From ESPRESSO to PLATO: Detecting and Characterizing Earth-Like Planets in the Presence of Stellar Noise

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From ESPRESSO to PLATO: detecting 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

March 2020 Dedication

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; heis 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.

ii Abstract

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 oftwo planetary parameters, the albedo and the spin-orbit angle.

The albedo of an exoplanet represents the fraction of stellar light reflected by the planetary atmosphere. 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 optical 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. Itinvolves, 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 areneg- 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.

iii This result, independent of the level of noise, is a consequence of the adopted methodology to model the stellar activity, the Gaussian process, which needs to detect the full stellar rotation to describe the 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, asthey 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 whitenoise 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

iv stress the importance of exoplanets in the frame of stellar analysis and vice-versa.

v Resumo

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 fluxouma 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, aluzre- fletida pode dominar o flux fora-de-trânsito se não existir mais nenhum fonte de ruído. Apresença 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 emmédia 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

vi de atividade estelar possam ser modelizados com um processo Gaussiano. A principal conclusão desta análise é que pelo menos uma rotação estelar completa é necessária para obter o albedo planetário. Este 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

vii num futuro distante.

Para concluir, podemos verificar que a descrição detalhada de um planeta, especialmente com actuais 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.

viii Riassunto

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 esopianeti 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 ifotoniat- 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, senon 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 perogni 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 riflessa planetaria e rumore bianco gaussiano, mediamente del livello predetto per CHEOPS, per diverse

ix magnitudini stellari. Assumendo che l’attività stellare potesse essere riprodotta con un processo Gaus- siano, abbiamo poi effettuato un fit delle simulazioni, con lo scopo di determinare la componente di 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 stellare per riprodurre fedelmente l’effetto delle macchie solari. Abbiamo anche dimostrato che, peruna 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 poteressere 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 2ms−1 o migliore è essenziale in tal senso. Tuttavia, le passate osservazioni di WASP-7b mostrano ancora residui insolitamente elevati, che non possono

x essere del tutto giustificati con nessuna delle sorgenti di rumore da noi esplorate e per questo richiedono un’ulteriore analisi. Tale analisi potrebbe essere fattibile, per sistemi simili a WASP-7, con lo spettrografo

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.

xi Contents

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

xii 4 Detecting the albedo of exoplanets accounting for stellar activity 66 4.1 Synthetic light curves ...... 67 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 detectable α ...... 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

xiii List of Tables

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, Fmean is 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 theright 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)including instrumental noise and differential rotation (DR); 2) including 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 for α = 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 injected α was 0.3 and the fit was performed accounting for differential rotation...... 97 5.6 Same as in Table 5.5 but for α = 0.6...... 98

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

xv List of Figures

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.Onthe 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 caseof

purely rotational broadening, which means that the net broadening WP due to all other

mechanisms is much less than the rotational broadening VS sin 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 with λ close to 45◦, on the right a system with λ 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

xvi 2.3 An image of a sunspot taken with the SDO, (Solar Dynamic Observatory), with clear Umbra and Penumbra areas. The photo is stored in the Debrecen Heliographic Data . . . 30 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. Inthe 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) andona 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−1 in 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

xvii 3.7 Top: a comparison between the transit feature as modelled by SOAP-T code (same results as SOAP3.0) and the theoretical model of a transiting planet over a non-spotted star (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 parameter α. 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 −1 −1 panel: v∗ sin i∗ = 5 km s and ζ = 3.0, 4.3 and 5.6 km s . Bottom panel: v∗ sin i∗ = 10 km s−1 and ζ = 4.3, 6.2 and 8.1 km s−1...... 65

xviii 4.1 Typical phase light curve used in our work. It shows the normalized stellar flux as a function of the stellar phase. The green line is the instrumental noise, the blue line 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 fluxin 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

xix 4.9 Recovered albedo values as a function of the input values for the 39 day-long simulation and a stellar rotation of 19 days. The red data points represent a Jupiter-sized planet, 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 rotation. 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 of α, 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 ofthetop 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 fit α 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 to α = 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 RMwith respect to the best fit...... 99

xx 5.5 Best fit α 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), granulation (gran) and instrumental noise (σ), in the second row for simulations including the oscillations (oscill) too. On the left side, plots relative to α = 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

xxi Chapter 1

Introduction

The field of extrasolar planets is a relatively young scientific subject, which attempts toanswersome 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 identifyingan 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

1 Chapter 1. Introduction 2

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 star γ 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 duetoan M = 1.7MJ planet

(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 casewasthe

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 of3ms−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 in2004: 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 209458b(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 observations 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, Chapter 1. Introduction 3

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 ofplanets 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):

−8 • sub-Earths (10 M⊕-0.1 M⊕)

• Earths (0.1 M⊕-2 M⊕)

• super-Earths (2 M⊕-10 M⊕)

• Neptunes (10 M⊕-100 M⊕)

3 • Jupiters (100 M⊕-10 M⊕)

3 • super-Jupiters (10 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 −2 −1 corresponds to hot-Jupiters, characterized by MP > 0.3MJup, semi-major axis a = [10 ; 10 ] 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. Theyare 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 thoseof 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 Chapter 1. Introduction 4

104

103

] 102 M [ 1 p 10 M Radial Velocity 100 Transit, TTV

10 1 Imaging Other 10 2 10 2 10 1 100 101 102 103 104 a [A. U. ]

104

103

2 ] 10 M [ 1

P 10 M Radial Velocity 100 Transit, TTV

10 1 Imaging Other 10 2 100 101 RP [R ]

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. Chapter 1. Introduction 5

1.2 Radial Velocity Transit, TTV 1.0 Imaging

0.8 Other

e 0.6

0.4

0.2

0.0 10 1 100 101 RP [R ]

Radial Velocity

0.8 Transit, TTV Imaging Other 0.6 e 0.4

0.2

0.0 10 1 100 101 102 103 104 MP [M ]

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. Chapter 1. Introduction 6 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 awayfrom 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 v v sin i = r = P P (1.1) λ c c where vP sin 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, vr is 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 Chapter 1. Introduction 7

Figure 1.3: A scheme of the radial velocity method. Credit: Las Cumbres Observatory Chapter 1. Introduction 8 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, while ν 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):

2π a∗ sin i K = ( )P (1.3) 2 PP 1 − e2 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: 3 3 G 1 M sin iP K2 = P (1.4) − 2 2 (1 e ) a∗ sin iP (M∗ + MP) 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:

( ) / 2πG 1 3 M sin i 1 K = P P (1.5) 2/3 − 2 1/2 PP M∗ (1 e )

By expressing the value of the gravitational constant, we can reformulate K in the following shape (Torres et al. 2008): ( ) ( ) −1/3 −2/3 P M sin i M∗ K = 28.4 m s−1 P P P (1.6) 1yr MJup M⊙ 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 MP sin 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 aP can 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 Chapter 1. Introduction 9

500

0 RV (m/sec) 500 200

O-C (m/sec) 300 400 500 600 Time - 2458000 (BJD)

500

0 RV (m/sec) 500 200

O-C (m/sec) 0.4 0.2 0.0 0.2 0.4 Phase

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). Chapter 1. Introduction 10

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). Chapter 1. Introduction 11

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 tothefirst 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 ofapplied 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 Chapter 1. Introduction 12 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 theplanet and it can be expressed as: ( ) F R 2 T1 = ∆F = P (1.7) F∗ R∗

F where T1 is the normalized flux during transit, R∗ is the stellar radius and R is the planetary radius. F∗ P 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:  ( )   2 1/2 P R∗ (1 + R /R∗) − (a/R∗ cos i )  T = P  P P  (1.8) d arcsin  2  π a 1 − cos iP

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:  ( ) /   ( ) 2 2 1 2   (1 − R /R∗) − (a/R∗ cos i )  PP  πTd P P  T = arcsin sin ( )  (1.9) f  1/2  π PP 2 2 (1 + RP/R∗) − (a/R∗ cos iP)

Together with these equations, we can estimate PP, through the detection of at least 2 transits. Measuring FT1 , T , T , P allows to estimate several physical parameters. From equation 1.7, we F∗ d f P 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: a b = cos iP (1.10) R∗ 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:

R + R∗ P = P (1.11) a 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◦. Chapter 1. Introduction 13

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) arethe 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. Chapter 1. Introduction 14

Agol & Deck 2016; Holman & Murray 2005; Miralda-Escudé 2002; Agol et al. 2005) and they showed how TTVs are stronger for systems with resonant planets (Mazeh et al. 2013). Examples of detected TTVs are Kepler-9 (Holman et al. 2010), Kepler-30 (Panichi et al. 2018) and Kepler-88 (Nesvorný et al. 2013). In all of these cases, measuring the TTV allowed a characterization of the planetary masses, because the amplitude of the TTV depends on the gravitational interaction between the planets. Moreover, from the detection of TTVs it was possible to discover new non-transiting planets. The earliest discovery dates to Ballard et al. (2011), who found a second planet in the Kepler-19 system. Several other discoveries followed, as for Kepler-115 (Panichi et al. 2018) and Kepler-47 (Becker et al. 2015). This detection method can reach the precision required to measure the TTVs induced by an Earth-sized planet on a Jupiter-sized planet, as predicted in Miralda-Escudé (2002). TTVs might as well be generated by Trojans (Leleu et al. 2017; Haghighipour et al. 2013), which are asteroids or even moons sharing the same orbit of a planet. Until now, no such signal was confirmed, however Janson (2013); Ford & Gaudi (2006) and Madhusudhan & Winn (2009) attempted a first detection. On top of this, exomoons (moons orbiting extrasolar planets) can generate TTVs (Heller 2016; Kipping 2009a,b). Until now, no cases were confirmed; however Szabó et al. (2013) found a number of hot-Jupiters which could be accompanied by exo-moons.

1.1.3 The Rossiter-McLaughlin effect

The Rossiter-McLaughlin effect represents the spectroscopic observation of a transit and it is aconse- quence of the stellar rotation. To have an idea of how this phenomenon works, it can be interesting to inspect figure 1.8 reported in Gaudi & Winn (2007). The top panel represents the transit of a planet in front of the parent star, at different phases. The second row shows the stellar surface with acolor passing from blue to red to resemble the shift in wavelength due to the stellar rotation. A planet is also drawn on top, to show which areas of the stellar surface are being transited. The third and fourth rows represent the effect of the transit on the stellar spectral lines in two cases: the first one when thestellar rotation is dominating, the second when other stellar noise sources tends to prevail. Unless the stellar rotational axis is pointing towards the observer, the stellar disk will appear as divided in two sides (second row in Figure 1.8). In the case represented in the Figure, the star is counterclockwise rotating and the left side of the star is approaching the observer, while the right side is receding. Due to the Doppler effect, the emitted light by the approaching side of the star isblue- shifted, resulting in a shift to lower wavelengths of all the observed spectral lines. On the other hand, the receding side of the star will show a red-shift, thus the wavelengths of the spectral lines are shifted to higher values. During the transit, when the planet projects its shadow on the blue-shifted side of the star, the total observed spectrum, which is the disk integrated of all disk components, will miss part of the blue-shifted light. All spectral lines show a bump in correspondence of the planet shadow. To determine the average RV shift, we estimate the Cross Correlation Function (CCF) and we fit it with a Gaussian. The bump will still appear in the CCF, forcing the fit towards redder wavelengths. Thus, the estimated RV is red-shifted. In the opposite situation, when the planet crosses the red-shifted side of the star, the bump appears on the right side of the spectral lines and the RVs will be negative. If we only account for the geometry of the system, the maximum amplitude of the signal can be Chapter 1. Introduction 15

Figure 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 VS sin 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. Chapter 1. Introduction 16

Figure 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 with λ close to 45◦, on the right a system with λ close to 90◦. analytically expressed as in Winn et al. (2010): ( ) 2 √ RP 2 RMAmp ≃ 1 − b v∗ sin i∗ (1.12) R∗ where v∗ sin i∗ is the projected equatorial rotation velocity of the star along the line of sight. Inspecting the Equation 1.12, we can deduce that the RM signal has larger amplitude as the planet is larger, the impact parameter smaller and the star is a fast rotator. For this reason, the RM effect can be more easily detected for giant planets orbiting fast-rotating stars. Moreover, the shape of the signal strongly depends on the trajectory of the planet and, in particular, on the angle between the stellar axis and the planetary orbit, the spin-orbit angle λ. In Figure 1.9, we show examples of RM signal for different values of λ. For example, if λ is close to 0◦, the planet is transiting the stellar disk in a perpendicular direction with respect to the stellar spin axis. The planet covers both the red–shifted and blue-shifted areas and the RM has a symmetric shape. If λ ∼ 90◦, the transit happens in parallel to the stellar axis. The planet only covers the receding or approaching side of the star and the RM signal is just one sided (second frame of Figure 1.9). If λ is close to 180◦, the orbit is retrograde, and the signal would be inverted with respect to the case with λ = 0◦. Inspecting again the equation of the RM amplitude, we can conclude that, as the photometric transit already offers the information about the planet-to-star radii ratio, the RM effect allows estimate, v∗ sin i∗ and λ. First discovered by Holt (1893), this phenomenon is named after Rossiter (Rossiter 1924) and McLaughlin (McLaughlin 1924), who extensively described it for a system of binary stars. Later on, the Chapter 1. Introduction 17 observations of the RM effect for binaries significantly increased and, in a more recent time, Schneider (2000) proposed to observe the same effect for a planetary transit. Queloz et al. (2000) observed for the first time an RM signal, generated by the planet HD209458 b. In recent years, the RM signal has been observed for an increasing number of planets, allowing to estimate λ for more and more systems. The TEPCAT catalog (Southworth 2011) collects all the planets for which the RM signal has been measured so far and it shows a wide range between the λ values measured, from aligned (Winn et al. 2011) to highly misaligned systems (Addison et al. 2018). Even cases of retrograde planets have been reported (e.g. Hébrard et al. 2011). Knowing λ for a significant number of systems allows us to test theories on planet formation and evolution. For example, Triaud (2011) suggested that λ might decrease as the system becomes older. On top of this, Winn et al. (2010) Albrecht et al. (and 2012b) argued that planets orbiting hot stars might have random λ values, while those around cooler stars tend to be on aligned orbits. These results propose that for a hot star the stellar winds during formation and in early stages of life of a planetary system affect the geometry, forcing planetary companions to be strongly misaligned with respect to the stellar equator.

1.2 Exoplanet atmospheres

The study of exoplanetary atmospheres is nowadays a growing field, especially focusing on the understanding of the chemical composition and of the planetary albedo. After the first discoveries, the number of explored atmospheres increased significantly in the last years, with a considerable amount of detections for hot-Jupiters (e.g. David et al. 2019; Steinrueck et al. 2019; Hoeijmakers et al. 2019; Kreidberg et al. 2015; Barstow et al. 2017)) and Neptune-sized planets (Benneke et al. 2019; MacDonald & Madhusud- han 2019; Fraine et al. 2014). The improvements of the analysis techniques and adopted instruments for observations allowed as well to measure the atmospheres of some super-Earths (e.g. GJ1132b, Diamond- Lowe et al. (2018); LHC1140c, (Spinelli et al. 2019); LHS3844b, (Kreidberg et al. 2019)). Though, a higher precision is necessary to improve the statistics on small size planets. The adopted techniques for such analyses are improving and they involve both spectroscopic and photometric observations. In the following, I will list some of the methods, distinguishing them between photometric and spectroscopic techniques.

1.2.1 Photometric techniques

Transmission spectroscopy

The transmission spectroscopy is one of the most successful techniques and it allowed to detect an increasing number of molecules in atmospheres of exoplanets. The transmission spectrum can be obtained during a primary transit. When a planet occults the stellar disk, the starlight crosses the planetary atmosphere, leaving a spectral imprint of its atmospheric composition. As the wavelength changes, the atmospheric optical thickness varies due to its molecular composition, determining a variation of the transit depth. As a result the estimated planetary radius depends on the wavelength of observation (Burrows 2014). An example of such dependence is reported in Figure 1.10. The transmission spectrum may show peaks of RP/R∗ at specific wavelengths, due to the absorption by atoms or molecules composing the atmosphere of the exoplanet. Considering the high precision necessary in photometric observation to distinguish the different transit depths due to the planetary atmosphere, the transmission spectroscopy Chapter 1. Introduction 18

0.152 5

CO2 4 0.150 Na

) 3 s K CO2 /R pl 0.148 2

H2O 1

0.146 -H2O- —H2O— 0 Scale Height

Transit Depth (R -1 0.144 H2O H2O -2

0.142 -3 0.3 0.4 0.5 1 1.5 2 3 4 5 Wavelength (µm)

Figure 1.10: The complete spectrum of WASP-39b atmosphere, with evident water features (Wakeford et al. 2018) is generally performed with space photometry. In this way, Sing et al. (2011) identified potassium in the XO-2b atmosphere, Charbonneau et al. (2002) discovered sodium in HD 209458 b and Pont et al. (2008) found signatures from both potassium and sodium in HD189733 b. Valeev et al. (2019) discovered the same elements in the atmosphere of the planet WASP-32b. Strong water absorptions were identified in several hot-Jupiters (Deming et al. 2013; Kreidberg et al. 2015) and Neptune-sized (Howe & Burrows 2012; Fraine et al. 2014); as an example, Deming et al. (2013) revealed a water feature at 1.4µm in the atmosphere of both HD 209458 b and

XO-1 b. Finally, molecules such as CO, CO2 and CH4 were recovered (Benneke 2015; Barstow et al. 2013). Mancini et al. (2017) analyzed the transmission spectrum of WASP-52b, discovering several molecules. Similar studies were performed by several authors in the last two years. Among them, we cite Mackebrandt et al. (2017) for TrES-3 b, Rackham et al. (2017) for GJ 1214-b, Wakeford et al. (2018) for WASP-39b, Southworth et al 2018 for XO1-b, Hoeijmakers et al. (2019) for KELT-9b.

Occultation spectroscopy

When a transiting planet passes behind its host star, we can observe a drop in flux because we stop receiving light from it. This phenomenon is called occultation or secondary eclipse and it represents the only moment of the entire planetary orbit in which we only observe the stellar emission. Immediately before and after the transit, though, we receive both the flux from the star and the planet. Thus, comparing the eclipse to the out of transit flux, we can isolate the planetary flux. Based on the wavelength band of photometric observations, with occultation spectroscopy we can detect the thermal emission or the reflected light of the planet. For optical observations, the planetary flux is dominated by reflection. So, the analysis will allow to measure thealbedo(Alonso et al. 2009). For infrared observations, the flux will be dominated by thermal emission (Deming et al. 2005). Chapter 1. Introduction 19

Phase curve analysis

The atmosphere of a planet can be probed as well with photometric phase light curves, which represent flux variations during the whole orbital period (Angerhausen et al. 2015; Esteves et al. 2013; Lillo-Box et al. 2014). A phase light curve, apart from the planetary transit and the secondary eclipse, includes several additional features: the reflected light by the planet, the planetary thermal emission, the beaming effect and the ellipsoidal modulation. The planetary phase curve analysis can be performed with two different objectives. The first oneis measuring the albedo of exoplanets in optical observations or analyzing the atmospheric emission in infrared data. In particular, measuring the thermal emission allows to obtain the equilibrium temperature of the planet as well. The second application consists of using the phase curve analysis as an alternative technique to detect non-transiting planets (Crossfield et al. 2010). In this section, we will mainly focus on the optical phase curves, since they are an important focus for this Ph.D. thesis. The ellipsoidal modulation is caused by the tidal distortion of the star due to the planetary orbit. It arises when the planet has at least Jupiter mass and it is close to its host star. In this conditions, its induced tidal deformation is no longer negligible. In its most simple way, it can be expressed as it follows: ( )3 FE MP R∗ 2 = −αe sin iP cos 2θ (1.13) F∗ M∗ r where iP is the orbital inclination and r is the planet-star distance (Lillo-Box et al. 2014). Since from now on we will only consider circular orbits, r is equal to the semi-major axis a, so that R∗/r becomes

R∗/a. theta is phase angle and αe is the ellipsoidal parameter expressed as:

(15 + u1)(1 + g) αe = 0.15 (1.14) 3 − u1 with u1 the stellar linear limb darkening coefficient estimated by Claret (2000) adopting a linear law. g is the gravity darkening coefficient. A complete formulation for the ellipsoidal modulation and the estimate of αe is reported in Morris et al. (2013), who used a Fourier series to reproduce the quasi-sinusoidal shape of this effect as observed in binary stars. By analyzing the equation 1.13, we can deduce that this effect is not dominating in phase curves. While for a Jupiter-sized planet with a 3-days orbit around a Sun-like star it contributes by less than 2 ppm, for a planet on a closer orbit and higher mass, it could increase to tens of ppm. The ellipsoidal modulation of a planet was detected for the first time by Welsh et al. (2010) for the planet HAT-P-7 b. Later on, this contribution continued to be accounted for in planetary analysis, though it was only detected in a low number of cases. (see Angerhausen et al. 2015, for some examples). The Doppler Beaming is a relativistic effect and it represents the photometric counterpart of theRV technique. When the star recedes from the observer, the dominant wavelengths are red-shifted and the overall stellar brightness decreases. On the contrary, if the star approaches the observer, the brightness increases. This effect is much stronger in ultra-compact binary stars(Zucker et al. 2007) and it was used to detect several non-eclipsing binaries. Loeb & Gaudi (2003) determined an analytic expression for this component: F K B = (3 − Γ) (sin θ + e cos ω) (1.15) F∗ c Here, K is the radial velocity semi-amplitude. The parameter Γ, with the hypothesis of a blackbody, is Chapter 1. Introduction 20 expressed as it follows: ex(3 − x) − 3 Γ = (1.16) ex − 1 The parameter x is derived through the Wien Law and it is equal to:

hc x = (1.17) (kBλTeﬀ) where h is the Planck constant, c is the light speed, kB is the Boltzmann constant and Teﬀ is the effective temperature of the star. Finally, λ is the average wavelength of photometric observations. When x ≪ 1, Γ = 2, otherwise Γ = 3 − x. The factor (3 − Γ) is called photon-weighted bandpass-integrated factor. So, in the case of optical photometric observations, we may get negative values of Γ. This expression of the beaming effect is extensively used (Jackson et al. 2012; Mazeh et al. 2013; Barclay et al. 2012). Nonetheless, the reported intensities for Jupiter-sized planets are on the level of ∼ 1 ppm, increasing to much higher values only for brown dwarfs. The reflected light component represents the fraction of stellar flux reflected by the planetary atmosphere and surface along the orbit. It can be expressed as:

FR = AP f (z) (1.18) F∗

Here, Fp is the planetary reflected flux, F∗ is the stellar flux. The parameter AP is the amplitude of the reflected light in equation 1.18 is ( ) R 2 A = A P (1.19) P g r where Ag is the geometric albedo and, again, r becomes a for circular orbits. Finally, in equation 1.18 f (z) is the planetary phase function. z depends on the orbital inclination i and on the orbital phase as it follows:

cos z = sin(ω + ν) sin iP (1.20)

The parameter ω is the argument of periastron, while ν is the true anomaly connected to the phase angle θ. The choice of how to model f (z) depends on the hypothesis we impose on the atmosphere of the planet. Madhusudhan & Burrows (2012) performed a comprehensive study on the different models of f (z), including the one associated with Rayleigh scattering from the atmosphere. However, most of the works in literature use a more simple model, as it was done in Esteves et al. (2013) and Angerhausen et al. (2015)). It describes the planet as a Lambertian sphere (Russell 1916), meaning we assume it as a perfect sphere with an atmosphere reflecting isotropically the stellar flux. In this case, the planetary phase function is given as (see e.g. Angerhausen et al. 2015; Esteves et al. 2013):

+ π − = sin(z) ( z) cos(z) f (z) π (1.21)

This equation is defined between 0 and 1 and it reaches it maximum when the planet is at the secondary eclipse, because this is the moment in which the reflected light of the star is totally reflected towards the observer. The minimum happens at the primary transit, when the reflection is totally directed towards the parent star. In general, planets in the Solar System do not behave as fully diffusive bodies, rendering the Lambertian approximation improper for them (Mallama 2009; Dyudina et al. 2005). On the other hand, they tend to fully radiate straight back to the source. For this reason other works proposed Chapter 1. Introduction 21 to adopt the Hilton phase function, which accounts for the visual magnitude of the planet ∆m(z) and incorporates more back scattering due to the cloud-cover (Rodler et al. 2010):

f (z) = 10−0.4∆m(z) (1.22) with ∆m(z) the planet visual magnitude: ( ) ( ) ( ) z z 2 z 3 ∆m(z) = 0.09 + 2.39 − 0.65 (1.23) 100◦ 100◦ 100◦

This model derives from Venus and Jupiter observations and it has a non-sinusoidal shape. Several phenomena can affect the planetary reflected light. The presence of small atmospheric particles determines a non-isotropic scattering, which changes the shape of the phase curve (Seager & Sasselov 2000). Additionally, the atmospheric circulation affects the temperature of the planet and determines cloud coverage movements. For this reason the phase curve will suffer by asymmetries and phase shifts (Sudarsky et al. 2005). Note that we will not present the thermal emission of the planet, which is extremely relevant for giant planets, especially in the infrared emission. Since my thesis work considers optical observations, this last component can be ignored. To have a complete analysis of the thermal emission refer to Carter (2019).

1.2.2 Spectroscopic methods

While photometric methods rely on space photometry, the spectroscopic techniques are mainly based on the ground high-resolution spectrographs. In general, the planet and the star cannot be spatially resolved, in which case the methodology will follow similar principles as for the photometric methods. One of the most common method is again the transmission spectroscopy, performed as in the case of photometric observations, by analyzing the transit depth for different wavelengths. This methodology, applied by Wyttenbach et al. (2015) and Khalafinejad et al. (2017), reveals to be more efficient in spectroscopy than in photometry, due to the large aperture of the ground telescopes used to perform spectroscopic observations. Moreover, with ground telescopes, it is possible to spectroscopically resolve Earth’s atmospheric absorption lines and the equivalent spectral lines in the planetary atmosphere. In this way, we can recover the spectral profile of several elements with minimum telluric contamination. An alternative technique consists of cross-correlating high-resolution observations with model planetary templates, to recover the exoplanetary signal. Again, depending on the observed bands, we can retrieve the reflected spectrum (as in Charbonneau et al. 1999; Collier Cameron et al. 1999) and the thermal spectrum (as for Snellen et al. 2010; Birkby et al. 2013). Though, since in infrared the planet- star contrast is stronger, the technique resulted to be more effective to detect the thermal spectrum, while it allowed to impose an upper limit on the reflected spectrum. When the planet and the star can be spatially resolved, the planetary spectrum can be fully isolated. This technique, successfully applied by Brogi et al. (2012) and (Snellen et al. 2014), works well especially for planetary systems close to us and for exoplanets distant from the host star. Finally, a new technique to retrieve the reflected light of an exoplanet was proposed by Martins et al. (2015), who used the CCF of the spectrum to enhance the signal-to-noise ratio and estimate the planetary albedo. Chapter 1. Introduction 22

1.3 Instruments

In this section, we present in detail some of the instruments which allowed and will allow a fine characterization of exoplanets and their atmospheres.

