Low Mass Contact Binaries: Orbital Stability at Extreme Low Mass Ratios.

W Ursa Majoris binary systems usually comprise of a low mass (spectral class F to K) primary with an even lower mass secondary. Due to their proximity the components are distorted, and both fill their Roche lobes leading to mass exchange between the components and the development of a common atmosphere. The systems are also referred to as contact binaries or over-contact systems. Contact binaries are quite common with estimates as high as 1 in 500 stars in the Galactic disk (Rucinski, 2006). The primary behaves like a main sequence star of the corresponding mass while the secondary has a larger radius and is more luminous than a matched main sequence counterpart due to expansion beyond its Roche lobe and the energy transfer from the primary through the common atmosphere. The presence of the common atmosphere results in almost identical photospheric temperatures for both components near the main sequence temperature of the primary component (Yildiz & Doğan, 2013). The median orbital period is near 0.35 days although it can range from about 0.2 to over 1 day (Latković et al., 2021).

Confirming that the transients termed red novae were the likely result of contact binary merger events (Tylenda et al., 2011) has led to greater interest in the study of orbital stability of such systems. Although there is only a single confirmed merger event, that of V1309 Sco (Tylenda et al., 2011), other galactic transients such as V4432 Sgr (Martini et al., 1999), V838 Mon (Brown et al., 2002) and OGLE2002-BLG-360 (Tylenda et al., 2013) also likely represent merger events; however, the pre-transient progenitors are as yet not confirmed contact binaries. Estimates of the galactic frequency of such events are as high as once every two to three years (Kochanek et al., 2014), although observable events may be less frequent. Theoretical developments over the past couple of decades have demonstrated that orbital instability leading to merger is critically dependent on the observed mass ratio between the components (Arbutina, 2007, 2009; Jiang et al., 2010b; Wadhwa et al., 2021; Arbutina & Wadhwa, 2024). The instability mass ratio itself is critically linked to the internal structure of the primary component and hence influenced by metallicity (Wadhwa et al., 2024) and potentially by the evolution of the system (Rasio & Shapiro, 1995; Li & Zhang, 2006; Jiang et al., 2010b).

The present work presents a comprehensive overview of the orbital stability of contact binaries taking full account of metallicity and main sequence evolution. The instability mass ratio is influenced by two variables: one physical and one structural. The mass ratio is essentially proportional to the structural parameter called the gyration radius ($k=\sqrt{I/MR^{2}})$, where $I$ is the moment of inertia, $M$ the mass and $R$ the radius) of a star and weakly influenced by the degree of contact or fillout factor($f$). We model the gyration radii of low mass stars from 0.6 $\rm M_{\odot}$ to 1.4 $\rm M_{\odot}$ taking into account rotation and binary tidal effects at different metallicities from zero-age main sequence (ZAMS) to terminal-age main sequence (TAMS). We apply them to the mass ratio instability equations from (Arbutina & Wadhwa, 2024) to derive the instability mass ratios for contact binaries with primaries ranging from 0.6 $\rm M_{\odot}$ to 1.4 $\rm M_{\odot}$ for various metallicities ranging from $-1.25\leq\rm[Fe/H]\leq 0.5$ and over a timescale from ZAMS to TAMS. The instability mass ratios were determined at marginal contact ($f=0$) and full over contact ($f=1$). The process yields an instability mass ratio range at various metallicities for the main sequence timescale for low mass contact binaries.

Finally, we reviewed the cataloged contact binary light curve solutions from Latković et al. (2021) and carried out literature search up to the end of March 2024 to compile a list of potential red nova progenitors meeting the revised criteria accounting for metallicity and age.

We base our analysis of orbital stability on the recent updated criteria by (Arbutina & Wadhwa, 2024). Full derivation of the instability mass ratio equations at marginal contact ($f=0$) and full over-contact ($f=1$) can be found in the cited reference. As suggested by Arbutina (2009) and modeled by (Wadhwa et al., 2024), at the onset of instability the mass of the secondary is well below the mass where a star becomes fully convective, and it is reasonable to adopt the $n=1.5$ polytrope value ($k_{2}=0.4527$) as the gyration radius value for the secondary component. Although initial investigators employed fixed values for the gyration radius of the primary, it is now clear that individually modeled values for the gyration radius of the primary have significant influence on the instability mass ratio (Wadhwa et al., 2021).

The updated instability mass ratio criteria from Arbutina & Wadhwa (2024) when combined with the fixed value of $k_{2}$ can be approximately reduced to two simple quadratic relations linking the gyration radius of the primary ($k_{1}$) to the instability mass ratio at marginal ($f=0$) and full over-contact ($f=1$) as follows:

and

The instability mass ratio moves in line with the gyration radius of the primary. The smaller the gyration radius the smaller the instability mass ratio. Internal structural constants have been modeled for some time and metallicity and age have a significant influence on the gyration radius given the evolving density and radius profiles (Claret, 2007). In the next section we will systematically explore the variation in the gyration radius of a solar mass star incorporating rotation, binary tidal distortions, metallicity and evolving age. We then look at effects of metallicity and age separately on the instability mass ratio and finally we examine the instability mass ratio with combined effects of metallicity and age. We extend our modeling to cover systems with primary components ranging from $0.6\rm M_{\odot}$ to $1.4\rm M_{\odot}$, for metallicities $\rm-1.25\leq\rm[Fe/H]\leq 0.5$ and ZAMS to TAMS ages.

