\documentstyle[twoside,epsf,hyperref]{article}
\input ibvs2.sty
\input ibvst.sty

\newcommand\uv{uvby-\beta \ }
\begin{document}
\IBVSheadDOI{62}{6186}{2 November 2016}

\IBVStitle{Pulsational stability of the SX P\lowercase{he} star AE UM\lowercase{a}}

\IBVSauth{Pe\~na, J.H.$^{1,2,3}$; Renter\'ia, A.$^{1,3}$; Villarreal, C.$^{1,3}$; Pi\~na, D.S.$^{1,3}$; Soni, A.A.$^3$; Guillen,~J.$^3$;  Vargas, K.$^3$; Trejo, O.$^1$}

\IBVSinst{Instituto de Astronom\'{\i}a, Universidad Nacional Aut\'onoma de M\'exico, M\'exico D.F. \\ e-mail: jhpena@astro.unam.mx}
\IBVSinst{Observatorio Astron\'omico Nacional, Tonantzintla}
\IBVSinst{Facultad de Ciencias, Universidad Nacional Aut\'onoma de M\'exico}

\IBVSobj{R CrB}

\IBVSabs{From newly determined times of maxima of the SX Phe star AE}
\IBVSabs{UMa and a compilation of previous times of maxima, we were able to}
\IBVSabs{determine the nature of this star. With uv photometry we}
\IBVSabs{determined its physical parameters.}


\begintext

\section{Motivation}

Soon after we started this research program into high amplitude
Delta Scuti stars (HADS) in 2011, we presented our ideas at the
Obergurgl Conference held in Austria the following year. There we
stated that our motivation was to find out how far we could go with
the observation of faint stellar objects utilizing the Astronomical
Observatory of Tonantzintla, Mexico, since the site no longer has
optimum conditions for astronomical observations. This observatory
has a 1--m diameter telescope and several smaller 10-inch telescopes
all provided with CCD cameras. All are used mostly for the training
of students and we are currently seeking scientific observational
programs to increase their educational experience. This was the
motivation for the present study. The star AE~UMa was chosen because
of the following characteristics: short period of variation (0.086~d), 
high-amplitude variation (0.44 mag) and relative brightness
(11.35 mag). These make it an ideal test object for both the
telescopes and the instrumentation.

AE Ursae Majoris was first reported to be a variable star by Geyer et al.\ (1955). 
Tsesevich and Filatov reported some results in 1956 and 1960,
but the type of variability  could not be determined from their
visual observations. In 1973 Tsesevich determined that this star was
a dwarf cepheid. Other studies were later carried out by several
authors (Szeidl 1974, Broglia \& Conconi 1975, Rodr\'iguez et al.\
1992, Hintz et al.\ 1997).


AE UMa is listed as an SX Phoenicis star (Garcia et al.\ 1995 and the
GCVS). Hence it is considered to be a metal-poor, Population II star
(unlike the most common dwarf cepheids which are Population I).
Before this classification AE~UMa was considered to be a double-mode
dwarf Cepheid or a high-amplitude $\delta$~Scuti star (HADS), which
are not Population II type stars. Rodr\'iguez et al. (1992), using a
$\delta m_1$ metal calibration for metal abundance, reported
[M/H]$=-0.3$, and Hintz et al.\ (1997) give values of [M/H] between
$-0.1$ and $-0.4$. He also mentions that ``no kinematic information
is available for AE~UMa; however, its galactic coordinates, might
lead one to conclude that it is possible a halo object and should
therefore be grouped with the SX~Phe stars".


The most relevant studies of AE~UMa are those of Szeidl (2001), Pocs
\& Szeidl (2001) and Zhou (2001) who found that the fundamental
period of AE~UMa had been essentially constant for 60 years in
accordance with the theoretical expectation. They stated that the
constancy of the fundamental period suggested that the star was in
the post-main sequence evolutionary state. They also found that the
first overtone period was decreasing at a rate of $(1/P_1)(dP_1/dt)
= -7.3 \times 10^{-8}{\rm y}^{-1}$.

Szeidl (2001) found that small long-term variations in the
amplitudes had been present for the previous 25 years. After these
two studies the only significant analysis was done by Coates \&
Landes (2008) who, utilizing the same data set as Pocs \& Szeidl
(2001) and using the beat-curve approach, deduced detailed and quite
precise information about the fundamental and first overtone periods
of AE~UMa and confirmed the values found by Pocs \& Szeidl (2001)
for the rates of change (assumed constant) of the periods, which are
similar in magnitude to those of other Pop.~I radially pulsating
$\delta$~Scuti stars (Breger \& Pamyatnykh 1998). In addition,
because they had access to times of maxima for the star post-1998,
they were able to extend the work of Pocs \& Szeidl (2001) and
deduce that there were possible sudden jumps in both the periods in
approximately 1996, thus adding AE~UMa to VZ~Cnc (Arellano Ferro et
al.\ 1994) as a radially pulsating $\delta$~Scuti star which has
possible sudden jumps in period.

In addition, without any analysis, AE~UMa has been extensively observed
in monitoring campaigns to acquire new times of maximum light of
pulsating stars. These times of maximum light of AE~UMa have been
compiled for behavior monitoring.

