On planetary mass determination in the case of super-Earths orbiting active stars. The case of the CoRoT-7 system
S.Ferraz-Mello, M.Tadeu dos Santos, C.Beauge, T.A.Michtchenko, A. Rodriguez
aa r X i v : . [ a s t r o - ph . E P ] M a r Astronomy&Astrophysicsmanuscript no. c7bc-2 c (cid:13)
ESO 2018May 20, 2018
On planetary mass determination in the case of super-Earthsorbiting active stars. The case of the CoRoT-7 system.
S. Ferraz-Mello , M. Tadeu dos Santos , C. Beaug´e , T.A. Michtchenko , and A. Rodr´ıguez Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas (IAG) – Universidade de S˜ao PauloRua do Mat˜ao, 1226 Cep: 05508–090 - S˜ao Paulo-Brasile-mail: [email protected] Observat´orio Astron´omico de C´ordoba, Universidad Nacional de C´ordobaArgentinaPreprint online version: May 20, 2018
ABSTRACT
Context.
Due to the star activity, the masses of CoRoT-7b and CoRoT 7c are uncertain. Investigators of the CoRoT team have proposedseveral solutions, all but one of them larger than the initial determinations of 4 . ± . M Earth for CoRoT-7b and 8 . ± . M Earth forCoRoT 7c.
Aims.
This investigation uses the excellent HARPS radial velocity measurements of CoRoT-7 to re-determine the planet masses andto explore techniques able to determine mass and elements of planets discovered around active stars when the relative variation of theradial velocity due to the star activity cannot be considered as just noise and can exceed the variation due to the planets.
Methods.
The main technique used here is a self-consistent version of the high-pass filter used by Queloz et al. (2009) in the first massdetermination of CoRoT-7b and CoRoT-7c. The results are compared to those given by two alternative techniques: (1) The approachproposed by Hatzes et al. (2010) using only those nights in which 2 or 3 observations were done; (2) A pure Fourier analysis. In allcases, the eccentricities are taken equal to zero as indicated by the study of the tidal evolution of the system; the periods are also keptfixed at the values given by Queloz et al. Only the observations done in the time interval BJD 2,454,847 – 873 are used because theyinclude many nights with multiple observations; otherwise it is not possible to separate the e ff ects of the rotation fourth harmonic(5.91 d = P rot /
4) from the alias of the orbital period of CoRoT-7b (0.853585 d).
Results.
The results of the various approaches are combined to give for the planet masses the values 8 . ± . M Earth for CoRoT-7band 13 . ± . M Earth for CoRoT 7c. An estimation of the variation of the radial velocity of the star due to its activity is also given.
Conclusions.
The results obtained with 3 di ff erent approaches agree to give masses larger than those in previous determinations. Fromthe existing internal structure models they indicate that CoRoT-7b is a much denser super-Earth. The bulk density is 11 ± . . cm − .CoRoT-7b may be rocky with a large iron core. Key words. star:individual:CoRoT 7 – planetary systems – star:activity – methods: statistical – techniques: radial velocities – massdetermination – CoRoT 7b – CoRoT-7c – exoplanets: hot super-Earths
1. Introduction
CoRoT-7b was the first super-Earth for which mass and radiushave been determined. CoRoT-7b and the recently discoveredGJ 1214b (Charbonneau et al. (2009)) and Kepler-10b (Batalhaet al. (2011)) are paradigms for the study of the physics of whatexo-Earths, super-Earths and / or mini-Neptunes can be. They setthe only real constraints available to models of the formation andevolution of hot telluric planets. For this reason, it is very impor-tant to have good radius and mass determinations. The radius ofCoRoT-7b, determined from the transits observed by CoRoT , is10,100 ±
600 km (Bruntt et al. (2010)), a value that may be im-proved, but whose magnitude is nevertheless definitively estab-lished. The mass, determined from the radial velocity measure-ments (4 . ± . . ± . M ⊙ (see Lanza et al. (2010)), 1-2 Gyr-oldG9V star CoRoT-7 ( = TYCHO 4799-1733-1) is too active. The The CoRoT space mission, launched on December 27th 2006, hasbeen developed and is operated by CNES, with the partnership ofAustria, Belgium, Brazil, ESA, Germany and Spain. variation in the measured radial velocity comes mainly from theactivity of the star whose spots determine the value of the radialvelocity integrated over its disk. This activity follows roughlythe rotation period of the star (23.64 days), but in a very irregu-lar way: the 150-day photometric observations done by CoRoTshow the variation displaying full span in some times, but almostdisappearing in others (see L´eger et al. (2009)). Because of thisactivity, CoRoT-7b is also the paradigm of the kind of problemsthat may be often found when planets with Earth-like masses arediscovered. One of the aims of the investigation reported in thispaper is to use CoRoT-7, for which an excellent set with 109HARPS radial velocity measurements exists, to explore tech-niques that may be used when dealing with low-mass planets.We may wish that future discoveries are done around more quietstars, but Earth-like planets are too important and we cannot dis-card any of them because of the central star activity. We hope thatCoRoT, KEPLER and ground based instruments will discovernew Earths, super-Earths and mini-Neptunes which, as Kepler-10b, may be in orbit around quiet stars. However, as CoRoT-7b,they may be found around non-quiet stars and the improvementof the techniques of mass and orbit determination used to study
1. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars
780 800 820 840 860 880
BJD-2,454,000 -2002040 R ad i a l V e l o c i t y ( m / s ) Fig. 1.
