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Εργαλεία Θεμάτων | Τρόποι εμφάνισης |
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#11
10-11-2012, 20:41
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
Το αθρωπινο γενος ειναι οτι χειροτερο......δεν θα μα γλυτωσει τιποτα και πουθενα...   
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#12
10-11-2012, 21:21
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
42 ετη φωτος ...........
μακρυα ειναι λιγο..... τωρα εχουμε μνημονιο 2 η 3 (το εχασα!!) θα μας βρει το μνημονιο Νο .............
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#13
10-11-2012, 21:24
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
 Τελικά Γιώργο μου,,,, είσαι μεγάλη....  
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#14
11-11-2012, 00:18
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
Τον εμαθες και συ εεεεε...
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#16
11-11-2012, 03:49
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
Καλά που έβαλες και την πηγή, γιατί διαφορετικά θα αμφισβητούσαμε την γνησιότητα του άρθρου, του οποίου παραθέτω τον πρόλογο παρακάτω, για κάποιους δύσπιστους ...  Habitable-zone super-Earth candidate in a six-planet system around the K2.5V star HD 40307 Mikko Tuomi ⋆1,2, Guillem Anglada-Escud´e⋆⋆3, Enrico Gerlach4, Hugh R. A. Jones1, Ansgar Reiners3, Eugenio J. Rivera 5, Steven S. Vogt5, and R. Paul Butler6 1 University of Hertfordshire, Centre for Astrophysics Research, Science and Technology Research Institute, College Lane, AL10 9AB, Hatfield, UK 2 University of Turku, Tuorla Observatory, Department of Physics and Astronomy, V¨ais¨al¨antie 20, FI-21500, Piikki¨o, Finland 3 Universit¨at G¨ottingen, Institut f¨ur Astrophysik, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany 4 Lohrmann Observatory, Technical University Dresden, D-01062 Dresden, Germany 5 UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA 6 Department of Terrestrial Magnetism, Carnegie Institute of Washington, Washington, DC 20015, USA Received XX.XX.2012 / Accepted XX.XX.XXXX ABSTRACT Context. The K2.5 dwarf HD 40307 has been reported to host three super-Earths. The system lacks massive planets and is therefore a potential candidate for having additional low-mass planetary companions. Aims. We re-derive Doppler measurements from public HARPS spectra of HD 40307 to confirm the significance of the reported signals using independent data analysis methods. We also investigate these measurements for additional low-amplitude signals. Methods. We used Bayesian analysis of our radial velocities to estimate the probability densities of different model parameters. We also estimated the relative probabilities of models with di ffering numbers of Keplerian signals and verified their significance using periodogram analyses. We investigated the relation of the detected signals with the chromospheric emission of the star. As previously reported for other objects, we found that radial velocity signals correlated with the S-index are strongly wavelength dependent. Results. We identify two additional clear signals with periods of 34 and 51 days, both corresponding to planet candidates with minimum masses a few times that of the Earth. An additional sixth candidate is initially found at a period of 320 days. However, this signal correlates strongly with the chromospheric emission from the star and is also strongly wavelength dependent. When analysing the red half of the spectra only, the five putative planetary signals are recovered together with a very significant periodicity at about 200 days. This signal has a similar amplitude as the other new signals reported in the current work and corresponds to a planet candidate with M sin i ∼ 7 M⊕ (HD 40307 g). Conclusions. We show that Doppler measurements can be filtered for activity-induced signals if enough photons and a sufficient wavelength interval are available. If the signal corresponding to HD 40307 g is a genuine Doppler signal of planetary origin, this candidate planet might be capable of supporting liquid water on its surface according to the current definition of the liquid water habitable zone around a star and is not likely to su ffer from tidal locking. Also, at an angular separation of ∼ 46 mas, HD 40307 g would be a primary target for a future space-based direct-imaging mission. Key words. Methods: Statistical, Numerical – Techniques: Radial velocities – Stars: Individual: HD 40307 1. Introduction Current high-precision spectrographs, such as the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al., 2003) and the High Resolution Echelle Spectrograph (HIRES; Vogt et al., 1994), enable detections of low-mass planets orbiting nearby stars. During recent years, radial velocity (RV) planet searches have revealed several systems of super- Earths and /or Neptune-mass planets around nearby stars (e.g. Mayor et al., 2009a,b; Lovis et al., 2011a; Pepe et al., 2011; Tuomi, 2012). The system of three super-Earths orbiting HD 40307 has received much attention because the planets appear in dynamically packed orbits close to mean motion resonances (Mayor et al., 2009a). This has been used as an argument to suggest that lowmass planets may be found in highly compact multiple systems ⋆ e-mail: guillem.anglada@gmail.com that are still stable in long-term, e.g. a possibility of having ten planets with masses of 17 M ⊕ within a distance of 0.26 AU on stable orbits (Funk et al., 2010). However, the physical nature of these companions as scaled-up versions of the Earths is not entirely clear (Barnes et al., 2009a). Their masses, between those of Earth and Neptune, suggest they are Neptune-like proto-gas giants that could not accumulate enough gas before it was blown away by the newly born star. On the other hand, recent transit observations of hot super-Earths around bright nearby stars (L´eger et al., 2009; Batalha et al., 2011;Winn et al., 2011) indicate that a good fraction of these hot super-Earth mass objects can have rocky compositions. In this article we re-analyse the 345 HARPS spectra publicly available through the ESO archive using a newly developed software tool called HARPS-TERRA (template-enhanced radial velocity re-analysis application; Anglada-Escud´e& Butler, 2012). Instead of the classic cross-correlation function method (CCF) implemented by the standard HARPS-ESO data reduction software (HARPS-DRS), we derive Dopplermeasurements by leastsquares matching of each observed spectrum to a high signal-tonoise ratio template built from the same observations. A description of the method and the implementation details are given in Anglada-Escud´e & Butler (2012). In addition to an increase in precision (especially for K andMdwarfs), this method allows us to performadditional analyses and tests beyond those enabled by the CCF data products provided by the HARPS-DRS. As an example, it allows us to re-obtain the RV measurements using only a restricted wavelength range. As we show in Section 5, this capability can be instrumental in ruling out the planetary nature of prominent signals correlated with stellar activity. We rely on the Bayesian framework when estimating the orbital parameters supported by the data, determining the significances of the signals, and the modelling of the noise in the measurements. In previous studies, radial velocities received within an interval of an hour or so have been commonly binned together in an attempt to reduce the noise caused by stellar- surfacerelated e ffects, i.e. stellar oscillations and granulation, and other factors within this timescale (Dumusque et al., 2010). In principle, this would enable the detections of planets smaller than roughly 5 M ⊕ with HARPS over a variety of orbital distances, even at or near the stellar habitable zone (Dumusque et al., 2010, 2011a). In our approach, and instead of binning, we apply a selfconsistent scheme to account for and quantify correlated noise in the Bayesian framework and use Bayesian model probabilities to show that a solution containing up to six planets is clearly favoured by the data, especially when the redmost part of the stellar spectrum is used in the RV analysis. Only the confluence of refinements in these data analysis methods (re-analysis of the spectra and Bayesian inference) allows the detection and verification of these low-amplitude signals. We start with a brief description of the stellar properties of HD 40307 (Section 2) and describe the statistical modelling of the observations and the data analysis techniques we used (Section 3). In Section 4 we describe the properties of the RV measurements and perform a detailed Bayesian analysis that identifies up to three new candidate signals. We discuss the stellar activity indicators and their possible correlations with the RV signals in Section 5. In this same section, we find that one of the candidates is spuriously induced by stellar activity by showing that the corresponding periodic signal (P ∼ 320 days) is strongly wavelength dependent. When the RVs obtained on the redmost part of the spectrum are analysed (Section 6), the 320-day signal is replaced by a signal of a super-Earth-mass candidate with a 200-day period with a minimum mass of about ∼ 7M⊕ orbiting within the liquid water habitable zone of HD 40307. The analysis of the dynamical stability of the system (Section 7) shows that stable solutions compatible with the data are feasible and the potential habitability of the candidate at 200 days (HD 40307 g) is discussed in Section 8.We give some concluding remarks and discuss the prospects of future work in Section 9. K2.5 V star is a nearby dwarf with a Hipparcos parallax of 77.95 2. Stellar properties of HD 40307 We list the basic stellar properties of HD 40307 in Table 1. This ?? 