\section{Observations}

Although the new times of maximum light of this star have been
reported elsewhere (Pe\~na et al., 2015) here we present the
detailed procedure for acquiring the data. These were all taken at
both sites of the Observatorio Astron\'omico Nacional, at
Tonantzintla (TNT) and San Pedro M\'artir, M\'exico (SPM). At
TNT the  1\,m telescope was utilized provided with a CCD SBIG
STL-1001E camera. At SPM two detectors were employed, a CCD camera and a
spectrophotometer in the $\uv$ system was attached to the 0.84\,m
telescope. The log of observation is presented in
Table~1.


\begin{table}[!t]
\begin{center}
 \caption{Log of observing seasons of AE UMa.}
  \begin{tabular}{lrc}
  \hline\hline
  Date(yr/mo/day)      & Telescope & Observers\\
\hline
11/12/0506   &  0.84 m    & arl, jhp\\
11/12/1011   &  0.84 m    & arl, jhp\\
12/01/2324   &  1.0 m & ESAOBELA12\\
12/01/2425   &  1.0 m & ESAOBELA12\\
12/03/0203   &  1.0 m & AoA12\\
12/03/0304   &  1.0 m & AoA12\\
13/03/0910   &  1.0 m & aas, err\\
13/04/1617   &  1.0 m & aas, ota\\
16/04/0102   &  1.0 m & jg \\
\hline
\end{tabular}
\end{center}
{\small Notes: arl, A. Renter\'ia; jhp, J. H. Pe\~na; aas, A. A. Soni; ota,
O. Trejo; err, E. Ram\'irez; jg, J. Guillen; ESAOBELA12: V. Abril,
C. Alberti, L. Ar\'eas, L. Batista, J. Buitrago, O. Garc\'ia, M,
Gonz\'alez, C. Ram\'irez, B. Recinos, A. Rodr\'iguez, J. Ruiz, F.
Soriano; AoA12: D. Aguilera, D. Deras, M. Espinosa, K. Valencia, P.
Vessi, C. Villarreal}
\end{table}



\subsection{Data acquisition and reduction at TNT}

During all the observational nights the following procedure was utilized.
Sequence strings in the $V$ filter were obtained. The integration time
for the 1\,m telescope was 3 min;  there were around 40,000 counts
for the 1\,m telescope, enough to secure high precision. The
reduction work was done with \textsc{AstroImageJ} (Collins 2012). This
software is easy to use. It is
free and works well on the most common computing
platforms. With the CCD photometry two reference stars were utilized
whenever possible in a differential photometry mode. The results
were obtained from the difference $V_{{\rm variable}} - V_{{\rm
reference}}$ and the scatter calculated from the difference $V_{{\rm
reference1}} - V_{{\rm reference2}}$. Light curves were also
obtained. The  newly obtained times of maximum light are presented in
Table~2.


\begin{table}
\begin{center}
\caption{New times of maxima of AE UMa. }
\begin{tabular}{cccc}
\hline\hline
Time of Maximum (HJD) & Telescope & Filter & Observatory\\
\hline
  2455901.9574 & 0.84 m & $y$ & SPM\\
  2455902.0394 & 0.84 m & $y$ & SPM\\
  2455906.9400 & 0.84 m & $V$ & SPM\\
  2455907.0230 & 0.84 m & $V$ & SPM\\
  2455950.8077 & 1.0  m & $V$ & TNT\\
  2455951.7985 &  1.0 m & $V$ & TNT\\
  2455989.8581 &  1.0 m & $V$ & TNT\\
  2455990.7191 &  1.0 m & $V$ & TNT\\
  2455990.8104 &  1.0 m & $V$ & TNT\\
  2456361.7207 &  1.0 m & $V$ & TNT\\
  2456361.8010 &  1.0 m & $V$ & TNT\\
  2457129.3859 &  1.0 m & $G$ & TNT\\
  2457480.7170 &  1.0 m & $G$ & TNT\\
  2457480.7995 &  1.0 m & $G$ & TNT\\
\hline
\end{tabular}
\end{center}
\end{table}



\begin{table}
 \caption{Transformation coefficients obtained for the observed season.}
\begin{center}
  \begin{tabular}{lccccccc}
\hline\hline
season          &  B       &   D   &   F   &   J    &  H     &   I   &   L\\
\hline
Dec 2011 &  0.054  & 0.967 & 1.058 & 0.041 & 1.030  & 0.161 & $-$0.888\\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Data acquisition and reduction at SPM}

During the night of the observations at SPM the following procedure
was utilized: each measurement consisted of at least five ten-second
integrations of each star and one ten-second integration of the sky
for the $uvby$ filters and the narrow and wide filters that define
H$\beta$. Individual uncertainties were determined by calculating
the standard deviations of the fluxes in each filter for each star.
The percentual error in each measurement is, of course, a function
of both the spectral type and the brightness of each star, but they
were observed long enough to secure sufficient photons to get a S/N
ratio of accuracy of $N/\sqrt{N}$ of $0.01$ mag in all cases. A
series of standard stars was also observed on this night to
transform the data into the standard system. The reduction procedure
was done with the numerical packages NABAPHOT (Arellano-Ferro \&
Parrao 1988).  The chosen standard system was that defined by the
standard values of Olsen (1983) although for the standard bright
stars some were also taken from the Astronomical Almanac (2006). The
transformation equations are those defined by Crawford \& Barnes
(1970) and by Crawford \& Mander (1966).

The coefficients defined by the following equations and that
adjusted the data to the standard system are:


\begin{eqnarray}
V_{{\rm std}}  &  =  &  A + B(b-y) _{{\rm inst}} + y_{{\rm inst}} \nonumber \\
(b-y)_{{\rm std}}  &  =  &  C + D(b-y) _{{\rm inst}} \nonumber \\
m_{1_{\rm std}}  &  =  &  E + F(m_1) _{{\rm inst}} + J(b-y)_{{\rm inst}} \nonumber \\
c_{1_{\rm std}}  &  =  &  G + H(c_1) _{{\rm inst}} + I(b-y)_{{\rm inst}} \nonumber \\
H\beta_{{\rm std}}  &  =  &  K + L(H\beta) _{{\rm inst}} \nonumber
\end{eqnarray}


In these equations the coefficients D, F, H and L are the slope
coefficients for $(b-y)$, $ m_1$,  $c_1$ and $\beta$, respectively.
The coefficients B, J and I are the color terms  of $V$, $m_1$, and
$c_1$. The averaged transformation coefficients of each night are
listed in Table~3. Errors were evaluated using of the
twenty-three standard stars observed. These uncertainties were
calculated through the differences in magnitude and colors, for ($V$,
$b-y$, $m_1$, $c_1$ and $\beta$) as (0.052, 0.0075, 0.0094, 0.025,
0.032), respectively which provide a numerical evaluation of our
uncertainties. Emphasis is made on the large range of the standard
stars in the magnitude and color values: $V$ (5.45, 8.40);
$(b-y)$ ($-$0.19, 0.63); $m_1$ (0.10, 0.66); $c_1$ (0.19, 0.91) and
$\beta$ (2.609, 2.828).

