.82 -6.00 -41.50 8.64 23 4974.79 0.53 0.57 -13.70 94.33 19.72 -135.81 24 4975.69 0.82 0.86 -21.32 -78.66 30.70 113.25 25 4976.79 0.18 0.22 28.82 -10.02 -41.49 14.42 26 4977.79 0.50 0.54 -9.27 102.72 13.34 -147.88 27 4978.68 0.79 0.83 -24.79 -60.07 35.69 86.48 28 4979.80 0.15 0.19 28.15 -30.70 -40.53 44.20 29 4980.80 0.47 0.51 -4.57 107.77 6.58 -155.16 30 4981.67 0.76 0.80 -27.10 -39.15 39.02 56.36 31 4982.80 0.12 0.16 26.42 -50.73 -38.04 73.03 o 1 Jun 4983.67 0.40 0.44 7.02 105.62 -10.11 -152.06 2 4984.66 0.72 0.76 -28.20 -16.85 40.60 24.26 3 4985.80 0.09 0.13 23.65 -69.10 -34.04 99.48 o 4 4986.66 0.37 0.41 12.16 97.88 -17.50 -140.92 x 5 4987.72 0.71 0.75 -28.31 -8.41 40.75 12.11 6 4988.80 0.06 0.10 19.96 -84.92 -28.73 122.26 x 7 4989.65 0.34 0.38 16.91 86.16 -24.34 -124.04 x 8 4990.80 0.71 0.75 -28.30 -5.26 40.75 7.57 9 4991.79 0.03 0.07 15.21 -98.32 -21.90 141.55 x 10 4992.64 0.30 0.34 21.08 70.91 -30.35 -102.09 11 4993.79 0.68 0.72 -27.65 16.97 39.81 -24.43 12 4994.79 0.00 0.04 9.46 -108.26 -13.61 155.87 13 4995.64 0.27 0.31 24.53 52.51 -35.31 -75.60 14 4996.78 0.64 0.68 -25.91 38.57 37.31 -55.53 15 Jun 4997.63 0.92 0.96 -5.96 -111.77 8.58 160.92 16 4998.63 0.24 0.28 27.06 31.71 -38.95 -45.65 Continued on next page

51 Table C.1 – continued from previous page

Calendar JD start phase phase ∆vs v¯s ∆vp v¯p selected day −2450000 start end m s−1 ms−1 km s−1 km s−1 night (1) (2) (3) (4) (5) (6) (7) (8) (9) 17 Jun 4999.77 0.61 0.65 -23.21 58.39 33.42 -84.06 18 5000.62 0.88 0.93 -11.92 -104.57 17.16 150.54 19 5001.62 0.21 0.25 28.51 9.30 -41.05 -13.39 20 5002.77 0.58 0.62 -19.70 75.61 28.36 -108.85 21 5003.61 0.85 0.89 -17.24 -92.81 24.83 133.62 22 5004.63 0.18 0.22 28.82 -8.99 -41.49 12.94 23 5005.76 0.55 0.59 -15.48 89.80 22.29 -129.28 x 24 5006.61 0.82 0.86 -21.68 -77.12 31.21 111.03 25 5007.73 0.18 0.22 28.82 -8.18 -41.50 11.78 26 5008.75 0.51 0.55 -10.73 100.37 15.45 -144.50 27 5009.60 0.79 0.83 -25.05 -58.29 36.06 83.91 28 5010.74 0.16 0.20 28.43 -25.30 -40.93 36.42 29 5011.74 0.48 0.52 -5.61 106.96 8.08 -153.99 30 5012.59 0.75 0.79 -27.24 -37.31 39.22 53.72 1 Jul 5013.74 0.12 0.17 26.79 -47.45 -38.57 68.31 2 5014.59 0.40 0.44 7.49 105.10 -10.78 -151.31 3 5015.58 0.72 0.76 -28.24 -14.94 40.66 21.51 4 5016.73 0.09 0.13 23.88 -67.87 -34.37 97.71 5 5017.58 0.37 0.41 12.57 97.04 -18.10 -139.71 x 6 5018.64 0.71 0.75 -28.31 -6.27 40.75 9.03 7 5019.72 0.06 0.10 19.80 -85.47 -28.50 123.06 8 5020.57 0.33 0.37 17.29 85.00 -24.89 -122.37 9 5021.71 0.70 0.74 -28.28 -2.56 40.72 3.68 10 5022.71 0.03 0.07 14.73 -99.39 -21.20 143.09 11 5023.56 0.30 0.34 21.42 69.39 -30.84 -99.89 12 5024.70 0.67 0.71 -27.50 19.60 39.59 -28.22 13 5025.70 0.00 0.03 8.95 -108.88 -12.88 156.76 14 5026.55 0.27 0.31 24.79 50.75 -35.69 -73.06 15 Jul 5027.70 0.64 0.68 -25.63 41.14 36.90 -59.22 16 5028.55 0.91 0.96 -6.51 -111.33 9.37 160.27 17 5029.54 0.24 0.28 27.23 29.77 -39.20 -42.85 18 5030.69 0.61 0.65 -22.94 59.95 33.03 -86.31 19 5031.54 0.88 0.92 -12.43 -103.70 17.89 149.29 20 5032.53 0.20 0.24 28.58 7.37 -41.15 -10.61 21 5033.68 0.57 0.62 -19.35 77.01 27.85 -110.86 22 5034.53 0.85 0.89 -17.68 -91.56 25.45 131.82 23 5035.57 0.18 0.22 28.82 -6.23 -41.50 8.97 24 5036.67 0.54 0.58 -15.08 90.89 21.70 -130.85 25 5037.52 0.82 0.86 -22.00 -75.65 31.68 108.91 26 5038.66 0.19 0.23 28.82 -5.43 -41.50 7.82 27 5039.67 0.51 0.55 -10.29 101.12 14.81 -145.58 28 5040.51 0.78 0.82 -25.28 -56.59 36.39 81.47 29 5041.66 0.15 0.19 28.29 -28.23 -40.73 40.64

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