From ESPRESSO to PLATO: detecting and characterizing Earth-like planets in the presence of stellar noise

and characterizing Earth-like planets

in the presence of stellar noise

Tese de Doutouramento

Luisa Maria Serrano

Departamento de Fisica e Astronomia do Porto,

Faculdade de Ciências da Universidade do Porto

Orientador: Nuno Cardoso Santos,

Co-Orientadora: Susana Cristina Cabral Barros

This Ph.D. thesis is the result of 4 years of work, stress, anxiety, but, over all, fun, curiosity and desire of exploring the most hidden scientific discoveries deserved by Astrophysics. Working in Exoplanets was the beginning of the realization of a life-lasting dream, it has allowed me to enter an extremely active and productive group. For this reason my thanks go, first of all, to the ’boss’ and my Ph.D. supervisor, Nuno Santos. He allowed me to be here and introduced me in this world, a distant mirage for the master student from a university where there was no exoplanets thematic line. I also have to thank him for his humanity, not a common quality among professors. The second thank goes to Susana, who was always there for me when I had issues, not necessarily scientific ones. I finally have to thank Mahmoud; he is not listed as supervisor here, but he guided me, teaching me how to do research and giving me precious life lessons, which made me growing.

There is also a long series of people I am thankful to, for rendering this years extremely interesting and sustaining me in the deepest moments. My first thought goes to my parents: they were thousands of kilometers far away from me, though they never left me alone and they listened to my complaints, joy, sadness...everything. Thank you, without your sustain I would not be here writing this thesis. I also have to thank my historical friends, Federico and Silvia. I went away from Trieste, still they kept on being always present and getting updated with my life.

A special thank goes to other Ph.D. students and researchers who shared with me nice moments: Akin, Raquel, Solene, João, Fatima and Elisa more than everybody, but I should mention a long list of people here. For this reason, I will just say: thank you CAUP, for the friendly environment you offered me. Thank you Nuno, Júlia and Jorge for sharing with me my other passion, archery, my best stress relies. And thanks to Alessia, Nicoló, Irene and all those, who in the last 2 years ’stucked with me until the very end’.

Without all of you these years would have been completely different and, probably, less interesting. Finally, I have to thank someone who entered my life silently and slowly, becoming unexpectedly important to me. Zé, you were there as a friend, you are still here as my love, and you sustained me through these last months of thesis. I hope our future is going to be bright.

The search for extra-solar planets dates back to the mid 20th century, when the Doppler effect was proposed as a possible detection method (Struve 1952). As the time passed, a deeper understanding of the stellar physics and its manifestation and the spectroscopic improvements, allowed the discovery of the first exoplanet, by Mayor & Queloz (1995). Their work represented a fundamental milestone for the field, which grew faster as new detection methods were adopted and the instrumental precision improved. Nowadays, this field, among the other objectives, heads towards a precise characterization of exoplanets and their atmospheres and the identification of an Earth-twin. Reaching these aims can only be possible by adopting very precise instruments and accounting for several sources of stellar noise. In this thesis, we specifically analyze the measurability with the current and future instruments of two planetary parameters, the albedo and the spin-orbit angle.

The albedo of an exoplanet represents the fraction of stellar light reflected by the planetary atmo-sphere. Since reflection depends on the structure and composition of the layers crossed by photons, knowing the albedo helps to probe the presence of clouds and specific molecules in the atmosphere. Measuring this parameter is challenging and it requires the detection of the reflected light through opti-cal photometric observations. This detection is possible in the context of phase curve analysis. A phase curve is the flux variation from the target star and its orbiting planets as a function of time. It involves, in optical wavelengths, the primary transit, the secondary eclipse and 3 more effects, the beaming effect, the ellipsoidal modulation and the reflected light component. While the beaming and ellipsoidal are neg-ligible, the reflected light might dominate the out-of-transit flux if there were no additional noise. The presence of instrumental noise and stellar activity may cause difficulties and obstacles in the detection of the planetary signal, even accounting for a precise knowledge of the planetary properties derivable from the transit feature. While the instrumental noise can be reduced with better instruments or even by binning the data, the stellar activity cannot be removed, especially in the context of a space telescope, such as CHEOPS, which will offer a limited time-span for observations (20 days).

In our work, we explored how the stellar activity could limit the detection of the planetary albedo, accounting for an increasing observational time and imposing CHEOPS precision as instrumental noise. In detail, we built mock light curves, including a realistic stellar activity pattern, the reflected light component of the planet and white noise, averagely on the level of CHEOPS noise for different stellar magnitudes. Afterwards, we fit our simulations with the aim of recovering the reflected light component and assuming the activity patterns could be modeled with a Gaussian process. The main conclusion of such analysis was that at least one full stellar rotation is necessary to retrieve the planetary albedo.

activity pattern. We found as well that, for a 6.5 magnitude star and accounting for CHEOPS noise level, it is possible to detect the planetary albedo up to a lower limit of RP = 0.03R∗. These results

can represent a starting point for phase curve analysis not only with CHEOPS, but also with future photometric missions, such as PLATO and TESS. They also show that detecting the albedo for Earth-like planets will only be possible with an increased photometric precision and long observations, as they will be offered by PLATO.

The projected spin-orbit angle is the angle between the planetary orbit and the stellar rotational axis. It can be measured through the Rossiter-McLaughlin (RM) effect, the radial velocity signal generated when a planet transits a rotating star. Stars rotate differentially and this affects the shape and amplitude of the RM signal, on a level that can no longer be ignored with precise spectrographs. Highly misaligned planets provide a unique opportunity to probe stellar differential rotation via the RM effect, as they cross several stellar latitudes. In this sense, WASP-7, and its hot Jupiter with a projected misalignment of ∼ 90◦, is one of the most promising targets. Although Albrecht et al. (2012a) measured the RM of WASP-7b, they found no strong detection of the stellar differential rotation, which suggests us the possibility of an imprecise measurement of the spin-orbit misalignment as well.

For this reason, we decided to explore the main hurdles which prevented the determination of WASP-7 differential rotation, adopting the tool SOAP3.0, updated in way it accounted as well for non-rigid stellar rotation. Furthermore, we investigated whether the adoption of the new generation spectrographs, like ESPRESSO, would solve these issues. We finally assessed how instrumental and stellar noise influence this effect and the derived geometry of the system. We found that, for WASP-7, the white noise represents an important hurdle in the detection of the stellar differential rotation, and that a precision of at least 2 m s−1 or better is essential. However, we noticed that the past observations of WASP-7b show unusually high residuals, which cannot be justified with any of the additional stellar noise sources explored in our analysis and, thus, they require further exploration. Such exploration would be well suited to the ESPRESSO spectrograph for WASP-7-like systems, as it will provide the radial velocity precision necessary to disentangle the instrumental and stellar noise sources. Unluckily this kind of measurement in the case of Earth-like planets appears to be a quite far achievement.

As an overall result we can conclude that the detailed description of the planet, especially with the current and new instruments, is only possible when properly accounting for the stellar noise sources. Moreover, the presence of a planet can help as well in understanding better certain stellar properties, as it is the case of stellar rotational pattern explored in our works. With this thesis, we can thus strongly

A pesquisa por exoplanetas foi pela primeira vez considera no século 20, quando o efeito Dopler foi proposto como um possível método de detecção (Struve 1952). Assim que o nosso conhecimento de Física estelar evoluiu e o campo da espectroscopia avançou o suficiente, foi possível descobrir o primeiro exoplaneta, por Mayor & Queloz (1995). Esta descoberta marcou o nascer do campo de exoplanetas, que cresceu cada vez mais rápido assim que novos métodos de detecção e análise foram descobertos e a precisão instrumental melhorou. Hoje em dia, um dos vários objectivos deste campo consiste em realizar caracterizações precisas dos exoplanetas e suas atmosferas, assim como a identificação de planeta gémeo da Terra. Atingir estes objectivos só possível adotando instrumentos de elevada precisão e tendo em conta várias fontes de ruído estelar. Neste trabalho, analisamos especificamente como dois parâmetros planetários, o albedo e o ângulo spin-órbita são exequíveis de serem medidos por instrumentos actuais e futuros.