0.53 mas, which implies a distance of 12.83 ?? 0.09 pc. It is somewhat smaller (M ⋆ = 0.77 ?? 0.05 M⊕; Sousa et al., 2008) and less luminous (log L star/L⊙ = −0.639 ?? 0.060; Ghezzi et al., 2010) than the Sun. The star is quiescent (log R′ HK < −4.99; Mayor et al., 2009a) and relatively metal-poor with [Fe /H] = - 0.31 ??0.03 (Sousa et al., 2008). It also lacks massive planetary companions, which makes it an ideal target for high-precision Table 1. Stellar properties of HD 40307. Parameter Estimate Reference Spectral Type K2.5 V Gray et al. (2006) log R′ HK -4.99 Mayor et al. (2009a) π [mas] 77.95??0.53 van Leeuwen (2007) log L star/L⊙ -0.639??0.060 Ghezzi et al. (2010) log g 4.47??0.16 Sousa et al. (2008) M star [M⊙] 0.77??0.05 Sousa et al. (2008) T eff [K] 4956??50 Ghezzi et al. (2010) [Fe /H] -0.31??0.03 Sousa et al. (2008) v sin i [kms−1] <1 Mayor et al. (2009a) P rot [days] ∼ 48 Mayor et al. (2009a) Age [Gyr] ∼ 4.5 Barnes (2007) RV surveys aiming at finding low-mass planets. According to the calibration of Barnes (2007), HD 40307 likely has an age similar to that of the Sun ( ∼ 4.5 Gyr). 3. Statistical analyses 3.1. Statistical models We modelled the HARPS RVs using a statistical model with a moving average (MA) term and two additional Gaussian white noise components consisting of two independent random variables. The choice of anMA approach instead of binning is based on the results of Tuomi et al. (2012b) and accounts for the fact that uncertainties of subsequent measurements likely correlate with one another at time-scales of an hour in an unknownmanner.We limit our analysis to MA models of third order (MA(3) models) because higher order choices did not improve the noise model significantly. E ffectively, the MA(3) component in our noise model corresponds to binning. However, unlike when binning measurements and artificially decreasing the size of the data set, this approach better preserves information on possible signals in the data. The two Gaussian components of the noise model are the estimated instrument noise with zero-mean and known variance (nominal uncertainties in the RVs) and another with zeromean but unknown variance corresponding to all excess noise in the data. The latter contains the white noise component of the stellar surface, usually referred to as stellar “jitter”, and any additional instrumental systematic e ffects not accounted for in the nominal uncertainties. Keplerian signals and white noise component were modelled as in Tuomi et al. (2011). In mathematical terms, an MA( p) model is implemented on measurement mi as m i = rk (ti) + γ + ǫi + p X j =1 φ jhmi−j − rk(ti−j) − γi exp ti−j − ti, (1) where rk(ti) is the superposition of k Keplerian signals at epoch t i and γ is the reference velocity. The random variable ǫi is the Gaussian white noise component of the noise model with zero-mean and variance σ2 = σ2i + σ2J , where σ2i is the (fixed) nominal uncertainty of the ith measurements and σ2J is a free parameter describing the magnitude of the jitter component. Finally, the free parameters of the MA( p) model are denoted as φ j, j = 1, ..., p – they describe the amount of correlation between the noise of the ith and i − jth measurements. The exponential term in Equation (1) ensures that correlations in the noise are modelled on the correct time-scale. Specifically, using hours as units of time, the exponential term (that is always < 1 because t i > ti −j) vanishes in few hours. To demonstrate the impact of binning on this data set, Section 4.1 shows the analyses of binned RVs with the common assumption that the excess noise is purely Gaussian. This corresponds to using the nightly average as the individual RV measurements and setting φj = 0 for all j in Eq. (1). 3.2. Bayesian analyses and detection thresholds To estimate the model parameters and, especially, their uncertainties as reliably as possible, we drew random samples from the parameter posterior densities using posterior sampling algorithms. We used the adaptive Metropolis algorithm of Haario et al. (2001) because it can be used to receive robust samples from the posterior density of the parameter vector when applied to models with multiple Keplerian signals (e.g. Tuomi et al., 2011; Tuomi, 2012). This algorithmis simply a modified version of the famous Metropolis-Hastings Markov chain Monte Carlo (MCMC) algorithm (Metropolis et al., 1953; Hastings, 1970), which adapts the proposal density to the information gathered from the posterior density. We performed samplings of models with k = 0, 1, ...7 Keplerian signals. The samples from the posterior densities were then used to perform comparisons of the di fferent models. We used the oneblock Metropolis-Hastings (OBMH) method (Chib & Jeliazkov, 2001; Clyde et al., 2007) to calculate the relative posterior probabilities of models with di ffering numbers of Keplerian signals (e.g. Tuomi & Kotiranta, 2009; Tuomi, 2011; Tuomi et al., 2011; Tuomi, 2012). We performed several samplings using di fferent initial values of the parameter vector and calculated the means and the corresponding deviations as measures of uncertainties of our Bayesian evidence numbers P(m|Mk), where m is the measurement vector and Mk denotes the model with k Keplerian signals. The prior probability densities in our analyses were essentially uniform densities. As in Tuomi (2012), we adopted the priors of the RV amplitude π(Ki) = U(0, aRV), reference velocity π (γ) = U(−aRV, aRV), and jitter π(σJ) = U(0, aRV), where U (a, b) denotes a uniform density in the interval [a, b]. Since the observed peak-to-peak di fference in the raw RVs is lower than 10 m s −1, the hyperparameter aRV was conservatively selected to have a value of 20 ms −1. The priors of the longitude of pericentre ( ω) and the mean anomaly (M0) were set to U(0, 2π), in accordance with the choice of Ford & Gregory (2007) and Tuomi (2012). We used the logarithm of the orbital period as a parameter of our model because, unlike the period as such, it is a scale-invariant parameter. The prior of this parameter was set uniform such that the two cut-o ff periodicities were Tmin and T max . These hyperparameters were selected as Tmin = 1.0 days and Tmax = 10Tobs because we did not expect to find signals with periods less than 1 day. Also, we did not limit the period space to the length of the baseline of the HARPS time series ( Tobs), because signals in excess of that can be detected in RV data (Tuomi et al., 2009) and because there might be long-period signals apparent as a trend with or without curvature in the data set. Unlike in traditional Bayesian analyses of RV data, we did not use uniform prior densities for the orbital eccentricities. Instead, we used a semi-Gaussian as π(ei) ∝ N(0, σ2e ) with the corresponding normalisation, where the hyperparameter σe was chosen to have a value of 0.3. This value decreases the posterior probabilities of very high eccentricities in practice, but still enables themif they explain the data better than lower ones (Tuomi, 2012; Tuomi et al., 2012a). For the MA components φj, we selected uniform priors as π (φj) = U(−1, 1), for all j = 1, 2, 3. This choice was made to ensure that theMA model was stationary, i.e. time-shift invariant – a condition that is satisfied exactly when the values are in the interval [-1, 1]. Finally, we did not use equal prior probabilities for the models with di ffering numbers of Keplerian signals. Instead, following Tuomi (2012), we set them as P(Mk) = 2P(Mk+1), which means that the model with k Keplerian signals was always twice as probable prior to the analyses than the model with k + 1 Keplerian signals. While this choice makes our results more robust in the sense that a posterior probability that exceeds our detection threshold is actually already underestimated with respect to equal prior probabilities, there is a physical motivation as well. We expect that the dynamical interactions of planets in any given system make the existence of an additional planet less probable because there are fewer dynamically stable orbits. This also justifies the qualitative form of our prior probabilities for the eccentricities. Our criterion for a positive detection of k Keplerian signals is as follows. First, we require that the posterior probability of a k-Keplerian model is at least 150 times greater than that of the k − 1-Keplerian model (Kass & Raftery, 1995; Tuomi, 2011, 2012; Tuomi et al., 2011; Feroz et al., 2011). Second, we require that the radial velocity amplitudes of all signals are statistically significantly di fferent from zero. Third, we also require that the periods of all signals are well-constrained fromabove and below. These criteria were also applied in Tuomi (2012). We describe the parameter posterior densities using three numbers, namely, the maximum a posteriori (MAP) estimate and the limits of the corresponding 99%Bayesian credibility sets (BCSs) or intervals in one dimension (e.g. Tuomi & Kotiranta, 2009). 