Table 4  lists the photometric values of the observed star. In this
table, column 1 reports the time of the observation in HJD, columns
2 to 5 the Str\"omgren values $V$, $(b-y)$, $ m_1$ and $c_1$,
respectively; column 6, H$\beta$.

\setcounter{table}{3}
\begin{table}
\begin{center}
 \caption{$\uv$ photoelectric photometry of AE UMa.}
  \begin{tabular}{lccccc}
  \hline  \hline
HJD & $V$ & $(b-y)$ & $m_1$ & $c_1$ & $\beta$ \\
\hline

$-$2455900 &   &       &       &   &    \\
\hline
6.8611 & 11.035 & 0.130 & 0.148 &   & 2.708  \\
6.8636 & 11.048 & 0.155 & 0.137 & 0.888& 2.778 \\
6.8684 & 11.100 & 0.143 & 0.132 & 0.846& 2.770 \\
6.8709 & 11.122 & 0.159 & 0.113 &      & 2.793 \\
6.8728 & 11.110 & 0.185 & 0.130 & 0.923&  \\
6.8738 & 11.142 & 0.174 & 0.153 & 0.870 & \\
6.8753 & 11.192 & 0.149 & 0.166 & 0.941 &  \\
6.8764 & 11.222 & 0.161 & 0.142 & 0.906 & \\
6.8776 & 11.182 & 0.202 & 0.095 & 0.981 & \\
6.8784 & 11.235 & 0.162 & 0.130 &       & \\
6.8794 & 11.203 & 0.226 & 0.071 & 0.972 & \\
6.8804 & 11.233 & 0.201 & 0.113 & 0.929 & \\
6.8847 & 11.271 & 0.195 & 0.153 & 0.834 & \\
6.8857 & 11.297 & 0.199 & 0.118 & 0.795 & \\
6.8871 & 11.297 & 0.216 & 0.142 & 0.767 & \\
6.8878 & 11.303 & 0.212 & 0.104 & 0.897 & \\
6.8889 & 11.330 &       & 0.198 & 0.681 & \\
6.8902 & 11.339 &       & 0.209 & 0.847 & \\
6.8913 & 11.351 & 0.162 & 0.221 & 0.764 & \\
6.8920 & 11.323 & 0.200 & 0.145 & 0.811 & \\
6.8930 & 11.388 & 0.218 & 0.697  &      & \\
6.8940 & 11.383 &       & 0.196 & 0.718 & \\
6.8969 & 11.354 & 0.233 & 0.073 &       & \\
6.8977 & 11.387 & 0.197 & 0.183 & 0.804 & \\
6.8991 & 11.365 & 0.212 & 0.125 & 0.831 & \\
6.9003 & 11.390 & 0.188 & 0.203 & 0.732 & \\
6.9015 & 11.391 & 0.194 & 0.197 & 0.715 & \\
6.9022 & 11.382 & 0.225 & 0.077 & 0.857 & \\
6.9033 & 11.415 & 0.197 & 0.193 & 0.704 & \\
6.9047 &        & 0.211 & 0.156 & 0.740 & \\
6.9057 & 11.400 & 0.229 & 0.152 & 0.731 & \\
6.9063 & 11.417 & 0.209 & 0.185 &       & \\
6.9076 & 11.433 & 0.209 & 0.131 & 0.841 & \\
6.9087 & 11.406 &       & 0.062 & 0.815 & \\
6.9109 & 11.430 & 0.221 & 0.120 & 0.859 & \\
6.9115 & 11.435 & 0.191 & 0.191 & 0.847 & \\
6.9125 & 11.413 & 0.224 & 0.125 &       & \\
6.9134 & 11.418 & 0.233 & 0.108 & 0.796 & \\
6.9155 & 11.440 & 0.230 & 0.080 & 0.825 & \\
6.9170 & 11.436 & 0.210 & 0.117 & 0.767 & \\
6.9178 & 11.415 & 0.242 & 0.099 & 0.751 &\\
6.9191 & 11.446 & 0.218 & 0.091 & 0.768 & \\
6.9206 & 11.414 & 0.254 & 0.045 & 0.832 & \\
6.9246 & 11.369 & 0.219 & 0.148 & 0.700 & \\
6.9254 & 11.361 & 0.201 & 0.189 & 0.795 & \\
\hline
\end{tabular}
\end{center}
\end{table}

\setcounter{table}{3}
\begin{table}
\begin{center}
 \caption{Continued.}
  \begin{tabular}{lccccc}
    \hline\hline
HJD & $V$ & $(b-y)$ & $m_1$ & $c_1$ & $\beta$  \\
\hline
$-$2455900 &   &   &   &   &   \\
\hline

