ZZ Piscis Austrinus (ZZ PsA): A bright red nova progenitor and the instability mass ratio of contact binary stars

Our observations yielded a single time of primary minimum, determined to be HJD 2458776.9669 (0.0002) using the method of Kwee & van Woerden (1956). A new period estimate of 0.37405 (0.00003) days was also determined using the PSearch V1.5 (Nelson 2006) software package available through the AAVSO. No meaningful period change (O-C) analysis was possible given the lack of previous observations as noted above.

The normalized and fitted light curves for all bands are shown in Figures 1, 2 and 3. The light curves were folded according to the elements stated above. The maximum magnitudes at phase 0.25 and 0.75 in B were estimated as $9.83\pm 0.01$ and $9.84\pm 0.01$; in V $9.26\pm 0.01$ and $9.26\pm 0.01$ and in R $8.71\pm 0.01$ and $8.72\pm 0.01$. Surprisingly, given the low mass ratio (see below) no significant variation in the maxima (O’Connell effect, Milone, 1968) was evident in any of the bands observed. The eclipses are of essentially equal depth (variation < 0.02 mag) and last approximately 37 minutes. The $V$ band amplitude of the light curve is 0.29 mag which agrees with the coarse ASAS survey amplitude of 0.27 mag. Rucinski (2001) developed a theoretical relationship between maximum eclipse amplitude, fill-out factor and the mass ratio of contact binary systems. Using values from Table 2 of that study we deduce that the system is likely to have a mass ratio below 0.1.

The system shows complete eclipses and therefore it is suitable for photometric analysis with high confidence that the correct mass ratio can be determined (Terrell & Wilson, 2005). The light curve was analyzed using the 2009 version of the Wilson-Devinney (WD) code (Wilson & Devinney, 1971). As there is no appreciable O’Connell effect, only unspotted modelling was carried out. The SIMBAD database records the spectrum of the system as F6V (Wenger et al., 2000). Based on calibrations from Cox (2000) the effective temperature of the primary component was fixed at 6514 K. The near equal depth of the eclipses suggests similar temperatures of the components and a common convective envelope, as such we fixed the gravity darkening coefficients $g_{1}=g_{2}=0.32$ (Lucy, 1967), bolometric albedos $A_{1}=A_{2}=0.5$, and applied simple reflection treatment (Rucinski, 1969). Limb darkening co-coefficients for each passband were obtained from van Hamme (1993) and logarithmic law for the coefficients was implemented as per Nelson & Robb (2015).

No spectroscopic mass ratio is available, only the visual classification (F6V) by Houk (1982) appears in the literature. The complete eclipses however mean that the photometric mass ratio can be more reliably discovered and is likely to be close to the actual mass ratio (Terrell & Wilson, 2005). Rucinski (1993) showed that the brightness variations of the contact binary stars are essentially completely dominated by geometrical elements, with reflection and gravity/limb darkening playing a minimal role. He showed that the light curve depends practically on only three parameters; the mass ratio $(q)$, inclination $(i)$ and the degree of contact or fill-out $(f)$. Unfortunately, there is a high degree of correlation between these parameters. The correlation is most marked between mass ratio and inclination, and between mass ratio and degree of contact. Reliable photometric solutions cannot be obtained when either of these two combinations are treated as free. Russo & Sollazzo (1982) described the “$q$” search or “grid” method whereby solutions are obtained for various fixed values of $q$, with the potential (fill-out) and inclination free and then compared to the observed light curve to find the best solution. Wadhwa (2006) obtained a preliminary mass ratio of 0.08 based on the analysis of ASAS data.

Using the WDWint package V5.6 (Nelson 2009) we analyzed the light curves of all three passbands simultaneously over the mass ratio range 0.05 to 0.13 at intervals of 0.01. Preliminary solution was found at $q=0.08$. To further refine the mass ratio a finer search at intervals of 0.001 for the mass ratio was made over the interval $0.075<q<0.085$. Throughout the analysis the inclination, potential (fill-out), temperature of the secondary and relative value of $L_{1}$ were adjusted for each fixed value of the mass ratio. Iterations were carried out until the suggested variation was less than the reported standard deviation. The normalised residual error (as reported by the analysis software) between the observed and modelled light curve was plotted against the mass ratio. The mass ratio with the smallest residual error, i.e. the best fit between observed and modelled light curve, was selected as the true mass ratio of the system. The “$q$” search grid is shown in Figure 4 and the critical parameters from the WD solution are presented in Table 1. A 3D representation of the system is shown in Figure 5.

