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\IBVShead{4549}{28 January 1998}

\IBVStitletl{The Nature of V829 A\lowercase{ql} -- A Triple-Mode Radially}{ Pulsating Post-Main-Sequence Delta Scuti Star}

\IBVSauth{G. Handler$^{1}$, H. Pikall$^{1}$, R. Diethelm$^2$}
\IBVSinst{Institut f\"ur Astronomie, Universit\"at Wien, 
T\"urkenschanzstra\ss e 17, A-1180 Wien, Austria}
\vspace{-1.5mm}
\IBVSinsto{e-mail: gerald@dsn.astro.univie.ac.at, holger@dsn.astro.univie.ac.at}
\IBVSinst{BBSAG \& Astron. Institut der Universit\"at Basel, 
Rennweg 1, CH-4118 Rodersdorf, Switzerland}
\vskip -1.5mm
\IBVSinsto{e-mail: diethelm@astro.unibas.ch}

\SIMBADobj{V829 Aql}
\IBVStyp{ DSCT, RR }

\begintext

Diethelm (1997, hereafter D97) obtained time-resolved CCD photometry of
V829\,Aql during 25 nights in 1995 and 1997 (see his paper for more
information). He suggested that this star is a multimode radial pulsator.

These observations have been analysed with respect to the underlying
pulsational frequencies. Starting with his
1997 measurements (which are of better quality and more numerous than the
1995 data), we computed amplitude spectra and performed multifrequency
sine-wave fits using the program package Period 98 (Sperl 1998). 
Our resulting 
amplitude spectra before and after successive prewhitening of simultaneously
optimized frequencies are shown in Figure~1.

\IBVSfig{9cm}{4549-f1.ps}{Amplitude spectra for the 1997 measurements of D97. Four frequencies can be identified and some further linear combinations of them (indicated with arrows) may be present}
\vskip -4mm
From Figure~1 it can be seen clearly
that (at least) four frequencies are present
in the light variations of V829\,Aql. Furthermore, some harmonics of these
frequencies (2f$_1$, 2f$_2$, 3f$_2$) can be discerned after prewhitening the
four dominant peaks. Because of the small amount of data we did not attempt
to fit these frequencies to push the analysis further. 

With that knowledge, we attempted a frequency analysis of the 1995
measurements. The frequencies f$_1$, f$_2$ and f$_3$ are present in these data 
as well, but because of severe aliasing and the higher noise level,
they do not dominate the
amplitude spectra. After prewhitening these three frequencies, the first
harmonics of them can be seen as well (2f$_1$, 2f$_2$, 2f$_3$). Again, we did
not try to calculate a higher order fit at this point. 

The results of our frequency analysis are summarized in Table 1. 
The error bars have been determined following
Kov\'acs (1981) and should be taken only as formal values; the real errors may
be higher by up to a factor of 2.
\vskip 3mm

\centerline{Table 1. Frequencies detected in the light curves of V829\,Aql}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Ident. & Freq. (1995) & Ampl. (1995) & Freq. (1997) & Ampl. (1997)\\
 & (cycles/day) & (mmag) & (cycles/day) & (mmag)\\
\hline
f$_1$ & 4.529 $\pm$ 0.006 & 83 $\pm$ 20 & 4.5260 $\pm$ 0.0008 & 85 $\pm$ 4\\
f$_2$ & 3.415 $\pm$ 0.006 & 75 $\pm$ 20 & 3.4196 $\pm$ 0.0008 & 82 $\pm$ 4\\
f$_3$ & 5.664 $\pm$ 0.006 & 64 $\pm$ 21 & 5.6597 $\pm$ 0.0025 & 29 $\pm$ 4\\
f$_4$=f$_1$+f$_2$ & -- & -- & 7.9498 $\pm$ 0.0022 & 26 $\pm$ 4\\
\hline
\end{tabular}
\end{center}
\vskip 3mm

From Table 1 it may be suspected that the amplitude of f$_3$ was different
between the two seasons, which is supported by the presence of its first
harmonic in the 1995 data. However, keeping the error sizes in
mind, this statement is only preliminary. Because of the
significantly different quality of the two yearly subsets of data and because
of the large gap between them, we did not attempt to calculate multifrequency
fits for the whole data set. We rather base the following discussion on the
1997 data only.

What is the cause of variability of V829\,Aql? The frequency ratios
f$_2$/f$_1$=0.7555$\pm$0.0002 and f$_1$/f$_3$=0.7997$\pm$0.0004 together with
the presence of linear combinations and harmonic frequencies strongly point
towards the excitation of radial pulsations in the fundamental mode (F)
plus the first and second overtones (1H and 2H, respectively).

