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UNIVERSIDAD DE CHILEFACULTAD DE CIENCIAS FÍSICAS Y MATEMÁTICASDEPARTAMENTO DE ASTRONOMÍA
A HIGH RESOLUTION SPECTROSCOPIC SEARCH FOR THE THERMAL EMISSIONOF THE EXTRASOLAR PLANET HD 217107 b
TESIS PARA OPTAR AL GRADO DE MAGÍSTER ENCIENCIAS, MENCIÓN ASTRONOMÍA
PATRICIO ERNESTO CUBILLOS VALLEJOS
PROFESOR GUÍA:PATRICIO ROJO RÜBKE
MIEMBROS DE LA COMISIÓN:María Teresa Ruiz González
Diego Mardones PérezJohn R. Barnes
SANTIAGO DE CHILEMAYO 2011
Resumen
En este trabajo hemos retomado y afinado un método de correlación para buscar directamente, en altaresolución, el espectro de planetas extrasolares sin tránsito. Nuestro objetivo principal es caracterizar laspropiedades físicas de estos objetos, específicamente la inclinación de su órbita, su masa y la proporción delos 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 tornoal centro de masa del sistema. Para observaciones lo suficientemente extensas, el espectro del planeta se va adesplazar con respecto al de la estrella lo suficiente para que sea detectable en observaciones espectroscópicasde alta resolución. Alineando y sumando los espectros de cada noche construimos un modelo del espectroestelar. Este es substraído a cada espectro, dejando un espectro residual compuesto por la emisión del planetainmerso 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 entrenuestros espectros residuales ymodelos de la emisión termal de la atmósfera del planeta. Evaluando para distintos valores de la inclinaciónde la órbita del modelo, obtenemos una curva de correlación. El valor de esta curva debe ser máximocuando la inclinación coincida con la inclinación del sistema. Para calcular elvalor de la proporción de losflujos entre el planeta y su estrella, recreamos observaciones inyectando espectros sintéticos del planeta conparámetros dados de inclinación y proporción de flujos. Luego, mediante un test deχ2 entre las curvas decorrelación, 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 2.14µm. Como resulatado, no logramos detectar el planetacon los datos disponibles, aunque determinamos límites superiores para su emisión termal, siendo menor a5×10−3 veces la emisión de su estrella, con 3–σ de certeza.
Además, exploramos el escenario ideal de observación para proyectos futuros, y describimos una es-trategia óptima de observación y selección de candidatos que maximice las probabilidades de detección.Finalmente, simulando observaciones realistas para Phoenix, generamos datos sintéticos de observacionesde otros candidatos para demostrar las ventajas de usar nuestra estrategia de observación. Calculamos límitesde 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 ofnon-transiting extrasolar planets. The main objective of this work is to characterize the physical propertiesof non transiting extrasolar planets, aiming to obtain the inclination of the orbit, themass of the planet, andthe 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 planetaround the center of mass of the system. For long enough observing runs, the spectral signals will shiftwith respect to each other, and thus will be detectable with high resolution spectroscopic observations. Byaligning 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 innoise.
The planetary spectrum is not readily detectable due to its much fainter signal.Therefore, we search forthe planet calculating the correlation between the residual data and thermal emission models of the planet’satmosphere, assigning different values to the inclination of the orbit of the models, expecting a peak in thecorrelation when we match the real value of the inclination. To asses the valueof the planet-to-star flux ratio,we reproduced the observations using synthetic spectra, injecting a scaled and shifted planetary spectrumaccording to given flux ratios and inclinations. Then, we determine the bestfitting parameters through aχ2
minimization between the data and the synthetic results.
We present the results of this technique on the planetary system HD 217107, observed with the highresolution spectrograph Phoenix, at 2.14µm. We could not detect the planet with our current data, but wepresent an upper limit to its thermal emission determined with a Monte Carlo Bootstrap method. With aconfidence 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 obser-vational and selection strategy to increase future probabilities of successby considering the best conditionsto observe and the best candidates using this method.
Finally, using realistic data sets for the Phoenix instrument we carried out simulations on other planet can-didates to demonstrate the improvements achieved when we use our optimal observing strategy. Detectabilitylimit of the method using this instrument and the simulated planets are given. We conclude that with oursame number of observations, it is possible to detect extrasolar planets with planet-to-star flux ratios of theorder 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 tra-bajo. Tanto su ayuda, guía, y experiencia, como constante apoyo anímico ypaciencia fueron fundamentalespara 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 mispropias decisiones, sin cuestionar, al momento de definir mi futuro. Gracias a ellos es que he tenido laoportunidad 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 quedecir 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 deMagíster, gracias por su disposición. De ellos adquirido un enorme conocimiento tanto en astronomía comoen las ciencias relacionadas. El nivel en dominio de la materia y capacidad de enseñar de la mayoría de ellosha 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 estosaños como estudiante de la Universidad de Chile, haciendo que los buenosmomentos hayan sido realmenteespectaculares y dando apoyo en los momentos mas críticos. Fue un agradorealmente compartir aquellosaños 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, MatiasJones, Viviana Guzmán, Felipe Murgas, Andrés Guzmán, Matias Vidal, entre tantos otros, han sido la mejorcompañía tanto como para pasar un buen momento como de ayuda en mis cursos ytrabajo.
Contents
1 Introduction 1
2 The Planetary System HD 217107 3
2.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 3
2.2 Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 4
2.3 Flux Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
3 Observations and Data Reduction 11
3.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 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 Data Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 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 Target Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 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 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 47
C Observing Log of HD 179949 50
II
List of Tables
2.1 HD 217107 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 3
3.1 Gemini South Telescope Characteristics . . . . . . . . . . . . . . . . . . . . . .. . . . . . 11
3.2 Observing log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 12
3.3 Infrared Lines Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 14
4.1 Theoretical Atmospheric Models. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 22
5.1 Favorable targets for Gemini South . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 31
5.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 33
C.1 Observation Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 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 Raw frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 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 Planetary Blurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 3 km s−1 Planetary Blurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Spectral Irradiances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 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 extrasolargiant planet in a close orbit (Hot-Jupiters)around the main sequence star 51 Pegasi [Mayor and Queloz, 1995] marked the beginning of a new researchfield in planetary sciences. Since then, the study of extrasolar planets hasbecame one of the most developingbranches in astronomy. As has constantly happened along the history, thisdiscovery brings more questionsthan answers. 51 Pegasi b, a Jupiter-Mass planet located at only 0.05 AU from his host star defied the existingformation theories [for example, Pollack, 1984], and broke the scheme ofplanetary systems with less massiverocky 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 mostsuccessful techniqueto discover extrasolar planets today, that the scenario has changed dramatically, knowing now hundreds ofother worlds outside the solar System. This technique, measures periodicalchanges in a star’s radial ve-locity, as the star and the planet orbit about their common center of mass, the star motion is detectablethrough the Doppler Effect. Along with other search methods like gravitational microlensing surveys, transitlight curve measurements, or pulsar timing monitoring, over 400 extrasolar planets have been discovered sofar. Their characterization, then, started to take place, most of these studies are carried out at optical andinfra-red wavelengths since it is there where the reflected light and the thermal emission reaches its maxi-mum 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 configurationpermits to directly observe theplanet, 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 planetaryatmospheres characterization has achieved the most exciting advances. Space telescopes have been able tomeasure secondary eclipses through the use of both spectroscopy andbroadband photometry. To mentionsome examples, it has been possible to identify molecules by fitting theoretical models of the planetary atmo-spheres, like water absorption in HD 189733b [Tinetti et al., 2007], as themost likely cause of the variationof 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 atmosphereand confirmed the existence of water using data from Hubble Space Telescope. Another interesting exam-ple is the measurement of the variation of the thermal emission with orbital phase as the fraction of theday-side surface we see is changing, for HD 189733b at 8µm distinguishing also the transit and secondaryeclipse [Knutson et al., 2007]. Just recently there have been successful results of measurements of secondaryeclipses 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 velocityextrasolar planets. For the rest of
1
the non-transiting extrasolar planets, the only way to be able to constrain physical parameters or characterizetheir atmospheres, is through the direct detection of their light, rather than observing the radial reflex motioninduced in the star. The very small flux ratios between the planet and their host stars, makes a direct detectiona very challenging goal. From secondary eclipse observations from Spitzer we know that planet-to-star fluxratios can be as high as 2.5×10−3 (measured at wavelengths between 3.6 and 24µm), at 2.4µm the expectedflux should be less than these values. From ground based telescopes, several authors have tried to observethe high resolution spectra of the planetary systems, separating the planetary and stellar spectra given theirDoppler-shift wobble. In the optical, Collier Cameron et al. [1999] performed a first attempt of the directdetection of the Doppler-shifted signature of starlight reflected from the giant exoplanet orbiting Tau Boötisimplementing a least-squares deconvolution technique, he set an upper limit to the albedo and radius, laterapplied the same technique in the search ofυ Andromeda b [Collier Cameron et al., 2002]. In the recentyears, 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 modelsof the reflected spectra), alsofinding upper limits for their albedos. While in the near-infrared, there is alsoan extensive list of attempts todetect 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 starswith 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 andHD 179949 b. LastlyBarnes 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 thefirst spectrum of the angularly resolved image of an extrasolar planet. Nonetheless, those resolved systemspresent 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-JupiterHD 217107 b using high resolution spectra, and models of its atmospheric spectrum in the infrared. Withpositive results this method could provide new information of non-transiting extrasolar planets and improvethe calibration of high-resolution atmospheric models. At the same time, it will also validate a method thatcould potentially be used to characterize atmospheres of non-transiting planets. Direct detection of extrasolarplanets’ emitted or reflected spectra, coupled with broadband photometry, would provide complementary in-formation on its characteristics, such as its temperature, chemical composition,and the presence of chemicaltracers 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 physicalproperties. 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 signaland search the Doppler-shifted signature of the planet, we also describethe theoretical planet atmosphericspectrum, and present the results of our observations of HD 217107. In Chapter 5 we discuss the sensitivityof the method and a strategy of the ideal data acquisition situation, then show simulations to illustrate theimprovements that can be achieved. And finally, in Chapter 6 we summarize the main conclusions of thiswork. In addition, in Appendix A take an in-depth analysis of the radial velocity method equations, men-tioned in Chapter 2. In Appendix B we detail the error propagation formulasused throughout this work. Andin Appendix C we complement the information of our simulations with an observing log of the planetarysystem 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 theSun in mass, radius, andeffective temperature, its spectral type, G8 IV, indicates that it is starting toevolve 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
Rs (R⊙) 1.08± 0.040.03 T07
Ks (m s−1) 140.6±0.7 W07
vg (km s−1) -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
mpsini (MJup) 1.33±0.05 W07
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: Perrymanand 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 measure-ments of the star at Lick and Keck observatories, the detection was then confirmed by Naef et al. [2001] usingCORALIE data. Later, Fischer et al. [2001] identified a linear trend in the residuals of the radial velocitycurve fit, and Vogt et al. [2005], hinted by a unusually large eccentricityalso, postulated the existence of asecond planetary companion (HD 217107 c) in an external orbit with a period of 8.5 years. The presenceof 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 minimummass of 2.6 MJupwith 10% of error for HD 217107 c, although a full orbit has not been observed yet, whileHD 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, isthe radial velocity technique (See figure 2.1). The basis of the method is relatively simple: As the planet andits host star inflict each other a gravitational tug, the spectra of that star is Doppler shifted as it revolves aroundthe center of mass of the system. The high precision achieved in radial velocity measurements has madethe plethora of planet detections possible. Nowadays, velocity variations down to 1 m s−1 can be achievewith HARPS (High-Accuracy Radial Velocity Planetary Searcher), the mostprecise Doppler-measurementsinstrument [Mayor et al., 2003].
