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\IBVShead{5163}{29 August 2001}

\IBVStitle{Autocorrelation Analysis of Two Pulsating Red Giants}

\IBVSauth{Percy, J. R.$^1$; Hussain, A.$^1$; Gomez-Forrellad, J. M.$^{2,3}$; Garcia-Melendo, E.$^3$}

\IBVSinst{Dept.\ of Astronomy, University of Toronto, Toronto ON Canada
M5S 3H8; e-mail: jpercy@erin.utoronto.ca}

\IBVSinst{Grup d'Estudis Astronomics, Apdo.\ 9481, 08080 Barcelona,
Spain; e-mail: jmgomez@astrogea.org}

\IBVSinst{Esteve Duran Observatory Foundation, Montseny 46 -- Urb.\ El
Montanya, 08553 Seva, Spain}

\SIMBADobj{HD 190152}
\SIMBADobjAlias{SX UMi}{HD 126409}

\IBVSabs{We have implemented the autocorrelation algorithm of Burki}
\IBVSabs{et al. (1978 AA, 65, 363), and used it to determine periods}
\IBVSabs{for SX UMi and the newly-discovered variable HD 190152.}

\begintext

About 10 per cent of the naked-eye stars are pulsating red giants (PRGs),
with amplitudes ranging from 0.01 to 10 magnitudes.  The first PRG ---
Mira --- was discovered over 400 years ago.  Small-amplitude PRGs
(SAPRGs), with amplitudes of 0.1 to 1 magnitude, and with early M spectral
type, were surveyed by Stebbins \& Huffer (1930).  In past studies (e.g.\
Percy et al.\ 1996, Percy et al.\  2001a,b), we found that autocorrelation
analysis was a useful adjunct to light curves and Fourier analysis for
determining the periods of these complex stars.  It determines
characteristic time scales by examining the cycle-to-cycle behaviour of
the star, averaged over the dataset.  The version of the autocorrelation
method that we use (written by Matt Szczesny and Adrien Desjardins, and
described by Percy \& Sen 1991) is very simple:  for each pair of
measurements, the difference in magnitude is plotted against the
difference in time, divided into appropriate ``bins"; this
``autocorrelation diagram" shows minima at integral multiples of the
characteristic time scale.  Each of the minima can be used to estimate the
characteristic time scale.  The height of the maxima is related to the
amplitude of the variations; the height of the minima, above the zero
point, is related to the average error of the measurements, and to the
degree of irregularity of the variations; if the variability is irregular,
then the minima do not persist with increasing $\Delta t$. 
Distortions or unequal depths of the minima may indicate the presence of
multiple periods.  Our algorithm is similar to the elegant ``variogram"
technique described by Eyer \& Genton (1999). 

Another form of autocorrelation analysis was published by Burki et al.\
(1978): the data in the light curve are moved sideways in time, and the
scatter is assessed; when the shift is equal to the characteristic time
scale (or an integral multiple thereof), then the fit is best.  The
purpose of this paper is to demonstrate the use of this second
algorithm on two PRGs --- one poorly-studied, and the other
newly-discovered.  Their light curves are shown in Figures~1 and~3.
Figures~2 and~4 illustrate this algorithm; the horizontal axis is the
sideways shift, in days; the vertical axis is a measure of the goodness
of fit (lower $\theta$ indicates a better fit).

\IBVSfig{8cm}{5163-f1.eps}{Differential $V$ light curve of SX UMi, relative to
HD 126048 ($V = 8.21$)}

\IBVSfig{8cm}{5163-f2.eps}{Autocorrelation diagram for the data in Figure 1}

\IBVSfig{8cm}{5163-f3.eps}{Differential $V$ light curve of HD 190152, relative
to HD 190323 ($V = 6.83$)}