O albedo de um exoplaneta representa a fração da luz estelar que é reflectida pela atmosfera plan-etária. Dado que a reflexão da luz depende da estrutura e composição das camadas atmosféricas atrav-essadas pelos fotões, saber o albedo ajuda a caraterizar a presença de nuvens e de moléculas específicas na atmosfera planetária. Medir este parâmetro é, no entanto, um desafio e requere a deteção da luz reflectida através de observações fotoelétricas no óptico. A curva de fase é a variação de fluxo de uma estrela alvo e dos planetas que a orbitam em função do tempo. Involve, no óptico, o trânsito primário, a elipse secundária and mais três efeitos, o efeito “beaming”, a modulação elipsoide e componente de luz reflectida. Apesar de o efeito de “beaming” e da modulação elipsoide serem negligíveis, a luz re-fletida pode dominar o flux fora-de-trânsito se não existir mais nenhum fonte de ruído. A presença de ruído instrumental e actividade estelar podem causar dificuldades e obstáculos na deteção do sinal planetário, mesmo tendo em conta um conhecimento preciso das propriedades planetárias derivadas do trânsito. Apesar do ruído instrumental poder ser reduzido com melhores instrumentos or até ao agru-par os dados, a actividade estelar ode ser removida, especialmente no contexto de telescópio espacial, como o CHEOPS, que oferecerá uma quantidade de tempo de observação limitada (20 dias). Neste trabalho, exploramos como a actividade estelar pode limitar a deteção do albedo planetário, tendo em conta o aumento do tempo observacional e impondo precisão do nível do instrumento CHEOPS como ruído instrumental. Mais precisamente, sintetizamos curvas de luz, incluindo um padrão de atividade estelar realista, a componente de luz refletida pelo planeta e ruído branco, este assumindo em média ruído perto do ruído instrumental do CHEOPS para diferentes magnitudes estelares. Depois fitamos as nossas simulações com o intuito de recuperar a componente de luz refletida e assumir que os padrões

resultado, independente do nível de ruído, é uma consequência da metodologia adoptada para modelar a atividade estelar, o processo Gaussiano, que necessita de detectar a rotação estelar completa to descrever o padrão de actividade. Encontramos também que, para uma estrela de magnitude 6.5 and tendo em conta um nível de ruído do nível do CHEOPS, é possível detectar o albedo planetário até um limite de Rp = 0.03R_∗. Estes resultados podem representar um ponto inicial para a análise de curvas de fase

não só para o CHEOPS, mas para futuras missões fotométricas, como por exemplo PLATO ou TESS. Também mostram que a detecção do albedo para planetas semelhantes à Terra só serem possíveis com uma precisão fotométrica superior e observações de longa duração, como as que serão possíveis com o PLATO.

O ângulo spin-orbita é o ângulo entre a órbita planetária e o eixo de rotação da estrela. Pode ser medido através do efeito Rossiter-McLaughlin (RM), o sinal de velocidade radial gerada quando um planeta transita em frente a uma estrela em rotação. Estrelas exibem rotação diferencial e isto afecta a forma e a amplitude do sinal RM, num nível que não já pode ser ignorado numa era de espectroscopia precisa. Planetas extremamente desalinhados oferecem então uma única oportunidade de caracterizar a rotação diferencial via o efeito RM, dado que os mesmos atravessam várias latitudes estelares. Assim sendo, WASP-7, o seu planeta Jupiter quente com desalinhamento de cerca de 90o, acaba por ser um alvo bastante promissor. Apesar de Albrecht et al. (2012a) mediram o efeito RM de WASP-7b, os mesmos não encontraram nenhuma indicação da rotação diferencial estelar, o que sugere a hipótese de uma medição imprecisa do desalinhamento spin-órbita. Por esta razão, decidimos explorar os entraves que impossibilitaram a determinação da rotação diferencial estelar de WASP-7, utilizando a ferramenta SOAP3.0, e modificando a mesma de maneira a ter em conta a rotação estelar não rígida. Adicionalmente, também investigamos como a adoção de espectrografos de nova geração, como o ESPRESSO, poderiam resolver oualiviar este problema. Avaliamos como ruído instrumental e estelar influenciam este efeito a resultante geometria do sistema. Encontramos que no caso de WASP-7, o ruído branco representa um importante obstáculo na detecção da rotação diferencial estelar, e que pelo menos uma precisão de 2 m s−1 ou melhor é essencial. No entanto, notamos que passadas observações de WASP-7b têm residuais extremamente elevadas, que não podem ser justificadas com as adicionais fontes de ruído estelar exploradas na nossa analise e que, como tal, é necessária futura exploração das mesmas. Tal exploração beneficiaria da utilização do espectrógrafo ESPRESSO para sistemas parecidos com de WASP-7, dado que iria fornecer a precisão em velocidade radial necessária para separar fontes de ruído instrumentais e estelar. Infelizmente este tipo de medição para planetas gémeos da Terra parece ser algo possível apenas

e novos instrumentos, só é possível quando se tem em conta a caracterização de contas de ruído estelar. Por outro lado, a presença de um planeta pode ajudar a a compreensão de certas propriedades estelares, como é o caso da rotação diferencial explorada no nosso trabalho. Com esta tese, podemos então frisar a importância do estudo de exoplanetas no contexto da análise estelar e vice-versa.

La ricerca dei pianeti extrasolari risale alla metà del XX secolo, quando l’effetto Doppler venne proposto come possibile metodo di rilevamento planetario (Struve 1952). Col passare del tempo, una comprensione più profonda della fisica stellare e delle sue manifestazioni, unita ai miglioramenti in ambito spettroscop-ico, permise la scoperta del primo esopianeta, ad opera di Mayor & Queloz (1995). Il lavoro di Mayor e Queloz rappresentò una pietra miliare fondamentale nel campo della ricerca planetaria, che crebbe esponenzialmente grazie all’adozione di nuove tecniche di rilevamento e ai notevoli sviluppi strumentali. Attualmente, i principali obiettivi della ricerca esoplanetaria sono una descrizione precisa degli esopi-aneti e della loro atmosfera e la scoperta di un gemello della Terra. Realizzare tali aspettative è possible solo attraverso l’uso di strumenti molto precisi e prendendo in considerazione diverse sorgenti di rumore stellare. In questa tesi, esaminiamo in dettaglio la misurabilità con gli attuali e futuri strumenti di due parametri planetari, l’albedo e il disallineamento orbitale.

L’albedo di un esopianeta rappresenta la frazione di luce stellare riflessa dall’atmosfera planetaria. Siccome la riflessione dipende dalla struttura e dalla composizione degli strati di materia che i fotoni at-traversano, conoscere l’albedo consente di verificare la presenza di nuvole e di rilevare specifiche molecole nell’atmosfera. Misurare questo parametro è una sfida e richiede il rilevamento di luce riflessa attraverso osservazioni fotometriche nell’ottico. Ciò è possibile nel contesto dell’analisi delle curve di luce. Una curva di luce è la variazione in funzione del tempo del flusso proveniente dalla stella e dai pianeti orbi-tanti attorno ad essa. Nell’ottico, include il transito primario, l’eclisse secondaria e 3 effetti aggiuntivi, l’effetto beaming, la modulazione ellissoidale e la luce riflessa. Mentre il beaming e la modulazione ellissoidale sono trascurabili, la luce riflessa potrebbe dominare il flusso all’esterno del transito, se non ci fosse alcun rumore aggiuntivo. La presenza del rumore strumentale e dell’attività stellare ostacola il rilevamento del segnale planetario, persino supponendo di conoscere con precisione tutti i parametri derivabili dal transito. Mentre il rumore strumentale può diminuire con l’adozione di strumenti migliori o anche effettuando un ’binning’ dei dati, l’attività stellare non può essere rimossa. Nel contesto delle osservazioni effettuate con un telescopio spaziale, come CHEOPS, che offrirà un tempo limitato per ogni target (20 giorni), l’attività stellar può rappresentare un problema.