Relative radial velocity measurement (dots) and the con-tribution of the activity as estimated with a filtering using the firstthree harmonics of the rotational period (solid line).(Adaptedfrom Fig.8 of QBM)such cases is important. This need is motivating a great deal ofinvestigations and the number of papers dealing with the massof CoRoT-7b (QBM; Boisse et al. (2011); Hatzes et al. (2010);Pont et al. (2011) ) is increasing. In addition to this interest, wehave to consider that HARPS, currently the only instrument ableto make measurements of CoRoT-7 with the required precision,is being on demand by a great deal of other targets and we can-not foresee when a new series of measurements with the samequality of the existing one will become possible.The nature and magnitude of the problem of the mass and orbitdetermination of the CoRoT-7 planets can be assessed from fig. 1(adapted from fig. 8 of QBM). It shows the 5 sets of measure-ments of the radial velocity of CoRoT-7 obtained with HARPS(dots) and the estimated part of the radial velocity due to the staractivity. The activity shown in fig. 1 (solid line) was obtainedin QBM by means of a filtering designed to eliminate periodicdisturbances corresponding to the first three harmonics of therotational period. These sets present patterns very di ff erent onefrom another showing that the construction of one single modelfor the activity in the whole interval is not possible. It also dis-courages the extensive use of Fourier tools (periodograms) overthe whole set. We have to search for techniques able to separatethe low-frequency rotation signals (periods 23.64 days and itsmain harmonics), from the higher frequency signal coming fromthe two planets (periods 0.853585 and 3.698 days cf. QBM). Thedi ff erence between the periods of the two components (rotationand planets) indicates that, in this case, we may filter the datafrom its low-frequency parts and consider separately the high-frequency information. In the case of one continuous signal, orat least of a long discrete evenly spaced time series, the problemthat we have to solve is classical and well known: We shouldconstruct and use a high-pass filter. However, the simple recipesto construct a high-pass filter in the frequency domain cannotbe used for a series with a limited amount of data and, worse,unevenly distributed. Because of the high correlation betweenthe absolute values of the Fourier transform (or spectral power)at di ff erent frequencies, these filters must necessarily be con-structed in the time domain.One high-pass filter in the time domain was used in QBM inthe first mass determination of CoRoT-7b and-7c. However, asdiscussed in Section 2 of this paper, the filter then used a ff ectedthe high-frequencies, the importance of which was downsizedwithout apparent reason, and set the cut-o ff below the frequency We follow the common usage in Physics where N th harmonic meansan oscillation whose frequency is N times the fundamental frequency.Thus the fundamental frequency 2 π/ P rot is the first harmonic, the sec-ond harmonic is the component whose period is P rot /
2, and so on. of the main alias of the period of CoRoT- 7b, thus a ff ecting alsothe amplitudes corresponding to this planet (cutting out the alias,the filter also considerably downsized the signal correspondingto the actual frequency of the planet).This paper starts with an analysis of the mass determination pub-lished in QBM’s CoRoT-7c discovery paper (section 2) and thenproposes a self-consistent agorithm founded on the filtering tech-nique used there (section 3), which is applied to the selected setof 52 observations made in 27 consecutive nights, between BJD2,454,847 and BJD 2,454,873 (section 4) to obtain the massesof the planets. The restriction of the analysis to this set of datesis due to the fact that, in 10 nights of this set, 3 observationswere done, spanning about 4 hours between the first and lastobservation in the night. This is not enough to completely de-stroy the aliasing due to the almost uniform spacing betweenobservations done in consecutive nights, but it allows us to dis-tinguish between two solutions with forced periods equal to thetransit period and its alias (see Hatzes et al. (2010) figure 7). Itis worth stressing the fact that in a series made of observationstaken always near the same hour in the night, no mathemati-cal tool exists able to distinguish between one frequency and itsaliases. In section 4, the dependence of the results on the filterparameters is also discussed. In sections 5 to 7, we present theresulting estimate of the star activity and analyze the residualsobtained by subtracting the activity from the observed radial ve-locities. Alternative techniques are discussed in sections 8 and9. textbf The approach discussed in section 8 follows a sug-estion by Hatzes et al. (2010) and uses only the observationsfrom nights where multiple observations were done. This is ofparticular importance because it allows an analysis independentof any explicit hypotheses on the behavior of the star activity.These observations are analysed here with the help of a biasedMonte-Carlo technique allowing confidence intervals to be ob-tained. The approach discussed in section 9 is a classical multi-period Fourier analysis. It di ff ers from other approaches using aFourier decomposition by the fact that, here, no a priori periodsare used. The periods of the solution found are those allowing usto get the best fit of the observations to a multi-periodic function.In section 10, we present some simulation results taking into ac-count tidal interactions showing the circularization of the orbitsand thus justifying the adoption of zero eccentricities for bothplanets. At last, we proceed with the discussion of the resultsand the conclusions.
2. Analysis of the first mass determination
The analysis of the mass determination published in the CoRoT-7c discovery paper (QBM) is the first step in this study and themain point to be considered concerns the filtering properties ofthe procedure used there. Is it equivalent to a high-pass filter? Inorder to know that, we compute the Fourier transforms of thegiven data and of the residuals obtained in QBM after subtract-ing the activity, respectively, and compare them one to another.The result presented in fig. 2 shows that, indeed, the used pro-cedure completely filtered the low-frequencies (the transform ofthe filtered series is close to zero for all frequencies below 0.22 d − . However, it also a ff ected the high-frequency components.The strong alias of the period of CoRoT-7b, at 5.925d (indicated Spectra obtained using date-compensated discrete Fourier trans-forms (DCDFT; cf Ferraz-Mello (1981)). DCDFT di ff ers from usualLomb periodograms because they also consider the constant compo-nent, whose neglect may a ff ect the height of the peaks (see discussionon floating-mean periodograms in Cumming et al. (1999) )2. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars FREQUENCY (1/d) F O UR I E R T R A N S F O R M B' C
Fig. 2.
Fourier transform (DCDFT) of the measured radial veloc-ities (black) and of the values obtained in QBM after subtractingthe estimated activity (red). Both transforms are presented us-ing the same units. C and B indicate the orbital frequency ofCoRoT-7c and the 5.925-day alias of the CoRoT-7b frequency,respectivelywith B’ in fig. 2), disappeared. The signal of CoRoT-7c was lessa ff ected by the filter but it was downsized without apparent rea-son, since no low-period terms were included in the filter. Themain reason for this result is the fact that the filtering was doneon the raw data, without taking into account the part of the signaldue to the planets. We may guess that the fact that these periodsare not at all commensurable with the Coherence Time (window)used in the filter (20 days) may have played a role in the deepsculpting done by the filter at the planet frequencies and theiraliases (the average of a periodic function over a time intervalnot commensurable with its period is di ff erent of zero.).A second factor may have been the fact that fittings are exactwhen the number of dates is smaller (as well known). Thus, nearthe borders of the interval, the estimated activity will be closerto the given data than in the middle of the period. The end e ff ectseems to be responsible for the fish-like appearance of the fil-tered velocities in the largest set (narrow in the extremities andwide in the middle; See the grey line and dots in fig. 8). Onecould think that such appearance might result from a particularbeat of the orbital frequency of CoRoT 7c and the nearby aliasof the orbital frequency of CoRoT-7b, but an a posteriori plot ofsynthetic velocities shows that this is not so. Because of the ac-tual phases and periods, the destructive interference of the twosine curves (at the actual discrete observation times) does nothappen in the borders of the interval, but near the middle of it.It is worth adding that one unconstrained 3-sinewave analy-sis of the filtered radial velocities in QBM, in the period BJD2,454,847 - 873, using a genetic algorithm completed with adownhill simplex gave as more important periods present in thefiltered data, 3.495d and 3.963d. The di ff erence in frequency ofthese two periods is 0 . d − , which is the inverse of 29.6 d(very close to the timespan of the observations used in the anal-ysis), clearly showing the interference of the timespan of the ob-servations in the considered subset.