1.3.1 Transit surveys

As mentioned in Section 1.1.2, given a single star, the probability of detecting a planet through the transit method is very low. For this reason, since the beginning it was clear that this method could be applied in two ways. The first one consisted of performing the follow-up of planets already discovered through the RV method and which had a high transit probability. The second application aimed towards the discovery of new planets and required wide angles surveys. Monitoring many stars at the same time, greatly increases the probability of detecting transiting planets around a significant percentage of targets. The first studies about planning wide sky surveys were proposing ground surveys and were very positive about their success (Gillon et al. 2005; Horne 2002). Nonetheless, some significant hurdles where encountered. As reported in Brown (2003), most of the ground surveys had a very low precision, for several reasons. Among the others, we recall a non-advanced photometric technology, an imprecise knowledge of the atmospheric noise in photometric surveys, and several instrumental instabilities. Moreover, these surveys resulted in numerous false positives, mainly due to eclipses from binary systems (Brown 2003). For example, the OGLE-III project (Udalski et al. 2002) found only 3 planets, all the others were false positives. The Vulcan Project identified 7 objects of interest, which revealed themselves to be false positive as well (Jenkins et al. 2002). Same results for STARE (Brown & Charbonneau 1999). Only after these first attempts, the technology for detecting planetary transits was improved and thedata analysis became finer, allowing for new discoveries with these data. Moreover, a new series ofground and space surveys were launched, allowing an increasing rate of discoveries until now. Among the ground surveys, we can highlight HATNet and WASP. The HATNet project includes HAT-North, with a precision of 4 mmag for a star with magnitude r= 9.5 (Bakos et al. 2004, 2002), and HAT-S, 5 mmag for a magnitude r= 10.5. It discovered more than 90 planets (Narita et al. 2009; Bakos et al. 2010) and contributed to characterize objects already identified with the RV technique (Bakos et al. 2009; Howard et al. 2012a). The WASP project reach a precision of 4 mmag for V= 9.5 and it monitored millions of stars, of which 130 where discovered to have planets (Smith et al. 2014; Triaud et al. 2013). Among the space-based survey, we highlight CoRoT, Kepler, TESS and Plato. CoRoT (Convection, Rotation and planetary Transits) was the first space mission designed to search for transiting planet (Moutou et al. 2013; Deleuil 2012). With a precision ranging between 75 ppm/h−1 to 1130 ppm/h−1 for stars with magnitudes 11 < R < 16, it discovered all the planets named from CoRoT-1b to CoRoT- 33b (see the catalog http://exoplanet.eu/catalog/). Kepler and TESS are the past and present of the exoplanet field, while Plato will be a future development of these two. Space surveys allow as well an analysis of planetary atmospheres, which has already been possible with Kepler. Phase curve analysis and transmission spectroscopy has been performed on Spitzer and Hubble Space Telescope data. A future development in this sense will be offered by Cheops, the first ESA space telescope. In this section we present the missions and telescopes which offered and will offer an important breakthrough in transit and phase curve analysis: Kepler, TESS, CHEOPS and PLATO. Chapter 1. Introduction 23

Kepler

The Kepler satellite (Borucki & for the Kepler Team 2010) represented a huge step in terms of photometric search of exoplanets. Launched in 2009, the mission entered an heliocentric orbit to avoid eclipses due to Earth and it used for observations a 0.95-m diameter telescope. It reached a precision of 30-40 ppm for a 12 mag star, with binned data over 6.5 h intervals (Christiansen et al. 2012). Initially programmed to last for 3.5 years, the mission was extended for a longer time. In 2013, the first phase of the mission finished, after having offered to community four continuous years of observation for145000 main sequence stars or close to main sequence, with a cadence of 30 min. As reported in Batalha et al.

(2010), among these stars (all with MV > 16), 90000 are G type, 3000 are M dwarfs, while 5000 are giants. The final aim of Kepler, exploring the structure and diversity of exoplanets Borucki et al. (2007), was largely fulfilled, with the discovery of more than 3000 exoplanets. Many of these new planets belongedto multi-planetary systems, with a completely different architecture than the Solar System (e.g. Kepler-62 Borucki et al. (2013), Kepler-89 (Weiss et al. 2013), Kepler-90 (Shallue & Vanderburg 2018; Cabrera et al. 2014; Batalha et al. 2013), etc.). Moreover, Kepler allowed the discovery of several terrestrial planets, some of them placed in the habitable zone of the parent star (for a detailed definition of the habitable zone Vladilo et al. 2015, 2013). Examples of potentially habitable planets are Kepler-186 f (Quintana et al. 2014), Kepler-438 b and Kepler-440 b (both by Armstrong et al. 2016). Kepler officially finished in May 2013, with the failure of the second reaction wheel. Because ofthis the telescope lost pointing stability, forcing the second phase of the mission, K2, to observe in a different field of view than Kepler. Thus, K2 was designed to observe along the ecliptic plane, to avoid theeffectof solar radiation pressure on the instrument pointing. As a result, the instrument became more unstable, with an instrumental noise of 80 ppm over 6 h of binning (Howell et al. 2014). K2 started in 2014, and lasted for more than 3 years. The new spacecraft systematic is being corrected in the data using several correction methods (e.g. Vanderburg et al. 2016, and references therein). In this way, Vanderburg et al. (2016) identified 234 candidates from the first year of observations, while Barros et al. (2016) selected 172 candidates from the initial 15 months. At the end of the mission, a total of 877 candidates has been reported (see more in Kepler website https://exoplanetarchive.ipac.caltech.edu/docs/counts_detail.html)

TESS

The Transiting Exoplanet Survey Satellite (TESS) was launched on April 18th 2018 and it is performing a 2yr transit search over the whole sky, monitoring more than 500000 stars with the magnitude limit of V ≤ 12. As mentioned in Ricker et al. (2015) and in Oelkers et al. (2018), TESS orbits Earth on an highly elliptic 2:1 lunar resonant orbit, allowing to observe both the northern and the southern hemispheres. Tess sky field is divided into 26 sectors, with an overlap at the ecliptic poles. These areas willthusbe scanned for the two full years of the mission, improving the sensitivity towards smaller and longer period planets. The satellite spends two 13.7 days orbits observing each field and in its first year of work itis mapping the southern sky. Year 2 will be dedicated to the northern sky. TESS aims to cover the voids in the planetary search by CoRoT and Kepler, which did not perform all sky observations and monitored V ≥ 12 stars. This survey provides a precision of 50 ppm every 6.5 hours, lower than Kepler precision, however still sufficient to discover between 1700 and 3000 exoplanets Sullivan et al. (2015). Chapter 1. Introduction 24

CHEOPS

CHaracterizing ExOPlanets Satellite (CHEOPS), by ESA, will allow phase curve analyses as well. The CHEOPS satellite (Fortier et al. 2014) aims to observe the planetary transits around bright stars, for a better characterization of already known systems. Launched by the end of 2019, it will orbit Earth on a low Earth Sun synchronous orbit, LEO-SSO, at 700 km of altitude. The telescope has a diameter of 0.32 m and it will observe a 6.5 V magnitude star with an instrumental noise of 155 ppm per minute (private communication).

PLATO

The PLAnetary Transits and Oscillations of stars (PLATO) is a satellite planned to launch in 2026 (Rauer et al. 2016) and to be located in the second Lagrangian point, as well as Kepler. The stability of the location in space will allow to reach a very high precision and to observe the targets for a long time. The mission aims to monitor very bright stars, with a magnitude ranging between 4 and 16. The choice of bright stars is justifed by the requirement to perform a complete characterization of the systems. In particular, PLATO will monitor stars for which an efficient ground based radial velocity follow-up is possible.

ARIEL

The Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL) is a space telescope planned for launch in 2028 as the fourth medium-class mission of the European Space Agency’s Cosmic Vision program. It will be located at second Lagrangian point and it will inherit the properties of Planck, with a collecting telescope area of 0.64m. The mission will last for 4 years and it aims to observe at least 1,000 known exoplanets distant from their stars, using the transit method. Moreover, it will possess a spectrometer which will allow to study and characterize the planetary chemical composition and their thermal structure.

1.3.2 Current and upcoming spectrographs

Currently, several spectrographs are being used to measure the RV of exoplanets and perform a detailed analysis of planetary atmospheres. In this section, we present HARPS and HARPS-N, which have been largely used for optical observations, and ESPRESSO, which represents the greatest result in terms of spectroscopic precision. For other optical spectrographs, such as the PFS, and infrared spectrographs, such as SPIRou and CARMENES we refer to Fischer et al. (2016).

KECK-HIRES

The HIGH REsolution Spectrograph is another echelle spectrograph installed on the Keck 10-m telescope in Mauna Kea, Hawaii and it has a resolution of R = 80000 (Vogt et al. 1994). It covers the spectral range 390-620 nm. HIRES can reach a precision of 2-3 m s−1 and it aims at analyzing the RVs of massive exoplanets. Chapter 1. Introduction 25

HARPS/HARPS-N

The High Accuracy Radial Velocity Planet Searcher (HARPS) and HARPS-N are two high-precision and high-accuracy échelle spectrographs with resolution R = 115000 and operating over a spectral range of 378-691 nm. HARPS is installed at the ESO 3.6m telescope, in La Silla (Chile), while HARPS-N is connected to the Telescopio Nazionale Galileo (TNG) in La Palma (Spain). The two spectrographs were designed to reach an accuracy of 1 m s−1 on slow rotating stars, with the final objective to observe the RV signature of a super-Earth like planet (Pepe et al. 2002).

ESPRESSO

The Echelle Spectrograph for PREcision Super Stable Observations (ESPRESSO) was planned to be an improvement of HARPS, with resolution R = 70000, 145000 and 190000 according to the different modality adopted (Pepe et al. 2013). It is installed in the Combined Coudè Laboratory at the ESO-VLT (Paranal) and it is linked to the four Unit Telescopes with optical coudè trains. It can operate with one or up to all the four telescopes of the VLT and it is expected to reach an RV precision of 0.1 m s for slow rotating stars. The final scope of ESPRESSO for RV analysis is to detect the signal of an Earth-like planet and study exoplanet atmospheres.

1.4 Scope and structure of the thesis

The improvements offered by Harps and Kepler changed and redirected the field of exoplanets towards three main goals:

• identifying an Earth sibling, a planet with mass and radius similar to our world and situated in the habitable zone of its star (Gillon et al. 2017; Ribas et al. 2016; Turbet et al. 2016; O’Malley-James & Kaltenegger 2017)

• understanding the process of formation of the Solar System through the comparison with other planetary systems Matsumura et al. (2017); Morbidelli & Raymond (2016)

• analyzing the exoplanetary atmosphere, towards the characterization of a potential Earth twin’s atmosphere.

Although Kepler could discover many Earth-size planets, characterizing their masses and confirming them was not possible. A better spectrograph than HARPS, which was the most precise at the time of KEPLER, was required to reach Earth-mass precision in radial velocity. Additionally, Kepler scanned only a part of the sky, selecting relatively faint star. Improving the sample will be possible with the new photometric surveys and missions CHEOPS, TESS and PLATO, while estimating the mass of Earth-sized planets is the final goal of the spectrograph ESPRESSO. As the instrumental precision improves, a detailed characterization of the planetary signal requires a more accurate knowledge of the additional noise sources due to the star. For instance, the presence of starspots on the stellar disk induces a radial velocity periodic signal and photometric modulations capable of affecting the planet detection and characterization, especially in the case of Earth-sized planet. As a consequence, for active stars, the detection of phase curves may be significantly hurdled by spots features, when the data are strongly limited in time (as they will be in the case of CHEOPS). It is not Chapter 1. Introduction 26 a case that, so far, the reflected light component of exoplanets has been detected for data covering long observational periods and a high number of planetary orbits (Angerhausen et al. 2015; Esteves et al. 2013; Lillo-Box et al. 2014). Another intrinsic property of the star is the stellar differential rotation, which can change the amplitude of the Rossiter-McLaughlin modulation induced by planets ((Hirano et al. 2011; Albrecht et al. 2012a; Cegla et al. 2016a, 2018). The granulation introduces an extra red noise in both photometric and spectroscopic observations. A high-precision characterization of an exoplanet will thus require to account for these phenomena. On the other hand, a good description of the planetary signal allows a precise characterization of the stellar properties. The aim of the present thesis is to assess how the stellar noise sources and the exoplanet properties can be distinguished one from the other with the new instruments (CHEOPS, ESPRESSO). As a final objective, we specifically point towards a detailed exploration of the detection limits posed by the different stellar noise sources to two planetary parameters: the spin-orbit angle and the planetary albedo. In Chapter 2, we introduce the main sources of stellar noise, in particular the effects of convection, the stellar differential rotation and the stellar activity. We describe as well how they affect the exoplanet signal. Chapter 3 introduces SOAP3.0, a numerical code which can produce the photometric and spectroscopic transit of a planet in front of a rotating star. Therein, we will also report our improvements to the code. Chapter 4 presents the analysis on the detectability of the planetary albedo with CHEOPS, when the star is active. In Chapter 5, we report an analysis on the possibility of measuring the stellar differential rotation through the Rossiter-McLaughlin effect for misaligned planetary systems. We explore, in details, which noise sources can hurdle such detection. In Chapter 6, we present the conclusions of this Ph.D. project and its future perspectives. Chapter 2

The influence of stellar physics on the planetary detection and characterization

This chapter is dedicated to a description of different stellar noise sources and properties, capable of affecting the detection and characterization of exoplanets. We will analyze, in detail, the granulation, the magnetic cycle, the stellar differential rotation and the limb darkening. Although, we will not explore in detail the stellar oscillations, which are a consequence of the dynamical equilibrium between gravity and pressure in the stellar body. Oscillations have the smallest timescales among the stellar noise sources (several minutes) and, for this reason, they can easily be averaged out by performing observations with a time of integration longer than their predicted time scale.

2.1 The stellar magnetic activity

On the solar surface, we can identify several activity features: among the others, spots, plages and faculae. They form due to the emergence of magnetic field lines, generated by the dynamo effect. Dynamo arises because the stellar rotation is coupled with the convection and conduction happening in the stellar plasma. To understand how it works, we can follow the explanation in Figure 2.1. The convective motions generate a magnetic field which is poloidal in its initial configuration. Duetothe stellar differential rotation, the magnetic lines are stretched all around the stellar body, forcing thefield into a toroidal shape. This effect is known as Ω-effect. In a second moment, the field lines are twisted due to an effect known as α-effect (Choudhuri 2013) and they are unrolled, until the field returns to be poloidal. The new shape has inverted magnetic poles with respect to the original ones. For this reason, returning to the initial magnetic field shape requires another cycle, opposite to the previous one.The entire dynamo cycle is defined as the time necessary for the magnetic poles to resume to theinitial configuration.

2.1.1 The magnetic cycle

Several studies on the duration of the solar cycle showed that it lasts for 22 years, since in 11 years the magnetic poles get inverted and more 11 years are required to return to the initial condition. The knowledge of the solar cycle dates back to the work by Schwabe (1844), who reported it for the first time,

27 Chapter 2. The influence of stellar physics on the planetary detection and characterization 28

Figure 2.1: The dynamo effect, taken from http://konkoly.hu/solstart/stellar_activity.html

Figure 2.2: Butterfly Diagram since 1874 until 2016, as reported by Hathaway (solar- science.msfc.nasa.gov/images/bfly.gif ) Chapter 2. The influence of stellar physics on the planetary detection and characterization 29 though only Hale et al. (1918) described it in detail. Until now, the Hale cycle, as it is also named, has been observed for more than two dozens times (Hathaway 2015). Though, trying to measure it for other stars is challenging because it requires long period observations. The only case in which the activity cycle of a star other than the Sun was accurately measured is that of KIC 806161. Karoff et al. (2018a) used long data observations performed as part of the Mount Wilson HK project (Baliunas et al. 1994), the California Planet Search program (Wright 2005) and the Kepler data and they estimated the cycle timescale of this star to be 7.41 ys. Several authors attempted the measurement of the stellar activity through Kepler data, though no other cycle time scale with a similar precision can be reported. As mentioned in Reinhold et al. (2017), 4 years of Kepler data are not sufficient to measure a stellar cycle. Mathur et al. (2014) studied the activity cycle of 22 Kepler F stars. F stars, in average, rotate faster than G or K dwarfs. Fast-rotating F stars represent a promising target for measuring the stellar cycle, because they are expected to show the shortest cycle periods (Metcalfe et al. 2007). Such expectations were suggested by observations as well (García et al. 2010; Salabert et al. 2011). Nonetheless, the analysis they performed on activity features could not result in a clear measurement of any stellar cycle. In one case, KIC 3733735, they managed to estimate a minimum timescale of at least 1400 days. Later, Montet et al. (2017) employed full-frame Kepler images to reconstruct the photometric long-term variability of more than solar-like 4000 stars. They could detect cycles in 10% of the cases and only in 28 systems among them they observed complete cycles, whose period increases from 2 to 4 years as P∗ increases. In a similar way, Reinhold et al. (2017) analyzed the variability patterns of more than 3000 Kepler stars. They managed to identify cycles for the 13% of the cases, especially F stars. For stars with 5 < P∗ < 25 days, they confirmed the correlation identified by Montet et al. (2017) between the rotational period and the stellar cycle. Though they could not detect any cycle longer than 2-4 years and they could not define cycles for rotators slower than 25 days. Short cycles, mainly for F stars, are not necessarily their main activity cycles. For instance, Fletcher et al. (2010) identified the presence of an additional biennial cycle on the Sun, which was confirmedby Simoniello et al. (2013), Simoniello et al. (2012) and Broomhall et al. (2012). As explained in Fletcher et al. (2010) and confirmed in Jeffers et al. (2018), this phenomenon could be connected to a polarity inversion in solar-like activity cycles. Evidence of an additional cycle was identified as well on other stars, such as HD114710, HD190406 and HD78366 (Oláh et al. 2016). Though no confirmation that they actually exist on F stars was ever identified. Reinhold et al. (2017) were the first to propose these detected short cycles of up to 4 years to be due to a secondary dynamo for the F-stars. An activity cycle shows the predominance of spots or faculae according to the age. Shapiro et al. (2014) showed that:

• less active stars are usually older (Soderblom et al. 1991) and they seem to show an increment of the stellar brightness with the stellar chromospheric activity. This suggests that they are plages dominated.

• more active stars are younger and they show an anti-correlation between the brightness and the chromospheric activity.

The Sun belongs to the old stars group, as discovered by Lockwood et al. (2007) and confirmed by Hall et al. (2009), thus it has a strong chromospheric activity (Shapiro et al. 2013). On top of this, the analysis of both photospheric and chromospheric activity shows a strong correlation between the two, meaning that a higher plages coverage is accompanied by a higher spot coverage as well (Shapiro Chapter 2. The influence of stellar physics on the planetary detection and characterization 30

Figure 2.3: An image of a sunspot taken with the SDO, (Solar Dynamic Observatory), with clear Umbra and Penumbra areas. The photo is stored in the Debrecen Heliographic Data, http://fenyi.solarobs.csfk.mta.hu/en/databases/DPD/ et al. 2014; Radick et al. 2018). On the other hand, the Sun seems to be an unusual example of dwarf, with a low photospheric activity if compared to the other solar analogs. Though, stars similar to the Sun were observed, as noted in Böhm-Vitense (2007). An example is 18 Scorpii which seems to be the closest analog to our star in terms of activity (Petit et al. 2009). More recently, Adibekyan et al. (2018) identified a solar sibling in HD 186302. During the course of this cycle, as the field lines twist,several activity features form on the solar surface, e.g. spots, plages, faculae.

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

When the magnetic lines are twisted, the magnetic flux tubes emerge from the photosphere, increasing the local pressure and inhibiting the convection. In absence of convection, there is no more heat exchange between the surface and the internal layers of the star. Thus, the surface, in correspondence of the emerging flux tubes, becomes colder than the average of the photosphere (Kitiashvili et al. 2013) and a stellar-spot is formed. Since the divergence of the magnetic field is equal to zero:

−→ −→ ∇ · B = 0 (2.1) to each spot of a certain polarity, another spot with the opposite sign is associated. Spots appear darker than the stellar surface due to their lower temperature. In particular, as evidenced in Figure 2.3, each spot has an external ring, called penumbra and an internal area called umbra. The umbra is the coolest and darkest area of the spot, while the penumbra is hotter and its contrast is closer to the stellar surface. While the magnetic field reaches its strongest level at the umbra, understanding whether or not the penumbral magnetic field is always different than zero remains an open question. Borrero et al. Chapter 2. The influence of stellar physics on the planetary detection and characterization 31

(2016) and Spruit et al. (2010) exclude it, even for the deepest layers of penumbra, while Bharti et al. (2012) obtained opposite result. Another consequence of twisting magnetic lines are plages, which can be observed in stellar chromo- sphere. Plages are characterized by a higher temperature than the stellar surface. They are supposed to originate in a similar way to the spots and they are generally associated to a spot. Sometimes plages without spots can be identified. The photospheric counterpart of plages are faculae, bright areas ofthe stellar photosphere, which form due to the concentration of magnetic field lines. A plage and a facula are usually described in similar ways, even though they appear in different observational wavelengths and different atmospheric layers.

2.1.3 The physical properties of activity features

In this section, we explore in detail the discoveries so far reported regarding to the properties of activity features: the latitudinal and longitudinal distributions, the size, the temperature contrast and the lifetime.

Latitudinal distribution

The latitudinal distribution of activity features was analyzed for the Sun over many decades. More recently, different techniques, among them Doppler Imaging and Doppler-Zeeman (Donati & Landstreet 2009; Strassmeier 2009) allowed to perform deeper analyses in this sense. Solar observations showed that the features are generally placed at two different latitudinal bands around the equator. The firstone emerges in the Northern Hemisphere between 7◦ and 30◦ of latitude, and the other one in the Southern Hemisphere between 7◦ and 45◦. At the beginning of the cycle, the feature appears at high latitudes. As the star rotates, it moves towards the equator. This behaviour is known as Spörer’s law of zones (Maunder 1903). For the spot case, the representation of the latitude distribution as a function of time gives the so-called Butterfly Diagram (Maunder 1904). A recent version of such plot is in figure 2.2. Chang (2012) showed that the spot distribution in the northern and southern hemispheres can be reproduced with a double Gaussian, with peaks on the latitudes ∼ 11◦ and ∼ 20◦. Studies on solar-like stars showed that the mean latitude for activity features increases with the rotation rate (Waite et al. 2015; Järvinen et al. 2007; Marsden et al. 2006). Fast-rotating stars are characterized by polar spots (Waite et al. 2015; Carroll et al. 2012, see e.g.). Several attempts to justify polar spots formation are reported in literature (Schuessler et al. 1996; Schrijver & Title 2001; Ișık et al. 2011; Yadav et al. 2015a, e.g.). Ișık et al. (2018) proposed one the most recent models, showing that the Coriolis force acting in the internal stellar layers moves the flux tubes towards the surface. The faster the star rotates, the more the tube is shifted upward, to higher latitudes. For evolved stars, the latitudinal distribution is much larger than the Sun, with spots located at very high and low latitudes (Künstler et al. 2015; Dunford et al. 2012). Furthermore, while in some cases a migration of spots was identified (Berdyugina & Henry 2007), in other cases it was excluded, as in (Yadav et al. 2015b).

Longitudinal Distribution

The longitudinal distribution of sunspots is very sparse and unpredictable (a possible model can come from (Yadav et al. 2015b)). The most evident characteristic is that spots generally appear in groups Chapter 2. The influence of stellar physics on the planetary detection and characterization 32

(Jiang et al. 2011). Furthermore, it has been noticed that spots tend to appear more frequently at some longitudes, with respect to others. Such longitudes are generally called active longitudes and they increase in number passing from zero at the solar minimum to four and more at the solar maximum (de Toma et al. 2000; Malik & Bohm 2009).

Spot size

With the hypothesis of a flat disk, the dimension of a stellar activity feature can be estimated bydefining the so-called filling factor: ( ) 2 Aspot Rspot f = = (2.2) A∗ R∗ where A∗ is the area of the visible stellar disk, Aspot is the area of the feature, and R∗ and Rspot are the radii of the star and feature. Sunspots have diameters from 6000 km to 60000 km (Solanki & Rüedi 2003), with corresponding filling factors between 0.001% and 1%. Other stars show even larger spots, capable of covering a large area of the stellar disk (Tas & Evren 2000). For this reason, in a spot simulation it is important to take into account the probability of detecting very large sunspots. Bogdan et al. (1998) analyzed the size distribution of sunspots and proposed a lognormal function to reproduce it: ( ) ( ) − ⟨ ⟩ dN = dN (lnA ln A ) exp σ (2.3) dA dA m 2ln A ( ) dN −1 6 2 22 2 where, is in units of MSH (millions of solar hemisphere, 10 MSH = 2πR⊙ = 3.1 ∗ 10 cm ) and dA m it represents the minimum of the distribution and is equal to 9.4. The parameter σA = 4 is the width −6 of the distribution, ⟨A⟩ is the mean of the data, in units of 10 A⊙. In general, this suggests that the number of observed spots tends to decrease as the area of the spots increases. The plage and faculae filling factor is still not well known. Though Chatzistergos et al. (2019) showed that for the Sun the filling factor is higher than the spot case, while Radick et al. (2018) pointed out that faculae seem to balance the spot coverage.

Temperature contrast

A sunspot has a varying temperature as we depart from its center. The penumbra has a higher temperature than the umbra, though cooler than the solar photosphere Solanki & Rüedi (2003). Egeland et al. (2017) estimated the sunspots temperature as limited to the interval 3900 K-5500 K. Plages have an average smaller contrast (Meunier et al. 2015) and the difference in temperature, with respect to Te f f , is predicted to be lower than the spot case. While for the Sun it has been possible to distinguish between umbra and penumbra, doing the same for spots on other stars requires a high instrumental precision, currently not at disposal. Large spot areas and spot temperature contrasts recovered on active stars are generally photometrically dominated by the effect of the star-spot umbra (Berdyugina 2005). The spot temperature can be estimated through the flux ratio, defined as the ratio between theflux emitted by the spot, Fspot, and the quiet star flux, Finactive:

Fspot fratio = (2.4) Finactive

The flux is expressed through the black body law, which relates it to the temperature oftheanomaly. Chapter 2. The influence of stellar physics on the planetary detection and characterization 33

Lifetime

The formation of an activity feature is a relatively long process, whose details have been deeply understood in the case of spots, while they are not well known for plages and faculae. At the beginning, the magnetic flux emerges from the solar surface. A pore with the size of a single granule forms andgrows in time, it becomes darker and kit finally forms a spot. In a similar way, the spot disappears (Kitiashvili et al. 2013; Loughhead & Bray 1961; Felipe et al. 2016; Lagg et al. 2014; Robustini et al. 2016; Toriumi et al. 2015). The evolution of a spot can be described taking into account four different times (Kipping 2012): I,

E, L and the time of maximum, tmax. In particular, the lifetime L is related to the spot area by:

A L = (2.5) W

−6 −1 with A the spot area in units of A1/2,⊙ and W = 10.89 0.18 in units of 10 A1/2,⊙ day (Petrovay & van Driel-Gesztelyi 1997). This is known as Gnevyshev-Walldemeier law and it is generally applied to other dwarves. Several studies were performed to analyze the lifetime of starspots. Bradshaw & Hartigan (2014) found that in solar type stars spots can have lifetimes of 10-200 days, if their size is 10,000– 100,000 MSH. Namekata et al. (2019) performed a deep analysis on L, estimating a range between 10 and 350 days for spots covering maximum 2.3% of the stellar heliosphere (though cases of long spot lifetimes remain rare). They measured as well the timescales for the formation and decay of spots, which resulted to be longer than the stable period of a spot.

