Issue 
A&A
Volume 588, April 2016



Article Number  A31  
Number of page(s)  7  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201527899  
Published online  14 March 2016 
Uncovering the planets and stellar activity of CoRoT7 using only radial velocities ^{⋆}
^{1}
Instituto de Astrofísica e Ciências do Espaço, Universidade do
Porto,
CAUP, Rua das Estrelas,
4150762
Porto,
Portugal
email:
joao.faria@astro.up.pt
^{2}
Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua Campo
Alegre, 4169007
Porto,
Portugal
^{3}
HarvardSmithsonian Center for Astrophysics, 60 Garden
Street, Cambridge,
MA
02138,
USA
^{4}
Department of Statistics, The University of
Auckland, Private Bag
92019, 1142
Auckland, New
Zealand
^{5}
Institut für Astrophysik, GeorgAugustUniversität,
FriedrichHundPlatz 1,
37077
Göttingen,
Germany
Received: 4 December 2015
Accepted: 27 January 2016
Stellar activity can induce signals in the radial velocities of stars, complicating the detection of orbiting lowmass planets. We present a method to determine the number of planetary signals present in radialvelocity datasets of active stars, using only radialvelocity observations. Instead of considering separate fits with different number of planets, we use a birthdeath Markov chain Monte Carlo algorithm to infer the posterior distribution for the number of planets in a single run. In a natural way, the marginal distributions for the orbital parameters of all planets are also inferred. This method is applied to HARPS data of CoRoT7. We confidently recover the orbits of both CoRoT7b and CoRoT7c although the data show evidence for the presence of additional signals.
Key words: methods: data analysis / planetary systems / stars: individual: CoRoT7 / techniques: radial velocities
All data and software presented in this article are available online at https://github.com/jfaria/exoBDCoRoT7
© ESO, 2016
1. Introduction
Imagine we have at our disposal a set of spectroscopic observations of an unknown star, and we can obtain precise radialvelocity (RV) measurements, using the crosscorrelation function (CCF) technique (Baranne et al. 1996; Pepe et al. 2002). The socalled line profile indicators (such as the full width at half maximum of the CCF, its bisector span, and associated quantities, e.g. Figueira et al. 2013) and activity indicators, like the (Noyes et al. 1984), are also usually available but are not necessary for our analysis. And without other instruments at hand, we cannot measure the star’s photometric variations, as is the case for most targets in RV surveys.
Armed with some statistical artillery, we aim to answer the following question: how many orbiting planets can be confidently detected in these data, and what are their orbital parameters and minimum masses?
Besides planets, other physical processes can induce variations in the radial velocities of a star. These include stellar oscillations, granulation, spots and faculae/plages, and longterm magnetic activity cycles (see, e.g. Saar & Donahue 1997; Santos et al. 2010; Boisse et al. 2011; Dumusque et al. 2011a). Some of these activityinduced signals can be mitigated or averaged out by adapting the observational strategy (Dumusque et al. 2011b). But signals caused by the presence of active regions on the stellar surface can show periodicities and amplitudes similar to the ones induced by real planetary signals, and may be harder to disentangle. Indeed, these signals can even mimic planetary signals (e.g. Figueira et al. 2010; Santos et al. 2014; Robertson et al. 2015).
Simultaneous photometric and RV observations have been successfully used to constrain activityinduced RV signals. However, this approach requires either a joint model for photometric and RV variations, which can be statistical (e.g. Rajpaul et al. 2015) or based on a description of the stellar features inducing the signal (e.g. Lanza et al. 2010), or a conversion from photometric variations to RV variations (e.g. Haywood et al. 2014, hereafter H14).
