Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof...

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Hare-and-hound exercises of the Data Analysis Team of the Seismology Working Group by Appourchaux and Berthomieu May 2005

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Page 1: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Hare-and-hound exercises ofthe Data Analysis Team of

the Seismology Working Group

by

Appourchaux and Berthomieu

May 2005

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HH#1

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Report of the hare-and-hound exercise of the Data Analysis Team

T.Appourchaux

Space Science Department of ESA, P.O. Box 299, 2200AG, Noordwijk, The Netherlands

T.ToutainObservatoire de la Cote d’Azur, Departement Cassini, UMR CNRS 6529, F-06304 Nice, France

C.BarbanObservatoire de Paris, DASGAL, CNRS UMR 8633, F-92195 Meudon, France

ABSTRACT

Here we report on the results of a hare-and-hound exercise performed by the Data ReductionGroup. Spectra of 6 different stars have been simulated and fitted for extracting the small andlarge separation of the stars. Rotational splittings were also fitted. The ability of the groupfor detecting low-degree stellar p modes is at least proven on high signal-to-noise ratio spectra.Further work will be needed for making the simulation more real and less easy. A new hare-and-hound exercise is already planned.

1. Introduction

Hare-and-hounds (H-H) exercises are commonly performed in helioseismology. These kinds of blindtests are heavily linked towards testing inversions. In the recent years, the emphasis has slowly shifted totesting any alleged technique thought to detect g modes. A similar approach for detecting stellar p modes isnow envisaged since the strongest modes might be at the limit of detection.

Hereafter, we describe first how the stellar time series are simulated, second how the synthetic spectraare fitted and/or the stellar p-mode parameter extracted. Then, we discuss the different techniques ofdetecting stellar p modes. We close the report by looking how the simulation can be improved, and setthe scene for the next round of H-H.

2. Spectrum simulation

In helioseismology, the solar p modes are stochastically excited by convection. It results that thestatistics of the spectra is well known: it is a � � with 2 degrees of freedom (Woodard, 1984; Anderson et al.,1990; Appourchaux et al., 1998). For stellar p modes, we have taken the same assumption. As soon as theFourier spectra can be calculated, the inverse Fourier transform provides the time series. It is not necessaryto excite the p modes in the time domain as performed by Barban et al. (1999). The former is more efficientthan the latter.

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Fig. 1.— Linewidth and amplitude of the modeled stellar p modes as a function of frequency for differentstars. They are here independent of age.

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Fig. 2.— Modeled background stellar noise as a function of frequency for different stars. The time scalesare a linear function of mass. The amplitude of the 3 different noise are independent of age and mass.

In both cases, we need to know how one can simulate properly the characteristics of the stellarspectrum, namely: Mass, Age, Temperature, Fine internal structure, Internal dynamics, Mode excitationand damping, Stellar and Instrumental noise.

The first three items determine the central frequency (��), the large (���), the small separation (��)and the smaller separation (��) (Christensen-Dalsgaard, 1984). The essential data were taken from Audard& Provost (1994) and were interpolated from 1�� to 2��, and from a grid of hydrogen content� rangingroughly from 0. to 0.7. For mass lower than 1 �� we extrapolated the interpolation down to 0.8 ��. Theformula was used for getting the frequencies of modes � � � to 3; modes of higher degree were not includedin the simulation.

The mode excitation and damping was derived from the work of Houdek (1996) and of Houdek et al.(1999). The amplitudes and linewidths were often extremely simplified. Figure 1 gives an example of thevariation of the linewidth with frequency for various stellar masses. Since the model was done very quickly,it is quite likely that these figures do not exemplify fine details as reported by Houdek (1996); it should betaken only as a starting point for a finer modeling.

For simulating the internal dynamics of the stars, we assumed that the stars rotate rigidly with theirrotation axis perpendicular to the line of sight. As for the distribution of amplitude with the degrees, weassume that the relative amplitudes follow the geometrical visibilities usually used for the VIRGO data andIPHIR data (Toutain & Frohlich, 1992).

The noise was derived from that of the solar noise. For the stellar noise, we arbitrary decided that thetime scales defining the various super-, meso- and granulation noise be scaled like the stellar mass (i.e. the

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heavier the star the longer the time scales) like shown on Fig. 2. Of course, a better work needs to be donefor the noise if we want to make the simulations more real.

The simulated time series are 1 month long with a 30-s sampling time. Six time series were simulatedfor six different stars which main characteristics are given in Table 1. The data were blindly generated sothat even the hare could play the role of the hounds.

Star Mass (in ��� �� ��� �� (in � Hz)

1 1.33 0.23 84.90 1.002 1.23 0.39 106.80 0.563 1.86 0.69 81.20 1.504 1.26 0.13 87.50 1.505 1.21 0.58 120.10 0.256 0.85 0.17 151.40 0.69

Table 1: Main characteristics of the simulated stars. �� is the hydogen content, and �� is the rotation rateof the star as a rigid body.

3. Data analysis

3.1. Echelle diagram

One of the simplest, quickest and dirtiest method to detect the large separation and other details of thestellar p-mode spectra is to produce an Echelle diagramme. The original idea was from Grec (1981) whoused the asymptotic property of the solar p modes for showing how each degree would line up in such adiagramme. The Echelle diagramme consists in cutting in pieces of �� � the spectra of the star, and then topile up these pieces on top of each other. If modes are indeed detected vertical lines should appear clearlyin the diagramme. The trick is really to find the ��� that would produce such ‘vertical’ lines. With someexperience, and with simulated data with high signal-to-noise ratio, it takes just a few tens of minutes to getthe proper separation. Trial and error will do especially on a Friday afternoon before the week-end.

Figure 4 shows the results of this procedure, and Table 2 gives the large separation obtained accordingly.

3.2. Histogram

This technique was developed by Barban et al. (1999). Figure 5 shows the result of this procedure.Table 2 gives the large separation obtained with this technique.

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Fig. 3.— Power spectra (smoothed to 1 �Hz) for the 6 synthetic time series. The signal-to-noise ratio is atbest about 100.

3.3. Maximum likelihood fitting

The technique used for fitting stellar spectra is the same as for full-disk solar spectra Toutain & Frohlich(1992). We use Maximum Likelihood Estimators known as MLE, for a review see Appourchaux et al.(1998). Table 2 shows the result obtained; all the mode frequencies are available upon request (ask ThierryAor Toutain for details).

4. Discussion

It is quite clear that the helioseismologists were at an advantage mainly because they were at the end ofthe processes: synthetic data production (ThierryA) and MLE (Toutain). As a matter of fact, the analysis andsimulation performed are too closely related to solar-type stars. Nevertheless, this bias created an interesting

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Fig. 4.— Echelle diagrammes for the 6 synthetic stellar spectra. The modes can be identified according tothe way the ridges are split. For instance, the top left diagramme shows clearly from left to right, the �=3, 1,2, 0 ridges. The parabolic curvature of the ridges is due to the asymptotic formula used.

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Fig. 5.— Histogrammes of the stellar spectra as computed by Barban et al. (1999). The largest peak indicatesthe location of the large separation. The second two largest peaks display the mean structure of the modelocated at ���� � �. The sub-structure of within the largest peak is a manifestation of the splitting (See timeseries 0 and 3, i.e. stars 1 and 4).

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Star Mass (in��� ��� (in � Hz) �� (in � Hz)Echelle MLE Histo. Echelle MLE Histo.

1 1.33 85.0 85.4 90.0 - 0.963 � 0.034 -2 1.23 106.0 107.0 108.0 - 0.507 � 0.016 -3 1.86 81.0 81.5 80.0 - 1.502 � 0.013 -4 1.26 88.0 88.5 87.0 - 1.520 � 0.012 -5 1.21 120.0 120.0 120.0 - 0.196 � 0.022 -6 0.85 152.0 152.0 151.0 - 0.659 � 0.019 -

Table 2: Main characteristics of parameters derived from the 3 main analysis techniques.

‘bug’: the psychological approach used by one of the fitter made him thought that the data producer mayhave forgotten to produce synthetic data for a star with an inclined rotation axis. The simulated data were abit too easy to play with: a more realistic and a more noisy spectrum should be simulated in the future.

As far as the simulation is concerned, several improvements should be included such as:

� Inclined rotation axis

� Differential rotation

� Better linewidths and amplitudes

� More realistic stellar noise

� Lower signal-to-noise ratio

As far as the H-H exercise is concerned, we will likely make a full H-H involving 2 complete teams ofModelers-Observers doing a double blind H-H using simulated data from the other team. This activity hasbeen started and will be reported during the next team meeting.

Acknowledgments

Many thanks to our beloved PI, Annie Baglin, for keeping alive the flame of the COROT project, andfor keeping on fighting: ‘You’ll never walk alone’. For anyone reading who is that quote from? Send youranswer to [email protected].

References

Anderson E.R., Duvall J. T. L., Jefferies S.M., 1990, ApJ, 364, 699

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Appourchaux T., Gizon L., Rabello-Soares M.C., 1998, A&AS, 132, 107

Audard N., Provost J., 1994, A&A, 282, 73

Barban C., Michel E., Martic M., et al., 1999, A&A, 350, 617

Christensen-Dalsgaard J., 1984, In: A.Mangeney, F.Praderie (eds.) Space Research in Stellar Activity andVariability, 11, Observatoire de Meudon

Grec G., 1981, Ph.D. thesis, Universite de Nice

Houdek G., 1996, ‘Pulsation of solar-type stars’, Ph.D. thesis, Univeristat Wien

Houdek G., Balmforth N.J., Christensen-Dalsgaard J., Gough D.O., 1999, A&A, 351, 582

Toutain T., Frohlich C., 1992, A&A, 257, 287

Woodard M., 1984, ‘Short-period oscillations in the total solar irradiance’, Ph.D. thesis, University ofCalifornia, San Diego

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Seismology of � Scuti stars with COROT: prospects for mode identification

C. Barban, M.-J. Goupil, C. Van’t Veer

Observatoire de Paris, DASGAL, UMR 8633, 92195 Meudon, France

R. Garrido

Instituto de Astrofisica de Andalucia, C.S.I.C., Apdo. 3004, 18080, Granada , Spain

ABSTRACT

� Scuti stars are high amplitude, nonradial multiperiodic oscillators. The major problem isthe identification of the observed frequencies as specific oscillation modes, however this step isimportant to provide constraints upon the internal structure of the star.

We show here that despite the complicated frequency distributions which can be expectedfrom COROT space mission for a typical � Scuti star, it will be possible to identify at least somemodes or at least the degree for low-� modes from space data.

1. Introduction

The � Scuti stars are pulsating variables located in the lower part of the classical instability strip withspectral type from A2 to F0. They are multiperiodic pulsators oscillating with large amplitude comparedto solar-like oscillations (amplitudes from ground-based observations range between a few mmag to 1/10mag) and periods less than 0.3 days. Only a few 5 to 10 frequencies in general and in some cases up to 30frequencies in the case of FG Vir (Breger et al. 1998) have been detected from the ground. The observedfrequencies span an interval in which models predict much more frequencies than observed. Either thecorresponding modes are not excited or they are excited up to much smaller amplitudes that can be detectedfrom ground-based observations. We do not yet understand how modal selection operates, this prevents toidentify firmly modes associated with individual frequencies (e.g. to determine the degree, azimuthal orderand radial order of each mode) and therefore to take full advantage of the available seismological tools. Abetter knowledge of the global structure of the frequency spectrum would provide precious indications aboutmodal selection processes. This requires to detect oscillations with much smaller amplitudes and thereforeto increase significantly the signal-to-noise ratio.