2.1.4 The effect of activity features on the detection and characterization of exoplanets

The presence of spots, plages and faculae represents an important source of noise when we detect and characterize exoplanets. They can affect the transit, RV and the RM effect signal. In the following,I present the main results so far obtained in literature.

The influence of activity features on the transit lightcurve

As mentioned earlier, when a spot appears on a stellar disk, the total flux decreases because the spot temperature is lower than the rest of the stellar surface. In contrast, when a plage appears, the flux increases and the light curve shows an increment. On top of this, spots and faculae can also appear together with a planetary transits. Figure 2.4, reported in Llama & Shkolnik (2015), shows how activity feature can affect the transit feature. The top panel of Figure 2.4 reports images of the Sun acquired by the SDO spacecraft in 10 different wavelengths and in two different moments, one associated to a lower activity phase and the otherone with higher activity. In this figure, the authors represented as well the simulation of a Jupiter-sized planet transit, with a radius of Rp = 0.1R∗ and an impact parameter of b = −0.3. In this way, in the right side of the Figure, the transits happen in front of the biggest active area on the Sun’s surface. The bottom panel of Figure 2.4 shows simulations of transit light curves. They are the results of summing the planetary transit to the stellar flux integrated from the real frames. The plots show the normalized flux as a function of the orbital phase and they report deformations induced on the stellar emittedflux due to both the stellar activity and the planetary transit. Since both the photosphere and the activity contrasts are wavelength dependent, the photometrically Chapter 2. The influence of stellar physics on the planetary detection and characterization 34

Figure 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). Chapter 2. The influence of stellar physics on the planetary detection and characterization 35 observed activity features vary according to the pass-band. A transiting planet, which crosses a spot as well, shadows a lower fraction of the stellar flux. As a consequence, the transit signal shows a bumpin correspondence of the covered spot. On the contrary, when the planet occults a plage, it blocks a higher fraction of the stellar flux. Thus, the transit feature becomes deeper (see for instance Llama & Shkolnik 2015). Furthermore, as we vary the passband of observation, the intensity of the spots and plages effect changes. As an example, in Figure 2.4, the deepest plage effect is in the X-rays, while in the infrared the photospheric emission is dominated by spots. The presence of a non-occulted spot or plage on the stellar disk affects the transit depth. In detail a spot increases the transit depth, while a plage decreases it (Czesla et al. 2009). Such activity noise has a strong influence on the planetary radius measurements (Czesla et al. 2009). As showed in Barros et al. (2013), the presence of an occulted spot can lead to an overall underestimation of the planet radius and Oshagh et al. (2013b) estimated this underestimation to be of the 4% for a Jupiter-sized planet. Such variation changes according to the wavelength of observation, as described in Llama & Shkolnik (2015), Llama & Shkolnik (2016) and Oshagh et al. (2014) and becomes stronger if the size of the planet is smaller (especially Earth-sized planets). Similar results are reported in Bruno et al. (2016) and Désert et al. (2011), who show as well an overestimation of the stellar radius in presence of occulted faculae. Spot occultation was identified in several planets; some examples are WASP-10b (Barros et al. 2013), CoRoT-7b (Barros et al. 2014), HAT-P-11b (Southworth et al. 2011b), GJ 436 (Ballard et al. 2010), Kepler-17 (Bonomo & Lanza 2012), Kepler-30 (Fabrycky et al. 2012). Following several transits and finding consecutive signatures of the same spot allows an estimate of the stellar rotational period,asit was done by Béky et al. (2014); Csizmadia et al. (2015). Several works managed to identify spot signatures in multiband transit analysis. For instance, by analyzing WASP-52 transits Mancini et al. (2017) identified several spot occultation events, which are most likely associated to the same spot. A similar result is reported for infrared observations of WASP-52 in Bruno et al. (2018) and for multiband observations of other systems by Mancini et al. (2015, 2014); Sanchis-Ojeda et al. (2013); Huitson et al. (2013). Additionally, the spot features are usually stronger than eclipses and the planetary phase curves. As a consequence identifying the secondary eclipses is very hard (Lillo-Box et al. 2014) and quiet stars were usually selected to identify planetary phase curves (Angerhausen et al. 2015). Alternatively, the phase curve is filtered out to remove the major effect of spots and later on it is phase-folded toaverageout the remaining stellar activity and the granulation (as it was done by Esteves et al. 2013, and others). An alternative way to separate the stellar activity from the planetary signal is to adopt a Markov Chain MonteCarlo method with Gaussian Process (MCMC with GP) (Barclay et al. 2012), which allows to separate the periodicities of the planet and of the stellar rotation, treating the stellar activity as an additional source of noise. For more details on this methodology see Chapter 3.

The effect of activity features inRV

In RV time series, a spot can generate two effects. The first one is known as flux effect(Dumusque et al. 2014). The stellar rotation moves spots from their initial configuration. Due to the temperature contrast, the movement of the spots on the stellar surface injects a velocity shift, and changes the line shape (Lagrange et al. 2010; Meunier & Lagrange 2013). The second effect is the inhibition of the convective blue-shift. Since spots are magnetically active, the convection is inhibited inside them Chapter 2. The influence of stellar physics on the planetary detection and characterization 36

Figure 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) Chapter 2. The influence of stellar physics on the planetary detection and characterization 37

(Dumusque et al. 2014; Aigrain et al. 2012; Lanza et al. 2011; Meunier & Lagrange 2013). As the CB is locally blocked, it determines a change in the total RV of the star and it no longer affects the flux as in the rest of the stellar surface. This can contaminate the RV. For instance, not accounting for a spot in RV analysis might affect the planetary RV, and modify the shape and amplitude of the RMsignal (Dumusque et al. 2017; Feng et al. 2017; Robertson et al. 2016). Plages determine the same effects as spots. A comparison between them shows that for plages the flux effect is inverted and less intense (Barros et al. 2014). Additionally, the area of plages is averagely larger than spots. As a consequence, overlapping the flux effect signals of spots and plages showsa dominance of spots over plages (Meunier & Lagrange 2013). The presence of a spot may cause a false positive detection in RV analysis. As described in Sec- tion 1.1.1, the presence of planets can also be confirmed applying the Lomb-Scargle periodogram to the RV signal (Baluev 2015) and looking for peaks in the resulting diagram. Unluckily, the signal caused by a dark spot is identical to the one generated by a planet, if the stellar rotation is equal to the orbital period. This has caused several false positives in the planetary search (Figueira et al. 2010; Santos et al. 2014). The only way to distinguish planets from spots requires to adopt activity indicators and perform several RV observations. If the intensity of the signal varies in time and it finally disappears, it is generated by a spot. In RM analysis, a spot can be crossed during the transit, generating an anomaly in the RM signal (for plages the effect would be inverted). How an occulted spot can influence the RM analysis was explored by Oshagh et al. (2016), who found that not accounting for it in the analysis can change the spin-orbit angle estimate by at least 30◦. Figure 2.5 shows an example of how an occulted spot and plage would look like both in a transit feature and in the RM signal (Oshagh et al. 2016). Moreover, Oshagh et al. (2018) demonstrated that a fit performed on several RMs deformed by stellar activity can result in spin-orbit angle variations of at least 40◦ for the same systems. They suggest two solutions to overcome this issue. The first one consists of folding the RMs one on top of the other, so to reducethe activity noise. The second one requires simultaneous transit observations, to isolate the properties of the occulted spot. On top of this, unocculted spots can mimic several planetary atmospheric phenomena. As an example, it can mimic the rayleigh scattering, causing different best fits on consecutive transits (Rackham et al. 2017; Oshagh et al. 2014; Mackebrandt et al. 2017). To overcome this problem Louden et al. (2017) and Sedaghati et al. (2017) correct for the brightness with specific stellar models.

2.2 The convective motions and granulation

F and G stars are characterized by internal convective motions, which may rise to the stellar surface, generating a surface pattern formed by granules. This phenomenon is named granulation (see Figure 2.6). Each granules forms in correspondence of a stellar convection cell approximating the stellar surface. The granulation pattern has been observed on the Sun with two different scales and sizes: granulation and supergranulation. The granulation has typical horizontal length scales around 1500 km (Rieutord & Rincon 2010), velocities ranging from 0.5 to 1.5 km s−1 (Title et al. 1989), and a lifetime/renovation time of 5-10 min. The supergranulation, situated below the photosphere, has a typical horizontal scale of up to 30000 km, a dynamical evolution time of 24-48 h, a strong 300-400 m s−1 (rms) horizontal flow component and a significantly weaker 20-30 ms−1 vertical component (Rincon et al. 2017; Del Moro Chapter 2. The influence of stellar physics on the planetary detection and characterization 38

Figure 2.6: Image of an area of the solar surface by the SDO. In evidence, the photospheric granules

2004; Christensen-Dalsgaard 2004). For a long time, several works debated about the existence of an additional time scale pattern between granulation and supergranulation. Known as mesogranulation, this additional feature appeared to have a lifetime of several hours and a horizontal scale of 5000- 10000 km (November et al. 1981; Title et al. 1989). Though, subsequent Doppler observations excluded mesogranulation could actually exist (Yelles Chaouche et al. 2011; Berrilli et al. 2013), showing it was a ghost feature generated by averaging procedures. Granulation and supergranulation are known to exist on stars other than the Sun, with time scales varying according to the stellar type. We know, for instance, that G-stars behave in average similarly to the Sun, while the granules of F-type stars are larger and longer-lasting than those of G-stars (Dravins 1982). Though, the precise time-scales and amplitudes are specific for each stars. Dumusque et al. (2011) performed such analysis for a sample of 4 G-stars and one K-star, a result which is far from being generalized to all stars and to all stellar types. Meunier et al. (2015) showed that the granulation and super-granulation generate a stochastic noise on the stellar photometry and on the RV time series measurements, which hardens the identification of Earth-sized planets in the habitable zone. Furthermore, analyzing the Lomb-Scargle periodogram of the RV signal, Meunier et al. (2015) found that the granulation and super-granulation generate a forest of short peaks. The periodicity of an Earth-sized planet generates a peak in the periodogram stronger than those associated to granulation, though, Meunier & Lagrange (2019) shows that the super-granulation peaks are way higher than the false positive limit, especially for Earth-sized planets. Moreover, averaging out the granulation and supergranulation in RV analysis may not be straightforward especially when we Chapter 2. The influence of stellar physics on the planetary detection and characterization 39 need to reach a precision of 0.28 m s−1, required to observe an Earth-sized planet. Granulation can be mitigated with a large number of observations, covering 500−1000 hours. In the case of super-granulation the noise injected is much larger. Meunier & Lagrange (2019) calculate a 40-60% detection rate for an Earth mass planet with a period of a 300-days, when observations cover 3600 hours spread over ten years. Granulation represents as well an extra correlation noise in transit analysis. Chiavassa et al. (2017) showed that for G and K stars the granulation pattern varies within a timescale which is lower than the usual planet transit duration. They also showed that, during a planetary transit, the occulted regions of the stellar surface differ in local surface brightness as a result of convective-related surface structures. Consequently, the transit shape is related to the specific behaviour of the granulation in the occulted area, not always precisely deduced based on the granulation behaviour in the rest of the stellar surface. This effect may hurdle the precise characterization of an exoplanet during transit, injecting, inthecase of a G-star, an uncertainty of 0.90% on the estimation of the radius of Earth-like planets and of 0.40% for Neptune-sized planets. More recently, Sarkar et al. (2018) show how granulation affects a transit on different wavelengths of observation and this can affect the precision with which each transit parameter is estimated. Convective motion causes as well two additional effects, the macro-turbulence and the center-to-limb convective blue-shift (CB).

2.2.1 The macro-turbulence

Spectroscopically observing a single convective cell, one can detect a velocity shift. The super-position of all the convective cells affects the spectral lines broadening them. As a consequence also the spectral line is broadened (Mucciarelli 2011; Gray 1984). The macro-turbulence is defined as the line broadening induced by convective cells larger than the photons mean free path. For G stars they affect on a level −1 −1 of vmac = 4 km s and they increase to more than 6 km s for F type stars (Doyle et al. 2014). A law to estimate the macro-turbulence intensity is reported in Doyle et al. (2014) and depends on the stellar surface gravity and the stellar effective temperature. The macro-turbulence can affect RV analysis (and the RM effect) and they must be takeninto account when modelling the spectral line, thus the cross-correlation function (CCF) of a star (as it was done in Hirano et al. 2011; Cegla et al. 2016a).

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

The convective blue-shift (CB) is an additional difference between the measured stellar line positions and their laboratory counterparts (Adam et al. 1976). It is a consequence of granulation. The emerging granules are brighter, they move towards the observer (they are blue-shifted) and they cover a greater fraction of the stellar surface with respect to the inter-granular lanes. This generates an almost constant spectral blue-shift, which affects the RV by 300 m s−1 for Sun-like stars. As the precision increases, an additional variation appears, the CB is not any more a constant effect, because it varies with the limb angle. This phenomenon is called center-to-limb CB (Shporer & Brown 2011) and it is specific of each star. Shporer & Brown (2011) modelled the center-to-limb CB, varying it with the limb darkening and the projected area. They argued that ignoring such effect should influence the estimation of the spin-orbit Chapter 2. The influence of stellar physics on the planetary detection and characterization 40

Figure 2.7: The optical depth according to the stellar surface area we are looking at (from http://spiff.rit.edu/classes/phys440/lectures/limb/limb.html) angle. Cegla et al. (2016b) improved on this by adopting a 3D magneto-hydrodynamic (MHD) solar simulation to determine the center-to-limb variation of the CB effect. They also included the impact of an asymmetric line profile on the stellar disk. In particular, Cegla et al. (2016b) report that ignoring −1 the center-to-limb CB on moderately rotating stars (e.g. v∗ sin i∗ = 6 km s ) could potentially inject systematic biases of ∼ 20◦ or more in the projected obliquities. The effect of CB (several ms−1) is stronger on slow rotating stars, while for fast-rotating star Cegla et al. (2016b) identified a significant influence due to the asymmetric line profile (non Gaussian). Though, there is no clear evidence forthe spectral lines not to be Gaussians, which is why in real observations it is still a good approximation using the Gaussian profile for modeling the CCF. An updated version of their MHD simulation, and corresponding CB predictions, was presented in Cegla et al. (2018). The center-to-limb CB affects RM analysis, injecting variations along the transit modulation. An attempt of accounting for it was performed by Cegla et al. (2016b), who showed that for a precision of 0.1 m s−1, the center-to-limb CB impacts the estimation of the spin-orbit angle by 10-20◦ when the impact parameter is null and the spin-orbit angle is 0◦. They suggest this is a possible consequence of a degeneracy between λ and the convective blue-shift. For other impact parameters, the uncertainties decreased to λ = 1 − 3◦.

2.3 The limb-darkening

The flux emitted by a star can be simplified as the stellar surface was a flat disk. Although,dueto the actual spherical shape, the surface luminosity is not uniform over the whole disk and it varies from the center to the limbs. This phenomenon is known as limb darkening and it depends on the optical depth and on the increment of the stellar temperature with the stellar depth from the photosphere. The optical depth is the natural logarithm of the ratio between the incident and transmitted radiant power through the stellar body: ( ) Φ τ = ln i (2.6) Φe Chapter 2. The influence of stellar physics on the planetary detection and characterization 41

where Φi is the incident radiant flux and Φt is the transmitted one. Figure 2.7 shows the behaviour of the optical depth according to the area of the stellar surface the flux comes from. The maximum stellar depth visible to the observer corresponds to the layer at which τ = 1 and a fraction of 1/e photons escapes. We can suppose that the intensity of radiation varies linearly with τ. If the line of sight is directed towards the center of the stellar disk, it will cross a deeper layer of the stellar surface before τ = 1, when compared to the stellar limb. Moreover, since the temperature of the star decreases with the distance from the center, on the limbs the line of sight crosses stellar layers colder than at the disk center. As a consequence the limbs appear darker than the rest of the stellar surface. Claret (2000) performed an in depth study on the limb darkening and proposed the following law to describe how the flux varies throughout the stellar disk:

∑4 n/2 I(x, y) = 1 − un(1 − µ ) (2.7) n=1 √ where x, y are the coordinates of the single point on the stellar disk and µ = cos θ = 1 − x2 − y2 is the angle between the normal to the stellar surface and the observer line of sight. It is also called center- to-limb angle, because it changes with the distance from the center of the selected point on the stellar surface. Its values range between 0 and 1. Nevertheless, several expressions can still be used to express the limb darkening. Claret & Bloemen (2011) lists the linear law:

I(x, y) = 1 − u(1 − µ) (2.8) the square root law: √ I(x, y) = 1 − c(1 − µ) − d(1 − (µ)) (2.9) and the quadratic law: 2 I(x, y) = 1 − u1(1 − µ) − u2(1 − µ) (2.10)

While equation 2.7 is the most complete one and the closest to the solar case (Claret 2000), it needs 4 coefficients, which are hard to estimate for most of the stars. As a consequence, the most usedexpression is the quadratic law, for which the two coefficients, u1 and u2, are constrained by two conditions u1 +u2 < 1 and u1 + 2u2 > 0 (Kipping 2013; Mandel & Agol 2002). These boundaries depend on two properties of limb darkening, as reported in Kipping (2013):

• the intensity profile I(x, y), needs to be everywhere positive

• I(x, y) decreases from the stellar center to the limb

Additionally, the limb darkening coefficients depend on the photometric passband. Claret & Bloemen (2011) proposed the limb darkening coefficients applicable for observations with specific filters andfor the Kepler bandpass. More recently the TESS un values were published (Claret 2017). The limb darkening affects significantly the transit shape both in photometry and in spectroscopy. Figure 2.8 shows a transiting planet on a limb darkened star. In detail, the black transit represents the behaviour of the light curve in the absence of limb darkening. In this case, the borders of the transit are straight lines. In contrast, the red line accounts for limb darkening. The stellar surface luminosity varies as we approach the limbs of the star and, correspondingly, the transit ingress and egress are rounded. Moreover the limb darkening affects the bottom of the transit rounding it as well. Adopting theentire Chapter 2. The influence of stellar physics on the planetary detection and characterization 42

Figure 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. Chapter 2. The influence of stellar physics on the planetary detection and characterization 43

expression 2.7 is hard to implement, without a knowledge of the four coefficients un. For this reason, in transit analysis the law 2.7 is simplified to a quadratic limb darkening law. Soderhjelm (1999) were the first to include the limb darkening in the analysis of the transitof HD 209458 b, following the suggestions in Deeg et al. (1998). Since then, it was never neglected in transit analyses. In some cases u1 and u2 were free parameter, in other cases they were fixed to the values proposed in Claret & Bloemen (2011). Though, Barros et al. (2012) and Neilson et al. (2017) showed that fitting the limb darkening coefficients in transit analysis may lead tovaluesfor u1 and u2 different from the results obtained through stellar evolution models. Especially for high-precision transits, not accounting for the correct values of u1 and u2 can strongly bias the result, therefore precise modelling of the limb darkening law is necessary to perform for the planetary characterization to be precise (Csizmadia et al. 2013). Parviainen & Aigrain (2015) propose to constrain the limb darkening coefficients with informative priors based on modern tabulated values (as estimated through spherical stellar atmosphere models Sing 2010; Claret et al. 2014; Husser et al. 2013). On top of this, since the limb darkening influences the RM signal as well, it is always accounted for in the RM model, again fixing or varying the coefficients (see e.g. Albrecht et al. 2012a; Hirano et al. 2011). Nonetheless, to include it, it is important to have a first precise estimation through the transit analysis.

2.4 The stellar differential rotation

Stars form after the collapse of a gas cloud. During their formation, the clouds lose part of the angular momentum. As a consequence, when a star is born, it inherits part of the original gas cloud rotational velocity. The rotational velocity decreases as the star becomes older. This rotation is non-rigid, because the star is made of high-temperature plasma which is subjected to convection. Due to the interaction between the stellar rotation and the convective motions, stars differentially rotate, which means that their rotational velocity varies with latitude (Kichatinov & Rudiger 1995; Kitchatinov & Rüdiger 1999; Küker & Stix 2001; Collier Cameron 2007). In detail, the convective motions bring hotter stellar fluid closer to the cooler surface, carrying a part of the stellar angular momentum. The interaction between the radiative and the convection zones generates a redistribution of the angular momentum among the different latitudes through the so-called meridional flow. Kitchatinov & Rüdiger (1999) suggested the first theoretical model to describe the stellar differential rotation in the case of late type dwarfs and giants. They showed that the amplitude of the differential rotation changes with the stellar type. An improvement of such modelby Küker & Stix (2001) showed that, if compared to late-type stars (Collier Cameron 2007), early type stars have shallower convective zones which leads to stronger differential rotation variations at a given equatorial velocity. For the Sun, the rotational velocity decreases from the equator to the poles and many stars, mostly main sequence, have shown a solar-like differential rotation pattern (Karoff et al. 2018b; Reiners 2012, and references therein). In some cases, even young pre-main sequence stars exhibit this behavior (Donati et al. 2000). For some stars, an anti-solar differential rotation (i.e. faster rotation at the poles than the equator) has been measured. These are usually evolved post-main sequence stars, e.g. K giants (Kóvári et al. 2017; Kővári et al. 2015; Weber et al. 2005; Strassmeier et al. 2003) and subgiants (Harutyunyan et al. 2016). Anti-solar differential rotation arises when the radial motions, including the laminar ones, prevail Chapter 2. The influence of stellar physics on the planetary detection and characterization 44 over the horizontal turbulent velocities (Kitchatinov & Rüdiger 2005; Karak et al. 2015). The stellar differential rotation is a key ingredient in stellar dynamo models, because it contributes to the generation and maintenance of the stellar magnetic fields. Underlying mechanisms which generate and sustain differential rotation are poorly understood, and measuring it for a large sample of stars (different spectral types and ages) can provide a more comprehensive view of the stellar magnetic activity and activity cycles. The stellar differential rotation can be modelled using the following law, derived from solar observations: 2 4 Ω(θ) = Ωeq(1 − α sin θ − β sin θ) (2.11) where Ωeq is the equatorial angular velocity, Ω(θ) is the surface shear, defined as the angular velocity of rotation as a function of the stellar latitude θ. α and β are the quadratic and quartic relative differential rotation. With the hypothesis of observing the star as a flat disk, with the z axis along the line of sight, the latitude θ can be expressed as:

θ = arcsin (y sin i∗ + µ cos i∗) (2.12) with x the horizontal axis on the stellar disk, varying between -1 and +1, y the vertical axis on the stellar disc (also between -1 and +1) ad i∗ the stellar inclination with respect to the line of sight (Hirano et al. 2011; Cegla et al. 2016a). Nonetheless, the law 2.13 requires to calculate both of the parameters α and β. While for the Sun applying this law is possible and α and β have been calculated, for the other stars reaching a precision high enough to estimate both of the parameters is hard. As a consequence the generally applied law is: 2 Ω(θ) = Ωeq(1 − α sin θ) (2.13) where α is called relative differential rotation and it is defined as: ∆Ω α = , (2.14) Ωeq where ∆Ω is the rotational frequency difference between the poles and the equator. The relative differential rotation α has been measured for numerous stars. For main sequence A-F types stars, no value above 0.45 were measured and the average results are between 0.1 and 0.2 (Reiners 2012). For G-stars, higher values of α were obtained; for instance, Karoff et al. (2018b) estimated α ∼ 0.53 for the sun-like star HD173701. For the Sun, Snodgrass (1983) estimated α ∼ 0.19, though the value universally accepted is 0.20. For K dwarf stars, Balona & Abedigamba (2016) reported an average value of α = 0.2 and an upper limit of 0.4; they also suggested that α has a higher average value for G-stars and then decreases for earlier and later stellar types.

The effect of stellar differential rotation onRM

The stellar differential rotation can affect the RM analysis of an exoplanet. In the case inwhichthe star is characterized by a solar-like differential rotation, since the rotational velocity decreases aswe approach the poles, the amplitude of the RM signal decreases for any spin-orbit angle (Hirano et al. 2011; Gaudi & Winn 2007), though the effect is stronger when the planet is inclined with respect to the stellar axis. Gaudi & Winn (2007) adopted an analytic model of the RM signal which took into Chapter 2. The influence of stellar physics on the planetary detection and characterization 45 account the stellar differential rotation and they concluded that its effect was negligible in comparison to the RV precision of spectrographs at that time. Later, Hirano et al. (2011) showed that, for stars −1 rotating faster than v∗ sin i∗ =10 km s , the contribution of differential rotation in the RM signal could be crucial with the upcoming instruments. More recently, there have been some attempts of measuring the stellar differential rotation through the RM signal. Cegla et al. (2016a) ruled out the possibility of rigid body stellar rotation for HD 189733, although they could not closely constrain the level of α (which was found to be between 0.28 and 0.86 Cegla et al. 2016a). Albrecht et al. (2012a) attempted to measure the stellar differential rotation of WASP-7, which is an F star. Though, they estimated a high value (α, 0.45), if compared to the average of the stellar type accounted for. However, these authors explained that their results should be taken with caution. Some systematic biases might affect their model fits resulting in unphysical solutions. Nonetheless, a stable and more precise spectrograph, working on a larger aperture telescope, (e.g. ESPRESSO), offers the opportunity of a clear detection of stellar differential rotation through RM. The stellar differential rotation modifies the shape oftheRMas the stellar inclination changes, as shown in Hirano et al. (2011). Thus, a well-constrained measurement of the relative differential rotation α will allows to break the degeneracy between v∗ and sin i∗ and to measure the stellar inclination through the RM effect analysis (Hirano et al. 2011). This means that while analyzing the RM it will be possible to use as free parameters the stellar inclination and the equatorial stellar rotational velocity, together with α. On top of this, the stellar differential rotation affects as well the position of a spot in time,sincea spot rotates synchronously with the stellar surface. Models accounting for the stellar differential rotation effect on spots have been produced (Kipping 2012; Herrero et al. 2014). Kipping (2012) also explored the effect in photometry of spots differentially rotating. Though, a deep analysis of the effect of differential rotation on the spot signature in radial velocity has not been performed so far. Chapter 3

SOAP3.0: a tool for simulating a spectroscopic and planetary transit in presence of stellar activity

In an epoch in which the instrumental precision is improving extremely fast, it becomes important to account for stellar noise sources in transit simulation for both photometry and spectroscopy. In this chapter, we present an updated version of the tool SOAP3.0, which was presented to the public as SOAP in 2012 (Boisse et al. 2012). The original version of this code produced the effect of spots and plages on both the photometric and spectroscopic observations of a rotating star. It could also compute the bisector span (BIS), a spectroscopic diagnostic modulation which allows to monitor the stellar activity. Finally, it could perform an estimate of the Cross-Correlation Function (CCF), which is equivalent to the mean line of the spectrum. In 2013, this code was updated into SOAP-T, so that it could produce the spectroscopic and photometric transit of a planet in presence of spots and plages (Oshagh et al. 2013a). On top of this, it could as well account for up to 10 activity features, while before just 4 of them were included. After that, Dumusque et al. (2014) updated the sofware by Boisse et al. (2012) into the new version SOAP2.0, which used the solar CCF to produce the photometric and spectroscopic modulation of the star in presence of stellar activity. In this way, the tool could account for the inhibition of the convective blue-shift. Akinsanmi et al. (2018) released the final version of the code, SOAP3.0, which combines both SOAP2.0 and SOAP-T and introduces the ring features around exoplanets. In this chapter, we roughly present SOAP3.0 and the updates we performed during the Ph.D. period to account for additional stellar features: the stellar differential rotation, the center-to-limb variation of the convective blue-shift and the macro-turbulence.