3. A self-consistent high-pass algorithm
The high-pass filter used in QBM may be shortly described asfollows. First, we define one time window (the Coherence Time),fixed as being a guess on the number of days in which it maybe reasonable to fit the given harmonic function to the activity.Then, we construct N time windows of the chosen size, eachcentered on one of the N dates of the observations and including all observations inside the window. In every window, the dataare Fourier analyzed and represented by the first terms of a har-monic series whose fundamental period is the star rotation pe-riod (23.64 d) plus a constant. The activity at a given date is es-timated as the (weighted) average of the values given to it by theharmonic representations of the signal in all subsets includingthe given date. For the sake of clearness, let us add the followinginformation: (i) The actual window size is fixed in such a waythat the ends of the window do not separate observations done inthe same night (if necessary, the actual window is taken slightlylarger than the nominal Coherence Time); (ii) Near the borders,the subsets are incompletely filled as the windows extend to be-yond the considered interval covering some nights where no ob-servations were done.In this paper, we propose an improvement of this procedure. Themain change is that, now, a predicted signal corresponding to thetwo planets is subtracted from the radial velocities before the fit-ting of the harmonic function. Next, the activity is estimated asdescribed above and subtracted from the observed radial veloc-ities; the resulting residuals are used to determine the masses.The new masses may then be used as first guesses in a new runof the algorithm and leads to an improved prediction of the sig-nal corresponding to the two planets, and so on. The procedureis iterated as many times as necessary up to reach a satisfactoryconvergence. Formally, we may say that the whole procedure de-fines a map x n + = F ( x n ), (here, x represents the masses) whichis iterated up to get x n + = x n . One problem appearing in theactual application of this scheme is a possible slow convergenceof this map. For this reason, to obtain the results discussed inthis paper, we have rather used an alternative accelerated map: x n + = F ( x n + λ ( x n − x n − )) with λ = m ) is almost constant. To havedi ff erent convergence ratios along two orthogonal directions isa common feature in maps; in this case it seems to be due tothe slow separation between the planet CoRoT 7b and the fourthrotation harmonic. However, notwithstanding the slow conver-gence, the five runs converged to the same point P , representedby a star in Fig. 3 The reliability of the map was checked usingsome synthetic sets of data constructed using the first, secondand fourth rotation harmonics with the amplitudes indicated insome Fourier analyses (see section 9), two sine curves of half-amplitude 6 m.s − , corresponding to two planets, and a Gaussiannoise. The results are as follows: (1) When the noise is not in-cluded, the map reproduces the two planets exactly as given;(2) When noise in the range 1.7–2.5 m.s − is added, the re-sults for CoRoT-7b fall around the given value within 0.3 m.s − .However, the results for CoRoT-7c fall systematically ∼ − below the given value. We have taken these results into accountin estimating the error bars of the final results.The analysis of the spectrographic parameters related to thestar activity (FWHM, Rhk, bisector span) shows the contribu-
3. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars
MASS 7c X sin i (Earth=1) M ASS ( E a r t h = ) Q1QBM P Fig. 3.
Evolution of the masses in five chains of results iterativelyobtained from five di ff erent sets of initial conditions (diamonds)using the 4-harmonic high-pass filter and observations done inthe period BJD 2,454,847 – 873. All of them converge to thesame point P (star). One of the chains started at (0,0). The figurealso shows the result of the first iteration in QBM (point Q1) andthe mass values adopted in that paper.tion of the rotation higher harmonics to the observed activity.However, they were of little help to decide on the kind of filter-ing to be used. The power spectra of these parameters in the pe-riod BJD 2,454,847 – 873 are shown in the fig. 4. Unfortunately,the features of the spectra of the bisector span and FWHM showonly small bumps at the position of the higher-order harmon-ics, but similar bumps were found when using scrambled datathus showing that the observed one cannot be distinguished frombumps generated by white noise and are thus meaningless. Thecorresponding data seems to be a ff ected by the low brightnessof the star. The only power spectrum showing significant peaksabove the minimum level of significance is the power spec-trum of the index log R hk where peaks corresponding to higher-order harmonics are clearly seen even if some o ff set due to theshort time span of the observations used can be noted. A simi-lar analysis using the photometric observations done by Quelozet al.(2009) (QBM) showed almost no influence of the 4 th har-monic and led them to neglect it in the construction of the high-pass filter. We will use both filters with and without the fourthharmonic and consider them in the composition leading to theconclusions of this paper on the mass of the planets. In additionwe mention that the power spectra clearly show some higher-order harmonics which may a ff ect our results. The considerationof them would require new improvements, di ff erent of those pre-sented in this paper.We shall mention that the use of the rotation’s fourth harmonicraises some critical questions. Indeed its period is one fourth of23.64d, that is 5.917 d, and one of the main aliases of the pe-riod of 7b is 5.925 d. However, aliases are defined for uniformlyspaced time series with observations separated by multiples ofone constant value (e.g. one sidereal day) and cannot be avoided PERIOD (days) P O W E R FWHMBSlogRhkP/4 P/3 P/2
Fig. 4.
Power spectra of the spectrographic parameters associ-ated with the star activity: bisector span, FWHM and log R hk , inthe period BJD 2,454,847 – 873. The exact location of the mainrotation harmonics is indicated. The gray rectangle is below theminimum significance level. Peaks inside this rectangle can beproduced by random data.as far as the separation between observations is kept unaltered.This is a classically known fact and it has been taken into ac-count in the scheduling of the spectroscopic observations as soonas the observations showed this coincidence. The observationsdone in the period JD 2,454,847 –873 include 52 data obtainedin 27 consecutive nights and in 10 of these nights 3 observa-tions were done covering about 4 hours. This was not su ffi cientto completely eliminate aliasing problems, but power spectra ex-tended beyond the nominal Nyquist frequency showed that theperfect mirroring of the power spectrum has been avoided ( seefig. 7; see fig. 10 of QBM). Several other tests were done. Leastsquares determinations involving simultaneously the photomet-ric period 0.853585 d and the fourth rotation harmonic lead tocorrelation values in the range 0.70 – 0.88 depending on the de-sign of the experiment done and the observations used. Some ofthese results are worrisome, but the final test is provided by thecoincidence of the convergence points in chains starting at verydi ff erent mass values (as the test ones shown in fig. 3). One char-acteristic of least-squares procedures involving highly correlatedparameters is that the results become erratic. This has never beenthe case here. However, it is clear that without the multiple-dataper night policy adopted in the last periods of observations, itwould be absolutely impossible to separate the rotations fourthharmonic from the alias of the period of CoRoT-7b.In what concerns the higher harmonics, we mention that theproximity of the period of the 6 th harmonic (3.94 days) to theperiod of CoRoT-7c (3.698 days) is a problem of di ffi cult solu-tion. They are far enough to be separated one from another, butthe beat period of the two components (about 60 days).is muchlarger than the timespan of the used subset (27 days). When thealgorithm used in this paper is extended to include these har-monics, its convergence becomes excessively slow. The Fourieranalysis of the residuals obtained with the 4-harmonic high-passfilter and an extended time interval (BJD 2,454,847 – 884) indi-cated that the amplitude of the radial velocities due to CoRoT-7cmay be a ff ected in up to 0.5 m.s − . This possible o ff set will betaken into account in the final results.