Our light curve solution, with an estimated fill-out of >95%, represents one of the highest recorded degrees of contact for a binary system and suggests probable near-merger. In addition to good physical contact, there is solid thermal contact between the systems. The secondary, although smaller than the primary, is hotter by less than 200 K and as such the system is similar to other low mass ratio systems with hotter secondaries such as V857 Her (Qian & Yang, 2005), ASAS J083241+2332.4 (Sriram et al., 2016), V1187 Her (Wadhwa, 2017), LO And (Nelson & Robb, 2015) and TY Pup (Sarotsakulchai et al., 2018).

Apart from the mass ratio, determination of the absolute magnitude $\text{M}_{\text{v}}$ of the primary is essential to accurately determine the absolute parameters of the system. Based on previously published colour-period and colour-period-$\text{M}_{\text{v}}$ relationships, Gazeas & Stȩpień (2008) derived the relationship $\text{M}_{\text{v}}=-8.4\log P+0.31$ (where $P$ is the period in days), giving $\text{M}_{\text{v}}=3.90$ for the ZZ PsA primary. However, the distance to ZZ PsA is known from parallax to be $135.48\pm 0.99$ pc (Gaia Collaboration, 2018). During the eclipse at phase 0.5 (lasting about 37 minutes), only the light from the primary contributes to the light curve at mid-eclipse. The apparent visual magnitude at mid-eclipse was estimated as $9.53\pm 0.01$, giving a distance-based absolute magnitude of $3.97\pm 0.02$ for the primary, after applying the extinction correction of $E(B-V)=0.034$ (Schlegel et al., 1998), as described by Wadhwa (2019). The distance-based results are used in all subsequent calculations.

From the primary $\text{M}_{\text{v}}$, we can find the primary luminosity. At maximum brightness (phase 0.25 or 0.75), the system has $V=9.26\pm 0.01$ which yields system absolute magnitude M${}_{V}=3.70\pm 0.02$, and hence the system luminosity, secondary luminosity and secondary absolute magnitude. Rovithis-Livaniou et al. (2001) found that the primary components of contact binary systems follow in general the zero-age main sequence (ZAMS) mass-luminosity relation. Using this relation (Demircan & Kahraman, 1991) for ZAMS stars with mass $>$ 0.7$M_{\sun}$, we can derive the mass of the primary; from the mass ratio we then estimate the mass of the secondary. The estimated $M_{\text{1}}$ of 1.213$M_{\sun}$ is consistent with the F6 spectral classification. The separation of the components $A$ is then estimated from Kepler’s third law, expressed as $(A/R_{\sun})^{3}=74.5(P/\mathrm{days})^{2}((M_{1}/M_{\sun})+(M_{2}/M_{\sun}))$.

As the two components are enclosed in a common envelope, their relative radii are not independent, but dependent on the Roche geometry and mass ratio. The WD program solution provides the fractional radii of the primary and secondary in three orientations. From the geometric mean we estimate the fractional radius of the primary $r_{1}=0.5958$, and that of the secondary $r_{2}=0.2341$. From ${R}_{1}/{R}_{\sun}=r_{1}A$ and ${R}_{2}/{R}_{\sun}=r_{2}A$ (Awadalla & Hanna, 2005), we derive the primary and secondary radii. The relative radius and luminosity of the secondary are considerably higher than those predicted for stars either at ZAMS or terminal-age main sequence (TAMS) stages of evolution. This is a common finding among contact binary systems (Stępień & Gazeas, 2012b). The absolute parameters for ZZ PsA are given in Table 2.

Our derivation of the instability mass ratio is similar to that of Arbutina (2007), expanded to more clearly show the contribution of the secondary component. We adopt the stability criterion $dJ_{T}=0$, assume that the gyration radii of the primary and secondary are not equal, and further assume that, as both stars are within a common envelope, their radii are not independent, but the radius of the secondary is dependent on the separation, mass ratio and the radius of the primary.

