{'fraction_non_alphanumeric': 0.03981264637002342, 'fraction_numerical': 0.021818610685222315, 'mean_word_length': 4.354442940911142, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The demographics of neutron star - white dwarf mergers: rates, delay-time distributions and progenitors

The mergers of neutron stars (NSs) and white dwarfs (WDs) could give rise to explosive transients, potentially observable with current and future transient surveys. However, the expected properties and distribution of such events is not well understood. Here we characterize the rates of such events, their delay time distribution, their progenitors and the distribution of their properties. We use binary populations synthesis models and consider a wide range of initial conditions and physical processes. In particular we consider different common-envelope evolution models and different NS natal kick distributions. We provide detailed predictions arising from each of the models considered. We find that the majority of NS-WD mergers are born in systems in which mass-transfer played an important role, and the WD formed before the NS. For the majority of the mergers the WDs have a carbon-oxygen composition (60-80%) and most of the rest are with oxygen-neon WDs. The rates of NS-WD mergers are in the range of 3-15% of the type Ia supernovae (SNe) rate. Their delay time distribution is very similar to that of type Ia SNe, but slightly biased towards earlier times. They typically explode in young 0.1-1Gyr environments, but have a tail distribution extending to long, Gyrs-timescales. Models including significant kicks give rise to relatively wide offset distribution extending to hundreds of kpcs. The demographic and physical properties of NS-WD mergers suggest they are likely to be peculiar type Ic-like SNe, mostly exploding in late type galaxies. Their overall properties could be related to a class of rapidly evolving SNe recently observed, while they are less likely to be related to the class of Ca-rich SNe.


Introduction
The mergers of double compact objects binaries can give rise to explosive and/or transient events and the formation of exotic objects, which otherwise do not from through the evolution of single stars. Numerous studies explored the mergers of double white dwarf (WD) binaries, mostly in the context of type Ia supernova (SN) progenitors (e.g. Wang & Han 2012;Wang 2018, for reviews). The mergers of black holes (BHs) and neutron stars (NSs) and their combinations (BH-BH, BH-NS, NS-NS) have also been explored extensively, both as potential gamma-ray bursts (GRB) progenitors, as well as gravitational wave (GW) sources, such as those detected by the advanced LIGO-VIRGO consortium (Abbott et al. 2016(Abbott et al. , 2017. Nevertheless, the mergers of other double compact-object binaries, such as NS-WD binaries (which are likely the most common type of double compact object binaries, besides WD-WD binaries; Nelemans et al. 2001a) received less attention. King et al. (2007) suggested that the accretion of the debris of a disrupted WD following a NS-WD merger can give rise to unique type of a long gamma ray burst. Recently, Metzger (2012) have studied the outcomes of NS-WD mergers and suggested the early phases of accretion give rise to faint thermonuclear explosion occurring in the accretion disk (see also Margalit & Metzger 2016. Our recent study (Zenati et al., in prep.) fur-ther explored such mergers through more detailed models, finding that they are mostly driven by accretion with only very little contribution from thermonuclear sources, mostly consistent with the previous results. Such events may also contribute to the chemical evolution of galaxies, as their nucleosynthetic products somewhat differ from those of ordinary supernovae (Margalit & Metzger 2016. Although NS-WD mergers may have various observable explosive outcomes, the demographics of such events and their progenitors had been little explored. Here we use extensive population synthesis models of binary stellar evolution to characterize the demographics and rates of NS-WD mergers. Note that NS binaries may also form in clusters and in the field through dynamical captures (Postnov & Yungelson 2014;Klencki et al. 2017); here we only consider the unperturbed evolution of isolated primordial binaries. Given the many uncertainties involved in the modeling of binary evolution, we explore several different models, which differ in their approach to the main uncertainties (e.g. the distribution of NS natal-kicks, the properties of common envelope evolution etc.). Our models provide us with detailed predictions for the overall rates of WD-NS mergers; their delay time distribution (DTD); expected host galaxies; and the distribution of the properties of their progenitors (e.g. WD masses and composition). In addition we briefly Article number, page 1 of 15 arXiv:1804.01538v1 [astro-ph.HE] 4 Apr 2018 A&A proofs: manuscript no. NSWD discuss the role of such mergers in producing GW sources observable by next-generation GW detectors.
The last two decades of observational progress and the advent and development of large-scale automated supernovae (SNe) surveys have revealed the existence of many novel types of "peculiar" transients, including several families of nonstandard type Ia and other likely thermonuclear SNe (Li et al. 2001(Li et al. , 2011. Future and upcoming surveys such as ZTF, ATLAS and LSST are likely to discover many more peculiar transients, and in particular faint and/or fast-evolving transients not observable by current shallower/low-cadence surveys. If NS-WD mergers result in peculiar transients as predicted by current models, they might be observable over the coming decade. The characterization of such events and their distribution is therefore a critical step in identifying them. Moreover, it opens the window for connecting the theory with observations and constraining the models for NS-WD mergers, their outcomes and even their potential role in affecting the chemical evolution of galaxies. In the following we first discuss the population synthesis models we employ and their various properties (Sect. 2), we then describe our results (Sect. 3) regarding their evolution, rates & delay time distribution, galaxy offset and gravitational wave signatures, and finally discuss their implications and conclude.

Population synthesis
The formation and evolution of compact binary mergers is simulated with the binary population synthesis (BPS) code SeBa (Portegies Zwart & Verbunt 1996;Toonen et al. 2012;. SeBa is a fast code for simulating binary evolution based on parametrized stellar evolution, including processes such as stellar winds, mass transfer episodes, gravitational wave emission and supernova (SN) kicks. We employ SeBa to generate a large population of binaries on the zero-age main-sequence (MS), we simulate their subsequent evolution, and extract those that lead to a merger between a NS and a WD.
It was shown in Toonen et al. (2014a) that the main sources of differences between different BPS codes is due to the choice of input physics and initial conditions. To assess the systematic uncertainties in our predictions we construct different models that differ with respect to the SN kick and the common-envelope (CE) phase, as described by the main models in Tbl. 2. Furthermore, we construct an additional set of models that differs with respect to the population of primordial binaries and assumptions regarding stable mass transfer processes. For these additional models we assume a single distribution of SN kicks (based on Hobbs et al. 2005, Sect. 2.2) and consider two approaches for modeling the CE-phase (model αα& αα2, since model γα gives similar results as model αα, see Sect. 2.3). The different models are described in the following sections.
In this paper, the term primary (secondary) refers to the initially more (less) massive component of a given binary.