3.3. Periodogram analysis As is traditionally the case when searching for periodic signals in time series, we used least-squares periodograms (Lomb, 1976; Scargle, 1982) to probe the next most significant periods left in the data. In particular, we used the least-squares periodograms described in Cumming (2004), which adjust for a sine wave and an o ffset at each test period and plot each test period against the F-ratio statistic (or power) of the fit. While strong powers likely indicate the existence of a periodic signal (though strong powers may be caused by sampling-related features in the data as well), the lack of them does not necessarily mean that there are no significant periodicities left (e.g. Tuomi, 2012). This is especially so in multi-Keplerian fits due to strong correlations and aliases between clearly detected signals and yet-undetected lower amplitude companions (e.g. Anglada-Escud´e, et al., 2010). The reason is that residuals must necessarily be calculated with respect to a model that is assumed to be correct, which is clearly not the case when adding additional degrees of freedom, i.e. additional planetary signals, to the model. Therefore, determining the reliability of a new detection based on goodness-of-fit comparisons is prone to biases, which e ffectively reduces the sensitivity and reliability of these detections (Tuomi, 2012). While periodograms of the residuals are very useful, they do not properly quantify the significance of the possibly remaining periodicities and, therefore, we used them as a secondary rather than a primary tool to assess the significance of new signals. The analytic false-alarm probability (FAP) thresholds as derived by Cumming (2004) are provided in the figures as a reference and for illustrative purposes only. We used the same periodogram tools to assess the presence of periodicities in the time series of a few activity indicators. 4. Analysis of the RV data The 345 measurements taken on 135 separate nights were obtained over a baseline of ∼ 1900 days. In contrast to the discussion in Mayor et al. (2009a), we could not confirm a long-period trend using our new RVs and we did not detect evidence for a trend in the new CCF RVs obtained using the HARPS-DRS. As shown in Anglada-Escud´e & Butler (2012) (Fig. 3), changes in the continuum flux accross each echelle order (also called blaze function) induce RV shifts of several ms −1 if not properly accounted for. This e ffect was reported to affect HARPS measurements in Pepe (2010) and appears to have been fixed by HARPS-DRS v3.5 based on the release notes 1 (issued on 29 October, 2010). With respect to HD 40307 in particular, we found that when RVs were derived without blaze function correction, a strong positive drift ( ∼ 2 ma−1yr−1) was left in the Doppler time series. We speculate that the trend reported earlier may be caused by this blaze function variability. A notable feature of the HD 40307 data set is that the epochs have a long gap of 638 days between 3055 - 3693 [JD-2450000].This feature can e ffectively decrease the phase-coverage of the data on longer periods and complicates the interpretation of periodograms due to severe aliases. 4.1. Analysis of binned data In a first quicklook analysis, we worked with the nightly averages of the radial velocities as obtained from HARPS-TERRA using the standard setup for K dwarfs. In this setup, all HARPS echelle apertures are used and a cubic polynomial is fitted to correct for the blaze function of each aperture. After this correction, the weighted means ˆ ve of all RV measurements within each night e are calculated. As a result, we obtain internal uncertainties of the order of 0.3-0.4 ms −1 for the HARPS-TERRA RVs. Because of stellar and /or instrumental systematic errors, we observed that these individual uncertainties are not representative of the real scatter within most nights with five or more measurements. Using three of those nights we estimate that at least 0.6 m s−1 must be added in quadrature to each individual uncertainty estimate. After this, the uncertainty of a given epoch is obtained as σ−1 e = PNe i (σei ) −1, where the sum is calculated over all exposures obtained during a given night. Finally, based on their longterm monitoring of inactive stars, Pepe et al. (2011) inferred a noise level of 0.7 ms −1 to account for instrumental and stellar noise. After some tests, we found that adding 0.5ms −1 in quadrature to the uncertainties of the nightly averages ensured that none of the epochs had uncertainties below the 0.7 ms −1 level. The typical uncertainties of a single night derived this way were of the order of 0.8 ms −1. These corrections are basically only welleducated guesses based on the prior experiencewith RV data and reported stability of the instrument. Therefore, onemust be especially careful not to over-interpret the results derived from them (e.g., powers in periodograms and significance of the signals). In the fully Bayesian approach, we treat the excess noise as a free parameter of the model, therefore the Bayesian estimates of the noise properties should in principle also be more reliable. First, we re-analysed the nighly binned RVs to see whether we could independently reproduce the results of Mayor et al. 1 www.eso.org/sci/facilities/lasilla/instruments/harps/tools/drs.html 5 10 15 20 0 5 10 15 20 Power 0 5 10 15 20 Power 0 Power 10 100 1000 5 10 15 20 Period [days] 0 Power Nightly binned RVs 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 51-d 34-d 320-d 4-th signal 5-th signal 6-th signal 7-th signal 200-d 200-d 200-d Fig. 1. Least-squares periodograms of the binned HD 40307 radial velocities for the residuals of the models with three (top) to six (bottom) periodic signals. The analytic 10%, 1%, and 0.1% FAPs are shown as horizontal lines. (2009a) when HARPS-TERRA measurements and our Bayesian methods were used. Assuming an unknown Gaussian noise parameter (e.g. Tuomi et al., 2011) in addition to the estimated measurement uncertainties, the posterior samplings and the corresponding model probabilities easily revealed the three strong signals corresponding to periods of 4.3, 9.6, and 20.4 days. The residual periodogram of the three-Keplerian model revealed additional strong periodicities exceeding the 1% FAP level (Fig. 1, top panel) and we tested more complicated models with up to six Keplerian signals. Especially, we tested whether the additional power present in the three-Keplerianmodel residuals (Fig. 1, top panel) at periods of 28.6, 34.8, 51.3, and 308 days, peaking above the 10% FAP level, are statistically significant by starting our MCMC samplings at nearby seed periods. The global four-Keplerian solution was found to correspond to the three previously known super-Earth signals and an additional signal with an MAP period of 320 days. This period was bounded from above and below and its amplitude was strictly positive – in accordance with our detection criteria. The corresponding posterior probability of the four-Keplerian model was 1.9 ??105 times greater than that of a three-Keplerian one, making the 320-day signal significant. In addition to this signal, we could identify a 51-day periodicity (Fig. 1, second panel) that satisfied the detection criteria as well. Including this fifth signal in the model further increased the model probabilitity by a factor of 6.6 ??106. We could furthermore identify a sixth signal with our six-Keplerian model, correponding to a period of 34.4 days. However, even though the samplings converged well and the solution looked well-constrained, the six-Keplerian model was only five times more probable than the five-Keplerian one and would not be detected using our criteria in Section 3.2. Both of the two new significant signals had MAP estimates of their radial velocity amplitudes slightly lower than 1.0 ms −1 – the signals at 51 and 320 days had amplitudes of 0.70 [0.31, 1.09] ms −1 and 0.75 [0.38, 1.12] ms −1, respectively, where the uncertainties are denoted using the intervals corresponding to the 99% BCSs. We note that the periodogram of sampling does not have strong powers at the periods we detect (see Fig. 1 in Mayor et al., 2009a). Given the uncertain nature of the signal at 34.4 days and the potential loss of information when using the nightly averaged RVs (artificial reduction of the number of measurements), we performed a complete Bayesian reanalysis of the full dataset (345 RVs), now including the aforementioned moving average approach to model the velocities. 