2.1 S Ant

Spectroscopic data on S Ant were gathered over 5 nights during the period January 21–26 2021 with the High Efficiency and Resolution Canterbury University Large Échelle Spectrograph (HERCULES) of the Department of Physics and Astronomy, University of Canterbury Hearnshaw et al. Reference Hearnshaw, Barnes, Kershaw, Frost, Graham, Ritchie and Nankivell2002). This was attached to the 1m McLellan telescope at the Mt John Observatory (UCMJO). Images were collected with a 4 k $\times$ 4 k Spectral Instruments (SITe) camera (Skuljan Reference Skuljan, Kurtz and Pollard2004). The 100 $\mu$ m fibre, which is suited to typical seeing conditions at Mt John, enables a theoretical resolution of $\sim$ 40 000. Exposure times were usually $\sim$ 900 s for this 6th mag star. Raw observations were reduced with the latest version of the software hrsp (Skuljan Reference Skuljan2021), that produces wavelength calibrated and normalised output conveniently in ‘FITS’Footnote ^c formatted files. Each recording covers the wavelength range 4 520–6 810 Å. The log of observations is given in Table B1 and a descriptive summary of the observed spectra, giving line identifications and suitability for measurement, is presented as Table B3 (which is placed into the appendix so as not to disrupt the paper flow). Many metallic absorption lines are present in these spectra, though they are considerably widened by rotational effects and often blended. While the primary star’s lines are readily located it is often difficult to identify clearly those of the secondary that have low signal-to-noise ratios and may be relatively more affected by material motions in the source medium (Ruciński 2020).

We checked the spectral type designation using our measurements of the H $_{\beta}$ equivalent width. The value was 4.9 Å – which would place the spectral class a little cooler than F3V, according to Jaschek & Jaschek (Reference Jaschek and Jaschek1987), but within one type-unit. Measurement of the B–V colour (Section 3.1) agrees with this spectral type, after a small correction for interstellar reddening.

For two of the observations close to orbital phase 0.5 (w9236044s, w9240025s, see Table B1), the secondary component was totally eclipsed, allowing us to treat those spectral images as the primary component only. They could thus be directly modelled with synthetic spectra calculated directly with appropriate model atmosphere parameters. Fixing the surface gravitational acceleration of the primary component at log $g=3.98$ (cgs), and using mass, radius, and composition estimates anticipated for the primary component, synthetic spectra from Kurucz model atmospheres could be constructed for a grid of effective temperature and projected rotation values to compare with the observations. Preliminary findings imply a temperature of the primary component of 7 100 K and its projected equatorial surface rotational velocity of 150 km/s. The temperature of the primary component is in good agreement with its B–V colour (Section 3.1) and its H $_{\beta}$ equivalent width obtained above. The models are plotted on the observations in Fig. B1.

Sixteen spectra around the elongation phases of S Ant, from observations collected in clear weather, have been selected for RV analysis. The highly broadened lines did not completely separate, even at the greatest observed elongation.

As a check on procedure, the data of Duerbeck & Ruciński (2007) were analysed using the ELISa (Čokina, Fedurco, & Parimucha 2021) package for parametrisation, and a Markov Chain Monte Carlo (MCMC) technique for the curve-fitting process. ELISa includes both LC and RV analytical components. The ELISa search for an optimal parameter set depends on Markov Chain sampling of parameter hyperspace, while the probability estimation invokes an essentially random, ‘Monte Carlo’, distribution of observational errors.

With preliminary orbital parameters obtained by ELISa, we applied the spectral disentangling technique korel (Hadrava Reference Hadrava1995) to the observed spectra of S Ant. Spectra during eclipses were excluded, except for one mid-eclipse spectrum, where the Rossiter-McLoughlin effect (Rossiter Reference Rossiter1924; McLaughlin Reference McLaughlin1924) vanishes. The mean relative light contributions of the components were taken from the results of our initial LC analysis (next Section). A total of 8 spectral orders (93, 97–103) were processed with korel. The derived RVs from each order have been averaged to produce the final set. The final RVs are presented in Table 1 and displayed in Fig. 2. In Table 2 we present orbital parameters from the spectral disentangling procedure, where symbols have their usual meanings. Uncertainty estimates are produced as part of the output of the fitting program.

Table 1.

Radial velocity measurements of the components of S Ant from Korel applied to UCMJO data. The values in the $O-C$ columns represent the deviations of the individual measurements from the fitted RV curves. Orbital phases were calculated using the quadratic ephemeris given in Table 8.

