On the Detectability of Post-common-envelope Binary Central Stars of Planetary Nebulae

Binary interactions are now the favored mechanism by which aspherical planetary nebulae (PNe) are thought to form (Jones & Boffin 2017, and references therein). More than 80% of PNe have been found to be aspherical (Jacoby et al. 2010), however, photometric surveys have shown that only some 20% of all PNe host a post-common-envelope (post-CE) binary central star (Miszalski et al. 2009; Jacoby et al. 2021). This raises an important question: is binarity really responsible for the shaping of all these aspherical PNe? And, if so, why have these systems gone undetected?

Some binaries will merge during the CE, perhaps still leading to an aspherical PN, but more likely some other form of astrophysical transient (Kamiński et al. 2018). Others will be so wide that they avoid the CE altogether. Another intriguing possibility to resolve the apparent shortfall is that there is a significant population of post-CE central stars which avoid detection due to a lack of high amplitude photometric variability (Sowicka et al. 2017). Double-degenerate post-CE central stars generally will not display high levels of photometric variability (e.g., Boffin et al. 2012), but are not expected to comprise such a significant fraction of all post-CE binaries. Instead, these "missing" post-CE binaries would need to be systems where the orbital period was too long, or the main-sequence companion too small for an appreciable irradiation effect to be observed (Jones & Boffin 2017). To investigate this hypothesis, we simulated a sample of post-CE binary light curves covering a range of orbital periods and companion spectral types, in order to probe which regions of this parameter space would be likely to evade detection.

Model light curves for the i-band (matching the most successful ground-based survey for binary central stars, OGLE; Miszalski et al. 2009) were simulated using the PHOEBE2 package. The parameters of the hot pre-white-dwarf primary were taken from the evolutionary tracks of Bertolami & Miguel (2016) for a typical central star mass (0.57 M_⊙) and age (7300 yr), while TMAP model atmospheres were used to represent its emergent spectrum (see example in Jones et al. 2022).

The properties of the main-sequence secondaries were varied to cover spectral types from K5 to L2 (Cifuentes et al. 2020). Unfortunately, the model atmospheres included in PHOEBE do not reach such low temperatures, so the emergent spectra of the secondaries were modeled as blackbodies with limb-darkening coefficients taken from the tables of Claret et al. (2013).

The orbital period (0.1 ≤ P ≤ 10 days) and inclination (0° ≤ i ≤ 90°) were also varied, resulting in a total of 5700 different combinations of model parameters.

Of the 5700 models, 93% ran successfully while the remaining 7% failed due to the periods of those models being so short that one or both of the stars was overflowing its Roche lobe. The derived amplitudes of the irradiation effect (i.e., excluding the additional variability due to eclipses) for the successful models are plotted in Figure 1.

From the models, it is clear that essentially all configurations are detectable at the very shortest periods (i.e., 0.1–0.4 day) except at the very lowest inclinations. Earlier type companions are, as expected, detectable at even longer periods. Mid-M-type companions are generally detectable out to periods of several days (beyond the typical orbital periods of the currently known sample of post-CE binary central stars; Jones & Boffin 2017). Earlier spectral types are detectable at even longer periods.

Some authors have previously suggested that there may be a "missing" population of long-period (a few days or more) post-CE binary central stars of PNe, which could have evaded detection by ground-based surveys. Preliminary modeling by De Marco et al. (2008) and Jones & Boffin (2017) offered some support for this hypothesis, but here we have undertaken a more complete exploration of the parameter space. Consistent with those previous studies, we find that essentially all binaries with secondaries earlier than M5 should be detected, as should binaries with later-type secondaries at periods of less than ∼0.3 day. As such, any post-CE binaries evading detection would need to be relatively low mass and/or long period. Population synthesis studies do predict an appreciable number of systems in this parameter space (e.g., Davis et al. 2010; Camacho et al. 2014), but certainly not enough to explain the difference between the 20% observed central binary fraction and the 80% asphericity fraction for PNe.

For binarity to be responsible for the shaping of all aspherical PNe there needs to be a significant contribution from wide stellar binaries (Jones et al. 2017), central stars with planetary mass companions (Boyle 2018) or CE mergers (De Marco et al. 2015). The former is seemingly the most likely given that the majority of main sequence binaries are too wide to experience a CE and that a large fraction of central stars show an infrared excess consistent with a companion at an undetermined separation (Douchin et al. 2015).

Software: PHOEBE2 (Prša et al. 2016; Conroy et al. 2020), TMAP (Rauch & Deetjen 2003; Werner et al. 2003; Reindl et al. 2016), Plotly (Plotly Technologies Inc., 2015).