4.2. Analysis using all RV measurements The analyses of the unbinned data immediately showed the three previously announced signals (Mayor et al., 2009a) with periods of 4.3, 9.6, and 20.4 days. Modelling the data with the superposition of k Keplerian signals and an MA(3) noise model plus the two Gaussian white noise components, our posterior samplings and periodogram analyses identified these signals very rapidly, enabling us to draw statistically representative samples from the corresponding parameter densities. The residual periodogram of this model (three Keplerians and MA(3) components of the noise removed) revealed some significant powers exceeding the 0.1% and 1% FAP level at 320 and 50.8 days, respectively (Fig. 2, top panel). Samplings of the parameter space of a four-Keplerian model indicated that the global solution contained the 320-day periodicity as the fourth signal and yielded a posterior probability for the four-Keplerian model roughly 1 .5??106 times higher than for the three-Keplerian one. The nature of this signal and its relation to the stellar activity (Section 5) is discussed in Section 5.3. We continued by calculating the periodogram of the residuals of our four-Keplerian model (Fig. 2, second panel) and observed a periodogram power that almost reached the 1% FAP level at a period of 50.8 days. Including this fifth signal further increased the posterior probability of our model by a factor of 6 .4 ?? 105. The residuals of the five-Keplerian model contained a periodicity at 34.7 days (Fig. 2, third panel) exceeding the 1% FAP level. The corresponding six-Keplerian model with this candidate received the highest posterior probability – roughly 5 .0??108 times higher than that of the five-Keplerian model. Since the parameters of this sixth candidate were also well-constrained, we conclude that including the 34.7-day signal in the statistical model is fully justified by the data. We also attempted to sample the parameter space of a seven- Keplerian model but failed to find a clear probability maximum for a seventh signal (see also the residual periodogram of the six-Keplerian model in Fig. 2, bottom panel). Although the periodicity of the seventh signal did not converge to a wellconstrained probability maximum, all periodicities in the six- Keplerian model at 4.3, 9.6, 20.4, 34.7, 50.8, and 320 days were still well-constrained, i.e. their radial velocity amplitudes were statistically distinguishable from zero and their periods had clear 5 10 15 20 0 5 10 15 20 Power 0 5 10 15 20 Power 0 Power 10 100 1000 5 10 15 20 Period [days] 0 Power Full spectrum RVs 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 51-d 34-d 320-d 4-th signal 5-th signal 6-th signal 7-th signal 200-d 200-d 200-d Fig. 2. Least-squares periodograms of all 345 RVs of HD 40307 for the residuals of the models with three (top) to six (bottom) periodic signals together with the analytic 10%, 1%, and 0.1% FAPs. parameters of a seventh signal using the posterior samplings, we cannot be sure whether the corresponding Markov chains had converged to the posterior density and cannot reliably calculate an estimate for the posterior probability of the seven-Keplerian model. Therefore we stopped looking for additional signals. From this analysis, we can state confidently that there are six significant periodicities in the HARPS-TERRA radial velocities of HD 40307 when the whole spectral range of HARPS is used. As we show in the next section, one of them has the same period as the chromospheric activity indicator (S-index) and requires more detailed investigation. The analysis of all 345 RVs indicates that for these data binning appears to be a retrograde step in extracting periodic signals from the RV data. We infer that binning serves to alter measurement uncertainties and damp the significance levels of the periodicities in the data. probability maxima. Because we were unable to constrain the 5. Stellar activity We examined the time series of two activity indicators derived from the cross correlation function properties as provided by the HARPS-DRS. They are the bisector span (BIS) and the fullwidth at half-maximum (FWHM) of the CCF. These indices monitor di fferent features of the average stellar line. Briefly, BIS is a measure of the stellar line asymmetry and should correlate with the RVs if the observed o ffsets are caused by spots or plages rotating with the star (Queloz et al., 2001). The FWHM is a measure of the mean spectral line width. Its variability (when not instrumental) is usually associated with changes in the convective patterns on the surface of the star. A third index, the so-called S-index in theMountWilson system (Baliunas et al., 1995), is automatically measured by HARPS-TERRA on the blaze-corrected one-dimensional spectra provided by the HARPS-DRS. The S-index is a measure of the flux of the CaII H and K lines ( λH = 3933.664 Å and λK = 3968.470 Å, respectively) relative to a locally defined continuum (Lovis et al., 2011a) and is an indirect measurement of the total chromospheric activity of the star. For simplicity, the analysis of the activity indicators was performed throughout for the 135 nightly averaged values using sequential least-squares fitting of periodic signals that are each described by a sine-wave model (period, amplitude and phase). 5.1. Analysis of the FWHM and BIS The BIS was remarkably stable (RMS ∼ 0.5 ms−1) and the periodogram of its time series did not show any significant powers. Visual inspection of the time series for the FWHM already shows a very significant trend of 5.3 ms −1 yr−1. The 345 measurements of the FWHM are listed in Table A.2. A sinusoidal fit to this trend suggested a period of 5000 days or more (see top panel in Fig. 3). After removing the trend, two more signals strongly show up in the residuals. The first one was found at 23 days and had an analytic FAP of 0.005%. After fitting a sinusoid to this signal and calculating the residuals, an extremely significant peak appeared at 1170 days with an analytic FAP of 0.002%. After including this in a model with three sinusoids, no additional signals could be seen in the periodogram of the residuals with analytic FAP estimates lower than 10% (Fig. 3, bottom panel). We also show the FWHM values together with the fitted periodic curves in Fig. 4. While the signals in the FWHM were significant, we did not clearly detect their counterparts in the RVs. Given that BIS does not show any obvious signals either, we suspect that the periodicities in the FWHM might be caused by instrumental e ffects, e.g. tiny changes of the focus inside the spectrograph, rather than intrinsic variability of the stellar lines. The instrumental origin would reconcile the absence of correspondent drifts in the BIS and in the RVs. A similar indication of drifts and sensitivity of the FWHM to instrumental issues has been reported in e.g. Lovis et al. (2011a).While this adds some caveats on the long-term stability of the HARPS instrumental profile (and therefore its long-term precision), unless it is found to have similar periods, we see no reason to suspect that any of the signals in the RVs are spuriously induced by changes in the FWHM (intrinsic or instrumental). 5.2. Analysis of the S-index of 345 measurements of the S-index are provided in Table A.2. Again, this analysis was performed for the nightly binned measurements using least-squares periodograms and the sequential inclusion of sinusoidal signals. The last two S-index measurements were well above the average and could not be reproduced by any smooth function. Even if they are representative of a physical process, such outlying points cannot be easily modelled by a series of a few sinusoids because these would add many ambiguities to the interpretation of the results. When these two points were removed, the periodograms looked much The S-index also shows strong coherent variablity. The full set 5 10 15 20 25 30 0 5 10 15 20 25 30 Power 0 5 10 15 20 25 30 Power 0 Power 10 100 1000 5 10 15 20 25 30 Period [days] 0 Power FWHM 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 5000+ d 23-d 1170-d 1st signal 2nd signal 3rd signal 4th signal Fig. 3. Periodogram series of the signals detected in the FWHM, from most significant to less significant (top to bottom). 