Figure 2.

Radial velocity curves of S Ant from Korel applied to UCMJO data. Residuals to the RV model were plotted in the bottom figure. RVs of the primary and the secondary components are marked as filled and hollow symbols, respectively. Orbital phases were calculated using the quadratic ephemeris given in Table 8.

Table 2.

S Ant: Fitting parameters for RV data from UCMJO. $V_{\gamma}$ value of Korel solution has been adopted from individual RV measurements.

In Fig. 3, we display reconstructed mean spectra of the strong, measurable features in the $\lambda$ 552-559 nm region of the component stars of S Ant, together with residuals from the korel fits. It appears from the residual plots, that there is no significant additional component in these spectra. As well, profile fitting using the program prof (see, e.g. Oláh et al. Reference Oláh, Budding, Butler, Houdebine, Gimenez and Zeilik1992) enabled us to determine Doppler shifts of the relatively well-defined $\lambda$ 4824.13 Cr II line to be determined in a consistent way. In Table 2 we also present orbital parameters from the RV data obtained with the program prof.

Figure 3

. Disentangled spectra of the S Ant components (top), together with residuals (bottom). Numbers on the right side of the residual plot refer to the orbital phase.

The parameters listed in Table 2, apart from $K_1$ that derives from direct measurements, depend on constraints imposed by the physical circumstances. Thus, the mass function f (neglecting the eccentricity term and with the constant $C = 1.03615\times 10^{-7}$ ) yields:

(1) \begin{equation} f = CP K_1^3 = \frac{M_1 q^3 \sin^3 i}{(1+q)^2} .\end{equation}

This can be re-written as:

(2) \begin{equation} K_2 = \sin i \left( \frac{M_1}{CP} \right)^{1/3} ( 1 + q)^{-2/3},\end{equation}

where the right side becomes slowly varying for low q, with $M_1$ constrained by the spectral type. The result is that $K_2 \approx 235 \pm 5$ km s $^{-1}$ , for plausible values of q in the range 0.25–0.35 and primary RV amplitudes in the range 65–85 km s $^{-1}$ . Values of $K_2$ inferred from direct measurements are compatible with this but have a much greater probability of error.

2.2 $\varepsilon$ CrA

The spectrographic data for $\varepsilon$ CrA were obtained with essentially similar UCMJO arrangements as reported above for S Ant, except that the data for $\varepsilon$ CrA were obtained in August 2006. The log of observations is given in Table B2. See Table B4 (in the appendix) for a discussion of key features in the spectra.

We started the analysis by trial-fitting the radial velocities (RVs) of Ruciński (2020), adopting Ruciński’s period and applying again the ELISa program (Čokina, Fedurco, &Parimucha 2021). We found very similar results to those of Ruciński (2020), i.e. mass ratio $q = 0.128^{+0.005}_{-0.003}$ , $a \sin{i} = 3.54 \pm 0.04$ R $_{\odot}$ , and centre of mass velocity $V_{\gamma} = 61.88^{+0.48}_{-2.59}$ km s $^{-1}$ , giving confidence to the parameterisation procedures. The high systemic velocity (Fuchs &Wielen, Reference Fuchs and Wielen1987) suggests a great age to the binary, in keeping with other ideas about low mass-ratio A-class contact binaries.

As in the case of S Ant, the secondary component of $\varepsilon$ CrA was totally eclipsed at phase 0.5. For two of the observations close to orbital phase 0.5 (w3966049s, w3966050s, see Table B2), we modelled with the observed spectra with synthetic spectra calculated directly with appropriate model atmosphere parameters. This time the surface gravitational acceleration of the primary component was fixed at log g=4.05 (cgs), and using mass, radius and composition estimates anticipated for the primary component, synthetic spectra from Kurucz model atmospheres could be constructed for a grid of effective temperature and projected rotation values to compare with the observations. Preliminary findings are of rather low $T_\mathrm{eff}$ and high $V_\mathrm{rot}$ sini values and we defer further discussion of this work in the present paper. Concerning high projected equatorial rotation velocities, however, this may align with the discussion of Ruciński (2020) on the complex velocity field in $\varepsilon$ CrA associated with the effects of mass transfer. The synthetic spectra plotted over the observed spectra are shown in Fig. B2.