4. The mass of the planets
The technique described in the previous sections is certainly thebest one we can devise to eliminate from a given series of un-evenly spaced observations the contributions of irregular longperiod terms and to get a remaining part which may be used to
4. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars determine the parameters of a short-period signal (the planets).It is now applied to the observations. However, for the reasonsdiscussed above, we give up using all observations, but concen-trate on the set of observations done in the period BJD 2,454,847– 873. In addition to the aliasing problem, the consideration ofthe whole set – formed by 5 di ff erent subsets spanning 106 daysplus 3 isolated observations one year before – is made di ffi cultby the irregular variation of the stellar activity from the epoch ofone set to the next.Another important setting in this determination is that we willconcentrate on the masses and fix the periods in the values pre-viously determined (L´eger et al. (2009) and QBM). One of thereasons is that having restricted the interval under study to a setof only 27 days, there is no possibility of improving a perioddetermination resulting from observations taken from a time in-terval 4 times larger. The analysis of the covariance matrix withthe 27- day data shows very high correlation (0.97) between pe-riods and phases for both planets what means that this short setcannot be used to simultaneously determine periods and phases.The results of this algorithm depend on the model used in thehigh-pass filter. Two main parameters were investigated. One ofthem is the model used in the interpolation to determine the ac-tivity at a given date: here, we considered both the 3- and the 4-rotation harmonics models. In a lesser extent, the 6-harmonicsmodels has also been considered but the beat of the periods ofCoRoT-7c and that of the 6 th rotation harmonic impairs the pro-cedure convergence. The other parameter is the Coherence Time,which sets the size of the window. We investigated several ofthem starting with the 20-day interval as used by QMB, but con-sidered also some other values in the range 8–22 days. The re-sults are shown in fig. 5. As far as the Coherence Time is kept ina limited interval, the results do not show large variations. Also,since the codes themselves depend on some operational param-eters that might a ff ect the results, the procedures of filtering andmass determination were done with two very di ff erent codes:one, a lengthy steepest descent (diamonds) and the other a two-part code using a genetic algorithm completed with a downhillsimplex (crosses). Labels indicate the corresponding CoherenceTimes. For the set shown by crosses, only the highest label (22)was shown to avoid excessive overlap of symbols and labels; thesolutions with Coherence Times between 8 and 12 days clusteraround m B = m Earth . The others lie between these two limits.The green rectangle indicate a joint interval of confidence. Theindividual statistical errors of the self-consistent determinationswere estimated as 0.5 m.s − that is, 0.7 and 1.1 Earth masses, for7b and 7c respectively.The sets of masses obtained for both planets show significantdependence on the used high-pass filter. The mass of CoRoT-7bshows some variations, following the 4 th rotation harmonic is in-cluded or not in the model. When the 4 th harmonic is included,we obtain for CoRoT-7b a mass above 8 Earth masses, while theresults with only 3 harmonics is smaller than 8 Earth masses, Onits turn, the models with 3 and 4 harmonics indicate, for CoRoT-7c, a mass around 14 Earth masses, but the results becomesless well determined when higher harmonics are included, ina way leading to believe that the actual mass of CoRoT-7c issmaller than the obtained value. The results of some runs usingmore observations (the 72 observations done in the interval BJD2454845 – 873), also result for CoRoT-7c a mass smaller than 13Earth masses. The results depend also on the period adopted forthe planets. Because of its very short period, CoRoT-7b showsa larger sensitivity. Fortunately, the period of CoRoT-7b is verywell known from the CoRoT photometry and this source of errorcan be discarded. Q1 Q2 B 5P AH 16
MASS 7c X sin i (Earth =1) M ASS ( E a r t h = )
18 16 FF Fig. 5.
Planet masses resulting from several runs of the iterativehigh-pass filtering with 3-, 4-harmonic filters (sets 3H, 4H). Theother labels indicate the Coherence Time in days (see details inthe text). The green square represents the interval of confidenceof the solution given in Table 2. The large blue cross indicatesthe adopted solution in QBM and the red squares Q1 and Q2the solutions corresponding to the published values of K , K , inQBM. The solutions A (alternative; only CoRoT 7b), B(Boisse),H(Hatzes), 5P (Fourier with 3 harmonics), FF(free Fourier) willbe discussed in sections 8 and 9.The experiments done have also shown that the results are sensi-tive to the adopted weighting rules. All results in this paper wereobtained using, at the beginning, the standard errors publishedby the observers (L´eger et al.(2009)), which were propagatedfollowing the classical rules of the least-squares formulas forunequally weighted observations (see Linnik, (1963)) . In thesequence, variances were obtained for each subset used in thefiltering and were used to weight the averages on each date giv-ing the estimate of the activity. At last, let it be reminded that themass determination from the filtered radial velocities cannot usethe same weights as the given observations. The filtered RVs aredi ff erences between the observed RV and the estimated activityand are, therefore, a ff ected by the errors in both these quantities.The classical formulas after which the variance ( σ ) of the dif-ference is the sum of the variances of the two quantities enteringin the subtraction is used and the new weights are defined as theinverse of the resulting variances.
5. The star activity
Figure 6 shows the activity determined using the 3-, 4-harmonichigh-pass filters (bottom) and the di ff erence between them (top).. The used classical weighting rules are able to take into account thefact that the observations done at the dates 2454860.75 and 2454864.63have a quality worse than the others; the weight associated to them wassome 10 times less than the weight given to the more precise observa-tions in the set. 5. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars [4H] [3H]
25 30 35 40 45 50 55BJD - 2454820-20020 R V ( m / s ) -404 D R V ( H - H ) Fig. 6.