3.1 The initial version of SOAP3.0

SOAP3.0 is a numerical code which organizes the stellar surface into a grid of small squared cells and estimates the spectroscopic and photometric signal over the entire disk by summing the contributions of each cell.

46 Chapter 3. Updated SOAP3.0 47

As a first step, it requires to initialize a CCF, which can be input in two different ways:asa Gaussian or as an observed CCF. The choice of which CCF to use is left to the user. The Gaussian CCF in SOAP3.0 uses the instrumental widening as Full-Width-Half-Maximum FWHM, which correlates to the width σ of the CCF by: FWHM σ = √ (3.1) 2 2ln2 As a first estimate, the FWHM is calculated as:

c FWHM = (3.2) R with c the light speed and R the instrumental resolution. On the contrary, when the user chooses to account for the observed CCF, the code calls a file containing the CCF data. As the stellar CCF isesti- mated, it is convoluted, to account for the instrumental widening induced by the selected spectrograph. The injected CCF is shifted cell by cell on the stellar surface, to account for the local stellar projected rotational velocity v∗ sin i∗. An explanation of the CCF treatment is in Figure 3.1. According to the position of each cell, the code estimates the local observed velocity and uses it to Doppler shift the CCF. Then, all the CCFs are weighted by a quadratic limb darkening law and summed one on top of the others to estimate the CCF of the quiet star. The flux is then fixed to 1 and weighted by the limb darkening law. To account for spots and plages, the code initializes them as circles placed at the disk center. The circumferences of the inhomogeneities are then divided into a finer grid and each grid point is rotated to estimate their correct location on the stellar disk at the starting time. A second rotation, based on the stellar rotational phase, is applied, thus moving the inhomogeneity on the photosphere. During this movement, the feature will not appear any more as a disc with respect to the observer. It will appear as an ellipsoidal feature. Phase after phase, the code estimates the inhomogeneity visibility by checking if it is located inside the stellar disk. If it is visible, an inverse rotation is performed to configure the feature again as equatorial. For each grid point located inside the spot/plage, the code Doppler shifts the input CCF according to the local velocity and weights it for a limb darkening law and for the intensity associated to the inhomogeneity (accounting for the flux effect). If the inhomogeneity is a spot, thefinal CCF contribution (determined as sum of the local CCF) is subtracted to the one of a quite star. If it is a plage, the estimated CCF results being added to the one of a quite star. In a similar way, the flux contribution is estimated weighting the flux of a quiet star for the intensity of the active region. To account for the planet, SOAP3.0 calculates the planetary trajectory using the equations of motion as described in Oshagh et al. (2013a). Subsequently, it determines if the planet is in the foreground or in the background with respect to the star. To do so, it checks whether or not the projected distance between the planet center and the stellar center is smaller than the stellar radius. If the planet is inside the stellar disk, the code identifies the area of the grid where the planet is located and scans it to determine whether each grid cell is within the stellar disk. If so, the local CCF is modelled Doppler-shifting the Gaussian according to the projected stellar rotation velocity and weighting it by the quadratic limb-darkening law. The code estimates as well the flux covered by the planet, applying the limb darkening law totheflux of the quiet star in the areas covered by the planet. The planet contribution is finally subtracted to the flux and CCF of the star. Additionally, SOAP3.0 can model the case in which spots/plages and the planet appear at the same time in the system. There are two possibilities. In the first one, the spot is not covered by the planet. Chapter 3. Updated SOAP3.0 48

−2.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0 rotation RV Figure 3.1: An example of how the CCF is Doppler shifted on the different regions of the stellar surface. Chapter 3. Updated SOAP3.0 49

In this situation, the code simply combines the procedures applied to account for the two components, inhomogeneities and planet. If the spot/plage is covered by the planet, modeling the overall contribution is more tricky. SOAP3.0 excludes from the calculation the parts of the spot whose distances to the center of the planet are smaller than the planet’s radius. Such points will not be scanned during the spot scanning process and they will not contribute to the CCF and flux of the star. In this way, SOAP3.0 can produce the “bump” anomalies inside the transit light curve generated by transited spots. Finally, for the details about the ring features, we redirect to Akinsanmi et al. (2018), as rings are out of the scope of the thesis.

3.1.1 Input and output parameters before the updates

To summarize, the code requires a series of parameters to implement the photometric and spectroscopic transit of a planet in front of a spotted star:

• to model the gaussian CCF: σ, the width of the CCF for a non rotating star (in km s−1); the window of velocities in which the CCF has to be estimated (typically 20 km s−1 for slow rotators, to be increased for fast rotators); the depth of the Gaussian and the step in which the CCF has to be sampled.

• to model the star: u1 and u2, the linear and quadratic limb darkening coefficients; the stellar radius R∗ expressed in units of solar radii; i∗, the stellar inclination with respect to the line of sight (in degrees); P∗, the stellar rotational period (in days); T∗, the stellar temperature; ψ, the initial phase for the simulations.

• to model the planet: PP, the orbital period (in days); RP, the planet radius in units of stellar radii;

a, the semi-major axis in units of stellar radii; iP, the orbital inclination with respect to the line sight (in degrees); e, the eccentricity; ω, the argument of periastron (in degrees); the spin-orbit

angle (in degrees); T0, the time of passage at the periastron (in days); t, the difference between the phase of the star and the phase of the planet.

• to model each spot/plage: latitude and longitude of the spot; Rspot, the size of the active region in

units of stellar radii; Tdi f f , the temperature contrast with respect to the effective temperature of the star; a flag to switch on the active feature and another flag which is set to 0 if the inhomogeneity is a spot, to 1 if it is a plage.

• additionally, the code calls a parameter grid, which represents the linear resolution of the stellar disk. The star is thus divided into grid2 cells. It also calls nrho, the resolution of the circumference of each spot.

As output, the code gives:

• FLUXstar_quiet, the flux of the quiet star;

• CCFstar_quiet, the CCF of the quiet star;

• FLUXstar, the overall flux of the system;

• CCFstar_ﬂux, the overall CCF of the system; Chapter 3. Updated SOAP3.0 50

Figure 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).

• CCFstar_bconv, the CCF of the star just with convective blue-shift;

• CCFstar_tot, the overall CCF of the star

• rvﬂux, the radial velocities only due to the flux effect;

• rvbconv, the RVs only due to the convective blueshift;

• rvtot, the RVs of the transit spectroscopy, including possible spots

3.1.2 SOAP3.0 performance before the updates

Spots/plages and their effects on the stellar signal

SOAP3.0 can reproduce the effect of inhomogeneities on stars, perfectly in line with the observations. Figure 3.2 shows the effect of a dark spot on the photometry and spectroscopy of thestar,as reported in Boisse et al. (2012). The last row shows the BIS. On the left, the Figure reports the case of an equatorial spot: photometrically, the spot causes a decrement in the light curve. Spectroscopically, as the star rotates, when the spot appears on the red-shifted side of the star, it subtracts its contribution to the stellar CCF. As a consequence, the overall RV is blue-shifted. If the spot is on the blue-shifted side of the star, the overall RV is red-shifted. The right panel of figure 3.2 reports the effect of varying the latitude of the spo. Since the star is initialized as edged on, increasing the latitude causes the spot to slowly disappear from the observer’s view. As a consequence, the effect of the spots slowly decreases as the latitude increases. Dumusque et al. (2014) explored the effect of limb-darkening, convective blue-shift, flux effect and spectroscopic resolution on spots. In Figure 3.3, Dumusque et al. (2014) shows the effect of varying the limb darkening law. In the top panel, we see the changes induced on the flux, for both a spot (top frame) and a plage (bottom frame). In the bottom panel, we have the effect on RV. The effect ofthe limb darkening law is stronger for photometry than for spectroscopy. In photometry, the linear limb Chapter 3. Updated SOAP3.0 51

Spot 0

−2000

−4000

−6000

−8000 Flux [ppm] −10000

−12000 Plage 800 700 600 500 400 300

Flux [ppm] 200 100 0 -¼/2 -3¼/8 -¼/4 -¼/8 0 ¼/8 ¼/4 3¼/8 ¼/2 µ Spot 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10 Plage 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10

-¼/2 -3¼/8 -¼/4 -¼/8 0 ¼/8 ¼/4 3¼/8 ¼/2 µ

Figure 3.3: In the top panel, the flux effect of the limb darkening on a spot (top frame) and onaplage (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). Chapter 3. Updated SOAP3.0 52

Spot 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10 Plage 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10

-¼/2 -3¼/8 -¼/4 -¼/8 0 ¼/8 ¼/4 3¼/8 ¼/2 µ

Figure 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).

Spot 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10 Plage 15

10 ] 1

¡ 5 s . m

[ 0

R −5

−10

-¼/2 -3¼/8 -¼/4 -¼/8 0 ¼/8 ¼/4 3¼/8 ¼/2 µ

Figure 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−1 in the active region. The red line adopts the observed solar CCF. The Figure is in Dumusque et al. (2014). Chapter 3. Updated SOAP3.0 53

Convective blueshift effect Flux effect 1.0 0.0 ¼/2 ¼/2 −0.2 0.5 3¼/8 −0.4 3¼/8 0.0 ¼/4 −0.6 ¼/4

Delta CCF −0.5 Delta CCF ¼/8 −0.8 ¼/8

−1.0 0 −1.0 0 1.0 µ 1.0 µ -¼/8 -¼/8 0.8 0.5 -¼/4 0.6 -¼/4 0.0 -3¼/8 0.4 -3¼/8

Delta CCF −0.5 Delta CCF -¼/2 0.2 -¼/2

−1.0 0.0 −10 −5 0 5 10 −10 −5 0 5 10 1 1 RV [km.s¡ ] RV [km.s¡ ]

Figure 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). darkening law increases the effect of spots/plages by hundreds of ppm. In RV the effect is smaller, though on the limit with the precision of ESPRESSO (< 1 m s−1). In Figure 3.4, we can analyze the effect of the resolution on RV(Dumusque et al. (2014) demonstrates that no bug in the code exists for the resolution to affect the flux). In general, a decrement in resolution decreases the amplitude of the effect for both spots and plages. The differences are on the levelofthe m s−1, though for plages, whose contrast is smaller, the effect is less relevant in terms of intensity. Since the plage intensity depends on the center-to-limb angle, we note a generally different behaviour overall the curve. In Figure 3.5, Dumusque et al. (2014) explores the effect of the inhibition of the convective blue-shift on RV, by varying the CCF called by the code to model the spots and plages. The red line uses the solar CCF for a spot, the blue dashed line adopts a Gaussian CCF (the same for the quiet photosphere and for the inhomogeneity), while the blue line corresponds to the case in which the Gaussian CCF is shifted by 350 m s−1 as a possible approximations of the convective blue-shift effect. They demonstrate that, in general, a shifted Gaussian CCF is not sufficient to account for the inhibition of the convective blue-shift. The result obtained with the observed CCF is different than the one with the shifted Gaussian. The effect is more relevant for plages than for spots. To complete, in the left side offigure 3.6 Dumusque et al. (2014) shows that the CCF correction varies with the limb angle, with a stronger effect when assuming the observed CCF (lower panel) than when using the shifted CCF for the spot. Finally, the right side of Figure 3.5, shows the flux effect on the CCF, for both spots (top panel)and plages (bottom panel). Again the CCF correction varies with the center-to-limb angle, with a stronger effect for spots than for plages. For spots, the differences vary on a higher extent with the center-to-limb angle (on a level of 0.8 between the center and the edge of the star). For plages, the differences are maximum 0.25.

Modeling a planetary transit with SOAP3.0

SOAP3.0 can model the spectroscopic and photometric transit of a planet in front of a rotating star. As the code uses still the implementation by SOAP-T, we comment on its performance with the results reported in Oshagh et al. (2013a). Chapter 3. Updated SOAP3.0 54

Figure 3.7: Top: a comparison between the transit feature as modelled by SOAP-T code (same results as SOAP3.0) and the theoretical model of a transiting planet over a non-spotted star (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). Chapter 3. Updated SOAP3.0 55

000

00 0

0 00 0 0 0 0 0 000 0000 0000 00000 0000 0000 000 0000 000 00 0 0 0 0 0

Figure 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).

The top panel of Figure 3.7 shows a comparison between three different transits obtained without limb darkening law (cyan line), with linear limb darkening law (red line) and with quadratic law (yellow). The dotted lines correspond to the results obtained with the model by Mandel & Agol (2002). The continuous lines are the simulations produced for the same planet with SOAP-T. The two models show strong compatibility, on a level of 10−6. The bottom panel of figure 3.7 reports the best fit on RM observations of WASP-3b, obtained with SOAP-T. The best-fit results for the spin-orbit angle, λ = 20◦ 3.3◦ is λ = ◦+9◦ compatible with the result by Simpson et al. (2010), 13−7◦ . The projected stellar rotational velocity −1 v∗ sin i∗ = 13.3 0.45 km s shows stronger compatibility with the spectroscopic broadening result, 13.4 km s−1, than with the value by Simpson et al. (2010), 19.6 km s−1. These results highlight the strong reliability of the code in reproducing both photometric and spectroscopic transits. Finally, Figure 3.8 reports the best fit result obtained with an edge-on geometry for the transit photometry of HAT-P-11b (Sanchis-Ojeda & Winn 2011), which shows the typical features of occulted spots. As we can observe from the plot in the bottom frame, the residuals are on a level of 10−4 ppm, confirming that the code can reproduce occulted spots.

3.2 Updates to SOAP3.0

To improve the code, we decided to add some additional stellar physics to it, previously not accounted for. We enlist the introduced features:

• we updated the estimate of the stellar rotational velocity to account for the differential rotation. Now the code calculates the contribution of the velocity for each cell, accounting as well for the Chapter 3. Updated SOAP3.0 56

latitude of the star. As law, we adopted the formalism in equation 2.13. The final CCF of the system will thus suffer by a general decrement of the stellar velocity as the latitude increase, when the star has solar-like differential rotation.

• we changed the estimate of the σ of the CCF, by accounting for the law in Doyle et al. (2014):

2 vmac = 3.21 + 0.00233(T∗ − 5777) + 0.000002(T∗ − 5777) − 2(log g − 4.44). (3.3)

where vmac is the convective broadening of the CCF, while log g is the surface gravity of the star. Now, the final CCF should be the convolution of two Gaussians, one with FWHMequal to the instrumental broadening and the other whose FWHM is the convective broadening. The convolution of two Gaussian is still a Gaussian whose FWHM is: √( ) c 2 FWHM = + v2 (3.4) R mac

As a consequence, uses this value when modeling the FWHM of the initial stellar gaussian. Con- vection could as well change the profile of the Gaussian, making it asymmetric. Not accounting for the correct shape could change the retrieved geometry of the system, as it was shown in Cegla et al. (2016b). The reloaded RM, (Cegla et al. 2016a), might be an alternative technique to isolate the CCF of F stars in the future. Nonetheless, even applying this method, Bourrier et al. (2018), Bourrier et al. (2017) and Cegla et al. (2016a) found no evidence for the local stellar CCF of G, K and M stars to be non-Gaussian within their level of precision/SNR. No measurements were performed so far regarding to the CCF of an F star.

• we introduced another term to the local velocity of each grid cell, given by the center-to-limb variation of the convective blue-shift. In particular, we summed a velocity estimated with:

2 3 4 vCB = c1 + c2acos(µ) + c3acos(µ) + c4acos(µ) + c5 ∗ acos(µ) (3.5)

where acos(µ) is the center-to-limb angle, expressed in degrees. The parameters of this polynomial,

c1, c2, c3, c4 and c5 were obtained by fitting the results of the MHD simulations for theSun described in Cegla et al. (2016b) and in Cegla et al. (2018). In particular, for the 200G case in

Cegla et al. (2016a), we obtained: c1 = 9.026, c2 = −0.1866, c3 = 0.1119, c4 = −0.004172 and −5 c5 = 4.107 ∗ 10 . For the 0G case, we used the model in Cegla et al. (2018), with c1 = 0.1403, −5 c2 = −4.7495, c3 = 0.2623, c4 = −0.0065 and c5 = 6 ∗ 10 .

3.2.1 New input parameters in SOAP3.0

With the new changes, we added the following parameters in SOAP3.0:

• α, the relative differential rotation. To remove the stellar differential rotation, this parameter can be fixed to 0.

• c1, c2, c3, c4 and c5, the five parameters for the center-to-limb CB.

• log g, the surface gravity of the star. Chapter 3. Updated SOAP3.0 57

3.2.2 Testing the updated SOAP3.0

The stellar differential rotation

We performed a series of tests with two main objectives:

• to verify that changes injected gave the same results, for rigid rotation, as in the first version of SOAP3.0;

• to check if the shape of the RM signal with differential rotation is analogous to the predictions performed in previous papers;

• to determine how the differential rotation changes the RM signal once we vary the rotational pattern of the star and the planetary system geometry.

We produced simulations with the following, fixed, initial conditions: a stellar rotation of P∗ = 6 days ◦ (to choose averagely fast rotators), a stellar inclination of i∗ = 90 (i.e. the spin-axis is perpendicular to ◦ the line of sight), and a planet inclination of iP = 88 (so the planet transits close to a stellar latitude of ◦ 45 ). We fixed as well the planet radius to Rp = 0.1R⊙, to account for Jovian planets whose RM signal should be stronger than that by Earth- or Neptune-like planets. We finally fixed a spin-orbit angle of λ = 60◦, forcing the planet to mainly occult the red-shifted or blue-shifted side of the planet. In this way, the RM assumes an asymmetric shape. We then performed a first series of tests by producing RM simulations with our updated version of SOAP3.0 assuming α = 0.0, 0.2, 0.4, 0.6, 0.8, and 1. To test that the code worked properly, we simulated the same system with the previous version of SOAP3.0 (Akinsanmi et al. 2018), which should retrieve the same RM signal as the α = 0.0 case. The simulations are plotted in the top panel of Figure 3.9. For the α = 0 case, the residuals (shown in the bottom plot) are equal to zero. This supports the reliability of our updates to the code. Furthermore, we found that, as the relative differential rotation increases, the amplitude of the RM signal decreases, with a maximum difference of 12 m s−1 between α = 1 and α = 0. This happens because we used as constant the rotational velocity at the equator, by fixing P∗. As α increases, the rotation decreases at a faster rate with the stellar latitude. Hence, the planet blocks a stellar area that is less red-shifted or less blue-shifted. This result further confirms the findings from Hirano et al. (2011). The shape of the RM signals and residuals in Figure 8 by Hirano et al. (2011) are in strong agreement with ours. Note that in Hirano et al. (2011) the residuals at latitude 0 are null. In our case, they are just closer to 0 than in the rest of the curve. This is normal, because Hirano et al. (2011) and other works used an analytical model to produce the RM signal. The planet is treated as a point and at the equator there is no difference between the rigid rotation case and the case with stellar differential rotation. On the contrary, SOAP3.0 estimates the RV considering all the stellar areas shadowed by the planet during the transit. At the passage through the equator, the planet will additionally cover areas with stellar differential rotation and the overall signal decreases with respect to the rigid rotation case. In the bottom panel of Figure 3.9, we show the results obtained by fixing α = 0.2 and varying the spin-orbit angle. We found no significant difference between the cases with λ = 60◦, 30◦ and 0◦. However, for λ = 90◦ , the RM signal is totally blue-shifted, meaning the planet only crosses the red-shifted side of the star. The residuals with respect to the modulation in the absence of differential rotation show similar minima on the stellar limbs. As additional tests, we also fixed α = 0.2 and modified other parameters that are expected to influence the RM amplitude. We first varied the planet radius choosing the values Rp = 0.1R∗, 0.07R∗ and 0.04R∗. Chapter 3. Updated SOAP3.0 58

P * = 6 days, Rp = 0.1R , i * = 90 , ip = 88 , = 60

0 ] s /

m = 0.0 [

V 40 = 0.2 R = 0.4 = 0.6 = 0.8 80 = 1.0 Old SOAP3.0 ] s

/ 20 m [

10 s l

a 0 u d

i 10 s e r 0.075 0.050 0.025 0.000 0.025 0.050 0.075 time [days]

P = 6 days, Rp = 0. 1R , ip = 88◦, i = 90◦, α = 0. 2 ∗ ¯ ∗

λ = 90◦

λ = 60◦ 120 λ = 30◦

λ = 0◦ 80 λ = 90◦ SOAP3. 0 ]

s λ = 60 SOAP3. 0

/ ◦

m 40

[ λ = 30◦ SOAP3. 0

V λ = 0◦ SOAP3. 0 R 0

80 ] s /

m 2 [

s 0 l a

u 2 d i

s 4 e

r 0.05 0.00 0.05 time [days]

Figure 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. Chapter 3. Updated SOAP3.0 59

P = 6 days, ip = 88◦, i = 90◦, λ = 60◦, α = 0. 2 ∗ ∗

0 ] s / 40 m [ Rp = 0. 1R V ¯ R Rp = 0. 07R ¯ Rp = 0. 04R 80 ¯ Rp = 0. 1R , SOAP3. 0 ¯ Rp = 0. 07R , SOAP3. 0 ¯ Rp = 0. 04R , SOAP3. 0 ¯ ] s / m

[ 2

s l

a 0 u d i 2 s e

r 0.05 0.00 0.05 time [days]

P = 6 days, Rp = 0. 1R , i = 90◦, λ = 60◦, α = 0. 2 ∗ ¯ ∗

] 40 s / m [ 80 V

R ip = 90◦ ip = 90◦, SOAP3. 0

ip = 89◦ ip = 89◦, SOAP3. 0

ip = 88◦ ip = 88◦, SOAP3. 0

ip = 87◦ ip = 87◦, SOAP3. 0

ip = 86◦ ip = 86◦, SOAP3. 0 ] s

/ 4 m [

s l

a 0 u d i

s 4 e

r 0.05 0.00 0.05 time [days]

Figure 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. Chapter 3. Updated SOAP3.0 60

P = 6 days, Rp = 0. 1R , ip = 88◦, λ = 60◦, α = 0. 2 ∗ ¯ 40

0 ] s / m [

V 40 i = 90◦

R ∗ i = 45◦ ∗ i = 30◦ ∗ i = 90◦, SOAP3. 0 80 ∗ i = 45◦, SOAP3. 0 ∗ i = 30◦, SOAP3. 0 ∗ ] s /

m 2 [

s 0 l a

u 2 d i

s 4 e

r 0.05 0.00 0.05 time [days]

◦ Figure 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.

The results are in the top panel side of Figure 3.10. As Rp decreases, the planet occults a smaller area of stellar disk and the intensity of the RM signal decreases. The presence of the solar differential rotation changes the shape of the RM signal and decreases its amplitude. When the planetary radius becomes smaller the planet covers smaller and smaller areas and the effect of the differential rotation decreases.

In the case of Rp = 0.04R⊙, the residuals are almost equal to zero. ◦ ◦ ◦ ◦ ◦ As final step we assigned to iP the values 90 , 89 , 88 , 87 and 86 and produced simulations for each case. The results are reported in the bottom panel of Figure 3.10. As iP decreases, the amplitude of the residuals tends to decrease as well, because the planet occults an area of the stellar disk with a slower rotation than the equator. On top of this, we wanted to explore the effect of changing the inclination of the stellar rotational ◦ ◦ axis in the model. We first simulated the case with P∗ = 6 days, iP = 88 , RP = 0.1R⊙ , λ = 60 and α = 0.2 and we imposed to the inclination of the stellar rotational axis the values 90◦ , 45◦ , 30◦ and 0◦. The results are in Figure 3.11. Changing the stellar inclination, the planet crosses different areas of the star. As the latitude increases, the stellar surface rotates more slowly and and the amplitude of the residuals tend to increase. Thus, we decided to explore in detail how the residuals change if the stellar inclination is close to 0◦ or 90◦ and the spin-orbit angle is λ = 90◦ . In these conditions, the stellar axis and the planetary orbit are perpendicular one with respect to the other. The plots in Figure 3.12 ◦ ◦ ◦ report the two cases with i∗ = 5 in the top frame, and 90 in the bottom one. The case with i∗ = 5 is an almost pole-on configuration, in which the planet crosses areas of the stellar surface that rotate very slowly. The differential rotation causes the formation of a bump in the center of theRM.As α Chapter 3. Updated SOAP3.0 61

P = 6 days, Rp = 0. 1R , i = 90◦, ip = 88◦, λ = 90◦ ∗ ¯ ∗ 0

40 ] s /

m α = 0. 0 [

V 80 α = 0. 2 R α = 0. 4 α = 0. 6 α = 0. 8 α = 1. 0 Old SOAP3. 0 ]

s 0 / 0.05 0.00 0.05 m [

10 s l a

u 20 d i s e

r 0.05 0.00 0.05 time [days]

P = 6 days, Rp = 0. 1R , i = 5◦, ip = 88◦, λ = 90◦ ∗ ¯ ∗

2 ] s /

m α = 0. 0 [

V α = 0. 2 R α = 0. 4 α = 0. 6 α = 0. 8 α = 1. 0 Old SOAP3. 0 ]

s 0 / 0.05 0.00 0.05 m [

s l a u d

i 2 s e

r 0.05 0.00 0.05 time [days]

◦ ◦ Figure 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 parameter α. 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. Chapter 3. Updated SOAP3.0 62 increases, the bump increases in amplitude, determining a decrement of the RM signal almost to zero as α = 1. This bump happens in correspondence to the point of the transit where the planet passes closer ◦ to the pole of the star. In the case, of i∗ = 90 , the residuals tend to increase on the stellar limbs, where the planet crosses areas of the star which rotate more slowly. No significant change can be observed in correspondence to the center of the transit, where the planet obscures an equatorial area. By analyzing these configurations, we can deduce that the differential rotation allows to break the degeneracy between the stellar period of rotation at the equator and the inclination of the stellar rotational axis, because for different i∗ we obtain different residuals due to the geometry of the system. This result confirms those by Hirano et al. (2011) and encourages a deeper analysis on the opportunity we have to analyze the rotational pattern of stars through the RM signal.

