arXiv:astro-ph/0303055v1 3 Mar 2003 tp://cdsweb.u-strasbrg.fr/A+A.htx (130.79.128.5) http cdsarc.u-strasbrg.fr to ftp anonymous via S Cie,wt h OAI pcrgaha h 1.2-m the at spectrograph telescope CORALIE Swiss the Euler with (Chile), ESO [email protected] ihdffrn hfs h euti w-iesoa corre- two-dimensional a is result The shifts. templates different the two with of two of combination a the correlation against of spectrum the observed velocities calculates radial TODCOR the two components, by of combination shifted a spectra resolved. is known easily spectrum be observed cannot with the components Assuming deal two when the to of spectra, lines 1979) double-lined the Davis in encountered gener- & difficulties a Tonry the 1974; as tech- cross-correlation (Simkin introduced (one-dimensional) nique was the It of 1994). alization Mazeh spectra binary & double-lined (Zucker of ve- components two- radial both the of a derive locities to is technique correlation CORrelation) dimensional (TwO-Dimensional TODCOR Introduction 1. edopitrqet to requests offprint Send ⋆⋆ ⋆ DI ilb netdb adlater) hand by inserted be will (DOI: Astrophysics & Astronomy ut-re OCR plcto oosrain ae wit taken observations to application TODCOR: Multi-order al sol vial neetoi oma h CDS the at form electronic in available only is 4 Table ae nosrain olce tteL il Observatory Silla La the at collected observations on Based 3 2 1 te fvr an opnos h ehiu sfis applie first is technique The 3 companions. is faint enable very orders many of of ities combination The spectra. multi-order Abstract. 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V the in primary the than fainter magnitudes 68 hspprpeet napiaino h w-iesoa C TwO-Dimensional the of application an presents paper This ehd:dt nlss–tcnqe:rda eoiis–bi – velocities radial techniques: – analysis data methods: .Zucker S. h OAI cel spectrograph echelle CORALIE the aucitn.zuckernew no. manuscript hyZce,e-mail: Zucker, Shay : 1 , 2 .Mazeh T. , .Tesse D41004 HD system The I. 1 .C Santos C. N. , rvia or , − 1 loigtemaueeto ogtr rn nthe in trend long-term a of measurement the allowing , mzn h rcso ftemaue eoiis eneed we velocities, measured max- the en- while of companion, to precision faint the a order detect imizing In to ability spectra. our hance multi-order the spec- to modern produce the Due trographs of companions. many technology, faint detector detect in progress to ability the hance cooler detecting for favorable is ratio spec- stars. IR flux on the searched TODCOR where who apply (2002), tra, to al. had et companions, Prato faint and for Therefore, (2002) lines. al. spectral using et of Mazeh number signal- secondaries the the and on faint ratio mainly 1994). to-noise depending detect limited, Zucker still to is & radial TODCOR ability (Mazeh the the secondaries deriving However, faint when of of important feature velocities most This ob- system. is the combined in TODCOR the contained of information spectrum spectral served templates the different two all of utilizes use spec- different simultaneous of the are types, secondary tral the secondary. and the primary and the primary When the for templates different use secondary. the and primary the the both identifies of simultaneously velocities radial peak whose function, lation eet h aeo D404 hr h secondary the where 41004, HD of case the to here d h eeto n esrmn fterda veloc- radial the of measurement and detection the s so f0 of ision h oensetorpsoe nte aht en- to path another offer spectrographs modern The to ability its is TODCOR of advantages the of One 3 en,Switzerland verny, ,RyodadBvrySclrFclyo Exact of Faculty Sackler Beverly and Raymond s, .Udry S. , aut fEatSine,TlAi University, Aviv Tel Sciences, Exact of Faculty r clr ihacmaino 19 a of companion a with rcular, . ms km 6 idt OAI pcr fti system, this of spectra CORALIE to lied ais pcrsoi tr:individual: stars: – spectroscopic naries: ⋆,⋆⋆ 3 Reain(OCR loih to algorithm (TODCOR) ORrelation n .Mayor M. and , − 1 n aiiae nobtlsolution orbital an facilitates and 3 M J minimum coe 7 2018 27, October h 2 S. Zucker et al.: Multi-order TODCOR: application to CORALIE & ELODIE to combine the spectral information in all the relevant or- Table 1. Stellar parameters of HD 41004 A (reproduced from ders. Originally, TODCOR was devised to analyse only Santos et al. 2002) single-order spectra, and further generalization is there- fore needed in order to use the information in multiple Parameter Value orders. Spectral Type K1V/K2V When applying TODCOR to each order separately, Parallax [mas] 23.24 the weak signal of the secondary may produce a certain Distance [pc] 43 mv 8.65 local peak of the correlation function at the correct sec- B − V 0.887 ondary velocity. However, this peak can be easily