Preliminary mean relative light contributions of the components are essential for the proper decomposition of the component spectra from the composites in the source data (Hadrava Reference Hadrava1995, Reference Hadrava1997). The LC and RV analysis is thus, in principle, an iterative procedure, although this would seldom involve more than one parametrisation cycle.

The korel program, used in our study, applies Fourier transforms for spectral decomposition. Spectra obtained during eclipses (except the totalities) were not used for this, as complications, such as the Rossiter-McLaughlin effect (Rossiter Reference Rossiter1924; McLaughlin Reference McLaughlin1924), come into play. The korel code does not address such contingencies. Our RV measurements of the components of $\varepsilon$ CrA from Korel are presented in Table 3 and displayed in Fig. 4.

Table 3.

Radial velocity measurements of the components of $\varepsilon$ CrA from Korel applied to UCMJO data. The values in the $O-C$ columns represent the deviations of the individual measurements from the fitted RV curves. Orbital phases were calculated using the linear ephemeris given in Table 3.

Figure 4.

Radial velocity curves of $\varepsilon$ CrA from Korel applied to UCMJO data. Residuals to the RV model were plotted in the bottom figure. RVs of the primary and the secondary components are marked as filled and hollow symbols, respectively. Orbital phases were calculated using the linear ephemeris given in Table 4.

Figure 5.

Disentangled spectra of the $\varepsilon$ CrA components (top), together with residuals (bottom). Numbers on the right side of the O–C plot refer to the orbital phase.

In this way, korel led to the spectroscopic orbital parameters given in Table 4, where symbols have their usual meanings and the uncertainties of parameters are calculated using a simplex procedure. Our velocity semi-amplitudes for the components turned out to be a few km/s higher than those listed in Ruciński (2020), the mass ratio retaining a similar value ( $q = 0.133$ , see Table 4). Reconstructed primary and secondary spectra from order 103 ( $\Delta\lambda\sim$ 5 000–5 550 Å) are shown in the top panel of Fig. 5, where the lower panel shows the O–C residuals of the spectrum fittings. Order 103 shows relatively strong Th I and Fe I features at $\lambda$ 5528 and $\lambda$ 5535, respectively.

Table 4.

Comparison of korel orbital parameters with those of Ruciński (2020) for $\varepsilon$ CrA.

3.1 S Ant photometry

BVI LCs of S Ant were collected by MB at the Congarinni Observatory (NSW 2447) using an 80 mm f6 refractor, stopped down to 50 mm to avoid saturation effects, and an Atik One 6.0 CCD camera equipped with Johnson-Cousins photometric filters. MaxIm DLTM software was used for image calibration and aperture photometry. An ensemble of 3 comparison stars was used (TYC 6613 678, TYC 6613 1556, and TYC 6613 130). The adopted values for the reference were V = 6.725, B–V = 0.200 and V–I = 0.139. From this, we determined for S Ant: V = 6.295, B–V = 0.326, V–I = 0.196; V = 6.304, B–V - 0.326, V–I = 0.197 at the first and second maxima respectively. At the secondary occultation minimum, we found: V = 6.746, B–V = 0.325, V–I = 0.198. These measures are in keeping with the literature values cited in Section 1.1, but the colours appear a little more blue than typical Main Sequence values of type F3 (Eker et al. Reference Eker2018).

The LCs are presented in Fig. 6, together with the modelling results shown as smooth curves. We applied the optimal curve fitting program WinFitter (WF, Rhodes Reference Rhodes2023) for the LC modelling. The resulting parameter estimates are collected in Table 5.

Table 5.

Optimal parameters for the WinFitter photometric model fits to the ground-based BVI and TESS Sector 9 data for S Ant. The conservative uncertainty estimates reflect the spread of numerous curve-fitting results, rather than the formal errors of particular fits.

