Electromagnetic transients and gravitational waves from white dwarf disruptions by stellar black holes in triple systems

Mergers of binaries comprised of two compact objects have been under recent intense scrutiny. Such mergers can give rise to explosive transient events, heralding the birth of exotic objects which cannot be formed through single star evolution. Using a large number of direct N-body simulations, in this paper we explore the possibility that a white dwarf (WD) is dynamically driven to tidal disruption by a stellar-mass black hole (BH) as a consequence of the joint effects of gravitational wave (GW) emission and Lidov-Kozai oscillations imposed by the tidal field of a outer tertiary companion orbiting the inner BH-WD binary. We explore the sensitivity of our results to the distributions of natal kick velocities imparted to the BH and WD upon formation, semi-major axes and eccentricities of the triples, and stellar mass ratios. Under the assumption of momentum-conserving natal kicks, we find rates of WD-TDEs through the triple channel in the range $1.2\times 10^{-3}-1.4$ Gpc$^{-3}$ yr$^{-1}$ for $z\leq 0.1$, rarer than stellar TDEs in triples by a factor of $\sim 3$--$30$, depending on the magnitude of the natal kicks. The uncertainty in the TDE rates may be greatly reduced in the future using gravitational wave (GW) observations of Galactic binaries and triples with LISA. WD-TDEs may give rise to high energy X-ray or gamma-ray transients of duration similar to long gamma-ray bursts but lacking the signatures of a core-collapse supernova, while being accompanied by a supernova-like optical transient which lasts for only days. WD--BH and WD--NS binaries will also emit GWs in the LISA band before the TDE. The discovery and identification of triple-induced WD-TDE events by future time domain surveys and/or GWs could enable the study of the demographics of BHs in nearby galaxies.

Collaboration 2018). Thanks to the discovery of gamma-ray and non-thermal afterglow emission in coincidence with the LIGO-detected merger GW170817 (Abbott et al. 2017), binary NS mergers are now the confirmed progenitor of at least one short gamma-ray burst (GRB). NS-BH mergers may also produce short GRBs, at least for small mass ratios and high BH spin such that the NS is tidally disrupted outside of the BH horizon (e.g. Foucart et al. 2018). The coalescence of binary white dwarfs (WDs) provide a likely pathway to produce Type Ia supernovae (SNe; Katz & Dong 2012;Maoz et al. 2014;Livio & Mazzali 2018;Toonen et al. 2018;Hamers 2018).
Mergers of NS-WD and BH-WD binaries are expected to occur as well. To date, ∼ 20 NS-WD binaries have been confirmed in our Galaxy (van Kerkwijk et al. 2005), while only one field BH-WD binary candidate is presently known (Bahramian et al. 2017). In such binaries, the WD may approach the NS or BH close enough to be disrupted as a tidal disruption event (WD-TDE). For example, such coalescence events could result from GW emission in isolation (Metzger 2012) or as a consequence of non-coherent scatterings in star clusters (Leigh et al. 2014;Kremer et al. 2019).
What makes the mergers of NS-WD and BH-WD binaries of particular interest is the possibility that they could generate peculiar transients. Several works have characterized the possible electromagnetic (EM) signatures of the tidal disruption of a WD by a NS or a BH (Fryer et al. 1999;King et al. 2007;Metzger 2012;Fernández & Metzger 2013;Margalit & Metzger 2016;Toonen et al. 2018;Zenati et al. 2019;Fernández et al. 2019). In particular, such events may produce a high energy transient similar to a GRB, or thermal emission similar to a short-lived supernova (Metzger 2012; Zenati et al. 2019). Interesting is also the case in which the WD is disrupted by an intermediate-mass black hole (IMBH; Rosswog et al. 2008;MacLeod et al. 2016;Fragione et al. 2018); tidal compression during such WD-IMBH events could generate a large quantity of 56 Ni capable of powering a peculiar Type Ia-like supernova. NS-WD, BH-WD, and IMBH-WD encounters also produce GW emission up to the point of disruption, observable by the planned LISA detector.
In this paper, given the preponderance of triples in the Galaxy (e.g. Leigh & Geller 2013), we explore a new triple channel of WD-BH merger events, in which the WD is driven sufficiently close to the BH to be tidally disrupted as a consequence of the joint effect of GW emission and Lidov-Kozai (LK) evolution imposed by the tidal field of a third companion that orbits the BH-WD binary. We start from the progenitors of the BH and WD and model the effects of natal kicks during the formation of the compact objects (e.g. BH birth in a supernova) and the survival of the triples. Given the many uncertainties involved in the modelling of binary evolution, we explore a variety of models which make different assumptions about the distributions of natal kicks, semi-major axes and eccentricities of the triple, and initial stellar mass ratios. In total, we run ∼ 10 4 direct N -body simulations to explore the prospects for BH-WD systems. We determine how the probability of a WD-TDE depends on these assumptions, map the parameter distributions of merging systems back to the initial distributions, and compute the WD-TDE rate in the Universe through the triple channel.
The paper is organized as follows. In Section 2, we present our numerical methods and describe the properties of the triple population that we evolve. In Section 3, we discuss the parameters of the merging systems. The implications for possible electromagnetic counterparts are presented in Section 4, and for gravitational waves in Section 5. Finally, in Section 6, we discuss the implications of our findings and draw our conclusions.