53000 53500 54000 54500 55000 -20 -10 0 10 20 30 JD-2400000 [days] -30 FWHM - 5910 [m s -1] Fig. 4. Habitable-zone super-Earth candidate in a six-planet system around the K2.5V star HD 40307 Time series of the FWHM activity index. The solid black line represents the best fit to a model containing three sinusoids (periods of Mikko Tuomi ⋆1,2, Guillem Anglada-Escud´e⋆⋆3, Enrico Gerlach4, Hugh R. A. Jones1, Ansgar Reiners3, Eugenio J. Rivera5, Steven S. Vogt5, and R. Paul Butler6 1 University of Hertfordshire, Centre for Astrophysics Research, Science and Technology Research Institute, College Lane, AL10 9AB, Hatfield, UK 2 University of Turku, Tuorla Observatory, Department of Physics and Astronomy, V¨ais¨al¨antie 20, FI-21500, Piikki¨o, Finland 3 Universit¨at G¨ottingen, Institut f¨ur Astrophysik, Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany 4 Lohrmann Observatory, Technical University Dresden, D-01062 Dresden, Germany 5 UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA ABSTRACT Context. Aims. Methods. 6 Department of Terrestrial Magnetism, Carnegie Institute of Washington, Washington, DC 20015, USA Received XX.XX.2012 / Accepted XX.XX.XXXX The K2.5 dwarf HD 40307 has been reported to host three super-Earths. The system lacks massive planets and is therefore a potential candidate for having additional low-mass planetary companions. We re-derive Doppler measurements from public HARPS spectra of HD 40307 to confirm the significance of the reported signals using independent data analysis methods. We also investigate these measurements for additional low-amplitude signals. We used Bayesian analysis of our radial velocities to estimate the probability densities of different model parameters. We also estimated the relative probabilities of models with differing numbers of Keplerian signals and verified their significance using periodogram analyses. We investigated the relation of the detected signals with the chromospheric emission of the star. As previously reported for other objects, we found that radial velocity signals correlated with the S-index are strongly wavelength dependent. Results. Conclusions. We identify two additional clear signals with periods of 34 and 51 days, both corresponding to planet candidates with minimum masses a few times that of the Earth. An additional sixth candidate is initially found at a period of 320 days. However, this signal correlates strongly with the chromospheric emission from the star and is also strongly wavelength dependent. When analysing the red half of the spectra only, the five putative planetary signals are recovered together with a very significant periodicity at about 200 days. This signal has a similar amplitude as the other new signals reported in the current work and corresponds to a planet candidate with M sin i ∼ 7 M⊕ (HD 40307 g). We show that Doppler measurements can be filtered for activity-induced signals if enough photons and a sufficient wavelength interval are available. If the signal corresponding to HD 40307 g is a genuine Doppler signal of planetary origin, this candidate planet might be capable of supporting liquid water on its surface according to the current definition of the liquid water habitable zone around a star and is not likely to su ffer from tidal locking. Also, at an angular separation of ∼ 46 mas, HD 40307 g would be a primary target for a future space-based direct-imaging mission. Key words. 1. Introduction Methods: Statistical, Numerical – Techniques: Radial velocities – Stars: Individual: HD 40307 Current high-precision spectrographs, such as the High Accuracy Radial Velocity Planet Searcher (HARPS; Mayor et al., 2003) and the High Resolution Echelle Spectrograph (HIRES; Vogt et al., 1994), enable detections of low-mass planets orbiting nearby stars. During recent years, radial velocity (RV) planet searches have revealed several systems of super- Earths and /or Neptune-mass planets around nearby stars (e.g. Mayor et al., 2009a,b; Lovis et al., 2011a; Pepe et al., 2011; Tuomi, 2012). The system of three super-Earths orbiting HD 40307 has received much attention because the planets appear in dynamically packed orbits close to mean motion resonances (Mayor et al., 2009a). This has been used as an argument to suggest that lowmass planets may be found in highly compact multiple systems ⋆ ⋆⋆ e-mail: mikko.tuomi@utu.fi; m.tuomi@herts.ac.uk e-mail: guillem.anglada@gmail.com that are still stable in long-term, e.g. a possibility of having ten planets with masses of 17 M ⊕ within a distance of 0.26 AU on stable orbits (Funk et al., 2010). However, the physical nature of these companions as scaled-up versions of the Earths is not entirely clear (Barnes et al., 2009a). Their masses, between those of Earth and Neptune, suggest they are Neptune-like proto-gas giants that could not accumulate enough gas before it was blown away by the newly born star. On the other hand, recent transit observations of hot super-Earths around bright nearby stars (L´eger et al., 2009; Batalha et al., 2011;Winn et al., 2011) indicate that a good fraction of these hot super-Earth mass objects can have rocky compositions. In this article we re-analyse the 345 HARPS spectra publicly available through the ESO archive using a newly developed software tool called HARPS-TERRA (template-enhanced radial velocity re-analysis application; Anglada-Escud´e& Butler, 2012). Instead of the classic cross-correlation function method (CCF) implemented by the standard HARPS-ESO data reduction software (HARPS-DRS), we derive Dopplermeasurements by leastsquares matching of each observed spectrum to a high signal-tonoise ratio template built from the same observations. A description of the method and the implementation details are given in Anglada-Escud´e & Butler (2012). In addition to an increase in precision (especially for K andMdwarfs), this method allows us to performadditional analyses and tests beyond those enabled by the CCF data products provided by the HARPS-DRS. As an example, it allows us to re-obtain the RV measurements using only a restricted wavelength range. As we show in Section 5, this capability can be instrumental in ruling out the planetary nature of prominent signals correlated with stellar activity. We rely on the Bayesian framework when estimating the orbital parameters supported by the data, determining the significances of the signals, and the modelling of the noise in the measurements. In previous studies, radial velocities received within an interval of an hour or so have been commonly binned together in an attempt to reduce the noise caused by stellar- surfacerelated e ffects, i.e. stellar oscillations and granulation, and other factors within this timescale (Dumusque et al., 2010). In principle, this would enable the detections of planets smaller than roughly 5 M ⊕ with HARPS over a variety of orbital distances, even at or near the stellar habitable zone (Dumusque et al., 2010, 2011a). In our approach, and instead of binning, we apply a selfconsistent scheme to account for and quantify correlated noise in the Bayesian framework and use Bayesian model probabilities to show that a solution containing up to six planets is clearly favoured by the data, especially when the redmost part of the stellar spectrum is used in the RV analysis. Only the confluence of refinements in these data analysis methods (re-analysis of the spectra and Bayesian inference) allows the detection and verification of these low-amplitude signals. We start with a brief description of the stellar properties of HD 40307 (Section 2) and describe the statistical modelling of the observations and the data analysis techniques we used (Section 3). In Section 4 we describe the properties of the RV measurements and perform a detailed Bayesian analysis that identifies up to three new candidate signals. We discuss the stellar activity indicators and their possible correlations with the RV signals in Section 5. In this same section, we find that one of the candidates is spuriously induced by stellar activity by showing that the corresponding periodic signal (P ∼ 320 days) is strongly wavelength dependent. When the RVs obtained on the redmost part of the spectrum are analysed (Section 6), the 320-day signal is replaced by a signal of a super-Earth-mass candidate with a 200-day period with a minimum mass of about ∼ 7M⊕ orbiting within the liquid water habitable zone of HD 40307. The analysis of the dynamical stability of the system (Section 7) shows that stable solutions compatible with the data are feasible and the potential habitability of the candidate at 200 days (HD 40307 g) is discussed in Section 8.We give some concluding remarks and discuss the prospects of future work in Section 9. 2. Stellar properties of HD 40307 We list the basic stellar properties of HD 40307 in Table 1. This K2.5 V star is a nearby dwarf with a Hipparcos parallax of 77.95 ?? 0.53 mas, which implies a distance of 12.83 ?? 0.09 pc. It is somewhat smaller (M ⋆ = 0.77 ?? 0.05 M⊕; Sousa et al., 2008) and less luminous (log L star/L⊙ = −0.639 ?? 