Here we present a framework that models activityinduced signals as correlated noise in the RV observations and does not require simultaneous photometric observations. The method is based on the fact that, for data of sufficient quality, it is possible to distinguish if an oscillating signal has the Keplerian shape that is expected from a planet, or some other approximatelyperiodic shape as expected from stellar activity. Below we describe our model and apply it to High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al. 2003) observations of CoRoT7.
2. A Bayesian model for RV data
In this section we present in detail our model for RV data and describe the two types of signals we consider: planetary signals and activityinduced noise.
2.1. RV planetary signals
The dynamical evolution of a planetary system is governed by the gravitational interactions of its constituent bodies. For most systems with multiple planets one can assume, to a good approximation, that the mutual planetary perturbations are negligible on timescales that are comparable to the duration of observations. The stellar RV perturbations that are due to a multipleplanet system can then be modelled as a linear superposition of Keplerian orbits.
Each Keplerian can be described with five RV observables: the semiamplitude K, the orbital period P, the eccentricity e, an orbital phase φ, and the longitude of the lineofsight, ω. We also consider a systemic velocity, v_{sys}, of the centre of mass of the system, which corresponds to an RV zeropoint measured by HARPS.
2.2. Gaussian processes to model stellar activity
Even if activity signals cannot be easily described analytically (a complete description would require knowledge of the active region distribution and evolution, temperature contrast and stellar parameters, such as inclination and limb darkening), we can make some assumptions about their form. Besides assuming the signals will be continuous and smooth, stellar rotation induces a periodicity or quasiperiodicity, as active regions evolve and cycle in and out of view on the stellar surface.
For the purposes of detecting planets using RV measurements, the signals caused by stellar activity can then be seen as correlated quasiperiodic noise. Gaussian processes (GP) are an increasingly common tool to deal with correlated noise (e.g. Roberts et al. 2012). In their application to regression problems, GPs can be seen as prior distributions over functions, which will be constrained by the data (e.g. Rasmussen & Williams 2006). For our purposes, we use the GP to model the stochastic component of our signal – that is, the stellar noise.
A GP is defined by its mean function, the deterministic component of the signal, and its covariance function, which defines the overall behaviour of the functions under the GP distribution. When the covariance function is evaluated at the observed times, the covariance matrix is obtained. From the many possible choices for a covariance function, the quasiperiodic kernel is the most widely used in the exoplanet literature (e.g. H14; Grunblatt et al. 2015; Rajpaul et al. 2015), which results in a covariance matrix of the form (1)This represents some of our expectations for the activityinduced RV signals: the correlations decay on a timescale of η_{2} days and have a periodic component with period η_{3} days. The parameter η_{4} controls the relative importance of the periodic and decaying components. The parameter η_{1} represents the amplitude of the correlations. This covariance matrix also takes additional uncorrelated noise into account, added quadratically to the diagonal, where σ_{i} are the reported RV uncertainties and s is a free parameter.
2.3. The complete model
Fig. 1 Representation of the relations between parameters and observations in our RV model, as a probabilistic graphical model. An arrow between two nodes indicates the direction of conditional dependence. The circled nodes are the parameters of the model, whose joint distribution is sampled by the Markov chain Monte Carlo (MCMC) algorithm. The double circled node v_{i} represents the observed RVs. The filled nodes represent deterministic variables: if these variables have parent nodes ( and Σ), they are given by a deterministic function of those parents; if they do not have parents (t_{i} and σ_{i}), they are assumed as being given and thus fixed. The variables inside boxes are repeated a number of times, as shown in the bottom left corner of each box. 