Observations from space will be characterized by a much higher signal-to-noise ratio than from theground. This will result in the detection of a much larger set of modes for each � Scuti star, by detectingnew small amplitude modes and high degree modes. Indeed, high � modes are expected to be excited butvisibility effects i.e. averaging geometrical effects when observing in integrated light, decrease the observedamplitudes to much smaller values than can be observed from the ground. But the detection of a much largernumber of frequencies will yield frequency spectra very much complicated and then it is legitimate to wonderwhether mode identificationwill not be even more difficult to obtain (Dziembowski et al. 1998)? The purpose

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of this work is to show how, in space data, it is possible to identify at least some modes or at least the degreeof the low-� modes, and this despite the complications due to the presence of high � modes.

Spectroscopic or photometric techniques have been deviced to identify modes (Garrido et al. 2000).This work is a progress report on our effort to develop an alternative method which does not involve anyknowledge of a model and using very precise data, i.e. frequencies, rather than amplitude and phase. Anideal case was presented in Baglin et al. (2000). In this paper, modes up to �= 10 were assumed to be detectedwith the low detection threshold of space data. The detected range of frequencies was assumed to be broadenough toward high frequencies and then it included lower frequency part of the asymptotic p-mode regime.It was shown there that although the frequency spectrum is more complex than the one from ground-basedobservations, it is still possible to obtain the mean value of the large separation and to proceed further inattempting to determine the degree of at least some modes which will constitute the cornerstone of a modeidentification.

However from ground-based observations the detected frequencies do not include part of the asymptoticregime. We do not know whether space data will provide frequencies in the asymptotic regime. In the presentpaper, we therefore discuss the case where these frequencies are not detected. Section 2 describes how thefrequency spectrum is built. In Section 3, patterns are searched using frequency spacing histograms. Thenechelle diagrams are computed. Discussion and perspectives are given in Section 4.

2. Amplitude distribution: visibility effect

We consider a 1.85 M� model (see Baglin et al. 2000 for more details) and retain modes with degreefrom 0 to 10 with frequencies in the theoretical unstable range [80-410]�Hz given by the oscillation codewritten by W. A. Dziembowski (1982). High frequencies which falls in the asymptotic regime are notincluded. The oscillation amplitudes are computed in the COROT photometric system (370-950 nm),according to Watson (1988) for the flux change for a nonradial pulsation. Stellar model atmospheres are builtfor that specific purpose with ATLAS9 (Heiter et al. 2001 in prep.) and consistent limb darkening coefficientscomputed (Barban et al. 2001, in prep.). For the non adiabatic phase lag between temperature and pressurevariations, we use the values given by Dziembowski’s oscillation code. The intrinsic amplitude is assumedto be the same for all the modes and is chosen so that �= 0 modes have an amplitude of 1 mmag as typicallyobserved from the ground. For the amplitudes of modes with higher degrees, we used the amplitude ratiowith �= 0. The amplitude ratio depends on the inclination to the line of sight. We averaged the amplitudesobtained for values of the inclination to the line of sight of 0, 20, 40, 60, and 80 degrees.

Figure 1 with a typical result illustrates the visibility effect. In the case of Fig. 1, from the ground with amaximum amplitude of ���� in relative flux, we are able to detect some �= 0, 1 and 2 modes and from spacewith COROT, we will see some �= 0,1,2,3,4,6,8 and 10 modes. So depending on the threshold, we mustexpect to detect more or less modes with different �. Whether odd highest � modes (5,7,9) will be detecteddepend on intrinsic amplitude and detection threshold. The different feature for even and odd modes from �=4 is attributed to geometric and limb-darkening terms of the expression of flux change. Similar results have

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Fig. 1.— Oscillation amplitudes in intensity as a function of �. This graph is obtained for a frequency 210�Hz. The line marked “GROUND” corresponds to a typical detection threshold obtained from the ground:����. The line marked “COROT” corresponds to the expected space detection threshold of � ����.

been obtained by Balona & Dziembowski (1999) and by I. Roxburgh (private communication).

To be more realistic, we have adopted an amplitude distribution which looks like FG Vir data (Bregeret al. 1998). We arbitrarily defined an amplitude distribution with gaps which yields a frequency spectrumlooking like the one of FG Vir. (Barban et al. 2000).

3. Search for patterns in the frequency spectrum

Although the considered oscillation modes are not in the asymptotic regime, We nevertheless searchin the frequency spectrum built in Section 2 for some frequency equidistances. Differencies between twofrequencies are computed to build a histogram of these differencies which we call a frequency spacing

histogram (e.g. Breger et al. 1999).

When all the unstable mode frequencies are included, no specific structure can be detected. Thehistogram is perturbed by high � modes which do not respect the equidistance and by mixed and g modes(Barban et al. 2000).

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If now we turn to the COROT case, no significant improvement is obtained when we keep the unstablemodes with amplitude above � ����. The histogram is still perturbed by nonequidistant modes as in theprecedent case. The same is true when we remove high � modes which perturb the histogram by increasingthe amplitude threshold. That means that now we keep modes with amplitude higher to ���� (Fig. 2)

Because mixed modes of low degree contaminate low frequency regime, we keep the same amplitudethreshold of ���� and remove the low frequency modes which are mostly mixed modes and perturb thehistogram. Figure 3 shows:

- an excess of signal between approximately 40 and 50 �Hz with a strong peak at 40 �Hz. This peakis interpreted as a mean value of the large separation (��, �� � �n�� � �n����), which differs from theasymptotic value, i.e. 47 �Hz.

- a low frequency decrease. The decrease at low frequencies is attributed to the remaining mixed modes.

From the ground with a detection threshold of ����, we would obtain Fig. 4.

We have shown here that from space data it is possible, at least, to recover a mean value of the largeseparation which is not the asymptotic one but which can help to identify modes as we will see in the nextSection. Ground based observations can serve to confirm space observations and vice-versa.

3.1. Towards mode identification

Once �� is known either as described above (or with another method), it is possible to built an echellediagram (Unno et al. 1989). This diagram emphasizes the signature of the small separation predicted by theasymptotic theory in function of the degree l of the modes:

�n���� � �n���� � ����� ��D���� (1)

where D���� is a quantity which depends on the derivative of the sound speed, and is particularlysensitive to the stellar core (Unno et al. 1989). This can be used as a first step towards identifying the detectedmodes as � place themselves in different patterns. In the case where we are not in the asymptotic regime, weknow here that remains such trend at lower frequencies. Then this method can also be used where we arenot in the asymptotic regime using the mean value of the large separation corresponding to the consideredregion.

From simulated ground-based data, it is difficult to say something very precise because of the too smallnumber of modes (Fig. 5). Without models it is difficult to know if the pattern in the echelle diagram is dueto �= 0 or �= 1.

With the set of modes corresponding to modes with frequencies greater than 200 �Hz and with modeswith amplitude greater than ����, we saw in the last Section that we can determine a mean value of the largeseparation. This set of modes is now used to build an echelle diagram (Fig. 6) corresponding to the mean

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Fig. 2.— Frequency spacing histogram computed in the frequency unstable domain for modes withamplitude greater than or equal to ����

Fig. 3.— Frequency spacing histogram computed from the modes frequencies in [200-409]�Hz and formodes with amplitude greater or equal than ����.

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Fig. 4.— Frequency spacing histogram computed from the modes frequencies in [200-409] �Hz and formodes with amplitude greater or equal than ����.

Fig. 5.— Echelle diagram with k as an integer with �� ==40 �Hz for �==0 to �==10 modes between 200and 409 �Hz and amplitudes greater than ����.

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value of the large separation found (i.e. 40 �Hz).

Fig. 6.— Echelle diagram with k as an integer. Left: with �� = 40 �Hz for modes between 200 and 409�Hz and amplitudes greater than ����. Right: with �� ==47 �Hz for modes between 50 and 800 �Hz.

The unstable modes we consider in this present case have frequencies in the range where everything ismixed up in the echelle diagram of the “ideal case” (Fig. 6, Right). However selecting higher frequencies inthe unstable domain and using the value of the large separation given by the histogram (Fig. 6, Left), we canrecover nice structures associated to �==0,1 and 2. We are not far from mode identification.

4. Conclusion and perspectives

We have built a frequency data set as can be expected from space data and have shown that it is stillpossible to recover patterns in the complicated simulated space data frequency spectrum and then to be notfar from modes identification. Althoughdependent on the amplitude pattern chosen here, it is still reasonableto conclude that there exists a regime in frequency and amplitude which is not reacheable from the groundand which remains simple enough to determine �= 0, 1, 2 modes. Once these modes are determined, the othermodes can be easily identified.

The frequency data does not include rotational splittings. The study of the effect of rotation is a work inprogress. Another next issue will be extracting structure associated to high degree modes and mixed modes.

Acknowledgements

We thank M. Auvergne for giving us the characteristics of the COROT photometric system and fordiscussions.

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REFERENCES

Baglin A., Barban C., Goupil M.J., Michel E., Auvergne M., “Delta Scuti and Related Stars”, ASPConference Series, Vol. 210, 2000, M. Breger & M.H. Montgomery, eds., p. 359

Balona L.A., Dziembowski W.A., 1999, MNRAS 309, 221

Barban,C., Goupil, M.J., Van’tVeer C., Garrido, R. , 2000, Soho/Gong workshop, Spain

Breger M., Pamyatnykh A.A, Pikall H., Garrido R., 1999, A&A 341, 151

Breger M. Zima W., Handler G., Poretti E., Shobbrook R.R., Nitta A., Prouton O.R., Garrido R., RodriguezE., Thomassen T., 1998, A&A 331, 271

Dziembowski W.A., Balona L.A., Goupil M.-J., Pamyatnykh A.A., 1998, in: Proc. The First MONSWorkshop: Science with a Small Space Telescope, eds.: H. Kjeldsen, T.R. Bedding, AarhusUniversitet, p. 127

Dziembowski W.A., 1982, Acta astron. 32, 147

Garrido R., “Delta Scuti and Related Stars”, ASP Conference Series, Vol. 210, 2000, M. Breger & M.H.Montgomery, eds., p.67

Unno W., Osaki Y., Ando H. Saio H. Shibahashi H., 1989, Nonradial Oscillations of Stars, University ofTokyo Press, 2nd ed.