The orbital angular momentum of a binary can be written as

where $M_{1}$ and $M_{2}$ are masses of the primary and secondary component, respectively, total mass $M=M_{1}+M_{2}$, $\mu=M_{1}M_{2}/M$ and $q=M_{2}/M_{1}<1$. $\omega$ is the orbital angular velocity. With synchronization assumed, the spin angular momentum of a binary

where $R_{1}$ and $R_{2}$ are taken to be the volume radii. The fill-out factor is defined as

where $\Omega$ is the dimensionless surface equipotential. We have adopted linear dependence of $\Omega$ on the volume radius (Arbutina, 2007). Volume radii for the inner Roche lobe are (Eggleton, 1983):

and for the outer Roche lobe (Yakut & Eggleton, 2005):

where $i=1$ and $2$ refer to the primary and secondary star.

Since the surfaces of components in the contact system are at the same potential (the same $f$), from Eq. 3:

where

and

The total angular momentum of a contact binary system $J_{\mathrm{T}}=J_{\mathrm{O}}+J_{\mathrm{S}}$ can then be expressed as

From the condition $\frac{\mathrm{d}J_{\mathrm{T}}}{\mathrm{d}A}=0$, the critical separation ($A_{inst}$) is given by

If the angular momentum of the secondary has been neglected ($k_{2}=0$), the above condition for critical separation reduces to (Rasio, 1995)

By combining Eq. 3 for the primary

and Eq. 10, for a fixed fill-out factor ($0\leq f\leq 1$) one can also find the instability mass ratio given in the implicit form by an algebraic equation that needs to be solved numerically for $q\equiv q_{\mathrm{\scriptscriptstyle inst}}$:

Given the low mass ratio of contact binary systems approaching merger, the secondary is a very low mass star, for which the totally convective $n=1.5$ polytrope would be appropriate, with $k^{2}\approx 0.205$. The value is almost identical to the value obtained for all ZAMS stars below 0.4 M_☉ modelled by Landin et al. (2009). Our relationships are derived by adopting this fixed value of $k$ for the secondary. We use the values of $k$ for 0.5 M_☉ to 1.6 M_☉ from Landin et al. (2009) to derive the instability mass ratio ($q_{inst}$), for a range of fill-out fractions, of contact binary systems with primary component of spectral classes from M0 to F0. The results are summarised in Table 3. The fill-out fraction $f$ has a significant effect on $q_{inst}$ for systems with primary components below one solar mass. On the other hand, there is minimal difference in $q_{inst}$ between $f=1$ and $f=0$ for systems with primaries in the upper F spectral class. Figure 6 shows the relationship between primary mass and $q_{inst}$ at $f=1$ and at $f=0$. Both can be fitted with quadratic relationships (correlation coefficients >0.99):

and

Table 3 shows a linear relationship between $f$ and $q_{inst}$ for stars of different mass. One can therefore easily obtain the instability mass ratio for any given fill-out, after obtaining the values for $f=1$ and $f=0$ using the above quadratic relationships.

Using Eqs. 14 and 15, we estimate the instability mass ratio of ZZ PsA to be 0.0813. The photometric mass ratio is marginally below this level, suggesting that the system has entered an unstable period. From Eq 10 the theoretical instability separation can be expressed as a function of the mass ratio, primary radius and the gyration radii. Using the photometric mass ratio for ZZ PsA and Eq 12, we determine the theoretical fractional radius of the primary as 0.630 which at the current separation would yield a radius for the primary as 1.50 $R_{\sun}$. Using this estimate of the radius of the primary and 0.2432 as the value of $k$ from Landin et al. (2009) for a 1.2 $M_{\sun}$ star, from equation 10 we derive the theoretical instability separation as 2.487 $R_{\sun}$. Combining the theoretical instability separation with our component mass estimates, we would expect that the system reach its instability separation at a period of 0.397 days. ZZ PsA is well within the instability parameters on a number of fronts and represents a very bright potential red nova progenitor worthy of further study and regular monitoring.

Since the observed merger event of V1309 Sco, there has been considerable interest in the analysis of contact binary light curves. We examined the list of 46 deep low mass ratio contact systems from Yang & Qian (2015) and searched the literature for more recent examples of potential merger candidates. We used the reported values of the light curve solutions and mass of the primary to exclude all systems where the reported value of $q$ was more than 5% above $q_{inst}$ as calculated using the relationships above. For the remaining examples the instability separation and estimated period at onset of instability were compared to the current reported values. From an initial sample of more than 60 systems we were left with two other potential merger candidates and three others that were difficult to classify. We discuss these briefly.