MODE TYPING OF THE DOMINANT MODE IN THE DELTA SCUTI STARX CAE BY LINE PROFILE MOMENTS

L. Mantegazza1,2, E. Poretti1

1 Osservatorio Astronomico di Brera, Via Bianchi 46, I–22055 Merate, Italy2 Dip. Fisica Nucleare e Teorica, Universita di Pavia, Italy

ABSTRACT

We use moment method developed by Balona (1987)to identify the dominant mode of the multimode pul-sating δ Scuti star X Cae. It is shown as a ` = 1and m = −1 non-radial mode gives a satisfactory fitof the amplitudes and phases of light and line mo-ments variations provided that the rotational axishas i ≥ 45o. The physical parameters correspondingto this mode are derived and discussed.

1. INTRODUCTION

One fundamental step in order to perform astero-seismology with δ Scuti stars is the typing of theirpulsation modes. The recent multisite campaignshave shown that these objects have very rich pul-sational spectra, however the photometric studies,while allow accurate mode detections, are inadequateto identify them. One powerful tool to this end is thestudy of line profile variations which are suitable tobe analyzed with different techniques. In particularthe technique of analysis of the low order momentsdeveloped by Balona (1986a, 1986b, 1987b, 1996) isvery promising for the study of low order non radialmodes, which are those responsible of light varia-tions.

In this paper we will show the result supplied by itto the study of the dominant mode of the star XCae. Our photometric and spectroscopic studies ofthis star (Mantegazza and Poretti, 1992 (PaperI),1996 (PaperII)) have shown the presence of at least14 pulsation modes with one of them far strongerthan the others (ν = 7.393 c/d, AB = 0.049 mag;all the others have amplitudes below 0.01 mag). Apreliminary analysis of the line moments has been al-ready performed in Paper II; here we present an im-proved analysis applied to the moments of the domi-nant mode, made following Balona et al. (1996). Allthe data relevant for this analysis are in Paper II.We remind that spectra with 50000 resolution wereobtained at La Silla Observatory with the CAT tele-scope on November 26–29, 1992.

In the meantime, with the availability of the Hippar-cos parallaxes (Antonello, private communication),it has been possible to fix the stellar absolute magni-tude, which has been matter of discussion in PaperII since values obtained from different uvbyβ cali-brations were in partial disagreement. The valueobtained from Hipparcos data is MV = 1.27 ± 0.15

which agrees with that derived from Crawford (1979)calibration (1.05±0.30) in Paper II. The ubvyβ pho-tometry allows to derive also Teff = 7000oK, so bycombining it with the Hipparcos absolute magnitudewe get R = 3.5R.

2. APPLICATION OF THE METHOD OFMOMENTS

The Balona’s method assumes that the pulsation ve-locity amplitude is much smaller than the projectedrotational velocity and that the ratio between rota-tion and pulsation frequencies (Ω/ω) is small. Theseconditions are both verified by the dominant modeof X Cae which has a radial velocity amplitude of3.6 km/s to be compared with ve sin i = 70 km/s,and, by combining this with the radius, we getProt/ sin i = 2.55d and thus Ω/ω ≥ 0.05.

The amplitudes and phases of line moment varia-tions and light variations can be expressed in termsof three velocity fields vr (vertical velocity), vh (hor-izontal velocity) and vf . This last is a fictitiousvelocity field which quantifies the distortion of theline profiles generated by the variation of the tem-perature distribution on the photosphere due to thenon–radial pulsations and is defined as vf = ∆F/F ·ve sin i, where F is the flux. For each possible non–radial mode (identified by ` and m ) and a particu-lar inclination of the rotational axis i the amplitudesand phases of the light curve and of the first few mo-ments can be fitted to the predicted ones and thepulsational parameters computed. Since there are 6unknowns (amplitudes and phases of the three veloc-ity fields) by using at least three moments and thelight curve, it is possible to solve the system with theleast squares, and in this case the residual defines adiscriminant that allows to compare solutions withdifferent (`,m, i) sets and obtain the most likely ones.The discriminat has dimension of velocity since allthe equations of condition in the algorith are nor-malized to velocities.

Since the moment accuracy decreases as the order in-creases only the first few moments can be used, so inorder to have a significant discriminant it is useful toemploy as many constraints as possible. In the caseof δ Scuti stars vh < 0.08·vr ( since vh ' 74.4·Q2 ·vr,Q pulsational constant), so that it is reasonable toconstrain vh = 0.08vr and with the same phase, andso to reduce the unknown velocity fields to two. An-other constraint is that vf < ve sin i, since larger val-ues imply relative flux variations greater than unity.

For our application to X Cae we have used amplitudes and phases of the first 5 moments and of theB light curve (this last has been chosen because thelines used to compute the moments are at about4500A). Light and moments equations have beenweighed according to the standard errors of theiramplitudes and phases as obtained from the Fourierfit. The computation included all the modes with to0 ≤ ` ≤ 4 and −` ≤ m ≤ `. The discriminant isplotted againts angle of inclination in Fig.1 for themodes which best fit the data.

Figure 1: Discriminant (arbitrary units) as a function ofangle of inclination (in degrees) for various modes (`,m)

3. DISCUSSION

It is apparent that (1,−1) and (2,−1) are the bestcandidates. Both are prograde modes (m < 0) andthis is in agreement with a visual inspection of theline profiles which show that the dominant perturba-tion is moving in the prograde direction (see Fig.5 ofPaper II). Moreover if we look at the velocity ampli-tudes we see that for (2,−1) as its discriminant ap-proaches its minimum values vf approaches ve sin iand overcomes it for i > 75o, so, while this modesupply a good mathematical fit of the observed quan-tities, it is not physically meaningful. Therefore themost promising identification for ν = 7.393 c/d is(1,−1). This mode matches almost perfectly ampli-tude and phase of the light curve. The minimumdiscriminant is attained at i = 70o and for this in-clination vr = 19 km/s and vf = 14 km/s, this ve-

locity implies an amplitude of the temperature variation of about 300oK. The ratio between flux andradius variations results f = 15 and the phase dif-ference between these two quantities is ψ = 118o

and ∆R/R = 0.013. As a comparison, the Cepheids,which are radial pulsators, have < ψ >= 124o ± 12and 5 ≤ f ≤ 15 (Balona & Stobie, 1979).

Obviously we can see from the figure that the in-clination is not strongly constrained and any valuewith i > 45o it could be acceptable. In this range18 < vr < 24, and 13 < vf < 18. In order to fullyconstrain i could be useful to get similar solutions forthe other observed modes. Unfortunately the presentdata do not allow to estimate amplitudes and phasesof their moments with the required accuracies. Tothis end a new set of spectroscopic data covering alonger time baseline is necessary. A 8 – day run isscheduled on November 1996 always at the ESO–CAT telescope.

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