6.9271 & 11.339 & 0.210 & 0.089 & 0.848 & \\
6.9283 & 11.277 & 0.247 & 0.065 & 0.869 & \\
6.9294 & 11.270 & 0.221 & 0.090 & 0.864 & \\
6.9301 & 11.222 & 0.203 & 0.131 & 0.777 & \\
6.9320 & 11.167 & 0.163 & 0.133 & 0.822 & \\
6.9332 & 11.164 & 0.102 & 0.221 & 0.735 & \\
6.9344 & 11.065 & 0.115 & 0.194 & 0.824 & \\
6.9350 & 10.989 & 0.151 & 0.115 & 0.914 & \\
6.9364 & 10.909 & 0.127 & 0.140 & 1.054 & \\
6.9378 & 10.845 & 0.138 & 0.105 & 1.114 & \\
6.9402 & 10.849 & 0.093 & 0.130 & 1.058 & \\
6.9409 & 10.830 & 0.084 & 0.192 & 0.911 & \\
6.9420 & 10.852 & 0.070 & 0.209 & 1.015 & \\
6.9427 & 10.847 & 0.086 & 0.177 & 1.086 & \\
6.9438 & 10.881 & 0.076 & 0.179 & 0.979 & \\
6.9454 & 10.899 & 0.092 & 0.167 & 1.042 & \\
6.9465 & 10.919 & 0.092 & 0.173 & 0.995 & \\
6.9472 & 10.912 & 0.125 & 0.141 & 1.005 & \\
6.9484 & 10.952 & 0.100 & 0.177 & 1.049 & \\
6.9498 & 10.987 & 0.091 & 0.196 & 1.037 & \\
6.9523 & 11.038 & 0.138 & 0.153 & 0.969 & \\
6.9531 & 11.061 & 0.111 & 0.168 & 0.953 & \\
6.9544 & 11.057 & 0.160 & 0.116 & 0.943 & \\
6.9555 & 11.083 & 0.146 & 0.153 & 0.997 & \\
6.9568 & 11.109 & 0.146 & 0.160 & 1.049 & \\
6.9575 & 11.112 & 0.161 & 0.126 & 1.014 & \\
6.9587 & 11.122 & 0.185 & 0.110 & 0.967 & \\
6.9603 & 11.172 & 0.183 & 0.064 & 1.100 & \\
6.9620 & 11.165 & 0.215 & 0.048 & 1.043 & \\
6.9627 & 11.165 & 0.183 & 0.150 & 0.975 & \\
6.9643 & 11.227 & 0.173 & 0.143 & 0.863 & \\
6.9656 & 11.212 & 0.204 & 0.121 & 0.806 & \\
6.9681 & 11.263 & 0.198 & 0.150 & 0.767 & \\
6.9688 & 11.284 & 0.182 & 0.179 & 0.736 & \\
6.9699 & 11.325 &       & 0.258 & 0.742 & \\
6.9706 & 11.279 & 0.205 & 0.108 & 0.842 & \\
6.9719 & 11.295 & 0.213 & 0.120 & 0.814 & \\
6.9732 & 11.322 & 0.206 & 0.156 & 0.656 & \\
6.9744 & 11.307 & 0.220 & 0.132 & 0.711 & \\
6.9751 & 11.314 & 0.222 & 0.142 & 0.754 & \\
6.9763 & 11.393 &       & 0.227 & 0.776 & \\
6.9775 & 11.399 &       & 0.200 & 0.736 & \\
6.9800 & 11.415 &       & 0.214 & 0.650 & \\
6.9808 & 11.380 & 0.219 & 0.117 &       & \\
6.9836 & 11.377 & 0.218 & 0.126 & 0.787 & \\
\hline
\end{tabular}
\end{center}
\end{table}

\setcounter{table}{3}
\begin{table}
\begin{center}
 \caption{Continued.}
  \begin{tabular}{lccccc}
    \hline
    \hline
HJD & $V$ & $(b-y)$ & $m_1$ & $c_1$ & $\beta$  \\
\hline
$-$2455900 &   &   &   &   &   \\
\hline

6.9849 & 11.396 & 0.216 & 0.141 & 0.714 & \\
6.9858 & 11.402 & 0.212 & 0.165 & 0.670 & \\
6.9870 & 11.426 & 0.213 & 0.180 & 0.608 & \\
6.9883 & 11.397 &       & 0.119 & 0.767 &\\
6.9895 & 11.380 &       & 0.077 & 0.748 & \\
6.9905 & 11.452 & 0.205 & 0.116 &       & \\
6.9919 & 11.478 & 0.197 & 0.130 & 0.606 & \\
6.9931 & 11.488 & 0.174 & 0.194 & 0.640 & \\
6.9963 & 11.414 &       & 0.080 & 0.726 & \\
6.9970 & 11.479 & 0.194 & 0.165 & 0.763 & \\
6.9984 & 11.457 & 0.199 & 0.193 & 0.682 & \\
6.9991 & 11.428 &       & 0.058 & 0.725 & \\
7.0002 & 11.430 &       & 0.096 & 0.812 & \\
7.0012 & 11.432 &       & 0.067 & 0.739 &\\
7.0121 & 11.294 & 0.142 & 0.185 & 0.818 & \\
7.0198 & 10.980 & 0.082 & 0.162 & 0.940 & \\
7.0214 & 10.944 & 0.065 & 0.211 & 0.946 & \\
7.0232 & 10.925 & 0.075 & 0.178 & 1.018 & \\
7.0256 & 10.939 & 0.082 & 0.181 & 0.992 & \\
7.0270 & 10.953 & 0.093 & 0.172 & 1.038 & \\
\hline
\end{tabular}
\end{center}
\end{table}


\section{Frequency Content}

The frequency content of the pulsation of AE~UMa was determined utilizing two
different data sets: i) the light curves of Rodr\'iguez et al.\
(1992) and those presented in the current paper and ii) the compiled
list of times of maximum light, including the newly acquired ones.


\subsection{Time Series Analysis}

We were lucky that this star was observed in the $\uv$ photometric system in 1986 and 1987
by Rodr\'iguez et al.\ (1992) at the Sierra
Nevada Observatory, Spain, with a twin spectrophotometer like the one
we used in 2011 at SPM Observatory, M\'exico, some
twenty-five years later. Rodr\'iguez et al.\ (1992) published their
observations in magnitude differences ${\rm AE~UMa} - {\rm BD} + 44^\circ$ 1898 so,
transformation into the standard system was immediate. Furthermore, observations
were reported in $V$ and $B$ filters by Broglia \&
Conconi (1975). In section 2.2 we described in detail the
transformation followed to obtain the reported observations of the present study. The whole $V$ sample is constituted of 1299 data points in the $V$ filter in four seasons
covering a time span of 38 years.