Watson R.D., 1988, Ap&SS 140, 225

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HH#2

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Results

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���� ������� � ��� ���� ����

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Page 34: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Hare � Hound exercise with simulated COROT data

G� Berthomieu �� T� Appourchaux� on behalf of the COROTSeismology Working Groupy� D�epartement Cassini� UMR CNRS ����� Observatoire de la C�ote d�Azur� BP

����� �� Nice Cedex�� France�� Research and Science Support Department of ESA� ESTEC� P�O�Box ����

NL���AG Noordwijk� The Netherlands�

Abstract� With the aim of preparing the interpretation of future COROT observa�tions� a Hare�and�Hound exercise has been performed on a solar�like oscillating star�The methods used to construct simulated time series and to recover the propertiesof the star using the frequencies extracted from these time series are described andthe comparison between the results and the theoretical inputs are presented�

�� Construction of simulated time series

Two hare�and�hound exercises �H�H� have been performed� by theteams of the COROT sismogroup� The di�erent steps of the H�Hexercise are the following�� Construct a stellar model M �mass�� Y �helium�� Z�X �metallic�

ity�� �ov �overshoot� with constraints� on luminosity �L�L��� e�ectivetemperature �Teff� and metallicity �Fe�H�� Compute the frequenciesof the models �n��� Construct a simulated time series which representswhat the observation of the pulsating model by COROT would give�� Extract the frequencies� hereafter referenced as �observed frequen�

cies �n�� and rotational splittings �not yet exploited��� Interpret them in terms of internal structure and rotation of the

star� Hereafter results concern the direct approach that is the search forthe closest model to the �observations� No inversions are presented�

Two input models �respectively Ex� and Ex��� satisfying the con�straints� have been constructed by the Nice and Meudon groups usingthe CESAM code their frequencies computed and provided to theteams in charge of constructing the time series �referenced hereafterby Appourchaux� Toutain� IAS�� Simulated time series are constructedassuming amplitudes and damping rates according to Houdek et al������ A�A� ���� ����� stellar noise according to solar noise �or even�at� � COROT noise and a given inclination of the stellar rotationaxis�y

T� Appourchaux� C� Barban� F� Baudin� G� Berthomieu� M� Bossi� P� Boumier� M�J� Goupil� Y�

Lebreton� P� Morel� B�L� Popielski� J� Provost� T� Toutain� I� Roxburgh�

The constraints on the models are� ���� � log�L�L�� � ���� ����� � log�Teff � ������ ����� � Z�X � ���

c� ���� Kluwer Academic Publishers� Printed in the Netherlands�

poster�porto������pub�p�tex� ���������� ���� p�

Page 35: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Figure � The echelle diagram for Ex�

Figure �� a�Di�erence between extracted frequencies and theoretical input frequen�cies for Ex� �� � in black�� � in blue� relatively to the frequency�b� Di�erencebetween extracted frequencies by three independent groups and theoretical inputfrequencies for Ex�� � � relatively to the frequency�

The identi�cation of the modes � degree l and azimuthal orderm� ismade with the help of the echelle diagram� The degrees are identi�edaccording to the splitting structure of the �l � �� l � �� pair versus thatof the �l � �� l � �� pair�

The determination of the mode parameters �here principally fre�quencies� is made using Maximum likelihood estimators� The modelused for �tting assumes a Lorentzian pro�le of the mode� a degree�dependent visibility� a rotational splitting� a star inclination and a �atbackground noise� The statistics of the power spectra is a �� with �d�o�f� The modes are �tted by pairs over a ���� Hz �or so� window�Figure � give the di�erences between the extracted frequencies andtheoretical input frequencies�

�� Interpretation of the �Observed frequencies�

In Ex�� according to Berthomieu et al ����� Provost et al� ����� theaim is to select the models which �t the �observed large spacing

poster�porto������pub�p�tex� ���������� ���� p��

Page 36: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Figure � Small frequency spacings ���� and ���� for di�erent models satisfyingobserved� large spacing� observations� �full dots� and input model �open circle�with M� �heavy line�� M� �dashed line��

600 800 1000 1200 1400

-2

-1

1

2

3

∆ν

[µΗΖ

]

ν [µΗΖ]

1

2

3

4

Figure �� Reduced echelle diagram for degree l � for a model �solid curve� andobservations �dashed��

�n�� � �n�� � �n���� and the small �observed spacings de�ned by����� � �n���� � �n������ ���� � ��n���� � ��n���� � �n�������� Smallspacings are sensitive to the core overshoot parameters� with a highestsensitivity at high frequency�

Among all the models which �t the large spacing� the �t with �ob�served small frequency spacings in large frequency range favours mod�els with low core overshoot M� �considered as �rst choice� output� intable ��� while the �t in the low frequency range where the error barsare smaller favour models M� �output� in table ���

In Ex�� the frequencies for l � � are developped according to� �n�� ���cut�k � D� ���offset� where � D� is a mean large spacing� ��offsetis a third order polynomial �t of �observed ��offset The reduced echelle

diagram � �gure �� is de�ned by the di�erence ��offset � ��offset� Fourextreme points are choosen to characterize the spectrum� These pointsand the mean large spacing are used to constrain the �ve parametersinvolved in the model computations �see Popielski et al ������ Severalmodels can �t well the constraints� The �best �t model is choosen asthat which reproduces the most of frequencies�

poster�porto������pub�p�tex� ���������� ���� p��

Page 37: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Table I� Comparison between input� and output� models for the two exercises�

Ex� Input Output Ex� Input Output� Output�

Nice Meudon Meudon Nice M� Nice M�

M�M� ��� ���� �� � ���� ����

X ����� ����� ����� ���� �����

Z�X ����� ���� ������ ����� ������

�ov ��� ��� ���� ���� ����

log�L�L�� ��� � ����� ���� ���� �����

log�Teff � ����� ���� ����� ���� �����

Age �My� ���� ���� ���� ���� ����

Xc ����� ����� ���� ���� ����

rcore�R� ���� ���� ���� ���� ��� ���� ��� ���� ��� ����

rZC�R� ����� ����� ����� �����

�� Conclusion

The data analysis of the simulated time series has produced observedfrequencies in good agreement with the input theoretical frequenciesexcept in low and moderate frequency range� The errors are of theorder of ��� to ����Hz except in the large frequency domain� Usingthese �observed frequencies we are able to �nd models close to theinput models but the solution is not unique� In Ex�� the results stressthe importance of the small spacing ����� very sensitive to the coreovershooting� for discriminating the models� However the �observed���� have large errors in the high frequency domain� This may alter thechoice of best �t� In Ex�� despite the use of di�erent stellar evolutioncodes and oscillation codes the input and output models are ratherclose� For future works� we need to improve the criteria of model selec�tion and to study the sensitivity of the models to stellar parameters� tonumerical codes and to the physics� The next step will be to performthe inverse problem with the best models as reference model�

References

Berthomieu G�� Toutain T�� Gonczi G�� Corbard T�� Provost J�� Morel P�� �����Proceedings of SOHO���GONG����� ESA SP� � � ��

Popielski B�� Goupil M�J�� ����� Proceedings of the COROT Week � Viennahttp���www�astrsp�mrs�fr�projets�corot�meeting�CW��html

Provost J�� Berthomieu G�� Morel P�� ����� COROT H�H� results and conclusions�http���www�astrsp�mrs�fr�projets�corot�meeting�CW��html

poster�porto������pub�p�tex� ���������� ���� p��

Page 38: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Fitting and interpretation

Page 39: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 68: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 69: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

Generation

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Sept� ��� �����

Simulation of COROT time series

Hare and Hound Exercise

C� BARBAN� NSO�DASGAL

� Simulated Stellar signal Power Spectrum

I assume that the peak associated to an oscillation mode in the power spectrum has aLorentzian pro�le and I follow the recommendation of Anderson et al� ���� to build thepower spectrum�

��� Oscillation frequencies

Oscillation frequencies are provided by Ian Roxburgh�

��� Amplitudes

I use the estimates of the maximum oscillations amplitude of Houdek et al� �������data found on MONS web site� http���www�obs�aau�dk� jcd�MONS�solarlike�modeparam�html��

Figure �� Amplitude as a function of frequency for ��� the multiplet ��� the multiplet�� and the multiplet �� before correction�

I adopt as the amplitude distribution a bell shape similar to what is observed for theSun with a maximum of � ppm around the predicted maximum amplitude frequency�Then� for �� modes� I mimic this amplitude distribution by a Gaussian shape� I assume

Page 71: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

the same amplitude distribution for modes � � �� To mimic whole disk intensity observations� I apply to � � � modes visibility coe�cients� These coe�cients are calculatedfor the COROT photometric system using the equation of relative variations of �ux ofWatson ����� I made a mistake in the computation of the visibility coe�cients� then the�rst time series sent to the data �tter corresponds to Figure �� after correction I obtainFigure � and this corresponds to the second time series sent�

Figure �� Amplitude as a function of frequency for ��� the multiplet ��� the multiplet�� and the multiplet �� after correction�

��� Linewidths

Figure �� Linewidth as a function of frequency for a stellar model of M���M��L������L� � Te��� ���D� K� Xc���D��

The linewidths are taken from Houdek et al� ������ �data from MONS web site� �Figure ���

Page 72: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

��� Background stellar noise

I adopt a stellar background noise similar to that observed for the Sun �Figure � andbased on Appourchaux et al� �������

Figure � Modeled background stellar noise a a function of frequency �deduced from Fig� �of Appourchaux et al� ��������

� Simulated Time Series

��� Stellar signal� ss�t�

The time series data is obtained by computing the inverse Fourier Transform of the builtFourier spectrum�

��� Instrumental noise Time Series� si�t�

A white noise of ��� ppm for � observations days is introduced �si��t���

The orbital noise �f�������� Hz� amplitude � ppm� is also introduced �si��t���

Then the �nal instrumental time series is� si�t� si��t� � si��t�

��� Final Time Series� s�t�

The �nal time series with a sampling of �� s and a total length of ��� days is�

s�t� �ss�t� � si�t�� � w�t�

where w�t� is function of time� w�t� � for some points and w�t� � for the otherpoints�

I introduce in w�t�� the e�ect of the solar panels rotation �for the �rst two months of the total � months�

I remove data each �� days over a period corresponding to the length of the orbit� thesame thing is done for the last � months of the total � months��

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a breakdown� I remove data over � days� the e�ect of the activation of the �magneto coupleurs�� I remove points in the time

series spread out over the �rst half of the length of the orbit� I did the same thing for the�nd half of the orbit�

Acknowledgments� F� Hill for the �stellar part� and M� Auvergne for the �instrumentalpart�� R� Garrido and C� Van�t Veer for providing model atmosphere and other parametersused for the computation of the visibility coe�cients�

References�

Appourchaux et al� ����� COROT SWG meeting� Sept� ����Anderson et al�� ����� A�A �� � ���Houdek et al�� ����� A�A ���� ���Watson � ����� Ap�SS � �� ���

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XQY*]�K,XUJ6K�SU�\NBbWPUK=PQR\RTX,RWV/J`NO]a�6i*K ��¦ �/n �\¨�£6n��C� � ��©�n � RTL/K XQJ6Y*S,[�ihbWPUY V���XUJ6K SQY*XQNBb4XQY*RT^^6RWm$n� � v�6 @ G�v,t������),i*K�bWSUK$�/K�bWP�m=YjXUJ�]�K�b4S=XQJ6Y*S�PQK�L/RTPUXMm=bWS=m=PQY*XUXQK�^«NO^6Z6K�P�XQJOK$Y*^\·BN6K�^O[�K$RWV�b�·BN?�BbW^Y*^6SQR�]�^6Yhb6��bW^�b4]�^6K_SQYhb�b4^6Z�R�XQJ6K_PM^6K�NOPQRTXUY*[��/K�JBb9�\Y*RTNOPQS�n

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On expressing mode splitting with

Clebsh-Gordan coe�cients and related issues

T.Appourchaux

September 14, 2001

1 Goal

The goal of this document is to express the splittings given in terms of (�m �

�0)=m as a function of the ai expansion based on Clebsh-Gordan coe�cients.