TRIPLE POPULATION
The stellar triples in our simulations are initialized as follows. In total, we consider nine different sets of initial conditions (see Table 1).
In all models, we adopt the Kroupa (2001) initial mass function in the relevant mass range where the constant coefficient takes into account the fraction of stars with m < 0.5 M such that the integral of ∞ 0 f (m)dm = 1. We draw the stellar progenitor of the most massive star in the inner binary m1 from the mass range 20 M -150 M , which we assume collapses to a BH. The exact value of the BH mass depends on details of the stellar evolution related to, for example, metallicity, stellar winds and rotation. However, for simplicity, we assume that MBH = m1/3 (Silsbee & Tremaine 2017;Fragione et al. 2019). In our fiducial model, we adopt a flat mass ratio distribution for both the inner and outer orbit (Sana et al. 2012;Duchêne & Kraus 2013). 1 The mass of the secondary in the inner binary is sampled within the range 1-8 M . We assume that this star gives birth to a WD of mass (Hurley et al. 2000) and radius where M ch = 1.44 M is the Chandrasekhar mass. The mass of the third companion (m3) is drawn from the range 0.5-150 M . We note that we assume that if the mass of the tertiary is in the range 1-8 M it generates a WD, if in the range 8 M -20 M it collapses to a NS of mass 1.3 M , and if in the range 20 M -150 M it collapses to a BH of mass m3/3 (Silsbee & Tremaine 2017). We run one model where all the masses are drawn independently from each other from Eq. 1. For comparison, we also estimate how the final WD-TDE rate changes if the mass ratio distribution is assumed to be log-uniform (Sana et al. 2013). The distributions of the inner and outer semi-major axes, ain and aout (respectively), are assumed to be log-uniform (Kobulnicky et al. 2014), but we also consider a model with uniform distributions of inner Table 1. Models: name, mean of WD kick-velocity distribution (σ), eccentricity distribution (f (e)), maximum outer semi-major axis of the stellar progenitor triple (a 3,max ), fraction of WD-TDEs from the N -body simulations (f WD−TDE ). and outer semi-major axes. We set as a minimum initial orbital separation ain(1 − e 2 in ) ≈ 10 AU to avoid mass transfer (e.g. Antonini et al. 2017), and adopt different values for the initial maximum separation of the triple a3,max = 2000 AU-5000 AU-7000 AU (Sana et al. 2014). For what concerns the orbital eccentricities ein and eout, we assume flat distributions (e.g. Geller et al. 2019). For comparison, we run one additional model where we take a thermal distribution of eccentricities. Finally, the initial mutual inclination i between the inner and outer orbits is drawn from an isotropic distribution, while the other relevant angles are drawn from uniform distributions.
After sampling the relevant parameters, we check that the initial configuration satisfies the stability criterion of Mardling & Aarseth (2001) for stable hierarchical triples. If this is not the case, we sample again the triple parameters as explained above. Otherwise, we let the primary star in the inner binary undergo an SN explosion and instantaneously convert it to a BH. As a result of the mass loss, the exploding star is imparted a kick to its center of mass (Blaauw 1961), and the system receives a natal kick due to recoil from an asymmetric supernova explosion. We assume that the BH natal velocity kick is drawn from a Maxwellian distribution with a mean velocity σ. We implement momentumconserving kicks, in which we assume that the momentum imparted to a BH is the same as the momentum given to a NS (Fryer & Kalogera 2001). As a consequence, the kick velocities for the BHs are lowered by a factor of 1.4 M /MBH with respect to those of NSs. The value of σ is highly uncertain. We adopt σ = 260 km s −1 for NSs, consistent with the distribution deduced by Hobbs et al. (2005), but we also run an additional model where we set σ = 100 km s −1 , consistent with the distribution of natal kicks found by Arzoumanian et al. (2002). We also explore a model where no natal kick is imparted during BH formation. We update the orbital elements of the triple as appropriate (see e.g. Fragione et al. 2019), checking that the new configuration satisfies the stability criterion for stable hierarchical triples (Mardling & Aarseth 2001). If the system remains stable, we assume that the secondary forms a WD. Also in this case, we assume that the WD natal velocity kick is drawn from a Maxwellian distribution. We consider both models in which the WDs receive no natal velocity kick (σWD = 0 km s −1 ) and models where the mean velocity is σWD = 0.5 km s −1 , consistent with the new findings of El-Badry & Rix (2018). After the second SN event, we update again the orbital elements of the triple and again check that it is stable. Finally, if the third companion is more massive than 1 M , we let it undergo conversion into either a WD, NS or BH, of mass m fin Table 1 reports the fraction of systems that are stable after all the SNe have taken place; this is denoted by f stable for each of our models.
We integrate the triple systems by means of the AR-CHAIN code (Mikkola & Merritt 2006, including PN corrections up to order PN2.5. We perform ∼ 1000 simulations for each model in Table 1, and impose a number of stopping conditions as follows: • The system undergoes 1000 LK-cycles, i.e. the total time exceeds 1000 TLK, where the triple LK timescale is where mtot = MBH + MWD + m fin 3 and Pin and Pout are the inner and outer orbital periods, respectively.
• The WD is tidally disrupted by the BH in the inner binary due to a high orbital eccentricity. This occurs whenever their relative distance becomes smaller than the tidal disruption radius, • The system age exceeds 10 Gyr.