0.060; Ghezzi et al., 2010) than the Sun. The star is quiescent (log R′ HK < −4.99; Mayor et al., 2009a) and relatively metal-poor with [Fe /H] = - 0.31 ??0.03 (Sousa et al., 2008). It also lacks massive planetary companions, which makes it an ideal target for high-precision Table 1. Stellar properties of HD 40307. Parameter Estimate Reference Spectral Type K2.5 V Gray et al. (2006) log R′ HK -4.99 Mayor et al. (2009a) π v P [mas] 77.95??0.53 van Leeuwen (2007) log Lstar/L⊙ -0.639??0.060 Ghezzi et al. (2010) log g 4.47??0.16 Sousa et al. (2008) M star [M⊙] 0.77??0.05 Sousa et al. (2008) T eff [K] 4956??50 Ghezzi et al. (2010) [Fe /H] -0.31??0.03 Sousa et al. (2008) sin i [kms−1] <1 Mayor et al. (2009a) rot [days] ∼ 48 Mayor et al. (2009a) Age [Gyr] ∼ 4.5 Barnes (2007) 3. Statistical analyses 3.1. Statistical models m i RV surveys aiming at finding low-mass planets. According to the calibration of Barnes (2007), HD 40307 likely has an age similar to that of the Sun ( ∼ 4.5 Gyr). We modelled the HARPS RVs using a statistical model with a moving average (MA) term and two additional Gaussian white noise components consisting of two independent random variables. The choice of anMA approach instead of binning is based on the results of Tuomi et al. (2012b) and accounts for the fact that uncertainties of subsequent measurements likely correlate with one another at time-scales of an hour in an unknownmanner.We limit our analysis to MA models of third order (MA(3) models) because higher order choices did not improve the noise model significantly. E ffectively, the MA(3) component in our noise model corresponds to binning. However, unlike when binning measurements and artificially decreasing the size of the data set, this approach better preserves information on possible signals in the data. The two Gaussian components of the noise model are the estimated instrument noise with zero-mean and known variance (nominal uncertainties in the RVs) and another with zeromean but unknown variance corresponding to all excess noise in the data. The latter contains the white noise component of the stellar surface, usually referred to as stellar “jitter”, and any additional instrumental systematic e ffects not accounted for in the nominal uncertainties. Keplerian signals and white noise component were modelled as in Tuomi et al. (2011). In mathematical terms, an MA( p) model is implemented on measurement mi as = rk(ti) + γ + ǫi + p X j =1 φ jhmi−j − rk(ti−j) − γi exp ti−j − ti, (1) where rk(ti) is the superposition of k Keplerian signals at epoch t i + and γ is the reference velocity. The random variable ǫi is the Gaussian white noise component of the noise model with zero-mean and variance σ2 = σ2i σ2J , where σ2i is the (fixed) nominal uncertainty of the ith measurements and σ2J is a free parameter describing the magnitude of the jitter component. Finally, the free parameters of the MA( p) model are denoted as φ t i j, j = 1, ..., p – they describe the amount of correlation between the noise of the ith and i − jth measurements. The exponential term in Equation (1) ensures that correlations in the noise are modelled on the correct time-scale. Specifically, using hours as units of time, the exponential term (that is always < 1 because > ti−j) vanishes in few hours. To demonstrate the impact of binning on this data set, Section 4.1 shows the analyses of binned RVs with the common assumption that the excess noise is purely Gaussian. This corresponds to using the nightly average as the individual RV measurements and setting φj = 0 for all j in Eq. (1). 3.2. Bayesian analyses and detection thresholds To estimate the model parameters and, especially, their uncertainties as reliably as possible, we drew random samples from the parameter posterior densities using posterior sampling algorithms. We used the adaptive Metropolis algorithm of Haario et al. (2001) because it can be used to receive robust samples from the posterior density of the parameter vector when applied to models with multiple Keplerian signals (e.g. Tuomi et al., 2011; Tuomi, 2012). This algorithmis simply a modified version of the famous Metropolis-Hastings Markov chain Monte Carlo (MCMC) algorithm (Metropolis et al., 1953; Hastings, 1970), which adapts the proposal density to the information gathered from the posterior density. We performed samplings of models with k = 0, 1, ...7 Keplerian signals. The samples from the posterior densities were then used to perform comparisons of the di fferent models. We used the oneblock Metropolis-Hastings (OBMH) method (Chib & Jeliazkov, 2001; Clyde et al., 2007) to calculate the relative posterior probabilities of models with di ffering numbers of Keplerian signals (e.g. Tuomi & Kotiranta, 2009; Tuomi, 2011; Tuomi et al., 2011; Tuomi, 2012). We performed several samplings using di fferent initial values of the parameter vector and calculated the means and the corresponding deviations as measures of uncertainties of our Bayesian evidence numbers P(m|Mk), where m is the measurement vector and Mk denotes the model with k Keplerian signals. The prior probability densities in our analyses were essentially uniform densities. As in Tuomi (2012), we adopted the priors of the RV amplitude π(Ki) = U(0, aRV), reference velocity π U (γ) = U(−aRV, aRV), and jitter π(σJ) = U(0, aRV), where (a, b) denotes a uniform density in the interval [a, b]. Since the observed peak-to-peak difference in the raw RVs is lower than 10 m s −1, the hyperparameter aRV was conservatively selected to have a value of 20 ms −1. The priors of the longitude of pericentre ( ω) and the mean anomaly (M0) were set to U(0, 2π), in accordance with the choice of Ford & Gregory (2007) and Tuomi (2012). We used the logarithm of the orbital period as a parameter of our model because, unlike the period as such, it is a scale-invariant parameter. The prior of this parameter was set uniform such that the two cut-o ff periodicities were Tmin and T max . These hyperparameters were selected as Tmin = 1.0 days and Tmax = 10Tobs because we did not expect to find signals with periods less than 1 day. Also, we did not limit the period space to the length of the baseline of the HARPS time series ( Tobs), because signals in excess of that can be detected in RV data (Tuomi et al., 2009) and because there might be long-period signals apparent as a trend with or without curvature in the data set. Unlike in traditional Bayesian analyses of RV data, we did not use uniform prior densities for the orbital eccentricities. Instead, we used a semi-Gaussian as π(ei) ∝ N(0, σ2e ) with the corresponding normalisation, where the hyperparameter σe was chosen to have a value of 0.3. This value decreases the posterior probabilities of very high eccentricities in practice, but still enables themif they explain the data better than lower ones (Tuomi, 2012; Tuomi et al., 2012a). For the MA components φj, we selected uniform priors as π 3.3. Periodogram analysis (φj) = U(−1, 1), for all j = 1, 2, 3. This choice was made to ensure that theMA model was stationary, i.e. time-shift invariant – a condition that is satisfied exactly when the values are in the interval [-1, 1]. Finally, we did not use equal prior probabilities for the models with di ffering numbers of Keplerian signals. Instead, following Tuomi (2012), we set them as P(Mk) = 2P(Mk+1), which means that the model with k Keplerian signals was always twice as probable prior to the analyses than the model with k + 1 Keplerian signals. While this choice makes our results more robust in the sense that a posterior probability that exceeds our detection threshold is actually already underestimated with respect to equal prior probabilities, there is a physical motivation as well. We expect that the dynamical interactions of planets in any given system make the existence of an additional planet less probable because there are fewer dynamically stable orbits. This also justifies the qualitative form of our prior probabilities for the eccentricities. Our criterion for a positive detection of k Keplerian signals is as follows. First, we require that the posterior probability of a k-Keplerian model is at least 150 times greater than that of the k − 1-Keplerian model (Kass & Raftery, 1995; Tuomi, 2011, 2012; Tuomi et al., 2011; Feroz et al., 2011). Second, we require that the radial velocity amplitudes of all signals are statistically significantly di fferent from zero. Third, we also require that the periods of all signals are well-constrained fromabove and below. These criteria were also applied in Tuomi (2012). We describe the parameter posterior densities using three numbers, namely, the maximum a posteriori (MAP) estimate and the limits of the corresponding 99%Bayesian credibility sets (BCSs) or intervals in one dimension (e.g. Tuomi & Kotiranta, 2009). As is traditionally the case when searching for periodic