The non-explosive stellar merging origin of the ultra-massive carbon-rich white dwarfs

White dwarfs are the final stage of stellar evolution for the majority of stars and about a quarter of white dwarfs are found in binary systems (Hollands et al., 2018). However, about half of intermediate mass main-sequence stars (the progenitors of the white dwarfs) are in binaries (Ferrario, 2012; Duchêne & Kraus, 2013). This discrepancy provides evidence that some white dwarfs formed in stellar mergers (Briggs et al., 2015; Toonen et al., 2017).

The merger of double degenerates (DDs) provides one path toward Type Ia supernova explosions. The study of white dwarfs that lead to such events, but failed to explode, yield important constraints to the modelling of the mechanisms leading to a thermonuclear runaway (Ruiter, 2020). The expectation is that white dwarfs formed via DD mergers are more massive than white dwarfs descending from single stars and this was indeed shown to be the case by the population synthesis calculations of Briggs et al. (2015, 2018). Temmink et al. (2020) further investigated the impact of binary evolution on apparently single white dwarfs and found that about a quarter of them should have formed in a stellar merger, but that only about 3 per cent formed from a DD merger, which is consistent with the findings of Briggs et al. (2015). Schwab (2021) calculated evolutionary models for these white dwarf merger remnants and showed that they should be massive and fast rotating. Overall, the products of DD mergers should be rare.

The entire sub-class of hot, carbon-rich white dwarfs are candidate merger products (Dunlap & Clemens, 2015). White dwarfs with carbon lines or molecular carbon bands are generally classified as DQs (Dufour et al., 2005). The presence of carbon in most of these stars can be explained by the dredge-up of core carbon by a deep helium convection zone (Pelletier et al., 1986; Fontaine & Brassard, 2005). This model explains the observed increase in the carbon abundance with increasing temperatures up to $T_{\rm eff}\lesssim 10\,000$ K but fails to explain the surge in carbon observed in hotter DQ white dwarfs (Dufour et al., 2007, 2008b; Coutu et al., 2019). These hot DQ white dwarfs display neutral and ionized carbon line spectra in a carbon dominated atmosphere at effective temperatures ranging from $\sim 18\,000$ to $25\,000$ K (Dufour et al., 2008b). Cooler objects display a blend of neutral carbon lines and molecular carbon bands at temperature ranging from $\sim 10\,000$ to $16\,000$ K (Coutu et al., 2019) in a mixed carbon/helium atmosphere. Using Gaia parallaxes, Coutu et al. (2019) suggested that these hot DQs should also be very massive ($\gtrsim 0.8$ M_⊙). Furthermore, Dunlap & Clemens (2015) showed that these hot DQs have high transverse velocities. This means that despite their high temperatures and high masses, characteristic of a young white dwarf population, they would in fact belong to an old one. These properties, along with the presence of a magnetic field and high rotation rate (Dunlap & Clemens, 2015), strongly suggest that these objects formed through the merger of two white dwarfs.

This population of massive carbon-polluted merger products should extend below $\sim 10\,000$ K and could be distinguished from canonical DQ white dwarfs by having a higher carbon abundance at a given temperature. Therefore, we are dealing with an unusual white dwarf population that has a different origin from the general white dwarf population. To avoid confusion, we shall refer to the combined population of hot DQ white dwarfs and their cooler counterparts as “C-enriched DQs”.

Strong independent evidence in support of the merger hypothesis for the origin of the C-enriched DQs is provided by a kinematical study of the DQ white dwarf LP 93-21. Kawka et al. (2020) showed that the total age of LP 93-21, assuming single star evolution, is too short and inconsistent with its Galactic kinematics which place LP 93-21 in the halo hence bolstering a merger origin for the star.

In this paper we extend the kinematical approach to the now much larger sample of massive DQ white dwarfs and we investigate, through a population synthesis study, whether the stellar merger hypothesis for the origin of the C-enriched DQs is consistent with observed population properties (age, mass).