topped Teff [K] 5010 by spurious random peaks. This rules out, for example, log g [cgs] 4.42 calculation of the independently for each MV 5.48 order and then averaging the velocities, since many of [L⊙] 0.65 the secondary velocities would have resulted from wrong Mass [M⊙] ∼ 0.7 ′ peaks. Concatenation of the spectral orders to one single log RHK −4.66 spectrum would require special treatment to the gaps and Age [Gyr] 1.6 Prot [days] ∼ 27 overlaps between adjacent orders. Co-adding overlapping −1 regions and bridging the gaps both require interpolation, v sin i [km s ] 1.22 − which would introduce artificial noise into the analysed [Fe/H] 0.09/+0.10 spectrum. The approach we suggest here is to calculate the corre- 2. Characteristics of HD 41004 lation function for each order separately, and then combine the correlation functions of all the orders. The combina- HD 41004 is a visual double system, consisting of a K1V– M2V pair. According to the Hipparcos catalogue, the pair tion emphasizes the relevant correlation peak, and aver- ages out the spurious random peaks. A simple average is a common-proper-motion pair, with a V-magnitude dif- ′′ ± ′′ of the correlation functions may not be efficient enough, ference of 3.68 and a separation of 0.541 0.033 . The basic stellar parameters of HD 41004 A are summarized in since the combination scheme has to consider the differ- ent spectral information in the different orders, and weigh Table 1, reproducing Table 1 of Santos et al. (2002). The them accordingly. Zucker (2003) introduces such a scheme, effective temperature, surface and were calculated by Santos et al. through Str¨omgren photome- based on a few plausible statistical assumptions. try, which also showed that the star is photometrically Using this scheme, multi-order TODCOR is applied stable within the instrumental precision. An independent in this work to the CORALIE spectra of HD41004. As estimate of the metallicity was obtained from analysis of Santos et al. (2002) have shown, these spectra are com- the Cross-Correlation Function of CORALIE. The two in- posed of the spectra of the two visual components of the ′′ dependent estimates of the metallicity are quoted in the system, that are separated by 0.5 . Santos et al. found table. that the radial velocities measured by CORALIE for these The principal result of Santos et al. (2002) is the de- spectra showed a minute periodical variability. They sug- tection of a radial-velocity periodical variability in the gested that this apparent variation was actually related CORALIE spectra of HD41004. The variability pattern to a much larger variation in the velocity of the faint B presented in their paper is consistent with the presence component. The presence of the spectrum of HD41004B of a planet orbiting HD 41004 A with a 1.3- period. caused a variable asymmetry in the line shape of the com- However, Santos et al. rejected this possibility based on posite spectra, which was reflected in a minute variability the bisector shape analysis of the CORALIE spectra. This of the measured radial velocity. analysis revealed a periodic line shape variation, having TODCOR, when applied to the multi-order CORALIE the same period as the radial-velocity variation. spectra of HD 41004, derived the radial velocities of both The interpretation Santos et al. suggested is the pres- components without any assumptions regarding the orbit. ence of an object orbiting the M2V star HD41004Ba. In The derived velocities confirm the conjecture of Santos this model A (the K star) and B are orbiting each other et al.: the B component indeed moves periodically with a in a wide orbit, while B is further composed of two ob- 1.3-day period, and the A component shows a long-term jects, Ba (the M star) and Bb, in a very close orbit, with trend. We present here an orbital solution for the B com- a period of 1.3 days. According to this interpretation, the ponent and derive the slope of the long-term trend of the CORALIE spectra of HD41004 are a combination of the A component, based on the precise velocities derived by A and B components, because their separation is much TODCOR. smaller than the diameter of the CORALIE fiber. Since The next section briefly reviews the previous results the flux of the B component in the relevant wavelength concerning HD 41004. The analysis and its results are pre- range is only about 3% of the flux of the primary, large sented in Sections 3 and 4. Section 5 discusses the results radial-velocity variations of B cause only small variations and their implications. Section 6 concludes the paper with of the measured radial velocity, but their effect is mani- a few remarks. fested in the bisector shape. S. Zucker et al.: Multi-order TODCOR: application to CORALIE&ELODIE 3