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Our complete model for the RV observations is shown in Fig. 1, in the form of a probabilistic graphical model. The diagram relates all the parameters (see Table 1 for their description) and observables in the model, stating their interdependencies. From this graphical representation we can build an expression for the joint probability of the parameters of interest and the data.
Let Θ be the vector of all the parameters in the model: . The notation means that the subset { P,K,e,φ,ω } will be repeated N_{p} times (for each of the N_{p} planets). An observed dataset is composed of N radialvelocity observations v_{i}, at times t_{i} and with associated uncertainty estimates σ_{i}. The diagram in Fig. 1 is a representation of the joint probability density function (PDF) for all the variables: where we condition on the information ℐ^{1}. In the remainder of the paper we will include { t_{i} } and { σ_{i} } in ℐ, because they are assumed to be fixed. To ease the notation, we also group the RV values in a proposition D = { v_{i} }. The are obtained from evaluating the sum of Keplerian signals at the observed times.
The joint PDF can be factored and rearranged to give Bayes’s theorem, (2)providing an expression for the posterior distribution for all the parameters, conditioned on the observed data. The posterior distribution contains all the information about the parameters that is available to us. With this distribution to hand, we can then infer how many planets are supported by the data and their orbital parameters and masses.
2.4. Determining the number of planets
To calculate the joint posterior distribution, following Eq. (2), we need the three terms on the righthand side: the prior p(Θ  ℐ), likelihood p(D  Θ,ℐ), and evidence p(D  ℐ).
For the likelihood, the choice that reflects most genuinely our state of knowledge is a multivariate Gaussian distribution^{2}, with mean and covariance matrix Σ. The complete covariance matrix can be obtained, deterministically, from the values of η_{1},η_{2},η_{3},η_{4},σ_{i}, and s, according to Eq. (1). The loglikelihood is then given by: (3)where r is the vector given by for all data points.
The priors used for all the parameters are listed in Table 1. Most of them are the same as those used by Brewer & Donovan (2015) and were chosen to represent uninformative or vague prior knowledge. For the orbital periods and semiamplitudes we assign hierarchical priors that are conditional on the hyperparameters, μ_{P}, w_{P}, and μ_{K}, respectively (see Table 1). This reflects our belief that knowing the parameters of one planet provides a small amount of information about the parameters of another planet.
Meaning and prior distribution for the parameters in the model. In some cases we sample on the logarithm of the parameter.
For most of the parameters of the GP covariance kernel, we assigned loguniform priors in sensible ranges. For η_{3}, the parameter that can be interpreted as the stellar rotation period, we assumed a uniform prior between 10 and 40 days.
To sample the joint posterior distribution, we used the algorithm proposed by Brewer (2014). The particular difficulty here – since N_{p} is not known – is that the sampling algorithm needs to jump between candidate solutions with different numbers of planets. Brewer (2014) proposed a method that uses birthdeath Markov chain Monte Carlo (MCMC) moves to infer N_{p}, within the diffusive nested sampling framework (Brewer et al. 2011). This way, one can estimate the value of the evidence while improving the mixing in complex posteriors (affected by multimodality and phase transitions). Brewer & Donovan (2015) applied this method to RV data of ν Oph and Gliese 581, although these authors did not incorporate GPs in their analysis.
3. Application to HARPS data
We apply the method described in the previous section to HARPS observations of CoRoT7. The planet CoRoT7b was first announced by Léger et al. (2009) and was the first superEarth with a measured radius. Its orbital period is estimated from the transits in the CoRoT light curve as being P_{b} = 0.854 days (Léger et al. 2009). A second nontransiting planet, with P_{c} = 3.69 days, was detected in a followup RV campaign (Queloz et al. 2009) and a more disputed detection of a third planetary signal was reported by Hatzes et al. (2010).
Owing to the high activity levels of the host star, this system has since generated a wealth of discussion, resulting in different estimates for the masses of the planets (Lanza et al. 2010; Boisse et al. 2011; FerrazMello et al. 2011; Pont et al. 2011; Hatzes et al. 2011). Simultaneous observations from CoRoT and HARPS were obtained in 2012 to help settle these issues (Barros et al. 2014; H14). These observations were analysed by H14 with an RV model that is similar to ours but considering information from the simultaneous photometric observations.
We emphasise that, in the following, we analyse the full set of RV observations. In summary, the star was observed with HARPS in 2009 and 2012, with a total of 177 public RV measurements. The average error bar on these measurements is 2 m s^{1} (which includes photon and instrumental noise) and the RV dispersion is 10 m s^{1}, over the complete 3yr timespan.
A common procedure in (current) RV studies is to fix the number of planets and sample the posterior for the remaining parameters (and possibly calculate the evidence) in a stepbystep approach. If we fix N_{p} = 2, we recover the orbital parameters of the two known planets, CoRoT7b and CoRoT7c, within the errors reported in H14 and Barros et al. (2014). The method we present in this paper is, however, much more general and allows for the full posterior for N_{p} to be obtained from one run. We now proceed with this general method.
We ran our algorithm on the full set of RVs, using the priors in Table 1, and obtained 16 248 effective samples from the joint posterior distribution^{3}. The evidence for our model is log (p(D  ℐ)) = −530.9. The resulting marginal posterior distribution for N_{p} is shown in Fig. 2.
Fig. 2 Posterior distribution for the number of planets N_{p}. The counts are number of posterior samples in models with a given number of planets. The two ratios of probabilities between models with 1, 2, and 3 planets are highlighted; note that p(0) = p(1) = 0. 