We have gathered all known DQ white dwarfs from several sources including those from the recent studies by Coutu et al. (2019), Blouin & Dufour (2019) and Koester & Kepler (2019) and we have adopted the properties (effective temperature, surface gravity, mass and carbon abundance) from these sources. For stars that have not been analysed using Gaia measurements, we used available spectroscopy, photometric measurements and Gaia distance measurements to determine their properties. If the effective temperature and carbon abundance are known, we combined this information with the Gaia parallax and photometry and our model photometry to estimate the surface gravity and mass. We used the evolutionary mass-radius relations for helium atmospheres of Benvenuto & Althaus (1999) combined with our model DQ spectra (Kawka et al., 2020) to calculate absolute magnitudes.

From this sample we extracted white dwarfs with masses greater than 0.8 M_⊙ that we consider representative of potential merger products. We checked all massive candidates for their carbon abundance to ensure they lie in the enhanced carbon sequence and removed those that appear to lie in the lower carbon abundance sequence. To this sample we added two new massive DQs from the SkyMapper survey of high proper motion white dwarfs (Vennes et al., in preparation, see Appendix A). The spectra of these two stars were obtained with the Focal Reducer and low dispersion Spectrograph (FORS) at the European Southern Observatory and the Wide-Field Spectrograph (WiFeS) at Siding Spring Observatory (SSO). These massive and C-enriched DQs are listed in Table 2 with their atmospheric parameters and they are shown as the green circles in Fig. 1.

The characteristics of the C-enriched DQs are different from those of ordinary white dwarfs as summarised below.

1. 1.

   Their optical spectra show notable absorption lines of C i at 4270.2, 4933.4, 5053.6, 5181.8, 5381.8Å and/or C ii at 4267, 4300, 4370, 4860, 6578, and 6583Å  and weak He i lines. Weak Swan bands are detected at the cooler end of the population ($\approx 10\,000$K). The detection of hydrogen in C-enriched DQ white dwarfs is relatively rare, i.e., in less than $\approx 20$ percent of the population (Dufour et al., 2007, 2008a; Coutu et al., 2019), such as in the two massive DQ white dwarfs G35-26 (Thejll et al., 1990) and G227-5 (Wegner & Koester, 1985). An upper limit of $\log{\rm(H/C})\approx-2.7$ is otherwise achieved among the hottest objects from the absence of hydrogen lines in the spectra (Dufour et al., 2008a). Upper limits to the hydrogen abundance in cooler objects have yet to be determined relative to the dominant species, helium or carbon (Coutu et al., 2019).

2. 2.

   Their masses are substantially higher. The average mass of H-rich white dwarfs with effective temperatures below 60 000 K is $0.540$ M_⊙ while that of He-rich white dwarfs below 50 000 K is $0.575$ M_⊙ (Bédard et al., 2020). The average mass of the C-enriched DQs is $\langle M\rangle=1.027\,M_{\odot},\sigma_{M}=0.116\,M_{\odot}$ (see section 6), thus substantially larger. In fact, their average mass surpasses even that of the magnetic white dwarfs ($\langle M\rangle=0.87\,M_{\odot},\ \sigma_{M}=0.22\,M_{\odot}$; Kawka, 2020).

   This suggests that either the main sequence progenitors of the C-enriched DQs were substantially more massive or that they are the result of mergers.

3. 3.

   About 70 percent of hot DQs are magnetic (Dufour et al., 2013; Dunlap & Clemens, 2015) exhibiting field strengths from 0.3 MG up to 18 MG but with the majority of fields between 1 and 4 MG. This is a much higher percentage than among the general white dwarf population (14 - 20 percent in volume-limited samples, Kawka et al., 2007). This is again a strong indicator that stellar merger occurred and was responsible for the generation of their magnetic fields (Tout et al., 2008; García-Berro et al., 2012; Wickramasinghe et al., 2014; Briggs et al., 2015).

4. 4.