Santos et al. performed some simulations to test their Table 2. Known stellar characteristics of the templates finally interpretation and concluded that their results are con- chosen sistent with the presence of a orbiting Primary Secondary HD 41004 Ba, with a radial-velocity amplitude of the or- −1 Parameter Template Template der of 5kms . They also found a long-term linear trend in the radial velocities, which did not agree with the Name HD 52698 GJ 393 Spectral Type K11 M2.51 older measurements. They ascribed it to the motion of 1 1 B − V 0.89 1.52 HD41004A around HD41004B, but did not rule out an Teff [K] 5235 N/A additional component that may be involved in this varia- log g 4.69 N/A tion. [Fe/H] 0.21 N/A v sin i [km s−1] 2.0 < 2.92 3. Analysis 1. H¨unsch et al. 1999 The data we analysed comprised the 86 CORALIE spec- 2. Delfosse et al. 1998 tra of HD41004 used by Santos et al. (2002), which were obtained between November 2001 and February 2002, ex- cept for one spectrum obtained in December 2000. The dial velocities exhibited a clear nearly circular orbit with −1 system was further monitored and in this paper we add an amplitude of around 6 km s , and a linear trend of the 17 spectra obtained as of March 2002. primary. We finally chose the configuration which yielded CORALIE is a fiber-fed, cross-dispersed echelle spec- the smallest residuals in the secondary velocities, relative trograph, mounted on the 1.2-m Leonard Euler telescope to the best-fit orbital solution. Table 2 lists the known stel- at La Silla (Queloz et al. 1999). With a resolution of lar characteristics of HD52698 and GJ393 – the two tem- λ/∆λ = 50000, it covers the wavelength range 3800– plates finally chosen. The stellar parameters of HD 52698 6900 A,˚ with 68 echelle orders. The CORALIE system uses (effective temperature, and metallicity) a software code that produces automatically the radial ve- were derived as in Santos et al. (2001), whereas an es- locities, but for this work we used the reduced spectral timate of the v sin i was obtained using the calibration of orders. the CORALIE Cross-Correlation Function width (see the The analysis used 32 CORALIE orders within the Appendix in Santos et al. (2002)). spectral range 4780–6820A,˚ after having excluded the or- Table 2 quotes the original equatorial rotational ve- ders that are heavily polluted by telluric lines. Bluer orders locities for both templates. Even mild broadening of the −1 were excluded because the secondary signal is expected primary template (v sin i = 1kms ), on top of its orig- −1 to be too weak in these orders. Multi-order TODCOR inal v sin i of 2kms , degraded the solution consider- was used to combine the correlation functions from the ably. On the other hand, the secondary template could be −1 32 orders. Note that the pixel-to-radial-velocity scale is broadened by v sin i up to 5kms without significantly different for each order. Therefore, before combining the affecting the solution. Finally we used a broadening of −1 correlation functions we had to interpolate and re-sample v sin i = 1kms , which yielded the smallest residuals of them into a pre-determined scale. the secondary velocities. TODCOR requires two spectral templates as simi- Figure 1 shows the flux ratio as a function of wave- lar as possible to the expected primary and secondary length, corresponding to the chosen spectral types of K1V spectra, and an assumed value for the flux ratio be- and M2.5V, using the SED library of Pickles (1998). The tween the two spectra. We searched for the best tem- dashed line in the figure shows the flux ratio obtained by plates in two datasets of spectra obtained by ELODIE assuming a black-body radiation law for the two spectra, and CORALIE. The CORALIE templates, appropriate with a temperature of 3500K for the secondary. It is clear for solar-type stars, were obtained as part of a program that while the black-body model roughly fits the detailed to derive precise abundances of planet-hosting and non- flux-ratio, it is not accurate enough and there are large dif- planet-hosting stars (Santos et al. 2000, 2001). The tem- ferences, e.g., around 6500 A.˚ In any case, using a constant plates for the later stars were obtained using ELODIE as flux ratio as in the single-order application of TODCOR part of a program studying the binarity of close M dwarfs is clearly not sufficient for such a wide spectral range. (Delfosse et al. 1998, 1999). The flux-ratio information Figure 2 demonstrates the way multi-order TODCOR was taken from Pickles (1998), using his measurements of measures the radial velocity of HD41004B. The Figure typical stellar spectral energy distribution (SED), normal- presents one-dimensional “cuts” of the two-dimensional ized according to the V-magnitude difference of HD 41004. correlation function. In each “cut” the correlation is shown By convolving the templates with a rotational broadening as a function of the secondary velocity, while the pri- profile (e.g., Gray 1976), we created an additional degree mary velocity is fixed according to the location of the of freedom which expanded our template library, allowing two-dimensional maximum. Because of the small flux- a better fit of the templates to the observed spectra. ratio between the two templates, we expect the correla- In all the configurations of two templates that had tion value to change very little when the secondary tem- spectral types similar to K1V and M2V, the secondary ra- plate alone is shifted. However, in each “cut” we still ex- 4 S. Zucker et al.: Multi-order TODCOR: application to CORALIE & ELODIE