2 Background

In the antique age of helioseismology (before the 90's or so), mode splittings

were usually expressed in term of Legendre polynomials. Unfortunately, these

polynomials are orthogonal only on a continuous space (between [-1,1]) not on

a discrete set such as (-1,0,+1) for l=1, for exanple. Therefore, other expansion

are required that can either be computed by hand or derived from quantum

mechanics.

3 Clebsh-Gordan expansion

Riztwoller and Lavely (1991) derived the following polynomials from quantum

mechanics. The splittings are expressed as follows:

�(l;m) � �(l;0) =

i=nX

i=1

ailPi

l(m) (1)

where

P1l(m) =

m

l(2)

P2l(m) =

6m2� 2l(l + 1)

6l2 � 2l(l + 1)(3)

P3l(m) =

20m3� 4m(3l(l + 1)� 1)

20l3 � 4l(3l(l + 1)� 1)(4)

1

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P4l(m) =

70m4 � 10m2(6l(l + 1)� 5) + 6l(l + 1)(l(l + 1)� 2)

70l4 � 10l2(6l(l + 1)� 5) + 6l(l + 1)(l(l + 1) � 2)(5)

Please note that for all i we have Pi

l(l) = 1. The polynomials are derived from

Eqs (39) to (44) of Riztwoller and Lavely (1991). The derivation of (�m��0)=m

is then straightforward using Eqs (1) to (5).

Bon courage!

2

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HH#3

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Seismic interpretation of solar�like targetsProgress report

G� Berthomieu � Seismic interpretation group

Laboratoire Cassini� OCA� Nice

Participants� G� Berthomieu� J Ballot � CEA group� MJ� Goupil� Y� Lebreton� J Lochard�

A� Mazumdar� E� Michel� J� Montalban � Li�ege group� P� Morel� J� Provost� I� Roxburgh

The aim of the group was to give priorities to � stars as COROT targets� HD������HD������ HD���� HD���� These stars are solar�like stars with di erent masses andevolutionnary stages� Their position in the HR diagram is indicated in �gure � whereevolutionnary tracks with �dashed line� and without �full line� core overshoot are alsoplotted� HD����� is at the begining of the main sequence phase� HD����� and HD���may be either at the end of the main sequence phase or in a post main sequence phaseaccording to the physics used for stellar modelling� HD��� is always in a post mainsequence phase�

Figure �� Position of the four solar�like targets in the H�R diagram�

The work has been done in strong collaboration with the data analysis group througha Hare and Hound exercise� �see the report of T� Appourchaux�� Our aim was to usethe frequencies extracted from the simulated time series to infer the structure of the inputmodel� Following what has been done for the Sun� the method used by all the groups wasamodel calibration� that is trying to construct a stellar model which satis�ed as closely aspossible all the constraints�

Di erent criteria have been used to achieve this task� For the seismic interpretation�the frequencies are not used directly because we know from the solar case that they includecontributions of surface e ects� activity����� Thus linear combinations of the frequencies

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where these contributions cancel more or less are prefered� The criteria which have beenused are the following�

� Position in the HR diagram and chemical constraints� Large frequency di erences �� Mean density of the star� Small frequency spacing �� core structure �� age� evolutionary stage� Second di erence of frequencies

�� Acoustic depth of the convection zone and the helium ionisation zoneThe last criterium gives information on the star independently of the stellar model�The results and their interpretation have been discussed in a meeting in Paris� The

contributions of Lochard et al� Ballot et al� Montalban et al� Mazundar et al�� Provost et al��are available at the web address� http���virgo�so�estec�esa�nl�html�corot�datagroup�third hh�html�This report gives a summary of their work�

A� Inferences from the position in the HR diagram and metallicity constraints

To construct a stellar model� the unknowns parameters are the mass M�� the age t�initial helium and heavy elements abundances Y� Z�X� and the parameters describing theconvection� the mixing length parameter �� and the core overshoot parameter �ov�

Three constraints � the bolometric magnitude Mbol� the e ective temperature Teff �and the surface metallicity Fe�H are generally derived from astrometric� photometric andspectroscopic observations� They lead to select a range of parameters at given physicswhich may correspond to di erent evolutionary stages for the star as is seen in �gure ��for HD����� and HD����

B� Seismic analysis� Large frequency di�erences

The large frequency di erences are de�ned by� ��l�n � �l�n � �l�n�� The mean valueat high frequencies is related to the mean density of the star�

Figure �� Large frequency di erences for the input model �triangles�� the closest model�losanges� and data from simulated time series of HD����� �full dots��

As seen in �gure �� there is a systematic di erence between the large frequency dif�ferences from the input model and closest model in the case of HD������ This is due to

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the fact that the simulated time series for that star includes �surface and activity e ects�which have a quasi linear dependence on the frequency�

C� Seismic analysis� Small frequency di�erences

The small frequency di erences are sensitive to the properties of the stellar core andthus to the age and evolutionary stage� They are de�ned by�

����� � ���n � ���n������� � ���n � ����n � ���n�����

Figure �� Small frequency spacing ����� and ����� for two main sequence �heavy anddashed line� and one post main sequence model �thin line� of HD����

It is seen from �gure � that ����� is the more sensitive diagnistic of the age i�e� theevolution state of the star� With this indicator we can discriminate between a mainsequence and post main sequence model�

D Second di�erence of frequencies� Acoustic depth of the convection zone andHelium ionisation zone

Discontinuities in sound speed derivatives introduces oscillations in the frequencies�Such discontinuities occur at the basis of the convection zone where we have an adiabaticgradient in the convection zone and a radiative gradient in the radiative zone just below�There is also a rapid variation of the adiabatic exponent � in the ionization zone of He�lium II which induces a rapid variation of the sound speed� The oscillating terms in thefrequencies are visible in the second di erence in frequencies� Their periods are related tothe acoustic depth of the layer where the discontinuity takes place�

Table �� Comparison of the acoustic depths of the basis of the convection zone and locationof the helium II ionisation zone estimated from the simulated data and their theoreticalvalue from the input model�

� �CZ �CZ input �HeII �HeIIinput

HD����� ���� ���� ������ ���� � �� ������

HD����� ���� � �� ���� ��� � �� ��

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This determination of the acoustic depths �CZ and �HeII is important because it maybe used to select the closest model when some problems arise with the small spacings� Thishas been the case for HD����� due to the problems with rotational splitting �see reportof T� Appourchaux� and the values derived for �CZ and �HeII have been used to calibratedi erent global parameters of the model� Figure � illustrates the calibration of chemicalcomposition with acoustic depths�

Figure �� Calibration for HD����� of helium abundace and metallicity with acousticdepths of convection zone boundary and helium ionisation zone constraints versus meandensity constraint derived from the large frequency di erences�

E� Results of H�H� exercise

Table II illustrates the results of HH� exercise and the kind of problems encountered�All the results obtained by the di erent groups and discussed during the Paris meeting are

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not reproduced here but can be seen on the web�It must be kept in mind that the results�and thus their interpretation� depend on the assumptions which have been made �rst onthe construction of both theoretical models and simulated time series�

For the �rst star HD������ the input and output models agree despite the fact thatproblems have arised in the identi�cation of modes l � � and � turning the small spacingunusable constraint� �see T� Appourchaux�s report�� The results for the second starHD����� are more instructive� The input model was constructed including microscopicdi usion of helium and heavy elements and some additional mixing to prevent surfaceabundance anomalies� The search for closest model was made without including anydi usion mecanism� This explains the di erence in mass� surface abundaces and centralhydrogen� However the acoustic depths of convection zone base and heliumII ionisationzone are close and more surprisingly the convective core radius� It seems as if a di erencein overshoot parameter compensated the lack of di usion� This is being studied�

The case of HD��� is particular because the input model was constructed with ametallicity totally out of the range admitted for that star as it is seen in the table� Theluminosity too was derived with an old solar bolometric magnitude� The search of closestmodel has shown this di�culty to �t with constraints� Nevertheless� the use of frequenciesallowed to �nd acoustic depths of convection zone base and heliumII ionisation zone closeto the those of the input model and also the radius of the convective core�

No theoretical interpretation has been carried for the star HD����

Table II� Values of the mass �in solar unit� M�M�� the surface abundances in heliumand heavy elements Ys� Fe�Hs� the central hydrogen content Xc� the acoustic depth ofconvection zone �CZ and heliumII ionisation zone �HeII � the core radius rcore and the coreovershoot parameter�ov for the input and output �closest model� models�

� M�M� Ys Fe�Hs Xc �CZ �HeII rcore �ov

HD�����Input ��� �� ��� �� ���� �� ��� �

Output ��� �� �� ��� ���� ��� ��� �

HD���Input ���� ��� ����� ��� ���� ���� ��� ��

Output ���� �� ��� ��� ���� ���� �� ���

rCZ

HD��� �Input ���� �� ����� �� ��� ��� ��

Output ���� �� ��� ���� ��� ���� ���

� out of the observational range Fe�H � ����� ���

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F� Conclusion

The results of the exercise have shown our ability to recover the evolutionary stageof the star and properties of the structure� particularly the acoustic depth of the basis ofthe convection zone and of the HeliumII ionisation zone� independently of a model� Theradius of the convective core is also in ageement with input one� The introduction in thefrequencies of the surface e ects induces a shift in the values of the large di erences ��l�nand thus some uncertainty in the calibrationof the mean density of the star� For smallfrequency di erences this e ect can be suppressed by dividing them by ��l�n�

As far as the COROT targets are concerned� it is suggested to give a good priority tothe stars HD������ HD����� and HD���� A lower priority is given to the star HD���due to the complexity of its spectrum induced by the fact that it is a rather evolved star�

More work is to be done on the signature of di usion and core overshoot on thefrequencies� on the e ect of activity and surface e ects� on evolved stars� Moreover� onlythe direct problem of looking for a stellar model which satis�es all the constraints has beenconsidered� Inversion technics adapted to the COROT data must be developped�

Despite the fact that some rotational splitting has been added in the H�H exerciseno work has been done on the rotation� Such a work is planed for next exercise�

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1

MODE EXTRACTION FROM TIME SERIES: FROM THE CHALLENGES OF COROT TO

THOSE OF EDDINGTON

Paper presented by and with the courtesy of Jesper Schou

T.Appourchaux1, O.Moreira1, G.Berthomieu2, and T.Toutain2

1European Space Agency, P.O.Box 299, 2200AG Noordwijk, The Netherlands2D�epartement Cassini, UMR CNRS 6529, Observatoire de la Cote d'Azur, BP 4229, 06304 Nice Cedex 4, France

Abstract

With more than 30 years of experience in extractionof eigenmodes from power spectra of solar signals, we arenow almost ready to apply this knowledge onto the fore-coming missions: COROT and Eddington. Unfortunately,the �tting task di�ers by 3 orders of magnitude; COROTwill be able to get time series of stellar light for some 30stars, while Eddington will be able to gather such data forabout 50000 stars. While for COROT, our current toolscan be applied by hand, the case of Eddington is somewhat

more complex. We are looking forward having automatic�tting procedures that will allow to recover mode param-eters for about 90% of the solar-like stars. Unfortunately,about 10% of these stars will require some more delicateattention that will cost time to take care of. We will usethe example of the infamous HD57006, known to be quiteevolved with a diÆcult eigenmode spectrum, to explainhow a star can evolve from an easy-to-�t target (90% ofthe solar-like stars) to a diÆcult-to-�t target (10% of theremaining stars). In the latter case, new techniques fordetecting narrow peaks (g-mode like) out of broad peaks(p-mode like) has been devised in the context of the hare-and-hound exercise of COROT. This latter and other tech-niques will be used to implement the automatic �ttingprocedure for the remaining 10% of Eddington solar-likestars.