Figure 2.1: Chart of the different detection methods, with 399 extrasolar planets observed (up to February2010), the radial velocity detection method has allowed the detection of most ofthem.
The radial velocity curve of the star in a binary system as seen from Earth, can be expressed as (seeAppendix A for details):
vssini = γ + Ks(cos(ν +ω) + ecosω) (2.1)
4
Whereγ is the radial velocity of the center of mass of the system,Ks is the radial velocity amplitude ofthe star,ω is the angle between the line of nodes and the line from the star to the planet at periastron (theargument of the periastron), andν is the position angle measured from the periastron (the true anomaly).
The time dependence of the true anomaly is given implicitly, through the eccentricanomaly (E), by theset of equations:
tanν
2=
√
1+ e1− e
tanE2
(2.2)
2π
P(t − T) = E − esinE (2.3)
Equation 2.3 is known as the Kepler equation (see for example Murray and Dermott [1999]), whereT isthe time of periastron passage of the planet. Thus, the orbital parametersP, T, e, andω can be determinedthrough measurements of equation 2.1.
Then, by determining the value of these orbital parameters, it is possible to numerically solve equations2.2 and 2.3 and obtain the radial velocity curve of an object for any giventime. Figure 2.2 shows the radialvelocity 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 11.7years of period) in the radial velocity of the star of 37.5 m s−1 of amplitude [Wright et al., 2009], but theplanet’s smaller mass (2.6 MJup) and its greater distance from the star (5.32 AU) makes the interaction withHD 217107 b of a secondary order in importance.
From these orbital parameters, we can derive other physical properties of the orbit. From the velocityamplitude of the star, written in terms of the orbital parameters:
Ks =2π
Passini√
1− e2(2.4)
Whereas is the semi-major axis of the star’s orbits around the center of mass, and fromKepler’s third law:
a3 =
(
2π
P
)2
G(Ms+ mp) (2.5)
with a = as + ap, and making use of the relationasMs = apmp; the minimum mass of the planetary com-panion (mpsini) can be deduced (under the approximationM ≫ mp) as:
mpsini ≈(
P2πG
)1/3
KsM2/3s
√1− e2 (2.6)
From a quick analysis of equation 2.6, it is not surprising why the first extrasolar planets discovered are asmassive (or more) as Jupiter and in short period orbits, we see thatKs (the observable) is proportional tomp,
it is (weakly) inversely proportional to the period of the orbit, and it is proportional toM2/3s .
The radial velocity of the planet,vpsini is given by its reflex motion around the center of mass of thesystem, is proportional to the star’s radial velocity curve and it is shifted half of the phase:
5
0.0 0.2 0.4 0.6 0.8 1.0Orbital Phase
150
100
50
0
50
100
150
v ssin
i (
m s
1 )
Figure 2.2: Radial velocity curve of HD 217107vs. orbital phase. The crosses mark the observations ofWittenmyer et al. [2007] used to compute this orbital solution. The boxes overthe curve indicatethe coverage of our observations, the filled boxes represent the runsutilized in the analysis, whilethe open boxes represent the discarded runs (details in Chapter 4).
vp(t)sini = − vs(t)siniMs
mpsini×sini (2.7)
Unfortunately, one of the disadvantages of the radial-velocity technique,is that the sine of the inclinationremains 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 orbitalparameters obtained from equation 2.1 lets us determine its value for any desired time, solving the Keplerequation (eq. 2.3). The approximated value of the minimum mass of the planet is also known, and it is givenby 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 effectivetemperature (obtained from the spectroscopy). As the search of extrasolar planets is performed only in thevicinity of the Sun, all these planet host stars have a measurable parallax,thus their absolute magnitude canbe 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 curvein time, where the only unknownparameter is the inclination of the orbit. Other than transit observations, a direct detection of the radialvelocity of the planet seems to be the only way to determine the value of the inclination, which would allowto 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 comparisonto their host stars, this will give an idea of the expected order of magnitude of the planet to star flux ratio asa function of wavelength.
The total luminosity of non transiting extrasolar planets consist of three components: the directly reflectedstellar light (which does not contribute to the heating of the planet, hence it does not interfere in the energybalance), the thermally re-emitted flux of the absorbed energy from the star, and the intrinsic radiation dueto the object’s contraction and cooling.
Then, assuming that the total re-emitted power by the planet (Pout) is being balanced by the incident powerfrom 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 powerof the total incident radiation on the planet, determines the energy absorbedby the planet as 1− A times thestellar flux incident on the planet (Fs), integrated over the intersected cross section:
Pin = (1− A)FsπR2p (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 calledBlack-Body Radiation, this is the radiation of a cavity inthermodynamic equilibrium at a fixed temperatureT. Its radiation is isotropic and the power per unit area,per unit solid angle per unit frequency of a black body at temperatureT is:
Bλ(T) =2hc2
λ5
exp[
hc/(λkT)]
− 1 (2.10)
with k the Boltzmann constant,c the speed of light in vacuum, andh the Planck constant. If this isintegrated over all wavelengths, and over all angles we obtain the black body irradiance or flux density:
F =∫
dλ
∫
dΩBλ(T) = σT4 (2.11)
with σ the Stefan-Boltzmann constant. Evaluating at the surface of the star, the temperature is called theeffective temperature, this is the Stefan-Boltzmann law. Knowing this, we write the stellar flux at the planetdistance in terms of the stellar surface flux using the inverse square law of fluxes:
a2 ·Fs(a) = R2s ·Fs(Rs) (2.12)
And using the Stefan-Boltzmann law, combining the expressions 2.11 and 2.12we can rewrite equation2.9 for the incident power on the planet as:
Pin = (1− A)πR2pσT4
s
(
Rs
a
)2
(2.13)
7
Assuming that the energy absorbed by the planet is thermalized before being re-emitted, the planet willadopt a surface effective temperature,Te f f. Accordingly, the thermally emitted power is equal to its surfaceflux emitted integrated over the emitting area:
Pout = 4πR2pσT4
e f f (2.14)
The equilibrium temperature of a planet (Teq), is defined as its effective temperature when the planetbalances the energy received from its host star with its the thermal radiationemission. For highly irradiatedatmospheres, the internal energy in the energy balance equation is negligiblein comparison to the strongstellar irradiation, thus, neglectingPint from equation 2.8 and using equations 2.13 and 2.14 we have for theequilibrium temperature:
Teq =
(
1− A4
)1/4(
Rs
a
)1/2
Ts (2.15)
As mentioned at the beginning of this section, the total luminosity of the planet hasalso a directly reflectedlight 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 considerthe reflected star light on the planet.
Then, for a reference value of the bond albedo ofA= 0, we found an equilibrium temperature for HD 217107 bof Teq = 1040±19 K. This value allow us to have a broad idea of the spectral irradiance ofthe planet com-pared to that of the star as a function of wavelength when approximated as aPlanck function. The spectralirradiance observed, as seen from Earth, of each one is obtained integrating over the solid angle:
Fλ(T) =∫
dΩBλ(T)cosθ = πBλ(T)
(
Rd
)2
(2.16)
WhereR is the radius of the object, andd is the distance from the observer to the object. Figure 2.3 showsa plot of the spectral irradiance of HD 217107 and HD 217107 b. From the giant extrasolar planets withmeasured radius, most of the values lie within one and two Jupiter radii, we expect that the unknown valueof HD 217107 b’s radius is most likely within this range. We plot then, the planet flux for three differentradii, between one and two Jupiter radii.