Nel nostro lavoro, abbiamo esplorato il modo in cui l’attività stellare può limitare il rilevamento dell’albedo planetario, prendendo in considerazione una durata crescente delle osservazioni e imponendo come errore strumentale quello previsto per CHEOPS. In dettaglio, abbiamo prodotto curve di luce simulate, che includevano un configurazione realistica di attivtà stellare, una componente di luce rif-lessa planetaria e rumore bianco gaussiano, mediamente del livello predetto per CHEOPS, per diverse

luce riflessa. La conclusione più importante di tale analisi è stata che l’albedo può essere misurato se i dati coprono almeno un’intera rotazione stellare (e un intera orbita planetaria). Questo risultato, indipendente dal livello di rumore presente nei dati, rappresenta una conseguenza del metodo adottato per riprodurre l’attività stellare, il Processo Gaussiano, che richiede di rilevare un’intera rotazione stel-lare per riprodurre fedelmente l’effetto delle macchie solari. Abbiamo anche dimostrato che, per una stella di magnitudine 6.5 e considerando il livello di rumore di CHEOPS, è possibile rilevare l’albedo di pianeti con raggio maggiore o uguale a 0.03R∗, corrispondente ai più piccoli pianeti di tipo nettuniano. Questi risultati rappresentano un punto di partenza nell’analisi delle curve di fase, non solo con i dati di CHEOPS, ma anche con le future missioni fotometriche, quali PLATO e TESS. Dimostrano anche che rilevare l’albedo di pianeti di tipo terrestre sarà possibile solo con una maggiore precisione fotometrica e con osservazioni di lunga durata. PLATO rappresenterà una notevole occasione in questo senso.

Il disallineamento orbitale è l’angolo compreso tra il piano orbitale e l’asse di rotazione stellare. Può essere misurato tramite l’effetto Rossiter-McLaughlin (RM), che rappresenta il segnale di velocità radiale misurato quando un oggetto transita una stella in rotazione. Le stelle ruotano in modo differenziale e questa loro proprietà influisce sulla forma ed ampiezza del segnale in una maniera tale da non poter essere più ignorata con gli spettrografi di precisione. Pianeti fortemente disallineati forniscono un’opportunità unica di misurare la rotazione differenziale attraverso l’effetto RM, perchè transitano diverse latitudini stellari. In questo senso, WASP-7, con il suo Giove caldo caratterizzato da un disallineamento orbitale molto vicino a 90◦, è uno dei target più promettenti. Benchè Albrecht et al. (2012a) abbiano misurato l’RM di WASP-7b, non sono riusciti a rilevare con sicurezza la rotazione differenziale, riportandone una stima da essi stessi giudicata dubbia. Questo suggerisce anche la possibilità di una misura imprecisa del disallineamento orbitale.

Per questa ragione, abbiamo deciso di esplorare gli ostacoli principali che possano aver impedito la stima della rotazione differenziale di WASP-7. Per fare ciò abbiamo aggiornato il programma SOAP3.0, in modo che tenesse conto della possibilità di una rotazione differnziale della stella. Successivamente, abbiamo studiato la possibilità di adottare gli spettrografi di nuova generazione, come ESPRESSO, per risolvere questi problemi. Infine, abbiamo stabilito in che modo il rumore strumentale e stellare possano influenzare la stima della rotazione differenziale e della geometria del sistema. Abbiamo dimostrato che, per WASP-7, il rumore bianco rappresento un ostacolo importante per il rilevamento della rotazione differenziale, e che una precisione di almeno 2 m s−1 o migliore è essenziale in tal senso. Tuttavia, le passate osservazioni di WASP-7b mostrano ancora residui insolitamente elevati, che non possono

ESPRESSO, perchè offre nella misura della velocità radiale la precisione necessaria a isolare le sorgenti di rumore strumentale e stellare. Sfortunatamente, tale tipo di misura nel caso di pianeti di tipo terrestre sembra ancora lontano.

Come risultato complessivo, possiamo concludere che una descrizione dettagliata del pianeta, spe-cialmente con gli attuali e nuovi strumenti, è possibile solo se si tiene accuratamente in conto del rumore stellare. Inoltre, la presenza di un pianeta può anche aiutare a comprendere meglio certe proprietà stellari, come nel caso della rotazione della stella, esplorata nei nostri lavori. Con questa tesi, possiamo dunque sottolineare l’importanza degli esopianeti nell’ambito dell’analisi stellare e viceversa.

1 Introduction 1

1.1 Detection and characterization of extrasolar planets . . . 6

1.1.1 The Radial Velocity method . . . 6

1.1.2 The transit method . . . 11

1.1.3 The Rossiter-McLaughlin effect . . . 14

1.2 Exoplanet atmospheres . . . 17

1.2.1 Photometric techniques . . . 17

1.2.2 Spectroscopic methods . . . 21

1.3 Instruments . . . 22

1.3.1 Transit surveys . . . 22

1.3.2 Current and upcoming spectrographs . . . 24

1.4 Scope and structure of the thesis . . . 25

2 The influence of stellar physics on the planetary detection and characterization 27 2.1 The stellar magnetic activity . . . 27

2.1.1 The magnetic cycle . . . 27

2.1.2 The effects of stellar magnetism: starspots, plages and faculae . . . 30

2.1.3 The physical properties of activity features . . . 31

2.1.4 The effect of activity features on the detection and characterization of exoplanets . 33 2.2 The convective motions and granulation . . . 37

2.2.1 The macro-turbulence . . . 39

2.2.2 The center-to-limb variation of the convective blue-shift effect . . . 39

2.3 The limb-darkening . . . 40

2.4 The stellar differential rotation . . . 43

3 Updated SOAP3.0 46 3.1 The initial version of SOAP3.0 . . . 46

3.1.1 Input and output parameters before the updates . . . 49

3.1.2 SOAP3.0 performance before the updates . . . 50

3.2 Updates to SOAP3.0 . . . 55

3.2.1 New input parameters in SOAP3.0 . . . 56

3.2.2 Testing the updated SOAP3.0 . . . 57

4.1.1 Stellar activity . . . 68

4.1.2 Instrumental noise . . . 69

4.2 Data analysis method . . . 69

4.2.1 Gaussian process for modeling the stellar activity . . . 69

4.2.2 Analysis method . . . 71

4.3 Reliability test . . . 72

4.4 Results . . . 75

4.4.1 Simulation properties . . . 75

4.4.2 Lower limit for the observation length . . . 75

4.4.3 Variation with stellar magnitude . . . 78

4.4.4 Variation with orbital period . . . 78

4.4.5 Variation with planetary radius . . . 81

4.4.6 Variation with stellar activity level . . . 81

4.5 Towards a complete fitting model for phase light curves . . . 83

4.6 Test with CHEOPS gaps . . . 84

4.7 Tests on real data: Kepler-7 and KIC 3643000 . . . 86

4.8 Conclusions . . . 87

5 Stellar Differential Rotation 89 5.1 WASP-7 and its hot-Jupiter . . . 90

5.2 Simulations . . . 93

5.3 Results . . . 94

5.3.1 Minimum detectableα . . . 94

5.3.2 Varying the instrumental noise . . . 97

5.3.3 Varying granulation and oscillations . . . 101

5.3.4 Varying the exposure time . . . 101

5.3.5 The effect of convective broadening . . . 103

5.3.6 Limb darkening effect . . . 104

5.3.7 Spots . . . 105

5.4 Discussion and conclusions . . . 105

6 Conclusions and future works 108 6.1 Conclusions . . . 108

6.2 Future works . . . 111

6.2.1 Effect of spot evolution or differential rotation on the albedo estimation . . . 111

6.2.2 Detectability of planetary eclipses . . . 112

6.2.3 Effect of stellar differential rotation on the RM signal in presence of occulted and un-occulted spots . . . 112

6.2.4 Breaking the degeneracy between v_∗sin i_∗ and the stellar differential rotation for aligned systems . . . 112

Bibliography 113

4.1 CHEOPS standard deviations for stars of different magnitudes. Courtesy of the CHEOPS

consortium. . . 69

4.2 Adopted priors for the five parameters of the MCMC; P0,∗represents the original value of the stellar rotation used to build the simulation, Fmeanis the flux average, and ptp is the peak-to-peak variation of the light curve. . . 69