Primordial binaries
In our standard models the primordial binaries are generated in the following way: -Primary masses M 1 are drawn from a Kroupa IMF (Kroupa et al. 1993) with masses in the range 4-25M . For the normalization of the rates, primary masses between 0.1-100M are considered; -Secondary masses M 2 are drawn from a flat mass ratio distribution with 0 < q ≡ M 2 /M 1 < 1 (Raghavan et al. 2010;Duchêne & Kraus 2013); -The orbital separations a follow a flat distribution in log(a) (Abt 1983); -The initial eccentricities e follow a thermal distribution (Heggie 1975); -A constant binary fraction 1 binary fraction B of 75% is assumed which is appropriate for B-type primaries (Raghavan et al. 2010;Duchêne & Kraus 2013;Sana et al. 2014).

Supernova kicks
At the end of the life of a high-mass star, its core collapses under pressure of self-gravity. These core-collapse (CC) SN bring about some of the most energetic explosive events in the universe, in which the interstellar medium is enriched by processed materials that are expelled from the newly formed remnant. The sudden mass loss from the binary changes the gravitational potential of the system, such that the orbit can change dramatically (Hills 1983;Kalogera 1996;Tauris & Takens 1998); The orbit can become highly eccentric and even dissolve, giving rise to two single stars escaping in different directions (i.e. runaway and walkaway stars, e.g. Blaauw 1961;Eldridge et al. 2011, Renzo et al. in prep.). The kick experienced by the system due to the mass loss alone is referred to as the 'Blaauw-kick' (Blaauw 1961). Interactions in close binary systems that strip the stellar envelopes of the donor stars prior to the SN, limit the mass loss during the SN and so the effect of the Blaauw-kick (Huang 1963;Tutukov & Yungelson 1973;Leonard et al. 1994).
There is ample evidence for an additional velocity kicks imparted to the NS during the SN, such as pulsar scale heights (e.g. Gunn & Ostriker 1970), proper motions of pulsars (Cordes et al. 1993;Hobbs et al. 2005;Lyne & Lorimer 1994;Verbunt et al. 2017), and the high velocities of some single NS (Chatterjee et al. 2005;Becker et al. 2012). The formation mechanisms of these 'natal-kicks' is an unsolved problem (see e.g. Kusenko & Segrè 1996;Scheck et al. 2006;Wongwathanarat et al. 2013;Holland-Ashford et al. 2017;Katsuda et al. 2017), but is likely caused by asymmetries in the neutrino losses and/or in the mass loss in the SN ejecta, (Janka 2012, for a review). These asymmetries could be generated during the late phases of stellar evolution.
The natal kicks can be of order several hundreds of km/s (e.g. Hobbs et al. 2005). As such the kicks can easily disrupt binaries with periods larger than a few months, for which the kinetic energy imparted through the natal kick is higher than the binding energy of the orbit. However, smaller natal kicks, of the order of several tens of km/s, have been derived for other systems, such as high-mass X-ray binaries, double NS binaries, and pulsars retained in globular clusters (Pfahl et al. 2002;Schwab et al. 2010;Beniamini & Piran 2016). The origin of the reduced natal kicks has been linked to different CC scenarios (i.e. electroncapture SN versus iron CC), or linked to the specific masses of the core and ejecta (Podsiadlowski et al. 2004;van den Heuvel 2004;Knigge et al. 2011;Janka 2012;Tauris et al. 2015;Bray & Eldridge 2016;Janka 2017, and references therein).
In this paper we adopt three different models for the SNkick. In our main model we randomly draw a natal kick from the distribution of Hobbs et al. (2005) which is a Maxwellian distribution with a 1D rms of σ = 265km/s. As an alternative model we adopt the distribution of Arzoumanian et al. (2002) which has two Maxwellian components, one at high velocities (σ = 500km/s) and one at low velocities (σ = 90km/s). In our last model, there are no natal kicks, and so the SN-kick consists of only the effect of the instantaneous mass loss i.e. Blaauw-kick (Blaauw 1961).

The common-envelope phase
Common-envelope evolution is a mass-loss phase in the formation of compact binaries, which is thought to lead to a severe shrinkage of the binary orbit. Thus explaining the existence of compact binaries (a 1 − 10R ) containing one or two compact objects whose progenitors would not fit in that orbit during the giant phases. However, despite the importance of the CE-phase for many types of binaries (e.g. supernova type Ia progenitors, cataclysmic variables, low-mass X-ray binaries), and the tremendous effort of the community, the CE-phase is poorly understood (for a comprehensive review, see Ivanova et al. 2013).
The CE-phase is commonly modeled based on the conservation of energy (Paczynski 1976;Webbink 1984;Livio & Soker 1988): where M d is the mass of the donor star, M c the mass of its core, R its radius, λ the structure parameter of its envelope, and E bin is the binding energy of the the envelope. As a source of energy to unbind the envelope, classically the orbital energy E orbit is considered, with an efficiency of α CE . Other origins for the envelope ejection have also been considered, such as recombination energy (e.g. Livio 1989;Ivanova et al. 2015;Nandez et al. 2015;Clayton et al. 2017), dust-driven winds (Glanz & Perets 2018) and jets (Armitage & Livio 2000;Soker 2004;Chevalier 2012;Papish et al. 2015;Soker 2017). Whereas hydrodynamical simulations of the CE-phase generally are unable to expel the envelope when only orbital energy is considered, including these various other processes can potentially lead to a successful expulsion of the bulk of the envelope.
In this paper we focus on the classical energy balance of CEevolution, and vary the efficiency with which orbital energy can be used to expel the donor's envelope. In model αα we assume α CE λ = 2, whereas for model αα2, α CE λ = 0.25 is assumed. The prior is calibrated to formation of double WDs (i.e. second phase of mass transfer, see Nelemans et al. 2000Nelemans et al. , 2001b, whereas the latter is calibrated on the formation of compact WD-MS systems (Zorotovic et al. 2010;Camacho et al. 2014;Zorotovic et al. 2014).
An alternative model for CE-evolution is the γ-CE which is based on a balance of angular momentum in stead of energy as where J init resp. J final is the angular momentum of the pre-and post-mass transfer binary respectively, and M a is the mass of the companion. The γ-prescription was introduced to explain the first phase of mass transfer in the formation of double WDs (Nelemans et al. 2000;van der Sluys et al. 2006). In our model γα , we apply the γ-CE with γ = 1.75, unless the companion is a degenerate object or the mass transfer is dynamically unstable, for which the α-CE with αλ = 2 is applied (Nelemans et al. 2001b). Note that the model γα is of less importance here compared to the population of double WDs, as the first phase of mass transfer tends to be stable for the systems considered in this paper (Sect. 3.1).