Important conclusions can be made from Fig. 2.3, we see that the planetaryspectrum peaks in the infraredregion of the spectrum between 2 and 3µm, while the star spectrum peaks at visual wavelengths, near 0.4µm. Toward shorter wavelengths the fluxes drop exponentially (leftward of∼1 µm for the planet, while thestar’s flux starts to drop at even shorted wavelengths). On the other side, for wavelengths longer than 10µmit falls as a power law, proportional toλ−4.
This behavior can be deduced also from equation 2.10. For short wavelengths its called Wien’s approxi-mation, we haveλ ≪ hc/kT, and the exponential term dominates the emission:
Bλ(T) =2hc2
λ5 exp
(
−hc
λkT
)
(2.17)
While in the long wavelengths approximation, known as the Rayleigh-Jeans law, λ ≫ hc/kT, we have:
Bλ(T) =2ckTλ4 (2.18)
8
10-1 100 101 102
Wavelength (m)
10-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Flux (
Wm
2 m1 )HD 2171072.0 RJup planet
1.5 RJup planet
1.0 RJup planet
Figure 2.3: Log-log plot of the spectral irradiance of HD 217107 (black) and of HD 217107 b (simulatedfor three radii between 1 to 2 Jupiter radii) as function of wavelength, emittingas black body.The effective temperatures of the star and the planet are 5646 K and 1040 K, respectively. Thevertical dashed line marks the waveband of our data around 2.14µm.
More illustrating is the planet to star flux ratio as a function of wavelength, which is given by:
Flux ratio =Fλ(Tplanet)Fλ(Tstar)
=Bλ(T = 1040 K)Bλ(T = 5646 K)
(
Rp
Rs
)2
(2.19)
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, theflux ratio decreases, while for longer wavelengths the flux ratio increases, and from wavelengths greater than10 microns remains nearly constant. Although the best flux ratios are foundfor longer wavelengths, thenet flux from the star and the planet fall asBλ(T) ∝ λ−4.In consequence, the Signal to Noise ratio for longerwavelengths will be significantly lower than in the near infrared range. Thebottom panel shows the flux ratioaround our observing band (from 2.136 to 2.145µm). Within a narrow band like this, the flux ratio does notchange 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 importanteffect in the flux ratio.
A black-body approximation gives an estimate of the order of magnitude of theflux ratio. A better de-termination of the planet’s emergent spectrum involve the consideration of a number of other factors as theatmospheric chemistry or radiative transference (See Section 4.2).
9
100 101 102
Wavelength (m)
10-7
10-6
10-5
10-4
10-3
10-2
Pla
net
to s
tar
flux r
ati
o
2.0 RJup
1.5 RJup
1.0 RJup
2.130 2.135 2.140 2.145 2.150 2.155Wavelength (m)
10-6
10-5
10-4
10-3
10-2
Panet
to s
tar
flux r
ati
o
2.0 RJup
1.5 RJup
1.0 RJup
Figure 2.4: Top: Planet to star flux ratio of the system HD 217107 emitting as black bodies for the planetradii: 2.0, 1.5, and 1.0 Jupiter radius (red, green, and blue respectively). Bottom: Same as above,but zoomed on the region around 2.14µm, the two vertical lines enclose the wavelength rangeselected 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 spectralresolution near-infrared spectrometer. The instrument is mounted on a 8.1-meter diameter altitude-azimuthtelescope at Gemini South Observatory, in Cerro Pachón, Chile, at an altitude of 2722 meters. Phoenix is ahigh 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 AladdinII array. An argon hollow cathode wavelength calibration source is supplied with the instrument. Table 3.1summarizes 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 – 2.145µm
Slit length 14 arcseconds
Slit width 3 pixels
Dispersion ∼ 10−5µm
Spectral resolution 40 000
We observed 11 nights, between August 14 and November 28 of 2007, collecting over 950 frames of ourtarget, which represent 15.4 hours of observation (see Table 3.2). This wavelength band presented a fewnumber of telluric absorption lines, leaving most of the spectrum available forfurther processing, but theselines 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 toregister the sky at the same pixels of the target). After receiving the data from the first runs, we had to tuneour 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 saturationof the instrument. For the first four nights, the exposure time was set to 25 seconds, whereas for the rest ofthe nights it was set to 80 seconds. We requested as well, dark frames (10per night), flat-field frames (10per 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 anindependent data set in order to minimize any systematic effects that might arisedue to different atmosphericconditions 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 (typically10) dark frames, which we subtracted from each raw data frame, also built a master Flat-field image foreach night in a similar way. The routines, used the flat-field images to identify hot pixels, and mark themas bad pixels if they have a value beyond 3.5 sigma from the median of the values of the 9 pixels in itsneighborhood, iterating three times, masking the discarded pixels before thenext iteration. Bad pixels werenot considered in any further processing stages. We then divided the frames by the master flat-field to correctfor the pixel-to-pixel variation in the CCD response. Then, to remove the sky emission we subtract eachimage from their corresponding opposite A or B image. Appendix B.2 gives athoroughly description of thereduction.
Finally, we extracted the spectra from the frames using an IDL implementation1 of the Optimal SpectrumExtraction algorithm [Horne, 1986]. The algorithm produces the best attainable signal to noise ratio byapplying nonuniform pixel weights in the extraction, assigning lower weightsto noisy pixels containing asmall fraction of light included in the object spectrum, in this way, avoiding waste of information. Thealgorithm 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: Nodposition B. As seen in this image, the vertical axis represents the spatial position in sky alongwhich the slit is oriented (14 arc seconds length from top to bottom), while the light is beingdispersed along the horizontal axis. During the observations, the star was located at two differentnodding spatial positions, thus, for a pixel that captured the star light in a frame, in the next framewill 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) / (standardspectrum). 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 thenmade 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 nobad pixels are found. The optimal extraction achieves the highest possiblesignal to noise ratio byweighting 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 and6 to find the spatialprofile image, but do not use the bad pixel mask found in step 5 for the extraction. Rather, next iteratesteps 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 andstrengths in a high resolution ThAr line atlas [Hinkle et al., 2001]. Table 3.3 shows the lines and wave-lengths from the line catalog in the range of our data that matched the lines in ourlamps exposures. Figure3.2 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
0 100 200 300 400 500 600 700 800 900Pixel
0
100
200
300
400
500
600
Flu
x (c
ounts
)
Ar I
Ar I blend2.14528 m
Ar II
UnidAr II
Ar II
Th I2.13751 m
Ar I
Figure 3.2: Lamp calibration exposure for the observing run of November25th. Eight of the lines of the data(labeled lines) matched lines in the catalog, we labeled two of them with their correspondingwavelength as reference.
As we had available only one lamp for each night and nod position and (sub pixel) offsets in wavelengthare present in the data, this wavelength solution represented only a roughwavelength calibration. To reachthe high precision needed for this work, we fine tuned the calibration using ahigh resolution spectrum of theSun2 to identify the telluric lines and use them to perform a more precise calibration.
2http://bass2000.obspm.fr/solar_spect.php
14
2.138 2.140 2.142 2.144Wavelength (m)
0.80
0.85
0.90
0.95
1.00
1.05
Norm
aliz
ed f
lux
Sun spectrum
Data spectrum
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). Thelines cover the whole spectrum leaving few pixel where the solution is extrapolated rather thaninterpolated.
We identified the telluric lines as those lines appearing in both the solar spectrumas well as in an averagespectra of our data set. Considering that HD 217107 is a star of a spectral type similar to the Sun, bothstar should share similar stellar absorption lines, nevertheless, given the non-zero proper radial velocity ofthe target with respect to Earth and given the dispersion relation of the instrument of∼10−5, correspondingto a velocity difference of∼1.4 km s−1 per pixel, those lines should not be aligned, therefore, avoiding thepossibility of a misidentification of lines.
For this telluric calibration, first we calculated the relative shifts between the frames in each night andnod position set, we selected the first spectrum as reference, while the rest of the spectra was shifted (usingspline interpolations) in intervals of 0.01 pixels for a range of two pixels in each direction, then calculatedtheir correlation with the reference spectrum, and minimized the root-mean-square of that value to find theshift that returned the best match. Next, we co-align all of the spectra in theset, and construct an averagespectrum. We used this average spectrum to calibrate the wavelength by identifying the absorption linescommon to both spectra (see Figure 3.3). A total of fifteen lines across the spectrum were selected andrecorded 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 solutionof the form:
wavelength =c0 + c1 · p + c2 · p2 (3.1)
in this formula, the wavelength is expressed in microns, withp the value of the pixel position. Figure3.4 shows the fit and the corresponding residuals. We performed a third order polynomial fit, but it did notrepresent 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 knownto present a high variability in strength in short amounts of time, making the telluric absorption lines changefrom frame to frame, thus, in order to avoid systematic errors produced bya incorrect removal of the telluric
15
lines, we decided to mask the pixels at wavelengths dominated by the identified telluric absorptions, anddiscard them from all upcoming processing stages. As consequence, from the 1024 initial pixels of eachspectrum, on average, 670 pixels remained for the upcoming data analysis step.
0 100 200 300 400 500 600 700 800 900
Pixel position
2.136
2.138
2.140
2.142
2.144W
avele
ngth
(
m)
0 100 200 300 400 500 600 700 800 900Pixel position
-2e-5
-1e-5
0.0
1e-5
2e-5
Resi
duals
(
m)
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, allthe points have residuals less than 1×10−5
µm and no pattern is seen in the residuals. Thecoefficients of the best fitting polynomial (equation 3.1) are:c0 = 2.145407,c1 = −1.0305×10−5,andc2 = −1.8496×10−10, the dispersion relation of the residuals is:RMS= 4.21×10−6
µm.