The center-to-limb variation of CB

To test the effect of the center-to-limb CB, we produced simulations turning off the stellar differential ◦ rotation and imposing the following initial conditions: RP = 1RJ, PP = 4 days, R∗ = 1R⊙, i∗ = 90 and e = 0.0. We additionally imposed λ = 0., to produce the case of an aligned orbit, and varied v∗ sin i∗ from 2 km s−1 to 10 km s−1. We estimated the residuals of the simulations this way retrieved with respect to the cases without center-to-limb CB. We repeated this procedure for both of the cases with the solar 200G polynomial and the solar 0G polynomial. The results are in figure 3.12, where the top panel displays the 0G case, while the bottom shows the 200G case. The general effect that we observe is a stronger influence of the 0G polynomial on the RV residuals, perfectly in line with the predictions by Cegla et al. (2018). We highlight as well an inversion in the behaviour of the residuals as the stellar −1 rotational velocity increases. The inversion happens for both of the polynomials at v∗ sin i∗ > 4 km s . ◦ ◦ For a more detailed analysis, we consider as well two additional tests, one for λ = 90 and iP = 90 , ◦ ◦ the second one for λ = 90 and iP = 88 . We tested these cases just for the 0G polynomial, which has the strongest effects on the limbs. The residuals of the simulations with respect to the cases withoutCB ◦ are reported in Figure 3.14. The top panel corresponds to the case for iP = 90 , while the bottom one ◦ is valid for iP = 88 . It is evident that as iP decreases, the impact parameter increases and the effect on the limbs becomes smaller, because the planet transits areas of the star with a stronger radial velocity.

The macro-turbulence effect

To test the macro-turbulence effect, we performed two series of simulations with RP = 1RJ, PP = 4 days, ◦ −1 R∗ = 1R⊙, i∗ = 90 and e = 0.0. We chose two values of v∗ sin i∗, 5 and 10 km s . Then, for the first case, we input macro-turbulence values of ζ = 3.0, 4.3 and 5.6 km s−1, in line with a G star case. For the second case, we assigned ζ = 4.3, 6.2 and 8.1 km s−1, in line with an F star case. The results are in Figure −1 −1 3.15. The top panel corresponds to v∗ sin i∗ = 5 km s , while the bottom is for v∗ sin i∗ = 10 km s . The figure also shows the residuals, with respect to ζ = 4.3 km s−1 for the slow rotator, and ζ = 6.2 km s−1 for the fast rotator. The first conclusion we can make is that the residuals are stronger for stronger v∗ sin i∗. Moreover, we also note that increasing the macro-turbulence parameter, the depth of the RM effect increases. These results are in line with those reported in Hirano et al. (2011). Chapter 3. Updated SOAP3.0 63

Old CB polynomial Rp = 1RJ, ip = 90 , i * = 90 , = 0 vsini = 2 km/s 0.6 vsini = 3 km/s vsini = 4 km/s vsini = 5 km/s 0.4 vsini = 6 km/s vsini = 7 km/s ] s / vsini = 8 km/s

m 0.2 [

vsini = 9 km/s s l vsini = 10 km/s a u d i 0.0 s e r

0.2

0.4

1.5 1.0 0.5 0.0 0.5 1.0 time [days]

New CB polynomial Rp = 1RJ, ip = 90 , i * = 90 , = 0 vsini = 2 km/s 1.00 vsini = 3 km/s vsini = 4 km/s 0.75 vsini = 5 km/s vsini = 6 km/s 0.50 vsini = 7 km/s ] s / vsini = 8 km/s m

[ 0.25

vsini = 9 km/s s l vsini = 10 km/s a

u 0.00 d i s e r 0.25

0.50

0.75

1.00 1.5 1.0 0.5 0.0 0.5 1.0 time [days]

Figure 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). Chapter 3. Updated SOAP3.0 64

New CB polynomial Rp = 1RJ, ip = 90 , i * = 90 , = 90 1.0

0.5

0.0 ] s / m [

0.5 s l vsini = 2 km/s a

u vsini = 3 km/s d i 1.0 s vsini = 4 km/s e r vsini = 5 km/s 1.5 vsini = 6 km/s vsini = 7 km/s 2.0 vsini = 8 km/s vsini = 9 km/s vsini = 10 km/s 2.5 1.5 1.0 0.5 0.0 0.5 1.0 time [days]

New CB polynomial Rp = 1RJ, ip = 88 , i * = 90 , = 90

0.50

0.25

0.00 ] s / m

[ 0.25

s l vsini = 2 km/s a

u 0.50 vsini = 3 km/s d i

s vsini = 4 km/s e r 0.75 vsini = 5 km/s vsini = 6 km/s 1.00 vsini = 7 km/s vsini = 8 km/s 1.25 vsini = 9 km/s vsini = 10 km/s 1.50 1.5 1.0 0.5 0.0 0.5 1.0 time [days]

Figure 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 thestar. ◦ Top panel: tests for the solar 0G model and iP = 90 . Bottom panel: tests for the solar 0G model and ◦ iP = 88 . Chapter 3. Updated SOAP3.0 65

RM effect for Rp = 1RJ, ip = 90 , i * = 90 , = 0 40

20 ] s / m

[ 0

V R 20 = 4.3km/s = 3.0km/s 40 = 5.6km/s

4 ]

s 2 / m [

s l 0 a u d i

s = 4.3km/s e r 2 = 3.0km/s = 5.6km/s 4 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 time [days]

RM effect for Rp = 1RJ, ip = 90 , i * = 90 , = 0 100

50 ] s / m

[ 0

V R = 6.2km/s 50 = 8.1km/s = 4.3km/s 100

5.0 ] s / 2.5 m [

s l 0.0 a u d i

s 2.5 = 6.2km/s e r = 8.1km/s 5.0 = 4.3km/s

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 time [days]

Figure 3.15: RM simulations for an alingned planet, varying the macro-turbulence parameter. Top −1 −1 −1 panel: v∗ sin i∗ = 5 km s and ζ = 3.0, 4.3 and 5.6 km s . Bottom panel: v∗ sin i∗ = 10 km s and ζ = 4.3, 6.2 and 8.1 km s−1. Chapter 4

Detecting the albedo of exoplanets accounting for stellar activity

In this chapter, we present a work on the detection of the planetary reflected light component through photometric observations in the presence of activity features.

The reflected light component is a function of the orbital phase and it is directly proportional to the planetary albedo. Retrieving it from a photometric phase curve requires to properly account for the instrumental noise and the stellar activity modulation. While the instrumental noise can be significantly reduced with the adoption of high-precision instruments, the stellar activity cannot be avoided in general. For instance, Gilliland et al. (2011) showed that the stellar-induced noise is the largest noise contribution to Kepler stars observations.

Previous works studied real stellar light curves, modeling both the primary and the secondary transit and also the beaming effect, ellipsoidal modulation, and the reflected light component. Angerhausen et al. (2015) performed a study on a large sample of quiet Kepler stars known to host hot-Jupiters, with the aim to identify secondary transits and planetary albedo values. On the other hand, Basri et al. (2013) compared the activity level of Kepler stars with that of the active Sun, showing that 30% of the selected sample is more active than the Sun. The scope of our work was to determine the limits imposed by the instrumental noise and the stellar activity on the identification of the planetary albedo, inthe context of the observing conditions imposed by the new joint ESA-Switzerland optical photometric space mission CHEOPS (CHaracterizing ExOPlanet Satellite).

In Sect. 4.1, we introduce our phase curve model. In Sect. 4.2 we describe the adopted method to analyze the simulated light curves, and, in the following paragraph, we present blind tests which proved the reliability of this method. In Section 4.4 we report the results of the data analysis. In Sect. 4.5 we assess the consequences of introducing the beaming effect and the ellipsoidal modulation in the model and when additional priors are inserted around the planetary radius and the semi-major axis. Sect. 4.6 explores the effects on the measured albedo when we account for the predicted gaps inCHEOPS observations. In Sect. 4.7, we present the limits of our method of analysis, by testing it on real data. The last section presents the conclusions. The results of this work led to our first publication, Serrano et al. (2018).

66 Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 67

Figure 4.1: Typical phase light curve used in our work. It shows the normalized stellar flux as a function of the stellar phase. The green line is the instrumental noise, the blue line 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 inabsence of instrumental noise. This light curve is as well a representation of most of the tests performed in this paper.

4.1 Synthetic light curves

We simulated stellar light curves as a function of the stellar phase, Φ. Each light curve includes three different components:

• the reflected light modulation due to the planet, FP F∗

• the stellar activity, F∗,spotted F∗

• a white noise on the same level of the instrumental noise, Fnoise . F∗

These three components are summed together as it follows:

F F F∗, F total = P + spotted + noise (4.1) F∗ F∗ F∗ F∗ Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 68

The total flux has no units, because it is normalized with respect to the stellar flux. InFigure 4.1, we show a typical stellar light curve used for this work. In particular, we report the comparison between the instrumental noise, the planetary phase curve, the stellar activity pattern, the total flux, and the total flux without instrumental noise. All the curves are expressed as a function of the stellar phase. The spots are spread throughout the light curve, and the amplitude of the activity signal is much higher than the reflected light component of the planet, preventing visual identification. Previous works used the eclipse and the planetary phase curve at the same time to retrieve the albedo of exoplanets (e.g., Esteves et al. 2013; Angerhausen et al. 2015). Though, the secondary transit is not always detectable, due to the activity modulation. A notable example is planet Kepler-91b. Lillo-Box et al. (2014) analyzed the entire phase curve of this planet, including all the components (reflected light, ellipsoidal modulation, beaming effect, transit and eclipse). In detail, they managed to identify the position of the secondary transit in the phase curve, although raising doubts on which feature represented the eclipse. The presence of stellar activity in the phase curve caused hurdles in distinguishing the specific modulation due to the eclipse from the spots themselves. Due to theseveral difficulties encountered when trying to identify the secondary transit, we chose to exclude itfromthe model. In this way, we could as well verify whether the reflected light component was sufficient to identify the albedo after a full characterization of the planetary orbit. This suggests the possibility of a future development of our tool in the case of non-transiting planets. For example, Crossfield et al. (2010) discovered an additional non-transiting planet with the planetary phase curve analysis alone. The CHEOPS timing will be one minute, though its observations will have two types of gaps. The first one occurs when the satellite crosses the South Atlantic Anomaly. In this location, the terrestrial magnetic field generates loops which trap high-energy particle. The impact of such particles onthe satellite detector generates glitches in the observations. As a consequence, observations belonging to such moments are removed from the data. These gaps solely depend on the satellite position with respect to our planet. The second type of gaps is a consequence of the target occultation by Earth itself, the Sun. Each observational gap will last several minutes, though all together they are expected to cover a significant fraction of the data. To have a much longer sampling than the predicted gaps, we simulated curves with 2 hs timing, thus 12 points per day. In this way, the gaps are much shorter than the bin size, impacting the data with a decrement of the signal-to-noise ratio in each bin. The results of our tests are thus expected to be similar to what we would obtain with more realistic curves in presence of gaps. We explore this detail in Section 4.6. The reflected light was produced through an analytic model as presented in 1.2. For the stellar activity and the instrumental noise, we refer to the next two paragraphs.

4.1.1 Stellar activity

As we mentioned in the introductory chapter, with the expression stellar activity, we refer to a group of phenomena occurring on the stellar surface, spots and plages over all (Berdyugina 2005). As we already explored, spots inject a decrement in the stellar light curve, biasing the planetary characterization when not included in the analysis (e.g., Oshagh et al. 2013b; Barros et al. 2013). Plages induce an opposite effects on the stellar light curves, if compared to spots. Though, the influence of plages onphotometric observations are less significant, due to their lower temperature contrast when compared to spots (e.g., Berdyugina 2005; Meunier et al. 2010). For this reason, we modelled stellar light curves in presence of spots alone. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 69

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

Magnitude Noise for two-hour timing 6.5 14ppm 8 17ppm 10 29ppm

Table 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, Fmean is the flux average, and ptp is the peak-to-peak variation of the light curve.

Parameter Prior Interval Ag Uniform [0; 1] , P∗ Gaussian [N(P0,∗ 3)] −4 p1 Log-Uniform [ 10 ; 600 ] × −5 × 4 p2 Log-Uniform [ 5 10 ; 2 10 ] Offset Uniform Fmean − 2 ptp; Fmean + 2 ptp

When trying to measure the planetary albedo, the stellar activity can represent an important noise source, capable of hiding the reflected light component. Even for quiet stars, the signal of thephase curves can be much lower than the stellar activity. For instance, the planetary phase modulation is usually lower than 200 ppm (e.g., Angerhausen et al. 2015; Lillo-Box et al. 2014). On the other hand, the stellar activity can reach much higher levels, even 104 ppm, as observed for Corot-7 (Léger et al. 2009). For our tests, we modeled the stellar activity with the tool SOAP-T (Oshagh et al. 2013a), whose updated version is described in Chapter 3.

4.1.2 Instrumental noise

We modeled the noise component as Gaussian noise with a standard deviation comparable to the value achieved by the satellite CHEOPS for different stellar magnitudes. In Table 4.1, we report the values of CHEOPS noises for the stellar magnitudes we accounted for.

4.2 Data analysis method

In this section, we present the analysis method applied on the simulated light curves. Our choice was to use a Markov Chain Monte Carlo (MCMC), with the aim of establishing whether we were able to recover the planetary albedo in presence of both stellar activity and instrumental noise. We assumed as already well known all the planetary parameters, apart from the albedo itself. Furthermore, to distinguish the stellar activity from the planetary reflected light, we adopted a Gaussian Process, whose details are given in the following section.

4.2.1 Gaussian process for modeling the stellar activity

Understanding the properties of spots which appear in a photometric curve is a challenge, due to the high degeneracy among spot properties. As we discussed in Chapter 2, a spot generates a feature, Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 70

+0.01656 albedo = 0.29720 0.01625 −

0.24 0.29 0.34 +0.00151 p1 = 0.00688 0.00107 − 014 0. 012 0. .010 1 0 p 008 0. 006 0. .004 0 0.0045 0.0100 0.0155 +0.00770 P = 18.99432 0.00822 ∗ − .02 19 .01 19 .00 19 .99

[days] 18 .98

∗ 18 .97 P 18 18.96 18.99 19.02 +0.02357 p2 = 0.35023 0.01941 − 45 0. 40 0. 2

p 35 0. 30 0. 0.28 0.35 0.42 0.49 +0.00008 oﬀset = 0.99867 0.00009 − 9990 0. 9989 0. 9988 0. 9987 0. 9986 0. .9985 oﬀset 0 9984 0. 9983 0. 0.9985 0.9989 .24 .26 .28 .30 .32 .34 .97 .98 .99 .00 .01 .02 .30 .35 .40 .45 0 0 0 0 0 0 .006 .008 .010 .012 .014 0 0 0 0 0 0 0 0 0 18 18 18 19 19 19

albedo p1 P [days] p2 oﬀset ∗

Figure 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. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 71 whose depth depends on both the temperature contrast and the spot size. On the other hand, the same feature might likewise appear as a consequence of an overlap between two or more different spots. Such possibility dramatically increases the number of possible solutions for modeling the same stellar light curve. Performing an imaging observation of the star, to locate and describe the spots, is not a possibility nowadays due to the strong instrumental limitations in this sense. An alternative technique to reproduce the spots effect in photometric (and RV) observations isthe so-called Gaussian process (GP), which was successfully applied in several works (Faria et al. 2016; Haywood et al. 2014; Rajpaul et al. 2015). This method treats the stellar activity as a correlated noise, because it accounts for the spots movement on the stellar surface as a consequence of the stellar rotation. A GP is in part defined by its covariance function. For activity-induced signals, a quasi- periodic covariance function is the most common choice, allowing to account for the spot evolution and the stellar differential rotation (Rajpaul et al. 2015). Though, within short observations (for the lifetime of spots, see Berdyugina 2005) these additional spot properties should be non-observable. SOAP-T models star-spots as periodic features, which forced us to adopt a fully periodic covariance, thus ignoring the aperiodic component. The applied covariance is given by   π −  2 (ti t j)   2 sin ∗  Σ = p exp − P  + σ2δ . (4.2) i j 1  2  i j p2

The parameter p1 is the amplitude of the correlations. The exponent is the periodic correlation, which describes the dependence of star-spots on the stellar rotation and on a decaying timescale, p2. The last part of this covariance function includes a diagonal component which accounts for the instrumental noise σ (δi j is the Kronecker delta). From now on, p1, p2 and P∗ are also called hyper-parameters of the GP. This covariance function is compatible with the complete periodicity of our simulated light curves, though it might be less adequate for the analysis of more general light curves. Furthermore, we fixed the level of instrumental noise, σ, as equal to the predicted CHEOPS noise for a given stellar magnitude. Unlike similar works, which applied as well GP on real data, we performed the tests on synthetic light curves without accounting for a jitter parameter. Even though we attempted to include it in the analysis, the first tests showed a close to zero jitter. For this reason wefound meaningless introducing it as an additional parameter in these special cases. To perform the GP regression we adopted the george package (Ambikasaran et al. 2015).

4.2.2 Analysis method

To sample from the posterior distributions for the parameters of our model, we performed an MCMC, adopting as tool emcee (Foreman-Mackey et al. 2013). The MCMC explored in total 5 parameters: the geometric albedo Ag, the stellar rotation P∗, the amplitude of the correlation p1, the timescale decay of the periodic modulation p2, and an offset to fit the average value of the light curve. Table 4.2 lists the selected priors for each parameter. For the albedo, we adopted a uniform prior between 0 and 1, the range of possible values for this parameter. This choice is justified by the fact that Ag is otherwise unconstrained and it rarely becomes higher than 1 (not the case of the modelled exoplanets in our work). We assumed the stellar rotational period as already known with an uncertainty of 3 days, with the hypothesis it had already been measured through other techniques in previous papers. For this reason, we imposed a Gaussian prior, centered on the injected value of P∗ and with a standard Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 72

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

Stellar radius R∗ 1 R⊙ Stellar inclination I 90◦ Stellar temperature T∗ 5778 K Linear limb-darkening coefficient c1 0.29 Quadratic limb-darkening coefficient c2 0.34 Planet radius RP 0.1 R∗ Time of mid-transit t0 0.3 days Eccentricity e 0 Argument of periastron w 0◦ Inclination of the orbital plane i 89◦ Projected spin-orbit misalignment angle λ 0◦ deviation of 3 days. The GP coupled with an MCMC allowed us to derive the stellar rotation with much better precision than in other methods, such as a Lomb-Scargle periodogram or an autocorrelation function technique, as demonstrated by Angus et al. (2018). Nonetheless, it is still possible that P∗ of the planet host is unknown. In this case, it becomes mandatory to first apply these techniques on the data, to estimate the stellar period prior for the MCMC. The other hyper-parameters, p1 and p2, could vary within wide log-uniform priors, whose intervals are listed in Table 4.2. To conclude, for the GP offset, we adopted a uniform prior, centered on the average flux level Fmean. The prior width was equal to four times the peak-to-peak variation of the light curve. The MCMC calls as model the same reflected planetary phase curve tool with which we produced the simulations. It calls as well the GP presented in the previous section to reproduce the activity features. The overall likelihood is expressed as a multivariate Gaussian distribution (Ambikasaran et al. 2015; Faria et al. 2016). We run the MCMC using 30 chains, whose initial parameters were randomly extracted from the prior distribution. For each simulation, we imposed a 500-step burn-in and, after, we sampled the chains for 1000 steps. In total, we retrieved 30000 samples from the posterior distribution function. As best fit values for each parameter, we adopted the medians of the posterior distributions, whenthe posteriors were Gaussians. Though detecting the albedo was not always possible. When the posterior peaked at 0, the minimum of the Ag interval, the shape of the distribution was hyperbolic. For these cases, identified with a visual check, we adopted the mode as best-fit parameter. AS 1σ uncertainties we estimated the differences between the best-fit value and the 16th and 84th percentiles, respectively. In Figure 4.2, we report an example of the parameter posterior distributions retrieved in the case of a star with rotational period equal to 19 days, orbited by a planet with RP = 0.1R∗. The system is modeled for a total of 13 orbital periods.

4.3 Reliability test

To test the reliability of our code, we performed 9 blind tests. The mock light curves were prepared by one of the supervisors and they were passed to us together with all planet parameters, except for the albedo. Additionally, we knew an initial range of possible rotational period values. We performed the MCMC analysis with the scope of checking whether the injected albedo and stellar rotational period could be properly estimated. Table 4.3 reports the initial parameters, in common among all the blind tests. The Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 73

Table 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.

Pattern Spot Longitude Brightness Size (R∗) 1 0◦ 0.50 0.080 2 55◦ 0.48 0.075 a 3 120◦ 0.52 0.081 4 174◦ 0.48 0.079 5 227◦ 0.50 0.083 6 290◦ 0.49 0.076 1 0◦ 0.50 0.040 b 2 20◦ 0.48 0.045 3 35◦ 0.52 0.041 4 121◦ 0.48 0.049 1 0◦ 0.50 0.02 c 2 34◦ 0.48 0.025

Figure 4.3: Comparison between patterns a, b, and c adopted in the blind tests. Their properties are reported in Table 4.4. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 74

Table 4.5: Input properties and recovered parameters for the blind tests.

# Input properties Recovered parameters Duration Pattern PAg P∗ Ag P∗ p1 p2 (days) (days) (days) (days)

. . . . . [+0.037] . [+0.003] . [+0.0009] . [+0.018] 1 10 66 a 2 2 0 8 7 00 0 828[−0.035] 7 000[−0.003] 0 0049[−0.0006] 0 319[−0.016]

. . . . . [+0.018] . [+0.003] . [+0.0009] . [+0.016] 2 10 25 a 1 5 0 2 7 00 0 212[−0.017] 7 000[−0.003] 0 0048[−0.0007] 0 302[−0.017]

. . . . . [+0.020] . [+0.003] . [+0.0009] . [+0.017] 3 10 25 a 1 2 0 1 7 00 0 129[−0.019] 7 000[−0.003] 0 0049[−0.0007] 0 301[−0.017]

. . . . . [+0.012] . [+0.003] . [+0.0028] . [+0.109] 4 17 08 b 2 2 0 3 12 30 0 319[−0.012] 12 299[−0.003] 0 0091[−0.0021] 0 729[−0.105]

. . . . . [+0.013] . [+0.005] . [+0.0030] . [+0.361] 5 18 75 b 1 5 0 5 12 30 0 469[−0.011] 12 295[−0.004] 0 0092[−0.0020] 0 790[−0.327]

. . . . . [+0.006] . [+0.005] . [+0.0023] . [+0.041] 6 20 42 b 1 2 0 4 12 30 0 400[−0.006] 12 296[−0.005] 0 0085[−0.0016] 0 616[−0.037]

. . . . . [+0.012] . [+0.965] . [+0.0015] . [+0.341] 7 19 41 c 2 2 0 6 19 74 0 609[−0.012] 20 780[−0.777] 0 0029[−0.0011] 1 134[−0.212]

. . . . . [+0.015] . [+0.028] . [+0.0024] . [+0.319] 8 26 08 c 2 7 0 35 19 74 0 340[−0.015] 19 739[−0.030] 0 0041[−0.0018] 1 511[−0.250]

. . . . . [+0.005] . [+0.022] . [+0.0027] . [+0.469] 9 24 42 c 1 2 0 15 19 74 0 155[−0.005] 19 732[−0.025] 0 0044[−0.0025] 1 601[−0.347]

host star had the same properties of the Sun, with stellar rotational axis inclination i∗ perpendicular to the line of sight. As a consequence, the stellar equator was seen edge-on. As limb-darkening coefficients (Claret 2000), we chose values compatible with the Kepler bandpass tables in Claret & Bloemen (2011). The orbiting planet was a Jupiter-sized planet, whose orbital period and albedo varied case after case. The stellar activity patterns were 3, equally distributed among the 9 mock curves. Their star-spot properties are summarized in Table 4.4. We note that these characteristics were not known before applying the MCMC, in way to make the tests unbiased. Figure 4.3 displays a comparison among the activity patterns. The most active is pattern a, used in tests 1, 2 and 3. The least active, c, was applied to the last three simulations. Table 4.5 reports the input values which varied among the mock curves, together with the best fit results retrieved with the MCMC. In the second column, we enlist the observational length, in the third the adopted activity pattern. The fourth column shows the period of the simulated planet, expressed in days. The fifth and sixth columns report the input albedo and stellar rotation, the onlyunknown parameters before the MCMC. In the remaining four columns we show the output of the MCMC analysis,

Ag P∗, p1 and p2. We do not report here the offset and the relative errors, since it resulted tobe 0.999 for all the blind tests. As a general result, we found a strong compatibility, within 1σ, between the input and output albedo. We can state the same for almost all the stellar rotational periods, compatible with the injected values within 2σ. We report an exception, the seventh simulation, for which P∗ was overestimated and the error was about 1 day. This incompatibility can be justified through an observational length close to the rotational period. Although the reason behind this will be clarified in the next section. For the Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 75

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

Spot Longitude ∆T (K) Size (R∗) 1 270◦ 400 0.045 2 80◦ 500 0.045 3 250◦ 663 0.045 4 340◦ 700 0.045

Table 4.7: Fixed planetary properties.

Time of mid-transit t0 0.2 days Eccentricity e 0 Argument of periastron w 0◦ Inclination of the orbital plane i 90◦ Projected spin-orbit misalignment angle λ 0◦ tests with the same activity pattern, the hyper-parameters and the offset were very similar between each other, showing that the GP regression works properly. These tests support the reliability of our analysis tool, sustaining how it can properly disentangle the planetary reflected light from the stellar noise for a wide range of planetary and stellar properties.

4.4 Results

To explore to what extent our method could retrieve P∗ and Ag, we used our MCMC analysis to fit several series of 60-day-long simulations which differed by the injected P∗. Depending on the simulation or planetary system property we wished to investigate, we modified the observation length, the planetary radius, the albedo, the orbital period, and the spot dimension.

4.4.1 Simulation properties

As initial condition, we imposed the case of a 6.5 magnitude Sun-like star, with the same properties as in Table 4.3). The injected stellar activity pattern included four spots, modeled with different longitudes. In this way, the activity features managed to cover the entire stellar light curve, properly mimicking a realistic spot modulation. Table 4.6 reports the spots properties. All the spots were placed at the stellar equator, meaning that their latitudes were equal 0. The spot sizes were fixed as well. The planetary physical quantities which remained fixed for all the simulations are listed in Table 4.7.