   The kinematics characteristics of the C-enriched DQs indicate that they belong to an older population (see Section 5 and Dunlap & Clemens, 2015). Thus these C-enriched DQs are highly unlikely to be the progenies of single massive stars since white dwarfs with masses of $0.8-1.2$ M_⊙ are the descendants of main sequence stars of $3.5-8$ M_⊙ (e.g., Romero et al., 2015) whose evolution to the compact phase is very fast ($\leq 0.5$ Gyrs). Such warm/hot white dwarfs would be confined to the thin Galactic disd. If, instead, they evolved from single stars and became white dwarfs billions of years ago, they would be very cool and dim, unlike the population of the currently observed C-enriched DQs. Therefore, it is much more likely that these DQs are the outcome of merging events many of which occurred with long delay times from the formation of the progenitor binary.

The points highlighted above strongly support a merger hypothesis for the C-enriched DQ white dwarfs (Dunlap & Clemens, 2015). However, not all mergers are expected to produce C-enriched DQ white dwarfs. The studies of Briggs et al. (2015) have shown that high field magnetic white dwarfs are the result of merger events during common envelope evolution with the major contributors coming from low-mass main-sequence stars merging with the degenerate cores of red giant branch (RGB) and asymptotic giant branch (AGB) stars. The DD mergers only represented a very small fraction of the population of magnetic white dwarfs and populated the high-mass tail of the magnetic white dwarf mass distribution. The latter are the most likely progenitor’s candidates of the C-enriched DQs. A second possible channel may consist in the merger of a white dwarf with a naked helium star or in the merger of two naked helium stars. The common property that characterises these two channels is the absence of hydrogen in their envelopes.

In order to test the stellar merger hypothesis as an explanation for the origin of the C-enriched DQs, we have used the rapid Binary Stellar Evolution (BSE) code of Hurley et al. (2002) which includes the updates of Kiel & Hurley (2006). Common envelope (CE) evolution, as first proposed by (Paczynski, 1976), is necessary to explain compact binaries whose size is smaller than the initial radius of the primary star. The outcome of CE evolution is either a merged object or a close binary. Since the processes that govern the CE phase are still not fully understood and the ejection of the envelope may or may not be complete, two parameters are usually introduced. The first is the CE efficiency parameter $\alpha_{\rm CE}$ which was introduced to parametrise the efficiency of the injection of orbital energy into the envelope (Livio & Soker, 1988) and the other is $\lambda_{\rm b}$ which depends on the structure of the donor star and on how tightly bound to the core the envelope is. Here we use the BSE option whereby $\lambda_{\rm b}$ is calculated from the detailed stellar evolution models of Pols et al. (1995) obtained with the Cambridge STARS code (Eggleton, 1971). An extensive explanation of the nature of this parameter and the range of values it can attain are in Loveridge et al. (2011).

Since the envelope clearance efficiency is low at small $\alpha_{\rm CE}$’s, the envelope has a longer time to exert a drag on the orbit and consequently the number of stars that merge during CE increases as $\alpha_{\rm CE}$ decreases. Most of these merger events occur during the RGB or AGB phases of the primary star. Those systems that do not coalesce emerge from CE evolution at smaller orbital separations. Ivanova et al. (2013) have shown that the post-CE orbital separation is directly proportional to $\alpha_{\rm CE}$. Ruiter et al. (2011) found that in order to obtain a number of events that is consistent with the predicted rate of SNe Ia from the DD merger channel one has to assume complete CE efficiency, that is, $\alpha_{\rm CE}\lambda_{\rm b}=1$. Similarly, Ruiter et al. (2019) investigated the various pathways to neutron star formation via the accretion induced collapse (AIC) of oxygen-neon white dwarfs in interacting binaries or via merger induced collapse. They explored their results using two different approaches for CE evolution. In one they have $\alpha_{\rm CE}\lambda_{\rm b}=1$. In the other they keep $\alpha_{\rm CE}=1$ but with the donor binding energy parameter based on the stellar evolution calculations of Xu & Li (2010) and on the evolutionary state of the donor at the onset of CE evolution. They find that the AIC birthrates are similar in both cases. In the present work, we also require $\alpha_{\rm CE}=1$, since low $\alpha_{\rm CE}$’s yield too many merged objects with $M\lesssim 0.95$. Because of our choice of $\lambda_{b}$, our simulations are closer to those labelled Model 2 in Ruiter et al. (2019).