LC modelling for close binary stars often refers to the numerical integration procedure of Wilson & Devinney (Reference Wilson and Devinney1971) (hereafter referred to as WD), which represents the distorted component surfaces as equipotentials, according to the classical point-mass formulation attributed to Roche (Reference Roche1873) recalled in Ch. 3 of Kopal (Reference Kopal1959). Both the WD and WF methods converge to the same approximation for the surface perturbation when the internal structural constants $k_j$ are neglected, implying neglect also of the effects of tides on tides. The relevant formula, Eqn 1-11 in Ch.1, or 2-6 in Ch. 3 of Kopal (Reference Kopal1959), is:

(3) \begin{equation} \frac{ \Delta{\prime} r }{r_0} = q \sum^{4}_{j = 2} r_0^{j+1} (1 + 2k_j) P_j(\lambda) + n r_o^3(1 - \nu^2), \end{equation}

where r is the local stellar radius expressed as a fraction of the orbital separation of the components with mean value $r_0$ . The direction cosine of the angle between the radius vector $\hat{r}$ and the line of centres is here $\lambda$ , and $\nu$ is the direction cosine of the angle between $\hat{r}$ and the rotation axis. The coefficients $k_j$ (in WF) can be taken from suitable stellar models, e.g. Inlek, Budding, & Demircan (Reference Inlek, Budding and Demircan2017). They are set to zero in WD.

The surface perturbations ${\Delta}{\prime} r$ are here serial harmonic functions, that start with terms of order $r^3$ . The inclusion of terms of order $r^6$ and higher would imply the gravitational interaction of tides on tides being taken into account for self-consistency. Parallelism of WD and WF fitting functions, in which the mutual interactions of the perturbations are neglected, therefore continues through to terms of order $r^5$ , implying three additive tidal terms. Rotation with constant, synchronised, angular velocity, as usually assumed, and required in the Roche approximation, calls for only one source term.

Figure 6.

Ground-based BVI and TESS photometry of S Ant with the optimal model from WinFitter fitting. Residuals to the model are plotted in the lower figure.

We can see from Table 5, and the contact criterion $r_1 + r_2 \approx 0.75$ (Kopal Reference Kopal1959; Table 3.3), that the components of S Ant cannot be far from the idealised configuration of Roche lobes osculating at the L $_1$ point. The mean radii approximate to values corresponding to the mass ratio $q \sim 0.3$ (see Table 3-1 of Kopal Reference Kopal1959). This prior value for q tallies with a consensus of previous literature estimates and our own spectroscopic RVs discussed in Section 2.1. The WF posteriors are thus consistent with a near-Roche configuration.

We see directly from Fig. 6 and Table 5 that all four LCs show roughly the same ratio for the depths of the two minima ( $\sim$ 0.8). The lack of variation in relative depths with colour implies the two stars must have close photospheric temperatures. The LCs, by indicating a flatter central region of the secondary eclipse, point to S Ant being of the ‘A-type’ Binnendijk (Reference Binnendijk1970), where the larger, slightly brighter, star eclipses its companion at the secondary minimum. In keeping with this lack of colour variation, the ratio of $L_1/L_2$ is $\sim$ 4 in the BVI wavebands. This result aligns with a coarse estimate for a ratio of radii $\sim$ 0.6, in fair accord with the mass ratio, according to Kuiper (Reference Kuiper1941)’s approximation $r_2/r_1 \sim q^{0.46}$ . In turn, this points to similar effective surface temperatures.

This preliminary assessment thus raises the classical question for contact binaries (Kuiper Reference Kuiper1941): why should two stars of such differing masses show closely similar surface temperatures? Note that this point applies to the EW systems as a whole, and it bespeaks some special conditions that apply to the group generally.

3.2 S Ant: O’Connell effect

The asymmetry in the LCs in Fig. 6 – in this case, the first maximum being higher than the second – is easily seen. The TESS LCs (Sector 9) confirm the ground-based data on this point. This is an example of the well-known O’Connell effect (Roberts Reference Roberts1906; O’Connell Reference O’Connell1951; Milone Reference Milone1968). In what follows, we interpret this in terms of a locally heated region on the primary photosphere. This idea can be reconciled with other evidence. Before following up on this, however, we should note that LC asymmetry is hardly present in classical photometric models. An initial LC fitting using a standard model can thus be expected to yield only approximate parameters. These may be improved in an iterative procedure if our interpretation of the spot’s effects is adequate.

Fig. 7 shows the fitting of a ‘hot spot’ to the residuals from the standard model match, using the fitspot application of the WinFitter suite. fitspot requires the specification of a circular spot’s longitude $\lambda$ , latitude $\beta$ , angular size $\gamma$ , inclination of the spin axis i, flux calibration unit U, the spot-bearing star’s luminosity as a fraction of that of the whole system $L_1$ , spot central relative flux, $\kappa$ and the limb-darkening coefficient $u_1$ . The appearance of Fig. 7 – i.e. the data scatter and simple form of the fitting function – indicate the information content, or number of independently specifiable parameters, to be low. But we can reasonably adopt $\beta = 0$ , $\gamma = 0.175$ ( $\sim$ 10 deg), $i = 1.29$ (74 deg), $L_1 = 0.85$ , $u_1 = 0.5$ . The remaining 3 parameters ( $U, \lambda, \kappa $ ) were adjusted so as to minimise $\chi^2$ .

