\documentstyle[twoside,ibvs2,epsf]{article}
 
\begin{document}
 
\IBVShead{5951}{21 September 2010}
 
\IBVStitletl{AC Bootis -- An Unevolved W-Type Overcontact}{ Eclipsing Binary with a High Mass Transfer Rate}

\IBVSauth{Nelson, Robert H.$^{1,2}$} 

\IBVSinst{1393 Garvin Street, Prince George, BC, Canada, V2M 3Z1 email: b-o-b.nelson@shaw.ca [remove dashes]}
\IBVSinst{Guest investigator, Dominion Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council of Canada}

\SIMBADobj{AC Boo}

\begintext

The variability of AC Boo (= BD +46\deg2004 = TYC 3474-905-1 = HIP 73103, 
Sp.~F8V) was discovered by Geyer (1955, as reported in Mauder, 1964).  
Zessewitsch (1956) first identified the system as a W-UMa-type and attempted to 
determine its period, getting values of 0.38514 and 0.4278192 days.  
Mauder (1964) secured light curves in $B$ and $V$ photoelectrically and 
determined the correct period of 0.352429 days.  He went on to do an analysis 
of the light curves using the rectification method.  Lacking radial velocities, 
he was forced to estimate the mass ratio by indirect methods.  
Binnendijk (1965) used the 28-inch reflector at the Flower and Cook Observatory 
to secure a photoelectric light curves in two colours and derived orbital 
elements from the rectification method.  Mancuso et al. (1974, 1977, 1978) 
determined times of minima and full light curves photoelectrically.  
They classified the system as W-type and went on to use the methods of Wood 
(1971) and Wilson-Devinney (1971, 1990, hereafter WD) to analyze their light 
curves, taken in 1972 and 1973, and also those of Binnendijk (1965).  
Assuming a spectral type of F0 (attributed to Mauder, 1964), they used fixed 
parameters $i = 85\fdeg47$, $\Omega_1 = 7.330$, $T_1 = 6100 $~K, 
and mass ratio $q = M_2/M_1 = 3.57$.  (The temperature was estimated from the 
colour index, and the others indirectly from Russell-Merrill (1952) analysis.)  
Noting the variation in the light curves over time, they obtained temperatures 
of the secondary varying from $T_2 = 5735$ to 6055 K.  Schieven et al. (1983) 
secured further light curves photoelectrically, analyzed some period changes, 
and performed a light curve synthesis using the methods of Rucinski 
(1976a, b, and c).  They also noted the variability of the light curves 
(on a time scale of days) and also the presence of an asymmetry in the portion 
of the light curve following the primary (flat-bottomed) minimum.  
They attributed the latter to the presence of starspots.  They obtained two 
values of the mass ratio: 0.31 and 0.28.  (Note that these values $<$1 imply 
an A-type system.)  Linnell, et al. (1990) report modelling with their 
(unpublished) photometric data and the mass ratio of 
\hbox{0.41 (A-type) = 2.44~(W-type)} of Hrivnak (1993) requiring a complex set 
of dark spots and third light.  No error estimate was provided.  Finally, 
Hrivnak (1993), using cross-correlation techniques, derived RV values which 
fit a double sine curve somewhat poorly, resulting in an rms deviation 
of 16.7 km/s and the above reported mass ratio. 

With improved versions of the WD code that support star spots, and the 
opportunity to obtain new radial velocities (RVs) using analysis by the modern 
broadening functions due to Rucinski (1976a, 1976b, 1976c, 1992, 2004), an 
updated study was deemed important.