Stable mass transfer
Stable mass transfer between two hydrogen-rich stars is a frequent phenomenon in the evolution of the NS-WD merger progenitors (see also Sect. 3.1). Therefore, it is important to assess how our standard assumptions impact the synthetic rates. We construct three models which differ in several aspects: the angular-momentum loss mode (Pols & Marinus 1994;Soberman et al. 1997;Toonen et al. 2014a). When the mass transfer is not completely conservative, not only mass but also angular momentum is lost from the system. The effect of this on the orbit can be severe, and depends crucially on how the mass is lost; for example whether the mass is lost directly from the primary, or whether is first crosses into the Roche lobe of the secondary, and subsequently is expelled from close to the surface of the secondary. The standard assumption in SeBa when the accretor is a non-degenerate star, is that the specific angular momentum of the lost matter is 2.5 times the specific angular momentum of the orbit (Portegies Zwart 1995;Nelemans et al. 2001b). Here we test the effect of this; in an alternative model we assume that the specific angular momentum is equal to that of the secondary. This mode of angular momentum loss is also known as 'isotropic re-emission'. the stability of mass transfer is dependent on the reaction of the stellar radii R and the corresponding Roche lobes R RL to the transfer of mass and angular momentum. When mass transfer proceeds on timescales roughly equal or shorter than the thermal timescale of the donor star, the donor cannot achieve thermal equilibrium anymore (and possibly even hydrostatic equilibrium). Modeling of such mass transfer phases and determining the stability of mass transfer for different types of binaries is not trivial. Regarding BPS codes, as they rely on stellar evolution models of stars in thermal and hydrostatic equilibrium, they cannot adequately calculate the donor properties and the stability of mass transfer. Instead, BPS codes rely on parameterizations or interpolations to determine the stability of mass transfer. These are based on simplified stellar models (e.g. polytropes) or calculations from detailed stellar evolution codes (e.g. Hjellming & Webbink 1987;de Mink et al. 2007;Ge et al. 2010Ge et al. , 2015. It is important to realize that even the detailed stellar evolution codes are not adequate in simulating mass transfer on timescales sufficiently above the donor's thermal timescale (e.g. Pavlovskii & Ivanova 2015 As an alternative model, we test how sensitive are NS-WD mergers to our assumptions regarding the stability of mass transfer. We do this by adjusting the stability criterion in SeBa. This is based on the adiabatic response of the star : and of the Roche lobe. If ζ RL < ζ ad we assume mass transfer proceeds in a stable manner (e.g. Webbink 1985;Pols & Marinus 1994). For every Roche lobe filling system, ζ RL is calculated numerically by transferring a test mass of 10 −5 M at every time-step. ζ ad depends on the type of star considered and is tabulated in Toonen et al. (2012), appendix A3 therein. In the alternative model, we decrease ζ ad from 4 to 2 for all stellar types with radiative or shallow convective envelopes. the accretion efficiency β is the degree to which mass transfer is conservative, that is the fraction of mass lost by the donor star that can be accreted by the companion (i.e. β ≡ M d /Ṁ a ). Modeling of the evolution of some binaries (such as φ Per) indicates that mass transfer can proceed fairly conservative (Pols 2007), however for other systems spin-up of the accretor is expected to limit the amount of accretion (e.g. Packet 1981). Attempts to constrain the efficiency have remained inconclusive so far (e.g. de Mink et al. 2007, and references therein).
As the viability of some of the evolutionary channels (i.e. 'semi-reversed WDNS' and 'reversed NSWD', see Sect. 3.1) is directly related to the amount of accretion onto the secondary, we constructed two alternative models that assume completely conservative and 50% conservative mass transfer onto non-degenerate companion stars (i.e. β = 1 and β = 0.5). In comparison, in the standard models we assume that the accretion rate onto these stars is limited to the thermal timescale of the accretor times a factor that is dependent on the ratio of Roch-lobe radius of the accretor radius to its effective radius (Pols & Marinus 1994;Portegies Zwart & Verbunt 1996;Toonen et al. 2012). This inhibits accretion in systems with low mass ratios q = M 2 /M 1 .

Evolution
In this section, we describe the evolutionary path for progenitors of binary mergers between a NS and a WD. In Sect. 3.1.1-3.1.5 we discuss the formation of the NS-WD binary, whereas Sect. 3.1.6 focuses on the subsequent evolution leading to the merger. The BPS simulations let us assess which are the dominant formation channels, their rates and properties. We identify the four main channels, of which the first is a dominant channel for most models and consistently a major contributer. See also Tbl. 2 for corresponding rates. We distinguish between systems in which the NS forms first (hereafter NSWD), and where the WD forms first (hereafter WDNS). The latter is possible due to binary mass transfer, in which an initial low-mass secondary accretes material from its companion star, such that it becomes massive enough to end its life as a NS star. See below for more details.
When we refer to all mergers, combining those of NSWD and WDNS, we use the term NS-WD mergers.  Fig. 1. Example of the evolution of a system in the 'direct WDNS'channel. The primary initiates two phases of mass transfer before contracting to the WD stages of evolution. Afterwards the secondary fills its Roche lobe. Later, as the secondary collapses to a SN, the NS-WD -binary is formed. Orbital and stellar parameters are given in the columns besides the illustration of the binary. The abbreviations of the stellar types are defined in Tbl.1. The extent of the Roche lobes and stars in the illustration are approximate and are not drawn to scale.

Pathway 1: direct WDNS
The most prominent channel is shown in Fig. 1. In this channel the primary star becomes a WD before the secondary becomes a NS. In particular the WD is formed before the NS-progenitor has evolved significantly, and before is has initiated a mass transfer event.
In more detail: The primordial binaries of this channel have primary & secondary masses in the range of ∼7-11M , and ∼4.5-11M , respectively. The initial orbits are relatively small (semilatus recti in the range a SLR = a(1 − e 2 ) ∼ 20 − 100R ). With these parameters the primary star fills its Roche lobe during the Hertzsprung gap, giving rise to a phase of stable mass transfer. Mass transfer continues until the primary star has lost its hydrogen envelope, and the star becomes a hydrogen-poor helium-rich star of about 1.-2.3M . As the primary star evolves further, it fills its Roche lobe again, and another phase of (stable) mass transfer ensues. Afterwards, the primary contracts to become a WD. During the prior mass transfer phases, the secondary has accreted a significant amount of mass, rendering it possible to collapse to a NS at the end of its life. Before it does so, the secondary star initiates one or two mass transfer phases as it evolves. Firstly, the secondary fills its Roche lobe as a hydrogen-rich star. As the mass ratio with the WD companion is low, the mass transfer is unstable, and leads to a CE-phase in which the orbit shrinks significantly. Any companion is not expected to accrete much during the CE MacLeod & Ramirez-Ruiz 2015b,a). Given the reduced orbit, the secondary fills its Roche lobe a second time as a hydrogen-rich helium-poor star. The WD companion is not expected to accrete much as the retention efficiency of WDs is very strongly dependent on the mass transfer rate (e.g. Bours et al. 2013;Toonen et al. 2014b).
When the secondary collapses to a NS, it undergoes a SN explosion which drastically alters the binary orbit (Sect.2.2). Whereas the pre-SN orbit is expected to have been circularized at the onset of the mass transfer phases, the eccentricities of the post-SN orbits span the full range of [0,1) for our models with natal kicks, and up to about 0.6 due to the mass-loss kick alone (For more details see Sect. 3.3).
To conclude, we find that the most common channel of NS-WD mergers is the one in which the WD is formed before the NS (WDNS channel). At first sight, this seems counterintuitive, as one might expect the progenitor of the NS to be more massive than that of the WD initially (on the zero-age MS), and therefore to evolve faster. However, due to the transfer of mass onto the secondary during the stable mass transfer phases, the secondary becomes massive enough as to undergo a supernova explosion and form a NS.