Bottom: Star activity resulting from the high-pass filter.The labels [4H] and [3H] indicate the results obtained respec-tively with 4 and 3 harmonics. The color coding of the com-ponents of these curves, when visible, is the same as in Fig. 7.The dots indicate the measured radial velocities with their errorbars. The error bars of the activity estimated with the 4 harmon-ics high-pass filter are also shown. Top: Di ff erence between theresults labeled 4H and 3HThe activities determined with both filters and various values ofthe Coherence Time are shown. For a given model, they do notshow visible di ff erences, notwithstanding the fact that these dif-ferences exist and a ff ect the mass determination. The analysis ofthis figure may be summarized in a few words: (1) The filteringis very robust with respect to the chosen Coherence Time andmodel; (2) The results are more smooth when less harmonicsare included in the filter.The activity estimated here may be compared to the one esti-mated by Pont et al. (2011) from the analysis of the bisectorspan measurements. It is remarkable that the features of the ac-tivity given if Fig. 2 of Pont et al. are very similar to those shownin fig. 6 and corresponding to the 4-harmonic filtering; howeverone may note that the total span of the RV due to the star activityis, there, about half of that shown in fig. 6.
6. Quality of the new high-pass filter
As done for the QBM determination, we may compute theFourier transforms of the residuals V obs − V activ and compare themto that of the observations. Some Fourier transforms (DCDFT)are shown in fig. 7 (top) for the two models and several val-ues of the Coherence Time. In order to avoid the complicationarising from many almost overlapping curves, we present onlythose transforms for which the height of the peak near the fre-quency labeled B’ (alias of the orbital frequency of CoRoT-7b)nearly matches the peak of the transform of the observed data.It is worth stressing that the frequency B’ is not separated fromthe side lobes of the rotation period (which are broad because ofthe short time span of the used observation set). Because of thissuperposition, it is di ffi cult to assess the quality of the filtering B C'
Frequency (1/d) F O UR I E R T R A N S F O R M ( e x t en s i on ) B' C
Frequency (1/d) F O UR I E R T R A N S F O R M P r o t P r o t/ P r o t/ P r o t/ Fig. 7.
Top: Fourier Transforms (DCDFT) of the observed(black) and filtered data with the 3-and 4-harmonic high-passfilters in the period BJD 2,454,847 873. Bottom: Extension ofthe transforms to beyond the Nyquist frequency to show theirbehavior below 1 day. B , C are the orbital frequencies of CoRoT7b and 7c; B ′ , C ′ are aliases of B , C . The rotation period and itsfirst harmonics are indicated on the top axis.by inspection of the filtered spectra at this frequency. We may re-member that the composition of two frequencies in a spectrum isnot just an addition since each of them carries one phase and thee ff ect of the superposition cannot be assessed only by comparingtheir moduli. In this case, the superposition is reinforced by theshort time span of the considered data. With a longer timespan,peaks would be sharper (as in fig. 2) and could be separated, butusing all available data in the analysis would mean to work witha discontinuous set of observations, which introduces additional(and in some extent unsolvable) di ffi culties in the estimation ofthe activity. One noteworthy e ff ect of the superposition is the ap-parent enhancement of the peaks at B and B’.The frequency of CoRoT-7c, however, is less a ff ected, at least asfar as higher-order harmonics are not included.For the sake of giving an additional information on the aliasinge ff ects, we present in fig. 7 (bottom) an extension of the trans-forms to an interval of frequencies including the actual orbitalfrequency of CoRoT-7b (labeled B) and one alias of the orbitalfrequency of CoRoT-7c (labeled C’). As expected, the trans-forms in the given intervals are almost identical in both plots, butnot perfectly identical because of the large proportion of nightswith multiple observations in the selected time interval.The comparison of the Fourier transforms shows that the bestquality filtering were obtained with Coherence Times 12-14 dayswhen using the 4-harmonic filter and 18-20 days when using the3-harmonic filter (the curves for 18 and 20 days are almost iden-tical).
6. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars
30 40 50
TIME (BJD - 2,454,820) -15-10-50510 F I L T E R E D R A D I A L VE L O C I T Y ( m / s ) -10-5051015 H4 t =14 dQBM Fig. 8.
Filtered radial velocities obtained with the 4-harmonichigh-pass filter and Coherence Time τ = d (top) compared tothose given in QBM (bottom).
7. The filtered radial velocities
Figure 8 (Top) shows the radial velocities obtained subtractingfrom the measured radial velocities the activity resulting fromthe use of the 4-harmonic filter and Coherence Time τ = d .They may be compared to the ones used in the discovery paper(Fig. 8 Bottom). The main di ff erences appear in the beginningof the interval, where the filtered radial velocities appear muchlarger in our results than in QBM and in the middle of the inter-val where the contrary occurs. It can be easily seen that aroundthe date 42, in our results, a destructive beat occurs between theRV sine curves corresponding to the two planets, at the actualobserving times.We should stress the fact that the epoch BJD 2,454,820 used inthe plots and calculations of this paper is not the same epochused in other papers. It is an arbitrary epoch close to the ac-tual dates and allows the phases (i.e. the longitudes at the fixedepoch) to depend less strongly on the periods, thus making thenumerical procedures more robust. In adition, we note that inall steps one additive constant representing the radial velocity ofthe planetary system is determined together with the other un-knowns.
8. A model-independent approach
The use of a high-pass filter allowed us to separate parts ofthe RV measurements due to high and low frequency compo-nents. The results are, however, model-dependent. The high-pass filtering has given statistically coherent estimates for themass of CoRoT-7c, but that result can carry some systematice ff ects due to some higher-order rotation harmonics. In whatconcerns CoRoT-7b, the filtering led to two di ff erent solutions.From the purely statistical point of view, the 4-harmonic filter-ing should be better than the 3-harmonic filtering. However be-cause of the aliasing involving the rotation 4 th harmonic and theorbital period of CoRoT-7b, it is convenient to confirm the re- sults with some alternative technique independent of assump-tions involving the rotation period and its harmonics. An al-ternative approach was suggested by Hatzes et al. (2010) (andpaper in preparation) in which only data from nights in whichmultiple observations were done are used, and assuming that inthe 4 hours time span of one observation night, the activity ofthe star does not change and may be fixed as the same for thethree observations . We note that there are 10 nights in the in-terval 2,454,847 - 873, in which three consecutive observationswere done. Then, the problem is to fit those observations with 2planetary sine waves (the eccentricities may be assumed equalto zero as discussed in section 10) and a set of 10 independentconstants added to the data in the corresponding dates. The mainproblem with this approach is the small number of degrees offreedom in the best-fit problem. We have 30 data and 14 un-knowns. The number of degrees of freedom is 16. In analogywith a chi-square distribution, we may guess that every solutionleading to a w.r.m.s ≤ − higher than the minimum shallbe considered as belonging to the standard interval of confidence(see Press et al. (1986)). In order to improve the determination,we have chosen to add to these observations those from 16 othernights in which 2 observations were done. We have then 62 dataand 30 unknowns. The number of degrees of freedom becomes32. This is twice of what we would have if only using the 10nights with 3 observations each and means a less broad confi-dence interval, including only the solutions leading to a w.r.m.s. ≤ − larger than the minimum.To assess the set of all these ’good-fit’ solutions, we may usethe same biased Monte-Carlo technique used to obtain good fitsfor the planets HD 82943 b,c (Ferraz-Mello et al. (2005)). As be-fore, the frequencies were fixed at the values given in QBM (theyare well determined) and only 4 unknown planetary parameterswere considered: The two masses and the two phases. The (bi-ased) random search produced thousands of solutions. Thosewith w . r . m . s . ≤ .