16
Chapter 4
Data Analysis and Results
A quick estimation of the photon noise limit in our datavs. the estimated planet to star flux ratio, makesus notice that it is impossible to directly distinguish the planet’s signature from thestellar one in a singlespectrum. Because of this, we use all the spectra collected in each run to subtract the stellar component, andsearch for the planetary Doppler-shift signature through a correlationmethod between the remaining residualspectra and synthetic models of the thermal emission spectrum of the planet following the idea of Deminget 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 orfrom the source of a front of waves will detect a change in the wavelength of the received signal, the lightobserved from stars and planets is affected by this phenomenon. For non relativistic velocities (vr ≪ c), thechange in wavelength for electromagnetic waves can be expressed as:
∆λ
λ=
λobs−λrest
λrest=
vr
c(4.1)
whereλobs andλrest are the observed and rest frame wavelength, andvr the radial velocity of the source(as a convention, the velocity is positive if the source is moving away from theobserver and negative if thesource is moving towards the observer). Then, for a beam of light emitted at a wavelengthλem, from Earth itwill be observed at a wavelength:
λobs= λem
(
1+∆vc
)
(4.2)
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 ofthe system (γ), the orbital motion of the object projected along the line of sight (vsini), and the barycentricvelocity 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 nodposition, we align the spectrain a reference system in which the star remains at rest (Doppler-shifting them according to the formula 4.3for the Star radial velocity, using a spline interpolation). Having the spectralined up, we construct a stellartemplate from the average of the set. Then, the stellar templates and the spectraare normalized such theircontinuum is set to a median of one. Finally, the stellar templates are shifted backto the star’s radial velocityof each spectrum before being subtracted to each frame. Although a combination of the different nights toobtain the stellar template would smear more the planetary component, we avoided this possibility, since itis highly probable that other systematics would be introduced, for example, from the different instrumentalset up or different atmospheric conditions.
Since the planet is approximately a thousand times less massive than its host star, the planetary Dopplerwobble is the same order of magnitude greater (see equation 2.7), added to the out of phase motion of theplanet with respect to its star (as they are orbiting each other), the planetary signature will not be addedcoherently, and thus will be blurred in the stellar template. Figure 4.1 shows theeffects of the averaging onthe planetary spectrum if we could see the planetary spectrum alone.
2.136 2.138 2.140 2.142 2.144Wavelength (m)
0.0
0.2
0.4
0.6
0.8
1.0
Norm
aliz
ed p
lanet
flux
firstlast
smear 30 km s1Figure 4.1: Simulated blurring of the planetary spectrum over one run usingsynthetic spectra of
HD 217107 b. The blue curve represents the planetary spectrum as seen in the first exposureof 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 planetaryspectrum would look after the average of all the spectra comprised between the initial and finalexposure. The planetary spectrum is clearly blurred loosing its original shape (for an instrumentalspectral resolution of 40 000).
As seen in figure 4.1, the planet spectrum is blurred in the stellar template due tothe relative Dopplershifts, the bigger the shifts are, the more blurred the planetary spectrum willbe. But, when the velocity spanis not enough to produce a significant shift, the planetary signature in the stellar template will be similar to theoriginal spectra, then, when we subtract this template to each frame, the planet signature will be subtractedas well, as happened in some of our nights. from equation 2.7, the maximum radial velocity span of theplanet (sini = 1.0) given the radial velocity span of the star during the observing run is given by:
18
(∆vpsini)max≈Ms
mpsini∆vssini (4.4)
Plugging in the values from table 2.1, the maximum velocity span for HD 217107 bin a night is:
(
∆vpsini
km s−1
)
max≈ 0.803×
(
∆vssinims−1
)
(4.5)
Then, for five of our runs (see table 3.2) the planetary spectrum will notshift more than a few km s−1.Figure 4.2 shows the blurring of the planetary spectrum for small Doppler shifts.
2.136 2.138 2.140 2.142 2.144Wavelength (m)
0.0
0.2
0.4
0.6
0.8
1.0
Norm
aliz
ed p
lanet
flux
firstlast
smear 3 km s1Figure 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 extentof the run, a longer time span will yield a larger velocity span. A second, lessobvious but very importantfactor, is the timing of the observation, given the particular sinusoidal shape of the radial velocity curve, thereare moments when the relative motion between the planet and the star is minimal as seen from Earth (neargreater elongation of the orbit), while at the moment inferior or superior conjunction the radial velocity spanis maximal. Thus, for an equal time span, the orbital phase of the system whenthe system is being observedcan 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 3hours long observation),while the best systems (in this aspect) can shift up to 40 km s−1 for 3 hours observing windows. Figure 4.4contrasts the planetary blurring for different velocity spans.
As bottom line, the keyword here is velocity span, which depends on the length of each observation, thetiming of that observation, and the selection of an ideal candidate. All this variables must be considered in
19
0.20 0.25 0.30 0.35Orbital Phase
120
100
80
60
40v ssin
i (
m s
1 ) t=2.4 h
v=12.33 m/s
t=2.5 hv=3.84 m/s
Figure 4.3: Close up, of radial velocity curve of figure 2.2 on the nights ofthe 25th (gray) and 26th (white)of November. Although both observing runs span about the same time extent,the radial velocityspan 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 dataset useless, regardless of the length of the run. Notice also that the blurring depends on the (undeterminedyet) sine of the inclination.
The subtraction of the stellar template, leaves a residual spectrum consistingof the signature of the planet,attenuated in some degree during the averaging process and immersed in the poisson photon noise. Thisresidual 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 treatedthe problem and tried to understand and predict the effects of intense stellar insulation at such small orbitaldistances, developing theoretical models of the atmospheres of extrasolargiant planets.
Initial simplified models [Saumon et al., 1996, Guillot et al., 1996] assumed that afraction of the lightof the parent star is reflect as a gray-body (the reflected spectral emission is a copy of received black-bodydistribution, 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 ofcomplexity, and now integrateand consider a number of different parameters and their interactions, such as, the intense irradiation, theatmospheric structure, temperature profiles, chemistry, the planet’s cooling and contraction history, dynam-
20
2.136 2.138 2.140 2.142 2.144Wavelength (m)
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Norm
aliz
ed p
lanet
flux
smear 3 km s1smear 12 km s1smear 22 km s1smear 30 km s1
Figure 4.4: Planetary spectrum blurring for different radial velocity spans (see legend). We can comparehow effectively the planetary spectrum is lost in the stellar average, for very low radial velocityspans (∼3 km s−1) as in some of of our observing runs, for low spans (∼12 km s−1) as in the bestof our observing runs, for typical velocity spans of good targets (∼22 km s−1) and for good runsof good targets (∼30 km s−1). Not only lines start to blend, but also, the depth of the absorptionsis more reduced for larger shifts.
21
ics, and stability (e.g.: Fortney et al. [2008], Burrows et al. [2008], Madhusudhan and Seager [2009]). Forexample, 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 andthe 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 theirremoval from the gas phase(“rain-out”).
• The use of a large and constantly updated opacity database including also the opacity of clouds, suchas 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 planetspresent temperature inversions(hot stratospheres), appear bright in the mid-infrared secondary eclipse, and feature molecular bands inemission rather than absorption, they will have large day/night temperature contrasts and negligible phaseshifts between orbital phase and thermal emission light curves because radiative timescales are much shorterthan possible dynamical timescales. On the other side, those that are cooler,absorb incident flux deeperin the atmosphere, where atmospheric dynamics will more readily redistribute absorbed energy, leading tocooler 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 ther-mal emission models of its atmosphere [model described in Fortney et al., 2005,2006]. Since the orbit ofHD 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 param-eters of the three models available, while Figure 4.5 shows the orbital distanceof HD 217107 b along oneorbit, indicating the bounds for which we used each one of the models. These models are all solar metallicity,with gravity g = 20 m s−2, cloud-free, and use the molecular abundances that are appropriate for chemicalequilibrium. 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 mod-els as observed through the telescope, it is necessary to lower their resolution to the instrumental spectralresolution. We empirically characterized the instrument through the analysis of the emission lines in thecalibration lamps, measuring the full width at half maximum (FWHM) of an isolated line, from which wedetermined the spectral resolution (R) through the equation:
R=λ
∆λ=
λ
FWHM(4.6)
22
0.0 0.2 0.4 0.6 0.8 1.0Orbital Phase
0.060
0.065
0.070
0.075
0.080
0.085
Separa
tion (A
U)
T=1095 K
T=1029 K
T= 981 K
Figure 4.5: Orbital distance of HD 217107 b as a function of the orbital phase. The horizontal dashed linesdelimit the selected ranges for each one of the models, labeling on the left the model used, thegray and white boxes, as in Figure 2.2 denote the used and discarded observing runs.
We found a value ofR≈ 40000 corresponding to a velocity resolution∆v = 7.49 km s−1(∆λ = 5.35 µmin wavelength). Figure 4.7 shows a model after being convolved with a Gaussian function with aFWHMrepresenting the planetary spectrum as seen in our data.
4.3 Correlation
An important statistical test that can establish the “sameness” of two data sets isthe correlation function.The returned value of this function lies within−1 and 1. A positive value of this function indicates thatthe two data sets have some degree of correlation, that is, they are similar, while a negative one, denotesanti-correlation. A value near zero tells that the two sets are uncorrelated.
We apply this test, between our residual spectra and the theoretical models, todetermine if, for a certaininclination, the model spectra match our observations. In the final result wesum up the correlations from allour residual spectra, and divide it by the number of spectra (to keep theranges within the bounds−1 and 1).