Pathway 2: direct NSWD
For these systems the primary evolves into a NS first, and afterwards the secondary star initiates the mass transfer phases, loses its hydrogen and helium envelope, and ends its life as a WD. An example for such evolution is shown in Fig. 2.
In more detail: The initial stellar masses and orbital separations are larger compared to the previous channel. The primary masses are mostly in the range ∼11-19M (bu extending down to 8M ), the secondary masses are in the range ∼2-10M , and the semi latus rectii are in the range a S LR ∼ 50 − 2000R . The evolution of high-mass primaries is different from the lowmass ones in the previous channel. For the high-mass stars helium burning ignites before the star has reached the red (giant branch) regions in the Hertzsprung-Russell diagram as opposed to low-mass stars. The primary fills its Roche lobe after leaving the MS. The mass transfer is stable and the orbit widens. The primary may fill its Roche lobe again as a helium-star. After nuclear burning has ceased, the primary explodes in a SN and forms a NS. Subsequently, as the secondary star evolves, it fills its Roche lobe. Due to the high mass ratio with the NS component, the mass transfer is unstable and a CE-phase develops. Any eccentricity induced by the SN is likely rescinded leading up and during the CE-phase. Commonly the secondary star fills its Roche lobe once more as a helium star. When its helium envelope is lost, and nuclear burning ceases, the star settles on the WD track, and the NS-WD binary is formed.
During the second part of the evolution of these systems, that is after the formation of the NS, they could be observed as X-ray binaries through accretion onto the NS (e.g. Tauris & van den Heuvel 2006). Any mass transfer phase initiated by the

Pathway 3: semi-reversed WDNS
In this channel (Fig. 3.) the primary becomes a WD before the secondary becomes a NS, hence the name WDNS. 'Semireversed' reflects the neck-to-neck race that the stars are in to form the first compact object in the system. The secondary has already initiated a phase of mass transfer, and evolved to the hydrogen-poor helium-burning tracks, before the WD primary is formed. In more detail: In this evolutionary channel, the initial primary masses and semi-latus rectii are similar to those of channel 1 (direct WDNS), however the initial mass ratios are preferably close to one, q i ∼ 0.85 − 1.0. The primary fills its Roche lobe as a fairly unevolved star. Once the hydrogen envelope is removed, the mass transfer phase ceases and the primary becomes a helium-star. As the initial stellar masses and evolutionary timescales are similar, the secondary accretes a significant amount of mass. As a result its evolution is accelerated, the secondary expands and fills its Roche lobe while the primary is still a helium star. Due to the large mass ratio, the following mass transfer phase is likely unstable. The orbital separation shrinks significantly. The binary now consists of a double helium star. Subsequently the primary initiates a third phase of mass transfer in which the WD is formed. A fourth and final mass transfer event occurs when the secondary fills its Roche lobe. It collapses to a NS in a SN explosion. An eccentric WDNS is formed.  Fig. 3. Example of the evolution of a system in the 'semi-reversed WDNS'-channel. The primary initiates the first phase of mass transfer in which the secondary accretes a large amount of mass. As a result its evolution is accelerated, and it fills its Roche lobe subsequently. There are now two hydrogen-poor helium star (stripped stars) in the system. Afterwards the primary fills its Roche lobe a second time and forms a pre-WD. Lastly, the secondary initiates the final mass transfer event, and collapses to a NS. The layout and legend is similar to that of Fig. 1 Fig. 4. Example of the evolution of a system in the 'reversed NSWD'channel. The first five stages of the evolution are similar to that in the 'semi-reversed WDNS'-channel of Fig. 3, that is until the formation of the double helium star. In the current channel the evolution of the secondary is accelerated to the point where it will fills its Roche lobe not once, but twice before the primary has become a compact object. The secondary collapses to a NS before the primary becomes a WD. The evolution of the stars is therefore 'reversed'. The layout and legend are similar to that in Fig. 1.

Pathway 4: reversed NSWD
The evolution of this channel is shown in Fig. 4. The NS is formed before the WD, similar to channel 2 (direct NSWD). In the 'reversed NSWD' channel, the evolution of the secondary typically overtakes that of the primary, and the secondary becomes the first compact object in the binary. The detailed evolu-tion is similar to that of channel 3 (semi-reversed WDNS), with the order of the third and fourth mass transfer phases being reversed. Due to the strong accretion, the subsequent evolution of the secondary, both as a hydrogen-rich and helium-rich star, is shorter than that of the helium-rich primary.
Note that this evolutionary pathway shows similarities to that described in (Toonen et al. 2012, 'Formation reversal channel') for the formation of double WD mergers as progenitors of Type Ia supernovae.
In more detail: The initial masses of both stars are approximately in the range of 5-8M in this channel. The initial mass ratios are predominantly between 0.85 and 1, but they can extend up to 0.5. The initial semi-latus rectii a SLR are smaller than 200R ). When both stars have lost their hydrogen envelopes and are on the helium burning branch, the primary stars have low masses of ∼ 0.5−1.5M , whereas the secondaries are more massive, ∼ 2.5 − 3.5M . The lifetimes of such massive helium stars is a few Myr, whereas ∼10-150 Myr for the low-mass primaries.