signals in time series, we used least-squares periodograms (Lomb, 1976; Scargle, 1982) to probe the next most significant periods left in the data. In particular, we used the least-squares periodograms described in Cumming (2004), which adjust for a sine wave and an o ffset at each test period and plot each test period against the F-ratio statistic (or power) of the fit. While strong powers likely indicate the existence of a periodic signal (though strong powers may be caused by sampling-related features in the data as well), the lack of them does not necessarily mean that there are no significant periodicities left (e.g. Tuomi, 2012). This is especially so in multi-Keplerian fits due to strong correlations and aliases between clearly detected signals and yet-undetected lower amplitude companions (e.g. Anglada-Escud´e, et al., 2010). The reason is that residuals must necessarily be calculated with respect to a model that is assumed to be correct, which is clearly not the case when adding additional degrees of freedom, i.e. additional planetary signals, to the model. Therefore, determining the reliability of a new detection based on goodness-of-fit comparisons is prone to biases, which e ffectively reduces the sensitivity and reliability of these detections (Tuomi, 2012). While periodograms of the residuals are very useful, they do not properly quantify the significance of the possibly remaining periodicities and, therefore, we used them as a secondary rather than a primary tool to assess the significance of new signals. The analytic false-alarm probability (FAP) thresholds as derived by Cumming (2004) are provided in the figures as a reference and for illustrative purposes only. We used the same periodogram tools to assess the presence of periodicities in the time series of a few activity indicators. 4. Analysis of the RV data The 345 measurements taken on 135 separate nights were obtained over a baseline of ∼ 1900 days. In contrast to the discussion in Mayor et al. (2009a), we could not confirm a long-period trend using our new RVs and we did not detect evidence for a trend in the new CCF RVs obtained using the HARPS-DRS. As shown in Anglada-Escud´e & Butler (2012) (Fig. 3), changes in the continuum flux accross each echelle order (also called blaze function) induce RV shifts of several ms −1 if not properly accounted for. This e ffect was reported to affect HARPS measurements in Pepe (2010) and appears to have been fixed by HARPS-DRS v3.5 based on the release notes 1 (issued on 29 October, 2010). With respect to HD 40307 in particular, we found that when RVs were derived without blaze function correction, a strong positive drift ( ∼ 2 ma−1yr−1) was left in the Doppler time series. We speculate that the trend reported earlier may be caused by this blaze function variability. A notable feature of the HD 40307 data set is that the epochs have a long gap of 638 days between 3055 - 3693 [JD-2450000].This feature can e ffectively decrease the phase-coverage of the data on longer periods and complicates the interpretation of periodograms due to severe aliases. 4.1. Analysis of binned data e i In a first quicklook analysis, we worked with the nightly averages of the radial velocities as obtained from HARPS-TERRA using the standard setup for K dwarfs. In this setup, all HARPS echelle apertures are used and a cubic polynomial is fitted to correct for the blaze function of each aperture. After this correction, the weighted means ˆ ve of all RV measurements within each night e are calculated. As a result, we obtain internal uncertainties of the order of 0.3-0.4 ms −1 for the HARPS-TERRA RVs. Because of stellar and /or instrumental systematic errors, we observed that these individual uncertainties are not representative of the real scatter within most nights with five or more measurements. Using three of those nights we estimate that at least 0.6 m s−1 must be added in quadrature to each individual uncertainty estimate. After this, the uncertainty of a given epoch is obtained as σ−1 = PNe (σei ) −1, where the sum is calculated over all exposures obtained during a given night. Finally, based on their longterm monitoring of inactive stars, Pepe et al. (2011) inferred a noise level of 0.7 ms −1 to account for instrumental and stellar noise. After some tests, we found that adding 0.5ms −1 in quadrature to the uncertainties of the nightly averages ensured that none of the epochs had uncertainties below the 0.7 ms −1 level. The typical uncertainties of a single night derived this way were of the order of 0.8 ms −1. These corrections are basically only welleducated guesses based on the prior experiencewith RV data and reported stability of the instrument. Therefore, onemust be especially careful not to over-interpret the results derived from them (e.g., powers in periodograms and significance of the signals). In the fully Bayesian approach, we treat the excess noise as a free parameter of the model, therefore the Bayesian estimates of the noise properties should in principle also be more reliable. First, we re-analysed the nighly binned RVs to see whether we could independently reproduce the results of Mayor et al. 0 5 10 15 20 Power 0 5 10 15 20 Power 0 5 10 15 20 Power 10 100 1000 Period [days] 0 5 10 15 20 Power Nightly binned RVs 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 320-d 51-d 34-d 4-th signal 5-th signal 6-th signal 7-th signal 200-d 200-d 200-d Fig. 1. Least-squares periodograms of the binned HD 40307 radial velocities for the residuals of the models with three (top) to six (bottom) periodic signals. The analytic 10%, 1%, and 0.1% FAPs are shown as horizontal lines. (2009a) when HARPS-TERRA measurements and our Bayesian methods were used. Assuming an unknown Gaussian noise parameter (e.g. Tuomi et al., 2011) in addition to the estimated measurement uncertainties, the posterior samplings and the corresponding model probabilities easily revealed the three strong signals corresponding to periods of 4.3, 9.6, and 20.4 days. The residual periodogram of the three-Keplerian model revealed additional strong periodicities exceeding the 1% FAP level (Fig. 1, top panel) and we tested more complicated models with up to six Keplerian signals. Especially, we tested whether the additional power present in the three-Keplerianmodel residuals (Fig. 1, top panel) at periods of 28.6, 34.8, 51.3, and 308 days, peaking above the 10% FAP level, are statistically significant by starting our MCMC samplings at nearby seed periods. The global four-Keplerian solution was found to correspond to the three previously known super-Earth signals and an additional signal with an MAP period of 320 days. This period was bounded from above and below and its amplitude was strictly positive – in accordance with our detection criteria. The corresponding posterior probability of the four-Keplerian model was 1.9 ??105 times greater than that of a three-Keplerian one, making the 320-day signal significant. In addition to this signal, we could identify a 51-day periodicity (Fig. 1, second panel) that satisfied the detection criteria as well. Including this fifth signal in the model further increased the model probabilitity by a factor of 6.6 ??106. We could furthermore identify a sixth signal with our six-Keplerian model, correponding to a period of 34.4 days. However, even though the samplings converged well and the solution looked well-constrained, the six-Keplerian model was only five times more probable than the five-Keplerian one and would not be detected using our criteria in Section 3.2. Both of the two new significant signals had MAP estimates of their radial velocity amplitudes slightly lower than 1.0 ms −1 – the signals at 51 and 320 days had amplitudes of 0.70 [0.31, 1.09] ms −1 and 0.75 [0.38, 1.12] ms −1, respectively, where the uncertainties are denoted using the intervals corresponding to the 99% BCSs. We note that the periodogram of sampling does not have strong powers at the periods we detect (see Fig. 1 in Mayor et al., 2009a). Given the uncertain nature of the signal at 34.4 days and the potential loss of information when using the nightly averaged RVs (artificial reduction of the number of measurements), we performed a complete Bayesian reanalysis of the full dataset (345 RVs), now including the aforementioned moving average approach to model the velocities. 