0.08 JD = 2452312.722 JD = 2452343.590 −4 −4 x 10 x 10 4 4 0.07 2 2 0 0 −2 −2 0.06 order 31 −4 −4 −4 −3 x 10 x 10 1 0.05 0 0

0.04 order 57 −10 −1 −3 −3 Flux ratio x 10 x 10 2 0.03 2

0 0 0.02 order 67 −2 −2 −4 −4 x 10 x 10 4 0.01 4 2 2 0 0 0 −2 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 −2 combined −4 −4 Wavelength [Angstroms] −50 0 50 100 −50 0 50 100 RV [km s−1] RV [km s−1] Fig. 1. The flux ratio in the relevant wavelength range, calcu- 2 2 lated according to Pickles (1998). The dashed line represents Fig. 2. The upper three panels in each column show “cuts” the flux ratio corresponding to black-body radiation laws of (see text) of the two-dimensional correlation function for three 3500 K and 5240 K. selected orders. The dashed lines represent the best-fit sec- ondary velocities while the dotted lines represent the primary velocity. It is not shown on the right column where it almost coincides with the secondary velocity. The lower panel in each pect to see a relative peak around the correct secondary column shows the “cut” of the function obtained after combin- velocity. The Figure is divided into two columns, corre- ing the corresponding correlation functions of all the 32 orders. sponding to two different exposures (JD = 2452312.722 A third-order best-fit polynomial was subtracted from all the and JD = 2452343.590). Using the best-fit orbit, the ex- shown functions to accentuate the local peak. pected secondary velocity for the spectrum used in the left −1 column of the Figure is 35.2kms , while for the right −1 The listing of the radial velocities of A and B and the column it is 42.2kms . These velocities are marked on corresponding times, can be obtained at the CDS. the figure by dashed lines. The primary velocity is repre- sented by a dotted line in the left column, while in the right column it almost coincides with the secondary ve- 4. Results locity. For this graphic demonstration only, a third-order Figure 3 shows the resulting orbit of HD 41004 B, while best-fit polynomial was subtracted from all the plotted the orbital elements are summarized in Table 3. Applying functions in order to accentuate the local peak. multi-order TODCOR to many orders (32), combined The three upper panels in each column demonstrate with the large number of measurements (103) yielded a the problems in the single-order TODCOR. On the left, very precise orbital solution. Thus, the radial-velocity am- the first upper panel exhibits a very prominent peak in a plitude (K) was found with a precision of about 1%. This wrong velocity. The second and third panels show a mod- fine precision allowed also a very accurate estimate of the erate peak at about the expected velocity In almost all (e). Although very small, the eccentric- 32 orders, some local peaks appeared around the correct ity is still non-vanishing, with a significance level of 2·10−5 velocity. The lower panel shows the result of combining according to the Lucy & Sweeney (1971) test. Assuming a the correlation functions of all the analysed orders. The mass of ∼ 0.4 M⊙ for HD 41004 Ba, the companion min- correct peak is clearly emphasized relative to the spurious imum mass is 19 MJ. The uncertainty of 0.25 MJ does peaks. not take into account the uncertainty in the mass of Comparing the two columns of Figure 2, we see that we HD41004Ba – MBa. A 20% uncertainty in MBa would cannot know in advance which orders present the correct result in a 2.3 MJ uncertainty in MBb,min. peak. Thus, we have to combine all 32 orders in order to Figure 4 shows the velocities of HD41004A. The ve- have the correct peak emphasized. The right column of the locities measured after JD = 2, 452, 200 show a clear Figure demonstrates another advantage of TODCOR: in linear trend. The best-fit line has a slope of +105 ± this spectrum the secondary velocity is almost identical to 10ms−1 −1, and is also shown in the Figure. As Santos the primary velocity. In the conventional one-dimensional et al. (2002) have already noticed, the first isolated mea- cross-correlation, there is no way to measure the two ve- surement does not agree with the linear trend implied by locities, due to the blending of the two correlation peaks. the other measurements. A Lomb-Scargle periodogram of TODCOR, by using two different templates, allows the the de-trended velocities (Figure 5) shows no hint of the measurement of both velocities. 1.3-day periodicity, again proving that only the B compo- S. Zucker et al.: Multi-order TODCOR: application to CORALIE&ELODIE 5