The masses of the stars, $M_{1}$ for the primary and $M_{2}$ for the secondary, are assigned values in the range $0.8-10$ M_⊙ while the initial orbital period at the ZAMS, $P_{\rm i}$, varies in the range $10-10\,000$ days. The masses of the primary are randomly chosen according to Kroupa (2001) mass function and those of the secondary stars according to a flat mass distribution with $q=M_{2}/M_{1}\leq 1$ (e.g. Ferrario, 2012). The initial period distribution is assumed to be uniform in the logarithm (Kouwenhoven et al., 2007). The metallicity is near solar taking into consideration that we may slightly underestimate the number of merger events since their rates are higher at lower metallicity due to lower wind-mass-loss rates (Côté et al., 2018).

In our population synthesis calculations we have followed the evolution of $10^{7}$ binaries up to an age $t_{Gal}=13$ Gyrs. From this evolved population we then extracted all single white dwarfs that were the result of either the merger of two white dwarfs or of a white dwarf with a naked helium star or two naked helium stars. All such mergers yield white dwarfs with no hydrogen in their atmospheres. The evolutionary path that leads to these events requires one or two common envelope phases. If the two stars do not coalesce during common envelope evolution, they both evolve to the compact star stage and form a close binary system consisting of two white dwarfs. Because the merger of two white dwarfs is driven by gravitational wave radiation, their merging can be delayed substantially (see section 6). To gain more familiarity with these complex processes, we refer the reader to the thorough review of Ivanova et al. (2013) on binary evolution, on the role that common envelope evolution plays in bringing stars together, and on possible mergers or explosions.

Our population synthesis calculations correspond to a single starburst (one generation of stars). In order to model the currently observed C-enriched DQ population, which is the result of binaries that were born at different Galactic times, we have assigned various birth epochs to the starburst, in agreement with the Galactic disc star formation history (SFH) of Crocker et al. (2017) and shown in Fig 3. Briefly, this SFH is given by

where $z$ is the cosmological redshift as first proposed by van Dokkum et al. (2013) and Snaith et al. (2014). This form was then renormalized by Crocker et al. (2017) so that the integrated stellar mass of the disc is $(3.7\pm 0.5)\times 10^{10}$ M_⊙, in agreement with Bland-Hawthorn & Gerhard (2016). In this context we would like to remark that the star formation history that we have adopted does not take into account a possible inside-out assemblage history of our Galaxy (Xiang et al., 2018) and that the effect on the present day merger population remains to be investigated.

This SFH allowed us to scale up the progenitors of the DQs to a number (and thus mass) that makes this subgroup of binaries consistent with the total mass of the Galactic disc. The method consisted in producing many generations of DQs whose progenitor binaries were born at times that were randomly sampled from the SFH of equation (2) under the simplifying assumption that there is an equal number of single stars as number of binaries. Each of our merged white dwarf was then assigned a location in the Galaxy in the cylindrical coordinate system ($R$, $\Phi$, $z$) with origin in the Galactic centre. The distribution of stars in the $R$ and $z$ directions were taken to follow exponential laws (e.g. van der Kruit & Searle, 1982, and references therein) with radial and vertical scale-lengths of $r_{0}=2.6\pm 0.5$ kpc and $z_{0}=800\pm 180$ pc respectively at $R=8$ kpc (distance of the Sun from the Galactic centre Bland-Hawthorn & Gerhard, 2016). A multiplicative factor $(t/t_{Gal})^{1/2}$, where $t$ is the total age of the star (from the birth of the binary on the main sequence to the present time, Eggleton et al., 1989), has been applied to the distribution in the $z$ direction. This was done to take into consideration the vertical age gradient in the Milky Way disc. Effective temperature and magnitude were assigned to each synthetic object using the tables available at http://www.astro.umontreal.ca/~bergeron/CoolingModels (Bergeron et al., 2011; Bédard et al., 2020).