4.3 Stellar and planetary properties common for all the performed blind tests. . . 72

4.4 Spot properties used to generate the activity patterns of the blind tests with SOAP (Oshagh et al. 2013b). The pattern labeled a has been adopted for tests 1-3, the b pattern for tests 4-6, and the c pattern for the last three tests. This information was unknown by the person that performed the analysis. . . 73

4.5 Input properties and recovered parameters for the blind tests. . . 74

4.6 Spot properties introduced in SOAP-T (Oshagh et al. 2013a). . . 75

4.7 Fixed planetary properties. . . 75

5.1 Adopted parameters for simulating the RM of WASP-7 with SOAP3.0. The properties are taken from Albrecht et al. (2012a); Southworth et al. (2011a) and Hellier et al. (2009) 90 5.2 Results of our fitting procedure applied on the simulations of WASP-7 RM signal including instrumental noise (σ = 2 m s−1_{) and differential rotation. 0n the left side, we report the} results of the fit performed accounting for rigid rotation in the model, while on the right we show the results obtained as we inject the stellar differential rotation in the fitting model. . . 94

5.3 Results of our fitting procedure applied on the simulations of WASP-7 RM signal: 1) in-cluding instrumental noise and differential rotation (DR); 2) inin-cluding instrumental noise, center-to-limb convective blue-shift (CB) and differential rotation (DR). The injected α was 0.3 and the fit was performed accounting for differential rotation. . . 97

5.4 Same as in Table 5.3 but forα = 0.6. . . 97

5.5 Results of our fitting procedure applied on the simulations of WASP-7 RM signal: 1) including different levels of instrumental noise, DR and granulation; 2) including different levels of instrumental noise, DR, granulation and oscillation. The injectedα was 0.3 and the fit was performed accounting for differential rotation. . . 97

5.6 Same as in Table 5.5 but forα = 0.6. . . 98 xiv

instrumental noise (2 m s−1), DR and different levels of granulation (gran) and oscillation (oscill). The injectedα was 0.3 and the fit was performed accounting for differential rotation. 98 5.8 Same as in Table 5.7 but forα = 0.6. . . 98 5.9 Results of our fitting procedure applied on the simulations of WASP-7 RM signal which

include a different injected FWHM, instrumental noiseσ = 2 m s−1 _{and differential} ro-tation (α = 0.3 on the left side and 0.6 on the right side). The fit was performed fixing FWHM= 6.4 km s−1 _{. . . 103} 5.10 Results of our fitting procedure applied on the simulations of WASP-7 RM signal which

include a different limb darkening law, instrumental noise σ = 2 m s−1 _{and differential} rotation (α = 0.3). The fit was performed fixing u1 = 0.2 and u2 = 0.3. . . 104

1.1 Top: planetary mass as a function of the semi-major axis. Bottom: planetary mass as a function of the planetary radius. . . 4 1.2 Top: orbital eccentricity as a function of the planetary radius. Bottom: orbital eccentricity

as a function of the planetary mass. . . 5 1.3 A scheme of the radial velocity method. Credit: Las Cumbres Observatory . . . 7 1.4 Top row: RV observations for the planet MASCARA-3b. Bottom: RVs as a function of

the planetary phase (Hjorth et al. 2019). . . 9 1.5 Example of a Lomb Scargle analysis on the system HD2071 by Suárez et al. (2002). On

the left, the Lomb Scargle periodograms, with the peak due to the first planet detected, HD2071-b, in the first row, and to the second planet, HD2071-c, in the second row. On the right side the corresponding estimated RV laws (red lines) and the relative measurements and errorbars (black dots). . . 10 1.6 A schematic view of the orbit of a planet around its parent star and the relative light

curve. (Winn 2010) . . . 11 1.7 Transit light curves for some of the first exoplanets discovered by the satellite Kepler. . . 13 1.8 RM description as reported in Gaudi & Winn (2007). The top row shows three different

moments of an exoplanetary transit. The second row shows the same but the star is coloured to reproduce the stellar rotation speed, neglecting differential rotation. In the third and the fourth row, the authors show the observed stellar absorption line for each phase reported in the first and second rows. In particular, the third row shows the case of purely rotational broadening, which means that the net broadening WP due to all other

mechanisms is much less than the rotational broadening VSsin IS. The occultation of the

planet determines a time-variable bump in the line profile. The fourth row reports the same but for the case, in which other line-broadening mechanisms besides rotation are important. . . 15 1.9 A schematic representation of the RM signal as it appears for different spin-orbit angles.

On the left, the case of a completely aligned planet, at the center an example of misaligned system withλ close to 45◦_{, on the right a system withλ close to 90}◦_{. . . 16} 1.10 The complete spectrum of WASP-39b atmosphere, with evident water features (Wakeford

et al. 2018) . . . 18

2.1 The dynamo effect, taken from http://konkoly.hu/solstart/stellar_activity.html . . . 28 2.2 Butterfly Diagram since 1874 until 2016, as reported by Hathaway . . . 28

2.4 Two sets of simultaneous observations of the Sun obtained from NASA’s SDO spacecraft. On each image the trajectory of a simulated Rp= 0.1R∗and b= −0.3 hot-Jupiter (with b

the impact parameter) is plotted. On the left,there are images about a moment of the Sun with low activity, on the right a moment with high activity. The bottom panel reports the transit light curves as a function of the planetary phase. The transit are modelled on the simulated planet transiting the observed solar disk in different wavelengths Llama & Shkolnik (2015). . . 34 2.5 An example of a spot and plage crossing event during the transit of planet, as observed

in photometry and in spectroscopy (Oshagh et al. 2016) . . . 36 2.6 Image of an area of the solar surface by the SDO. In evidence, the photospheric granules . 38 2.7 The optical depth according to the stellar surface area we are looking at . . . 40 2.8 The transit feature without the limb darkening (black thick line) and with limb darkening

(red thick line). The star is coloured in way to show the variation of luminosity as the distance from the center increases. . . 42

3.2 On the left, a simple dark spot effect on the photometry and spectroscopy of a star. In the last frame we also see the BIS effect. On the right panel, same as before but for different latitudes (Boisse et al. 2012). . . 50 3.3 In the top panel, the flux effect of the limb darkening on a spot (top frame) and on a

plage (bottom frame). In the bottom panel, same, but for RV. The red lines are for a quadratic limb darkening law, the green lines for a linear limb darkening law. The size of the active region is 1%. The contrast of the active region is 0.54 in the case of a spot (663K cooler than the Sun), and it is estimated as in Meunier et al. (2010). The active region is located at the center of the stellar disk when the center to limb angle is 0, and on the limb when it is±π/2. The figure is in Dumusque et al. (2014). . . 51 3.4 Same as in 3.3, but for spectroscopy, to display the effect of the resolution. The blue

dashed lines correspond to R> 700000, the green dotted lines to R = 115000 (HARPS) and the red continuous lines to R= 55000 (CORALIE, red continuous line). The Figure is in Dumusque et al. (2014). . . 52 3.5 Same as in 3.3, but for the convective blue-shift (Dumusque et al. 2014). The blue dashed

line uses the same Gaussian CCF in the quiet photosphere and in the active region, the green line corresponds to a model with the same Gaussian CCF, shifted by 350 m s−1in the active region. The red line adopts the observed solar CCF. The Figure is in Dumusque et al. (2014). . . 52 3.6 CCF correction due to an equatorial spot or plage of size 1% for an edge-on star. On the

left side, the convective blue-shift correction when assuming a Gaussian CCF shifted by 350 m s−1(top panel) or when assuming the observed CCF (bottom panel). On the right side, the flux correction for an equatorial spot (top panel) and for a plage (bottom panel). The Figure is in Dumusque et al. (2014). . . 53

