\documentstyle[twoside, ibvs2, epsf]{article}

\begin{document}

\IBVShead{4713}{26 May 1999}

\IBVStitletl{On the Orbital Period Changes of AK Herculis}

\IBVSauth{H. Rovithis-Livaniou$^1$, A. Kranidiotis$^1$, E. Fragoulopoulou$^1$, N. Sergis$^1$,\\ P. Rovithis$^2$}

\IBVSinst{Section of Astrophysics, Astronomy and Mechanics, Dept.\ of Physics, University of Athens, Panepistimio\-polis, GR-157 84 Zografos, Athens, Greece, e-mail: elivan@cc.uoa.gr}

\IBVSinst{Astron.\ Inst.,  National Obs.\ of Athens, P.O. Box 20048, Athens 118 10, Greece, e-mail: rovithis@astro.noa.gr}

\SIMBADobj{AK Her}
\IBVStyp{ eclipsing binary }
\IBVSkey{photometry; times of minima; orbital period changes} 
\IBVSabs{We present 10 unpublished photoelectric minima of the}
\IBVSabs{eclipsing binary AK Her and examine its orbital period changes.}

\begintext

AK Her is a contact eclipsing binary of W UMa-type and the brighter 
component of the double star ADS 10408. Its companion is about 4 
magnitudes fainter than the eclipsing pair and lies very close to it: at 
a distance of 4\farcs7 in position angle 322\deg.

The variability of AK Her was reported by Pickering (1917) and the
system has been observed photoelectrically many times (for details see 
at Tunca et al.\ 1987 and the references therein). The system exhibits 
variable light curve and an obvious O'Connell effect. Besides, AK Her 
is an X-ray source (Cruddace  \& Dupree, 1984).

So far, many studies have been made concerning its period variations and 
the detected periodicities are extended from 58 to 78 years.
This is mainly due to the available observational material at
the time of analysis, and secondly to the fact that the
search for periodicities was based on the $O-C$ diagram, which  
depends on the ephemeris used. So, Schmidt \& Herczeg (1959) found a
periodicity of 64~yr; Woodward \& Wilson (1977) of 58~yr;
Barker \& Herczeg (1979) of 78.03~yr; Glownia (1985) of 65.95
and Tunca et al.\ (1987) of 75.72~years. In this report besides the unpublished times of minimum light of AK Her that we present, we also examine its orbital period changes, which was found not to follow the sinusoidal variation proposed some years ago.


Our photoelectric observations of AK Her were carried out during 3 
nights in 1985, 3 in 1986 and 4 in 1987 with
the 1.2-m Cassegrain reflector at the Kryonerion Astronomical Station
of the Athens National Observatory, Greece.  Standard $B$ and $V$ filters
and a two-beam, multi-mode, nebular-stellar photometer were used. 
The stars  BD +16\deg3123  and BD +16\deg3124
were used for comparison and checking respectively, and reduction
of the observations was made in the usual way (Hardie, 1962;
Henden \& Kaitchuk, 1987). The derived 10 new times of minimum light are presented in Table~1 the successive columns of which give the Hel.~JD., the type of minimum and the $O-C$, where the C's were computed using the Kwee \& van Woerden (1956) method; they are the mean values from the $B$ \& $V$ observations and Woodward's (1942) light elements were used:
$${\rm Min~I} = 2422977.254 + 0\fday42152207 \times E.$$


\begin{table}
\begin{center}
\caption{The times of minimum light of AK Her, as derived from our photoelectric observations.}
\begin{tabular}{lclc} \hline
Hel.\ JD       & Min. &   Epoch    &  $O-C$ \\
$244 0000.+$  & Type &            &   (days)  \\   \hline
$6210.3050$   &  I   &   55117    &  0.0191  \\ 
$6210.5153$   &  II  &   55117.5  &  0.0186  \\ 
$6212.4140$   &  I   &   55122    &  0.0205   \\
$6597.4725$   &  II  &   56035.5  &  0.0185   \\ 
$6598.3146$   &  II  &   56037.5  &  0.0175   \\ 
$6977.4748$   &  I   &   56937    &  0.0189   \\ 
$6978.5262$   &  II  &   56939.5  &  0.0163   \\ 
$6979.3711$   &  II  &   56941.5  &  0.0181   \\ 
$7011.4056$   &  II  &   57017.5  &  0.0170   \\   \hline
\end{tabular}
\end{center}
\end{table}


In order to construct the $O-C$ diagram of AK Her (Fig.~1), Tunca's et 
al.\ (1987) list of minimum light was used, together with our data (Table~1), and the list was completed with up to date data, which are given in Table~2. In Fig.~1, the best fitted polynomial (for details see Kalimeris et al., 1994), used to describe the data, is presented by the heavy continuous line, while the sinusoidal term found by Tunca et al.\ (1987) is denoted by the dashed line.