4.4.2 Lower limit for the observation length

Our first series of tests aimed at exploring how the observational length can affect the albedo detection together with the stellar period of rotation. We fixed the planetary radius RP = 0.1 R∗ and the orbital period PP = 3 days and we produced simulations with an increasing observational length. The first simulation covered an entire orbital period, the following twice PP, and so on, until we reached a maximum observation of 60 days. We produced, in this way, 20 different simulation for each of 5 different stellar rotational periods 7, 11, 19, 23 and 26 days. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 76

Figure 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. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 77

For each simulation we performed the MCMC and we retrieved an offset always around 0.999. The −3 hyper-parameters stabilized on p1 ∼ 10 and p2 ∼ 0.4, just in the moment in which the retrieved rotational period P∗ was compatible with the injected value. Though, such compatibility could be observed when the observations covered more than one entire stellar rotation. The top panel of Figure 4.4 reports the dependence of the recovered albedo with the number of stellar rotations. The bottom panel shows the same for the albedo uncertainties. The different data series correspond to a different injected P∗. The most evident result is that for observational lengths shorter than P∗, the MCMC is not capable of recovering the albedo. Inspecting the plots, we notice that this trend is stronger for shorter stellar rotational periods, 7 and 11 days. For these simulations, we observe the strongest uncertainties on the albedo, before 1 P∗. Right after this data length, the errors dramatically decrease. For the other tests, we can observe a similar though smoother trend. Another interesting detail is the stronger compatibility of the retrieved albedo with the input one for the 23 and 26 days cases. This is evident even for observational lengths shorter than 1 P∗ and it may depend on the number of data points with which the rotational period is sampled. As P∗ becomes longer, it covers a longer time span and the number of data points describing the activity pattern increases as well. This renders easier to isolate with the MCMC the reflected light component of the planet. When observations become longer, the albedo is detected more precisely, stabilizing itself around the input value 0.3. Between one and two stellar rotations, the albedo is compatible with the input within 2σ. For more than two stellar rotations, the values are compatible to within 1σ. As a final test, we adopted as MCMC prior on the stellar rotation period a uniform distribution between 1 to 28 days. We applied the updated fitting tool on a 19 days stellar rotation simulation, with data length of 39 days. The best fit albedo and P∗ were strongly compatible with the results we obtained when we used a restricted prior on the stellar rotation. We can thus conclude that, even when the stellar rotation is unkown, the MCMC with GP can still derive it directly from the photometric observations.

For simplicity, we decided to keep the Gaussian prior on P∗ for all the following tests.

Fast rotators

By inspecting Figure 4.4, we can stress a different behavior between fast and slow rotators. When compared to slow rotating stars, fast rotators show much larger errors on the albedo before one stellar rotation. As the observation becomes longer than 1 stellar rotation, the errors fall below 0.1. The top panel of Figure 4.4 requires a further analysis on the cases with P∗ = 7 and 11 days. For the 11 days rotator, the retrieved albedo after eight orbital observation periods stabilizes to a slightly underestimated value with respect to the input, 0.3. Though, the output and input Ag are compatible between each other within 2σ. A similar trend can be observed as well for the 7 days case, with the results compatible with Ag = 0.3 within 1σ. We conclude that for fast rotators, the albedo is underestimated after long observations, nonetheless for the 11 days tests, this trend seems more evident. An explanation for this behaviour might be connected to the smaller amount of data within one rotation. Reducing the timing of data might improve the reliability of the estimated albedo. To explore this possibility we produced three additional series of tests, with P∗ = 11 days and with different cadences, 110, 30 and 28 minutes. We chose the 28 and 110 min cadence to explore whether slightly varying the timing, from 28 to 30 min and from 110 to 120 min could somehow show a specific trend in the retrieved albedo. Figure 4.5 reports the albedo as a function of the number of stellar rotations for the four timings. The 120 min case is the worst, the albedo is always underestimated for longer observations. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 78

Figure 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.

The 110 min case shows a slightly overestimated albedo, compatible with Ag = 0.3 within 1σ as the observations become longer than two stellar rotations. The 30 and 28 min cases are both compatible with the initial albedo after two entire rotations, though the results are much more accurate than the other two series of tests. We can conclude that a smaller binning allows to retrieve a more accurate albedo. On the other hand, we found no correlation between the number of data points within the same data length and the possible underestimation or overestimation of the albedo.

4.4.3 Variation with stellar magnitude

As a second test, we produced two series of simulations with P∗ = 19 days, and varying the stellar magnitude, thus the instrumental noise. Table 4.1 reports the chosen stellar magnitudes and the associated noise values as they are predicted for CHEOPS. In Figure 4.6, we report the best fit albedo and the relative error bars as a function of the number of the observed stellar rotations. As a general result, we observed an increasing uncertainty as the light curve becomes more noisy. For the 14 and 17 ppm cases, we observe compatibility with Ag = 0.3 within 1σ. The 29 ppm trend shows several oscillations with respect to the input values. Though after 2.5 stellar rotations, it tends to stabilize on an albedo compatible with the injected one, again within 1σ.

4.4.4 Variation with orbital period

In this section, we test the effect of the orbital period on the albedo. As the orbital period increases, the semi-major axis increases as well, causing a decrement of the ratio RP/a. Moreover, the planetary phase curve amplitude is proportional to the square of RP/a. This suggests that increasing the semi-major axis decreases the intensity of the planetary signal. We now aim to estimate an upper limit on the orbital Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 79

Figure 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.

period, above which we can no longer detect the albedo.

We produced several series of tests, by fixing P∗ = 19 days and varying the orbital period from 3 to 15 days with steps of one day. The mock curves had three different data lengths, 30, 39 and 60 days. We performed the MCMC and report the recovered albedo and relative errors as a function of PP in Figure 4.7. The top panel shows the albedo trends, while the bottom one allows the inspection of the albedo error variation with PP. In the plots, we indicate as well RP/a as secondary horizontal axis. For the 30 day-long set of data, the estimated albedo is inaccurate for orbital periods longer than 4 days, when −3 RP/a is lower than 8 × 10 . In the 39 day case, longer observations allow measuring a reliable albedo −3 up to an orbital period of 8 days, corresponding to RP/a < 6 ∗ 10 . With 60 days of observation, the albedo is compatible with Ag = 0.3 within 1σ. The exception is represented by the cases with PP = 9, 11, and 12 days, when the best fit albedo and the input value are compatible solely within 3σ.

As an additional test we produced similar 39 days long mock curves without stellar activity. We performed again the MCMC and the albedo values are shown as well in Figure 4.7, as an orange line. Ag is always compatible, within 1σ, with the input value 0.3 until at PP = 8 days. We can deduce that in the 30 and 39 day-long observations the hurdle to the albedo detection is represented by the white noise.

When PP = 9 and 10 days, the albedo is again compatible with the injected value, though the error bar is higher and Ag is no longer exactly 0.3. Such behaviour is confirmed can be observed as well for all the series of tests with instrumental noise. We can justify it as a consequence of the stellar activity. When the orbital period is equal to 8, 9 and 10 days, it is almost equal to half the stellar rotational period, 19 days causing a degeneracy between the activity trend and reflected light component. To break this degeneracy, a higher number of observed stellar rotations becomes necessary. As a final statement, we also observe an increment of the albedo uncertainties with the orbital period. This occurs because the signal of the planet becomes lower, thus requiring longer observations to be identified. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 80

Figure 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. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 81

Figure 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.

4.4.5 Variation with planetary radius

In this section, we aim at exploring the effect of the planetary radius, RP on the albedo. We produced

39 days-long simulations again with Ag = 0.3 and P∗ = 19 days and we gradually decreased the planetary radius from 0.1 R∗ to 0.01 R∗. We applied the MCMC on the mock data and in Figure 4.8 we report the albedo as a function of the planetary radius. The retrieved Ag is well accurate if compared to the input value until RP = 0.02 R∗. On the other hand, the error-bars tend to increase as RP decreases, assuming a value of 0.12 when RP = 0.05R∗. We can conclude that the albedo is well measurable for planets bigger or as big as small Neptunes. For smaller planets, the uncertainties increase significantly and, when RP = 0.01R∗, the albedo is overestimated, with the best fit approaching the center of the prior. Thus, for Earth-like planets the signal is no longer detectable, being it well below the white noise.

As an additional test, we produced two series of mock curves, fixing RP = 0.1 R∗ and 0.05 R∗ and varying the albedo between 0.6 and 0. We then performed the MCMC to explore whether the albedo was always measurable for both a Jupiter and a Neptune-sized planets. Figure 4.9 reports the retrieved albedo compared to the input one. The red line shows the trend for Jupiter-sized planets and the blue one for Neptunes. While for Jupiters the albedo is precisely estimated with an error bar around 0.15, for

Neptunes Ag is slightly underestimated with respect to the injected value and the error-bars are larger,

0.1. Though, even for for this smaller planets the input and output Ag are well compatible between each others within 1σ. We can conclude that for a fixed radius, a variation of the albedo is almost hasno effect on the estimated error-bar. As the albedo is lower than 0.1, it no longer can be detected for a Neptune-sized planet. The posterior distribution assumes a Poissonian shape, with peak on 0.

4.4.6 Variation with stellar activity level

As a final test, we explored the dependence of the recovered albedo on stellar activity level. In doingso, we produced 39 days-long simulations with albedo 0.3, RP = 0.1 R∗ and P∗ = 19 days. We then varied the activity pattern, multiplying it by 100, 10, 5, and 0.1 and adding −99, −9, −4, and 0.9 respectively. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 82

Figure 4.9: Recovered albedo values as a function of the input values for the 39 day-long simulation and a stellar rotation of 19 days. The red data points represent a Jupiter-sized planet, 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. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 83

Figure 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.

In this way, we obtained realistic simulations with a wide range of activity levels. We performed again the MCMC and we report the albedo as a function of the logarithm of the activity level in Figure 4.10. In the plot, the first point corresponds to the test in absence of stellar noise. For all the cases, Ag is well detected, confirming the results obtained with the blind tests. The MCMC with GP can thus distinguish the reflected light component of a Jupiter-sized planet even for very active stars.

4.5 Towards a complete fitting model for phase light curves

In this section we aim at testing how our analysis would be affected in the case of a more realistic model. In doing so, we performed two different tests. The first one required to improve our planetary phase curve model by adding the previously ignored beaming effect and the ellipsoidal modulation. The new light curves become:

F F F F F∗, F total = P + B + E + spotted + noise . (4.3) F∗ F∗ F∗ F∗ F∗ F∗

The fit was performed with a modified version of the fitting tool presented insection 4.2. In detail, we introduced in the model both of the new effects and we added as additional free parameter the planetary mass. The selected prior for the mass was a Gaussian, with average 1 Jupiter mass and standard deviation 0.02. We produced a simulation which included all of the three effects in equation 4.3. The orbiting planet assumed the same properties as for the tests we presented in the previous section. Additionally, we imposed that the planet had MP = 1 Jupiter Mass. For the linear limb-darkening coefficient, u, and the gravity-darkening coefficient, g, we used typical values for Sun-like stars from Claret & Bloemen (2011) Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 84

(u = 0.6230, g = 0.3456). With this setup, we ran the MCMC using the new set of free parameters. We found as best fit albedo

Ag = 0.296 0.016. This result is strongly compatible, within 1σ, with the input albedo value, 0.3, and with the result obtained without including beaming effect and ellipsoidal modulation. Accounting for the two additional effects in the planetary phase curve model has no effect on the precision andaccuracy of the estimated albedo. Though, we highlight that the posterior distribution of the mass coincides with the prior, with the uncertainty on the best fit equal to the input 0.015. This means that with these data we have no possibility to constrain the planetary mass. The amplitudes of the beaming effect and the ellipsoidal modulation are, indeed, close to 2 ppm for a 1 Jupiter-mass planet, which is significantly lower than the instrumental noise we accounted for in the simulation. Therefore, in this analysis we cannot distinguish the beaming effect and the ellipsoidal modulation from the injected white noise. In the second test, we modified again the tool in Section 4.2 by accounting for the planetary radius and the semi-major axis (both in units of stellar radii) as free parameters. We imposed on them Gaussian priors, with standard deviations equal to their typical uncertainties in transit fits. We accounted for an uncertainty on the planetary radius of 0.005, while for the semi-major axis we used 0.05 (as reported in ground-based transits of hot-Jupiters around bright stars, see Turner et al. 2016). In this way, we properly accounted for the uncertainty on transit parameters entering the phase curve model which included again reflected light, beaming effect and ellipsoidal modulation. Afterwords, we fitthemock curve used for the first test using updated MCMC and we retrieved Ag = 0.302, slightly different from the output with RP and a fixed. The most significant difference was in the uncertainty, 0.040, higher than the previous value of 0.015. This is a reasonable result, because the reflected light component depends on both the planet radius 2 and the semi-major axis in the form of (RP/a) . Applying the theory of error propagation, we calculate that the ratio RP/a has itself a standard deviation of 0.0006, which decreases the precision of the albedo. Moreover, the posteriors on the planet radius and the semi-major axis are close to their priors. The reflected light component offers no contribution in improving RP and a. Moreover, even though the semi-major axis appears in the beaming effect and the ellipsoidal modulation, we showed that these phenomena are too small if compared to the instrumental noise. We have no opportunity of isolating them, in this specific case of a Jupiter planet, without knowing the mass. The error of the recovered albedo will thus depend on the uncertainties of both RP and a.

4.6 Test with CHEOPS gaps

In more realistic conditions, the satellite CHEOPS will perform observations with small gaps of several minutes. In this section, we aim at exploring how such characteristic of CHEOPS data might affect the detection of the albedo. We simulated a 39 days-long light curve accounting for the average percentage as predicted for CHEOPS observations, around 40% of the entire phase curve. The gaps were distributed in time in a realistic way (courtesy of M. Lendl). The new mock phase curve had a cadence of one minute and an instrumental noise of 155 ppm. The parameters of the planetary system were those reported in 4.7, for the planet, and in 4.3 for the star. For the stellar activity pattern, we considered again the parameters in Table 4.6. As we did in previous simulations, we additionally imposed Ag = 0.3,

RP = 0.1 R∗ and P∗ = 19 days. We binned the new simulation over two hours. Figure 4.11 compares the non-binned (black error- Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 85

Figure 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. Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 86

Figure 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. bars) and binned phase curves (blue error-bars). The plot shows that the gaps cover a time span of several minutes, a small fraction if compared to the orbital period of 3 days, and small as well with respect to the binning of 120 min. As a consequence, as we bin the phase curve, the error of the new data points are on average 40% larger than 14 ppm (the error of the 2 hr binned simulations with no . [+0.015] gaps). We performed the MCMC on this new mock curve, retrieving an albedo of 0 2898[−0.015]. This result is compatible within 1σ with the input value, 0.3, though lower than the result we obtained in = . [+0.003] σ absence of gaps. The stellar rotation is P∗ 19 001[−0.003] days, compatible with the input within 2 . We can conclude that the gaps in CHEOPS observations will not significantly change the reliability of the data analysis.

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

To verify that our code could work in more realistic conditions, we tested it on Kepler data. In particular, we chose a quiet star, Kepler-7, which has P∗ = 16.7 days and it is orbited by a 1.6RJ planet. After Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 87 inspecting all Kepler quarter observations, we chose the 10th quarter, for being one of the quietest. We later binned the data on two hours. A visual inspection of the light curve easily showed spot evolution, which suggested that the periodic kernel is most likely no longer a reliable model for the stellar activity.

Nonetheless, we fitted the simulation with the MCMC, retrieving Ag = 0.36. This result is strongly compatible with the value 0.35 reported in literature (Angerhausen et al. 2015). On the other hand, the GP is incapable of recovering the activity pattern. The rotational period of the star is 15.7 days, while in literature the estimated value is 16.7 days. The hyper-parameters p1 and p2 are as well physically poorly constrained. The final activity modulation looks aperiodic, even though the periodic kernel should just allow for periodic activity features. This result was predictable, since Kepler-7 shows evidence of an aperiodic activity distribution. To perform a more realistic test on the reliability of our MCMC, we thus needed to select a fully periodic star. We explored the McQuillan catalog of stars (McQuillan et al. 2014), which lists several Kepler stars with strong periodicities. Among them, we chose the star with Kepler identifier KIC 3643000, a rotationally variable star with a rotational period P∗ = 18.94 days. Among the data we extracted a fraction of Kepler observations in which almost no spot evolution could be visually observed. To these data, we added a planetary phase curve. The planet had the properties reported in table 4.7, as well as RP = 0.1 R∗, an orbital period of 3 days and an albedo of 0.3. We additionally binned the data again over 2 hs. The final phase curve had an average noise of 39 ppm, higher than the value we used in the tests so far. Though, being the curve almost 3 times longer than our usual simulations and since the star is a slow rotator, it should be possible to retrieve the albedo.

We performed the MCMC. The best fit albedo was Ag = 0.260.11, while the stellar rotational period +0.0071 ∗ = . resulted P 18 4818−0.00069. If compared to our simulations, we estimate an error-bar on the albedo twice as large. This could be due to the instrumental error, or even to the possibility of a not completely constrained stellar rotational period (which is lower than the period reported in the literature). Following the results of section 4.4.3 and with the opportunity of observing more than the two stellar rotations selected in this case, the precision could significantly increase. We can conclude that we could retrieve the injected albedo within 1σ and still with a reasonably small error-bar. Figure 4.12 shows the modified phase light curve in black, which includes boththe Kepler star observations and the planetary modulation. The red line is the best fit obtained with the MCMC, the orange represents the estimated activity modulation, while in green we report the planetary phase curve, shifted up by 1.

4.8 Conclusions

In this work, we explored how the stellar activity and the instrumental noise could affect the detection of the planet phase curve, thus the measurement of the albedo. To reach this scope, we produced simulated light curve, accounting for the planetary reflected light modulation, the stellar activity pattern andthe instrumental noise. To model the activity features we adopted the tool SOAP-T, with the planet switched off. The white noise had a standard deviation equal to CHEOPS average predicted noise, whichwill perform photometric observations of bright stars. To explore the measurability of the planetary albedo, we adopted an MCMC, which models the stellar activity pattern using a GP with periodic kernel. The free parameters of the MCMC are the albedo, an offset that represents the average of the stellar activity pattern, and the hyper-parameters oftheGP, Chapter 4. Detecting the albedo of exoplanets accounting for stellar activity 88 which include the stellar rotational period. As a first step, we performed 9 blind tests, which showed that our model can correctly recover the albedo for a wide range of stellar activity patterns. Then, we performed a series of tests, aimed at exploring the limits with which we could retrieve the albedo, while varying several parameters. Our main results were the following:

• To recover both of the parameters P∗ and Ag the MCMC with GP requires data covering at least one entire stellar rotation.

• Binning over 2 hs is sufficient to properly constrain slow rotators. In the case of fast rotators, our procedure requires a smaller sampling to retrieve more accurate results.

• For 6.5 mag stars, CHEOPS can measure the albedo of the exoplanets with a maximum orbital

period of 13 days for 39 days-long data. For shorter data the maximum PP will decrease.

• When the instrumental noise is 155 ppm per minute, we can measure the albedo of the smallest Neptunes.

We performed as well several additional tests, in which we varied the MCMC or modified the model for the simulations. We found that:

• Adding the beaming effect and the ellipsoidal modulation to the model does not affect the precision of the best fit albedo.

• Accounting for the transit fit uncertainties on RP and a enlarges the albedo error-bars.

• Our method works as well when we account for gaps in observations, as they are predicted for CHEOPS.

• We can estimate the albedo of a simulated exoplanet when we adopt a real activity modulation, in place of the simulated one with SOAP-T.

In summary, accounting for CHEOPS predicted noise level, even in presence of a high activity level, we can estimate the planetary albedo for short period planets and observations at least longer than one stellar rotation. This result can be extended to any photometric follow-up of a planetary system. In particular, once the stellar activity is properly modelled with the GP, the precision of the albedo measurement solely depends on the instrumental noise. Since our analysis tool excludes the transit feature (and eclipse), we can as-well apply it to observe non-transiting exoplanets (Crossfield et al. 2010), thus helping in characterizing their atmosphere. Chapter 5

Can we detect the stellar differential rotation of WASP-7 through the Rossiter-McLaughlin observations?

In this chapter we present the results of the second work we performed during our Ph.D, which was aimed at exploring the detectability of the stellar differential rotation through the RM effect and determine the precision of the retrieved geometry of the planetary system. As we explored in Chapter 2, stars rotate differentially rotate and such property can be measured through different techniques. One of the most popular methods requires to measure period variations from long term photometric observations (Reinhold & Gizon 2015) or through the analysis of chromospheric activity (Donahue et al. 1996) or through spots detection in transit observations (Zaleski et al. 2019). Alternatively, we can follow the time migration of individual features on Doppler maps (Donati et al. 2000; Barnes et al. 1998) or even study the effect of stellar differential rotation on line profiles (Reiners et al. 2000). Measuring the stellar differential rotation through the RM signal, the spectroscopic observation ofa transit, is an alternative methodology. As we described in Chapter 3, the stellar differential rotation, for stars rotating like the Sun, causes a decrement in the RM signal, which, when detectable, can affect the measurement of the projected spin-orbit angle (λ). The knowledge of λ is fundamental for testing theories on planet formation and evolution (see e.g. Triaud 2011; Albrecht et al. 2012b; Winn et al. 2010, among the others). So far, a wide diversity of values have been measured for such parameter, ranging from aligned (Winn et al. 2011) to highly misaligned systems (Addison et al. 2018), and also retrograde planets (e.g. Hébrard et al. 2011). Though, the possibility of an undetected stellar differential rotation might cause significant errors in the reported values in literature, which might become relevant asthe spectroscopic precision increases. In the past there have been attempts of modeling the stellar differential rotation effect on RM signal (Gaudi & Winn 2007; Hirano et al. 2011). More recently, some works tried to measure it through RM as well, though with no conclusive clear detection. Albrecht et al. (2012a) included the differential rotation in the model of WASP-7, though when fitting the RM signal they could not attribute much reliability to their estimate of the parameter α. Cegla et al. (2016a) did detect the stellar differential rotation of

89 Chapter 5. Stellar Differential Rotation 90

Table 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)

Stellar Properties

R∗ [R⊙] stellar radius 1.432 Teﬀ [K] effective temperature 6520 P∗ [ days] stellar rotational period 5.17 −1 v∗ sin i∗ [ km s ] projected rotational velocity 14 ◦ i∗ [ ] stellar axis inclination 90 u1 linear limb darkening 0.2 u2 quadratic limb darkening 0.3 Planetary Properties

Rp [RJ] planet radius 1.33 Pp[ days] planet orbital period 4.95 a [AU] semi-major axis 0.0617 ◦ iP [ ] orbital inclination 87.03 λ [◦] spin-orbit angle 86

HD 189733, though they reported a high level of uncertainty on α (between 0.28 and 0.86). If in the case of HD 189733 α could vaguely be determined, why this was not the case of WASP-7? And still, why the errorbars on α remain large even in the case of HARPS data (for HD 189733). To answer this question, as model for the RM signal we adopted the updated tool SOAP3.0, presented in Chapter 3, which included as well the stellar differential rotation. Then, we decided to explore which sources of noise prevented and still could hurdle the detection of stellar differential rotation through the RM signal and lead to an incorrect estimation of the system geometry. As reference example we decided to choose WASP-7b, which seemed to be a favorable target, since it transits several stellar latitudes. Apart from the instrumental noise, we analyzed the effect of several noise sources with stellar origins: the center-to-limb variation of the convective blue-shift (Dravins et al. 2017; Cegla et al. 2016a; Shporer & Brown 2011), granulation, oscillations and the convective broadening due to granulation (Kupka & Muthsam 2017, and references therein). On top of this we aimed at determining the advantages offered by a more stable and precise spectrograph, working on a larger aperture telescope. ESPRESSO (Pepe et al. 2014) could be the best approach for this, and we mainly focused on it. Though, other instruments might help, such as EXPRES (Louis et al. 2019) and NEID (Schwab et al. 2016). In Sect. 5.1 we present our analysis of the observed RM signal of WASP-7b, reported by Albrecht et al. (2012a). Sect. 5.2 describes the simulations of the synthetic WASP-7b RM signal. In Sect. 5.3, we explore the effect of several sources of noise on the detection of differential rotation. Finally, inSect. 5.4, we identify the potential noise sources which prevented the detection of the stellar differential rotation in the past, and comment on the improvements offered by future instruments. The results of this work were published in Serrano et al. (2020).

5.1 WASP-7 and its hot-Jupiter

When the spin-orbit angle is close to 90◦, it is more likely that a transiting planet occults many stellar latitudes, which makes it easier to determine the latitudinal dependence of the stellar differential rotation. Chapter 5. Stellar Differential Rotation 91

Figure 5.1: Fit of the observed data of WASP-7b using the updated SOAP3.0 with differential rotation. 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 respectto the best fit. DR stands for differential rotation. Chapter 5. Stellar Differential Rotation 92

Due to its close to polar orbit, the hot-Jupiter WASP-7b is one of the most promising target for measuring α through RM (see Table 5.1 for the main properties of the system). It orbits an F5V-type star with apparent visual magnitude 9.51. Albrecht et al. (2012a) performed observations of WASP-7b RM signal with a cadence of 10 minutes (min.), adopting the Planet Finder Spectrograph (PFS). This instruments is reported to have a resolution of R = 38000 and a maximum RV precision of 1.25 m s −1 (Crane 2010). In the case of WASP-7, the observed RV precision raises to 5.6 m s −1, due to the relatively high magnitude and the fast rotation of the star (Bouchy et al. 2001). Albrecht et al. (2012a) performed 2 ◦ ◦ a χ minimization on the observed RM, estimating a spin-orbit misalignment λ0 = 86 6 and a −1 projected rotational velocity (v∗ sin i∗)0 = 14.0 2.0 km s (Albrecht et al. 2012a). The authors found

α0 = 0.45 0.1 when the prior on α was restricted to 0.1 − 0.5. Afterwards, they significantly expanded the α prior and retrieved α = 0.9 (no error was reported in this case), which would indicate that the pole rotates extremely slower with respect to the equator. This condition is very extreme and has never been observed so far, especially for F stars (see Reiners et al. 2013; Balona & Abedigamba 2016), so they expressed doubts on the values they obtained for α. We analyzed WASP-7 data by Albrecht et al. (2012a) with the updated SOAP3.0. We first fitted a linear polynomial to the out-of-transit data to remove the effect of the Keplerian orbit and non-occulted star-spots from the RV data. We then performed a χ2 minimization on the RM signal, using the updated SOAP3.0 model. In our fit, we used the resolution of the PFS and as FWHM oftheCCF 10.3 km s−1, which accounts for both the instrumental and convective broadening. As limb darkening parameters, we approximated those reported in Claret (2000) for F-type stars (u1 = 0.2 and u2 = 0.3), considering that

(Albrecht et al. 2012a) expressed doubts on the coefficients they estimated (u1 = 0.17 and u2 = 0.38). As mentioned in Hirano et al. (2011), accounting for the stellar differential rotation allows to break the degeneracy between the stellar period of rotation and the stellar inclination. Fitting both P∗ and i∗ can ◦ be reasonable after α is well constrained. For i∗ , 90 , the amplitude of the RM decreases, because the planet transits a lower number of latitudes and, in the case of WASP-7, it will cross areas of the star which are rotating more slowly. This will degrade the precision with which α is retrieved. Nonetheless, ◦ the objective of our work is to verify if we can measure α. For this reason, we used i∗ = 90 , as Albrecht et al. (2012a) estimated using the method by Schlaufman (2010). In the fit, we fixed all the parameters to the values in Table 5.1, except for v∗ sin i∗, λ and α. We varied the free parameters in the intervals: λ ◦ ◦ ◦ −1 −1 from 61 to 111 with a step of 0.4 , v∗ sin i∗ from 8 to 20 km s with a step of 0.25 m s , and α between 0 and 1, with a 0.033 step. These steps were chosen because we saw that the fit remained unchanged if they were smaller. Throughout the work, we report the 1σ uncertainties on the best fit values. When the error-bar was large enough to go beyond the explored range of values for α ([0, 1]), we imposed as errors the differences of the best fit with respect to the upper or lower limit. To test the reliability of the χ2 minimization, we used it to fit a mock simulation of a WASP-7b RM with α = 0.6 and without instrumental noise. The best-fit parameters were exactly equal to the injected ones. α = . +0.25 = . . −1 λ = ◦ ◦ The fit on real data gave 0 75−0.41, v∗ sin i∗ 14 5 1 0 km s and 90 4 . This result is compatible with that of Albrecht et al. (2012a) within 1σ. In contrary to them, we find smaller error-bars on the v∗ sin i∗ and λ and a larger error on α. In Figure 5.1, we report the observed data and our best-fit model. In the lower frame, we also show the residuals, which have a maximum amplitudeof 25 m s−1 but an RMS of 11 m s−1, higher than the instrumental noise of 5.6 m s−1. This suggests the presence of an extra unexplained noise of 9.8 m s−1. At 3σ, we cannot constrain α and the estimated Chapter 5. Stellar Differential Rotation 93 result is higher than what is expected and has been observed for stars of this type.