This system has shown significant period change since 1929 when the first 
visual observations were recorded.  Altogether 246 times of minimum have been 
entered into the Excel worksheet that is part of the 
``Eclipsing Binary $O-C$ Files" (observed minus calculated) database at the 
AAVSO site that is maintained by the author (Nelson, 2010 -- updated annually).  
These include three new times of minimum determined during light curve 
acquisition and reported here -- see Table 1.

\begin{table}
\centerline{{\bf Table 1.} New times of minima for AC Boo}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Minima&Error (days)&Cycle&Type\\
\hline
\hline
55300.8561&0.0002&28893&I (occultation - flat bottom)\\
55301.7380&0.0005&28895.5&II (transit)\\
55312.8395&0.0002&28927&I (occultation - flat bottom)\\
\hline
\end{tabular}
\end{center}
\end{table}

Throughout the years, a total of 10 elements (epoch, period) have been used 
by various authors; some give the (incorrect) phase 0.5 of the deeper, or 
flat-bottomed minima (and of course, 0.0 for the shallower minima).  The 
elements of Schieven et al. (1983) seem to give the best results (but note 
that there seems to have been a typo in HJD$_0$ -- the correct value, as 
determined from his data, is given in Equation 1).  As corrected,  the elements 
give correct phases for the minima of Mauder (1964) and Binnendijk (1965), 
(subject to the requirement to adjust the cycle counts by + or $-$ 0.5 for 
various ranges to obtain a contiguous relationship):

\begin{equation}
{\rm J.D.\ Hel.\ min\ I} = 24 45117.781 (1) + 0.3524321 (2) E
\end{equation}

The results are plotted in Figure 1.  The reader will note, as mentioned by 
various authors, that the period seems to have been constant from 1929 to 1982 
(cycle counts from $-$54880 to 0); the best-fit period in this interval 
was 0.3524294 (1) days.  Close to cycle 0 there was a sudden rise in the period; 
after that, the period displayed a slow, steady increase over time.  The abrupt 
change of period can be explained by an episodal mass interchange, possibly as 
the two stars established contact (but see below).  The portion of the curve 
after cycle 200 (and especially after cycle 1400) can be fitted closely by a 
parabola -- denoting a constant rate of change.  (Assuming no other cause, this 
implies a constant mass transfer rate; this is calculated later.)  The elements 
best describing the curved section are given in Equation 2.  
(Then ${\rm d}P/{\rm d}E = 2\times$ the quadratic 
coefficient = $8.13 \times 10^{-10}$ days/cycle.)

\begin{equation}
{\rm J.D.\ Hel.\ min\ I} = 24 45117.8019(1) + 0.3524305(1) E + 4.1(3) \times 10^{-10} E^2
\end{equation}

\IBVSfig{6.8cm}{5951-f1.eps}{Observed - Computed ($O-C$) plot for AC Boo}
\IBVSfigKey{5951-f1.eps}{AC Boo}{O-C diagram}

During September of 2008 and 2009, the author took six spectra 
(10 \AA/mm reciprocal dispersion, resolving power 10,000) at the 
Dominion Astrophysical Observatory (DAO) in Victoria, British Columbia, Canada; 
he then used the Rucinski broadening functions (Rucinski, 2004) to obtain 
radial velocity (RV) curves (see Nelson et al., 2006 for details).  The spectral 
range was 5014-5261 \AA.  A log of DAO observations and RV results is presented 
in Table 2.
An anonymous referee has pointed out that -- due to the short period of the 
system -- the one hour exposures represent a significant fraction of a period 
(0.118) and that excessive ``phase smearing" of RVs might occur.  A simple 
analysis reveals that, for circular orbits and assuming that neither star is 
undergoing an eclipse during any part of the exposure, the derived (averaged) 
RV values are reduced by a factor $f = \sin X / X$ from the instantaneous 
values (where $X = \pi t/P$, with $t$ = exposure time and $P$ = period).  
Note that factor $f$ is independent of phase (subject to the above conditions).  
Therefore, corrected RVs were obtained by simply dividing all the derived RVs 
by $f$.  (Note that, for $t = 3600$ seconds, $f = 0.977$.  Note also that 
neglecting this correction factor cannot affect the derived value for the mass 
ratio but does affect the derived values of mass, etc.; therefore this 
correction is needed.)