We have limited our population analysis to objects that have a Gaia G-magnitude, $G$, less than 20. This choice was determined by the study of Boubert & Everall (2020) on the completeness of Gaia DR2. More specifically, these authors found that over $3<G<20$, Gaia is essentially complete and falls from 100 to 0 per cent over $20.0<G<21.5$. Thus, all our model data represent a magnitude-limited sample of mergers with $G\leq 20$.

The 2nd data release of Gaia revealed an enhancement of massive white dwarfs in a narrow temperature range (Q-branch; Gaia Collaboration et al., 2018) that cannot be explained with the current cooling models. Cheng et al. (2019) showed that this enhancement could be explained by a delay in the cooling of massive ($>1.08$ M_⊙) white dwarfs by ²²Ne settling in C/O white dwarfs. This delay could potentially increase the cooling age of the cooler white dwarfs by 8 Gyr. Recently, it was shown that this cooling delay can only occur in white dwarfs with C/O cores and not in O/Ne core white dwarfs because crystallization occurs much earlier in the evolution of O/Ne white dwarfs as compared to C/O core white dwarfs (Camisassa et al., 2021; Blouin et al., 2021). Schwab (2021) showed the massive white dwarfs that form from the merger of two C/O white dwarfs end up being O/Ne white dwarfs, and therefore these would not experience this additional delay. The properties of the C-enriched DQs indicate that they may fall into this category.

Before examining the implications of the population syntheses, we first establish the kinematic properties of the population, and in particular its age distribution.

We now examine the kinematical properties of the observed sample of C-enriched DQs to establish to which Galactic population they belong. We plot in the top panel of Fig. 4 the Galactic space velocity components as $\sqrt{U^{2}+W^{2}}$ versus $V$, where $U$ is positive in the direction of the Galactic centre, $V$ is positive in the direction of the Galactic rotation and $W$ is positive toward the North Galactic pole. The velocities are relative to the local standard of rest.

To a first approximation, stars with a total velocity $v_{t}=(U^{2}+V^{2}+W^{2})^{1/2}\lesssim 70$ km s^-1 belong to the thin disc, stars with $70\lesssim v_{t}\lesssim 180$ km s^-1 belong to the thick disc (Venn et al., 2004), while stars with $v_{t}\gtrsim 180$ km s^-1 are likely to be halo objects. There is a likely overlap of thin and thick disc white dwarfs between about $50$ and $70$ km s^-1.

We present the plot of $J_{z}$ against $e$ in the bottom panel of Fig. 4. According to Pauli et al. (2006) thin disc stars occupy region $A$ which is characterised by low eccentricities and $J_{z}$ in the range $1\,600-2\,000$ kpc km s^-1. In region $B$, stars have larger eccentricities and lower $J_{z}$ and are likely to belong to the thick disc population. Region $C$ is generally populated by halo stars. We can see that the location of the C-enriched DQs in the $J_{z}$ against eccentricity plot is consistent with that of the $\sqrt{U^{2}+W^{2}}$ versus $V$ diagram. In particular, we note that there are two DQs that belong to the Galactic halo, one of which is LP93-21 (Kawka et al., 2020) which is on a retro-grade orbit and the other, SDSS J0918+4843, appears to have a very eccentric orbit with near-zero $J_{z}$. The two other halo candidates identified in the $\sqrt{U^{2}+W^{2}}$ versus $V$ diagram fall in the thick-disc region in the $J_{z}$ versus $e$ diagram, however they have a much higher $J_{z}$ than the other thick-disc candidates. These are likely halo stars since they also have a maximum vertical amplitude $z_{\rm max}>1.5$ kpc (Martin et al., 2017).

We have found that about half of the observed sample of DQs have kinematic properties that are consistent with those attributed to the Galactic thick disc or halo. However this a lower limit because for about one quarter of the sample we had to assume radial velocities of $0$ km s^-1 (see section 2.1). Pauli et al. (2003) suggest that as much as 23 per cent of thick disc white dwarfs can be misclassified as thin disc when assuming a radial velocity of $0$ km s^-1. We revisit this problem by recalculating the kinematics of the sample for which we have radial velocity measurements and assuming a zero velocity instead. Fig. 5 shows the shifts in the Galactic velocity components $\sqrt{U^{2}+W^{2}}$ versus $V$ and $J_{z}$ versus $e$ which confirms that the inclusion of the radial velocity measurements increases the kinematical age, i.e., it pushes some thin disc stars to the thick disc and thick disc stars to the halo.