The models before being correlated with the data, must be Doppler-shifted tomimic the radial velocity ofthe planet at the time of the observation of each frame, To do this, we use the ephemeris from the spectrumto obtain the radial velocity of the star, using equation 2.1. As discussed in section 2.2, the radial velocity ofthe planet is parametrized by the inclination of the orbit, therefore, assuming avalue of sini, we derive thevelocity through equation 2.7. Then, to find the radial velocity with respect toEarth we use equation 4.3,we plug the that result into equation 4.2, and finally apply the correspondingshift to the wavelengths of themodel.
We calculate the correlation degree as a function of the inclination,C(i), according to the correlation
23
2.136 2.138 2.140 2.142 2.144Wavelength (m)
0
1
2
3
4
5Fl
ux (10
7 ergcm2 s1 Hz
1 )
Figure 4.6: Planetary atmospheric model at infinite resolution. The main absorbing molecules are H2O, CH4,CO, and CO2. The equilibrium temperature isT = 1029 K.
2.136 2.138 2.140 2.142 2.144Wavelength (m)
0
1
2
3
4
5
Flux (10
7 ergcm2 s1 Hz
1 )
Figure 4.7: Planetary atmospheric model at an instrumental resolution corresponding toR= 40000.
24
function [also known as Pearson’s correlation coefficient, Press et al., 1992]:
C(i) =1N
N∑
k=1
∑Nkj=1
(
rk j − rk)
·(
τk j(i) − τk(i))
√
∑Nkj=1
(
rk j − rk)2
∑Nkj=1
(
τk j(i) − τk(i))2
(4.7)
Where we have used the notationfk j for the value of the function at the pixelj of the spectrumk, fk is themean value of the function in the spectrumk:
fk =1Nk
Nk∑
j=1
fk j (4.8)
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 spectrumk, andN the total number of observedspectra.
We produce then, a curve of the correlation degreevs.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 Poissonnoise, the correlation of the residual spectra with the models should be closeto zero correlation, except whenthe adoptedi matches that of the planetary system. if the planet’s signature is strong enough in comparisonwith the noise, an appreciable peak in the correlation would represent a successful detection of the planetarysignature 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 (Seecolumn 6 of Table 3.2) since they do not represent any significant improvement in the results, as the shift ofthe 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 insection 4.3). The degree of correlation found is close to zero at all inclinations, and we do not distinguishany distinctive positive peak that indicated an atmospheres with absorption features resembling those of themodels.
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 thedegree of correlation does not tell if the observed correlation is statisticallysignificant per se. On account ofthis, in this section, we describe a further analysis of the data to assess the statistical significance of the valueof 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 noisein the residual spectra) there are little chances to detect the planet, as the correlation will be small at all
25
0.0 0.2 0.4 0.6 0.8 1.0sin i
10-5
10-4
10-3
10-2
Pla
net-
to-S
tar
flux r
ati
o
0.0 0.2 0.4 0.6 0.8 1.0sin i
0.0060.0040.0020.0000.0020.0040.006
Correlation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
exp(
2 )Figure 4.8: Top: Correlation results for our data set of HD 217107 b, asa function of sini. The correlation
remains flat along all inclinations without any distinctive peak, the maximum valueis reached atsini = 0.71. Bottom: Goodness of fitχ2-map, the horizontal and vertical axes refer to the fittingparameters sini andFp/Fs of the synthetic planetary spectrum added. The gray scale denotesthe goodness of fit, as a function of chi-square relative to the best fit , ranging from black forthe best fit (atχ2
min) to white for the poorest fit. Additionally, with bootstrap procedures, wedetermined the solid lines that mark the 1, 2, 3, & 4–σ (from bottom to top) confidence levelsof the false alarm probability. The white cross marks the best fit, located at sin i = 0.84 andFp/Fs = 3.6×10−3, unfortunately below the 3–σ confidence level.
26
inclinations. On the other side, ifFp/Fs is large, we should expect a noticeable correlation at the inclinationof the planet. It is reasonable, then, to state that the planet-to-star flux ratiois the main physical parameterbounded to the correlation degree. Based on this, we will search for the best parameters in the parameterspace: [Fp/Fs , sini].
We, thus, recreate and process our observations using synthetic spectra, with known parameters for theplanet, to obtain “synthetic” correlation curves, which we can compare the data results, and determine whichare the most probable parameters that gave rise to our result, and also estimate what confidence can we putinto 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 eachnight, but keeping theoriginal order of the dates of the observations. As a consequence, any real planet signature woulddisappear, 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 andFp/Fs, respectively.For simplicity, we adopt a constantFp/Fs along the orbit.
Step 3: We process these synthetic data through the same routines used in our original data (sections 4.1through 4.3). We then iterate for a grid of values in the ranges: 0≤ i ≤ π/2 and 10−5 ≤ Fp/Fs ≤ 10−2,obtaining a set of synthetic correlations for sini andFp/Fs.
There are two effects that affect the intensity of the emitted flux over time. Firstly, there is expected aphase variation in the planet’s flux due to the changing distance between the planet and the star along theorbit, expecting a greater emission near the periastron of the orbit. Secondly, there should be a change inthe observed planet’s brightness as we observe a greater or smaller portion of the day side surface. Thissecond 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 classplanet) it is expected a lower day/night side temperature contrast since theyshould present a more importantredistribution of energy. None of these effects are simulated in this analysis, a refinement of the methodwould improve the accuracy of this technique in the future.
Once we have such models, we search for the best fit parameters through aχ2 minimization between thedata correlation curve and the synthetic correlation curves, generating agoodness-of-fit map. We plot, in agray scale,χ2 relative to the best fit (χ2
min) through the formula:
f (i, flux ratio) = exp
−α ·(
χ2i,fr −χ2
min
)
≡ exp(
−α ·∆χ2i,fr
)
(4.9)
This function takes values within the range: 0< f ≤ 1, at the position of the best fitting parameters hasthe value 1, and decreases as the fit get poorer. The scale goes then,from white for a value of 0 and it getsdarker as the value approaches to 1. The parameterα, it is just a plotting device that enables a good contrastin 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 thesynthetic correlations parametrized by the sine of the inclination and planet-to-star flux ratio of the injectedplanet’s spectrum (section 4.5).
Figure 4.9 shows an example of the synthetic correlations, for an injected planet with sini = 0.82. Assuspected, greater values ofFp/Fs enhance the peak of the correlation degree. Figure 4.10 shows slices ofFigure 4.9 for three values ofFp/Fs.
27
Figure 4.9: Example of synthetic correlation curves for an injected planetary spectrum with sini = 0.82, anddifferent givenFp/Fs (y–axis). The correlation curves then are plotted as a function of inclination(z−axisvs. x–axis), for example, the highlighted black curve. The correlation curve peaks at theinclination of the injected planet and takes values near zero far from that inclination.
0.0 0.2 0.4 0.6 0.8 1.0sin i
0.0050.000
0.005
0.010
0.015
0.020
0.025
0.030
Correlation
Fp /Fs =2.8103Fp /Fs =1.0103Fp /Fs =3.6104
Figure 4.10: Synthetic correlation curves from Figure 4.9, for three differentFp/Fs of the injected planetaryspectrum (see legend), as expected the correlation degree increaseswith largerFp/Fs.
28
As corollary, the stronger the planetary signal is compared to the noise, thegreater the correlation degreewill be, and the easier to detect the planet. Thus, our ability to detect the planetary spectrum in the photonnoise can be quantified by the planet-to-star flux ratio and the stellar flux, according to the expression (fluxesin number of photons):
Fp/Noise =(Fp/Fs) ·Fs√
|Fs|+ |Fp|≈ (Fp/Fs) ·
√Fs (4.10)
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 frequencywith 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 weusually iterated between 3 000 and 5 000 times), recording after each trial the correlation curve. This set ofcorrelation curves represent the correlation found in the absence of aplanetary signal, and, being createdfrom the data themselves, define an empirical probability distribution that include both the photon statisticsand the systematics errors. Then, at each inclination of the correlation curve, we stack the values of thecorrelation 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 thesignal strengths at which spurious detections occur with 35, 10, 1, and 0.1 per cent false alarm probabilityrespectively, 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 ofplanetary emission, and give a more accurate picture of the noise and systematic effects of our data set.