The real orbital period variation $P(E)$, and its rate of change, were 
also computed, as it is described by Kalimeris et al.\ (1994, 1995), and 
from the Fourier spectrum of the $P(E)$ function, two periodicities were 
detected: viz.\ $76.17 \pm 0.06$~yr and $38.1 \pm 0.1$~yr, with amplitudes of $0.071 \pm 0.002$~sec and $0.077 \pm 0.002$~sec, respectively. The first periodicity is close to this of 78.03~yr found by Barker \& Herczeg (1979) and to that of 75.72~yr found by Tunca et al.\ (1978); but, since it corresponds to the time interval for which observational data exist, it might not be true. 

From the present analysis, which includes the most recent available data, it is shown, (Fig.~1), that the orbital period of AK Her {\it does not follow a sinusoidal variation}. This could not be detected from the observational material available at the time of the previous analyses and shows clearly that the period variations cannot be predicted. As regards the second periodicity of 38.1~yr it had not been detected before; but, since it is half of the long one more data are needed to assure its existence.

\bigskip
{\it Acknowledgements:} This work was partly financial supported by 
Athens University
(grant no.\ 70/4/3305).

\begin{table}
\begin{center}
\caption{Photoelectric times of minimum light of AK Her, from the 
literature.}
\begin{tabular}{lrll} 
\hline  
Hel.\ JD.       & \multicolumn{1}{c}{$O-C$}  & Epoch & Reference  \\ 
                & \multicolumn{1}{c}{(days)} &       &       \\  
\hline
$2435933.576$   &    0.0019  &  30737   & AA 12, 200, 1962  \\ 
$2437168.412$   & $-$0.0148  &  33666.5 & BAC 14, 1963    \\ 
$2437172.418$   & $-$0.0132  &  33676   & BAC 14, 1963 \\
$2437356.643$   &    0.0066  &  34113   & BAV No. 15  \\
$2437505.427$   & $-$0.0067  &  34466   & BAV No. 15 \\ 
$2437824.519$   & $-$0.0184  &  35358   & BAV No. 15 \\
$2437881.413$   & $-$0.0069  &  35823   & BAC 15, 1964 \\
$2438227.511$   &    0.0100  &  36179   & AN 188, 1964 \\
$2438590.454$   &    0.0225  &  37040   & BAC 16, 1965 \\ 
$2438595.503$   &    0.0133  &  37052   & BAC 16, 1965 \\ 
$2438614.479$   &    0.0208  &  37097   & BAC 16, 1965 \\   
$2438620.372$   &    0.0125  &  37111   & BAC 16, 1965 \\
$2443717.426$   &    0.0216  &  49203   & AN 302, 1981 \\ 
$2443744.412$   &    0.0302  &  49267   & AN 302, 1981 \\ 
$2444372.4713$  &    0.0216  &  50757   & BBSAG 48, 1980   \\
$2446224.4309$  &    0.0240  &  55150.5 & IBVS 3078, 1987 \\ 
$2446228.4330$  &    0.0215  &  55160   & IBVS 3078, 1987 \\ 
$2446230.332$   &    0.0238  &  55164.5 & IBVS 3078, 1987 \\ 
$2446234.334$   &    0.0213  &  55174   & IBVS 3078, 1987  \\
$2446243.3987$  &    0.0233  &  55195.5 & IBVS 3078, 1987  \\
$2446244.4510$  &    0.0218  &  55198   & IBVS 3078, 1987 \\ 
$2446612.4410$  &    0.0230  &  56071   & IBVS 3078, 1987  \\
$2448100.4112$  &    0.0203  &  59601   & IBVS 3615, 1991  \\
$2450248.4941$  &    0.0267  &  64697   &  IBVS 4472, 1997   \\  
$2450259.45353$ &    0.0266  &  64723   &  IBVS 4670, 1999  \\ 
$2450275.4751$  &    0.0303  &  64761   &  IBVS 4555, 1998  \\ 
$2450310.4578$  &    0.0267  &  64844   &  IBVS 4555, 1998  \\ 
$2450508.574$   &    0.0275  &  65314   &  IBVS 4555, 1998  \\ 
$2450512.5802$  &    0.0293  &  65323.5 &  IBVS 4555, 1998  \\ 
$2450635.6645$  &    0.0291  &  65615.5 &  BBSAG 115, 1997  \\   
$2450865.6038$  &    0.0281  &  66161   &  IBVS 4633, 1998  \\ 
$2450866.6681$  &    0.0386  &  66163.5 &  IBVS 4633, 1998  \\ 
$2450903.5413$  &    0.0286  &  66251   &  IBVS 4633, 1998  \\ 
$2450971.4060$  &    0.0282  &  66412   &  IBVS 4633, 1998  \\  
\hline
\end{tabular}
\end{center}
\end{table}

\newpage

\IBVSfig{10cm}{4713-f1.ps}{The $O-C$ diagram of AK Her and the best fitted polynomial (continuous line). The sinusoidal variation (dashed line), proposed by Tunca et al.\ (1987) is also given.}

\references 

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