\vskip-33pt
\medskip

{\bf SX UMi} (HD 126409, HIP 70245, SpT M) was initially observed by
chance by two of us (JMGF \& EGM) as part of another program, using a
Starlight Xpress CCD camera with a Sony ICX027B chip and a Johnson $V$
filter on two 6-cm refracting finder telescopes at Mollet and Esteve Duran
Observatories.  The Esteve Duran data were adjusted to match the Mollet
data.  In subsequent seasons, data were obtained with an 8-cm refractor at
Mollet Observatory;  no adjustment was necessary for these data.  The
comparison star was HD 126048 (HIP 70059, SpT K2) for which Perryman et
al.\ (1997) give $V = 8.21$.  The check star was HD 125917 (HIP 70006, SpT
A3). Although HD 126048 is NSV 06640, Perryman et al.\ (1997) do not report
any variability; the standard deviation is 0.013 (consistent with
non-variability in a star of this magnitude); the maxima and minima are
8.35 and 8.39, respectively.  Furthermore, our 335 measurements of HD
126048 relative to HD 125917 between JD 2450507 and 2451312 give a mean of
+0.013 with a standard deviation of 0.0097, which is consistent with the
observational error.  We conclude that NSV 06640 was non-variable during
the times that we and {\it Hipparcos} observed it. Synthetic aperture
differential photometry was carried out.  No correction for differential
extinction was necessary, since the comparison stars were within 36\arcm\ of
the variable. 

\medskip

{\bf HD 190152} (BD+15$\deg$ 4029, GSC 01617-02068, PPM 137505, SAO
105602, not in the {\it Hipparcos} catalogue, SpT M) is a
previously-unknown variable which was also observed by chance by two of us
(JMGF \& EGM) as part of another program, using an 8-cm refractor at
Mollet Observatory, and the same CCD camera and reduction techniques.  The
comparison star was HD 190323 (HIP 98788, SpT F8), for which Perryman et
al.\ (1997) give $V = 6.83$, and the check star was HD 190067 (HIP 98677, SpT
G5). 

\medskip

\IBVSfig{8cm}{5163-f4.eps}{Autocorrelation diagram for the data in Figure 3}

\vskip-11pt
Figure 1 shows the differential $V$ light curve of SX UMi, relative to
HD 126048 ($V = 8.21$).  The mean $V$ is about 8.1.  Semi-regular
variations, with a cycle-count period of about 35 days and a total
range of 0.27 magnitude, are apparent, as are long-term variations in
amplitude and mean magnitude.  Figure 2 shows the autocorrelation
diagram of the data, using the Burki et al.\ (1978) algorithm.  On the
horizontal axis, $\tau$ is the sideways shift in time; on the vertical
axis, $\theta$ is the goodness of fit (lower $\theta$ indicates better
fit).  The first two minima give a period of about 38 days, but the
shallowness and the distorted appearance of the first and third minima
suggest that a second period may also be present.  Using all
undistorted minima gives a period of $37 \pm 1$ days.  The same data were
used as a test of wavelet analysis of SAPRGs (Percy \& Kastrukoff
2001), and gave a mean period of 38 days.

Figure 3 shows the differential $V$ light curve of HD 190152, relative to HD
190323 ($V = 6.83$).  The mean $V$ is about 8.3.  Semi-regular variations,
with a cycle-count period of 34 days and a total range of 0.2 magnitude,
are apparent.  Figure 4 shows the autocorrelation diagram of the data in
Figure 3, using the Burki et al.\ (1978) algorithm.  The four distinct minima give
a period of $32 \pm 1$ days for this previously-unknown variable. 

We conclude that the Burki et al.\ (1978) algorithm can be useful for
autocorrelation analysis of small-amplitude pulsating red giants.  We
report period determinations, using this algorithm, for SX UMi and HD
190152 --- a newly discovered variable. 

\bigskip

{\bf Acknowledgements.}  JRP and AH thank the Ontario Work-Study Program
at the University of Toronto for support.

\references

Burki, G., Maeder, A., \& Rufener, F., 1978, {\it A\&A}, {\bf 65}, 363

Eyer, L., \& Genton, M.G., 1999, {\it A\&AS}, {\bf 136}, 421

Percy, J.R., Desjardins, A., Yu, L., \& Landis, H.J., 1996, {\it PASP},
{\bf 108}, 139 

Percy, J.R., Dunlap, H., Kassim, L., \& Thompson, R.R., 2001a, {\it IBVS}, No.\ 5041

Percy, J.R., \& Sen, L.V., 1991, {\it IBVS}, No.\ 3670

Percy, J.R., \& Kastrukoff, R., 2001, {\it JAAVSO}, in press

Percy, J.R., Wilson, J.B., \& Henry, G.W., 2001b, {\it PASP}, in press

Perryman, M.A.C., and the {\it Hipparcos} Science Team, 1997, {\it The
Hipparcos and Tycho Catalogues}, ESA SP-1200

Stebbins, J., \& Huffer, C.M., 1930, {\it Publ.\ Washburn Obs.}, {\bf 15}, 138 

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