Key words: Stars: structure { Data analysis: time series {Data analysis: spectra �tting

1. Introduction

In the very near future, there shall be a eet of spacemissions aiming at understanding the internal structureof the stars: MOST1, COROT2 and Eddington.

All these missions will observe global oscillations ofthe stars by measuring their tiny light uctuations. Thedetection and identi�cation of these modes of oscillationis the challenge of all these missions. This challenge is for

1 Microvariability and Oscillations of Stars, a Canadian mis-

sion to be launched in June 20032 COnvection and ROTation, a CNES mission to be launched

in mid 2006

most stars extremely diÆcult (e.g. Cepheids) but easier forsolar-like stars. When the identi�cation is achieved, peakbagging can be performed. The theory of mode identi�ca-tion and peak bagging has been reviewed by Appourchaux(2003); it is now believed to be well understood.

In the context of the COROT mission, the practice ofmode identi�cation and peak bagging is developed throughthe use of hare-and-hound exercise. During this exercise,it appears that a very challenging star (namely HD57006)brought a new dimension to the usual challenge 3. Here wewould like to take this star as an example of the kind ofdiÆculties that the Eddington mission may face. For thatpurpose, we will follow HD57006 throughout its lifetimefrom the ZAMS4 until its nowadays evolutionary state.

2. Mode identification and peak bagging

The reader may wish to read the review on the subject byAppourchaux (2003), we summarize that review for thesake of completeness.

The mode identi�cation for solar-like stars is performedusing the Echelle diagram. It is based on the fact that thelow-degree mode frequencies of successive orders (n) of agiven degree (l) are distant from each others by roughlythe acoustic diameter of the star (��0). The piling up ofsection of the power spectrum (of the stellar time series)cut into piece of length ��0 produces `ridges' of poweralong an ideal vertical line. The location of the ridges withrespect to each other provides the means of tagging theridges with a given l.

The next step is the peak bagging operation whichtheory can be found in Appourchaux et al. (1998) andAppourchaux (2003). It consists in �tting the power spec-trum using Maximum Likelihood Estimators (MLE) anda model of the mode pro�le including parameters suchas frequencies, linewidth, splittings, background noise andpro�le shape. Error bars of parameters can be derived aswell as the signi�cance of these parameters (Appourchauxet al., 1998).

3 See papers of the Third COROT week in

http://www.astro.ulg.ac.be/orientation/asterosis/week34 Zero Age Main Sequence

Proc. 2nd Eddington workshop \Stellar Structure and Habitable Planet Finding", Palermo, 9{11 April 2003 (ESA SP-538,

July 2003, F. Favata, S. Aigrain eds.)

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3. The challenges in theory

3.1. COROT vs Eddington

The COROTmission shall be able to observe up to ten pri-mary seismological targets for which there will be a highsignal-to-noise ratio as to perform proper mode identi�-cation and peak bagging. The COROT secondary targetscan amount to less than a hundred. The Eddington mis-sion shall be able to observe more than 50000 stars.

For COROT, the primary targets can be analyzed by asingle scientist. The mode identi�cation and �tting can becarefully analyzed. Modes out of the main stream can be�tted by hand, and each �tted mode can be assessed for itsvalidity. One could imagine that the secondary targets arealso analyzed by the same scientists; the task may startto be somewhat diÆcult to handle.

For Eddington, this hand crafted work is to be aban-doned. Automated ridge identi�cation, mode �tting and�t rejection has to be implemented. At the time of writ-ing, the �rst task is still done by hand using the Echelle

diagram and adjusting the large separation as to have ver-tical ridges. We could envisage a step where after comput-ing the power spectrum, one identi�es the excess power,�lter the excess power and compute the Fourier trans-form for extracting the large separation (or as a matterof fact ��0=2); this technique was used by Gelly et al.(1986) on the �-Cen data. The next step is to computethe Echelle diagram using the derived large separation.As for the proper ridge identi�cation, one could try to �t5

2 pairs of peaks (l = 0 � 2 and l = 1 � 3) over ��0; theamplitude of the �tted peaks and their location would au-tomatically provide the degree tagging. The mode �ttingcan then be done as usual. The last step is the validationof the �t. It is envisaged that each mode be assessed for itssigni�cance using the likelihood ratio test (Appourchauxet al., 1998).

3.2. The stellar evolution

But the real challenges of either COROT or Eddingtonmay lie beyond these mundane details. The identi�cationand �tting steps described above can be applied to well-behaved solar-like stars; that is about 90% of the solar-like stars to be observed by Eddington. The remaining10% are evolved solar-like stars for which the automatedtechniques fall apart. Here we should outline that we donot really know the proportion of evolved stars to that ofgood stars. Even if we were to have only 0.1% of evolvedsolar-like stars in the Eddington mission, we would havethe same challenging diÆculties. As we will outline, thispart of the challenge is not to be neglected.

Here we would like to follow the evolution of a star, toscan the many stars of Eddington. We start with a star

5 where the peak power is maximum

I

II III

IV

V

VI VII VIII

Figure 1. The Hertzsprung-Russell diagram of HD57006. We

computed the mode frequencies of models labeled from I to VIII

located at di�erent positions on the evolutionary track of the

H-R diagram. The �rst four locations follow the evolution of

the star until the central hydrogen content is about 5%. The

last four locations follow the evolution of the star during the

burning of hydrogen in shell. The transition from model IV to

V is too fast by stellar standard but likely long enough by human

standards for being observed.

at the ZAMS, and study how the Echelle diagram evolveswith the star, and how the diÆculties evolve.

Here we chose the COROT primary target HD57006which scienti�c value is being assessed by the Seismol-ogy Working Group of the eponymous mission. The as-sessment is being performed in the frame work of thethird hare-and-hound exercise of COROT (Appourchaux,2003).

The star HD57006 has been represented by a 1:65M�

stellar model; that is still a solar-like star, according to thede�nition of Appourchaux (2003), when not evolved. It isnowadays suÆciently evolved as to have entered the phaseof burning hydrogen in shells, i.e. it has an helium core.This produces a large peak in the Brunt-Va��sala frequencyat the core of the star. It leads to the existence of the so-called mixed modes that have a p-mode character in theouter stellar regions, and a g-mode character in the stellarcore. The mixed modes, if detected, are powerful tools (likethe g modes) for understanding the internal structure ofthe star. Unfortunately, their mixed character make themdiÆcult to detect for they do not follow the asymptoticrelationship given by Tassoul (1980); in other words theydo not line up for making ridges.

In order to see how the Echelle diagram changes withthe evolution of HD57006, we computed frequencies oflow-degree modes all along the evolution track of the staras outlined by Fig. 1. The Echelle diagrams for each evo-lution stage in the central hydrogen burning phase areshown in Fig. 2; and for the hydrogen burning phase inshell in Fig. 3.

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Figure 2. Echelle diagram for the �rst four stages of the evolution of HD57006 during the central hydrogen burning phase; I (left,

top), II (right, top), III (left, bottom), IV (right, bottom). The echelle diagrams display the order as a function of frequency; the

large separation is annotated atop the diagram. The star and triangle are the l=0 and l=2 modes; the plus and square are

the l = 1 and l = 3 modes.

Figure 3. Echelle diagram for the �rst four stage of the evolution of HD57006 during the hydrogen burning phase in shells; V

(left, top), VI (right, top), VII (left, bottom), VIII (right, bottom). The echelle diagrams displays the order as a function of

frequency; the large separation is annotated atop the diagram. The star and triangle are the l=0 and l=2 modes; the plus and

square are the l = 1 and l = 3 modes.

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In the �rst evolution stage, the mode identi�cation and�tting would be rather classic. The large separation de-creases due to the increase in the stellar radius. The ridgesget closer from each other; the l = 0 and l = 2 mode-ridgeseparation decreases from 8 �Hz to 4 �Hz in this phase.This stage of the evolution is likely to be similar to thatof 90% of Eddington's solar-like stars.

In the second evolution stage, the mode identi�cationand �tting gets more diÆcult and challenging. The sepa-ration between the l = 0 and l = 2 mode ridge decreaseseven more to a mere 2 �Hz. The tagging of each ridge couldbe made extremely diÆcult if the linewidth mode is about1 �Hz. Nevertheless, it is possible to �t the mode pair asa single mode giving very limited information about theinternal structure of the star. Mixed modes start to ap-pear especially for l = 1. The l = 1 mode ridge is pro-gressively destroyed and disappears completely, renderingtheir tagging almost impossible. The l = 1 modes startto appear everywhere and are even sometimes right nextto the other regular p modes. Due to their mixed char-acter, the l = 1 modes are likely to be long-lived modes.Due to their erratic location in the Echelle diagram themixed l = 1 modes are bound to have mode frequenciesclose to the short-lived modes; in which the identi�cationbecomes even more diÆcult. This stage of the evolutionis likely to be similar to that of the remaining solar-likestars of Eddington.

3.3. Stellar rotation

An additional diÆculty to the mode identi�cation is thein uence of the rotation upon the determination of themode frequencies. When the small separation (Æ02; Æ13) isabout 2-3 times the rotational splitting, the proper deriva-tion of the frequencies of the l = 0�2 and l = 1�3 modesget ambiguous. It leads to small separation being negative.This ambiguity is likely to happen for solar-like stars witha rotational splitting ranging from 1 to 5 �Hz. In this case,it is required to �t the mode parameters not locally aroundthe l = 0� 2 or l = 1� 3 pair but globally over the spec-trum as already implemented by Jim�enez et al. (2002) andGelly et al. (2002) (See Neiner & Appourchaux in theseproceedings).

4. HD57006 as an evolved star

The star HD57006 is one of the candidate primary targetof COROT. In the framework of the third hare-and-houndexercise of COROT Seismology working group, a 150-daylong time series was generated by one of the authors (TT)with frequencies provided by another author (GB). Thetask of the data �tter (the 2 other authors: OM and TA)was to derive the frequencies of the detected modes.