4.7 Data Results Discussion
Regarding the results of HD 217107 b, the best fit occurs at sini = 0.838 andFp/Fs = 3.6× 10−3. Thisvalue disagrees with the maximum value of the correlation curve (Figure 4.8 Top), besides, the relativeimprovement inχ2 against the surrounding parameters is negligible, and furthermore, the bootstrap resultsindicate that this values is below the 3–σ confident limit of the signal not being a false positive. Also, thisFp/Fs is much larger than the expected value (from Section 2), where the predicted flux ratio ranges from2×10−5 and 1×10−4. The natural conclusion is that we can not state the detection of HD 217107b. Thisfeature, two orders of magnitude above the expected flux ratio, should beresult of a systematic errors duringthe processing of the data. The 3–σ confidence limit allows us to establish an upper limit in the flux ratio ofthe 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 analysiswe have developed a strategy to maximize the chances of success in a futuresearch. In this chapter we willinvestigate the limits of the direct search of extrasolar planets with this method, and study how can wedevelop 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 byPhoenix spectrometer at Gemini South, for the time constrained by a proposal for the first semester of ayear (From the beginning of February to end of July). We start with a list of extrasolar planets from TheExtrasolar Planets Encyclopaedia1, limiting to the confirmed non-transiting extrasolar planets. Even thoughother (potentially better suited) planetary systems could arise, when we consider other observatories (forexample, high resolution infrared spectrometers like CRIRES at VLT or NIRSPEC at Mauna Kea), or if weconsider the observable extrasolar planets during the whole year, the maingoal of these simulations is todemonstrate that the detection of extrasolar planets with our routines is possible in a real case. Thus, wefollow the constraints set by these conditions. For this same reason, and acknowledging that other extrasolarplanets should present different spectral features due to their own physical properties, we will disregardthese specific spectral features, and rather focus on the broad relative features in the planet and host starspectra, as are the planet-to-star flux ratios. The consideration of the appropriate spectral features like specificabsorption 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. Ifthe difference is greater, the observable time span of a night would be too short (less that∼3 hours) due tothe larger air mass. Also, the longitude of the observatory limits the right ascension of the targets, since theymust 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 signalto noise ratio. In the case of Infrared observations, we look at the K-band magnitudes. Another considerationis 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 ifthe orbits had inclinations of 30, this limit corresponds to theFWHM 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 higherradial velocity spans (enabling a greater Doppler shift between the planet and the star spectra during theruns), 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 217107listed 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 ourdata, that is, a similaramount 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 theobservatory, we calculated the air mass as a function of time, therefore we restrict ourselves to thetime 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–hourrange 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 whenthe planet is observable under 1.5 air mass (diagonal black lines), and themost favorable observing runs(vertical black lines). This target has very favorable conditions, sinceit spends more than 6 hours under1.5 air mass, and more than one hundred days in the observable range from Gemini South, allowing severalnights 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 presentsthe time at the beginning of the observation, the phase coverage, the radialvelocity span and average radialvelocity 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 179949for the first period of 2009(x–axis) as a function of Universal Time (y–axis). The color scale reflects the air mass of thesystem as observed from Gemini South Observatory. The purple stripesmark the twilight (thetime between dawn and sunrise, and between sunset and dusk). The diagonal black lines delimitthe air mass range set for this system (less than∼1.5), and each of the vertical black lines markthe nightly observing runs (3–hours) when the radial velocity span is maximal.
32
To create the synthetic spectra, we used the solar spectrum (from section3.3) to simulate the stellarcomponent, while for the planetary component we used the atmospheric modelsof HD 217107 b. Afterselecting 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 appropriateradial 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 applyour 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 ofthe planetary system HD 179949 (See Table 5.2), for which we will contrast the results when our observingstrategy is applied and when it is not, and highlight the aspects in which our idea improves the results. Weused the values from this table to compute the orbital solution of the star radial velocity (See Figure 5.2 Toppanel).
Table 5.2: Parameters
Parameter HD 179949a Tau Booa HD 73256b References
Star:
Te f f(K) 6168±44 6387±44 5636±50 aV05, bS04
d (pc) 27.05±0.59 15.60±0.17 36.5±1.0 B05
Ms (M⊙) 1.14±0.08 1.33±0.11 1.05±0.05 aV05, bS04
Rs (R⊙) 1.19±0.03 1.42±0.02 0.89 aV07, bU03
Ks (m s−1) 112.6±1.8 461.1±7.6 269.0±8.0 B05
vg (km s−1) -24.4±0.5 -15.9±0.5 29.5±0.2 aV05, bN04
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
mpsini (MJup) 0.916±0.076 4.13±0.34 1.87±0.27 B05
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 bthe planet to star flux ratio should be of the order of 10−4. But, as the purpose of this section is to showthe improvement of our observing strategy in contrast with a regular observation, we adopted a larger valueof Fp/Fs. The system was then simulated with an inclination, sini = 0.77 and a planet-to-stat flux ratio ofFp/Fs = 3×10−3.
33
0.0 0.2 0.4 0.6 0.8 1.0HD 179949's Orbital Phase
150
100
50
0
50
100
150
v ssin
i (
m s
1 )
0.0 0.2 0.4 0.6 0.8 1.0Tau Boo's Orbital Phase
!600
!400
!200
0
200
400
600
v ssin
i (
m s
"1 )
0.0 0.2 0.4 0.6 0.8 1.0HD 73256's Orbital Phase
#300
#200
#100
0
100
200
300
v ssin
i (
m s
$1 )
Figure 5.2: Radial velocity curve of the extrasolar planet host stars HD 179949 (Top panel), Tau Boo (Centerpanel), and HD 73256 (Bottom panel)vs.orbital phase. Calculated from the orbital parametersof Table 5.2. The phase origins are set at the orbits’ periapsis.
34
10-1 100 101 102
Wavelength (%m)
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Flux (
Wm
&2 m&1 )HD 2171072.0 RJup planet
1.5 RJup planet
1.0 RJup planet
10-1 100 101 102
Wavelength ('m)
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Flux (
Wm
(2 m(1 )
HD 1799492.0 RJup planet
1.5 RJup planet
1.0 RJup planet
10-1 100 101 102
Wavelength ()m)
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Flux (
Wm
*2 m*1 )
Tau Boo2.0 RJup planet
1.5 RJup planet
1.0 RJup planet
10-1 100 101 102
Wavelength (+m)
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
Flux (
Wm
,2 m,1 )HD 732562.0 RJup planet
1.5 RJup planet
1.0 RJup planet
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 Rightpanel) as function of wavelength. The planets’ spectra are in black colorand their respectiveplanets, 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, findingTeq = 1040± 19 K forHD 217107 b,Teq = 1541±55 K for HD 179949 b,Teq = 1673±51 K. for Tau Boo b, andTeq =1331±42 K for HD 73256 b.
35
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Wavelength (-m)
10-6
10-5
10-4
10-3
10-2
Flux r
ati
os
2.0 RJup
1.5 RJup
1.0 RJup
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Wavelength (.m)
10-6
10-5
10-4
10-3
10-2
Flux r
ati
os
2.0 RJup
1.5 RJup
1.0 RJup
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Wavelength (/m)
10-6
10-5
10-4
10-3
10-2
Flux r
ati
os
2.0 RJup
1.5 RJup
1.0 RJup
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Wavelength (0m)
10-6
10-5
10-4
10-3
10-2
Flux r
ati
os
2.0 RJup
1.5 RJup
1.0 RJup
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 Rightpanel) as function of wavelength, emitting as black bodies for the planet radii: 2.0, 1.5, and 1.0Jupiter radius of the planet (red, green, and blue respectively).
36
We present, two simulations for this system, first following our observing strategy, using the coordinatesand radial velocity of the target we calculated the observability windows already shown in Figure 5.1 andTable C.1. For a good target as this one, there are several favorable nights when the planet can cover a largeradial 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 selectingthe nights with larger radial velocity span (nights marked with an “x” in row (9) of Table C.1). Figure 5.5show the results of these simulations.
0.0 0.2 0.4 0.6 0.8 1.0sin i
10-5
10-4
10-3
10-2
Pla
net-
to-S
tar
flux r
ati
o
0.0 0.2 0.4 0.6 0.8 1.0sin i
10.0050.0000.0050.0100.0150.0200.025
Correlation
0.0 0.2 0.4 0.6 0.8 1.0sin i
0.0 0.2 0.4 0.6 0.8 1.0sin i
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
exp(
23452 )
Figure 5.5: Same as Figure 4.8 but for synthetic spectra, based on the planetary system HD 179949. Theinjected planet was simulated with: sini = 0.77 andFp/Fs = 3× 10−3. Left panel: Resultsusing our observing strategy. Right panel: Results without our observing strategy. The routinesuccessfully recovered the signal at sini = 0.78 andFp/Fs = 2.8×10−3 in both cases, although,when using our strategy, the correlation degree is stronger, and theχ2 peak is better defined incomparison with the right panel.
In both cases the correlation curves (Top panels) mark the inclination of thesynthetic orbit with a notice-able increment in the correlation degree near sini = 0.77. While theχ2-maps (Bottom panels) effectivelyindicate the best fit at sini = 0.78 andFp/Fs = 2.8× 10−3. We identify the differences between these twosimulations:
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 blurredin a greater extent in the stellartemplate (section 4.1), and consequently, less reduced when the template is subtracted, the planetaryspectrum 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 generallower, since it is less probable to reach such correlation degree by chance in a no-planet spectrum.The 3–σ confident limit lies atFp/Fs = 3×10−4 when using our observing strategy (for sini > 0.5), inthe other case, is lies close toFp/Fs = 4×10−4. Therefore, in general we can reach and discern lowerplanet-to-star flux ratios.
3.– Lastly, the peak in theχ2-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 inthe results that make a detection more probable.
As a general remark, we note, from these and the otherχ2-maps, a sensitivity bias for this method, favoringthe detection of more inclined orbiting systems, thus, leaving a inclination-sensitivity dependence at eachplanet-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 tothe physical conditions of theplanetary systems Tau Boo and HD 73256, implementing our observing strategy, we examine the chances ofa successful direct detection.
The orbital configuration of the system Tau Boo allows a planetary radial velocity span similar to theone of HD 179949 b (See Tables 5.1 and 5.2), the star radial velocity curve is shown in Figure 5.2, Centerpanel. The calculation of the Equilibrium temperature of the planet (Teq = 1673± 51 K), sets a range ofFp/Fs between 2×10−4 and 7×10−4 (See Figures 5.3 and 5.4 Bottom left panels). Figure 5.6 shows theobservability window of this planet, as it is close to the cutoff limit in declination, theair mass is greater thanin the other simulations.
Figure 5.6: Similar to Figure 5.1, daily observability windows of the extrasolar system Tau Boo for the firstperiod of 2009vs.Universal Time.
38
Figure 5.7 shows the simulation of Tau Boo, the planetary parameters set are: sini = 0.82 andFp/Fs =4× 10−4. The correlation curve (Top panel) peaks near the injected orbital inclination, and our routinesreturn the best fit parameters (Bottom panel): sini = 0.79 andFp/Fs = 3.6× 10−4, underestimating by afew percent the values. Nevertheless, theχ2-map shows an improvement in the region near the injectedinclination and flux ratio, and the bootstrap results set the 3–σ confidence limit aroundFp/Fs 1.5×10−4 (forsini > 0.5), making this detection 99% of not being a spurious signal.