\IBVSfig{8cm}{6186-f1.eps}{Phase plot of the $\uv$ photometry of Rodr\'iguez et al.\
(1992) and that of the present paper. The time span between both sets is
25 years. The period considered is 0.086017053 d.}
\IBVSfigKey{6186-f1.eps}{AE UMa}{other}


\IBVSfig{10cm}{6186-f2.eps}{Periodogram of $V$ magnitude of $\uv$ photometry of
Rodr\'iguez et al.\ (1992) and data from the present paper in Period04. Top,
Window function; middle, original data; bottom,
residuals.}
\IBVSfigKey{6186-f2.eps}{AE UMa}{periodogram}


Utilizing the period proposed by Pocs \& Szeidl (2001) we calculated
the phase for each $\uv$ data point and plotted magnitude $V$ and
color indexes in a diagram, Figure~1. We should here
mention the impressive result of how well all the magnitude and
color indexes conform the phase with the period used here. A few
points were completely out of the general pattern; we discarded them
and were left with a sample of 356 $\uv$ data points.

\IBVSfig{10cm}{6186-f3.eps}{O--C distributions vs time. A large scatter is caused by
reduction or observation procedure and not inherent to behaviour of the star.}
\IBVSfigKey{6186-f3.eps}{AE UMa}{other}


The first method utilized to determine the frequency content of AE~UMa 
was {\sc Period04} (Lenz \& Breger 2005), a well-known method widely
utilized by the $\delta$~Scuti star community. For the 1299 $V$
points of the sample we selected a frequency range between 0 and 50\,c/d. 
We ran the method twice and obtained the results listed in
Table 5 and represented graphically in Figure~2, with these other
characteristics: Zero point: 11.2871229; Residuals: 0.0584757486.
The comparison of the observations with the
predictions were astonishing given the long time basis of 38 years.

The same data set was tested in {\sc Period04} with the
period proposed by Pocs \& Szeidl (2001). The obtained results are
frequency 11.6253822 c/d; amplitude, 0.22205636 mag; phase, 0.819373 mag with a
zero point 11.287086 and a residual of 0.079247, which
demonstrate the goodness of the {\sc Period04} proposed period.

An absolute verification of the constancy of the period for at least
the last 25 years was done through the  $\uv$ photometry carried out
by Rodr\'iguez et al. (1992, starting in 1986 and continuing in 1987
over twenty nights) and that observed by us in 2011, 25 years later.
A phase was calculated for both seasons with a total of 356 points;
the resulting diagram is shown in Figure~1. As can be seen this
discards the supposition of a double mode variable although there is
some evidence of a variable amplitude. This also rules out a secular
variation of the period.

\IBVSfig{7cm}{6186-f4.eps}{O--C distributions in an histogram.
The values at the wings were discarded.}
\IBVSfigKey{6186-f4.eps}{AE UMa}{other}


\begin{table}[tc]
\centering \caption{Output of the {\sc Period04} package with time series
of the $V$ data of AE UMa}
\begin{tabular}{lccc}
 \hline
 \hline
Nr      &  Frequency (c/d)   &  Amplitude (mag) & Phase     \\
 \hline
F1 & $11.6253822 \pm 5.9\times 10^{-7}$ & $0.214 \pm 0.004$ & $0.775 \pm 0.003$ \\
F2 & $23.2514162 \pm 2.0\times 10^{-7}$ & $0.077 \pm 0.004$ & $0.275 \pm 0.009$ \\
\hline
\end{tabular}
\end{table}


\subsection{O--C Methods}

Before calculating the coefficients of the ephemeris equation, we
studied the existing literature related to AE UMa. Several authors
have conducted studies of the O--C behaviour of this particular
object and, in this preliminary stage, we took the existing equation
and reproduced the diagrams with our updated list of times of maxima
taken from the literature plus the data that we acquired.

The principal works on AE~UMa are those of Pocs \& Szeidl (2001) and
Zhou (2001). They are presented in Table 6. In the first one the
authors find this star varying with a constant period for more than
half a century. On the other hand, Zhou (2001) finds a varying
period (his equation 14).



\begin{table}[tc]
\centering
\caption{AE UMa ephemeris equations.}
\begin{tabular}{lllc}
\hline\hline
Author & \multicolumn{1}{c}{$T_0$} & \multicolumn{1}{c}{P} & $\beta$ \\
\hline
Pocs and Szeidl (2001) & 2442062.5824 & 0.086017076 & \\
Zhou (2001)            & 2442062.5827 & 0.08601585  & 5.71034 $\times 10^{-12}$\\
{\sc Period04}               &              & 0.086018677 & \\
MSDR                   & 2442062.5824 & 0.086017072 & \\
PDDM                   & 2456746.6464 & 0.086017069 & \\
\hline
\end{tabular}
\end{table}



Pocs \& Szeidl (2001) took all the O--C values, into account with the
exception of Filatov's data (124 maxima, time interval of 61 years)
and they arrived at the following quadratic polynomial fit

\begin{eqnarray}
O-C & = & (2.2 \pm 2.6) \times 10^{-4}  - (1.71 \pm 0.53) \times 10^{-8}  E \nonumber \\
& + & (0.053 \pm 0.053) \times 10^{-12}  E^{2} \nonumber
\end{eqnarray}

Since then, the only studies of the analysis of the pulsation of AE~UMa 
were done by Coates \& Landes (2008), and Niu et al. (2013) which
merely confirmed the constancy of the period of pulsation of AE~UMa.
There have been numerous reports on the times of maxima of
this star. Those previously observed by us (Pe\~na et al., 2015) are presented in Table~2 along with the newly observed maxima.