(Mandel & Agol 2002). The cyan line shows the result for a star without limb darkening. The red line reports the case with linear limb darkening law (u1= 0.6). Finally, the yellow line reports the model for a star with quadratic limb darkening (u1= 0.29 and u2= 0.34). The dash-dotted line, the dashed line, and the dotted line refer to the same geometries, though using the model by Mandel & Agol (2002). Bottom: the blue dots correspond to the spectroscopic transit observed for WASP-3b (Simpson et al. 2010) and the best RV fit obtained with SOAP-T. From Oshagh et al. (2013a). . . 54 3.8 A direct comparison between the observed data for the transit photometry of HAT-P-11b,

the green dashed line, and the best fit model with SOAP-T, the red dashed-dotted line. The bottom panel reports the residuals. From Oshagh et al. (2013a). . . 55 3.9 Top: RM simulations for different values of α, the relative differential rotation, 0.0, 0.2,

0.4, 0.6, 0.8 and 1. The orange dashed line represents the same simulation, produced with SOAP3.0, which only accounts for rigid rotation. Bottom: RM simulations for different values of λ, spin-orbit angle, 90◦_{, 60}◦_{, 30}◦ _{and 0}◦_{. The dashed lines represent the same} simulations, produced with the old SOAP3.0. In the bottom frames, we show the residuals with respect to the rigid rotation case. . . 58 3.10 Top: RM simulations for different values of RP, the planet radius, 0.1R_⊙, 0.07R_⊙, and

0.04R_⊙. Bottom: RM simulations for different values of iP, the planet orbital inclination,

90◦, 89◦, 88◦, 87◦ and 86◦. The dashed lines represent the same simulations, produced with SOAP3.0. The bottom part of each frame reports the residuals of the RM simulation with respect to the one produced with SOAP3.0. . . 59 3.11 RM simulations for different values of i_∗, the inclination of the stellar rotational axis, 90◦,

45◦, 30◦. The dashed lines represent the same simulations, produced with SOAP3.0. The bottom frame reports the residuals of the RM simulation with respect to the one produced with SOAP3.0. . . 60 3.12 RM simulations for extreme values of i_∗, 90◦, in the top frame, and 5◦, in the bottom one,

to produce equator on and almost pole on configurations, varying the differential rotation parameterα. The dashed lines represent the same simulations, produced with SOAP3.0. The bottom part of each frame reports the residuals of the RM simulation with respect to the one produced with SOAP3.0. . . 61 3.13 residuals of RM simulations in presence of CB with respect to the case without CB for

planets in aligned orbits. The different lines correspond to different rotational velocities of the star. Top panel: tests for the solar 200G model from Cegla et al. (2016b). Bottom panel: tests for the solar 0G model from Cegla et al. (2018). . . 63 3.14 residuals of RM simulations in presence of CB with respect to the case without CB for

planets in misaligned orbits. The different lines correspond to different rotational velocities of the star. Top panel: tests for the solar 0G model and iP = 90◦. Bottom panel: tests for

the solar 0G model and iP = 88◦. . . 64

3.15 RM simulations for an alingned planet, varying the macro-turbulence parameter. Top panel: v_∗sin i_∗ = 5 km s−1 and ζ = 3.0, 4.3 and 5.6 km s−1. Bottom panel: v_∗sin i_∗ = 10 km s−1 andζ = 4.3, 6.2 and 8.1 km s−1_{. . . 65}

corresponds to the planet phase modulation. Both of these plots are shifted by 1. The planet phase modulation is built accounting for albedo Ag = 0.3 and planetary radius

RP= 0.1 R∗. For the other parameters, we refer to the properties listed in Table 4.7. The

red line is the stellar activity modulation, which includes 4 spots, with properties listed in Table 4.6, the black line shows the total flux, and the orange line is the total flux in absence of instrumental noise. This light curve is as well a representation of most of the tests performed in this paper. . . 67

4.2 1D and 2D posterior distributions for the parameters for a star rotating with a period of 19 days with an orbiting planet with radius 0.1 R_∗ , observed for 13 full orbital periods, in presence of the four-spot activity pattern in Table 4.6. The input albedo is 0.3. . . 70 4.3 Comparison between patterns a, b, and c adopted in the blind tests. Their properties are

reported in Table 4.4. . . 73

4.4 Plots of the albedo and relative errors for the simulations obtained with P_∗ = 7, 11, 19, 23, and 26 days and increasing observational lengths. The input stellar properties are reported in Table 4.3, while the planetary properties are listed in Table 4.7. The activity pattern is the one of Table 4.6. The initial albedo is 0.3. In the top panel, we report the albedo and the associated error bars as a function of the number of observed stellar rotations. In the bottom panel, we again plot the errors of the albedo as a function of the number of observed stellar rotations. . . 76

4.5 Comparison between albedo values obtained for the 11 days rotator and with increasing duration of the observations, but in simulations with four different timings, 120 minutes as usual, 110 minutes, 30 minutes, and 28 minutes. The x-axis is the number of observed stellar rotations. For all the analyzed light curves, the unmentioned input properties are the same as described in the caption of Figure 4.4. . . 78

4.6 Recovered albedo and relative error bars as a function of the number of stellar rotations for a 19 days rotator and with three different instrumental noises, 14, 17, and 29 ppm per 120 minutes of observations. All the unmentioned input properties of the simulations are the same as reported in the caption of Figure 4.4. . . 79

4.7 Top: recovered albedo and relative error bars as a function of the orbital period for a 19 days rotator for simulations with 39 days with and without stellar activity and with 30 and 60 days in presence of activity. Bottom: errors of the albedo as a function of the number of stellar rotation observed, for the simulations with P_∗ = 19 days and observational lengths of 30, 39 and 60 days. Here we also add the error of the 39 day long simulation, but without stellar activity. For all the considered light curves, the input unmentioned properties are the same as in the caption of Figure 4.4. In both plots we also added the quantity RP/a as secondary horizontal axis. . . 80

4.8 Recovered albedo as a function of the planetary radius for 39 day-long simulation, a stellar rotation of 19 days, and an albedo of 0.3. . . 81

and the blue points show a Neptunian case. The unmentioned input properties of the simulations are the same as in the caption of Figure 4.4. . . 82 4.10 Recovered albedo and relative error bars as a function of the activity level in percentage

for 39 day-long simulation, a stellar rotation of 19 days, an input albedo of 0.3 and a 0.1 R_∗ planetary radius. The unmentioned properties of the simulations are the same as in Figure 4.4. The horizontal axis is in a logarithmic scale. . . 83 4.11 Simulation of stellar light curve in presence of gaps and with a timing of 1 minute. In

black we report the generated simulation, and in red the binned simulation. The gaps only cover some minutes. Time is expressed in days. . . 85 4.12 Extraction of the 12th quarter of Kepler observations for the star KIC 3643000 after

adding a planet and a two-hour binning. The black error bars represent the data, the red line shows the fit, the orange line show the identified stellar activity, and the green line plots the planetary phase curve shifted by 1. The planet phase modulation is built with an albedo of 0.3, a planetary radius RP= 0.1 R∗ , and the same properties as in Table 4.7. 86