Comparison with previous studies
Naively one would presume that the NS should descend from the most massive star in the initial binary, and therefore form before the WD (i.e. the NS-WD formation channels). The possibility of a reversal of the end-states of the two components has been noted previously e.g. for NS-NS (Portegies Zwart & Verbunt 1996), NS-BH (Sipior et al. 2004) and NS-WD (Tutukov & Yungel'Son 1993;Portegies Zwart & Yungelson 1999;Tauris & Sennels 2000;Davies et al. 2002;Church et al. 2006). As noted by Portegies Zwart & Verbunt (1996), in order for the reversal to happen the mass transfer should proceed rather conservatively, which implies that the initial masses of the stars should be rather similar.
The formation of binaries with a NS and a WD has been studied previously in the context of three binary pulsars with WD companions (Portegies Zwart & Yungelson 1999;Tauris & Sennels 2000;Davies et al. 2002;Church et al. 2006); PSR B2303+46 (van Kerkwijk & Kulkarni 1999), PSR J1141-6545 (Kaspi et al. 2000;Manchester et al. 2000) and PSR B1820-11 (Lyne & McKenna 1989). These systems contain young NSs (i.e. non-recycled pulsars with large magnetic fields and short spin-down times) in eccentric orbits (see also Zhang et al. 2011). These systems demonstrate that it is indeed possible for the NS to form after the WD, not only in simulations but also in nature. The above mentioned studies predominantly consider the 'direct WDNS'-channel, however both Tutukov & Yungel'Son (1993) and Church et al. (2006) also discuss the 'semirev WDNS'channel for the formation of WDNS.
Regarding the evolution of NSWD, Tutukov & Yungel'Son (1993) discusses the 'direct NSWD'-channel, but do not mention the 'reversed NSWD'-channel. We expect that in their model most NSWD are formed directly instead of through a reversed evolution, as they do not consider natal kicks (i.e. only the Blaauw-kick, see Tbl. 2).

Evolution towards the merger
After the formation of the WDNS system, the orbit shrinks due to the emission of gravitational waves (GWs). If the orbit is eccentric (such as for the WD-NS binaries), the GW in-spiral time can be significantly reduced. For example a 1.3M NS and a 0.8M WD in a circular orbit of 1R merge after about 70 Myr, whereas an imposed eccentricity of 0.5 (0.75) at the birth of the NS, reduces the in-spiral time to 25 (14) Myr.
Once the orbital period has shrunk to a few minutes, the WD will fill its Roche lobe. If the subsequent mass transfer proceeds stably, an ultra-compact X-ray binary (UCSB) forms. The timeaveraged X-ray luminosity of the majority of UCXBs is consistent with that expected from WD donors (Heinke et al. 2013), although other types of donor stars have also been suggested, such as helium burning stars or evolved MS-star (see e.g. van Haaften et al. 2013, for a population synthesis study). If unstable mass transfer develops in the NS-WD system, the WD is quickly disrupted on a dynamical timescale, leading to a merger of the system.
The stability of mass transfer from a WD to a NS has typically been modeled under the assumption of isotopic re-emission for the loss of angular momentum. In other words, the fraction of the transfered matter that cannot be accreted by the NS, is ejected isotropically from the vicinity of the NS (Sect. 2.4). This gives rise to a critical WD mass M WD,crit ∼ 0.38M , above which Roche-lobe overflow leads to a merger. However, during the early stages of the mass transfer the mass transfer rate is often highly super-Eddington, such that disc winds become important. Bobrick et al. (2017) recently showed that the disc winds reduce the stability of the mass transfer. They determine a critical WD mass of M WD,crit = 0.2M , excluding all carbon-oxygen and oxygen-neon WDs as UCXB donor stars.
Assuming M WD,crit = 0.38M , over 97% of semi-detached NS-WD systems merge in all models (in good agreement with van Haaften et al. 2013), whereas for M WD,crit ≡= 0.2M over 99.9% merges. In this work we adopt the latter prescription from Bobrick et al. (2017).

NS-WD merger rates
We find that the time-integrated rate of NS-WD mergers is (4 − 6) · 10 −5 per solar mass of created stars in most models (Tbl. 2). We do not find significant differences in the merger rates between the different models of the CE-phase, the stable mass transfer phase, or the SN kick distributions of Hobbs et al. (2005) and Arzoumanian et al. (2002). However, if NSprogenitors in interacting binaries receive small SN kicks, the NS-WD merger rate increases up to (10 − 20) · 10 −5 M −1 under the assumption of a zero velocity natal-kick, that is only the mass-loss kick. On the other hand, if the accretion efficiency of mass transfer is reduced to a constant 50%, the merger rate decreases to (0.3 − 3) · 10 −5 M −1 .
In this work, we have assumed a constant binary fraction B of 75%, however there are indications that the binary fraction increases with the mass of the primary star ( In this case, the NS-WD merger rate only slightly increases; with 6-7.5% higher rates across all models. We have also tested the effect of a different initial period and mass ratio distribution. Firstly, instead of a uniform distribution of periods, we adopt a log-normal distribution of periods (in days) with parameters µ = 4 and σ = 1.3 (Duchêne & Kraus 2013). The main difference is that the number of short period binaries (such as the progenitors in the first pathway 'direct WDNS') is suppressed compared to our main BPS models. Secondly, we adopt a mass ratio distribution N(q) that is strongly biased towards low-mass companions, that is N(q) ∝ q −2 . This is motivated by observations of wide binaries with massive primaries (Moe & Di Stefano 2017). With these changes in the primordial binary population, the NS-WD merger rate decreases by a factor 4-5.
Tbl. 2 also shows the NS-WD merger rate in each of the evolutionary channels identified in Sect. 3.1. Channel 'direct WDNS' contributes to the total merger rate in a significant and consistent way across all models. The corresponding rate is (2 − 5) · 10 −5 M −1 in most models. The contribution from channel 'direct NSWD' is strongly dependent on the CE-modeling and the SN-kick. The rate varies over two orders of magnitude. And so, the strong increase in the total merger rate for smaller SN-kicks can be attributed to this channel.
The reason that channel 'direct NSWD' is more sensitive to the SN-kick compared with the 'direct WDNS' channel is related to the orbital separation of the pre-SN binary. In the latter channel, the pre-SN semi-major axes are several solar radii, but in the former they are several tens to hundreds of solar radii. This is because the systems in channel 'direct NSWD' have widened in response to the (stable) transfer of mass, whereas the other systems have undergone several mass transfer phases and further common-envelope evolution in which the orbit has contracted (see Fig. 1 & 2).
The third channel ('semi-reversed WDNS') provides ∼5-20% of all mergers, and the 'reverted NSWD'-channel gives rise to up to ∼10% of the mergers. An exception to this is found in the additional αα models regarding the stability of mass transfer and the angular momentum loss mode (see Sect. 2.4). The enhanced accretion onto the secondary makes it possible for more secondaries to overtake the primary during its evolutionary progress. As a result, the 'reverted NSWD'-channel gives rise to ∼20% of all mergers in the models with enhanced mass transfer stability and isotropic re-emission of angular momentum. On the other hand, in our additional model with reduced accretion, the rate of the 'reversed NSWD'-channel, as well as that of the 'semireversed WDNS'-channel are negligible.

