THE PHOTOMETRIC STUDY OF A NEGLECTED NEAR CONTACT BINARY: BS VULPECULAE

The basic version of our direct period analysis entails the modeling of observed orbitally modulated variations making use of suitable models containing the orbital period and its pertinent secular changes as free parameters. In the present study, we limit ourselves only to analysis of light changes.

The first step of our more or less phenomenological modeling is the finding of the appropriate model function, F(t), that correctly describes the variations observed in the light curve of BS Vul. We aimed to search for functions with as few free parameters as possible. Comparing shapes of BS Vul mean BVRI light curves during the last several decades we concluded that all of them are nearly symmetrical with deeper primary and shallower secondary minima centered on 0.0 and 0.5 phases.

The shapes of individual light curves of BS Vul were described by the depths of primary and secondary eclipses a_1b, a_2b in the b = {B, V, R, and I} photometric bands, as well as the mean magnitudes outside of eclipses for the jth observer $\overline{m}_{jb}$. Light changes caused by proximity effects can be simulated by a simple double-wave cosinusoid with a semi-amplitude a₃ unique for all photometric bands. The shapes of light curves during both eclipses could be described by the unique parameter C, the widths of primary/secondary minima d₁, d₂. In total, we analyzed 14 light curves, containing 8177 individual measurements obtained in four different photometric bands by eight different observers.

A detailed inspection of the observed $\textit {BVRI}$ light curves indicated that there was an additional persistent light contribution that occurred between phases 0.25 and 0.45. Using the WD code, we modeled this excess as a hot spot on the secondary star (see Section 4). This effect was approximated by the last term described by three free parameters a₄, d₄, φ_s.

The observed light curves were then approximated by the model

$$\begin{eqnarray} \displaystyle F_{jb}(t) & \simeq & \overline{m}_{jb} + \sum _{i=1}^\mathrm{2} a_{ib} \left\lbrace 1-\left[1-\exp \left(-\frac{\varphi _i^2}{d_i^2}\right)\right]^C\right\rbrace \nonumber\\ && + a_{3}\cos (4\,\pi \,\vartheta)+ a_4\exp \left[-\frac{(\varphi _1-\varphi _{\rm s})^2}{d_4^2}\right], \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &\varphi _{1}(t)=\vartheta (t)-\mathrm{round}(\vartheta);\ \varphi _{2}(t)=\vartheta (t)-\mathrm{floor}(\vartheta)- {\frac{1}{2}}, \quad \end{eqnarray} \tag{ 2 }$$

where φ₁, φ₂ are the phases and ϑ(t) is the time-dependent phase function. The operator "round" rounds the argument to the nearest integer, while "floor" rounds the argument to the nearest integer toward minus infinity. The phase function ϑ(t) (for details see, e.g., Mikulášek et al. 2008, 2011a) is the monotonically increasing function of time t of which the integer part is the epoch E and the fractional part is the common orbital phase φ with zero phase put at the center of primary minimum. The phase function ϑ(t) thus has its origin at the time of the epoch origin M₀. Knowing the phase function we are able to determine the phase for any time. Wanting to solve the reverse task: to predict the moment of zero phase (primary minimum timing) for any epoch E, we need to find the inversion function of the phase function θ(ϑ).

The phase function ϑ(t) is bound with the instantaneous period P(t) by the relation

$$\begin{equation} \dot{\vartheta (t)}=P(t)^{-1};\quad \vartheta (t=M_0)=0. \end{equation} \tag{ 3 }$$

We found that the instantaneous orbital period P(t) of BS Vul is monotonically decreasing. Using Equation (3) and assuming the linear decrease of the period (Equation (4)) we can derive the exact expression for the phase function ϑ(t) and its inversion function θ(ϑ)

$$\begin{eqnarray} &P(t)=P_0+\dot{P}\,(t-M_0), \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \displaystyle \vartheta (t,M_0,P_0,\dot{P}) &=& \int _0^{t-M_0}\frac{d\tau }{P(\tau)}=\int _0^{t-M_0}\frac{d\tau }{P_0+\dot{P}\,\tau } \nonumber\\ \ms &=& \frac{1}{\dot{P}}\ln \left(1+\dot{P}\,\frac{t-M_0}{P_0}\right), \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\displaystyle \theta (\vartheta,M_0,P_0,\dot{P})=M_0+\frac{P_0}{\dot{P}} (e^{\dot{P}\,\vartheta }-1), \end{eqnarray} \tag{ 6 }$$