5.2 Simulations

In this section, we present how we modelled the mock RM signals on which we tested the detectability of the stellar differential rotation through RM observations. We produced simulations of WASP-7b RM signal with a 6 min exposure time and spectral resolution R = 140000, as offered by the spectrograph ESPRESSO. This instrument promises to achieve a precision of 10 cm s−1 for bright stars. For the RM analysis, its greatest advantage comes from the collective power of the 8 m aperture Very Large Telescope (VLT). This allows us to obtain more data points at a given signal-to-noise ratio than a 4 m class telescope, when sampling the RM of bright stars. To have a first estimate of the ESPRESSO RV precision with good atmospheric conditions, we used the ESPRESSO Exposure Time Calculator (ETC), which performs estimates for G, K and M stars. To produce a more realistic simulation, we modelled the mock RM signals assuming as local CCF FWHM 6.4 km s−1. This width is smaller than the one adopted for fitting the observations, because the instrumental FWHM is lower.

All simulations include stellar differential rotation as modelled with the updated version of SOAP3.0. We also added instrumental noise modelled a white noise with standard deviation equal to the precision of the instrument. We considered several levels of white noise. The first two noise levels, 2 m s−1 and 3 m s−1, are determined considering the stellar type of WASP-7, which is an F star. We estimated these values following the method described in Bouchy et al. (2001). The instrumental noise depends on the stellar type and on the rotational velocity of the star, v∗ sin i∗. To estimate it, a fast way is to scale the predicted noise for a K star with the same magnitude as WASP-7, 0.32 m s−1, accounting for the quality factors of a non rotating K star and of a rotating F star (reported in Table 2 in Bouchy et al. (2001)). For averagely rotating F stars (8 km s−1), we estimated 2 m s−1, while for a star as fast as WASP-7 we obtained 3 m s−1. Besides these, we also considered three lower noise levels, 0.5, 1 and 1.5 m s−1, that would correspond to other stellar types, G and K. Finally, we also tested other scenarios with increased instrumental noise of 5 m s−1 and 7 m s−1. 5 m s−1 is the predicted precision for HARPS with a 6 min exposure time for a star similar to WASP-7 (e.g. Hellier et al. 2019), while 7 m s−1 is the noise for the PFS spectrograph and other less precise instruments.

Moreover, we performed some tests accounting for the center-to-limb variation of the convective blueshift, modelled as described for the updated version of SOAP3.0 in the solar 0G case (Chapter 3). To test the effect of granulation and pressure mode oscillations, we modelled them by generating synthetic RV measurements with the components proposed by Dumusque et al. (2011). We considered Harvey-like functions to model granulation, mesogranulation, and supergranulation (Harvey 1984) and a Lorentzian function to model the frequency bump due to oscillations (Lefebvre et al. 2008, e.g.). As time scales and amplitudes we used the values determined for β Hyi (Dumusque et al. 2011, see Table 2 in) as it has the closest spectral type to WASP-7. The model for the power spectral density is finally inverted to obtain RV values at the simulated times of the RM signal. In detail, we produced two RV sets, one only accounting for granulation, and the second one with both granulation and oscillations. Chapter 5. Stellar Differential Rotation 94

Table 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 ofthe 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.

Simulation Fit with α = 0 Fit varying α input α v∗ sin i∗ λ v∗ sin i∗ λ α . . . −1 . ◦ . ◦ . . −1 . ◦ . ◦ . +0.20 0 1 13 8 0 4 km s 84 4 0 8 14 0 0 5 km s 86 0 1 2 0 13−0.13 0.2 13.5 0.3 km s−1 84.6◦ 1.5◦ 14.0 0.3 km s−1 86.0◦ 1.8◦ 0.23 0.04 0.3 13.0 0.3 km s−1 84.2◦ 0.8◦ 14.0 0.3 km s−1 85.2◦ 1.2◦ 0.30 0.10 0.4 12.8 0.3 km s−1 84.2◦ 1.1◦ 14.0 0.8 km s−1 86.0◦ 1.6◦ 0.40 0.05 0.5 12.8 0.3 km s−1 81.3◦ 0.6◦ 14.0 0.8 km s−1 86.0◦ 1.8◦ 0.50 0.10 0.6 12.5 0.3 km s−1 80.7◦ 1.2◦ 14.0 0.5 km s−1 85.2◦ 1.6◦ 0.60 0.10

5.3 Results

In this section, we present all the tests we performed on mock RMs of WASP-7b and the main results we obtained.

5.3.1 Minimum detectable α

To visualize how the differential rotation alone affects the RM signal of a planet like WASP-7b, in Figure 5.2, we show different models of WASP-7b for α from 0.0 to 1.0. In the bottom, we represent the residuals of each model with respect to the rigid rotation case (α = 0). We also add a schematic view of the system inside the plot. This way we show which areas of the stellar disc are transited by the planet. Since the spin-orbit angle is close to polar configuration, the planet transits several latitudes, sothe stellar differential rotation significantly affects the RM signal. Inspecting the mock RMsinFigure 5.2, we see that the effect of the differential rotation is stronger on the transit ingress and egressandit decreases as we get closer to the mid-transit time. At mid-transit, the difference between α = 0 and 1 is 20 m s−1, whereas between α = 0 and α = 0.6, it is 10 m s−1. Our objective is to verify if this difference is detectable. From now on, we will consider as maximum value of α = 0.6, because a stronger stellar differential rotation is unlikely and the average α for F type stars is much lower, around 0.1 − 0.2. We produced RM simulations which included a noise level of 2 m s−1 and various α values, 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6. We fitted them twice, once assuming rigid rotation and the second time accounting for the stellar differential rotation. The results are in Table 5.2, on the left panel are the tests with α fixed to 0 in the fitting model, while on the right α is a free parameter of the χ2.

When we assume rigid rotation, as α increases, v∗ sin i∗ and λ estimates become more inaccurate and ◦ −1 less precise with respect to the input values, λ0 = 86 and (v∗ sin i∗)0 = 14 km s . As α > 0.3, both the recovered v∗ sin i∗ and λ become incompatible with their true input values by at least 2σ. For the extreme ◦ −1 case of α = 0.6, λ diverges from the input value by 5.3 , while the v∗ sin i∗ diverges by 1.5 km s . For lower α, the differences between the best fit and the input values are smaller; however, the twovalues are incompatible within the uncertainties. On the other hand, as we see in the right panel of Table 5.2, when α is a free parameter and the input α ≥ 0.2, all of the parameters are retrieved. For smaller input α, our recovery could not rule out rigid body rotation within 1σ. For input α < 0.2, a star with rigid rotation can fit the data just as well as that with differential rotation. These results show that if the differential rotation is not accounted for in the fitting model, we obtain Chapter 5. Stellar Differential Rotation 95

Figure 5.2: Top: RM simulations for the planet WASP-7b and six different values of α, 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. Chapter 5. Stellar Differential Rotation 96

Simulation with DR and Simulation with DR and 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 [m s 1] [m s 1]

Simulation with DR, CB, and Simulation with DR, CB and 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 [m s 1] [m s 1]

Simulation with DR, gran and Simulation with DR, gran and 1.0 1.2

0.8 1.0 0.8 0.6 0.6 0.4 0.4

0.2 0.2

0.0 0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 [m s 1] [m s 1]

Simulation with DR, gran, oscill and Simulation with DR, gran, oscill and 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 [m s 1] [m s 1] Figure 5.3: Best fit α 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 to α = 0.3, on the right side those for α = 0.6. The fit accounts only for the differential rotation in the model. Chapter 5. Stellar Differential Rotation 97

Table 5.3: Results of our fitting procedure applied on the simulations of WASP-7 RM signal: 1) including instrumental noise and differential rotation (DR); 2) including 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.

Simulation DR DR + CB noise v∗ sin i∗ λ α v∗ sin i∗ λ α 0.50 m s−1 14.0 0.1 km s−1 86.0◦ 0.4◦ 0.30 0.03 13.8 0.3 km s−1 86.0◦ 0.8◦ 0.23 0.03 1.00 m s−1 14.0 0.3 km s−1 86.0◦ 0.8◦ 0.30 0.03 13.8 0.3 km s−1 86.0◦ 0.8◦ 0.23 0.10 2.00 m s−1 14.0 0.3 km s−1 85.2◦ 1.2◦ 0.30 0.10 14.0 0.5 km s−1 85.2◦ 1.2◦ 0.30 0.13 3.00 m s−1 14.0 1.0 km s−1 84.2◦ 1.6◦ 0.27 0.13 14.0 1.0 km s−1 84.8◦ 1.6◦ 0.27 0.20 5.00 m s−1 14.0 1.5 km s−1 84.8◦ 1.6◦ 0.30 0.23 14.0 1.5 km s−1 86.6◦ 2.4◦ 0.30 0.23 . −1 . . −1 . ◦ . ◦ . . . . −1 . ◦ . ◦ . +0.40 7 00 m s 14 0 1 5 km s 86 0 4 0 0 40 0 24 13 3 2 3 km s 86 0 3 2 0 130.13

Table 5.4: Same as in Table 5.3 but for α = 0.6.

Simulation DR DR + CB noise v∗ sin i∗ λ α v∗ sin i∗ λ α 0.50 m s−1 14.0 0.3 km s−1 86.0◦ 0.4◦ 0.60 0.03 13.8 0.3 km s−1 86.0◦ 0.4◦ 0.53 0.03 1.00 m s−1 14.0 0.3 km s−1 86.0◦ 0.8◦ 0.60 0.03 13.8 0.3 km s−1 86.0◦ 0.8◦ 0.53 0.03 2.00 m s−1 14.0 0.8 km s−1 85.2◦ 1.6◦ 0.60 0.10 14.0 0.8 km s−1 85.2◦ 1.2◦ 0.60 0.13 3.00 m s−1 14.0 0.8 km s−1 84.8◦ 1.6◦ 0.60 0.17 14.0 1.3 km s−1 84.8◦ 1.6◦ 0.57 0.23 5.00 m s−1 14.0 1.5 km s−1 86.6◦ 1.6◦ 0.63 0.28 14.0 0.8 km s−1 85.6◦ 2.4◦ 0.60 0.27 7.00 m s−1 14.0 1.5 km s−1 86.0◦ 3.2◦ 0.67 0.42 13.3 1.5 km s−1 86.0◦ 3.2◦ 0.47 0.40

systematically biased estimates of the obliquity of the system. Moreover, without including stellar noise in the mock RMs, when the instrumental noise is 2 m s−1, the lowest measurable α for a planetary system like WASP-7 is 0.2.

5.3.2 Varying the instrumental noise

Afterwords, we performed four different series of tests as follows. We produced simulations ofWASP- 7b RM with α = 0.3 and 0.6. The choice of selecting both 0.3 and 0.6 is justified by the attempt to understand if the differential rotation can be detected at all noise levels for a common valueof α as well as in the case in which the signal is stronger. We then added different levels of white noise to test the detectability of α. In the first series of tests, no other noise source was included; in the second,we

Table 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 injected α was 0.3 and the fit was performed accounting for differential rotation.

Simulation DR + gran DR + gran + oscill noise v∗ sin i∗ λ α v∗ sin i∗ λ α 0.50 m s−1 13.8 0.3 km s−1 85.4◦ 0.4◦ 0.23 0.03 14.0 0.3 km s−1 85.8◦ 0.4◦ 0.30 0.07 1.00 m s−1 13.8 0.3 km s−1 85.0◦ 0.4◦ 0.23 0.10 14.0 0.5 km s−1 85.8◦ 0.8◦ 0.33 0.13 2.00 m s−1 14.0 0.5 km s−1 84.4◦ 1.2◦ 0.30 0.10 14.0 0.8 km s−1 84.2◦ 1.2◦ 0.30 0.17 . −1 . . −1 . ◦ . ◦ . +0.27 . . −1 . ◦ . ◦ . . 3 00 m s 13 5 0 8 km s 83 6 2 4 0 13−0.13 14 0 1 3 km s 84 4 2 4 0 30 0 23 . −1 . . −1 . ◦ . ◦ . . . . −1 . ◦ . ◦ . +0.33 5 00 m s 14 0 1 5 km s 84 8 3 2 0 30 0 30 14 0 1 3 km s 84 2 3 2 0 30−0.30 . −1 . . −1 . ◦ . ◦ . +0.30 . . −1 . ◦ . ◦ . +0.40 7 00 m s 13 3 1 5 km s 84 2 3 2 0 13−0.13 14 0 2 3 km s 84 2 3 2 0 13−0.13 Chapter 5. Stellar Differential Rotation 98

Table 5.6: Same as in Table 5.5 but for α = 0.6.

Simulation DR + gran DR + gran + oscill noise v∗ sin i∗ λ α v∗ sin i∗ λ α 0.50 m s−1 13.8 0.3 km s−1 85.4◦ 0.4◦ 0.53 0.03 14.0 0.3 km s−1 85.8◦ 0.4◦ 0.60 0.07 1.00 m s−1 14.0 0.3 km s−1 85.0◦ 0.8◦ 0.53 0.07 14.0 0.5 km s−1 85.6◦ 1.2◦ 0.63 0.10 2.00 m s−1 14.0 0.8 km s−1 84.8◦ 1.2◦ 0.60 0.13 14.0 0.8 km s−1 84.2◦ 1.2◦ 0.60 0.17 3.00 m s−1 14.0 0.5 km s−1 84.8◦ 2.4◦ 0.60 0.13 14.0 0.8 km s−1 84.2◦ 1.6◦ 0.60 0.17 5.00 m s−1 14.0 1.5 km s−1 84.8◦ 2.4◦ 0.60 0.30 14.0 1.5 km s−1 86.6◦ 2.4◦ 0.63 0.33 7.00 m s−1 14.0 1.5 km s−1 84.2◦ 3.2◦ 0.47 0.30 13.3 1.5 km s−1 84.2◦ 3.6◦ 0.47 0.30

Table 5.7: Results of our fitting procedure applied on the simulations of WASP-7 RM signal: 1) including instrumental noise (2 m s−1), DR and different levels of granulation; 2) including instrumental noise (2 m s−1), DR and different levels of granulation (gran) and oscillation (oscill). The injected α was 0.3 and the fit was performed accounting for differential rotation.

Simulation DR + gran DR + gran + oscill gran v∗ sin i∗ λ α v∗ sin i∗ λ α 0.33 14.0 0.5 km s−1 85.2◦ 1.2◦ 0.33 0.07 14.0 0.5 km s−1 85.2◦ 1.2◦ 0.30 0.13 0.5 14.0 0.5 km s−1 85.2◦ 1.2◦ 0.30 0.10 14.0 0.5 km s−1 85.2◦ 1.2◦ 0.30 0.13 1 14.0 0.5 km s−1 84.4◦ 1.2◦ 0.30 0.10 14.0 0.8 km s−1 84.2◦ 1.2◦ 0.30 0.17 2 13.5 0.5 km s−1 83.0◦ 2.4◦ 0.13 0.13 13.5 0.8 km s−1 83.6◦ 1.2◦ 0.33 0.17 3 13.3 0.8 km s−1 83.0◦ 2.0◦ 0.10 0.10 14.0 0.8 km s−1 83.6◦ 2.4◦ 0.33 0.20

Table 5.8: Same as in Table 5.7 but for α = 0.6.

Simulation DR + gran DR + gran + oscill gran v∗ sin i∗ λ α v∗ sin i∗ λ α 0.33 14.0 0.5 km s−1 85.2◦ 1.6◦ 0.60 0.10 14.0 0.5 km s−1 85.2◦ 1.6◦ 0.60 0.13 0.50 14.0 0.5 km s−1 85.2◦ 1.6◦ 0.60 0.13 14.0 0.5 km s−1 85.2◦ 1.6◦ 0.60 0.13 1 14.0 0.8 km s−1 84.8◦ 1.2◦ 0.60 0.13 14.0 0.8 km s−1 84.2◦ 1.2◦ 0.60 0.17 2 13.3 0.8 km s−1 83.6◦ 1.6◦ 0.43 0.10 14.0 0.8 km s−1 84.2◦ 1.6◦ 0.63 0.20 3 13.3 0.8 km s−1 82.4◦ 2.4◦ 0.40 0.10 14.0 0.8 km s−1 83.6◦ 2.4◦ 0.63 0.20 Chapter 5. Stellar Differential Rotation 99

WASP-7b

0 ] s / 50 m [

V R

100

Fit with DR Simulation with DR, gran and oscill ] s / 0.10 0.05 0.00 0.05 0.10

m 5 [

s l 0 a u

d 5 i s e r 0.10 0.05 0.00 0.05 0.10 time [days]

Figure 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. Chapter 5. Stellar Differential Rotation 100 accounted for the solar 0G center-to-limb CB in the mock RMs; in the third we replaced the CB with granulation (as estimated for β Hyi); while in the last we included both granulation and oscillations in the simulated data, since they are usually coupled. This way, we could analyze how each of the mentioned phenomena can influence the estimate of α as the instrumental precision varies.

The best fit values for the three free parameters, λ, α and v∗ sin i∗ are reported in Tables 5.3, 5.4, 5.5 and 5.6. In particular, in Tables 5.3 and 5.4, we compare the best fits of the simulations only with instrumental noise (in the left side) with respect to those in presence of CB. In the left panels of Tables 5.5 and 5.6, we show the results for the simulations only with granulation, while in the right we report the retrieved parameters for mock data with granulation and oscillations. Figure 5.3 displays the results in Tables 5.3, 5.4, 5.5 and 5.6. In particular, we report the dependence of the recovered α and the relative error-bars as a function of different instrumental noises. Inspecting the results, we can highlight that, without considering center-to-limb CB, granulation and oscillations, α is always measurable for instrumental noise ≤ 2 m s−1. When α = 0.3 and the white noise is 3 m s−1, the error-bar is already large if compared to the input value, rendering the detection less constrained. Injecting the center-to-limb CB, which is on a level of ∼ 1.75 m s−1, underestimates the recovered α as the instrumental precision is ≤ 1 m s−1. Nonetheless, in general the results are compatible with the input values by 1 − 2σ for the three parameters explored, α, λ and v∗ sin i∗. When the instrumental noise is 2 − 3 m s−1 the retrieved α is not any more affected by CB. A low center-to-limb CB is justified bythe fact that the star is a fast rotator and the effect of CB is diluted. This result is in line with the findings of Cegla et al. (2016a), who mentioned that the CB dominates the residuals only for slow rotators. The injection of βHyi granulation causes similar results to those we obtain for CB, since they are still on a level of 1.75 m s−1, while the addition of the oscillations to the granulation introduces a red noise of 2.95 m s−1 and increases the minimum χ2 with respect to the cases only with granulation. This impacts the accuracy and the uncertainty on the fit (see Tables 5.5 and 5.6). In particular, it enlarges the error-bar on α and changes the best-fit results for the spin-orbit angle. Nonetheless, the retrieved alpha is closer to the injected one. To understand why just accounting for granulation affects more the results than having both granulation and oscillations, we inspected the mock RMs for the two cases. We observed that the granulation signal shows a lower frequency variation when compared to the oscillations. As a result, this changes the shape of the data, favouring lower values of α. The additional signal from oscillations compensates for this effect, enlarging the uncertainty on α. Finally, for all the performed tests, as the instrumental noise reaches 5 m s−1 and 7 m s−1, the error on α becomes large and, especially for 7 m s−1, the detection of the differential rotation is no longer significant because the error-bar is equivalent to more than 50% of thebestfit α. The error-bars do not significantly change when we add the stellar noises, indicating that white noise is the most determining factor for constraining the level of stellar differential rotation. The results at 5 and 7 m s−1 are in line with the test we performed on WASP-7b observations, where the instrumental noise was 5.6 m s−1. For comparison, we performed another test considering the resolution of PFS. We modelled the WASP-7b RM at the same phases as the real observations by Albrecht et al. (2012a). We included the effect of differential rotation in the mock data with α = 0.45, according to the best estimate of Albrecht et al. (2012a). We also added a white noise with standard deviation equal to the average uncertainties in the real observations (5.6 m s−1). Then, we fitted this simulation and we obtained λ = 84.4◦ 1.6◦, −1 v∗ sin i∗ = 13.5 2.0 km s and α = 0.47 0.45. Comparing the error-bars of this test to those for the ESPRESSO case with 5 and 7 m s−1 and those for the test on real data, we observe a similar behaviour. Chapter 5. Stellar Differential Rotation 101

The different resolution does not significantly affect the uncertainties. We can conclude that α can be constrained if the white noise is 2 m s−1 and and the stellar noise sources are lower than the instrumental noise. With 3 m s−1, α is detectable only if it is higher than 0.3 but when the instrumental precision is 5 m s−1, the stellar differential rotation can no longer be constrained and a rigid body rotation cannot be ruled out. To visualize an example of the fits so far performed, we report in Figure 5.4 a plot of the simulation which included the differential rotation with α = 0.6 and β Hyi-like granulation and oscillations, compared to its best fit model. We see that considering these noises sources, the residuals are around8ms−1, lower than the previous observations of WASP-7b, where the residuals are 25 m s−1.

5.3.3 Varying granulation and oscillations

β Hyi is a G9 star and its granulation and oscillation-induced RV variability are predicted to be lower than those of WASP-7, because the frequencies of these phenomena depend on the spectral type and are higher for F-type stars (around 60% higher, Dravins 1990). In this section, we explore at what level the granulation and oscillations start to affect the detection of α if the instrumental noise is 2 m s−1. As a first test, to account for different levels of granulation, we varied its amplitude, multiplying the simulated RVs by an amplification factor chosen among 0.33, 0.5, 1, 2 and 3. This is equivalent to creating an extra red noise on a level of 0.58, 0.875, 1.75, 3.5 and 5.25 m s−1 in the case of granulation, and of 0.98, 1.475, 2.95, 5.9 and 8.85 m s−1. We produced RM simulations including instrumental noise of 2 m s−1 and differential rotation for both α = 0.3 and 0.6. We then added the granulation and fitted the seven mock RMs we obtained. The results are in the left sides of Tables 5.7 and 5.8. In the first row of Figure 5.5, we also show the dependence of the error-bar of the best fit α as a function of the amplification factor. As far as the amplification factor is lower than or equal to one, the granulation remains below the instrumental noise. The geometry of the system is slightly affected, with a maximum λ variation of 1.6◦ with respect to the input value. When the granulation is larger than the instrumental noise, all of the free parameters tend to decrease and the α significantly diverges from the starting value. We repeated these tests for simulations including both granulation and oscillations together. The results of the fits are in the right sides of Tables 5.7 and 5.8. In the second row of Figure 5.5, we show the dependence on the error-bar of the best fit α as a function of the amplification factor. As we noticed in the previous section, the addition of oscillations seems to compensate the changes induced by granulation. Although the error-bars on α are almost doubled and we registered a best χ2 higher than in the case only with granulation. The reasons for which adding the oscillations no longer affects the accuracy of α are similar to those described in section 5.3.2. We can conclude that the granulation alone starts to affect as it becomes higher than that estimated for β Hyi by Dumusque et al. (2011), while including both β Hyi-like granulation and oscillations affects the precision with which we recover the differential rotation. Note that when we multiply by3the β Hyi granulation and oscillations, the additional stellar noise is 8.85 m s−1, close to the extra RMS of 9.8 m s−1 we estimated fitting WASP-7b observations.