Delay time distributions
A useful quantity is the delay time of the NS-WD merger. This is the time between the merger and formation of the binary with two zero-age MS stars. The delay time is the sum of the formation time of the NS-WD system, and the in-spiral time of the orbit to shrink due to gravitational wave emission. The NS-WD merger rate as a function of the delay time is shown in Fig. 5 and 6. These delay time distributions (DTDs) peak at short delay times and than decrease towards longer delay times. The slope of the DTDs differs between different models of the CE-phase (Fig. 5). The magnitude of the SN-kick mostly affects the normalization of the DTD, and to a minor degree the slope (Fig. 6). Varying the assumptions regarding the stable mass transfer phase (Sect. 2.4) do not significantly affect the DTD.
The top two panels of Fig. 5 show very different slopes of the DTD. These represent models αα & αα2 and indicate that the slope depends strongly on the modeling of the CE-phase (line 6 in Fig. 1), in particular the efficiency with which orbital energy is used to unbind the envelope. If the common-envelope leads to a more significant shrinkage of the orbit (as in model αα2), the orbits of the NS-WD binaries tend to be smaller at formation, leading to shorter in-spiral times due to gravitational wave emission. A similar trend is seen in the DTDs of merging massive CO-WDs (Ruiter et al. 2009). Table 2. Merger rate of NS-WD binaries per 10 5 M of created stars for the different BPS models (Sect.2). The main models vary with respect to the SN kick (drawn from the distribution of Hobbs et al. (2005), the distribution of Arzoumanian et al. (2002), or only the mass-loss kick, see e.g. Blaauw (1961)) and the common-envelope phase (model αα, αα2, or γα). In the additional models several assumptions are varied in regards to the stable mass transfer process. Columns 5-7 give the fraction of mergers with CO, ONe or He WDs, respectively. Columns 8-11 give the merger rates from the four most common evolutionary channels described in Sect. 3.1.

BPS Model
Rates (  Finally, we turn our attention to those systems with the shortest GW in-spiral times. Recently, we have shown Michaely & Perets 2018) that the in-spiral times of BH-BH, BH-NS, and NS-NS mergers can be as short as years or decades. Therefore, a fraction of a few 10 −4 − 10 −1 of LIGO gravitational wave sources and short gamma-ray bursts could be preceded by observable SN explosions. The short in-spiral times are achieved in systems that experience a SN-kick with the right amplitude and direction, such that the stars are cast into a close and/or highly eccentric orbit. The same mechanism also works for mergers between NS and WDs. With the SN-kick based on the distributions of Hobbs et al. (2005) and Arzoumanian et al. (2002), the NS-WD merger rate with in-spiral times less than 100 years, is approximately 10 −7 M −1 , i.e. of the order of a few×0.1% of NS-WD mergers could be preceded by an observable SN explosion years to decades before the merger. For the models with only the mass-loss kick, the in-spiral times of all systems in the current simulations are longer than 100 year. Given our current resolution, this gives an upper limit of approximately 10 −8 M −1 .

Characteristics of the white dwarf component
The composition of the WD component typically consists of carbon and oxygen (Tbl. 2). In models αα and γα ∼80% of mergers occur with a CO WD. In models αα2, the contribution is reduced to about 60%. NS mergers with ONe WDs are of secondary importance; contributing about 20% for models αα and γα, and 40% for models αα2.
The reason for the increase of ONe WDs around NSs for lower common-envelope efficiencies (such as in model αα2 compared to model αα) is related to the orbital energy available for the progenitors of the WDNS systems. In a binary with a high-mass companion compared to a low-mass companion (and other parameters kept constant), there is more orbital energy available. As a result the orbital shrinkage is weaker, and the CEphase more readily results in a detached binary then in a merger of the two stars.  Mergers with He WDs are not frequent in our models. Similarly as for the ratio of ONe and CO WD companions, NS-WD binaries with less massive companions have less orbital energy and are more likely to merge during the CE-phase, which reduces the number of NS-WD with helium WDs. The majority of the He WDs in our simulation form through a different evolutionary channel than the ones discussed in Sect. 3.1. For these systems, the initial binary has a large mass ratio and a wide orbit, such that the primary star fills its Roche lobe on the asymptotic giant branch. The first phase of mass transfer is unstable in contrast with the other evolutionary channels. Eventually the primary star collapses to a NS. When the secondary star fills its Roche lobe, a last phase of mass transfer commences. Typically this mass transfer phase is unstable, but if not a low-mass X-ray binary is formed. After the secondary star has lost its hydrogen envelope, it forms a helium WD.
Recently Zenati et al. (2018) investigated the formation of hybrid WDs i.e. WDs with a carbon-oxygen core and a thick envelope of helium up to ∼ 0.1M . The existence of a significant mass in helium can catalyze thermonuclear explosions during a merger with another compact object, for example NSs (Metzger 2012;Margalit & Metzger 2016) or WDs (e.g. Pakmor et al. 2013). In the former case, the tidally disrupted WD forms an accretion disk around the NS of mixed He-CO composition, nuclear burning is expected in the disk, but in the case of the hybrid WD it proceeds dynamically (Margalit & Metzger 2016); further details of such explosions are explored in depth in a forthcoming publication (Zenati et al., in prep.). In our simulations we find that mergers between a NS and a hybrid WD are not common, that is less than 0.1% of the total merger rate. These mergers preferably happen at short delay times ( 500Myr).
The WD mass as a function of delay time is shown in Fig. 7. In model αα & γα, we do not find a significant evolution of the average WD mass with delay time. For lower CE-efficiencies (i.e. model αα2), the average WD mass can increase somewhat with delay time.