\begin{table}
\centerline{\bf Table 2.}
\begin{center}
\begin{tabular}{|l|c|c|c|r|r|}
\hline
DAO&Mid Time&Exposure&Phase at&$V_1$&$V_2$\\
Image \#&(HJD-2400000)&(sec)&Mid-exp&(km/s)&(km/s)\\
\hline
\hline
4786*&54568.8890&3600&0.192&$-$275.7&50.2\\
4854*&54571.9030&3600&0.743&232.1&$-$109.9\\
5369&54927.8462&3600&0.655&200.6&$-$96.9\\
5381&54928.0313&1693&0.180&$-$266.4&32.5\\
5402&54928.9480&1800&0.781&221.2&$-$95.6\\
5420&54929.8189&3600&0.252&$-$278.2&56.6\\
\hline
\end{tabular}
\end{center}
\hspace*{1.6cm}*Taken in 2008
\end{table}

On seven nights April of 2010, the author took a total of 241 CCD images of the 
field in $V$, 266 in $R_C$ and 238 in $I_C$ (both Cousins) at his private 
observatory in Prince George, British Columbia, Canada.  The telescope was 
a 33 cm f/4.5 Newtonian on a Paramount ME mount; the detector was 
a SBIG ST-7XME CCD cooled to $-$20\deg C.  Reduction software was MIRA 
by Mirametrics, Inc., and light-box flats were used.  A list of the 
Variable, Comparison and Check stars appears in Table 3.

\begin{table}
\centerline{\bf Table 3.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Type&TYC &R.A.&Dec.&$V$&$B-V$\\
&3474$-$&J2000&J2000&Mags&Mags\\
\hline
\hline
Variable&905&14\hr 56\mm 28.341s&46\deg21\arcm44\farcs1&10.0$-$10.62&0.592\\
Comparison&835&14\hr 56\mm 07.846s&46\deg21\arcm26\farcs2&11.18&0.549\\
Check&966&14\hr 56\mm 26.300s&46\deg26\arcm50\farcs7&9.39&0.358\\
\hline
\end{tabular}
\end{center}
\end{table}

The following elements were used for phasing the light curves 
(see Nelson, 2010 for the $O-C$ relation):
\begin{equation}
{\rm JD\ Hel\ Min\ I} = 24 55312.8409 (1) + 0.3524320 (1) E
\end{equation}

The author used the 2004 version of the Wilson-Devinney (WD) light curve and 
radial velocity analysis program with the Kurucz atmospheres 
(Wilson and Devinney, 1971, Wilson, 1990, Kallrath, et al., 1998) as implemented 
in the Windows software WDwint (Nelson, 2009) to analyze the data.  To get 
started, a spectral type F8 V (SIMBAD - no reference given) and a 
temperature {$\mathbf T_1 = 6250 \pm 240$~K} were used; interpolated tables 
from Cox (2000) which gave $\log g = 4.368$ were used; an interpolation 
program by Terrell (1994) gave the (van Hamme, 1993) limb darkening values; 
and finally, a square root (LD=3) law for the limb darkening coefficients was 
selected, appropriate for hotter stars (Bessell, 1979).  
(The logarithmic LD=2 coefficients were tried but gave identical results.)  
Radiative envelopes were chosen for both stars, again appropriate for hotter 
stars (convective envelopes were tried but gave much poorer results.)   
The parameters are listed in Table 4.

\begin{table}
\centerline{\bf Table 4.}
\begin{center}
\begin{tabular}{|c|r|r|}
\hline
Quantity&\multicolumn{2}{|c|}{Value}\\
\cline{2-3}
--- &Star 1&Star 2\\
\hline
$g$&0.500&0.500\\
$A$&1.000&1.000\\
$x_{\rm bol}$&0.137&0.137\\
$y_{\rm bol}$&0.581&0.581\\
$x_{\rm V}$&0.148&0.148\\
$y_{\rm V}$&0.665&0.665\\
$x_{\rm Rc}$&0.049&0.049\\
$y_{\rm Rc}$&0.696&0.596\\
$x_{\rm Ic}$&$-$0.018&$-$0.018\\
$y_{\rm Ic}$&0.678&0.678\\
\hline
\end{tabular}
\end{center}
\end{table}

Mode 3 (for overcontact binaries) was chosen, based on the general appearance of 
the light curves.   