5.1 Fit of the observed data of WASP-7b using the updated SOAP3.0 with differential rota-tion. Top: the blue error bars represent the observed data by Albrecht et al. (2012a)for WASP-7b, while the thick red line is the best fit for the RM signal. Bottom: residuals of the observed RM with respect to the best fit. DR stands for differential rotation. . . 91 5.2 Top: RM simulations for the planet WASP-7b and six different values ofα, the relative

differential rotation, 0.0, 0.2, 0.4, 0.6, 0.8 and 1. Bottom: residuals of the RM simulation in the top plot with respect to the model without differential rotation (α = 0). The vertical black lines in the right side of the two frames represent the ESPRESSO error for averagely fast-rotating F stars, which is 2 m s−1, and they are added to allow a visual comparison with the effect of the differential rotation on RM. In the blank area of the top frame we also show a schematic geometry of the system. The stellar disk is represented as an orange disk. As the latitude increases, the orange fades to white to give an idea of how the rotational velocity decreases. . . 95 5.3 Best fitα and relative error-bars as a function of the instrumental noise. In the first row,

results for simulations of RM which included differential rotation (DR) and instrumental noise (σ), in the second row for simulations also with center-to-limb variation of the convective blue-shift (CB), in the third row for simulations also with granulation (gran) and in the last row for simulations including the oscillations (oscill) too. On the left side, plots relative toα = 0.3, on the right side those for α = 0.6. The fit accounts only for the differential rotation in the model. . . 96 5.4 Fit of the mock data of WASP-7b which include differential rotation (DR) α = 0.6,

granulation (gran) oscillation (oscill) and a white noise of 2 m s−1. The fitting model accounts only for differential rotation. Top: the blue error-bars represent the simulated data, while the thick red line is the best fit. Bottom: residuals of the modelled RM with respect to the best fit. . . 99

and instrumental noise (σ), in the second row for simulations including the oscillations (oscill) too. On the left side, plots relative toα = 0.3, on the right side those for α = 0.6. The fit accounts only for the differential rotation in the model. . . 102

6.1 Example of effect of stellar differential rotation for an aligned system. . . 111

Introduction

The field of extrasolar planets is a relatively young scientific subject, which attempts to answer some of the most ancient questions humans asked themselves: Are we alone? Is there another planet with an advanced civilization like ours? Will we ever be able to communicate with them in the near or distant future? For centuries we looked at the stars, wondering whether or not someone else existed out there. Exploring the Space and observing deeper than the Solar System became possible just in the 20th Century. Before then, the Universe was an impenetrable mystery and so it was understanding why life on other known planets or satellites could not exist or at least be evolved. In such a context, in which knowledge was a privilege, it was easy to impress people with false discoveries. A not so well known example is the ’Great Moon Hoax’. In 1835, the reporter Richard Adams Locke published in the ’Sun’ a series of six articles, announcing and describing the discovery of life on the Moon. The discovery was attributed to John Herschel, one of the most famous astronomers at the time, son of William Herschel. Published as a satire, the articles generated great excitement in the public who believed in the story. Only weeks later, when the story was already known in the rest of the world, the ’Sun’ announced it was a hoax, disclaiming it.

Finding life on other planets or, at least, believing it exists became the main engine which allowed the beginning of the great space missions after the 2nd World War. Still, the answer remains unsolved. So far, no living being has been discovered out of Earth. In this frame, an additional question rises: is life so rare that it can only exist in some of the planets far away from the Solar System? The difficulty of finding life out of Earth led to the search of exoplanets, with the aim of identifying an Earth twin. Though, developing detection methods to discover them was a challenge, which required great technological advancements. The first idea of a method to detect exoplanets was the Doppler spectroscopy, proposed by Belorizky (1938) and, later on, explored by Struve (1952). This method is now known as radial velocity and it is a development of the well confirmed technique which allows to discover spectroscopic binaries. The possibility to treat the planet as a companion with a much smaller mass than the star seemed affordable. Nonetheless, almost 40 years more were necessary to perform the first attempts of detection.

Although the radial velocity method was being improved more and more towards the detection of exoplanets, the first exoplanets were confirmed through a surprising technique, the pulsar timing method. Wolszczan & Frail (1992) discovered a planetary system around the pulsar PSR B1257+12. Three years later, Mayor & Queloz (1995) reported the detection, through Doppler spectroscopy, of the planet 51

Pegasi b, a 0.47MJ planets (Mayor & Queloz 1995). This discovery represented the beginning of a

new era, with the excitement of looking for new worlds and the final challenge of identifying an Earth-like planet. Moreover, it allowed to re-evaluate signals which had already been observed, though not proposed to be exoplanets. The oldest example is the starγ Cephei. Campbell et al. (1988) identified a radial velocity periodic variation and they attributed it to stellar pulsation. Only later, doubts were risen regarding the effective nature of such signal, until it was recognized as due to an M= 1.7MJplanet (Hatzes et al. 2003). Another example is the star HD 114762, orbited by an 11MJ companion. Latham et al. (1989) proposed this object to be a brown dwarf, though a debate rose on its effective nature (Kane & Gelino 2012, confirmed it as a planet, while the community still has doubts). A similar case was the 2.9MJ companion orbiting the star HD 62509. First detected by Hatzes & Cochran (1993), the signal was attributed to intrinsic stellar modulation (pulsation or rotational modulation). A later analysis recognized its nature as an exoplanet (Hatzes et al. 2006). Moreover, new exoplanets were discovered immediately after 51 Pegasi b: 70 Virginis b (Marcy & Butler 1996) and 47 Ursae Maioris b (Butler & Marcy 1996). As the technique was being improved, the first revealed objects were all giants. Though, 51 Pegasi b was a rare type of gaseous planet: it was a close-in short period planet, the first of the class of hot-Jupiters (up to 10 days, see Wang et al. 2015).

At the time, the best radial velocity precision was 15 m s−1. In 1996, the Keck-HIRES spectrograph and the Hamilton spectrograph saw first light and they could already reach the precision of 3 m s−1(Butler et al. 1996). These and more recent instruments allowed to discover a large number of giants (Jupiter mass or higher), proving that hot-Jupiters represent a small sub-sample of exoplanets. In 2003, the spectrograph HARPS saw first light as well. This instrument represented a benchmark in exoplanetary science: with a standard radial velocity precision of 1 m s−1, it allowed to observe exoplanets with a progressively lower mass. The first three Neptune-mass planets were discovered with HARPS in 2004: µ Arae c (Santos et al. 2004), GJ 436b (Butler et al. 2004) and 55 Cancri e (McArthur et al. 2004). The first multiple planetary system with low mass planets on close-in orbits was discovered by (Lovis et al. 2006). Moreover, Udry et al. (2007) detected the first two rocky planets around a G star, Gl581c and Gl581d. The HARPS survey allowed to show already in 2008 that the Neptune-mass planets and rocky planets on short period orbits (M_∗sin iP < 30 M⊕, PP < 50 days) represented a quite numerous

population (Lovis et al. 2009). Until 2012, the radial velocity method represented the most productive technique to discover and characterize exoplanets.

Meanwhile, another detection method began to be applied and improved, the transit photometry. Adopted, at first, to observe the photometric flux of a known planetary companion, HD 209458 b (Char-bonneau et al. 2000), it later allowed to identify the planet OGLE-TR-56b (Konacki et al. 2003). Though, discovering more planets with such technique was a challenge, for several reasons (atmospheric noise, low photometric precision, low probability of physically observing a planetary transit). Photometric obser-vations from space could overcome the issues; though, several years were necessary for such a mission to be realized. With the launch of the wide sky photometric survey Kepler (Borucki & for the Kepler Team 2010), the number of known exoplanets exponentially increased from some tens to more than 3500. The first Kepler major discovery was the system Kepler-9 (Holman et al. 2010), formed by two giant planets in a 2:1 near resonance, with periods of 19.24 days and 38.91 days. The planetary transits thus show transit timing variations, TTV, of tens of minutes. Later on, Batalha et al. (2011) discovered Kepler first rocky planet, Kepler-10b. Lissauer et al. (2011) announced a system of six closely packed Earth-sized planets, Kepler-11, while Doyle et al. (2011) identified the first transiting circumbinary planet,

Kepler-16b. Fressin et al. (2012) found as well the first planet smaller than Earth, Kepler-20e. The huge amount of data offered by Kepler allowed to understand more about the physical properties of planets and their mechanism of formation and evolution.