Figure 7.

Hot spot model for O’Connell effect in S Ant LC.

The flux excess of about 1.3 in the hot spot, in terms of the Planck function, corresponds to a temperature excess of about 500 K at 7 000 K. In the IR, at a wavelength of about 1 000 nm, this ratio drops to about 1.17. We may therefore accept, referring to Table 6, that the O’Connell effect in these LCs of S Ant can be accounted for by a region of about 1.5% of the primary’s photospheric area heated to a temperature excess over the surrounding photosphere of several hundred kelvins. The implied energy release is $\sim$ $10^{32}$ erg s $^{-1}$ .

Table 6.

Optimal hot spot parameters for matching the O’Connell effect on S Ant.

This heating effect may be associated with the impact of a mass transferring stream, consistent with the general scenario of Roche Lobe overflow including the Coriolis deflection (Lubow & Shu Reference Lubow and Shu1975), its role in the semi-detached phase of the thermal relaxation oscillator (TRO) model (Lucy Reference Lucy1976), or the Algol-like version of this condition (Stepien Reference Stepien2009). In order to substantiate this picture, let us write the continuity of energy flow in the form

(4) \begin{equation} L_\mathrm{spot} \approx\frac{1}{2} \dot{M} v^2.\end{equation}

In the following section, we will derive a value for $\dot{M}$ for the mass-losing star of at least $\sim$ 10 $^{18}$ gm s $^{-1}$ . The impacting kinetic term $v^2/2$ , per unit mass, in Equation (4) comes from the drop in potential energy in moving from the region of the inner Lagrangian point to the impact location. This can be written, approximately, as $\sim \Delta \left( GM_1/r \right)$ , where $M_1$ is the mass of the present primary, and r is the radial separation from the primary centre of mass, relating to the stream’s origin and impact positions. Equation (4) will balance the luminosity of the spot with the dissipation of the stream’s kinetic energy in the impact region with a value of the infall distance $\Delta r$ of not more than $\sim$ 10 % of the primary mean radius. In this way, the O’Connell effect is explained consistently with the apparent secular change of period that will be discussed in section 4.1.

3.3 $\varepsilon$ CrA photometry

BVI photometry of $\varepsilon$ CrA was carried out at the Congarinni Observatory over 8 nights of reasonably stable weather in July and August 2021 using a 200 mm f2.8 Canon lens on an SBIG STT3200 CCD camera. The field of view was $4.38 \times 2.95$ deg, which enables the inclusion of a number of bright comparison stars. For data reduction, we used HIP 93825 ( $V=4.209$ , $B-V=0.522$ , $V-I=0.417$ ) as the main comparison star. Thence we determined for $\varepsilon$ CrA: $V = 4.766$ , $B-V = 0.387$ , $V-I = 0.260$ ; $V = 4.769$ , $B-V - 0.376$ , $V-I = 0.290$ ; at the first and second maxima, respectively. At the occultation minimum, we found: $V = 4.966$ , $B-V = 0.403$ , $V-I = 0.247$ . These measures are not far from the literature values cited in Section 1.2, but the colours are a little more blue than the Main Sequence values of type F4 given by Eker et al. (Reference Eker2018).

Photometric data from the Transiting Exoplanet Survey Satellite (TESS) was also used. TESS collected data on $\varepsilon$ CrA with a 2-min cadence in Sector 13 (June 20 to July 17, 2019). These data are publicly accessible (Ricker et al. Reference Ricker2015). We downloaded and processed relevant information using the Lightkurve Python package (Lightkurve Collaboration et al. Reference Collaboration2018).

Photometric analysis was pursued using the numerical integration code of Wilson & Devinney (Reference Wilson and Devinney1971) (WD), combined with a Monte Carlo (MC) optimisation procedure (see Zola et al. Reference Zola2004). This method models the LC of an eclipsing binary star, including proximity effects, adopting Roche equipotentials (Kopal Reference Kopal1959) for the component surfaces. The most important feature of the MC search procedure is its exploration of a grid of trial values for the input parameters (priors). The method thus finds the model that provides a best fit to the observed LC by conducting up to hundreds of thousands of iterations, depending on the number of free parameters, with priors distributed over the selected input ranges, determined from preliminary expectations of component values and their expected uncertainties.