We calculated kinematics for all known C-enriched DQs using Gaia parallaxes and proper motions, and radial velocities for about two thirds of the sample. Radial velocities for the remaining third should be acquired to confirm the population kinematics.

We extracted two samples from the population synthesis calculations. The first contains white dwarfs that formed via DD mergers. The second consists of He star mergers. We show in Fig. 6 the type of merger versus delay time, noting that these are simulated data that only pertain to a single starburst. It is obvious that white dwarf-white dwarf progenitors have the longest delay times ranging from a few hundred Myrs to a Hubble time. The delay times of the second sample, instead, are shorter and mostly confined to below a couple of Gyrs. After performing integration over time (see section 4), we applied the following selection criteria to compare theory to observations. We took all relevant merger products, the DD mergers and He star mergers, within a distance of 200 pc which is the distance encompassing the majority of known C-enriched DQs (see Fig. 2). We also assembled the mergers products brighter than (Gaia magnitude) $G<20$ noting that neither observed sample following such criteria is complete since a number of very dim objects would escape detection even within a distance of 200 pc. From these selections we extracted the number of objects with a temperature in the range $8\,000\leq T_{\rm eff}\leq 25\,000$ K, and, separately, in the range $T_{\rm eff}\geq 25\,000$ K. Table 1 shows the average and dispersion of the age and mass distributions for the two different samples as well as the number of objects selected under these criteria. Although the simulated mass distributions appear similar both in their average and dispersion, the projected age distribution of DD mergers corresponds to a much older population than that of He star mergers. This is not surprising since it is entirely consistent with the simulated delay time data portrayed in Fig. 6.

Fig. 7 (top panel) shows the observed mass distribution of the C-enriched DQs. The mean average of the sample is $1.026$ M_⊙ with a dispersion of $0.116$ M_⊙. This sample is compared to the mass distribution from the DD merger and He star merger population syntheses. The DD mergers produce more massive white dwarfs, whereas He star mergers can produce a higher fraction of lower mass white dwarfs when compared to DD mergers. Both simulated mass distributions peak at slightly higher mass than the observed distribution with relatively fewer objects than observed at the low end ($0.8\,M_{\odot}$). The mass distribution of DD mergers simulation peaks at a slightly lower mass than that of He star mergers. The He star channel appears to form twice as many objects as the DD channel in the $d<200$ pc sample.

Our population synthesis study does not provide information on space velocity, however we can use the observed kinematics, such as the eccentricity of the Galactic orbit, to estimate a total age for each white dwarf. Kordopatis et al. (2011) determined the average eccentricity for the thin disc, thick disc and halo, and showed that the eccentricity increases with age. We assigned an age of 8 Gyr for the thin disc, an age of 10 Gyr for the thick disc (Sharma et al., 2019) and an age of 11 Gyr for the halo (Kilic et al., 2019). We fitted a function (Age $=13.5-2.05e^{-0.5}$, where $e$ is the eccentricity of the Galactic orbit) to these three points that we can apply to the C-enriched DQ white dwarf sample. Fig. 8 (top panel) compares ages of the observed C-enriched DQs assuming single star evolution (blue histogram) to the age determined from the eccentricity (green histogram). The age assuming single star evolution includes the cooling age of the white dwarf that is based on its mass and temperature added to an average pre-white dwarf lifetime of 200 Myr. In the lower two panels we show the synthetic populations ($G<20$) that emerged from DD mergers (middle panel) and He star mergers (bottom panel). The age distribution of objects emerging from the DD channel is in qualitative agreement with the kinematic age of the C-enriched DQs, although our simulations suggest that there should be a larger number of C-enriched DQs than currently observed. As mentioned earlier, the C-enriched white dwarfs would only be a subset of all merger products alongside other merger candidates such as ultra-massive magnetic white dwarfs. It demonstrates that the much longer lifespan of C-enriched DQs relative to their apparent (cooling) age is the result of binary evolution and interaction in the form of DD mergers. The age of the He star channel products is much lower and corresponds to a thin Galactic disc population rather than the observed thick disc or halo populations.