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Fig. 3 Joint posterior distribution for the semiamplitudes (top panel) and the eccentricities (bottom panel) together with the orbital periods of the Keplerian signals. 

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The posterior distribution for N_{p} is one of the main outputs of our method. But to actually decide on what is the number of planets orbiting CoRoT7, and answer our initial question, we need to clarify what we mean by “confidently detected”. Based on a scale suggested by Jeffreys (1998; see also Kass & Raftery 1995), some authors (e.g. Tuomi 2011; Feroz et al. 2011) have proposed that, to claim a detection of N_{p} planets, the probability of N_{p} should be 150 times greater than the probability of N_{p} − 1. This criterion requires “strong” (Jeffreys 1998) evidence for detecting a planet, thus considering false positives to be much worse than false negatives. By applying this rule to our results, we choose N_{p} = 2 as the number of planets confidently detected in our dataset.
We emphasise that having confidently detected two planets is a different matter from knowing that the number of planets is actually two. According to the posterior distribution, it is more likely that there are four planets. But our detection criterion depends on the ratio of probabilities for consecutive values of N_{p} (see Fig. 2), not on the probability values themselves. Of course, the posterior distribution is sensitive to the prior distributions, in particular as to how small we believe K might be.
The joint posterior distributions for the orbital periods, semiamplitudes, and eccentricities of the signals are shown in Fig. 3, where the samples for all Keplerians were combined. The figure shows a 2dimensional histogram of the posterior samples, where the colourmap represents bin counts and is set in a logarithmic scale.
The two detected planets are seen as overdensity regions at P_{b} = 0.85 days and P_{c} = 3.69 days. Their amplitudes and eccentricities are well constrained. There is a clear posterior peak around two days with amplitude and eccentricity mostly unconstrained. The posterior also shows a smaller peak at 9 days, the period reported by Hatzes et al. (2010) as a possible third planet (see also Tuomi et al. 2014).
Marginal posterior distributions for the parameters of the GP and the extra white noise parameter s are shown in Fig. 4. The posterior for η_{3} is particularly interesting as it provides a constraint to CoRoT7’s rotation period, obtained exclusively from the RVs. Our inferred value for the stellar rotation period of days is obtained solely from the RV timeseries and is in agreement (see Table 2) with earlier estimates which used the CoRoT light curve (Léger et al. 2009; Lanza et al. 2010; H14).
Also interesting is the joint behaviour of η_{2}, η_{3}, and η_{4}. For higher values of η_{4}, the periodic component of the covariance function loses importance relative to the decaying component, η_{2} gets smaller and η_{3} becomes unconstrained. In this case, the GP smooths the RVs on a timescale of η_{2} ≈ 3 days. But when η_{4} is of order unity (meaning the periodic component is present), the decaying timescale is higher and η_{3} is constrained around 22 days. The values of η_{2} in this situation (20−30 days) are closer to the stellar rotation period and are also consistent with the average lifetime of active regions measured in the CoRoT 2012 photometry (20.6 ± 2.5 days, H14). Our results therefore validate the approach taken by, e.g. H14 and Grunblatt et al. (2015), of modelling the activityinduced RV variations with a GP that has the covariance properties of the light curve.
Fig. 4 Marginalised 1D and 2D posterior distributions for the parameters of the GP and the extra white noise. The samples for all values of N_{p} were combined. The titles above each column show the median of the posterior and the uncertainties calculated from the 16% and 84% quantiles. The solid lines are kernel density estimations of the marginal distributions. 