0.0 0.2 0.4 0.6 0.8 1.0sin i
10-5
10-4
10-3
10-2
Pla
net-
to-S
tar
flux r
ati
o
0.0 0.2 0.4 0.6 0.8 1.0sin i
60.00460.0020.0000.0020.0040.0060.008
Correlation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
exp(
789:2 )Figure 5.7: Same as Figure 4.8 but for synthetic spectra of the planetary system Tau Boo, using our observing
strategy. The injected planet was simulated with: sini = 0.82 andFp/Fs = 4×10−4. The best fitparameters found are: sini = 0.79 andFp/Fs = 3.6×10−4.
39
We repeated the analysis, this time for the planetary system HD 73256. This planet has one of the largestmaximum radial velocities (See Tables 5.1 and 5.2), it has also the greatest planet-to-star Flux ratio of oursimulated candidates: between 2×10−4 and 8×10−4 (See Figures 5.3 and 5.4 Bottom Right panels), but thestar has a larger K band magnitude than the other stars in Table 5.1, which will produce a lower signal tonoise ratio. The planet was added with parameters: sini = 0.69 andFp/Fs = 7×10−4. Figure 5.8 shows theobservability windows of this system.
Figure 5.8: Similar to Figure 5.1, daily observability windows of the extrasolar system HD 73256 for thefirst period of 2009vs.Universal Time.
Figure 5.9 shows the results of this simulation, the Top panel shows a faint peak near sini = 0.67, but itis not conclusive. The best fit (Bottom panel) is found at: sini = 0.63 andFp/Fs = 7.8×10−4. In this case,the contrast in theχ2-map between the peak and the surrounding is slighter than in the other cases, probablydue to the lower signal to noise of this system. With a 3–σ confident limit aroundFp/Fs 2 ×10−4, again, thedetection has 99% confidence.
40
0.0 0.2 0.4 0.6 0.8 1.0sin i
10-5
10-4
10-3
10-2
Pla
net-
to-S
tar
flux r
ati
o
0.0 0.2 0.4 0.6 0.8 1.0sin i
;0.004;0.0020.000
0.002
0.004
Correlation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
exp(
<=>?2 )Figure 5.9: Same as Figure 4.8 but for synthetic spectra of the planetary system HD 73256, using our ob-
serving strategy. The injected planet was simulated with: sini = 0.69 andFp/Fs = 7×10−4. Thebest fit parameters found are: sini = 0.63 andFp/Fs = 7.8×10−4.
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 eachnight on their own isimportant to minimize systematic effects, and applied it to HD 217107 b in the near infrared spectralrange. The correlation curve (as a function of the inclination of the orbit) did not present any distinctivepeak, and their values were close to zero for all inclinations, the maximum correlation degree waslocated at sini = 0.71. Furthermore, using the original data, we created synthetic expected spectrafor known planet’s inclinations and flux ratios, to fit the sine of the inclination and planet-to-star fluxratio through aχ2 square minimization routine. We found as best fitting parameters: sini = 0.84 andFp/Fs = 3.6×10−3, but this is below our 3-σ confidence limit of false alarm probability. Besides, as canbe observed in Figure 4.8 (bottom panel), there is difficulty in determining the best fitting parametersfor the planet, as the relative improvement inχ2 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 notclaim a detection of HD 217107 b with our current data. According to the sensitivity of our methodand our data, we can constrain the planet emission, rejecting the flux ratio of HD 217107 b to its hoststar to be over 5×10−3 (3–σ confidence, from our bootstrap method). We attribute these results to twomain reasons: First, since the instrument was not well characterized at thetime, the data was not assensitive as expected, and we had to tune the service-mode observationsafter the first few observingwindows. Second, the nonexistence of a proper observing strategy atthe time of the observations madeus 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, bycarefully defining a candidate selection criteria and adopting an optimized observational strategy thatenhances the capability of this method.
In terms of the ideal candidate, there are two fundamental considerations totake into account: Howmuch radial velocity span can a planet have during an observing run, and how well can we distinguishthe planetary spectrum after the stellar component removal. Thus, the systems that are best suitedfor this technique are those in very close orbits, allowing the planets to have high orbital velocityand 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 choiceof the observation dates can make our attempts futile. Then we propose an observing strategy wherefor each night of the period of observations, we calculated the radial velocity evolution in time, andspecifically 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 observa-tions of other planetary systems as observed by Phoenix spectrograph, with the same amount of hoursand an appropriate schedule of observations. Under these conditions,we recreated observations of syn-thetic data based on the extrasolar planet HD 179949 b, contrasting the useof our observing strategywith a regular observation schedule, we recovered the planetary companion signature with parameterssini = 0.77 andFp/Fs = 3× 10−3 showing that the improvement is reflected in a greater correlationdegree peak, better determined best-fit parameters in theχ2-map, and deeper false alarm probabilitylimits when using our observing schedule. Finally, we performed simulations for two other planetarysystems following the most reasonable physical properties, successfullydetermining the inclination ofthe 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 ofhigher inclination systems leaving an inclination-sensitivity dependence at a given planet-to-star fluxratio. This is expected as greater inclinations implies less massive planets, therefore, they appear moresmeared during the average to create the star template spectra.
Targets that have an important day/night side brightness contrast, would bias even more a detectionfor an edge on configuration, since a bigger portion of the day side surface would be observablefrom Earth. Observations at times of larger radial velocity spans (preferred by our optimal observingstrategy) coincides with inferior and superior conjunctions, i.e. when the day side of the planet ismaximum and minimum, respectively. Clearly, observations near superior conjunction would capturethe largest amount of light possible from the planet and at the same time coverthe largest radialvelocity span for a determined time extent.
We conclude that high resolution instruments as Phoenix at Gemini South Observatory, are capableof detecting extrasolar planets with this method. Our simulations, show that if we perform a carefultreatment of the systematics, approaching to the photon noise limit, if we count withappropriatetheoretical models for the thermal emission of these objects, and if we follow a smart scheme in thedata acquisition, Doppler-shifted signals of extrasolar planets with flux ratios, with respect to theirstars, as deep as 104 can be recovered.
A refinement of this technique would involve the optimization of the total exposure time, evaluatingthe optimal distribution of time designated to the length of an observing runvs.the number of nightsof observation. The use of others instruments, like CRIRES at VLT Observatory, which has a largerwavelength coverage, and thus, a larger effective sensitivity, would also decrease the total time ofobservation. While adding a phase-dependent function of the planet’s brightness, to account for thechanging observed portion of the day/night side of the planet, as well as aphase function, account-ing for different amounts of irradiation for an eccentric orbit, would increase the accurateness of theparameters 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; theargument of the periastron,ω; the semi-major axis of the orbit,a; the eccentricity,e; and the time of passagethrough the periastron,T.
In a Cartesian system of coordinates, xyz, with the orbital planet in the xy plane and the origin fixed inthe center of mass of the system. Let thex axis point toward the periastron of the reduced mass orbit, in thiscoordinate system the position and the velocity of the reduced mass is given by:
~r = r (cosν x + sinν y) (A.1)
~r = (r cosν − r ν sinν) x + (r sinν + r ν cosν) y (A.2)
Whereν is the true anomaly andr is the distance between the masses. In terms of the orbital parameters,the magnitude of the distance and velocity can be expressed as:
r = a1 − e2
1 + ecosν(A.3)
r =r ν esinν
1 + ecosν(A.4)
And making use of the relationr2ν = na2√
1− e2 with n the mean motion, we have:
r =na√1− e2
esinν (A.5)
r ν =na√1− e2
(1 + ecosν) (A.6)
Then the velocity of the system can be written as:
~r =na√1− e2
(−sinνx + (e+ cosν) y) (A.7)
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 ofω, i, andΩ the is given by asuccession of three rotations:
- A rotation around the z axis by an angleω
- A rotation around the x axis by an anglei, giving the inclination between the xy and XY planes.
- A rotation around the z axis by an angleΩ.
These rotations are represented by the rotation matrices:
P1 =
cosω −sin ω 0
sin ω cosω 0
0 0 1
P2 =
1 0 0
0 cosi −sin i
0 sin i cos i
P3 =
cosΩ −sin Ω 0
sin Ω cosΩ 0
0 0 1
(A.8)
Thus, the velocity vector in the XYZ system, is given by:
Vx
Vy
Vz
= P3P2P1
rx
ry
0
=
(
rxcosω − rysinω)
cosΩ −(
rxsinω + rycosω)
sinΩcosi(
rxcosω − rysinω)
sinΩ +(
rxsinω + rycosω)
cosΩcosi(
rxsinω + rycosω)
sinΩ
(A.9)
Plugging in the value of equation A.7, we have:
Vx
Vy
Vz
=
na√1− e2
−sin(ν +ω)cosΩ − cos(ν +ω)sinΩcosi − e(cosΩsinω + sinΩcosωcosi)
sin(ν +ω)sinΩ + cos(ν +ω)cosΩcosi − e(sinΩsinω − cosΩcosωcosi)
cos(ν +ω)sini + ecosωsini
(A.10)
We recognize now, from the Z axis, the amplitude of the velocity,K = nasini/√
1− e2, and the radialvelocity 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, wemake use of the Error Propagation Equation. For a quantityx that is a function of one or more measuredvariablesui (each with variancesσ2
u,i), the varianceσ2x for x is given by [Bevington and Robinson, 2003]:
σ2x =
∑
u
[
σ2u
(
∂x∂u
)2]
(B.1)
B.1 Equilibrium Temperature
In section 2.3 we calculated the equilibrium temperature of a extrasolar planet(Teq) as a function of the star’sradius (Rs) the semi-major axis of the orbit (a), and the star’s temperature (Ts) through the equation:
Teq =
(
1− A4
)1/4(
Rs
a
)1/2
Ts
We then calculated the variance of the equilibrium temperature (σT p), applying the error propagationformula:
σT p = Teq
√
(
σTs
Ts
)2
+(
σa
2a
)2
+(
σRs
2Rs
)2
(B.2)
with σTs the error in the star’s temperature,σa the error in the semi-major axis of the orbit, andσRs theerror 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 isknown that for a signal level ofN 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 countsADU,
47
which are related to the number of photons asADU = N/G, whereG is the gain of the instrument, thenfollowing equation B.1, the variance of a pixel is:
σ2ADU = (
√ADU ·G)2 ·
(
1G
)2
=|N|G
(B.3)
Considering the error introduced by the read out noiseRON (in electrons) that behaves as shot noise, itmust be added squared to the variance, we have for the variance of a pixel in the raw data:
σ2raw =
|N|G
+(
RONG
)2
(B.4)
Accordingly, the same formula applies to the variance of the pixels in the raw Dark images (σ2d) and of the
raw Flat-field images (σ2f ). Next we propagate the errors, during the data reduction (see section3.2).