Effects of the SN-kick
The effect of the SN-kick is dependent on the binding energy of the pre-SN orbit and the kinetic energy of the SN-kick exerted on the pre-SN orbit. If the SN-kick is large in comparison with the orbital velocity, the system is disrupted. This happens for example for wide (a SLR 2 · 10 3 R ) systems in our models with natal-kicks. These orbits are sufficiently wide such that the Roche lobe overflow is not expected and the stars live practically as if they were isolated stars (not taking into account any effects due to the SN-kick). If the mass of the primary stars is in the range of ∼ 8 − 20M , and the secondary mass is in the range of ∼ 1 − 8M , the system could evolve to become a NS-WD binary if the NS-kick does not disrupt it. We find that the formation rate of disrupted binaries with one NS-component and one WD-component after a Hubble time is about 10 −3 M −1 if natal kicks are taken from the Hobbs-or Arzoumanian-distribution of SN-kicks. This is about two orders of magnitudes above the NSWD-merger rate in the same models. For lower SN-kicks (i.e. our models without a natal kick, and only a mass-loss kick), the disruption rate is 5 · 10 −4 M −1 .
If the magnitude of the SN-kick is of the same order as the average orbital velocity, the post-SN stellar orbits are dependent on the direction of the SN-kick and the orbital phase for eccentric orbits. If the stars are at apocenter and the SN-kick is aligned with the orbital velocity, the binary disrupts readily. On the other hand if the SN-kick is anti-aligned, the orbit shrinks. This can give rise to NSWD binaries that merge on short timescales after the SN-explosion (Sect. 3.2.1), and to systems in which only the secondary star initiates a phase of mass transfer. The latter systems are sufficiently wide initially, such that the primary star does not fill its Roche lobe, the stars practically evolve as single stars, until the primary collapses in a SN-explosion, in which the orbit shrinks. From this point on, the system evolves in a similar fashion as in the 'NS-WD direct' channel (see Fig. 2 from line 5 onwards); the secondary initiates one or two mass transfer phases, and after the formation of the NSWD, the system merges due to the emission of gravitational waves. The rate of this channel is a few 10 −8 M −1 .

The offset distribution
The SN-kick imparts some velocity to the binary even if the NS receives no natal kick. Therefore, the spatial distribution of mergers can differ from the initial galaxy density distribution i.e. the merger takes place at an offset. This effect can be important for the identification of WD-NS mergers among the bulk of transient extragalactic events, and eventually even to distinguish between different natal kick distributions for NSs.
In order to estimate the expected offsets for the NS-WD mergers of our BPS models, we integrate the motion of each binary in various galaxy potentials. The binary motion is integrated starting from the formation of the NS until the final merger (using the python package galpy 3 (Bovy 2015) which makes use of 4th order symplectic integrator). Below we briefly describe the setup of these simulations and the resulting distributions of offsets.
Due to the wide delay time distribution of NS-WD mergers, these events can take place in both old (elliptical) and young (disk) host galaxies. The gravitational field of the host and its symmetry play an important role in establishing the offset distribution. We consider four types of galaxies (based on Belczynski et al. 2002Belczynski et al. , 2006: dwarf and giant ellipticals, as well as a typical and bulgeless disk galaxies. The giant elliptical is constructed out of two potentials, one for the stellar and one for the dark matter component. For the former we adopt the Hernquist potential (Hernquist 1990) with a scale length of a = 5 kpc and mass M = 5 × 10 11 M , and a Navarro-Frenk-White (NFW) potential (Navarro et al. 1996) with the same parameters. The model for the dwarf elliptical galaxy contains the same two potentials but with a = 0.5 kpc and M = 5 × 10 8 M . The disk galaxy is constructed out of three potentials; 1) a Miyamoto-Nagai potential (Miyamoto & Nagai 1975) for the disk taking a radial scale length of a d = 4.2 kpc, a vertical scale length of b d = 0.198 kpc, and mass M = 8.78 × 10 10 M , 2) a Hernquist potential for the bulge with a = 0.79 kpc and M = 1.12 × 10 10 M , and (3) a NFW potential for the dark matter halo with a = 6 kpc, M = 5 × 10 10 M and a cutoff at 100kpc.
The initial spatial distribution of the binaries follow the Hernquist density profile in the case of the elliptical galaxy. For the disk galaxy, the initial positions of the stars are drawn from the Miyamoto-Nagai density profile (disk origin) or the Hernquist density profile (bulge origin). The initial speed (before the kick is imparted due to SN explosion) of the disk-born binary is chosen to be the circular speed. For the initial speeds in the bulges, we follow the prescription from Belczynski et al. (2002Belczynski et al. ( , 2006 i.e., we use the circular speed and randomize the orientation of the angular momentum. We have tested an alternative approach, where we draw the initial speed using the stellar dis-3 http://github.com/jobovy/galpy tribution function obtained under the assumption that the stellar system is ergodic and has reached a dynamical equilibrium. Our simulations show that these two approaches lead to very similar offset distributions.
We take the star formation history of the different types of galaxies into account. We assume a constant star formation rate for 10 Gyr for the disk galaxy without a bulge. In addition, for the case of the disk galaxy with a bulge, we assume star formation is enhanced in the bulge by a factor 10 during the first Gyr. For elliptical galaxies, we assume a constant star formation rate for the first five Gyr.
We assume that the orientation of the recoil velocity has a uniform distribution on a sphere. The recoil velocity is added to the binary velocity in the galaxy as a vector. The energy conservation level of the integrations is checked for each individual orbit to remain below |(E(t merger ) − E 0 )/E 0 | < 10 −6 . If the energy is not well conserved, the orbit is integrated once again with 80 times smaller time steps and checked for energy conservation on level |(E(t merger ) − E 0 )/E 0 | < 10 −4 . If the integrator still cannot guaranty the energy conservation, the orbit is removed from consideration. Usually, 3-5 out of ∼ 10000 orbits are removed, which should not significantly affect our results.
We find that binaries receiving recoil speeds of ∼ 200 km/s escape from dwarf elliptical galaxies, whereas they remain bound in the case of giant elliptical hosts. The offset from the center of a giant elliptical galaxy is negligible, so the spatial distribution of mergers follows the same stellar density of the underlying giant elliptical host galaxies.
For a disk galaxy the stellar density decays faster in the vertical direction, so a binary with a recoil speed of ∼ 100 km/s in this direction can be seen at a noticeable offset (1-10 kpc) when viewed edge-on, but not in the case of face-on orientation. Our simulations show that the offset distribution is very similar for the case of the disk galaxy with and without a bulge.
In Figure 8, we compare the initial and final (at the time of the merger) offset distributions for the dwarf elliptical and the disk galaxy models. The general trend is that while larger kicks reduce the merger rate, they give rise to larger offsets from the host galaxy. For the disk galaxy, the mean offset (face-on orientation) is ∼20-40 kpc for the strong natal-kicks of the Hobbs distribution, ∼20-30 kpc for the Arzoumanian distribution, and ∼8 kpc without natal kicks. For the elliptical galaxy, the mean offset are ∼700-800 kpc, ∼250-600 kpc, and ∼20-250 kpc, for the Hobbs-, Arzoumanian-, and Blaauw-distribution respectively. The offsets are most pronounced in the dwarf galaxy; over 40% of systems are found at offsets around 0.1-1 Mpc for the Arzoumanian distribution, and it increases to 80% for the Hobbs distribution