The BVI LCs of $\varepsilon$ CrA show a slight O’Connell effect in the B band, but the discontinuity in the residuals near the first maximum disturbs the V LC proportionately more than the B. The asymmetry is more apparent in the I data, but the discontinuities seen in the residuals (Fig. 8, lower panel) indicate that the asymmetries should be regarded cautiously. Note that there is very little asymmetry between the maxima in the TESS LCs.

Figure 8.

Ground-based BVI and TESS photometry of $\varepsilon$ CrA with the optimal model from WD+MC (Wilson-Devinney plus Monte Carlo) fitting. Residuals to the model are plotted in the lower figures.

Based on the previously cited literature we adopted the spectral type of $\varepsilon$ CrA as F2V. Then, using the calibration data of both Pecaut & Mamajek (Reference Pecaut and Mamajek2013) and Eker et al. (Reference Eker2018), we adopted the primary temperature as 6 820 $\pm200$ K. The temperature of the secondary ( $T_2$ ) was adjustable on the range 5 000–7 000 K. The input range of the orbital inclination was set to $50^\circ \lt i \lt 90^\circ$ , The mass ratio (q) was fixed at the value of 0.13 obtained in the RV analysis in Section 2.2. The non-dimensional surface potentials were allowed to vary on the range 1.0–3.0.

Table 7.

Optimal parameters for the WD+MC photometric model fits to the ground-based BVI and TESS Sector 13 data for $\varepsilon$ CrA. According to the WD procedure for overcontact binaries, such as W UMa stars, these constraints were applied: the surface potential ( $\Omega$ ), gravity brightening (g), bolometric albedo (A) and limb darkening parameters of components are the same. Here $g_1=g_2=0.32$ and $A_1=A_2=0.5$ are adopted. A quadratic limb-darkening law was applied with limb-darkening coefficients (x, y) taken from Claret (2017).

Table 8.

Parameters derived from O – C analysis of S Ant.

For the WD+MC program, the input range of $30^\circ \lt \beta \lt -30^\circ$ was set for the spot latitude and $0^\circ \lt \lambda \lt 360^\circ$ for the longitude. The input range of $10^\circ \lt \gamma \lt 50^\circ$ was set for the angular radius and $0.80 \lt \kappa \lt 1.3$ for the temperature factor. Two approaches were used in dealing with the asymmetric LCs. In the first approach, the separate bands and nights were treated on an individual basis, but this did not produce a uniform representation. A second approach adopted the TESS data as definitive, the asymmetry being very small. Two main spots were assumed to account for the B and I band asymmetries, while spots were not used in matching the almost symmetric V LC. In these fittings, the geometric parameters were kept constant at the values from the TESS data analysis, while other parameters were adjusted to optimise the fitting.

Results are presented graphically in Fig. 8. A very small amplitude variation ( $\sim$ 0.003 in relative flux) remains in the TESS LC residuals. Such residual effects notwithstanding, it is the TESS LC fitting results that are adopted in the absolute parameter calculations in (Section 5). The uncertainties given in Table 7, that was subsequently used for calculating the formal error estimates of the absolute parameters in Table 10, were obtained from the WD+MC code set with a 90% confidence level.

Table 9.

Parameters derived from O – C analysis of $\varepsilon$ CrA.

Table 10.

Absolute parameters of S Ant and $\varepsilon$ CrA.

Reasons for the asymmetry in the LCs include either hot spots that may relate to mass transfer or cool maculation. The latter option was pursued for $\varepsilon$ CrA on an empirical basis, and the results are presented in Table 7.

4.1 S Ant

Times of minimum light (ToMs) have been compiled from various sources, especially the O – C Atlas data of Kreiner, Kim, & Nha (Reference Kreiner, Kim and Nha2001) and Kreiner (Reference Kreiner2004). Observed ToMs are compared with calculated ones on the basis of an adopted reference ephemeris that is linear in the ToM number or epoch E. The corresponding time differences T are used to form the familiar ‘O – C’ diagram (Fig. 9). For additional background see Ch. 8 of Budding & Demircan (Reference Budding and Demircan2022). The ToMs used in Fig. 8 are listed in Table A1 in the Appendix. O–C data are matched with a trial function in E to check for systematic departures from the adopted linear ephemeris. A parabola was fitted to the residuals for S Ant using Excel $^\mathrm{TM}$ . The results are given in Table 8, and displayed in Fig. 9.