Because of the asymmetry of the light curves (unequal maxima and distorted 
light curve from about phase 0.25 to 0.50), noted by other authors, it was 
necessary to add a bright spot (a plage) on the more massive star (star 2).  
After a number of adjustments, the computed curved fit the observed values very 
closely indeed.  The fit for the same spot put on star 1 was poorer, and a dark 
spot anywhere on the system did not give any kind of fit at all.  It was also 
necessary to add third light; the values of $l_3$ are well above the 3~sigma 
values -- see Table 4.

Convergence by the method of multiple subsets was reached in a large number of 
iterations (owing to the many parameters to adjust -- 17).  It was noted that 
identical values (to 3 figures) for the mass ratio were obtained from the 
photometric only WD~solution, RV curve-fitting (alone), and the combined 
solution.  This agreement reinforces the validity of the solution.

A plot of the $VR_{C}I_{C}$ light curves and WD fit are shown in Figure 2; the 
RVs are shown in Figure 3 (the rms deviation from the fitted curve was 3.7 km/s).  
A three dimensional representation from Binary Maker 3 (Bradstreet, 1993) 
is shown in Figure 4.

\IBVSfig{5.2cm}{5951-f2.eps}{AC Boo:  V, R, I Light Curves -- Data and WD Fit}
\IBVSfigKey{5951-f2.eps}{AC Boo}{light curve}

\IBVSfig{8cm}{5951-f3.eps}{AC Boo:  Radial Velocity Curves -- Data and WD Fit.}
\IBVSfigKey{5951-f3.eps}{AC Boo}{radial velocity curve}

\IBVSfig{4.5cm}{5951-f4.eps}{Binary Maker 3 representation of the system -- at phases 0.24 and 0.44.}
\IBVSfigKey{5951-f4.eps}{AC Boo}{system representation}

Final WD output parameters are listed in Table 5 with formal errors; the real
error of the temperatures are around $\pm 250 K$ as noted above.

\begin{table}
\centerline{\bf Table 5.}
\begin{center}
\begin{tabular}{|l|c|c||l|c|c|}
\hline
Quantity.&Value&Error&Quantity&Value&Error\\
\hline
\hline
$T_1$ (K)&6250&---&$a$ (solar radii)&2.43&0.04\\
$T_2$ (K)&6241&6&$V_{\gamma}$ (km/s)&$-$25.8&0.1\\
$\Omega_1 = \Omega_2$&7.034&0.004&Spot Co-latitude&75&40\\
$q = M_2/M_1$&3.340&0.004&Spot Longitude&31&4\\
$i$ (deg)&86.3&0.5&Spot Radius&40&2\\
$L_1/(L_1+L_2) (V)$&0.251&0.0006&Spot Temp. Factor&1.019&.007\\
$L_1/(L_1+L_2) (R)$&0.250&0.0005&$R_1$ (pole)&0.263&0.001\\
$L_1/(L_1+L_2) (I)$&0.250&0.0004&$R_1$ (side)&0.274&0.001\\
$l_3/(L_1+L_2) (V)$&0.0013&0.0001&$R_1$ (back)&0.308&0.001\\
$l_3/(L_1+L_2) (R)$&0.0012&0.0001&$R_2$ (pole)&0.458&0.001\\
$l_3/(L_1+L_2) (I)$&0.0009&0.0001&$R_2$ (side)&0.493&0.001\\
$\Sigma\omega_{\rm res}^2$&0.00665&---&$R_2$ (back)&0.518&0.001\\
\hline
\end{tabular}
\end{center}
\end{table}

The WD output fundamental parameters are listed in Table 6 along with those from 
the properties of zero age main sequence stars (ZAMS; Cox, 2000).  The more 
massive star's mass and radius correspond closely to the tabular values 
(confirming its spectral type and its unevolved status), but it's overluminous 
by 3\%.  This may be due to the hot spot, or just to random error.  In 
estimating the distance, galactic extinction was ignored, owing to the high 
galactic latitude (58\deg). 