where P₀ is the instantaneous period at the moment t = M₀ and $\dot{P}$ is the time derivative of the period. Since the decrease in period $\dot{P}$ is very small, the functions ϑ(t) and θ(ϑ) can be replaced with their Maclaurin series expansions as follows:

$$\begin{eqnarray} \vartheta (t)&=&\frac{1}{\dot{P}}\ln \left(1+\dot{P}\,\frac{t-M_0}{P_0}\right) \nonumber\\ \ms & \doteq & \frac{t-M_0}{P_0}-\frac{\dot{P}}{2}\left(\frac{t-M_0}{P_0}\right)^2+ \frac{\dot{P}^2}{3}\left(\frac{t-M_0}{P_0}\right)^3-\cdots, \nonumber\\ \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \displaystyle \theta (\vartheta)&=& M_0+{\frac{P_0}{\dot{P}}} (e^{\dot{P}\,\vartheta }-1) \nonumber\\ & \doteq & M_0+P_0\,\vartheta + {\frac{1}{2!}}\,P_0\,\dot{P}\,\vartheta ^2+ {\frac{1}{3!}}\,P_0\,\dot{P}^2\,\vartheta ^3+\cdots. \end{eqnarray} \tag{ 8 }$$

Since the last terms in Equations (7) and (8) do not exceed 1.2 × 10⁻⁹ and 3 × 10⁻¹⁰ days, respectively, for any of the used measurements, we can restrict ourselves to only two (Equation (7)) or three (Equation (8)) first terms of the expansion.

All free parameters including $M_0,\ P_0,\ \mathrm{and}\ \dot{P}$ were computed by a standard weighted nonlinear least-squares method (LSM) regression applied to all the observational material (see, e.g., Mikulášek et al. 2011a). Weights of individual measurements were inversely proportional to the squares of standard deviations σ of their subsets with respect to the chosen regression model (Equation (1); see also Table 2).

Table 2. Virtual Times of Primary Minima and (O − C) of BS Vul versus the New Ephemeris Derived from Measurements Used for Its Orbital Period Determination, Number of Observations, Scatter in mag, and the Source of Photometry

HJDI − 2400000	$\overline{({O-C})}$	Type	N	σ	Source
				(mag)
17707.134(6)	−0.0075	pg	18	0.15	Grigorieva (1938)
28506.0026(17)	0.0051	vis	155	0.16	Szafraniec (1962) I
31725.4690(29)	−0.0011	vis	45	0.21	Szafraniec (1962) II
43291.0937(3)	−0.0006	V	626	0.040	de Bernardi & Scaltriti (1979)
53313.1292(4)	0.0006	V	104	0.019	ASAS I
54004.2354(3)	−0.0018	V	517	0.036	Mas-Hesse et al. (2004)
54639.6567(5)	−0.0006	V	113	0.029	ASAS II
55005.6753(18)	−0.0029	V	93	0.017	AAVSO I
55128.00292(5)	0.00042	BVRI	2568	0.014	This paper I
55146.5670(2)	0.0017	V	86	0.013	AAVSO II
55380.7416(4)	−0.0009	V	122	0.034	AAVSO III
55508.77826(5)	−0.00025	BVRI	3730	0.015	This paper II

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After several tens of iterations we derived the following parameters: C = 1.256(14), d₁ = 0.06901(3), d₂ = 0.064(2), a₃ = 0.0832(9) mag, a₄ = −0.0113(7) mag, φ_s = 0.344(7), and d₄ = 0.10. Depths of minima in the BVRI passbands are given in Table 3. The wavelength dependence of primary and secondary minima depths shows that the secondary component is smaller and cooler than the primary.