The master Dark frame is constructed from the median of a set of dark images, that is, at each pixelsposition, the value of the master dark frame (D) is the value from the set of dark imagesdi that has equallynumber of values that are greater and lower than. To estimate the error in themaster dark frame, we computethe standard deviation of the mean, not the median. The standard deviation ofthe median is complicatedand can only be estimated statistically using a bootstrap method, and it is more computationally expensive.Then, ifN is the number of dark images in the set, the formula of the mean is:
D =1N
N∑
i=1
di (B.5)
And, thus, the corresponding variance of a pixel of the master dark frame is:
σ2D =
1N2
N∑
i=1
σ2d,i (B.6)
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,
σ2F0 =
1N2
N∑
i=1
σ2f ,i (B.7)
Then, we must subtract the master dark frame (F1 = F0 − D) and the variance changes as:
σ2F1 = σ2
F0 + σ2D (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:
σ2F =
(
σF1
med(F1)
)2
(B.10)
Then, to correct the raw images (R), we subtract the dark frame (D) and divide biy the flat-field frame (F):
C =R− D
F(B.11)
Applying the error propagation formula for the sum and then for the division, the variance of a dark-flatcorrected image is:
σ2C =
1F2
σ2raw + σ2
D +(
R− DF
)2
σ2F
(B.12)
Now, the last step is the subtraction of the sky (S), which is just the dark-flat corrected image in the opposednodding position. The pixels in the final corrected sky subtracted image (I =C−S) have then a variance givenby:
σ2I = σ2
C + σ2S (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 sin i = 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 vs ∆vp vp selectedday −2450000 start end m s−1 m s−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 o25 4946.78 0.48 0.52 -4.86 107.57 6.99 -154.8626 4947.77 0.79 0.83 -24.41 -62.61 35.14 90.1427 4948.78 0.12 0.16 26.57 -49.46 -38.25 71.21 o28 4949.76 0.44 0.48 1.21 109.23 -1.74 -157.2629 4950.76 0.76 0.80 -26.87 -41.94 38.69 60.3830 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 -152.762 4953.75 0.73 0.77 -28.13 -19.87 40.49 28.61 o3 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 vs ∆vp vp selectedday −2450000 start end m s−1 m s−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 -142.155 4956.79 0.71 0.75 -28.31 -6.72 40.75 9.686 4957.79 0.03 0.07 15.77 -97.03 -22.71 139.69 x7 4958.74 0.34 0.38 16.51 87.35 -23.76 -125.76 x8 4959.79 0.68 0.72 -27.82 13.66 40.05 -19.669 4960.79 0.00 0.04 10.61 -106.71 -15.28 153.63 x10 4961.73 0.31 0.35 20.75 72.32 -29.88 -104.12 o11 4962.79 0.65 0.69 -26.42 33.57 38.03 -48.3312 4963.79 0.97 0.01 5.08 -112.40 -7.31 161.8213 4964.72 0.28 0.32 24.16 54.84 -34.79 -78.95 o14 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 7.82 161.4716 4967.72 0.24 0.28 26.78 34.60 -38.55 -49.82 o17 4968.79 0.59 0.63 -21.31 68.50 30.68 -98.6218 4969.71 0.89 0.93 -11.44 -105.36 16.47 151.6819 4970.71 0.21 0.25 28.38 12.36 -40.86 -17.7920 4971.79 0.56 0.60 -17.75 82.84 25.55 -119.2721 4972.70 0.86 0.90 -16.80 -94.03 24.19 135.3722 4973.72 0.18 0.23 28.82 -6.00 -41.50 8.6423 4974.79 0.53 0.57 -13.70 94.33 19.72 -135.8124 4975.69 0.82 0.86 -21.32 -78.66 30.70 113.2525 4976.79 0.18 0.22 28.82 -10.02 -41.49 14.4226 4977.79 0.50 0.54 -9.27 102.72 13.34 -147.8827 4978.68 0.79 0.83 -24.79 -60.07 35.69 86.4828 4979.80 0.15 0.19 28.15 -30.70 -40.53 44.2029 4980.80 0.47 0.51 -4.57 107.77 6.58 -155.1630 4981.67 0.76 0.80 -27.10 -39.15 39.02 56.3631 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 -152.062 4984.66 0.72 0.76 -28.20 -16.85 40.60 24.263 4985.80 0.09 0.13 23.65 -69.10 -34.04 99.48 o4 4986.66 0.37 0.41 12.16 97.88 -17.50 -140.92 x5 4987.72 0.71 0.75 -28.31 -8.41 40.75 12.116 4988.80 0.06 0.10 19.96 -84.92 -28.73 122.26 x7 4989.65 0.34 0.38 16.91 86.16 -24.34 -124.04 x8 4990.80 0.71 0.75 -28.30 -5.26 40.75 7.579 4991.79 0.03 0.07 15.21 -98.32 -21.90 141.55 x10 4992.64 0.30 0.34 21.08 70.91 -30.35 -102.0911 4993.79 0.68 0.72 -27.65 16.97 39.81 -24.4312 4994.79 0.00 0.04 9.46 -108.26 -13.61 155.8713 4995.64 0.27 0.31 24.53 52.51 -35.31 -75.6014 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 8.58 160.9216 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 vs ∆vp vp selectedday −2450000 start end m s−1 m s−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 -84.0618 5000.62 0.88 0.93 -11.92 -104.57 17.16 150.5419 5001.62 0.21 0.25 28.51 9.30 -41.05 -13.3920 5002.77 0.58 0.62 -19.70 75.61 28.36 -108.8521 5003.61 0.85 0.89 -17.24 -92.81 24.83 133.6222 5004.63 0.18 0.22 28.82 -8.99 -41.49 12.9423 5005.76 0.55 0.59 -15.48 89.80 22.29 -129.28 x24 5006.61 0.82 0.86 -21.68 -77.12 31.21 111.0325 5007.73 0.18 0.22 28.82 -8.18 -41.50 11.7826 5008.75 0.51 0.55 -10.73 100.37 15.45 -144.5027 5009.60 0.79 0.83 -25.05 -58.29 36.06 83.9128 5010.74 0.16 0.20 28.43 -25.30 -40.93 36.4229 5011.74 0.48 0.52 -5.61 106.96 8.08 -153.9930 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 68.312 5014.59 0.40 0.44 7.49 105.10 -10.78 -151.313 5015.58 0.72 0.76 -28.24 -14.94 40.66 21.514 5016.73 0.09 0.13 23.88 -67.87 -34.37 97.715 5017.58 0.37 0.41 12.57 97.04 -18.10 -139.71 x6 5018.64 0.71 0.75 -28.31 -6.27 40.75 9.037 5019.72 0.06 0.10 19.80 -85.47 -28.50 123.068 5020.57 0.33 0.37 17.29 85.00 -24.89 -122.379 5021.71 0.70 0.74 -28.28 -2.56 40.72 3.6810 5022.71 0.03 0.07 14.73 -99.39 -21.20 143.0911 5023.56 0.30 0.34 21.42 69.39 -30.84 -99.8912 5024.70 0.67 0.71 -27.50 19.60 39.59 -28.2213 5025.70 0.00 0.03 8.95 -108.88 -12.88 156.7614 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 -59.2216 5028.55 0.91 0.96 -6.51 -111.33 9.37 160.2717 5029.54 0.24 0.28 27.23 29.77 -39.20 -42.8518 5030.69 0.61 0.65 -22.94 59.95 33.03 -86.3119 5031.54 0.88 0.92 -12.43 -103.70 17.89 149.2920 5032.53 0.20 0.24 28.58 7.37 -41.15 -10.6121 5033.68 0.57 0.62 -19.35 77.01 27.85 -110.8622 5034.53 0.85 0.89 -17.68 -91.56 25.45 131.8223 5035.57 0.18 0.22 28.82 -6.23 -41.50 8.9724 5036.67 0.54 0.58 -15.08 90.89 21.70 -130.8525 5037.52 0.82 0.86 -22.00 -75.65 31.68 108.9126 5038.66 0.19 0.23 28.82 -5.43 -41.50 7.8227 5039.67 0.51 0.55 -10.29 101.12 14.81 -145.5828 5040.51 0.78 0.82 -25.28 -56.59 36.39 81.4729 5041.66 0.15 0.19 28.29 -28.23 -40.73 40.64
52
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