It is widely accepted that the primary component of a low mass contact binary system behaves like a main sequence star of similar mass. It is, however, impossible to know how old the primary component is as pre-contact evolution of the components is unknown. If one accepts the proposed origin of contact binaries from semi-detached systems where the designation of the primary and secondary have switched due to mass transfer during the semi-detached stage (Stepien, 2006), then it is possible that the system as a whole could be quite old; however, the "higher mass" phase of the current primary may not be that old. To this end we propose an update for estimating the orbital stability of contact binary systems.

Firstly, the mass of the primary component needs to be determined as accurately as possible. The method of choice would be through radial velocity measurements; however, this is rarely possible. We suggest that methods that do not involve the use of the assigned temperature of the primary component be used. We say this because various calibrations for the temperature can yield very wide estimates (Wadhwa et al., 2023b) and even small variations in the assigned temperature can result in a significant difference in the estimate of the mass. Using simple black body calculations, a change in temperature of $250\rm K$ can result in a greater than 5% change in the mass estimate. We prefer direct observational methods, with accurately determined distances and extinctions or colour calibrations where the magnitude estimates in different bands have been acquired simultaneously (Wadhwa et al., 2023a). Secondly, the metallicity should be determined spectroscopically or potentially through the GAIA data release 3 (Anders et al., 2022). Note that the photometric GAIA metallicity needs to be corrected as described in (Wadhwa et al., 2023a). Lastly, we propose that the gyration radius of the primary should be estimated at a set age of half the TAMS age or 10 Gyr where half TAMS age exceeds 10 Gyr. This is a somewhat arbitrary set point; however, as stated above it is impossible to know the age of the primary in its current form and as such the selection, we feel, is a reasonable compromise.

Below we present a set of associations for determining the instability mass ratio range for contact binary systems with primaries ranging from $0.6\rm M_{\odot}$ to $1.4\rm M_{\odot}$ (in steps of 0.1 $\rm M_{\odot}$) and metallicities from $\rm-1.25\leq\rm[Fe/H]\leq 0.5$ in steps of $\rm[Fe/H]=0.25$.

In this section we review the catalog of contact binary solutions (648 systems) from Latković et al. (2021) which covers up to the end of 2020. In addition, we performed a literature search for published contact binary solutions from 2021 to the end March 2024. We accepted the authors estimate of the mass of the primary, regardless of the methodology. As our modeling is for systems from $0.6\rm M_{\odot}$ to $1.4\rm M_{\odot}$ we excluded all solutions where the estimated mass of the primary was outside this range. Similarly, if the metallicity was not available or not within our modeled range the system was excluded. As noted above, the instability mass ratio is higher with low mass high metallicity primaries and the maximum instability mass ratio for our select systems is 0.226 ($M_{1}=0.6\rm M_{\odot}$ and $f=1$) and as such all solutions with an estimated mass ratio above 0.226 were excluded. Based on the estimated mass of the primary, we next determined the systems potential maximum instability mass ratio, i.e at $f=1$ and $\rm[Fe/H]=0.5$. If the mass ratio was in excess of the maximum potential instability mass ratio the system was excluded. We then estimated the metallicity of the system based on either the corrected GAIA photometry estimate (Wadhwa et al., 2023a) or spectroscopic estimate from the seventh data release (DR7) of the LAMOST survey (Qian et al., 2020). Where more than one estimate was available from the LAMOST survey we selected the estimate based on the higher signal to noise ratio. Next, we determined the instability mass ratio range at the estimated metallicity as described above.

The recognition of V1309 Sco transient as a red nova and likely merger event of contact binary components combined with identification of literally hundreds of thousands of contact binaries from various surveys has ignited the field of contact binary light curve modeling and identification of red nova progenitors. Regardless of the number of new contact binary identified, V1309 Sco remains the only confirmed merger case. Recent theoretical and practical refinements (Wadhwa et al., 2021, 2022b) were envisaged to assist in the identification of potential red nova progenitors and many investigators (see citations in section 4.1) used the updated criteria with confident predictions of unstable systems.

The most critical parameters determining orbital stability are the mass ratio and the gyration radius of the primary. The fill-out (degree of contact) plays a minor but crucial role. Where the light curve displays a total eclipse the mass ratio and fill-out can be reliably obtained from the light curve solution; however, care must be taken to avoid local degenerated solutions and potential third light contributions should be adequately addressed. As the gyration radius is dependent on the internal structure of the star, it was adjusted in the earlier stages to match the observations rather than formally modeled (Rasio, 1995). When the gyration radius was modeled (Rucinski, 1988; Landin et al., 2009; Jiang et al., 2010b) the results were to some extent overlooked by researchers even though they pointed to a significantly different understanding to the prevailing view of a universal minimum mass ratio for all contact binaries. Wadhwa et al. (2021) refined the orbital stability criteria and using previously modeled gyration radii values for rotating and tidally distorted stars of different mass (at the ZAMS) and showed that there was no one universal instability mass ratio, in fact, each system has its own instability mass ratio.