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Considering only the posterior samples with N_{p} = 2, Table 2 lists the median values of some orbital parameters for the two planets, and the maximum likelihood RV curves are shown in Fig. 5. Our estimates for the orbital parameters are in agreement, within the uncertainties, with the ones obtained by H14.
Fig. 5 Panel a): RV measurements of CoRoT7 from 2009 and 2012 and the twoplanet bestfit model (black curve). Note the different scales in the abscissae on the left and right parts of the plot. Panels b) and c) show the phased RV curves after subtracting each planet signal and the GP. 

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4. Discussion and conclusions
Building on the work of Brewer & Donovan (2015), we have developed a simple method to estimate the number of orbiting planets around active stars, using only RV measurements. We applied this method to HARPS observations of CoRoT7 and confidently detect CoRoT7b and CoRoT7c, while finding weaker evidence for two additional signals. In this framework, there is no need to use photometric observations, information from transit detections, or auxiliary activity indicators.
The posterior distribution for N_{p} shows evidence for the presence of extra signals, even if they do not meet our detection criteria. We note that the effects of considering a uniform prior for N_{p} (thus giving considerable prior weight to large numbers of planets) and a hierarchical prior for the semiamplitudes (which changes how the Occam’s razor penalty is taken into account) can be important and will be studied in the future.
Our choices for the likelihood, covariance function, and priors provide the model with a robustness against possible outliers which is similar to most analyses of RV data. If we had reason to suspect the presence of outliers, it would be straightforward to extend our model, e.g. by scaling each error bar with a common value or with individual values whose common prior is defined hierarchically.
Stepping back to appreciate our results, we find that there is a lot of information contained in the RVs, both about planetary and (arguably more importantly) activity signals, and that this information can be recovered. The GP provides a flexible and accommodating model for activityinduced signals, allowing us to infer the planetary masses and orbital parameters with more realistic uncertainties.
In the analysis of H14, the authors modelled the outoftransit photometry using a GP (as an interpolator) and applied the FF’ method of Aigrain et al. (2012) to obtain the RV signal that is due to activity. By comparing the evidence of models containing this activity signal plus zero, one, two, or three planets, they asserted that the twoplanet model was the most probable. To model all the quasiperiodic signals in the data, H14 included a GP (with the covariance properties of the CoRoT light curve) as part of the RV model. This was justified because “the FF’ method is likely to provide an incomplete representation of activityinduced RV variations” (H14). We also note that these authors use a very strong Gaussian prior for P_{b} (and the time of periastron), which was nevertheless completely justified by the transit observations.
Our analysis rests on much fewer assumptions – only the RV measurements were used, the rotation period of the star was allowed to vary, the prior for the orbital periods is much less stringent – but we are still able to recover the same number of confidently detected planets and reliable estimates of the orbital parameters and the stellar rotation period (see Table 2). This shows that all this information is contained in the RV measurements and can be recovered with the flexibility of the GP, if we better account for the uncertainty associated with it.
We should finally note two important properties of the CoRoT7 system, which made it ideal for this analysis. First, the amplitudes of the planet signals are much higher than the mean error bar of the HARPS observations, regardless of the stellar activity contamination. Second, the time sampling of the observations is almost ideal for the detection of shortperiod planets, and is very difficult to obtain as part of a typical RV survey. We highlight here the importance of further tests, using other wellstudied datasets of active host stars as well as simulated datasets, for assessing the limits of applicability of our method.