\begin{table}
\centerline{\bf Table 6.}
\begin{center}
\begin{tabular}{|l||c|c|c||c|c|c|}
\hline
Fundamental&Star 1&Star 1&Star 1&Star 2&Star 2&Star 2\\
Quantity&Tabular&WD&error&Tabular&WD&Error\\
\hline
\hline
Sp. Type&F8 V&---&---&F8 V&---&---\\
Mass ($M_\odot$)&---&0.36&0.03&1.18&1.20&0.05\\
Radius ($R_\odot$)&---&0.69&0.01&1.18&1.19&0.01\\
M$_{\rm bol}$&---&5.26&0.17&3.84&4.07&0.17\\
Log g (cgs)&---&4.32&0.004&4.388&4.36&0.004\\
Luminosity ($L_\odot$)&---&0.65&0.09&1.89&1.94&0.28\\
Distance (pc)&---&182&13&---&---&---\\
\hline
\end{tabular}
\end{center}
\end{table}

In an effort to understand the period behaviour of this system, the author 
scanned the data of Binnendijk (1965) [early data only -- later data required 
a spot] and Schieven (1983), and ran the same WD package, starting at the above 
solution set of parameters.  The results were (somewhat gratifyingly) almost 
identical with the above.  The system in 1965 and 1983 was overcontact then as 
well.  However, that does nothing to explain why the period could be constant 
for so long, suffer some kind of jump around cycle 0 (1982) and then display a 
constant rate of increase after that.

The calculation of the mass transfer rate for the curved part of the $O-C$ curve 
(i.e.~after cycle 0) proceeded as follows:

It is not difficult to show that, under the assumptions of mass and angular 
momentum conservation of the system as a whole, and using basic physics:

\begin{equation}
(M_1M_2)^3 P = {\rm constant}
\end{equation}

where $M_1$ and $M_2$  = masses (any units), and $P$= period.  Taking the 
natural logarithm of both sides of the equation and differentiating by time, 
and using the fact that \hbox{${\rm d}M_2 = -{\rm d}M_1$}, one gets:

\IBVSeqfig{
\begin{equation}
\frac{{\rm d}M_1}{{\rm d}t} = \frac{1}{3P}\left(\frac{1}{M_2} - \frac{1}{M_1}\right)^{-1} \frac{{\rm d}P}{{\rm d}t}
\end{equation}
}{\IBVSalteqfig{5951-e5}}

Substituting the values from Table 5, and using the derived value of 
\linebreak\hbox{${\rm d}P/{\rm d}t = +8.16 (6) \times 10^{-7}$~days/year} 
(taken from fitting the $O-C$ curve with a parabola), one gets a value 
of $-3.9 (3) \times 10^{-7}$~solar masses/year for the mass transfer rate of 
the primary (less massive) star (i.e., mass is transferred from the less to the 
more massive star).  This value is consistent with those of a number of other 
overcontact binaries (Schieven et al., 1983; Pribulla and Vanko, 2002).

In conclusion, AC Bootis is a subtype W overcontact binary comprised of 
unevolved stars.  A full solution for its fundamental parameters has been 
obtained.  It has been shown that it is undergoing mass transfer from the less 
massive to the more massive star at a rate of $3.9 \times 10^{-7}$ solar masses 
per year.

{\bf Acknowledgements:}
It is a pleasure to thank the staff members at the DAO (especially Dr. Dmitry 
Monin and Les Saddlemyer) for their usual splendid help and assistance.  
Thanks also to Dr. Dirk Terrell for his suggestions and comments.  An anonymous 
referee made a number of useful suggestions plus provided a reference 
(Hrivnak, 1993) that was, for some reason, not given in SIMBAD.

\references

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Cox, A.N., ed, 2000, {\it Allen's Astrophysical Quantities}, 4$^{\rm th}$ ed., (Athlone Press, London).

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\endreferences

\IBVSedata{5951-t7.txt}
\IBVSdataKey{5951-t7.txt}{AC Boo}{photometry}

\end{document}