HD41004B 42.62 42.6

] 42.58 −1 48 42.56 42.54 42.52 46 RV [km s 42.5 42.48 ] 42.46 −1 44 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300 2350 2400 JD−2450000 42 Fig. 4. Radial velocities of HD 41004 A. The dashed line is the

40 best fit to the velocities, ignoring the first isolated velocity.

38 Radial Velocity [km s 2

36 1.8

34 1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.4 Phase

Fig. 3. Phased radial velocities of HD 41004 B 1.2

1 Table 3. Best-fit orbital solution of HD 41004 B 0.8

P [days] 1.328199 ± 0.000081 0.6 T [JD] 2 452 339.212 ± 0.040 e 0.065 ± 0.014 0.4 γ [km s−1] 41.631 ± 0.057 0.2 ω [◦] 171 ± 11 −1 K [km s ] 6.192 ± 0.081 0 −3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a sin i [10 AU] 0.7544 ± 0.0098 −1 6 frequency [day ] f(m) [10− M⊙] 32.5 ± 1.3 † MBb,min [MJ] 18.64 ± 0.26 Fig. 5. Lomb-Scargle periodogram of the velocities of N 103 HD 41004 A after subtracting the linear trend. The dashed line −1 marks the frequency corresponding to the 1.3-day period of σO−C [km s ] 0.56 HD 41004 B. † Assuming MBa = 0.4 M⊙.

nent participates in the periodic motion, as suggested by 5.1. The long-term trend of the radial velocity of Santos et al. (2002). In light of the trend in the veloci- HD 41004 A ties of component A, we tried also to fit the B velocities with an additional trend but the resulting trend was not As we have demonstrated in the previous section (see statistically significant. Figure 4), the radial velocities of HD 41004 A show a clear long-term variation over time. For the period November The mean radial velocity of A during the linear part is 2001 to February 2002 this variation can be nicely fitted ± −1 42.5768 0.0009kms , which is very close to the center- by a linear increase. Note, however, that one single point, of-mass velocity of B – the difference is only ∆RV = at JD = 2451902.774, which was observed one year before ± −1 0.95 0.06kms . However, this difference depends also all the others – on December 2000, deviates substantially on the estimated velocities we used for the templates, from this linear fit. This point may indicate that the ex- and therefore we adopt a conservative error estimate of −1 trapolation of the linear approximation is no longer valid 0.1kms for ∆RV . for the time of that point. Thus, we may actually be seeing the curved part of a long-term orbit, caused by a fourth component in the system. Such an object would have an 5. Discussion orbital motion with a much shorter period than that of the AB system, which is at least 100 , as inferred from The radial velocities derived in the previous section sup- the observed separation. In fact, follow-up measurements port the triple-system model suggested by Santos et al. we are currently performing hint that this may indeed be for HD41004. Our results allow, nevertheless, a somewhat the case, but a detailed solution of the orbit would still be more detailed study of the system. premature. 6 S. Zucker et al.: Multi-order TODCOR: application to CORALIE & ELODIE

