université paris-saclay...report of the hare-and-hound exerciseof the data analysis team...

Hare-and-hound exercises ofthe Data Analysis Team of

the Seismology Working Group

by

Appourchaux and Berthomieu

May 2005

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 Côte d’Azur, Département 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.

Appourchaux et al.- COROT/SWG/Milestone 2000 2

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


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 & Fröhlich, 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


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.


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 & Fröhlich(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


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.


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).


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


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, Université de Nice

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

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

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

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

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

C. Barban et al. - COROT/SWG/Milestone 2000 2

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


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 spacinghistogram (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).


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


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 ��.


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 ��.


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.


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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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�ective

temperature �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�

�

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��

�

Figure � Small frequency spacings �� and �� for di�erent models satisfying

observed� 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� ��

offset

is 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�


�

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


Fitting and interpretation

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université paris-saclay...report of the hare-and-hound exerciseof the data analysis team...

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