Figure 4 shows the power spectrum of the time series.The distribution of the modes in the Echelle diagram inFig 4 does not permit to take any conclusion about any

l but l=0, in addition for higher orders it starts to bescrambled. As a matter of fact we realize a posteriori thatthe single ridge attributed to the l = 0 modes was mixedwith the l = 2 modes (See Fig 3). The detected modes fallin 2 categories:

{ short-lived modes (l tagged or unknown l){ long-lived modes (unknown l)

A speci�c strategy for each category is described hereafter.

4.1. Short-lived mode detection

The l = 0 modes were �tted as single modes using MLE.The other detected short-lived modes for which there wasno possible l tagging were also �tted in the same manner.

4.2. Long-lived mode detection

As mentioned in the previous section, long-lived modesappear because of the mixed character of the modes. Theycan be seen in Fig 4 as sharp peaks. There are two maincases to be considered:

{ long-lived mode alone{ long-lived mode embedded in short-lived mode

The �rst case happens at low order (low frequency below300 �Hz). The second case happens at higher frequency(typically order 13).

Narrow peaks alone: If one considers a pure noise signalwith a �2 statistical distribution, the probability that thepower within one bin is greater than m times the mean ofthe noise power, �, is:

PN (m) � Ne�m (1)

By setting a given value for PN (m), for instance 10%(which means 10% probability that a peak due to noiseis above m), choosing a window range in our spectrumthat contains N bins, and estimating �, one can deriveusing the equation (1) the correct value for m. This way,we have a statistical test for detecting the peaks that canbe considered as having a low probability of being due tonoise. This classical test was used by Appourchaux et al.(2000) for detecting long-lived p modes and g modes inthe SOHO data.

Narrow peaks mixed with short-lived modes: Above300 �Hz the peaks that we wanted to analyze are amongbroad modes (See Fig 4). Therefore, the application of theaforementioned test directly to the power spectrum is notvery useful because it detects peaks that are part of a sin-gle broad mode. In this case we cannot assume that thedetected peaks are all individual modes. Furthermore insome cases the detected peaks are very close to the broadmodes; it becomes impossible to assume reliably if they aresharp modes or if they are just part of the broad mode. We

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5

Figure 4. On the left hand side is the spectrum of the star HD57006 for model VIII, and on the right hand side the corresponding

Echelle diagram with ��0=30�Hz. The many peaks visible for higher orders are due to the larger linewidth of the modes at

higher frequencies.

need to �nd a test that can distinguish the broad modesfrom the sharp modes.

In order to solve this problem, we devised a techniquethat:

1. Fit the short-lived modes using MLE2. Correct the spectrum for the �tted model3. Apply the aforementioned test as if we had only narrow

peaks

Step 1: Assuming that p modes are stochasticallyexcited oscillator, one can derive that the power spec-trum of p modes oscillator is distributed around a meanLorentzian pro�les with a �2 probability distribution (An-derson et al., 1990; Appourchaux et al., 1998). The powerspectrum of the p modes can be described as:

P (�) =M(�)F (�); (2)

where F (�) is a random function with a �2 statisti-cal distribution with 2 d.o.f, and M(�) is the model ofthe �tted mode made of a single Lorentzian pro�le plusnoise. One can �t this model to the observed power spec-tra using the Maximum Likelihood Estimators technique(Anderson et al., 1990; Toutain and Appourchaux, 1994);this is the classic and well-known approach used for short-lived modes.

Step 2: After having done the �tting, one can dividethe power spectrum by the �tted pro�le M 0(�), one ob-tains:

P 0(�) =P (�)

M 0(�)� F (�) (3)

In a �rst approximation P 0(�) has a �2 statistical distribu-tion with 2 d.o.f. This is an approximation because M 0(�)is derived from data and has also a statistical distributionthat should be taken into account. We have performed aMonte-Carlo analysis con�rming that indeed P 0(�) hassuch a statistical distribution. This way we solved theproblem of the mixing between the sharp modes and thebroad modes.

Step 3: Applying the �2 test to HD57006 spectrum,we set PN (m) = 10%, � = 1, and a window size of 30 �Hz(corresponding to 389 bins). This last step is the same asfor the long-lived modes alone.

For each window we use the stepped approach. Theresult of which can be seen in Fig. 5.

The comparison with the original frequencies6 gavethat only about 1 peak was misidenti�ed over the band200�Hz-500�Hz. On average we should statistically havehad 1 peak (� 1) due to noise. It validates the approachtaken for detecting long-lived modes either singled put orembedded in short-lived modes.

4.3. l tagging of the frequencies

The l tagging of the frequencies of the modes could not

be derived from the Echelle diagram apart for the l = 0modes. As a matter of fact, due to the width of thesep modes, we may have �tted the mean location of thel = 0� 2 ridge.

The l tagging for the non-l = 0 p modes would need tobe derived from the splitting. After having derived the fre-quencies of the modes for the 2 categories, we performeda correlation analysis on the extracted frequencies for get-ting the signature of a possible splitting. Since we didnot �nd any splitting signature, we could not tag furtherthe identi�ed modes. At least we were able to derive thefrequency of the modes for the 2 categories as explainedabove.

Our inability to properly tag the frequencies of themodes has serious consequences for the usefulness of sucha star for understanding its internal structure. We couldimagine a scheme where we could search for the stellarmodel for which mode frequencies would match the �ttedmode freqeuncies. Given the erratic behaviour of the l = 1mode frequencies, this exercise could be `easily' achieved

6 Comparison performed a posteriori

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6

Figure 5. Result of the test for two di�erent ranges of frequencies. The �tting is presented on the top of the �gure, and the power

spectrum divided by the �tted model, as well as the bins above the 10% probability level, are presented on the bottom of the �gure.

for comparing input frequencies and output frequencies ofa stellar model coming out of a single routine (e.g. theCESAM code). This is only of pure academic interests asthe stars do not follow the CESAM code. . .

The only way out would be to use color informationfor tagging the frequencies of the modes. This methodis not used for solar-like stars but is very useful for say

Cepheids. It will involve proper computing of the radiativetransfer that is likely not to be easy. Nevertheless, a simpleapproximation to that problem is well known (Toutain andGouttebroze, 1993). This additional color information islikely to be useful for COROT, and should become a mustfor Eddington.

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5. Conclusion

In this paper, we concentrated mainly on the �tting chal-lenges that the COROT and Eddington missions are goingto face. For each stage of the stellar evolution there aretwo challenges to be solved:

{ Challenge I: power spectra �tting for solar-like starsin their hydrogen burning phase

{ Challenge II: power spectra �tting for solar-like starsin their hydrogen-shell burning phase

Challenge I provides no major problem for �ttingmost of star spectra; this can even be automated.

Challenge II is mainly related to properly identifythe degree of the modes. This challenge is much more dif-�cult and could cost a lot of time as it is only solvable byhand at the time of writing. For this purpose, we devisedan automated method for identifying long-lived modes em-bedded in short-lived modes. So we have been able to solvepart of Challenge II. The bulk of this challenge may onlybe solved by covering a given evolutionary track with starsof similar masses; thereby trying to follow with di�erentstars how the mixed modes could appear. Additional infor-mation related to color could be brought in for the degreeidenti�cation. This could make the automation feasible.The automation of the mode identi�cation and �tting forChallenge II is still in its infancy but 2 research trackshave been laid out. They shall be tested with COROT andperfected for Eddington.

Acknowledgements

This review bene�ted from the collaboration with the Data

Analysis Team of the Seismology Working Group of COROT.

Thanks to Ian Roxburgh for interesting discussions, and to

Coralie Neiner for commenting on the manuscript.

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Berthomieu, W. Chaplin, Y. Elsworth, W. Fin-sterle, D. Gough, J. T. Hoeksema, G. Isaak, A.Kosovichev, J. Provost, P. Scherrer, T. Sekii, and T.Toutain: 2000. ApJ 538, 401.

Appourchaux, T., L. Gizon, and M. C. Rabello-Soares:1998. A&A Sup. Series 132, 107.

Gelly, B., G. Grec, and E. Fossat: 1986. A&A 164, 383{394.

Gelly, B., M. Lazrek, G. Grec, A. Ayad, F. X. Schmider,C. Renaud, D. Salabert, and E. Fossat: 2002. A&A

394, 285{297.Jim�enez, A., T. Roca Cort�es, and S. J. Jim�enez-Reyes:

2002. Solar Physics 209, 247{263.Tassoul, M.: 1980. ApJ Sup. Series 43, 469.Toutain, T. and T. Appourchaux: 1994. A&A 289, 649.Toutain, T. and P. Gouttebroze: 1993. A&A 268, 309.

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HD43318

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50

52

54

56

58

60

600 700 800 900 1000 1100 1200 1300 1400 1500

∆ν

frequency

No overshooting

M=1.2,X=0.70,Z=0.011M=1.2,X=0.70,Z=0.012M=1.2,X=0.72,Z=0.011

Page 101: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

50

52

54

56

58

60

600 700 800 900 1000 1100 1200 1300 1400 1500

∆ν

frequency

overshooting = 0.2

M=1.2,X=0.70,Z=0.011M=1.3,X=0.70,Z=0.011M=1.3,X=0.70,Z=0.012M=1.3,X=0.70,Z=0.013

Page 102: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

2

2.5

3

3.5

4

4.5

5

5.5

6

600 700 800 900 1000 1100 1200 1300 1400 1500

δν02

frequency

M=1.2,X=0.70,Z=0.011,ov=0M=1.2,X=0.70,Z=0.012,ov=0M=1.2,X=0.72,Z=0.011,ov=0

M=1.2,X=0.70,Z=0.011,ov=0.2M=1.3,X=0.70,Z=0.011,ov=0.2M=1.3,X=0.70,Z=0.012,ov=0.2M=1.3,X=0.70,Z=0.013,ov=0.2

Page 103: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

4

5

6

7

8

9

10

600 700 800 900 1000 1100 1200 1300 1400 1500

δν13

frequency

M=1.2,X=0.70,Z=0.011,ov=0M=1.2,X=0.70,Z=0.012,ov=0M=1.2,X=0.72,Z=0.011,ov=0

M=1.2,X=0.70,Z=0.011,ov=0.2M=1.3,X=0.70,Z=0.011,ov=0.2M=1.3,X=0.70,Z=0.012,ov=0.2M=1.3,X=0.70,Z=0.013,ov=0.2

Page 104: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

0

2

4

6

8

10

600 700 800 900 1000 1100 1200 1300 1400 1500

δν01

frequency

M=1.2,X=0.70,Z=0.012, ov=0M=1.3,X=0.70,Z=0.011, ov=0.2

Page 105: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 109: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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He = 1517.78

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Page 110: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 111: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 112: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 113: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 114: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 145: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 146: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

byR.A. García

Service d'Astrophysique CEA Saclay

Hare & Hound # 3

Saclay Team:

S. Turck-Chièze & J. Ballot

Page 147: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

• 4 different estimators have been computed:– Periodogram (FFT)– Multitaper (4 Sine tapers)– Zero Padded Periodogram (4 times)– Averaged cross spectrum