With an increased time basis (more than 28114 days (77 years) or 326845 cycles
and 512 times of maxima compiled) we tested the reported
calculations. What was immediately obvious was the exceedingly large
spread in the O--C values (Figure~3). Those in the earlier stages had
already been discarded by Pocs \& Szeidl (2001) and Zhou (2001).
However, a large scatter in the $O-C$ values was found with more
recent data which did not exist on the earlier studies.
To diminish the scatter the following was done: a histogram of the
O--C values was calculated and the standard deviation was evaluated (Figure~4). From the graphical representation of the O--C differences
vs.\ time those values with large spread from the mean became
conspicuous. These were discarded. Some, the old ones, were those
that the previous authors had also eliminated. If they
are not considered, this eliminates the large spread and makes it
possible to determine a more logical behaviour of the star.


\subsection{Minimization of the standard deviation of the O--C residuals (MSDR)}

The second method for period determination utilizes as criteria of goodness, the minimization of the standard deviation of the $O-C$ residuals (see Pe\~na et al. 2016 for a detailed description).

This method is based on O--C standard deviation minimization. We
considered a set 508 data points of $T_{\rm max}$. These are presented in
Figure~5. This data set covers a time span of 77 years.

A mean period was determined in the differences of two or three times
of maxima that were observed on the same night with an associated
standard deviation. We swept the period between this limits. The output was a straight line with some values of the deviation of the O--C residuals clearly diminishing. We swept again for a closer period range around this feature calculating 5000 steps which gives the
sufficient accuracy provided by the time span of the observations.
The obtained precision of one millionth provides the new period and
the limits for the iteration. In each iteration, the O--C standard
deviation was calculated. We chose to be the best period that which showed
the minimum standard deviation (Figure~5). The resulting equation
is:


\begin{eqnarray}
T_{max} = (2442062.5824 \pm 3 \times 10^{-3})  + (0.086017072 \pm 2 \times 10^{-9}) \times E\nonumber
\end{eqnarray}


\IBVSfig{10cm}{6186-f5.eps}{Standard deviation vs. Period. The minimum, clearly discerned, served to determine the best period.}
\IBVSfigKey{6186-f5.eps}{AE UMa}{other}


\subsection{Period determination through an O--C differences minimization (PDDM)}


As in the case of BO~Lyn (Pe\~na et al.\ 2016), we employed a method
based on the idea of searching the minimization of the chord length
which links all the points in the O--C diagram for different values
of changing periods, looking for the best period which corresponds
to the minimum chord length.

A set of 508 times of maxima was considered to perform this
analysis. These times are those remaining after the analysis of the
histogram in Section~3.2. Two hundred and sixty-six consecutive
pairs of times of maxima were used to calculate the average period
and the standard deviation, being the resulting period and deviation
0.0857 d and 0.0039 d, respectively. Taking this into consideration,
we can fix an interval span in which the period is located in
$0.0856 \pm 0.0039$ days. Maintaining a period precision of a
billionth and taking the interval span period into consideration,
$7.9 \times 10^{6}$ periods were used to perform this method. The
$T_0$ used to calculate the O--C diagrams is 2456746.6464, the final
of H\"ubscher \& Lehmann (2015). Then the best period is the one with
the smallest chord length and it is shown in Figure~6. The resulting
linear ephemerides equation is

\begin{eqnarray}
T_{\rm max} = 2456746.6464 + 0.086017069 \times E \nonumber
\end{eqnarray}


\IBVSfig{10cm}{6186-f6.eps}{Period determination through an O--C
differences minimization, PDDM.}
\IBVSfigKey{6186-f6.eps}{AE UMa}{other}



\begin{table}[tc]
\begin{center}
\caption{Reddening and unreddened parameters of AE UMa}
\begin{tabular}{ccccccrrrrr}
\hline\hline
Phase & $E(b-y)$ & $(b-y)_0$ & $m_0$ & $c_0$ & H$\beta$ & $V_0$ & $M_V$ & DM & Dist & [Fe/H]\\
\hline
0.05  &  0.000  & 0.131 &  0.160 & 0.966  & 2.796 & 11.00 & 1.03 &  9.98 &  989 & \\
0.15  &  0.000  & 0.158 &  0.144 & 0.941  & 2.765 & 11.12 & 0.79 & 10.33 & 1165 & \\
0.25  &  0.015  & 0.161 &  0.161 & 0.864  & 2.771 & 11.16 & 1.54 &  9.62 &  838 & \\
0.35  &  0.009  & 0.187 &  0.153 & 0.805  & 2.746 & 11.28 & 1.72 &  9.56 &  815 & \\
0.45  &  0.000  & 0.211 &  0.151 & 0.768  & 2.715 & 11.38 & 1.38 & 10.00 & 1001 & $-$0.246 \\
0.55  &  0.000  & 0.216 &  0.146 & 0.768  & 2.709 & 11.43 & 1.20 & 10.22 & 1108 & $-$0.287 \\
0.65  &  0.007  & 0.209 &  0.139 & 0.764  & 2.716 & 11.41 & 1.44 &  9.97 &  988 & $-$0.430\\
0.75  &  0.000  & 0.207 &  0.154 & 0.769  & 2.721 & 11.41 & 1.53 &  9.88 &  946 & \\
0.85  &  0.017  & 0.140 &  0.168 & 0.833  & 2.801 & 11.18 & 2.27 &  8.91 &  606 & \\
0.95  &  0.000  & 0.132 &  0.168 & 0.949  & 2.797 & 11.00 & 1.19 &  9.81 &  915 & \\
\hline
\end{tabular}
\end{center} 
\end{table}


\IBVSfig{10cm}{6186-f7.eps}{Cycle variation of AE UMa in the theoretical grids of LGK86.}
\IBVSfigKey{6186-f7.eps}{AE UMa}{other}


\section{Physical Parameters}
Determining physical parameters for the stars is not a simple
task.
The advantage of intermediate band photometry is that, if it is well
calibrated and used together with the theoretical models, it can be
utilized to infer the physical conditions of the stars.


The whole $\uv$ sample consisted of some 360 data points. In both
data sets the weak point was the subset of H$\beta$ photometry.
Rodr\'iguez et al.\ (1992) observed only 58 data points and in
2011 our H$\beta$ sample consisted of only four data points.


