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\centerline{\ft RADIAL VELOCITY CURVES AND FIRST CALCULATIONS}
\centerline{\ft OF THE RADII FOR FOUR DOUBLE-MODE CEPHEIDS}
\vskip 8mm
\begintext

Double-mode Cepheids form a specific group of
Cepheids which includes a limited number of stars. The light
and radial velocity curves of these stars show double-mode
variations, whereas ordinary Cepheids show only one period.
The ratio of periods is almost the same for most of these
stars and is close to 0.71, in agreement with the theoretical ratio of
periods of the first overtone $P_1$ to fundamental tone $P_0$.
For CO Aur, unlike other stars, this ratio is close to 0.8; one can
imagine (based on the theory of stellar pulsations) that this
star pulsates both in the second and first overtone modes.

Berdnikov (1992, 1993) used a large number of original
photoelectric observations of 14 double-mode Cepheids to decompose
their light and color curves into two oscillations. Since 1987,
we have carried out systematic measurements of radial velocities of
northern Cepheids with a correlation spectrometer designed and made
by Tokovinin (1987). Most part of these observations were
included in our two catalogues (Gorynya et al., 1992, 1996).
These data, combined with our unpublished observations, allowed us
to derive separate radial velocity curves for two oscillations
in five photometrically well-studied Cepheids (V367 Sct, EW Sct,
BQ Ser, TU Cas, CO Aur).

Note that clear separation of radial velocity curves into two
oscillations was made possible by long sets of observations
resulting in good coverage of radial velocity curves.

Figures 1-5 show the decomposed radial velocity curves for each mode.

We used these curves to estimate the radii of the four double-mode
Cepheids using Balona's method (1977), which is a modification of the
well-known Baade--Wesselink technique (Wesselink, 1946). We were forced
to simplify our analysis because the number of radial velocity
observations is much less than that of photometric measurements, and
radial velocities alone do not permit us to find the relation
between the amplitudes and phases of the two modes found earlier in
photometric data (Berdnikov, 1992, 1993). We therefore assumed
that the two oscillations are independent of each other.

In this case the main least squares equation can be
written as
$$V = A(B-V)-5\lg(<R> + r_0 + r_1 ) + C$$
where $V$ and $(B-V)$ are current magnitude and colour; $<R>$, mean
Cepheid radius in $R_\odot$; and $A$ and $C$, the constants to be found.
The total pulsational radius variation $r=r_0+r_1$ can be found by
direct integration of radial velocity curves for two modes:

  $$r = - p P_0/R_\odot \int V_{r0} d\phi_0 - p P_1/R_\odot  \int V_{r1} d\phi_1$$

We use two color indices, $(B-V)$ and $(V-R)$, as the effective
temperature indicators.

Table 1 gives the Cepheid radii and their formal errors.
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\centerline{2}
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\centerline{\epsffile{4484-f1.ps}}
\centerline{Figure 1}
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\centerline{Figure 2}
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\epsfysize=7cm
\centerline{\epsffile{4484-f3.ps}}
\centerline{Figure 3}
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\centerline{3}
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\centerline{\epsffile{4484-f4.ps}}
\centerline{Figure 4}
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\centerline{\epsffile{4484-f5.ps}}
\centerline{Figure 5}
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\centerline{Table 1}
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\begin{center}
\begin{tabular}{ccccccc}
\hline
\noalign{\medskip}
Star & $P_0$ & $P_1$ & $R_{B-V}/R_\odot$ & $\sigma_R$ & $R_{V-R}/R_\odot$ & $\sigma_R$\\
\hline
\\
  EW   Sct &  $5\fd 8233$ & $4\fd 06714$ & 57 &   14 &       50 &    10  \\
  BQ   Ser &   4.2756 &   3.01191 &      56 &   19 &       35 &    14  \\
  TU   Cas &   2.1393 &   1.51827 &      31 &    2 &       25 &     3  \\
  CO   Aur &   2.5113 &   1.78300 &      40 &   10 &       30 &     5  \\
\\
\hline
\end{tabular}
\end{center}
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Radial velocity data for the faintest star, V367 Sct, cannot
be used for calculations because of large observational errors.
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\centerline{4}
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We derived the following period--radius relations for the four double-mode
Cepheids assuming that CO Aur oscillates in the first and second
overtones.
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\centerline{lg R = 1.33 + 0.59 lg P$_{0}$ \ \ \ \ \ \ \ \ for\ (B$-$V)}
\hspace*{4.8cm} $\pm$.08  $\pm$\ \ .14

\centerline{lg R = 1.21 + 0.61 lg P$_{0}$ \ \ \ \ \ \ \ \ for (V$-$R)}
\hspace*{4.8cm} $\pm$.07 $\pm$\ \ .12
\vskip 3mm

These relations agree well with that for single-mode
Cepheids (Ripepi et al., 1997; Sachkov et al., 1997); we consider
this agreement to be a justification of our technique applied to
the double-mode Cepheids.

The author is grateful to Drs. A.S. Rastorgouev, N.A. Gorynya
and N.N. Samus for helpful discussions.

This work was partially supported by the Russian Foundation for
Basic Research through grant 96-02-18491.

\vskip .7cm

\beginauthors
\authorsrightcol
M.E. SACHKOV
Institute of Astronomy of Russian
Acad. Sci.,
48, Pyatnitskaya Str.,
Moscow 109017, Russia
{\sl e-mail:} sachkov@sai.msu.su;
msachkov@ra.inasan.ac.ru                   %email

\endauthors



\references

  Balona, L.A., 1977, {\it M.N.R.A.S.}, {\bf 178}, 231

  Berdnikov, L.N., 1992, {\it Sov. Astron. Lett.}, {\bf 18}, No.\,8

  Berdnikov, L.N., 1993, {\it Sov. Astron. Lett.}, {\bf 19}, No.\,7

  Gorynya, N.A., Irsmambetova, T.R., Rastorgouev, A.S.,
      and Samus, N.N., 1992, {\it Sov. Astron. Lett.}, {\bf 18}, No.\,5

  Gorynya N.A., Samus, N.N., Rastorgouev, A.S., and Sachkov, M.E., 1996,
    {\it Astronomy Letters}, {\bf 22}, 175

  Ripepi, V., Barone, F., Milano, L., and Russo, G., 1997,
     {\it Astron. and Astrophys.}, {\bf 318}, 797

  Sachkov, M.E., Rastorgouev, A.S., Gorynya, N.A., and Samus, N.N., 1997,
     {\it Astron. and Astrophys.}, in preparation

  Tokovinin, A.A., 1987, {\it Soviet Astronomy}, {\bf 64}, 196

  Wesselink, A.J., 1946, {\it Bull. Astr. Inst. Netherland}, {\bf 10}, 91


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