Gravitational waves
NS-WD binaries eventually merge due to the final inspiral phase driven by GW emission. The inspiral continues down to the point where the WD is disrupted by the NS, likely forming an accretion disk around the NS. The tidal radius is of the order of (M NS /M WD ) 1/3 R WD , i.e. somewhat larger than a WD radius. The typical GW frequencies corresponding to the GW inspiral of NS-WD binaries even at their final shortest period orbits are typically ∼ 0.1 Hz, far below the frequency range observable by current and planned aLIGO GW detectors. Nevertheless, such binaries might be detectable by future space-born GW-missions such as LISA or DECIGO. Using similar calculations as described in Michaely & Perets (2018) we show the distribution of the positions of NS-WD mergers in the GW strain-frequency plot (throughout their evolution) compared with the expected sensi-tivity curves for LISA (solid lines) and DECIGO (dashed lines), where the latter are shown for sources originating at three different distance scales, Galactic (at 10 kpc), close-by extragalactic (at 50 Mpc) and far extragalactic (at 500 Mpc). As can be seen in Fig. 9 typically most of the Galactic (10 kpc) mergers would be observable by LISA and DECIGO, while only a small fraction of the mergers could be observed at close-by extragalactic scales, and none at large distances. Generally the scaling is linear with distance, and these numbers can be easily scaled to arbitrary distances.

Discussion and summary
In this work we have made the first systematic study of the demographics of NS-WD mergers, their properties, rates, delaytime distributions and observable aspects. We considered a variety of initial conditions and studied the sensitivity of our results to the various uncertainties in the physical processes involved in the stellar evolution of NS-WD progenitors. In particular we explored the dependence on NS natal kicks, mass transfer in binaries and common-envelope evolution.
Evolutionary channels: We find that in the main channel for NS-WD mergers, the white dwarf forms before the NS. The primary transfers mass to the secondary as it evolves off the main sequence, increasing the mass of the secondary. The latter then becomes more massive than the original primary and ends its life as a neutron star after the primary had already become a WD. Though the overall results depend on the various assumptions and initial conditions, the different models we consider provide qualitatively similar predictions, and quantitatively differ by at most a factor of 2-3 in the overall expected rates. Tables 2-3 summarize the main differences.
Rates, delay time distributions and host-galaxies: The observed rate of supernova Type Ia in field galaxies is about 10 −3 M −1 (e.g. Maoz et al. 2014;Maoz & Graur 2017). The rates of NS-WD mergers depend on the specific assumptions and initial conditions considered, but are generally in the range of 3.7 − 18 · 10 −5 , about 1.5 − 5% of the predicted rate of WD-WD mergers and 2 − 13% of the observed type Ia SN rate (see Tables  2-3). The delay time distribution peaks at early times (< 1 − 2 Gyrs), but the tail distribution extends up to a Hubble time. In a small fraction (a few 0.1%) of the cases, the SN producing the NS precedes the NS-WD merger by less than 100 yrs, and is therefore potentially observable. The delay time distribution peaking at early times suggest that NS-WD mergers are most likely to be found in late-type, disk and star-forming galaxies with only small fractions expected to be found in early type elliptical/S0 galaxies.
Composition: In most models the majority of NS-WD mergers involve CO-WDs with smaller fractions of ONe, and very small fractions of He WDs (0.3 − 1.4%; and a negligible fraction of hybrid He-CO WDs).
Offsets: The offsets of the expected location of NS-WD mergers generally follow the stellar density of their host galaxies. The only exception is the case of dwarf galaxies which escape velocity is small. In models which include NS natal kicks, the amplitude of the kicks could be sufficiently large as to eject the NS-WD binaries from the host galaxy, leading to very large up to a few 100 kpc offsets.
Possible observational candidates: Given the predicted properties, one may consider the observational manifestation of NS-WD mergers. (Metzger 2012) suggested that NS-WD merges could be related to the class of faint type Ib Ca-rich SNe Perets et al. (2010). However, our results indicate that the DTD of NS-WD mergers is inconsistent with the observed distribution of Ca-rich SNe, which typically explode in old environments (Perets et al. 2010;Kasliwal et al. 2012;Lyman et al. 2013;Perets 2014). Given the WD compositions (lacking in helium WDs), and the little amounts of intermediate and iron elements produced in such explosions (Zenati et al., in prep.), these transients are likely most similar, spectrally, to type Ic SNe. This, again, is at odds with the observed helium and significant amounts of intermediate elements in the spectra of Ca-rich SNe (Perets et al. 2010). The overall rates we find could be consistent with the estimated rates of Ca-rich SNe (Perets et al. 2010), but more recent estimates based on the PTF survey data suggest that the rates of Ca-rich SNe could be significantly higher (30 − 90% of the Ia SNe rate (priv.comm. Chris Frohmaier); factors of 3 − 7 times than the highest rate estimated in any of our models, Tbl. 3).
Another possibility is that NS-WD mergers manifest observationally as rapidly evolving SNe, a class or several classes of rapidly declining and energetically weak likely type-Ic SNe (Chevalier & Plait 1988;Poznanski et al. 2010;Kasliwal et al. 2010;Perets et al. 2011;Drout et al. 2013Drout et al. , 2014, mostly exploding in late-type galaxies. The estimated rates and DTD of such SNe could be consistent with our demographic results of NS-WD mergers, but more detailed studies of the expected light curves and spectra of such explosions is required for direct comparison. Gravitational-waves signature: The GW-inspiral of NS-WD binaries can be observable only by future space-born GW missions, as they peak at ∼ 0.1 Hz frequencies, far below the sensitivity range of aLIGO. These could be observed by the future LISA mission for Galactic sources, but local extragalactic sources could only be marginally detected, and none would be detected at high red-shifts. The proposed DECIGO mission would, however, be highly sensitive to such mergers even at larger distances. White dwarf mass (M ) Fig. 7. White dwarf mass as a function of delay time for three different models of common-envelope evolution. From top to bottom, the main models αα, αα2, and γα are shown, respectively, for the SN-kick distribution of Hobbs et al. (2005). The grey scale corresponds to a density of objects on a linear scale.
Article number, page 13 of 15 A&A proofs: manuscript no. NSWD   Fig. 9. The distribution of GW-sources from NS-WD mergers in the strain-frequency domain. The sources are assumed to be at 10 kpc distances. The sensitivity curves of LISA (solids lines), DECIGO (dashed lines) and aLIGO (dotted lines) are shown. Also shown are scaled sensitivity curves, as to show detectability of the same GW sources if scaled to 50 and 500 Mpc distances. The different panels show four of our main models. On the top left and right model ααand αα2 are shown respectively with the natal kick distribution of (Hobbs et al. 2005). On the bottom left and right are shown model αα with the kick distribution based on (Arzoumanian et al. 2002) and (Blaauw 1961), respectively.