67 m.s − are shown in fig. 9. In that figurewe superpose the results for the half-amplitudes K of the twoplanets. The mass of CoRoT-7b is constrained to the interval m B = . ± . m Earth (i.e.4 . ≤ K ≤ − ) and its longi-tude at the epoch BJD 2,454,820.0 is 200 ± w . r . m . s . ≤ .
67 m.s − exist for masses in a broadinterval: m C sin i ≤ m Earth ( K C ≤ . − ).The poor constraining of the solution for CoRoT-7c deservessome discussion. The alternative technique proposed by Hatzeset al. (2010) is good to separate low and high-frequency contri-butions. The period of CoRoT-7c is not short enough to be con-sidered as low. In the more favorable situations, the variationsof the RV due to CoRoT-7c may reach 2 m.s − in 5 hours. It isgenerally less, and not distinguishable from contributions com-ing from low-amplitude long-period terms. We cannot avoid therandom process used from mimicking, in the activity, contribu-tions with the same period as CoRoT-7c and from combiningthem with the planet RV.The results are not accurate but they are important because oftheir independence with respect to assumptions on the activitybehavior.The best constrained fit, defined as the leftmost point in fig. 9corresponds to the mass m B = . The results from the used RV modeling indicate that variations aresmaller than the observational errors. However, in one extreme (andinfrequent) case, the estimated activity varied by 0.8 m.s − between thefirst and the last observation done in the night. 7. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars Fig. 9.
Good fits obtained with a biased Monte-Carlo under thecondition wrms ≤ .
67 m.s − . The blue dots correspond toCoRoT-7b and the red ones to CoRoT-7c.mthe solutions oin the considered interval is shown by point Ain fig. 5.It is important to stress that each set of masses and phases of thetwo planets obtained in this experiment corresponds to one solu-tion whose residuals fit the observations better than 1.67 m.s − (w.r.m.s.) and cannot be discarded. The density of the points infig. 9 has no meaning (it rather reflects the strategy used to con-struct the sets). We note that only the error bar for m B is given.It was not possible to get a good determination for m c sin i (theerror bar would extend more or less over the whole width of thefigure).
9. Fourier approaches
As a first guess, one could assume that the observations canbe represented by the sum of several periodic functions anduse conventional non-windowed Fourier analyses to determinetheir parameters (amplitudes, periods, phases). The solutionsindicated by B and H in fig. 5 are those obtained by Boisse etal. (2011) and Hatzes et al. (2010) from the Fourier analysis ofthe observations. Boisse et al’s solution is the best fit of 5 sinecurves to the radial velocities, 3 of them having the periods ofthe rotation and its first harmonics. For that solution, besidesthe 52 points used in this paper, the remaining 9 points of thelatest observational subset were also considered. Hatzes et al’ssolution was obtained using all observations and a pre-whiteningplus filtering procedure based on periodograms; 9 di ff erent sinecurves contributed to the construction of the solution.If we proceed in the same way as Boisse et al. (2011) butconsidering only the set of 52 observations done used inprevious sections, the result is that shown by the symbol5P in fig. 5. In fact, we have done a great deal of Fourieranalyses of the measurements, which remained unpublishedonly because they assume a periodic behavior in the activitywhich is very improbable. Every function in a finite intervalmay be represented by a Fourier series, but in the present case,the main period needs to be the rotation period. Such periodicstructure is not seen in the residuals published by QBM (shownin fig. 1 above), in our own activity curve (shown in fig. 6above), in the activity as determined by Pont et al.( (2011);their fig. 1) or in the photometric series produced by CoRoT(L´eger et al. (2009)). In all these plots there is some kind Table 1.
Fourier decomposition of the data in the time interval2,454,847 – 873. The resulting w.r.m.s of the residuals is 1.76m.s − Period (d) K (m.s − ) Mass (Earth = × sin i ± + . − . ± ± ± ± ± ± + . − . ± ± + . − . of repetition associated with the rotation period, but in somesections the curve appears dominated by the rotation period, inothers by the rotation harmonics, and so on. None of the partialcurves appearing in fig. 1 shows the 23.64 period indicatedby photometry. An irregular behavior is also seen in the lightcurve, whose periodogram clearly shows the rotation period andits harmonics, but which is such that in some sections of it novariations are seen, while in others the rotation period is wellmarked.We report here only one of the experiments which consistedof several steps: (a) The periods were initialized in relativelybroad intervals bracketing the rotation period and its harmonics,allowing for possible di ff erences due to either the physics of theactivity or the beat between the actual periodic signal and thesampling set (actual observation dates). (b) The approximatedmore probable elements (including the periods) were determinedvia a chain of best-fits to the data; (c) In each step, the elementswere determined by the simultaneous best-fit of N trigonometriccomponents. The process was stopped when the addition of onenew term was no longer able to improve significantly the results(F-test) and the scrambled data produced spectra with peaks ofthe same size as the ones obtained with the real data. In suchcase, the inclusion of more unknowns in the process may lead tohigh correlation between the unknowns and to results that maybecome undistinguishable from artifacts.The eventual results of the free Fourier approach are shownin fig. 5 (point FF) and detailed in Table 1. The error barswere estimated with a biased Monte Carlo sampling of theneighborhood of the solution, which showed very asymmetricdistributions in some cases.It is important to stress the fact that the weakest of the fiveterms is already very uncertain as both the spectrum of theresiduals in the previous step and the a posteriori F-test haveshown that no actual improvement was obtained by the additionof this term to the solution.One important by-product of this analysis concerns the con-jecture of the existence of one third planet at P = . d , whichcannot be studied with the other techniques described in thispaper, since they do not allow such a slow periodic variationto be distinguished from the star activity. Peaks correspondingto periods around 9 days are recurrent in all analyses done,since the beginning of this investigation. They can be seenfor instance in the spectra of the raw data in fig. 2 and fig. 7.However, always, when a monochromatic filter is used and asinus curve with the amplitude and period of the rotation issubtracted, that peak disappears. We conjecture that it resultsfrom a complex beat between the rotation period, the samplingdates and the planets themselves.
8. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars
TIME (Myr) E CC E N T R I C I T Y J(0)=40 deg
Fig. 10.
Orbit evolution of two planets with masses 8 and 15Earth masses in a CoRoT-7-like system with mutually inclinedorbits. The damping of the eccentricities is due to the tides onthe inner planet only.
10. The eccentricities
In the mass / orbit determination presented in this paper, the ec-centricities were taken equal to zero. Indeed, an analysis of thedynamical problem shows that the tidal dissipation in the planetCoRoT-7b and its gravitational interaction with CoRoT-7c dampboth eccentricities. Simulations were done where the forces dueto the tide raised in the planet (cf Mignard (1979)) were addedto the gravitational ones. In all of them, the osculating eccentric-ities of the two planets stabilize in values of the order of resp.10 − and 10 − in a few tens of Myr, whichever initial eccen-tricities and inclinations are considered. In the solution shown infig. 10, the planets are initially on circular orbits with semi-majoraxes 0.0175 and 0.0456 AU (like CoRoT-7c) in two planes witha mutual inclination of 40 degrees. The chosen initial eccentric-ities do not a ff ect the solution because, immediately after thebeginning of the simulation, the gravitational interaction of the2 planets forces the eccentricities to be larger than 0.1 and 0.01respectively (In the coplanar simulations they jump to the equi-librium values fixed by the mutual perturbations; cf. Mardling(2007); Rodr´ıguez (2010); Rodr´ıguez et al. (2011)); Thereafter,the tidal dissipation in the inner planet starts dissipating the en-ergy of the system, making the orbit of CoRoT-7b slowly spiraldown towards the planet and become circular while the semi-major axis of CoRoT-7c remains almost unchanged. In addition,the exchange of angular momentum between the two planets alsodamps the eccentricity of CoRoT-7c and drives the planets to anequilibrium configuration (see Mardling (2011)). It is worth re-calling that after the circularization of the orbit of CoRoT-7b, thetidal friction in this planet almost ceases (see Ferraz-Mello et al.(2008)) and no longer continues to significantly a ff ect the evolu-tion of the system. Let be added that in some coplanar runs start-ing with eccentric orbits, the final eccentricities are yet smallerthan the ones shown in fig. 10.
11. Discussion
Three independent approaches were used to assess the massesof the exoplanets CoRoT-7b and CoRoT-7c notwithstanding thedi ffi culties created by the activity of the star which contaminatesthe measured velocities not as a jitter, but as a dominant signalsome 2 – 3 times more intense than the planetary contributions.Before comparing the results, it is worth stressing the fact that
30 40 50
BJD - 2454820 -10010 R E DUC E D R V ( m / s ) -10010 O - C ( m / s ) Fig. 11.
Bottom: Filtered radial velocities with the 4-harmonichigh-pass filter and 14-day Coherence Time (dots) and RV curvecorresponding to the planets solution obtained with the same pa-rameters. Top: Di ff erences between the filtered RV and the RVcurve.the residuals of the 3 analyzed models have weighted r.m.s. lessthan 1.9 m.s − , what may be considered good given the irregularactivity of the star. However, we have to prevent against usingthe minimum w.r.m.s. of the models to mutually comparethem, because, by definition, these quantities are minima ofthe residuals with respect to a given model and therefore, maybe considered as model-dependent. They shall be used only tomake comparisons within a given model.The introduction of a self-consistent algorithm improved thefiltering used in QBM and resulted in masses considerablylarger than the ones previously obtained. One fact influencingthe results is the adoption of a 4-harmonic filter in addition to the3- harmonic filter. The analysis of CoRoT’s photometric seriesdid not reveal the need of using the fourth harmonic. However,the analysis of the spectrographic parameters associated withthe activity of the star (mainly log R hk ) showed the importanceof high-orders harmonics even beyond the fourth. In addition,the fact that the 4 th harmonic is very close to an alias of theperiod of CoRoT-7b, makes its introduction necessary.The coincidence between the period of the fourth rotationharmonic (5.91 d) and an alias of the orbital period of CoRoT-7b(5.925 d) is a problem of major concern in this study. We haveto stress first that in series of astronomical observations doneat the same hour angle it is impossible to solve any aliasingproblem. In such case, a linear relationship appears betweenthe two components. We may force at will the value of onecomponent and compensate with the value of the other. Thereis no mathematical tool able to solve this problem. In this case,given the slight di ff erence between the two frequencies (5.91and 5.925) and the complexity of the signal, the only way toget one solution is to break the uniformity of the time intervalsin the sample by means of observations done in very di ff erenthour angles. Unfortunately, with observation from only oneobservatory this is not easy. Di ff erences of a few hours withrespect to the mean are the maximum that could have beenobtained in this case. The set of observations selected for thisstudy has 52 observations done in 27 nights, including 10 nightswith 3 observations in a 4 hours interval and 5 nights with 2observations. This is the only hope that we have to solve thebeat of the two frequencies before new observations are done.They were indeed separated using a 4-harmonic high-pass filter
9. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars and the convergence of the self-consistent iteration routes tothe same result (see fig. 3) is an indication that, in the adoptedalgorithm, they may be considered as independent.One additional comment with respect to the beat comes from theanalysis of fig. 6 (top). That figure shows the di ff erence betweenthe estimated activity in the two cases: with 3 and 4 harmonics,respectively. It is an evaluation of the contribution of the fourthharmonic to the estimated activity of the star. The surprise, inthis case, is that the apparent period of this contribution is not5.91 d, but only 5.25 d. Assuming that the rotation period isthe same observed by CoRoT, this di ff erence would mean thatwe are not dealing with one frozen periodic signal, but with asignal that is just nearly periodic: as the evolution of a periodicprocess whose period and phase are continuously changing.The other possibility is that the period is not the same as thepublished one. We note that 5.25 is one fourth of the mainperiod found in the free Fourier analysis of the radial velocities,21.15 days, and that the frequency of the highest peak in theFourier spectrum of the data, in fig. 7 is near 0.5 d F − − , inverse of 23.64 d. Whatever the reason responsiblefor this di ff erence, it certainly contributed to the fact that the4-harmonic filter succeeded to get one solution with the twocomponents separated, while classical least-squares solutionand a covariance analysis using the filtered RV obtained with thepublished rotation was unable to separate the orbital period ofCoRoT-7b from the alias of 4 th harmonic. As a check, we havedone some runs of our codes using this lower period. The resultsare masses near the values obtained using the actual values ofthe rotation period. So they confirm those results and show thatthe filtering algorithm is robust with respect to variations in therotation period used in the filter, at least as far as it is close tothe adopted ones.The quality of the results may be assessed from the residuals(O-C) and from the fitting of the observations to the modelshown in fig. 11. The obtained (O-C) correspond to a w.r.m.s.1.9 m.s − . Two comments may be added here. (1) The errorbars in fig. 11 are larger than those of the RV measurements.They result from the addition of the variances of the RVmeasurements to that of the activity given by the used model.The large errors in some dates in the middle of the intervalcomes from the fact that, in those dates, only one observationwas done per night, leading to larger statistical errors in theestimation of the activity in these dates; (2) These results absorbnaturally RV jumps as those that led Pont et al (2010) to statethat HARPS systematic errors may be huge in some cases.One of the 20 m.s − jumps reported by them, which occurredbetween the dates BJD 2454868 /
69 (
BJD − = / − ) isadded to it, one may verify that the part of the RV increase nonexplained is still large, but just 1 / The variation from 23.64 to 21,1 days in the activity period is con-sistent with di ff erential rotation di ff erences expected in a solar-type star(see Thomas and Weiss, (2008)). However, the time span of the observa-tions here considered is not large enough to allow us to give full creditto this result and the question should be reconsidered when new databecome available.