Today, the exoplanets are classified according to their mass in the following way (as proposed by Stevens & Gaudi 2013):

• sub-Earths (10−8 M_⊕-0.1 M⊕) • Earths (0.1 M⊕-2 M⊕) • super-Earths (2 M_⊕-10 M_⊕) • Neptunes (10 M_⊕-100 M_⊕) • Jupiters (100 M_⊕-103 _M ⊕) • super-Jupiters (103 _M ⊕-13 MJ)

Moreover, the analysis of the exoplanets census allowed to identify 4 populations of exoplanets. To visually observe them, we produced a plot of the planetary mass as a function of the semi-major axis, adopting the catalog in https://exoplanet.eu/catalog. The distribution is reported in the top frame of figure 1.1, where the planets are distinguished according to the detection method. The first population corresponds to hot-Jupiters, characterized by MP > 0.3MJup, semi-major axis a = [10−2; 10−1] A.U. and orbital period PP = [3; 10] days. Since they are close to the parent star, they are strongly irradiated,

causing an expansion of their outer layers and high equilibrium temperatures, 1500− 2500 K (Komacek & Showman 2016). As a consequence, their radii are anomalously large (Guillot & Showman 2002). Under the population of hot-Jupiters, we can spot a second group of planets, clearly separated from the first one. It corresponds to the population of hot or warm Neptunes and super-Earths. They are characterized by PP < 50 days and MP < 30M⊕. On the right with respect to these populations, there

are warm gas giants, with orbital periods between 10 and 200 days and semi-major axis between 0.01 and 1 A.U. Finally, the last population includes Jupiters and Neptunes similar to the giant planets in the Solar System. It is important to note that in this plot we report the value of the mass, which can only be estimated with the radial velocity method once the orbital inclination of the planet is known. For this reason, the population of warm gas giants is less numerous than it should be. A huge number of discoveries made through radial velocities are indeed not reported, since for these planets we just know the lowest limit of the mass. This plot is strongly biased, as we will see in the following sections, by the adopted detection method. For instance, the most efficient method to detect giants similar to those of our Solar System is the imaging technique. Since this method has not been widely applied, the number of discoveries associated to it is very small and it corresponds to the planets on the top right of the plot. Moreover, no rocky planet similar to Earth or Mars has been detected so far: their signal is too low for the radial velocity or transit method to detect similar objects.

The bottom plot of figure 1.1 shows the mass-radius correlation, with a clear separation between Jupiter-sized (RP > 10R_⊕) planets and smaller objects. Less evident is the separation of Earths from

the other exoplanets, especially considering the low number of discoveries in this sense. An analysis of the eccentricities measured for extrasolar planets helps as well to understand their formation processes. In figure 1.2, we report a plot of the eccentricity as a function of the planetary radius in the top frame and as a function the planetary mass in the bottom frame. Both of the plots show an increment of the

10

₁₀

₁₀

₁₀

₁₀

₁₀

₁₀

a [A. U. ]

10

10

10

10

10

10

10

M

p

[M

]

Radial Velocity

Transit, TTV

Imaging

Other

10

₁₀

R

P

[R ]

10

10

10

10

10

10

10

M

P

[M

]

Radial Velocity

Transit, TTV

Imaging

Other

Figure 1.1: Top: planetary mass as a function of the semi-major axis. Bottom: planetary mass as a function of the planetary radius. The data are taken from exoplanet.eu/catalog.

10

₁₀

₁₀

R

P

[R ]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

e

Radial Velocity

Transit, TTV

Imaging

Other

10

₁₀

₁₀

₁₀

₁₀

₁₀

M

P

[M ]

0.0

0.2

0.4

0.6

0.8

e

Radial Velocity

Transit, TTV

Imaging

Other

Figure 1.2: Top: orbital eccentricity as a function of the planetary radius. Bottom: orbital eccentricity as a function of the planetary mass. The data are taken from exoplanet.eu/catalog.

eccentricity dispersion as the planets becomes bigger and heavier. The strongest dispersion, however, is evident for hot-Jupiters, while for Jupiter-sized planets the distribution is similar to the one we observe for Earth-like planets and Neptunes. Such analysis allowed to understand that hot-Jupiters and Jupiters are clearly distinguished in terms of formation process. For example, Kley & Nelson (2012) suggest that hot-Jupiters should have formed farther away from their hosts, and then they migrated towards the stars. Warm Jupiters, though, should have formed in situ (Boley et al. 2016). Finally, Neptunes and Earths encounter different destinies during formation, with a high probability of migrating away from their original location. It is still not clear with which frequency each planetary population should be encountered around stars. For now, rocky planets should represent 30% of the overall census around FGK stars (Howard et al. 2012b) and 40% around M dwarfs (Bonfils et al. 2013). The hot-Jupiters, more easily detected, are thought to orbit only 1.2% of FGK planet-hosts (Wright et al. 2012)

1.1 Detection and characterization of extrasolar planets

This chapter describes three of the currently most used techniques for detecting and characterizing an exoplanet: the radial velocity method, the transit method and the RM effect. These methodologies perform indirect planetary observations, because they reveal the effect of the planet on the stellar signal. The transit and radial velocity methods are nowadays the most applied techniques, allowing to detect more than 90% of the so far discovered exo-planets (exoplanet.eu).

1.1.1

The Radial Velocity method

The radial velocity (RV) or Doppler method measures the projected motion, along the line of sight, of the primary star as it orbits around the barycenter of the system. It can be applied both for binary stars and planetary system. Here, we will concentrate on the phenomenon as it happens for a single planet system.

To determine the velocity and mass of the planet, we measure the Doppler shifts it induces on the stellar spectral lines. An example of how the Doppler effect works in RV observations can be visualized in Figure 1.3. The Figure shows a binary system composed of a star and a planet. In absence of orbiting objects, the star would just move in the galactic frame. As the star is orbited by a planet, the two bodies gravitationally interact and the star moves around the barycenter of the system. Since the planet has a lower mass in comparison to its parent star, the barycenter of the system is placed close to the stellar surface, sometimes below it. When the planet gets away from the observer, the star approaches. In the opposite situation, the star recedes. The movement of the host star shifts the wavelength of the spectral lines with respect to their laboratory counterparts. This shift is proportional to the velocity shift induced on the source by its planet:

δλ λ = vr c = vPsin iP c (1.1)

where vPsin iP is the radial velocity of the planet, iP is the orbital inclination, λ is the laboratory

wavelength of the spectral line, δλ is the shift in wavelength induced by the gravitational interaction between the star and the planet, vris the radial velocity, defined as the stellar velocity along the observer

line of sight and c is the light speed in the vacuum. When the star approaches the observer, the shift is negative and we observe a blue-shift. If the star recedes from the observer the shift is positive and we

detect a red-shift. As a result the overall velocity pattern has a periodic shape. An example for the RV signal of K2-291 is reported in Figure 1.4.

The radial velocity of a planetary system is equal to:

vr= γ + K [cos(ω + ν) + e cos(ω)] (1.2)

Here, γ is the systemic velocity with respect to the observer. The parameter ω is the argument of periastron, whileν is the true anomaly and it depends on the orbital phase. K is the semi-amplitude of the RV signal, expressed as follows Hilditch (2001):

K= 2π PP

a_∗sin iP

(

1− e2)2 (1.3)

with a_∗ the semi-major axis of the stellar orbit around the barycenter, PP the planetary period, iP the

inclination of the orbit and e the eccentricity. By introducing the third Kepler law in Equation 1.3, we get: K2 = G (1− e2₎ 1 a_∗sin iP M_P3sin3iP (M_∗+ MP)2 (1.4)

with G the universal gravitational constant, MP and M_∗are the planetary and stellar mass, respectively.