Results obtained from the $G<20$ sample in the population synthesis show a large excess of He star merger products relative to the DD merger products. Again the age distribution of these objects does not match the observed distribution.

Neither observed samples, $d<200$ pc and $G<20$, are complete but the simulations show that many more objects should be identifiable at current and future survey limits and that He star mergers should dominate in the thin Galactic disc. However, only DD merger products take the appearance of the kinematically old population of C-enriched DQs.

Because our population synthesis study does not provide information on space velocity we have not addressed the excess of C-enriched DQs with a transverse velocity $>70$ km s^-1 in the Q-branch region of the H-R diagram that was attributed by Cheng et al. (2019) to some additional cooling delay mechanisms. However, we have concluded that the products of mergers are likely to produce O/Ne core white dwarfs which do not experience additional cooling since crystallization occurs at higher temperatures than those of white dwarfs on the Q-branch. Therefore, if white dwarfs produced from mergers do experience additional cooling as they pass through the Q-branch, it cannot be through ²²Ne settling and another mechanism is required.

We know that one of the evolutionary channels leading to Type Ia SNe, used as standard candles to measure cosmological distances, consists in the merger of two white dwarfs. We can therefore state that these massive C-enriched DQs are failed Type Ia SNe, as first noted by Dunlap & Clemens (2015). We have also shown that these mergers are rare events and that only a few C-enriched DQs are observed with an estimated space density that is between 0.2 to 0.7 percent of the local space density of white dwarfs. Nonetheless, they may constitute from a few to about $50$ percent of all DD merger products.

Our study shows that this population of white dwarfs is old, with nearly half of the observed objects having kinematic properties consistent with those of stars belonging to the Galactic thick disc and halo. Our population synthesis results largely support these findings and are compatible with a population of white dwarfs descending from the merger of two white dwarfs. We found that the merger of stars whose envelope was stripped of hydrogen during common envelope evolution (He star mergers) would leave remnants much younger than actually observed. Note that the simulated distributions were not actually fitted to observed distributions; the predicted and observed population numbers may differ by more than a factor of two but are generally of the same order of magnitude. The population synthesis predicts a large number of very hot white dwarfs ($25\,000<T_{\rm eff}\lesssim 110\,000$) that resulted from DD mergers. They would represent 15 percent of DD mergers in a volume-limited survey ($d<200$ pc), and up to 75 percent in a magnitude-limited survey ($G<20$). These white dwarfs will most likely have carbon rich atmospheres not unlike the hottest known objects in the C-enriched population. Few such objects are known, but the ultra-hot, massive DZQ H1504+65 and cooler siblings (Werner & Rauch, 2015) which show a mixed carbon-oxygen atmosphere are emerging as possible candidates. The observed trend in carbon abundance with temperature in these likely merger products remains to be explained.

To conclude, we note that binary white dwarfs are sources of low-frequency gravitational waves. Therefore, some of the progenitors of these merging binaries will be detectable with the space-based gravitational wave observatory LISA, which is an European Space Agency-led mission, scheduled to launch in the early 2030’s. Whilst most binary white dwarfs are invisible in the electromagnetic spectrum, LISA will be able to ‘hear’ thousands of them millions of years before they merge.

LF and SV would like to express their gratitude for the hospitality of the staff at the International Centre for Radio Astronomy Research. We thank D.T. Wickramasinghe for useful discussions. This study is partly based on observations made with ESO telescope at the La Silla Paranal Observatory under programmes 097.D-0694 and 097.D-0063 and 090.D-0536. We thank M.S. Bessell for sharing with us the spectrum of J2140$-$3637. Funding for the SDSS IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. This work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia Multi-Lateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia.

The SDSS spectra are available publicly from the SDSS Archive (https://www.sdss.org/). The FORS2 and UVES spectra are from the author (AK).