As noted above the gyration radius is dependent on the internal structure of the star and all stars have different composition (metallicity) so it was clear that the role of metallicity on the instability mass ratio needed exploration. Some very preliminary work (Jiang et al., 2010a) indicated that the effect on the "minimum" mass ratio from changing metallicity is likely to be not insignificant. Wadhwa et al. (2024) comprehensively modeled the gyration radii of low mass contact binary systems for various metallicities from $\rm-1.25\leq\rm[Fe/H]\leq 0.5$ and showed that regardless of the mass of the primary component the instability mass ratio was lower for lower metallicities, i.e. metal poor systems were more likely to be stable. A review of the LAMOST spectra (Qian et al., 2020) suggests that over 70% of over 7000 surveyed contact binaries have metallicities below solar metallicity.

Internal structure of the star changes considerably even during main sequence evolution and as such it is likely the gyration radius is likely to change and in turn have an influence on orbital instability. Just as in the case of metallicity some earlier works indicated a potential profound effect especially in the later parts of the main sequence lifespan (Jiang et al., 2010b). In the present study, for the first time as far as we are aware, we model the gyration radii of rotating, tidally distorted stars at various metallicities and for the entire main sequence (ZAMS to TAMS) lifespan. We can confirm the previous finding that for any given mass the gyration radius is smaller at lower metallicity. In addition, we see that for stars with mass $>0.9\rm M_{\odot}$ there is a drop in the gyration radii from 1 Gyr onwards at all metallicities. For smaller mass stars the gyration radius is more stable at least up to 10 Gyr. Most (if not all) previous estimates of orbital stability have been based on the modeled gyration radius at the ZAMS (and solar metallicity) and the current study shows that for any given system the gyration radius is smaller than that at the ZAMS for systems older than 1 Gyr, except for the very low mass primaries where it is similar to the ZAMS value.

Our modelling assumes a fixed mass of the primary. This is unlikely to be the case due to constant mass exchange between the components resulting in constant changing in the gyration radius. This is more likely early in the evolution of contact binaries when mass exchange is thought to be more significant. If one considers the transfer of mass from the secondary to the primary, the increasing mass of the primary will continuously reduce gyration radius and thus the instability mass ratio, (for $M_{1}<1.4M_{\odot}$), however, the same transfer of mass to the primary will reduce the mass ratio. Other influences on orbital instability include potential non conservative mass transfer and/or mass loss due to the secondary overflowing its outer Roche lobe. Molnar et al. (2022) and Kobulnicky et al. (2022) have modeled contact binary evolution from formation to orbital instability considering mass transfer profiles from the secondary to the primary as well as the influence of component rotation. Their modelling suggests that instability is likely to occur at low mass ratios but only when the primary rapidly increases its moment of inertia which is most likely in the immediate post main sequence stage. They estimate the instability mass ratio in the range of 0.045 to 0.15.

The end result of the current modelling is that if the primary component of the system is of lower metallicity and we consider it as being halfway through its main sequence lifespan then it is likely to have an instability mass ratio considerably lower than what current practice suggests. For many systems where the mass of the primary is $\geq 1\rm M_{\odot}$ the instability mass ratio is so low that the mass of the secondary would fall below the threshold for hydrogen fusion, which is likely to result in considerable increase in energy transfer requirements to maintain thermal equilibrium. Applying a liberal inclusion criteria we find that of the eight systems that satisfy the revised criteria for instability all have methodological or astrophysical issues leading to reduced confidence with respect to their instability status. As noted above, there is only one confirmed merger event. The current and previous work have confirmed orbital instability is likely at low mass ratios with stability improving with reduced metallicity and potentially during the main sequence lifespan. Although hundreds of relatively bright contact binaries have been observed with accurate light curves and spectroscopic solutions the number of identified potential merger candidates is small. We suspect two possible explanations. Firstly, as noted above, contact binaries in the solar neighborhood are metal poor and hence more stable and secondly, we suspect an observational bias of imaging stable systems. Imaging multi-band light curves of contact binaries is a time-consuming exercise and as such only smaller size instruments are available, limiting the observation to relatively bright targets. Brighter targets are more likely to have heavier primaries which are inherently more stable. Latković et al. (2021) catalogued the light curve solutions of over 600 systems with over 400 having an estimate of the mass of the primary component. Less than 150 of these systems have an estimated mass of the primary component below one solar mass. Lower mass contact binaries, which have relatively higher instability mass ratios, are relatively faint and more difficult to identify from the survey photometry and difficult to observe with available instruments. Photometric surveys using large instruments such as the in development Extremely Large Telescope offer an avenue to identify low mass contact binaries.

The data underlying this article will be shared on reasonable request to the corresponding author.