In a first approach to unravel the nature of V829\,Aql, we can state that it is
either an RR Lyrae star or an evolved $\delta$ Scuti star. Unfortunately, we have
no spectroscopic information on this object. Garcia-Melendo \& Clement (1997)
compiled the periods for field and cluster double-mode RR Lyrae (RRd) stars.
The radial fundamental mode periods for these stars are between 0.46--0.58 days,
much longer than the 0.292 days we determined for V829\,Aql. The F/1H
frequency ratios for RRd variables are confined to a narrow range between 0.742 
and 0.747, smaller than the value we derived for V829\,Aql. Moreover, if we
determine the location of the star in the $(P_1/P_0-P_2/P_0)$ vs. $P_0$ diagram
of Kov\'acs \& Buchler (1994, hereafter KB94), we see that it cannot be 
represented with any of their RR Lyrae star models. All this, together with
the low galactic latitude ($b=-10\fday 7$) of the star, strongly suggests that 
V829\,Aql is not an unusual RRd variable, but rather an evolved $\delta$ Scuti 
star.

To verify this suspicion, we computed a number of 
post-main-sequence $\delta$ 
Scuti star model sequences with the Warsaw-New Jersey stellar evolution and 
pulsation code (see Pamyatnykh et al. 1998 for more details). Since we do not 
have any information on the metallicity of the star, we assumed solar 
abundances.

We followed the work by KB94 and evaluated the behaviour of their parameter
$\Delta$, the ``period ratio distance'', which is a measure of the deviation
of the theoretical 1H/F and 2H/F period ratios from the corresponding observed 
period ratios. The ``best'' models are then characterized by a minimum of 
$\Delta$ at the radial fundamental mode period. This was done along several
evolutionary tracks for different masses, and the results of this investigation
are displayed in Figure~2.

\IBVSfig{8cm}{4549-f2.ps}{The period ratio distance $\Delta$ vs. the radial 
fundamental mode period along post main sequence evolutionary tracks of 
2.1 $M_{\odot}$ (squares), 2.3 $M_{\odot}$ (triangles) and 2.5 $M_{\odot}$ 
(diamonds). The 2.1 $M_{\odot}$ model resembles the observed quantities for
V829\,Aql very well}

As one can see, the 2.1 $M_{\odot}$ model is an excellent match to the
observed quantities. Intriguingly, this model has the radial fundamental mode 
at a frequency of 3.4162 cycles/day, while this mode has a frequency of
3.4196 cycles/day in the real star. However, we are unable to reach $\Delta=0$,
and this best model
is slightly outside the red edge of the instability strip. Both problems may be
resolved by varying the metallicity of the models. From the work of KB94 and
from this example it is clear that stars like V829\,Aql can considerably
contribute to the evaluation of the reliability of pulsational model
calculations: if a sufficient amount of information on these stars is available
to constrain their position in the HR diagram (i.e. $T_{\rm eff}$, log $g$, 
$[M/H]$), the models must be able to reproduce their pulsational behaviour.
This will also have impact on the model calculations presently used for
asteroseismology of multimode nonradial $\delta$ Scuti pulsators.

At this date, only two (or three) $\delta$ Scuti stars simultaneously pulsating 
in the radial fundamental as well as the first and second radial overtone
modes are known besides V829\,Aql. These are AC And 
(Fitch \& Szeidl 1976, KB94), GSC 4018.1807 (Antipin 1997) and perhaps HR 6434 
(Scheck 1990 and references therein).

Regrettably, at best only low-resolution spectroscopy is available for AC And,
GSC 4018.1807 and V829\,Aql and only single-color broad-band photometric 
observations have been analysed in the literature. While the faintness of the 
three stars ($V>10$ mag) requires a fairly large telescope to obtain 
high-resolution spectra, multicolor narrow and/or broad-band photometry can be
obtained by photometrists with a large amount of available telescope time, an 
APT or well-equipped amateur astronomers.

\references

Antipin, S., 1997, {\it A\&A}, {\bf 326}, L1

Diethelm, R., 1997, {\it IBVS}, No.\,4530 (D97)

Fitch, W.S., Szeidl, B., 1976, {\it ApJ}, {\bf 203}, 616

Garcia-Melendo, E., Clement, C.M., 1997, {\it AJ}, {\bf 114}, 1190

Kov\'acs, G., 1981, {\it Ap\&SS}, {\bf 78}, 175

Kov\'acs, G., Buchler, J. R., 1994, {\it A\&A}, {\bf 281}, 749 (KB94)

Pamyatnykh, A. A., Dziembowski, W. A., Handler, G., Pikall, H., 1998, {\it A\&A}, submitted

Scheck, M., 1990, Master's thesis, University of Vienna

Sperl, M., 1998, private communication

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