222 224 226 228 230
LONGITUDE 7c (deg) L O N G I T UD E ( deg ) Fig. 12.
Longitudes at the epoch BJD = + simplex code (crosses), only some results are labeled to avoidoverlaps. Table 2.
Masses and elements (Epoch BJD 2,454,820.0)
CoRot-7b CoRot-7cK (m.s − ) 5 . ± . . ± . † ) 8 ± . . ± . ± ± ± † The mass of CoRoT 7c is minimal ( × sin i ) and combined with a biased Monte Carlo approach. Theresulting mass of CoRoT-7b lies close to results obtained usingthe 3-harmonic filter. The resulting mass of CoRoT-7c is illdetermined because the period of CoRoT-7c is not small enoughto allow it to be fully separated from the star activity.
12. Conclusions
The final results were obtained by combining the results of the3- and 4-harmonic filtering and of the multiple-observations-per-night alternative approach. They are given in Table 2. The re-sults from filtering were considered when the two codes usedwere convergent and lead to nearby values. Otherwise they werenot considered in the final estimation of the results. In particu-lar we mention that the largest and least Coherence Times (22and 8 respectively) were not considered. The other criterion wasthe quality of the filtering as indicated by the comparison of theFourier Transforms before and after the filtering (fig. 7)We remind that, because of the small timespan of the obser-vations used, we renounced to determine the periods and used
10. Ferraz-Mello et al.: Planetary mass determination of super-Earths orbiting active stars those given in the discovery papers (L´eger et al. (2009) andQBM).With the period determined from the transits and the date of thefirst transit of CoRoT-7b observed by CoRoT, we obtain 196.4degrees for the expected longitude at the epoch BJD 2454820.0.The small o ff set with respect to the result shown in table2 corre-sponds to a slight correction in the period, which would then be0 . ± . ffi culty in the re-sults for CoRoT-7c comes from the fact that it is di ffi cult to dis-entangle the mass of this planet from the rotation higher har-monics (mainly the 6 th ). In the case of the beats between theperiod of CoRoT-7b and the 4 th harmonic, it was possible to dis-entangle them by making multiple observations in many of thenights of the latest observation periods. In the case of CoRoT-7c,the beats are not related to aliasing, and the problems they raisecannot be solved by the strategy of making multiple observa-tions per night nor by mathematical tools allowing low and high-frequency components to be separated. To solve this kind of en-tanglement we should have observations spanning over a largertime interval. We have done some calculations using all observa-tions available in the interval BJD 2554825 – 884, increasing thetime span from 27 to 60 nights. This means to include discon-tinuities in the observations set, which make the analysis moredi ffi cult and, in some sense, of less confidence. In these runs themass obtained for CoRoT-7c is below 13 Earth masses. Addingthis to the evidences from the Fourier analysis of the residualsused in the runs labeled as 4H, that the 6 th rotation harmonicmay contribute with 0.5 m.s − to K , we may guess that valuesbetween 13 and 14 Earth masses are the more probable ones.One may wish that the improvement of techniques as the onerecently proposed by Pont et al. (2011) applied to simultaneousphotometric and spectrographic observation done over long timespans be able to give a final answer to this question in future.However, with the existing observations, we are confident thatthe above estimates are the best ones we can obtain and that allconsistent estimations fall in the ranges given above or, at least,in its immediate neighborhood.The longitudes at BJD = ± . / cm . This ismuch more than the 6.6 g / cm resulting from the mass given inQBM and mean not only that CoRoT 7b is rocky, but also thatthe contribution due to the iron core must be higher (between 50and 65 percent in mass; see Fortney et al. (2007); Seager et al.(2007)) and that the density at its center may be close to 25 g / cm (see Seager et al. (2007)). These values are higher than thosecorresponding to other known rocky planets, but all consistentdeterminations of the mass of CoRoT-7b lead to bulk densitiesof at least 9 g / cm showing that the high-density of CoRoT-7bis a constraint to be taken into account in the modeling of theplanet.The comparison of the various methods used in this paper al-lows us to say that the high-pass filter used by QBM (Queloz et al. (2009)), embedded in a self-consistent algorithm, is the bestone we can devise to disentangle long- and short-period termsin a given series of unevenly spaced observations. Its superiorityover Fourier analyses with fixed frequencies comes from the factthat the used running window allows the method to treat a signalwhich is not periodic or has a period di ff erent of the period usedin the filter. The procedure proposed by Hatzes et al. (2010) ofusing only nights with multiple observations and including anadditional V for each date is an important complement. It suf-fers, however, from a shortcoming due to the great number of un-knowns that it involves. As a consequence the number of degreesof freedom is small and the resulting confidence intervals are toolarge. In the current case, it just allowed us to determine the massof CoRoT 7b. This was great because this was the ultimate goalof our work, but the impossibility of getting reasonable confi-dence intervals for the mass of CoRoT 7c shall be mentioned.The planning of new observations should take this into accountand have as many multiple night observations as possible. Acknowledgements.
SFM and CB acknowledge the fellowships of the IsaacNewton Institute for Mathematical Sciences, University of Cambridge (UK)where they have developed the tools used to investigate this problem. MTS ac-knowledge FAPESP. The continuous support of CNPq (grants 302783 / / References