Figure 9.

Observed minus calculated (O – C) times of minimum light (ToMs) for S Ant, and parabolic model fitting. Black points represent primary minima; blue open circles represent secondary minima. The sizes of the symbols are proportional to their weights. Residuals are shown in the lower panel.

It can be shown that a steady transfer of mass between the stars in a close binary system will produce such a parabolic trend i.e.

(5) \begin{equation} T(E) = A + BE + C E^2 .\end{equation}

The corresponding rate of mass transfer takes the form:

(6) \begin{equation} \dot{M_1} = 243.3 \frac{M_1 C}{g(x) P^2} , \end{equation}

where $M_1$ is the mass of the donor star, P the orbital period. The function g relates to the system’s angular momentum and depends on the relative mass $ x = M_1/(M_1 + M_2)$ . If we conserve the orbital angular momentum and neglect the contribution of the relatively small rotational terms, the function $g(x) = (2x - 1)/(1-x)$ .

The resulting value of $\dot{M_1}$ thus remains negative for an upturned parabola ( $C \gt 0$ ) with $x \lt 1/2$ – see Fig. 9, upper panel. The residuals, after including the parabolic trend, are shown in the lower panel of Fig. 9, No further matching of residuals was attempted, and the parabola accounting for systematic effects to the available accuracy of the data. Following the foregoing prescription, our estimate for the rate of mass loss, in solar masses per year, is given in Table 8.

4.2 $\varepsilon$ CrA

The orbital period of $\varepsilon$ CrA was examined along the same lines as S Ant. In this study, 4 primary and 4 secondary ToMs were obtained from the TESS data, while 3 primary and 3 secondary ToMs were obtained from the previously discussed BVI observations. These times of minima were combined with 23 primary and 8 secondary ToMs taken from Kreiner (Reference Kreiner2004). All ToMs used in our analysis are given in Table A2 in the Appendix.

The O – C data using the linear ephemeris from Kreiner (Reference Kreiner2004) are shown in Fig. 10, which shows a relatively strong parabolic component. The corresponding form in Equation (5) was fitted to our data by the least squares method. Results are given in Table 9. The residuals from this fitting, shown in the lower panel, suggest a further, low-amplitude systematic effect.

Figure 10.

LTE representation superimposed on parabolic form of O – C changes of $\varepsilon$ CrA (upper figure). Only LTE representation of O – C changes after subtraction from the fitted parabola’s residuals (lower figure).

This quasi-sinusoidal change in the O – C diagram is suggestive of a light-time effect (LTE) associated with a third body. In order to look into this, the first set of residuals was checked against the LTE formula given by Irwin (Reference Irwin1959):

(7) \begin{multline} T(E) = A + BE + C E^2 \\ + \frac{a_{12}\sin{i_3}}{c} \left( \frac{1-e_3^2}{1+e_3\cos{\nu_3}}\sin(\nu_{3} + \omega_{3}) + e_{3}\cos{\omega_3} \right),\end{multline}

where c is the speed of light, and the other symbols have their usual meanings, as given by Irwin (Reference Irwin1959). A weighted least-squares solution for $T_0$ , $P_{orb}$ , Q, $a_{12}$ , $i_3$ , $e_3$ , $\omega_3$ , $T_3$ and $P_{12}$ is presented in Table 9. The observational points and theoretical best-fit curve, and also the residuals, are plotted against epoch number and observation years in the lower part of Fig. 10.

The low value of the projected separation of the close binary from the centre of mass ( $a_{12} \sin i $ ) compared with the period and the known mass of the close binary stars indicate that the wide orbit axis is relatively close to the line of sight for third body masses comparable to that of the close pair. With a separation of order an arcsecond, such a companion star (that has to be more massive than $\sim$ 0.4 M $_{\odot}$ ) should be detectable astrometrically.

Alternatively, vagaries from the parabolic trend in the upper part of Fig. 10 may be reflecting short-term changes in the O’Connell asymmetry, that may have physical implications on flow or accretion irregularities as well as discrepancies in the timings of light minima (see lower sub-diagram in Fig. 10). Disparities in implied mass transfer rates associated with contact binaries are not uncommon (Qian Reference Qian2001).