Page 148: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

• Background fitted with a 2 elements Harvey model• 2 groups of modes are clearly identified• The individual multiplets are not directly recognized

Page 149: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

• Distance between the two left structures changing with frequency =>• Different modes not a “changing” splitting

• Amplitude ratio => doublet l=0-2 and l=1 • From the autocorrelation: Small and big differences

• Constructing guessing frequencies for the fit

Page 150: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

• Symmetric and asymmetric profiles used • No systematic differences observed on the resultant frequencies

• Fitted parameters: Frequency, amplitude, Line-width– First step:

• fitting only one lorentzian profile per mode– The asymmetry takes into account the asymmetry of the whole mode– Better determination of the central frequencies

– Second step:• Using the new guessing table

– free splitting and different fixed amplitude ratios of the components of the multiplets

– Fixed splitting to the most common value found (~800 nHz)

• This process has been repeated for the 4 Fourier estimators

Page 151: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

530.5170 0.13 600.1405 0.1227 667.6530 0.2413 734.4192 0.1332 765.3281 0.1320 802.8481 0.0893 833.1413 0.1070 866.4656 0.1602 872.3123 0.1149 902.5326 0.2750 937.2448 0.0804 942.6650 0.1042 973.8620 0.1706 1009.1386 0.2244 1014.3876 0.1930 1045.7911 0.1186 1081.0989 0.2060 1086.3512 0.1376 1117.3852 0.2129 1152.3626 0.1874 1181.6210 0.3340 1157.7844 0.2039 1188.2857 0.4551 1222.6506 0.2759 1252.1820 0.4431 1227.9085 0.1812 1258.9915 0.2774 1294.3077 0.6501 1324.2065 1.0540 1299.4447 0.2135 1330.8657 0.2434 1367.0392 0.2661 0 0 1371.8733 0.3462 1403.4696 0.5409 1439.2745 0.4578 1468.8972 1.2170 1444.3102 0.3877 1476.5164 0.6188 1513.4282 0.7430 1517.4924 0.6915 1549.8008 0.6593 0 0 1622.9331 0.6533

0 0 1696.2605 0.6527 0 0 1769.5830 0.6735

Resultant frequencies and Big difference* Candidate modes (not present in all the estimators)

*

Page 152: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

Page 153: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

Page 154: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

Page 155: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

Page 156: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

14-15 May 2003 Rafael A. García

• Improving the algorithm for :– Fully automatic research– Find a criteria to select the modes form the

different computed estimators– When a fit fails:

• To reduce the number of free parameters• Using general trends for Amplitude & Line-width

• How to determine the splitting????• Is it a key parameter for the Stellar models????

Page 157: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

HD49933

Page 158: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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Page 159: Hare-and-hound exercises of the Data Analysis Team of the ...Report of the hare-and-hound exerciseof the Data Analysis Team T.Appourchaux Space Science Departmentof ESA, P.O. Box 299,

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HD49933

Teff = 6700 ± 100 KMbol = [3.25 ; 3.45][Fe/H] = -0.32 ±0.1

Vsin i = 10 ± 4 km/s

3.8325 ≤ log Teff ≤ 3.81950.52 ≤ log L/L_ ≤ 0.6 ( with Mbol,_ = 4.75)0.009315 ≤ (Z/X) ≤ 0.01476 (with (Z/X)_ = 0.0245)0.00652 ≤ Z ≤ 0.01033 ( if X = 0.70)

<Dn n,0>n>10 = 90.1765 (OM+TA)<Dn n,0>n>10 = 90.1799 (TT)

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Y1.8 0.0 / 0.2 0.03 0.701.5

Y1.8 0.0 / 0.2 0.019 0.701.35, 1.38

N1.8 0.0 / 0.20.00857 0.7361.15 – 1.3

N1.8 0.0 / 0.20.01033 0.70 1.1 - 1.25

N1.8 0.0 / 0.2 0.0082 0.70 1.05 - 1.3

N1.8 0.0 / 0.2 0.00652 0.70 1.05 - 1.15

diff a ov Z X M

Models

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The best models fitting Dnn,ℓ are

1. Dnn,ℓ = nn,ℓ – nn-1,ℓ

2. dn0,2 = nn,0 – nn-1,2

3. dn0,1 = 2 nn,0 – (nn,1 + nn-1,1)4. dn1,3 = nn,1 – nn-1,3

M : 1.12 – 1.25 M_ Z : 0.00652 – 0.01033 X : 0.70 – 0.736 Age: 2 – 3 Gyr. aov: 0.0 – 0.2 No Diff

M > 1.35 M_ 0.03 > Zi > 0.019 X : 0.70 Age < 1. Gyr. aov : 0.0 – 0.2 Diff

but they do not satisfy 2, 3, 4

Introducing diffusion, the constraints onthe model parameter would be

It appears that dn0,1 decreases at high frequenciesif overshooting is included, and dn0,1 decreasesat low frequencies if microscopic diffusion is

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HD57006

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Report of the Hare-and-Hound exercise #3 (HH3)

The case of HD57006

May 13, 2003

Abstract

In this report we present the analysis done to the synthetic spectra of H57006. For this evolved star,

we could not identify the modes and extract their frequencies and splittings with their formal errors in

the same way as for a solar like star. For this reason, this star revealed to be very diÆcult from the data

analysis point of view.

The Echelle diagram only permitted us to identify the l = 0 modes, but the �tting could reliably be

made only below 500�Hz. The understanding of the high-order-mode behaviour was rather impossible;

we were restricted to the low order analysis.

For the study of long-lived modes, we had to devise a technique that could distinguish narrow peaks

(`g-mode like') among the wider modes (`p-mode like').

1 Introduction

In the context of the Hare-and-Hound exercises, HD57006 is one of the targets chosen to be studied interms of its seismology. It is a post main sequence star with about 1.6 M

�, an evolved star with a contracting

core. Because of that, it is very complex in terms of its frequencies, and rise some doubts about our abilityto identify the modes in such stars.

The oscillation frequencies are in uenced by the sound speed, c, and the Brunt-V�ais�ala frequency, N(Tassoul, 1980; Audard and Provost,1994). As both depend on the mean molecular weight, �, (hence onthe central hydrogen abundance, Xc) they don't change very much during the main sequence. However,when the main sequence phase of stellar evolution ends, Xc is exhausted and the star settles in a state inwhich the hydrogen burns in a shell surrounding a helium core. The mass of the helium core is increased bythe hydrogen burning in the shell leading to an expansion of the envelope of the star. Such stellar interiortransformation strongly a�ects �, consequently the oscillation frequencies.

The modes are trapped in propagation zones, whose extension and location depends on variation of Nand on the Lamb frequency Sl = Lc=r, with L2 = l(l + 1). The propagation of p modes happens when themode frequency is greater than N and Sl, while the propagation of g modes happens when the frequencyof the mode is smaller than N and Sl. At the beginning of the evolution N is relatively small in stellarinterior and varies smoothly with radius, therefore there is a well-de�ned separation between the modeswhose eigenfunctions have nodes in acoustic propagation zone and those whose eigenfunctions have nodes inthe gravity propagation zone. As the evolution proceeds a strong gradient of chemical composition developsat the convective core frontier, giving rise to a large peak in N close to the frontier changing the structure ofthe mode propagation zones. As a result the eigenfunctions of g modes of low radial order develop nodes inthe acoustic propagation zone, while eigenfunctions of the low order p modes develop nodes inside the zoneof the peak of N . These modes have a mixed character, they behave as g modes close to the core frontierand as a p modes in outer regions of the star (Dziembowski and Pamyatnykh, 1991; Audard, Provost andChristensen-Dalsgaard, 1995).

In this report, we describe an attempt of identi�cation of the modes and extraction of the frequencies ofa star that has the terrifying scenario of such mixed modes.

1

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2 Data Analysis

One fundamental quantity in seismology is the large separation, ��0. It represents nearly the uniformseparation between the two modes of same degree and successive order (��0 � �n;l � �n�1;l) for p modesof high radial order and low degree. The frequency of these modes can be described using an analyticalasymptotic way (Tassoul, 1980). The ��0 is proportional to the characteristic frequency g =

pGM=R3

therefore it can be a measure of the mean density of the star, and hence of its mass and radius if g is known.The ��0 is also proportional to the inverse of the acoustic radius expressed in seconds.

Useful information can also be obtained from the small frequency separation that measures the departuresfrom the uniformity. It corresponds to the di�erence between two modes with same n + l

2and is de�ned

as dn;l = �n;l � �n�1;l. This quantity is sensitive to conditions in the structure of the core and to chemicalcomposition and therefore can be an age indicator (Audard and Provost, 1994).

Another quantity that can be derived from the oscillation frequencies is the second frequency di�erenceÆ2� = �n;l + 2�n�1;l + �n�2;l that has an oscillatory behaviour. It is related with the rapid variation of theadiabatic exponent due to the HeII ionization, therefore this quantity may provide a diagnostic to determinethe helium abundance (Audard and Provost, 1994).

2.1 Echelle Diagram

The method used to detect the large separation, ��0, is the Echelle diagram. The Echelle diagram consistsin cutting the spectrum into pieces of width ��0 and pile them up atop of each other. If ��0 is the correctone, \vertical" ridges will appear clearly in the diagram.

In the case of HD57006 \clearly" is not the best word to describe it. The distribution of the modes inthe Echelle diagram in the Figure 1 does not permit to take any conclusion about any l but l=0, althoughfor high orders it starts to be messy.

Figure 1: On the left hand side is the spectrum of the star HD57006, and on the right hand side the Echellediagram with ��0=30�Hz.

Because of the oddness of the HD57006 Echelle diagram, additional theoretical information about thisstar was necessary. Two sets of frequencies were provided by Ian Roxburgh and by M�ario Monteiro with thecollaboration of Jo~ao Marques.

As one can see on Figure 2, l=1 in both sets of frequencies behaves almost chaotic. After the 15th order,l = 2 starts to have the familiar behaviour next to the l = 0 on the left hand side, as predicted by theasymptotic theory. However if one looks again at the Echelle diagram in Figure 1, it is very hard, if notimpossible to recognize or to guess its location, therefore it wasn't possible to �t it. We only �tted l = 0,the results can be seen in Figure 3 and in Table 2.1.

Above 500�Hz, since the l = 0 and l = 2 modes are very close to each other and apparently very wide,it is diÆcult to aÆrm that only l = 0 is �tted, because it is possible that some l = 2 were �tted with or

2

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Figure 2: The Echelle diagram of the set of frequency given by the two stellar models of the star HD57006.l=0 is represented by a square, l=1 by a triangle, l=2 by an asterisk and l=3 by a plus sign.

Figure 3: The result of the l=0 mode �tting. For high orders (n>15) the �t is rather erratic. Since the starspectrum is very messy, at least after n>15, the l=2 modes are suÆciently close to that of the l=0 modes(as it can be seen in the Echelle diagrams of the stellar models) for being mixed with the l=0 . It is possiblethat some l = 2 are among the �tted l=0.