5.2. The close orbit and the rotation of HD 41004 Ba circularization through dissipation in Ba has not taken place. As mentioned in Section 3, the process by which we chose Now let us examine the possible involvement of Bb the templates allows a very crude estimate of the equa- in circularization processes. If the object is substellar or torial rotational velocity. The best-fit orbit was attained maybe even a very late M-dwarf, radiative zones in the when the M2 template was convolved with a rotation pro- −1 atmosphere may form either due to the internal physics file of v sin i = 1kms . Our procedure is not accurate (Burrows et al. 1997) or due to the external heating by the enough to use this value as a true measurement of v sin i. primary (Guillot et al. 1996). These would necessitate an- However, trying large values for v sin i degraded the qual- other approach to calculate the circularization timescale, ity of the orbital solution. For example, while the qual- probably based on dissipation of the dynamical tide (Zahn ity of the solution remained almost unchanged for v sin i −1 1977). The theory of orbital evolution through tidal dissi- as large as 5kms , the O − C RMS was doubled at −1 pation is still debated, specially the mechanisms for dissi- v sin i = 10kms . Thus, we believe that the equatorial pation of the dynamical tide (e.g., Claret et al. 1995), and velocity of Ba is of the order of a few kms−1, and proba- therefore it is not clear whether it contributes significantly bly less than 10 km s−1. Even this crude estimate is suf- in the case of HD 41004 B. ficient to conclude that v sin i is substantially lower than −1 the expected equatorial velocity of ∼ 20kms , assuming synchronization (Santos et al. 2002). 6. Conclusion At such a close orbit, with a period of 1.33 days, it We have presented in this work the first application of is usually assumed that synchronization and alignment multi-order TODCOR to echelle spectra. The case of of the spin with the orbital motion had been established HD 41004 demonstrates the unique capabilities of this (Santos et al. 2002). An obvious explanation of the ap- technique. It utilizes all the available prior knowledge parently small v sin i would be small . regarding both spectral components (the different tem- Assuming alignment, small inclination implies that the plates), it is not limited to a fixed flux ratio (we used the mass of the unseen companion is considerably larger than SED to calculate it) and it incorporates optimally the data its derived minimum value. from all the relevant spectral orders. Based on the best-fit solutions of the present case, we estimate the precision of −1 5.3. The small eccentricity of the close orbit the secondary velocity of HD41004 to be 0.56kms and that of the primary velocities to be 10 m s−1. The radial A binary or a star-planet system with a period as short velocities yielded accurate orbital elements of the unseen as 1.33 days is na¨ıvely believed to have been circularized. companion of HD41004B, and an accurate measurement However, our radial-velocity solution shows that the close of the radial acceleration of HD41004A. The latter sug- ± orbit has a finite non-zero eccentricity – e =0.065 0.014, gested an additional component may be present in the which has to be explained. system. Follow-up observations, currently underway, tend Let us first estimate the timescale of circularization to confirm this hypothesis. due to processes occurring in Ba. The relevant process The combination of multi-order TODCOR together is dissipation of the equilibrium tide through interaction with the high signal-to-noise and high resolution of the with the convective envelope (Zahn 1989). We follow Rasio CORALIE spectra render this analysis a very promising et al. (1996) and write: path toward expanding the database of spectroscopic bi-

τc M 1 a 8 naries and multiple systems (like HD 41004). It may also τcirc = ( ) , facilitate the detection of planets in binary stellar systems, f Menv q(1 + q) R which are lately the focus of an increasing interest. 2 1 ≈ 3 where the parameter τc (MR /L) is the eddy turnover Acknowledgements. This research was supported by the Israeli timescale, and its numerical value is about 0.5yr. The nu- Science Foundation (grant no. 40/00). Support from Funda¸c˜ao merical value of f is obtained by integrating the viscous para a Ciˆencia e Tecnologia, Portugal, to N.C.S. in the form of dissipation of the tidal energy through the convective zone a scholarship is gratefully acknowledged. and is of order unity (Zahn 1977). In our case the tidal pumping period – P/2 – is much smaller than τc, and References therefore only convective eddies with turn-over time less than P/2 contribute to the dissipation. The value of f Burrows, A., Marley, M., Hubbard, W. B., et al. 1997, α is then reduced to (P/(2τc)) (Zahn 1989). The correct ApJ, 491, 856 value of α is debated but generally assumed to be either 1 Chabrier, G. & Baraffe, I. 2000, ARA&A, 38, 337 (Zahn 1992) or 2 (Goldreich & Keeley 1977; Goldman & Claret, A., Gimenez, A., & Cunha, N. C. S. 1995, A&A, Mazeh 1991). We assume that the mass of the convective 299, 724 envelope – Menv – is very close to the – M Delfosse, X., Forveille, T., Beuzit, J.-L., et al. 1999, A&A, (e.g., Chabrier & Baraffe 2000). The resulting timescale is 344, 897 about 1010 yr, assuming α = 1, and certainly much larger Delfosse, X., Forveille, T., Perrier, C., & Mayor, M. 1998, if α = 2. At the age of 1.5 Gyr, it therefore seems that A&A, 331, 581 S. Zucker et al.: Multi-order TODCOR: application to CORALIE&ELODIE 7

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