The moment of the center of the basic minimum M₀ and the initial period P₀ were determined as follows: M₀ = 2455168.93592(4), P₀ = 0.47597002(3) days. With an ironclad certainty of 39σ we found a very steady shortening of the orbital period ${\dot{P}=-6.70(17)\,{\times}\, 10^{-11}=-0.00211(6)}$ s yr⁻¹ (see Figure 1), which corresponds with the epoch of the quick mass exchange phase. The times of primary minima can be predicted using

$$\begin{eqnarray} \mathrm{JD}_\mathrm{hel,I}(E) &=& M_0+P_0\times E+{\frac{1}{2}}P_0\,\dot{P}\times E^2 = 2\,455\,168.93592 \nonumber\\ && +0.47597002 \times E-1.594\cdot 10^{-11} \times E^2, \end{eqnarray} \tag{ 9 }$$

where E is an integer (epoch). Equation (9) predicts the primary minima with a formal precision of less than 1.5 minutes for the entire observed history of this star (from 1895 to 2011).

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The approach we applied did not use the timings of minimum light. Nevertheless, for better visualization and presentation of our results we could derive the so-called virtual times of minima and the corresponding virtual O − C values (similar to commonly used seasonal or normal values). Using the residuals of the observed data Δy_i = m_i − F(t_i) and phase derivatives of the model function in the moment of individual observations, we can create individual values of the phase shifts. They are expressed in days (O − C)_j with adapted individual weighting W_j for each observed date. An average of the phase shifts defined for arbitrarily selected groups of measurements $\overline{({O-C})}_k$ is then

$$\begin{eqnarray} && ({O-C})_j = -P(t_j)\,\Delta y_j \left(\frac{\partial F}{\partial \vartheta }\right)^{-1};\quad W_j=\left(\frac{\partial F}{\partial \vartheta }\right)^2 \nonumber\\ \ms &&\quad w_j;\quad \overline{({O-C})}_k=\frac{\sum _{j=1}^{n_k}\,({O-C})_j\, W_j}{\sum _{j=1}^{n_k}\,W_j}, \end{eqnarray} \tag{ 10 }$$

$\overline{({O-C})}_k$ derived for each of eight observational sets. $\overline{({O-C})}_k$ is defined as the mean shift of an observed phase curve (expressed in days) versus its model template. We assume that such introduced $\overline{({O-C})}_k$ are more reliable than standard (O − C) values based on times of eclipse minima only.

The majority of eclipsing binary (EB) period studies done in the past were based on the analysis of timings of mid-eclipses. Though advanced versions of our direct period analysis method also allow for the exploitation of the radial velocity measurements and EB eclipse timings, we decided not to use these data because they did not offer anything new in the BS Vul ephemeris determination. On the contrary, using the robustly determined ephemeris given by Equation (9) we can straightforwardly discuss the properties of particular types of EB minima timings and their usability for fine period analyses.

All the BS Vul minima timings known to us are collected in Table 4, where "plate" represents the observations from photographic plates, "vis" refers to visual observations, "pe" indicates photoelectric data, and "cc" denotes CCD observations. We divided the timings, derived by several methods, into three groups: (1) timings derived from photographic plates where BS Vul seemed apparently fainter than usually (30 timings); (2) timings of minima, derived as a rule by the Kwee-van Woerden method (1956), and applied to data series acquired by objective photometric measurements (photoelectric photometry, CCD photometry, or acquiring of photographic snaps series) in a time vicinity of the predicted minima (46 timings); and (3) timings of minima derived from series of several visual estimates of the brightness of the star in the vicinity of the predicted primary minima (104 timings).

The scatter of times in the first group (1) is rather large: σ(O − C) = 0.020 days, nevertheless the mean value of (O − C) is close to zero: $\overline{({O-C})}=-0.005(4)$ days. The second group of observations (2) display an excellent agreement with our quadratic ephemeris (see Equation (9)). We found that $\overline{({O-C})}=0.0005(3)$ days, the standard deviation of the scatter σ(O − C) = 0.0019 days. The standard deviation of visually determined minima timings (3) is of a moderate value: σ(O − C) = 0.0085 days. The corresponding weights of these three types of determination of minimum light should be in the ratio 1:110:5.6!

However, too credulously handling visually determined EB minima timings proved to be sometimes a rather hazardous business. Such data suffer from a lot of bad attributes that effectively limit their suitability for fine period analysis purposes. We can enumerate only the most serious ones.