5.3.4 Varying the exposure time

All the previous simulations show that with an instrumental precision of 2 m s−1 we have the opportunity to remove the white noise as a significant constrain in detecting the stellar differential rotation. Still, Chapter 5. Stellar Differential Rotation 102

Simulation with DR, gran, and Simulation with DR, gran, and 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 Gran level Gran level

Simulation with DR, gran, oscill and Simulation with DR, gran, oscill and 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 Gran and oscill level Gran and oscill level Figure 5.5: Best fit α 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), granulation (gran) and instrumental noise (σ), in the second row for simulations including the oscillations (oscill) too. On the left side, plots relative to α = 0.3, on the right side those for α = 0.6. The fit accounts only for the differential rotation in the model. Chapter 5. Stellar Differential Rotation 103

Table 5.9: Results of our fitting procedure applied on the simulations of WASP-7 RM signal which include a different injected FWHM, instrumental noise σ = 2 m s−1 and differential rotation (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

Simulation α = 0.3 α = 0.6 FWHM v∗ sin i∗ λ α v∗ sin i∗ λ α 6.0 km s−1 14.3 0.3 km s−1 86.0◦ 1.2◦ 0.33 0.10 14.3 0.8 km s−1 86.0◦ 1.2◦ 0.63 0.13 6.8 km s−1 14.0 0.8 km s−1 85.2◦ 1.2◦ 0.33 0.13 14.0 0.5 km s−1 85.2◦ 2.4◦ 0.63 0.06 7.2 km s−1 13.8 0.3 km s−1 85.2◦ 0.8◦ 0.30 0.13 13.8 0.3 km s−1 85.2◦ 1.2◦ 0.60 0.07 7.6 km s−1 13.8 0.8 km s−1 85.2◦ 1.2◦ 0.33 0.10 13.5 0.5 km s−1 85.2◦ 1.2◦ 0.60 0.13 8.0 km s−1 13.5 0.3 km s−1 85.2◦ 1.2◦ 0.33 0.10 13.5 0.5 km s−1 85.2◦ 1.6◦ 0.63 0.06

we mentioned that with 6 min. of exposure, the ESPRESSO predicted precision for WASP-7 is 3 m s−1, due to its fast rotational velocity. One way to decrease the instrumental noise is observing for a longer exposure time. To examine how this could affect the final fit, we performed two tests, with an exposure time of 10 and 15 min, respectively. For the two cases, we estimated an ESPRESSO instrumental precision of 2.78 m s−1 and 2.27 m s−1 respectively. We produced again mock curves for WASP-7b, accounting for these uncertainties and decreasing the number of data points, to consider the longer exposure time. In the fits, we only accounted for the stellar differential rotation. We fitted themock −1 ◦ ◦ curves and obtained: 1) α = 0.590.22, v∗ sin i∗ = 14.02.0 km s and λ = 86.0 1.6 , 2) α = 0.500.09, −1 ◦ ◦ v∗ sin i∗ = 13.5 0.5 km s and λ = 86.0 2.4 . For both of the cases, measuring the differential rotation is possible. When the exposure time is 10 min., α has a larger error-bar, which makes the determination of the stellar differential rotation less precise, even though more accurate. With 15 min of exposure, α is not any more accurate, however still compatible with the input value within 2σ. The choice to account for longer exposure times is justified by the attempt of overcoming the presence of granulation and oscillations. This is not entirely true. For example Dumusque et al. (2011) say that for a G star 10 min. of integration are required to lower the oscillation to 50 cm s−1 and at least 30 min to reduce the significance of granulation to the same level. F stars have stronger granulation and oscillations than G stars (Dravins 1990), thus integrating over 10 or 15 min is not enough to get rid of these phenomena. As mentioned in Chaplin et al. (2019), for an F5V star as WASP-7 102 min. of exposure are required to reduce the oscillation to 0.1 m s−1. In the previous sections, we also find that as the granulation and oscillations contribution is higher than those estimated for β Hyi, these phenomena are no longer negligible with a precision of 2 m s−1, because they inject an extra RMS, and affect the estimation of λ and α. For all these reasons, when analyzing observations of WASP-7b, even with longer exposure times, ignoring these phenomena will determine systematic biases in the estimation of α.

5.3.5 The effect of convective broadening

Another consequence of convection is a broadening of the CCF FWHM, which amounts to ≥ 6 km s−1 for F type stars (Doyle et al. 2014). This phenomenon is referred to as macro-turbulence. To test whether underestimating or overestimating it may affect the measurement of differential rotation via theRM effect, we produced simulations of the RM signal for WASP-7b. In the mock RMs, we accounted forthe stellar differential rotation, fixing α = 0.3 and 0.6, and 2 m s−1 for the instrumental noise. We varied the FWHM of the injected CCF, assigning values of 6.0, 6.8, 7.2, 7.6 and 8.0 km s−1 and fitted the resulting mock data, assuming a FWHM = 6.4 km s−1 (considering the instrumental resolution and the predicted Chapter 5. Stellar Differential Rotation 104

Table 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.

Simulation α = 0.3 α = 0.6 u1 u2 v∗ sin i∗ λ α v∗ sin i∗ λ α 0.25 0.25 14.0 0.5 km s−1 86.0◦ 1.6◦ 0.30 0.13 14.0 0.5 km s−1 86.0◦ 1.6◦ 0.60 0.13 0.15 0.35 14.0 0.5 km s−1 86.0◦ 1.6◦ 0.30 0.13 14.0 0.5 km s−1 86.0◦ 1.6◦ 0.60 0.13 0.25 0.35 14.5 0.5 km s−1 86.0◦ 1.6◦ 0.43 0.13 14.5 0.5 km s−1 86.0◦ 1.6◦ 0.70 0.13

macro-turbulence for WASP-7). The results are in Table 5.9, which show that the change of the local FWHM slightly affects the parameters, although still the best fit values are compatible with theinjected ones within 1σ. As the FWHM increases, the best fit v∗ sin i∗ decreases. This result is reasonable because an enlargement of the CCF is compensated by a lower depth of the CCF itself and in good compatibility with the findings of Doyle et al. (2014), who showed that for fast rotators not accounting properly for −1 the macro-turbulence slightly affects the best-fit, injecting uncertainties of 0.5 km s on v∗ sin i∗.

5.3.6 Limb darkening effect

Injecting improper limb darkening coefficients might as well affect the measurement of α. To explore to which extend this could affect the estimated α, we produced simulations of the WASP-7b RM signal, accounting only for the instrumental noise of 2 m s−1 and the differential rotation (α = 0.3 and 0.6).

We modified the limb darkening coefficients with respect to the selected values, u1 = 0.2 and u2 = 0.3, considering typical uncertainties of 0.05 − 0.1 in line with the values reported in literature (e.g. Albrecht

et al. 2012a; Addison et al. 2018). In particular, we chose: u1 = 0.25 and u2 = 0.25, u1 = 0.15 and

u2 = 0.35, and also u1 = 0.25 and u2 = 0.35. We then fitted each simulation, using as limb darkening

parameters in the fitting model u1 = 0.2 and u2 = 0.3. The results are shown in Table 5.10. All the results we obtained show compatibility with the injected parameters within 1σ. As long as the limb darkening coefficients are changed so that if one increases the other decreases, therecovered

α is the same as the injected value. In the last test, we increased both u1 and u2; the estimated α was different than the input one, even if still compatible with the input value within 1σ. The fact that only α is affected is reasonable. Since the planet has a high spin-orbit misalignment, during thetransitit occults nearly the same longitude. In this sense, the greatest changes in v∗ sin i∗ occur on the transit ingress and egress, because here the planet is closer to the stellar pole, which rotates slower because of the differential rotation. For this reason varying the limb darkening coefficient mainly affectsthe retrieved α. This explanation is valid only for the case in which λ is close to 90◦ (see e.g. Cegla et al. 2016a, 2018, for a different geometrical configuration).

We can conclude that within reasonable uncertainties, the limb darkening does not significantly affect the determination of the stellar differential rotation - at least for highly inclined systems like WASP-7. Moreover, the limb darkening coefficients are usually well constrained from transit observations, already available when exploring the RM signal generated by an exoplanet. Chapter 5. Stellar Differential Rotation 105

5.3.7 Spots

The determination of differential rotation could be plagued by occulted spots or other magnetic activity related features (e.g. faculae). To explore this aspect, we isolated the residuals of the fit of WASP-7b RM with α = 0.6 and an instrumental noise of 2 m s−1 and tried to reproduce it with spots and plages crossing the transit chord. The only condition wherein the stellar activity features might resemble the residuals was by having at least two spots on the transit ingress and egress and several plages along the transit chord, which seems unlikely, also given that the plage are more often visible on the limbs. Nonetheless, if during the transit the planet crosses a spot on the stellar disc, the RM signal shows a bump, which cannot be reproduced with differential rotation and needs to be accounted for to getthe correct system geometry. Non-occulted spots change the shape and depth of the RM and their effect on the estimate of system geometry was already explored in Oshagh et al. (2018), to which we refer for more details.

5.4 Discussion and conclusions

In this work, we explored the possible reasons behind the hurdles in detecting the stellar differential rotation through RM observations. On top of this, we analyzed the improvements we could gain by adopting more precise and stable spectrographs, on larger aperture telescopes (e.g. ESPRESSO). We first fit the observed RM of WASP-7 with a model which accounted for the stellar differential rotation and the instrumental noise reported in Albrecht et al. (2012a). The result gave an α with a large errorbar, and a system geometry in full agreement with the result reported in Albrecht et al. (2012b). Moreover, we performed a series of tests, in which we modelled WASP-7 RM signal with different levels of instrumental noise, varying α and eventually accounting for certain sources of stellar noise (granulation, oscillation, center-to-limb CB, limb darkening and convective broadening/macro- turbulence). We obtained the following results:

• in the absence of any stellar noise source, if the instrumental noise is 5 m s−1 or more, α is no longer detectable. The addition of stellar noise affects the results by enlarging the error-bars. Moreover, for a white noise of 7 m s−1, the injection of stellar noise caused an underestimation of the retrieved α by at least 0.1.

• for an instrumental noise of 2-3 m s−1, a solar-like center-to-limb CB and a βHyi-like granulation do not bias the detection of the stellar differential rotation. The final results are compatible with those retrieved in the absence of stellar noise within 1σ.

• if the instrumental noise is lower than 2 m s−1, adding the center-to-limb CB or granulation leads to an underestimated α. Adding the oscillations to the simulations with granulation, the value of alpha is closer to the injected value, though with higher uncertainties. A closer inspections of the mock RMs shows that this happens because of a lower frequency variation in the granulation component, if compared to the case with both granulation and oscillations.

• for a white noise of 2 m s−1, if granulation and oscillations together are twice or three times the βHyi level, they cause an underestimation of the spin-orbit angle by at least 2.5◦ and enlargement of the error-bar on α. Hence, these noise sources need to be accounted for, when modelling data with 2 m s−1 or better precision. The granulation alone causes an underestimate of both α and Chapter 5. Stellar Differential Rotation 106

v∗ sin i∗. The reasons for these different results between the simulations only with granulation and those with both granulation and oscillations are similar to those explained in the previous point.

−1 • the convective broadening slightly affects the v∗ sin i∗ when the instrumental noise is 2 m s . In detail, it decreases the retrieved projected velocity as the FWHM of the CCF becomes larger

• varying the limb darkening parameters by 0.05 each overestimates α by maximum 0.1 when the instrumental noise is 2 m s−1.

• for a WASP-7 like system, in the absence of any stellar noise, at 2 m s−1 the minimum detectable α is 0.2. If the instrumental noise increases to 3 m s−1 and stellar noise is included in the mock data, the lowest detectable α increases to more than 0.3.

These results suggest that, even with a precise knowledge of the stellar noise of WASP-7, the PFS instrumental noise of 5.6 m s−1 by itself imposes a limit to the detection of the stellar differential rotation. On top of this, as we showed in Section 5.1, a fit of WASP-7b PFS observations showed an average residual level of 11 m s−1. A difference in quadrature between this and the reported instrumental noise (5.6 m s−1) gave an additional RMS of 9.95 m s−1. This high level is hard to justify with any of the stellar noise sources we explored in this paper. If they do happen to have a stellar origin, they will appear as well in the data retrieved with a more stable spectrograph and therefore they could still bias the detection of α. Nonetheless, a stable spectrograph could allow us to more easily disentangle stellar and instrumental variability and, therefore, we could account for both in the fitting model. A close inspection of the PFS residuals shows a similar behaviour between the in and out of transit signal, which allows us to exclude a few possible stellar noise sources not inspected in the present analysis:

• a higher center-to-limb CB than the solar-like one. The similarity between the in and out of transit residuals, excludes the possibility that a stronger CB might be the source of a significant fraction of the additional RMS.

• some occulted and non-occulted spots might change the shape and depth of the transit, and bias the measured geometry (Oshagh et al. 2018). Nonetheless, the rotational velocity of the star is 5 days. Thus spots could have not moved significantly over the course of a transit, nor evolved much.

This leaves three possibilities, which, combined, might generate the extra noise:

• a higher granulation than the βHyi case. For F stars the granulation is predicted to be 60% higher than in the case of G stars (Dravins 1990). Considering both granulation and residual oscillations, the observations of WASP-7 could have at least 5 m s−1 of extra noise. Still, this value is far from covering the entire RMS of 9.95 m s−1, unless WASP-7 has a very high stellar granulations, which so far has never been predicted to exist and has never been detected.

• unaccounted instrumental noise, maybe connected to intrinsic instrumental instabilities.

• weather condition changes (no information about SNR and seeing variation during observation is reported in Albrecht et al. (2012a) and they could lead to increase the RMS residuals).

We conclude that, while the white noise represents a bias in detecting stellar differential rotation, we still have no definitive answer about the source of additional residuals in WASP-7b observations. Chapter 5. Stellar Differential Rotation 107

Because of its stable and precise RVs, the new spectrograph ESPRESSO offers an important opportunity to understand if these residuals still remain in the observations and eventually understand their origins. Moreover, the large aperture of the Very Large Telescope (VLT) will allow an increased sampling in the RM observations, which will aid in constraining potential stellar models. In this paper, we showed that for WASP-7 an instrumental error of 2 m s−1 or lower is necessary for detecting the stellar differential rotation. This level of precision can be reached with ESPRESSO in the 1UT mode for a star rotating slower than 10 km s−1 with a 6 min exposure time, however for stars rotating faster than 10 km s−1 a longer exposure time is necessary to decrease the instrumental noise (Bouchy et al. 2001). We showed that, for WASP-7, an exposure of 15 min should allow to decrease the white noise to 2 m s−1. However, in Section 5.3.4 we highlight that, with this exposure time, the recovered α is underestimated by 0.1. ESPRESSO in the 4UT mode would be able to reach a precision of 2 m s−1 in much less time, although it is unlikely that this mode will be used for RM observations alone. An alternative and more practical method could be to observe more transits and fold them one on top of the other. This would give more data points and decrease the uncertainty on α. The best approach would be to apply this method on magnetically quiet stars. In the case of active stars, observing more transits would allow us to perform predictions on the spot contribution and account for them in the fit. Once the instrumental noise is as low as 2 m s−1, and, if we can mitigate the possible noise sources encountered with the PFS observations, we may be able to measure the stellar differential rotation for WASP-7 and all stars with planetary system geometries similar to WASP-7b (from the TEPCAT catalog Southworth et al. (2011a) we cite CoRoT-1b, WASP-79b, WASP-100b, WASP-109b and HAT-P-32b). WASP-7 is an F star, thus on average it should have α = 0.1−0.2. On one hand, if ESPRESSO-level RM observations in the future are unable to constrain the stellar differential rotation, we can conclude that this star has a maximum α = 0.2. On the other hand, if such observations revealed an α = 0.3 or higher, we could break the degeneracy between the stellar inclination and the stellar rotation, as anticipated in Hirano et al. (2011) and attempted in Cegla et al. (2016b). Chapter 6

Conclusions and future works

6.1 Conclusions

My Ph.D. thesis aimed at exploring how the stellar noise and the planetary signal affect each others when adopting the last generation of photometric and spectroscopic instruments. Given the multitude of new missions on exoplanets, it was fundamental to explore whether or not they could help in improving the knowledge about the star and its orbiting planet. As a first step, we wanted to explore if and to which extent the space mission CHEOPS coulddetect the reflected light of exoplanets, allowing to measure the albedo as it promises as secondary objective. The planetary phase curves had already been observed for several exoplanets, with Kepler, Spitzer and HST data. The advantage of these instruments was the long observational period, which allowed to follow the planetary systems for several orbital periods. All of the reported detections of a planetary phase curve were possible because of the analysis techniques applied. To decrease the instrumental noise, the data were phase folded, according to the orbital period, and subsequently binned. In this way, authors managed to measure the albedo of many Jupiter-sized planets (Angerhausen et al. 2015; Lillo-Box et al. 2014; Esteves et al. 2013, etc.). The additional hurdle to the detection of the planetary phase curves is represented by the stellar activity, which can be much higher than the reflected light. Past works rarely accounted for starspots, because by phase-folding the observations they considered the activity feature contributions to be averaged out. With CHEOPS, all these assumptions are ruled out and new analysis techniques become necessary. CHEOPS is a photometric mission which promises to observe each target for a maximum of 15-20 days, with an instrumental precision of 155 ppm per minute for a 6.5 magnitude star. Since a detection for a phase curve is possible only when several orbits are observed, the detection of the planetary reflected light is only possible for short period planets, with orbital period at least equal to half the observing time. Moreover, the number of observed orbits will be too low for the stellar activity to be totally removed by phase-folding. For these reasons, we performed an analysis to understand the limits of an albedo detection with CHEOPS in presence of activity features. We produced simulations of stellar light curves, including the reflected light modulation of a planet as a function of the geometric albedo, a stellar activity pattern, modeled with SOAP-T (Oshagh et al. 2013a), and instrumental noise. The instrumental noise was a white noise with standard deviation equal to the predicted noise for bright stars with CHEOPS. The simulated curves had one data point every

108 Chapter 6. Conclusions and future works 109

2 hours, with the idea that the 1 minute cadence CHEOPS data were subsequently binned, to reduce the noise level and the number of data points. As fitting tool, we adopted an MCMC which accounted for the stellar activity through aGP.The GP models the activity features as a periodic signal, since SOAP-T does not account for spot evolution or differential rotation. To do so we assigned to the GP a periodic kernel, with periodicity equalto the period of rotation of the star. Our analysis showed that, knowing the albedo of the planet without knowing the stellar rotation is not possible. The MCMC required at least more than one entire stellar rotation to recover the planetary albedo. The number of required stellar rotation for the detection of the reflected light component decreased as the stellar rotation increased. This result is particularly interesting, because it shows that recovering both the albedo and the stellar rotation is only possible when the stellar activity is properly sampled, e.g. enough data points over a full rotation. We showed as well that, for fast rotators, a smaller binning helps to improve the precision on the retrieved albedo. As additional results, we demonstrated that, for the brightest stars, CHEOPS can measure the albedo of the exoplanets with an orbital period maximum of 13 days under the simulated datasets, and that for a noise level of 155 ppm per minute, we can detect the albedo of all planets with size equal or higher than Neptune. The results of this analysis were published in our first paper, Serrano et al. (2018), and they can help when developing observational strategies with CHEOPS. Though, the fitting tool we used has some limitations for future applications on real data. For example, if the photometric activity is not fully periodic the periodic kernel will give wrong albedo detections. Nonetheless, our work shows applicability of the MCMC with GP for different activity levels and various levels of instrumental precision, suggesting that better results can be obtained with future and more precise surveys, such as PLATO. In the second part of this Ph.D thesis, we updated the tool SOAP3.0 adding three features, the stellar differential rotation, the macro-turbulence and the center-to-limb CB. Our aim was to understand how these features could affect the RM signal and whether such signal could allow to measure the stellar differential rotation. We thus addressed the question: can the presence of a planet help knowing more about the host star? As mentioned, the stellar differential rotation can be measured with several spectroscopic and photometric techniques. We can measure period variations from long term photometric observations (Reinhold & Gizon 2015) or through the analysis of chromospheric activity (Donahue et al. 1996) or through spots detection in transit observations (Zaleski et al. 2019). Other methods require to follow the time migration of individual features on Doppler maps (Donati et al. 2000; Barnes et al. 1998) or to study the effect of stellar differential rotation on line profiles (Reiners et al. 2000). The usage of the RM effect to determine the stellar differential rotation has been limited until now. We can report several theoretical works (Hirano et al. 2011; Gaudi & Winn 2007) and a couple of detection attempts (Albrecht et al. 2012a; Cegla et al. 2016a). In detail, Cegla et al. (2016a) managed to constrain the relative differential rotation α for the star HD189733, with a 1σ error of 0.3 (α = [0.28, 0.86]). Albrecht et al. (2012a) performed an analysis on WASP-7b data obtained with the PFS. Even though they estimated an α value, they expressed concerns about their results. For this reason, we decided to explore which were the limiting factors in measuring α for a planetary system like WASP-7, which is characterized by a close-to-90◦ spin-orbit angle. A system with such a configuration should actually be one of the most favorable for the detection of the stellar differential rotation, because the planet occults different stellar latitudes. Though such a detection was significantly hurdled by several sources of noise, whose nature was unclear. We used the new tool to fit WASP-7b data by Albrecht et al. (2012a) and we retrieved a high α with a large errorbar, covering almost all Chapter 6. Conclusions and future works 110 the interval of possible values for such parameter. Our result confirmed the perplexities expressed by Albrecht et al. (2012a) and encouraged us to perform a deeper analysis. We thus produced with the updated SOAP3.0 WASP-7b mock RMs, accounting for different levels of instrumental noise and exploring the detectability of α in absence of additional noise sources. We tested as well what happened once some sources of stellar noise were included. In particular, we produced additional mock curves considering a solar-like center-to-limb variation of the convective blue-shift, or including the granulation and oscillations modeled as in the case of the star βHyi (Dumusque et al. 2011). We finally analyzed how the recovered α was affected when we varied the injected limb darkening values and the convective CCF broadening. The main result of our work was that, in absence of any stellar noise source, if the instrumental noise is 5 m s−1 or more, α is no longer detectable. The addition of stellar noise affects the results by enlarging the error-bars. Considering that the data with the PFS had a nominal error-bar of 5.6 m s−1, retrieving the stellar differential rotation with them was tricky. A much higher instrumental precision is necessary, around 2 m s−1, at which level we identified the granulation as a possible source of uncertainties. Ourwork thus allows to impose observational constraints on the possible identification of the stellar differential rotation effect on the WASP-7b RM signal. The detection is probable with a white noiseof2ms−1, if α is higher than 0.2 and considering βHyi-like granulation and oscillations and solar-like center-to-limb CB. On top of this, we explored the source of an additional RMS present in the PFS data, which amounts around 9.95 m s−1. The similar behaviour between the in and out-of-transit residuals allowed us to exclude two stellar noises: a higher center-to-limb CB and the presence of occulted and non-occulted spots. The first one affects only the in-transit residuals. Spots could affect the RM generatingan additional RMS in case of changes or movements over the course of the transit, which is highly unlikely considering the stellar rotation of 5 days. For this reason we suggested three possibilities, which could affect the data at the same time: a higher granulation, non-accounted instrumental noise, due tointrinsic instrumental instabilities, and weather condition changes. Though, we could find no definitive answer about the exact origin of this extra RMS. We managed to demonstrate that past instruments such as the PFS or HARPS could not measure the stellar differential rotation through the RM signal for a WASP-7 like star. Unfortunately, wecould not establish as well the effective measurability of such effect even with the precision of ESPRESSO. Nonetheless, the high precision of this new spectrograph can allow to constrain better the stellar models and eventually understand if the PFS additional noise is due to the granulation (unlikely to be as high as almost 10 m s−1). The answer is possible through new WASP-7 observations, which might be performed with ESPRESSO in the 1UT mode. Since WASP-7b is an F star, reaching a precision of 2 m s−1 requires more than 15 minutes of exposure, which we showed can hamper the accuracy of α. For this reason, the best strategy is to observe at least two or more RMs, a method which would allow as well to resolve the presence of occulted and un-occulted spots. Though, the minimum detectable α in absence of instrumental noise is 0.2. F stars have an average relative differential rotation around 0.1 and for this reason it is still possible that α is measurable for a limited number of F stars. Our Ph.D. thesis allowed to put constrains on the measurability of some important parameters of the planet, accounting for the noise predicted for some advanced instruments. We could highlight how knowing the properties of the star is fundamental for understanding the characteristics of a planet. On the other hand, the star itself can be fully understood if the planet is properly described. Stars Chapter 6. Conclusions and future works 111

Figure 6.1: Example of effect of stellar differential rotation for an aligned system. and their orbiting planets require a parallel description for a precise knowledge of the overall planetary system. The results of our work represent an important reference for the future observability of the phase curve analysis, not only in the frame of CHEOPS, but also for other missions, such as TESS and PLATO. Knowing the constraints imposed by the stellar activity at certain levels of noise will help in choosing the most appropriate targets for the detection of the albedo. In a similar way, we also have an opportunity to put constraints on the detectability of the stellar differential rotation and on the precision on the estimate of the projected spin-orbit angle with several spectrographs, not just ESPRESSO, but HIRES, HARPS and other instruments as well. Considering our work as reference could allow to predict the measurability of the differential rotation of G stars transited by planets with geometries similarto WASP-7, explored in our work. Even though for these phenomena, a measurement in presence of an Earth-sized planet seems still hard, the reachable precision with the last generation of photometric and spectroscopic instruments will permit a remarkable improvements in the characterization of exoplanets.

6.2 Future works

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

Accounting for stellar activity in photometric light curves became more straightforward as we started to apply the MCMC with GP to account for it. In Serrano et al. (2018) we used a periodic kernel to reproduce the activity features, since SOAP-T only allows for periodic features. Though, we showed that for real data covering almost 100 days, as for Kepler-7, this simplification gave unphysical results. The hurdle is represented by the spot evolution and the stellar differential rotation, which remove the full periodicity from the activity features. Can we account for these properties and retrieve the injected albedo? Even though, several works included them in the fitting models, so far there was no demonstration about the possibility of properly getting rid of these noise sources. We will update SOAP3.0 to account for the stellar differential rotation on spots as well and for the spot evolution and Chapter 6. Conclusions and future works 112 test, as we did in Serrano et al. (2018), if we can properly recover the reflected light component of an exoplanet. We will as well test the applicability of our fitting tool on real data (especially Kepler and CHEOPS), towards the future mission PLATO.

6.2.2 Detectability of planetary eclipses

A planetary eclipse is a much dimmer feature if compared to the transit and several times it is not straightforward to detect it. For instance, Lillo-Box et al. (2014) found the location of the planetary eclipse in Kepler-91b, though he could not identify which specific feature was to be associated to the occultation. Other similar cases of unidentified eclipses are reported in literature. Can this difficulty in detecting the eclipse be solely connected to the low planetary albedo? Or could it be as well a consequence of the applied method?. Our scope is to understand whether or not phase-folding the light curves allows to completely average out the stellar activity. To do so, we plan to use TESS data of WASP-121b, where we managed to locate some planetary eclipses, and analyze each eclipse separately, to estimate the albedo. Then, we will perform the same analysis once the curve is phase-folded and compare the results. We will later explore, in case of strong differences between the resulting albedo for the two methods, whether the reason is connected to activity features or even other noise sources (e.g. granulation).

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

The effect of spots on the RM signal was explored by Oshagh et al. (2016) and Oshagh et al. (2018), who showed that they can affect the estimation of the spin-orbit angle by 20−30◦. In our work on stellar differential rotation detection through RM we explored the effect of several noise sources, thoughwedid not perform a deep analysis on the spots influence on α. Nonetheless, we already know that un-occulted spots change the RM depth, while the occulted spots additionally injects bumps in RM. We will perform simulations of RM signals in presence of spots, with the aim of exploring whether we can measure the stellar differential rotation and how the estimation would change when analyzing several RM signals for the same target. We will then develop a strategy to properly account for spots in RM analysis when looking for the stellar differential rotation.

6.2.4 Breaking the degeneracy between v∗ sin i∗ and the stellar differential rotation for aligned systems

When a planet is completely aligned with respect to the star, λ = 0◦, the RM signal is perfectly symmetric. Figure 6.1 reports an example of an aligned system, in which we just vary the relative differential rotation. When α increases, the signal decreases, though maintaining the symmetry. This suggests that for aligned and closely-aligned systems, we can detect a degeneracy between v∗ sin i∗ and α. How can we break such degeneracy? We will explore this issue, by coupling the RM analysis to the transit analysis. It might be possible, for instance, that once we estimate the stellar period of rotation with a transit feature, we can measure α through the RM signal. Bibliography

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