With the advantages of the empirical calibrations of Balona and Shobbrook
(1984) and Nissen (1988), based on the earlier papers of Crawford
for stars of different spectral type stars for $\uv$ photometry, we
determined reddening and unreddened indexes. These, combined with
theoretical models such as those of Lester, Gray \& Kurucz (1986,
hereinafter LGK86), determined physical parameters such as surface
gravity and effective temperature. These calibrations have already
been described and used in previous analyses (Pe\~na \& Peniche
1994; Pe\~na \& Sareyan 2006).

As was described at the end of section O--C Analysis, a remarkable
phasing was accomplished with the two $\uv$ photometric seasons
observed twenty-five years apart. In order to calculate the
reddening and unreddened colors and in view of the poorness of the
H$\beta$ data of both seasons, we calculated mean averages in phase
bins of steps of 0.1. To follow the behavior of H$\beta$  we
overlapped the $V$ curve and discarded seven points which were out of
the general trend of the variation.

The application of Nissen's prescription gave, for the values
averaged in phase bins, the corresponding unreddened values that were presented
in Table~7. This Table lists, in the first column, the phase.
Subsequent columns present the reddening, the unreddened indexes,
the unreddened magnitude, the absolute magnitude, the distance
modulus, and distance and, in the last column, the metallicity
[Fe/H] for the stages when the star was of spectral type F.


Mean values were calculated for $E(b-y)$, Distance Modulus and
distance for two cases: i) the whole data sample and ii) in phase
limits between 0.3 and 0.8, which is customary for pulsating stars
to avoid the maximum. It gave, for the whole cycle, values of $0.005
\pm 0.007$; $9.83 \pm 0.40$ and $937 \pm 158$ for $E(b-y)$, DM and
distance (in pc), respectively, whereas for the mentioned phase
limits we obtained, $0.03 \pm 0.04$; $9.92 \pm 0.24$ and $972 \pm
106$ respectively.  The uncertainty is merely the standard deviation.
In the case of the reddening, most of the values in the spectral
type in the F stage of AE~UMa produced negative $E(b-y)$ values
which is unphysical. In those cases we forced the reddening to be
zero in which case the $(b-y)$ index is the same.


Once the unreddened colors are known, it is possible to determine
some physical parameters ($\log~T_{e}$ and $\log~g$) from a direct
comparison with the models developed. A metallicity has to be
assumed. We mentioned in the Introduction that with respect to the
metallic content, Rodr\'iguez et al.\ (1992), using a $\delta m_1$
metal calibration for metal abundance, reported ${\rm [M/H]}=-0.3$, and
Hintz et al. (1997) give values of [M/H] between $-0.1$ and
$-0.4$. McNamara (1997) reports ${\rm [M/H]}=-0.5$ for AE UMa. We,
utilizing the same data as Rodr\'iguez et al.\ (1992) and with the
same technique, obviously arrived at the same results. In the phase
bins the star shows the F nature in three different phases, $0.45$,
$0.55$ and $0.65$ and a mean value ${\rm [Fe/H]}$ of $-0.321$ is
obtained. All of these determined values fit the ${\rm [Fe/H]}$ vs.
$\log P$ relationship of McNamara (1997) (his Figure 1) adequately,
corroborating the assumed metallicity of AE~UMa.

LGK86 calculated their model outputs for several metallicities.
Particularly in the case of AE~UMa, for which we determined mean
metallicity of ${\rm [Fe/H]} = -0.32 \pm 0.10$, there are two models
which were applicable, either ${\rm [Fe/H]}$ 0.0 or $-0.5$. We
tested both since our determined mean metallicity of ${\rm [Fe/H]}
-0.32 \pm 0.10$ lies in between.


The temperature of the star was determined from its positions in the
LGK86 grids (Figure 8) and is listed in Table 8 for both metallicities. The
reddening $E(B-V)$ was calculated utilizing the well-known relation
$E(b-y) = 0.7 \times E(B-V)$. Table~9 presents a summary of the compiled characteristics.



\setcounter{table}{7}
\begin{table*}[!t]
 \caption{Effective temperature of AE UMa.}
\begin{center}
  \begin{tabular}{lccccccc}
\hline\hline
phase &  & $T_e$ &   $T_e$   & &    OP\&J  & $\log g$\\
\hline
& ${\rm [Fe/H]}$ &   0.0 &     $-$0.5 &   mean       &    &   $-0.5$\\
\hline
0.05 &&  7800 &  7100 &  7450 &     7822 &  3.4\\
0.15 &&  7500 &  7200 &  7350 &     7553 &  3.5\\
0.25 &&  7500 &  7200 &  7350 &     7605 &  3.8\\
0.35 &&  7200 &  7000 &  7100 &     7388 &  3.8\\
0.45 &&  6800 &  7500 &  7150 &     7119 &  3.5\\
0.55 &&  6800 &  7500 &  7150 &     7067 &  3.5\\
0.65 &&  7100 &  7700 &  7600 &     7128 &  3.8\\
0.75 &&  7000 &  7500 &  7250 &     7171 &  3.5\\
0.85 &&  7700 &  7400 &  7550 &     7866 &  4.1\\
0.95 &&  7700 &  7400 &  7550 &     7831 &  3.8\\
\hline\hline
\end{tabular}
\end{center}
\end{table*}



\begin{table}[tc]
\centering \caption{Compiled characteristics for AE UMa.}
\begin{tabular}{lcc}
\hline \hline
Spectral Type            &  A9     \\
Galactic longitude       & 176.1978  \\
Galactic latitude        & +47.7071  \\
Distance [pc]            & 972      \\
Reddening [mag] $E(b-y)$ & $0.005 \pm 0.007$  \\
Reddening [mag] $E(B-V)$ & 0.034  \\
Parallaxes [mas]         & 5.74 \\
Distance modulus [mag]   & 7.92    \\
Log $T_e$                & 3.85    \\
Log $g$                  & 3.8   \\
\hline
\end{tabular}
\end{table}