• 1.  
  Because of the poor quality of visual observations themselves the corresponding accuracy of the eclipse time determination used to be much worse than in the case of other types of photometric observations. The quality of visual timings used to be further deteriorated by the fact that some determinations were based on only a few individual observations done only in the immediate vicinity of the predicted time of minima. This is typical for some observers and one should be careful when using such unreliable minima timings. See the numbers of estimates used for minima timings determinations in Table 4.
• 2.  
  The timings of minima of the time series of visual estimates often used to be determined by obscure, irreproducible methods. In the worst case, the original estimates were not published and they are now inaccessible.
• 3.  
  Almost all methods developed for the processing of astrophysical observations are based on an application of LSM. Unfortunately, the method strictly requires that subsequent estimates are independent. This requirement is not met in time series of visual observations. "Observed" visual light curves are subjectively smoothed to the ideal of theoretical or photoelectric light curves with clearly defined descending and ascending branches. This smoothing therefore prevents us from estimating the real uncertainty of the timings' determination from the time series of only one observing session. The uncertainties formally calculated and published from the smoothed light curve data are many times smaller than they should be.
• 4.  
  The most severe flaw in the majority of the visual observations is that observers obviously knew the predicted time of minimum light to an accuracy of minutes. If the ephemeris for the time of minimum light was incorrect, most observers were influenced into confirming the predicted, incorrect time of minimum light. This subjective effect, which is able to completely depreciate observations, equally afflicts both beginners and "experienced" visual observers.

The above general starting points are based on our long-standing experience with visual observations of EBs. Two of us (Z.M. and M.Z.) are now preparing an essential study on the reliability of visual EB timings and their usability for period analysis, where all the above-mentioned points of discussion will be studied and documented in detail for many assorted cases.

Analyzing 104 visual timings listed in Table 4 from 1970 to 2003 we found that they correlated very well with the linear ephemeris of de Bernardi & Scaltriti (1979) $\textit {JD}_\mathrm{I}=2443271.578+0.47597147\times E$, that was generally used by visual observers for their observation schedule (see Figure 2). The linear fit of visual observations is almost parallel with the line of prediction, while timings of minima systematically lag behind the minimum forecast by several minutes. Visual observations have only a very weak correlation with the real quadratic ephemeris. The conclusion is obviously valid for almost all visual observers who targeted the star. It is apparent that practically all visual observers subjectively adjusted their observations to more or less confirm the existing ephemeris.

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The apparent "delay" in visual observers' timings with respect to the prediction agrees well with the heliocentric corrections for the times of observation. Observers used geocentric time, but the predicted minimum light time was heliocentric time, which was delayed via geocentric time by several minutes on the night when observations were done.

The most active visual observer of BS Vul, G. Samolyk, also published on the AAVSO Web site 528 brightness estimates obtained during his observations of 39 primary eclipses of BS Vul in the time interval of 1984–2002. We analyzed all these data in detail and arrived, i.e., at the following conclusions.

• 1.  
  Observations of BS Vul were always optimally scheduled, typically 12 estimates per eclipse were able to describe the complete eclipse quite well. Both descending and ascending branches of the light curve during eclipses were covered nicely.
• 2.  
  The resulting light curves were perfectly symmetric and they corresponded very well with ideal photometrically acquired curves. Samolyk light curves were very smooth; their scatter derived from the form of the light curve is only 0.055 mag! However, the scatter of individual times of eclipses indicates that the real scatter in estimates should be at least five times larger!
• 3.  
  The scatter of timings published by Samolyk and their bad affinity to the real quadratic ephemeris might have been caused by the application of an inappropriate method for the derivation of light curve minima from estimate series. We apply our own rigorous method to determine the times of minima when we know the light curve shape (Mikulášek et al. 2011b). We obtained diverse times of minima. Nevertheless, all of them were clustered around the de Bernardi & Scaltriti (1979) ephemeris. Consequently, the estimates themselves were biased to accommodate the forecast.
• 4.  
  It is evident that our findings prove the heavily subjective character of visual observations of short-period EBs, which disqualifies them as a source of unbiased information apt for fine EB period analyses.

On the other hand, we could use 200 visual estimates of brightness made by S. Piotrowski in two time intervals (1935–1939 (155 estimates) and 1945–1946 (45 estimates)), because these estimates have a different character than later AAVSO and BBSAG visual observations. The distribution of these old visual estimates shows that they were obtained in "monitoring mode" without the primary aim to obtain minima timings. Furthermore, they passed our careful analysis.