3

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instead of l = 0. Therefore we only present in Table 2.1 the frequencies �tted for l = 0 below 580 �Hz.

l = 0159.19391�0.038500000189.08200�0.044807799219.14301�0.036794599249.03085�0.013279705279.10117�0.038500000309.23514�0.099702701338.49667�0.14239374366.45700�0.17383800396.97388�0.40610036427.58200�0.28380781459.62387�0.15837938490.79501�0.81077600522.04315�0.44192797550.83026�6.6242681582.78986�6.8992927e-06

Table 1: l = 0 mode frequencies and their formal errors in �Hz as determined by maximum likelihoodestimators. The errors determined for the frequency 582�Hz seems to be way too low to be reliable, thisidenti�ed mode belongs to the high-order mess that can be observed in the echelle diagram. This is thereason why we concentrated our attention on frequencies below 500�Hz.

2.2 Mode identi�cation challenge

The main goal in asteroseismology is to derive from the oscillation frequencies of the modes, the internalstructure and rotation of the stars. With l=0 only we can measure ��0, the information that can be derivedis the acoustic radius of the star. In order to know the internal structure and the rotational of HD57006 itis necessary to extract the frequencies of the other modes.

For the high orders, n > 15, the spectrum seems too messy to be able to understand it and extractuseful information from it. The existence of large power narrow peaks spread along the lower part of theEchelle diagram n < 15 can be noticed. These peaks may lead us to additional information about theinternal structure of the star. We believe that we may learn something from their frequency extraction. Weconcentrated our attention to the frequency range below the 500 �Hz.

Analyzing the power spectrum below 500 �Hz, one can notice two distinct features:

� the isolated narrow peaks and

� a bunch of peaks close to each other.

Two bunches of peaks can be well noticed on the left hand side of the Echelle diagram (Figure 1) withinthe 13th and 15th orders. These two kind of features can not correspond to the same physics, since thatthe isolated narrow peaks are probably modes which their energy fall only into one bin, therefore it cancorrespond to long-lived modes, while the bunches of peaks may correspond just to one broad mode, and itcan correspond to short-lived modes.

To extract the frequencies of the isolated narrow peaks we need to assess whether they may be due tonoise. In some cases they can be embedded in the bunches of peaks that are supposed to be short-livedmodes. Therefore we also need to assess whether we can detect and distinguish them among the broadmodes.

4

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2.2.1 Long-lived modes detection

Narrow peaks alone

If one considered a pure noise signal with a �2 statistical distribution, the probability that the powerwithin one bin is greater than m times the mean of the noise power, �, is:

P(m) = e�m (1)

For a frequency band containing N independent bins, the probability that there aren't any bins withpower greater than m becomes:

PN (m) = (1� e�m)N (2)

Therefore the probability that at least one bin has a power greater than m becomes:

PN (m) = 1� (1� e�m)N (3)

then if e�m � 1 the equation (3) can be approximated to:

PN (m) = Ne�m (4)

Setting a given value for PN (m), for instance 10% (which means 10% probability that one bin above mis due to the noise), choosing a window range in our spectrum which contains N bins, and estimating �, onecan derive using the equation (4) the correct value for m. This way, we can create a statistical test thatcan detect the bins that can be considered as not being due to the noise (see also Appourchaux et al., 2000;Gabriel et al.,2002).

Below 300 �Hz there aren't any broad modes and we applied the test directly to P (�) we �ltered out thebins corresponding to the identi�ed l = 0.

Narrow peaks embedded in short-lived modes

Above 300 �Hz the peaks that we wanted to analyze are among broad modes. Therefore, to apply theaforementionned test directly to the power spectrum is not very useful because it detects bins that are partof a single broad mode, in this case we can not assume that the detected bins are all individual modes.Furthermore in some cases the power peaks are very close to the broad modes, and is not possible to assumereliably if they are sharp modes or if they are just part of the broad mode. We need to �nd a test that candistinguish the broad modes from the sharp modes.

In order to solve this problem, we devised a technique that:

1. Fit the short-lived modes using MLE

2. Correct the spectrum for the �tted model

3. Apply the aforementionned test as if we had only narrow peaks

Step 1: Assuming that p modes are stochastically excited oscillator, one can derive that the powerspectrum of p modes oscillator is distributed around a mean Lorentzian pro�les with a �2 probabilitydistribution (Toutain and Fr�ohlich, 1992; Appourchaux et al., 1998), therefore it is possible to apply astatistical test. In our case we want to extract the frequency corresponding to the sharp peaks in the powerspectrum, within the frequency range below 500 �HZ, that have high probability not to be due to noise.The power spectrum of the p modes can be described as:

P (�) =M(�)F (�) (5)

Where F (�) is a random function with a �2 statistical distribution, and M(�) is the model of the �ttedmode made of a single Lorentzian pro�le plus noise.

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Step 2: One can �t this model to the observed power spectra using the Maximum Likelihood Estimatorstechnique (Toutain and Appourchaux, 1994). If one divide the power spectrum by the �tted pro�le, let'scall it M 0(�), one obtain:

P 0(�) =P (�)

M 0(�)� F (�) (6)

In a �rst approximation P 0(�) has a �2 statistical distribution. This is an approximation because M 0(�) isderived from data and has also a statistical distribution that should be taken into account1. In this way wesolved the problem of the mixing between the the sharp modes and the broad modes.

Step 3: Applying the �2 test to HD57006 spectrum, we set PN (m) = 10%, � = 1, and a window sizeof 30 �Hz (corresponding 389 bins). For each window we �tted the broad modes within using the MLEtechnique and after divided the power spectrum by the resulted �tting we applied the statistical test. Theresult can be seen in Figure 4.

Figure 4: Result of the test for two di�erent ranges of frequencies.The �tting is presented on the top of the�gure, and the division of the power spectrum by the �tting, as well as the bins above the 10% probabilitylevel, are presented on the bottom of the �gure.

The assumption that the g modes have long life-time implies that almost all of the energy from a single gmode will fall in one bin or, at most, in two neighbouring bins (Gabriel et al., 2002). Therefore we assumedthat the sharp modes could be the g modes (and of course the broad modes are the p modes). The resultsare in Tables 2 and 3, and the distribution of the modes frequencies in the Echelle diagram can be seen inFigure 5.

non�` = 0320.195�0.256729353.860�0.246180373.999�0.324451409.970�0.225900451.725�1.45902474.797�0.512622

Table 2: Frequencies and their formal errors in �Hz of the non-l = 0 modes as determined by the MLEtechnique for the broad modes corresponding to the unidenti�ed p modes below 500�Hz.

1This shall be the subject of a Monte-Carlo analysis

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Figure 5: The distribution of the modes in the Echelle diagram. The l = 0 modes are presented as squares,the g modes as asterisks, and the non-l = 0 modes as triangles.

g modes

226.08025 236.18827 237.19136 252.39198 253.85802 255.09259 259.25926260.49383 265.58642 268.59568 276.62037 278.31790 279.78395 281.09568282.25309 282.40741 285.87963 287.65432 289.19753 290.66358 296.68210301.69751 303.39504 311.11109 315.81788 318.20986 332.25308 345.75616347.53085 363.11728 364.27469 371.68210 377.00617 400.54011 402.31481433.71914 435.49383

Table 3: Frequencies of the g modes extracted using the statistical test described in the subsection 2.2.1

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2.2.2 Autocorrelation and correlation

The correlation determines the degree of similarity between two signals. If the signals are identical, then thecorrelation coeÆcient is 1 (or -1); if they are totally di�erent, the correlation coeÆcient is small and close to0.

The Autocorrelation is a method which is frequently used for the extraction of the fundamental frequency,in this context this is the l = 0 ridge. If a copy of the same signal is shifted, the distance between the centralpeak (corresponding to the non shifted) and the next correlation maximum is taken to be the fundamentalfrequency, in this case that this the large separation (Figure 6, left). Having this in mind, this can be appliedto the extracted frequencies in order to check if there is some hidden pattern, or some kind of periodicitywhich is not very clearly to human eye.

We determined the autocorrelation of the extracted frequencies corresponding to the g modes plus thosecorresponding to the non�l = 0 p modes (Figure 6, right). We didn't �nd out a clear evidence for theexistence of a correlation as we can �nd for the autocorrelation of the extracted frequencies of l = 0 modesalone.

Assuming that the shapes of l = 0 and l = 1 in Echelle diagram might be very similar, we correlated thel = 0 frequencies with the frequencies of g modes plus non-l = 0 p modes to check if one can �nd a repetitionof the l = 0 ridge (Figure 7). However, once more we didn't clearly �nd any evidence for that.

Figure 6: On the left hand side, we have the autocorrelation of the extracted frequencies corresponding thel = 0 modes; we can see a fundamental frequency corresponding to the large separation. On the right handside, the correlation between the frequencies corresponding to the g modes plus the frequencies correspondingto the unidenti�ed p modes; we can not see any fundamental frequency.

3 Conclusion

High-order low-degree p modes of the HD57006 do not behave in the same way as one could predict fromthe asymptotic theory for solar-like stars in the main sequence. Looking at the Echelle diagram, we cannotsee the \vertical " ridges corresponding to l = 0; 2 and l = 1; 3, therefore it is not possible to determine thedn;l and Æ2�, hence is not possible to deduce the internal structure of the star.

If one focus our attention to the low orders, the HD57006 power spectrum seems to be able to providemore information about the star; we can identify clearly the l = 0 modes that can be �tted without majordiÆculties. One can also notice the presence of sharp modes that can possibly correspond to g modes or pmodes with mixed character, since in this phase of evolution both type of modes have a larger propagationzone close to the convective core frontier due to the Brunt-v�ais�ala frequency peak.

For the extraction of the frequencies of the sharp modes we devised a technique that can distinguishthem from the broad modes. This technique can also be applied to stars where the mixed modes characterexists but not so strong as in HD57006.

8

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Figure 7: Correlation between the identi�ed l = 0 p modes and the unidenti�ed p modes plus the g modes.If one con�ne our attention to the interval [-30�Hz,+30�Hz] we can see four maximums but they are notsuÆcient prominents to aÆrm that there is a clearly repetition of the l = 0 shape.

Last but not least, we were not able to identify any rotational splittings from the non-l = 0 modes.

References

Appourchaux T., Fr�ohlich C., Andersen B. N., et al., 2000, Apj, 538, 401Appourchaux T., Gizon L., Rabello-Soares M.-C., 1998, A&A,132,107Audard N., Provost J., 1994, A&A, 282, 73Audard N., Provost J., Chirstensen-Dalsgaard J., 1995, A&A, 297, 427Dziembowski W. A., Pamyatnykh A. A., 1991, A&A, 248, L11Gabriel A. H., Baudin F., Boumier P., Garc��a R. A., Truck-Chi�eze S., Appourchaux T., Bertello L.,Berthomieu G., Charra J., Gough D. O., Pall�e , Provost J., Renaud C., Robillot J.-M., Roca Cort�es T.,Thiery S., Ulrich R. K., 2002, A&A, 390, 1119Tassoul M., 1980, ApJS 43, 469Toutain T., Appourchaux T., 1994, A&A, 289, 649Toutain T., Fr�ohich C., 1992, A&A, 257, 287

9

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