4.2. Analysis using all RV measurements The analyses of the unbinned data immediately showed the three previously announced signals (Mayor et al., 2009a) with periods of 4.3, 9.6, and 20.4 days. Modelling the data with the superposition of k Keplerian signals and an MA(3) noise model plus the two Gaussian white noise components, our posterior samplings and periodogram analyses identified these signals very rapidly, enabling us to draw statistically representative samples from the corresponding parameter densities. The residual periodogram of this model (three Keplerians and MA(3) components of the noise removed) revealed some significant powers exceeding the 0.1% and 1% FAP level at 320 and 50.8 days, respectively (Fig. 2, top panel). Samplings of the parameter space of a four-Keplerian model indicated that the global solution contained the 320-day periodicity as the fourth signal and yielded a posterior probability for the four-Keplerian model roughly 1 .5??106 times higher than for the three-Keplerian one. The nature of this signal and its relation to the stellar activity (Section 5) is discussed in Section 5.3. We continued by calculating the periodogram of the residuals of our four-Keplerian model (Fig. 2, second panel) and observed a periodogram power that almost reached the 1% FAP level at a period of 50.8 days. Including this fifth signal further increased the posterior probability of our model by a factor of 6 .4 ?? 105. The residuals of the five-Keplerian model contained a periodicity at 34.7 days (Fig. 2, third panel) exceeding the 1% FAP level. The corresponding six-Keplerian model with this candidate received the highest posterior probability – roughly 5 .0??108 0 5 10 15 20 Power 0 5 10 15 20 Power 0 5 10 15 20 Power 10 100 1000 Period [days] 0 5 10 15 20 Power Full spectrum RVs 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 320-d 51-d 34-d 4-th signal 5-th signal 6-th signal 7-th signal 200-d 200-d 200-d Fig. 2. probability maxima. Because we were unable to constrain the parameters of a seventh signal using the posterior samplings, we cannot be sure whether the corresponding Markov chains had converged to the posterior density and cannot reliably calculate an estimate for the posterior probability of the seven-Keplerian model. Therefore we stopped looking for additional signals. From this analysis, we can state confidently that there are six significant periodicities in the HARPS-TERRA radial velocities of HD 40307 when the whole spectral range of HARPS is used. As we show in the next section, one of them has the same period as the chromospheric activity indicator (S-index) and requires more detailed investigation. The analysis of all 345 RVs indicates that for these data binning appears to be a retrograde step in extracting periodic signals from the RV data. We infer that binning serves to alter measurement uncertainties and damp the significance levels of the periodicities in the data. 5. Stellar activity times higher than that of the five-Keplerian model. Since the parameters of this sixth candidate were also well-constrained, we conclude that including the 34.7-day signal in the statistical model is fully justified by the data. We also attempted to sample the parameter space of a seven- Keplerian model but failed to find a clear probability maximum for a seventh signal (see also the residual periodogram of the six-Keplerian model in Fig. 2, bottom panel). Although the periodicity of the seventh signal did not converge to a wellconstrained probability maximum, all periodicities in the six- Keplerian model at 4.3, 9.6, 20.4, 34.7, 50.8, and 320 days were still well-constrained, i.e. their radial velocity amplitudes were statistically distinguishable from zero and their periods had clear Least-squares periodograms of all 345 RVs of HD 40307 for the residuals of the models with three (top) to six (bottom) periodic signals together with the analytic 10%, 1%, and 0.1% FAPs. We examined the time series of two activity indicators derived from the cross correlation function properties as provided by the HARPS-DRS. They are the bisector span (BIS) and the fullwidth at half-maximum (FWHM) of the CCF. These indices monitor di fferent features of the average stellar line. Briefly, BIS is a measure of the stellar line asymmetry and should correlate with the RVs if the observed o ffsets are caused by spots or plages rotating with the star (Queloz et al., 2001). The FWHM is a measure of the mean spectral line width. Its variability (when not instrumental) is usually associated with changes in the convective patterns on the surface of the star. A third index, the so-called S-index in theMountWilson system (Baliunas et al., 1995), is automatically measured by HARPS-TERRA on the blaze-corrected one-dimensional spectra provided by the HARPS-DRS. The S-index is a measure of the flux of the CaII H and K lines ( λH = 3933.664 Å and λK = 3968.470 Å, respectively) relative to a locally defined continuum (Lovis et al., 2011a) and is an indirect measurement of the total chromospheric activity of the star. For simplicity, the analysis of the activity indicators was performed throughout for the 135 nightly averaged values using sequential least-squares fitting of periodic signals that are each described by a sine-wave model (period, amplitude and phase). 5.1. Analysis of the FWHM and BIS The BIS was remarkably stable (RMS ∼ 0.5 ms−1) and the periodogram of its time series did not show any significant powers. Visual inspection of the time series for the FWHM already shows a very significant trend of 5.3 ms −1 yr−1. The 345 measurements of the FWHM are listed in Table A.2. A sinusoidal fit to this trend suggested a period of 5000 days or more (see top panel in Fig. 3). After removing the trend, two more signals strongly show up in the residuals. The first one was found at 23 days and had an analytic FAP of 0.005%. After fitting a sinusoid to this signal and calculating the residuals, an extremely significant peak appeared at 1170 days with an analytic FAP of 0.002%. After including this in a model with three sinusoids, no additional signals could be seen in the periodogram of the residuals with analytic FAP estimates lower than 10% (Fig. 3, bottom panel). We also show the FWHM values together with the fitted periodic curves in Fig. 4. While the signals in the FWHM were significant, we did not clearly detect their counterparts in the RVs. Given that BIS does not show any obvious signals either, we suspect that the periodicities in the FWHM might be caused by instrumental e ffects, e.g. tiny changes of the focus inside the spectrograph, rather than intrinsic variability of the stellar lines. The instrumental origin would reconcile the absence of correspondent drifts in the BIS and in the RVs. A similar indication of drifts and sensitivity of the FWHM to instrumental issues has been reported in e.g. Lovis et al. (2011a).While this adds some caveats on the long-term stability of the HARPS instrumental profile (and therefore its long-term precision), unless it is found to have similar periods, we see no reason to suspect that any of the signals in the RVs are spuriously induced by changes in the FWHM (intrinsic or instrumental). 5.2. Analysis of the S-index The S-index also shows strong coherent variablity. The full set of 345 measurements of the S-index are provided in Table A.2. Again, this analysis was performed for the nightly binned measurements using least-squares periodograms and the sequential inclusion of sinusoidal signals. The last two S-index measurements were well above the average and could not be reproduced by any smooth function. Even if they are representative of a physical process, such outlying points cannot be easily modelled by a series of a few sinusoids because these would add many ambiguities to the interpretation of the results. When these two points were removed, the periodograms looked much 0 5 10 15 20 25 30 Power 0 5 10 15 20 25 30 Power 0 5 10 15 20 25 30 Power 10 100 1000 Period [days] 0 5 10 15 20 25 30 Power FWHM 10% 1% 0.1% 10% 10% 10% 1% 1% 1% 0.1% 0.1% 0.1% 5000+ d 23-d 1170-d 1st signal 2nd signal 3rd signal 4th signal Fig. 3. Periodogram series of the signals detected in the FWHM, from most significant to less significant (top to bottom). 53000 53500 54000 54500 55000 JD-2400000 [days] -30 -20 -10 0 10 20 30 FWHM - 5910 [m s -1] Fig. 4. Time series of the FWHM activity index. The solid black line represents the best fit to a model containing three sinusoids (periods of .
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Ο ανόητος ... ψάχνει τα λάθη των άλλων, ο έξυπνος ... του εαυτού του, αλλά ο σοφός ... όλους τους συγχωρεί !!! 
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#17
11-11-2012, 03:55
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
ναι αλλα ειδες τι λεει λιγο πιο κατω απο τη μεση??? εχουν απαγορευσει τα χοντα για παντα......α καλα ειναι πολυ μπροστα αυτοι οι γηινοι
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#18
11-11-2012, 18:22
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
Υπερβολες......τα λεφτα ηταν πολλα Τολη......εεε Αργυρη ενοουσα!!
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#20
11-11-2012, 18:29
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Απάντηση: Εντοπίστηκε μία σούπερ-γη, 42 έτη φωτός μακριά
Απόσπασμα:
Χρησταρα μην τους ακους εγω με σενα ειμαι απλος ο δαιμονας του πιεστηριου που λενε....
 
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