When MP≪ M∗ and accounting again for the third Kepler law the semi-amplitude becomes:

K= ( 2πG PP )1/3 MPsin iP M_∗2/3 1 (1− e2₎1/2 (1.5)

By expressing the value of the gravitational constant, we can reformulate K in the following shape (Torres et al. 2008): K= 28.4 m s−1 ( PP 1yr )_−1/3 MPsin iP MJup ( M_∗ M_⊙ )_−2/3 (1.6)

As seen from this equation, as the mass of the planet decreases, the star moves less and closer to the barycenter, causing a shift decrement. In the case of the so-called hot-Jupiters, K reaches several hundreds of m s−1. For an Earth mass planet, this value can easily fall below 1 m s−1. The radial velocity method favors the detection of hot-Jupiters (Mayor et al. 2014).

Fitting equation 1.2 on the data we can estimate the orbital period (if we can manage to observe the RV of a full planetary orbit) and a lower limit for the planetary mass MPsin iP, once the mass of the

star M_∗ is estimated through other techniques (spectroscopic and asteroseismic analysis for instance). Breaking the degeneracy between ip and MP is possible through an observation of the planetary transit,

thus required to complete the description of the planetary system (see as examples Kosiarek et al. 2019; Southworth et al. 2011a; Latham et al. 2010, but most of planetary characterizations are performed by coupling the radial velocity method with the transit technique). Among the other geometrical properties of the planetary system, e depends on the shape of the RV and aPcan be estimated with the third Kepler

law.

In presence of more than one planet, the stellar RV depends on the contribution of each of the orbiting objects. Separating all the planetary contributions requires to account for multiple periodicities when analyzing the data. As a first check, we can apply a lomb-scargle periodogram on the RV data. This method was first proposed by (Lomb 1976; Scargle 1982) and after improved by Zechmeister & Kürster

500

0

500

RV (m/sec)

300

400

500

600

Time - 2458000 (BJD)

0

200

O-C (m/sec)

500

0

500

RV (m/sec)

0.4

0.2

0.0

0.2

0.4

Phase

0

200

O-C (m/sec)

Figure 1.4: Top row: RV observations for the planet MASCARA-3b. Bottom: RVs as a function of the planetary phase (Hjorth et al. 2019).

Figure 1.5: Example of a Lomb Scargle analysis on the system HD2071 by Suárez et al. (2002). On the left, the Lomb Scargle periodograms, with the peak due to the first planet detected, HD2071-b, in the first row, and to the second planet, HD2071-c, in the second row. On the right side the corresponding estimated RV laws (red lines) and the relative measurements and errorbars (black dots).

Figure 1.6: A schematic view of the orbit of a planet around its parent star and the relative light curve. (Winn 2010)

(2009) and Mortier et al. (2015). It analyzes the periodicities presented in the data by performing a chain technique. As a first step, it identifies in the periodogram the peak correspondent to the first planet. Then, the RV generated by the first planet are subtracted from the RV observed data. The residuals are later used to generate a new periodogram. If it shows a peak higher than the false alarm probability, a new planet is detected. The procedure can be repeated iteratively, until the strongest peak of the periodogram becomes lower than the false alarm probability. In figure 1.5, we show an example of two planetary detections related to the same planetary system (HD2071 Suárez et al. 2002).

This method allows to measure the orbital periods of the planets. To model the entire planetary system and estimate the other properties of the planets, the fitting methodology has to account for all identified orbiting planets. Several techniques can be applied to perform it. An example of applied techniques is the Markov Chain MonteCarlo method (MCMC) (see e.g. Faria et al. 2016; Tuomi & Jones 2012; Clyde et al. 2007).

1.1.2

The transit method

The primary transit is a flux dimming in the stellar light curve, generated when a planet passes in front of the stellar disk, blocking the flux emitted by the shadowed regions. Figure 1.6 shows an example of light curve, as it evolves along the planetary orbit. For the description of the out-of-transit curve, we refer to Section 1.2.1. We now focus on the primary transit, to understand how it evolves in time: when

the planet begins to enter the stellar disk, the flux decreases. As the planetary disk is completely inside the stellar disk, the transit reaches the minimum and the signal is flat (unless we account for the limb darkening, see Section 2.7). When the planet leaves the stellar area, the flux increases again, until the transit ends.

A complete model for a planetary transit in the case of a circular orbit was introduced by Mandel & Agol (2002), who described the star and the planet as uniform spheres. The transit feature can be characterized by three physical parameters: the flux decrement, the transit duration and the duration of the full occultation. The flux decrement depends on the fraction of stellar disk obscured by the planet and it can be expressed as:

FT 1 F_∗ = ∆F = ( RP R_∗ )2 (1.7) where FT 1

F∗ is the normalized flux during transit, R∗ is the stellar radius and RP is the planetary radius.

The transit duration Td is the entire period of time between the beginning and the end of the transit

and it can be shown to be expressed by:

Td= PP π arcsin   R_a∗ ( (1+ RP/R∗)− (a/R∗cos iP)2 1− cos2_i P )1/2   (1.8)

The duration of the full transit is the period of time in which the planet is completely inside the stellar disk. It corresponds to the deeper part of the transit feature, between the ingress and egress. It can be expressed as: Tf = PP π arcsin   sin ( πTd PP ) (_{(1− R} P/R_∗)2− (a/R_∗cos iP)2 )1/2 ( (1+ RP/R∗)2− (a/R∗cos iP)2 )1/2    (1.9)

Together with these equations, we can estimate PP, through the detection of at least 2 transits.

Measuring FT 1

F_∗, Td, Tf, PP allows to estimate several physical parameters. From equation 1.7, we

estimate the planet radius in units of the stellar radius. Additionally, by combining equations 1.7, 1.8 and 1.9, we can calculate the semi-major axis in units of stellar radii, the stellar density, and the impact parameter, b:

b= a

R_∗cos iP (1.10)

The impact parameter represents the projected distance between the planetary and stellar centers at mid-transit time. From b we can estimate the orbital inclination iP.

As explained in Kipping & Sandford (2016), the transit detection has strong geometrical biases. The geometric probability of observing a transit can be defined as:

P=RP+ R∗

a (1.11)

for circular orbits (Winn 2010). To understand how the different parameters affect the transit feature and how this changes the detection probability, we can inspect Figure 1.7. From the left to the right, the Figure shows transit light curves of five different planets, Kepler-4b, Kepler-5b, Kepler-6b, Kepler-7b and Kepler-8b. As implied by equation 1.7, the transit depth increases as the planet-to-star radii ration increases. Thus, fixing the planet radius and increasing the stellar radius renders the transit shallower. Equation 1.8 suggests that the transit becomes longer if the semi-major axis in units of stellar radii decreases, the orbital period of the planet is longer and the orbital inclination is close to 90◦.

Figure 1.7: Transit light curves for some of the first exoplanets discovered by the satellite Kepler.Taken from the website http://www.nasa.gov/content/light-curves-of-keplers-first-5-discoveries

On top of this, the overall probability of detecting a transit is very low and it strongly depends on the orbital period of the planet. The longer is the period, the lower is the probability of confirming a detection. In particular, it is hard to confirm a planet just with one transit event. Moreover, to detect transits, the observation needs to be longer than the orbital period. The chances of discovering a planet with photometric observations are close to zero, and they decrease as the orbital period becomes longer. To increase the detection rate, it is necessary to perform all sky observations.

All the mentioned biases affect the parameter space of the exoplanets discovered with this method. As mentioned in Winn (2010), before the Kepler mission, the transit method allowed to find planets with a radius much larger than Jupiter and placed in close-in orbits. With Kepler, the rate of discovered small size planets significantly increased and it is predicted to increase even more with the two missions TESS and PLATO, thanks to the improved photometric accuracy.

Transit timing variation

If the planetary system consists of a single planet, the orbit is a Keplerian as described in Section 1.3. In this case, the transit will happen with a perfectly periodic timing and always with the same duration. The presence of a close-in body might introduce additional gravitational interactions. This affects the system stability and the planetary transit suffers a time shift. Transit timing variations (TTVs) are the description of the deviation from the linear ephemeris of a planetary orbit.

The so far measured TTVs were generated by a second planet, which in some cases transited the parent star as well. Several works explored the possible planetary configurations leading to TTVs (e.g.