Nevertheless, the importance of this work is not on the specific application to CoRoT7, but instead on providing a simple and fast method to infer the number of planetary signals in the presence of stellar activity. With small modifications, this method can be used to search for planets around stars with different activity levels.
Figure 1 shows that all we know about the distribution of the { v_{i} } are its first and second moments. The multivariate Gaussian follows from the principle of maximum entropy when the distribution is constrained to having a specified covariance matrix (e.g. Cover & Thomas 2006).
The computer used to run the simulations was equipped with an Intel Core i54460 CPU running at 3.20 GHz and 4 GB of RAM. The running time to obtain 50 000 samples (from diffusive nested sampling’s target mixture distribution) was four days. Since the computational cost of the algorithm is dominated by the inversion of the covariance matrix, it is expected to scale roughly with N^{3}.
Acknowledgments
The work presented here grew directly from a collaboration that started at the Astro Hack Week 2015. We thank Andrew Collier Cameron for insightful discussions. We acknowledge the excellent opensource Python packages made available to the community (in particular DAFT and George). This work was supported by Fundação para a Ciência e a Tecnologia (FCT) through the research grant UID/FIS/04434/2013 and the grant PTDC/FISAST/1526/2014. J.P.F. acknowledges support from FCT through the grant reference SFRH/BD/93848/2013. R.D.H. gratefully acknowledges a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. B.J.B. is supported by a Fast Start grant from the Royal Society of New Zealand. P.F. and N.C.S. acknowledge support by FCT through Investigador FCT contracts (IF/01037/2013 and IF/00169/2012, respectively), and POPH/FSE (EC) by FEDER funding through the program Programa Operacional de Factores de Competitividade – COMPETE. P.F. further acknowledges support from FCT in the form of an exploratory project of reference IF/01037/2013CP1191/CT0001. M.O. acknowledges research funding from the Deutsche Forschungsgemeinschft (DFG, German Research Foundation) – OS 508/11. A.S. is supported by the European Union under a Marie Curie IntraEuropean Fellowship for Career Development with reference FP7PEOPLE2013IEF, No. 627202.
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All Tables
Meaning and prior distribution for the parameters in the model. In some cases we sample on the logarithm of the parameter.
All Figures
Fig. 1 Representation of the relations between parameters and observations in our RV model, as a probabilistic graphical model. An arrow between two nodes indicates the direction of conditional dependence. The circled nodes are the parameters of the model, whose joint distribution is sampled by the Markov chain Monte Carlo (MCMC) algorithm. The double circled node v_{i} represents the observed RVs. The filled nodes represent deterministic variables: if these variables have parent nodes ( and Σ), they are given by a deterministic function of those parents; if they do not have parents (t_{i} and σ_{i}), they are assumed as being given and thus fixed. The variables inside boxes are repeated a number of times, as shown in the bottom left corner of each box. 

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In the text 
Fig. 2 Posterior distribution for the number of planets N_{p}. The counts are number of posterior samples in models with a given number of planets. The two ratios of probabilities between models with 1, 2, and 3 planets are highlighted; note that p(0) = p(1) = 0. 

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In the text 
Fig. 3 Joint posterior distribution for the semiamplitudes (top panel) and the eccentricities (bottom panel) together with the orbital periods of the Keplerian signals. 

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In the text 
Fig. 4 Marginalised 1D and 2D posterior distributions for the parameters of the GP and the extra white noise. The samples for all values of N_{p} were combined. The titles above each column show the median of the posterior and the uncertainties calculated from the 16% and 84% quantiles. The solid lines are kernel density estimations of the marginal distributions. 

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In the text 
Fig. 5 Panel a): RV measurements of CoRoT7 from 2009 and 2012 and the twoplanet bestfit model (black curve). Note the different scales in the abscissae on the left and right parts of the plot. Panels b) and c) show the phased RV curves after subtracting each planet signal and the GP. 

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In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.