\section{Conclusions}

New times of maximum light have been gathered
for the HADS  AE~UMa, from CCD photometry at the Tonantzintla and
$\uv$ photometry at the San Pedro M\'artir Observatories, Mexico. With
the inclusion of these maxima and those gathered from the
literature, the ephemerides proposed by Pocs \& Szeidl (2001) was
slightly modified because a larger time span was available. This
result is confirmed from the time series $\uv$ data separated 25 years
apart. Two other methods were employed to determine the frequency content of the star. A sinusoidal behaviour of the residuals can be discerned. Thus, the binary
nature of AE~UMa, indicated by other authors has been tightened up
with the new times of maxima determined. Hence, the solution for AE~UMa 
is valid since it is supported by a longer time string. The
residuals indicate that this star might have the same sinusoidal
behavior as AN~Lyn (Pe\~na et al.\ (2015) or BO Lyn (Pe\~na et al.\
2016) which would indicate a binary nature. Some physical parameters
were determined for AE~UMa from $\uv$ photometry from the literature
(Rodr\'iguez et al.\ 1992) and that presented in this study. The
obvious recommendation is to gather more observations that, in the
long run could prove or discard this binary assumption.

\IBVSTack{We thank the staff of the OAN at TNT and SPM for their assistance in securing the observations. This work was partially supported by PAPIIT IN106615
and PAPIME PE113016. AR thanks the IA for allowing the telescope
time at SPM. Typing and proofreading were done by J.~Orta, and J.~Miller, respectively. 
C.~Guzm\'an, F.~Ru\'iz and A.~D\'iaz assisted
us in the computing. This research has made use of the Simbad
databases operated at CDS, Strasbourg, France and NASA ADS Astronomy
Query Form.}




\references


Arellano Ferro, A.  Nu\~nez, N. S., Avila, J. J., Trejoluna, J. J., 1994, {\it RMxAA}, {\bf 29}, 218

Arellano Ferro, A., Parrao, L., 1988, IA-UNAM Reporte T\'ecnico, 57

Nautical Almanac Office, 2006, {\it Astronomical Almanac}

Balona, L. A., Shobbrook, R. R., 1984, {\it MNRAS}, {\bf 211}, 375 \DOI{10.1093/mnras/211.2.375}

Breger, M., Pamyatnykh A.A., 1998, {\it A\&A}, {\bf 332}, 958 

Broglia, P., Conconi, P. 1975, {\it A\&AS}, {\bf 22}, 243 

Coates, D.W., Landes, H., 2008, {\it Comm. in Asteroseismology}, {\bf 153}, 8 \DOI{10.1553/cia_153s8}

Collins, K., 2012, {\it Astrophysics Source Code Library}, ascl:1309.001 \BIBCODE{2013ascl.soft09001C} 

Crawford, D. L., Barnes, J. V., 1970, {\it AJ}, {\bf 75}, 978 \DOI{10.1086/111051}

Crawford, D. L., Mander, J., 1966, {\it AJ}, {\bf 71}, 114 \DOI{10.1086/109865}

Filatov, G. S., 1960, {\it Astron. Tsirk.}, {\bf 215}, 20 %\BIBCODE{1960ATsir.215...20F}

Garcia, J. R., Cebral, J. R., Scoccimarro, E. R., et al. 1995,
{\it A\&AS}, {\bf 109}, 201 

Geyer, E., Kippenhahn, R., Strohmeier, W., 1955, {\it Kleine Ver\"off.
Bamberg}, {\bf 11} 

Hintz, E. G., Hintz, M. L., Jones, M. D., 1997, {\it PASP}, {\bf 109}, 1073 \DOI{10.1086/133977}

H\"ubscher, J., Lehmann, P., 2015, {\it IBVS}, {\bf 6149}

Lenz, P., Breger, M., 2005, {\it Comm. in Asteroseismology}, {\bf 146}, 53 \DOI{10.1553/cia146s53}

Lester, J. B., Gray, R. O., \& Kurucz, R. L., 1986, {\it ApJS}, {\bf 61}, 509 \DOI{10.1086/191122}

Nissen, P., 1988, {\it A\&A}, {\bf 199}, 146 

Niu, J. S., Fu, J. N., Yang, X. H., \& Zong, W. K., 2013, arXiv:1304.3772v2

Olsen, E. H., 1983, {\it A\&AS}, {\bf 54}, 55 

Pe\~na, J. H., Peniche, R., 1994, {\it RMxAA}, {\bf 28}, 139 

Pe\~na, J. H., Sareyan, J. P., 2006, {\it RMxAA}, {\bf 42}, 179 

Pe\~na, J. H., Renter\'ia, A. Villarreal, C. et al., 2015, {\it IBVS}, {\bf 6154}

Pe\~na, J. H., et al., 2016, {\it RMxAA}, {\bf 52}, 385 

Pocs, M. D., Szeidl, B., 2001, {\it A\&A}, {\bf 368}, 880 \DOI{10.1051/0004-6361:20010043}

Rodriguez, E., Rolland, A., Lopez de Coca, P., Garcia-Lobo,
E., Sedano, J. L., 1992, {\it A\&AS}, {\bf 93}, 189 

Szeidl, B., 1974, {\it IBVS}, {\bf 903}

Szeidl, B., 2001, {\it Comm. in Asteroseismology}, {\bf 140}, 56 

Tsesevich, V. P., 1956, {\it Astron. Tsirk.}, {\bf 170} 

Tsesevich, V. P., 1973, {\it Astron. Tsirk.}, {\bf 775}, 2 %\BIBCODE{1973ATsir.775....2T}

Zhou, A.-Y., 2001, {\it A\&A}, {\bf 374}, 235 \DOI{10.1051/0004-6361:20010722}


\endreferences

\end{document}