Inclination
A BH-WD binary is expected to be significantly perturbed by the tidal field of the third companion whenever its orbital plane is sufficiently inclined with respect to the outer orbit (Lidov 1962;Kozai 1962). Figure 1 shows the inclination probability distribution function (PDF) of systems that lead to a WD-TDE. We show the distributions for a3,max = 5000 AU and different values of σBH and σWD, Models A1-A4 (see Table 1). Most of the WD-TDEs in triples occur when the inclination approaches ∼ 90 • , since in this case the LK mechanism is efficient at pumping ein up to unity.  Figure 2 shows the cumulative distribution function (CDF) of MBH (top panel) and MWD (bottom panel) for systems that produce a WD-TDE for Models A1-A4. Systems with high values of σBH prefer higher BH masses. This is explained by our assumption of momentum-conserving kicks, where BHs receive a kick scaled by 1/MBH. Thus, more massive BHs are imparted lower velocity kicks on average and are more likely to be retained in bound triples, which eventually produce a WD-TDE. The distribution of the mass of the WDs does not display a strong dependence on the assumed mean velocity kicks for BHs and WDs.

Inner and outer semi-major axes
The choice of σBH affects the distribution of the orbital parameters of BH-WD systems that lead to a WD-TDE. Figure 3 shows the CDF of the inner (left) and outer (right) semi-major axes (top) and eccentricities (bottom) of BH-WD binaries in triples that lead to a WD-TDE, for different values of σBH and σWD. As also shown in Fragione et al. (2019), we find that larger mean natal kicks imply smaller inner and outer semi-major axes. This is because high velocity kicks preferentially unbind triple systems with wide orbits. The inner and outer eccentricities, however, do not depend on the assumed value of σBH. Also, the value of σWD does not affect the distribution of the orbital elements of systems that produce a WD-TDE. Fig. 4 shows how the distributions of ain and aout of BH-WD systems that lead to a WD-TDE depend on the initial distribution of the orbital elements and a3,max. We find that larger values of a3,max lead to larger inner and outer semimajor axes, though the dependence on this parameter is not significant. Model C1, where an initial thermal distribution of ein and eout is assumed, predicts a distribution similar to Model D2, where a3,max = 7000 AU. The CDFs are significantly affected by the choice of the initial distribution for ain and aout. We find that ∼ 50% of the BH-WD systems that lead to a WD-TDE have ain 50 AU and aout 1000 AU in Model A1 (f (a) log-uniform) and ain 200 AU and aout 5000 AU in Model B1 (f (a) uniform). Also in this case, the distributions for ein and eout do not depend on the details of the initial conditions. Figure 5 reports the distribution of WD-TDE times for all models (see Tab. 1). The shape of these CDFs is quite universal and does not depend on the assumed value of the mean kick velocity for BHs and WDs nor on the initial distribution of semi-major axes and eccentricities. In order to compute the WD-TDE rate from BH-WD mergers in triples, we follow a similar calculation to that in Silsbee & Tremaine (2017) and in Fragione et al. (2019). We assume that the local star formation rate is ηSFR = 0.025 M Mpc −3 yr −1 , thus the number of stars formed per unit mass, volume, and time is given by (Bothwell et al. 2011),

Rates
where m = 0.38 M is the average stellar mass. Adopting a constant star-formation rate, the WD-TDE rate in triples is then, Here f3 is the fraction of stars in triples, f stable is the fraction of sampled systems that are stable after the SN events take place, and fWD−TDE is the fraction of systems that produce a WD-TDE (see Tab. 1). The factor η assures that, when sampling the mass ratio q12 of the inner binary, the secondary (1 M m2 = q12m1 8 M ) produces a WD, where fq(q12) is the mass ratio distribution of the inner binary. We get η = 0.21 and η = 0.25 for an uniform and log-uniform mass ratio distributions, respectively. The factors ζ and κ take into account two main processes during the earlier evolution of the system which prohibit a WD TDE (Shappee & Thompson 2013). The first comes from the fact that stellar triples can merge during their main sequence (MS) life before the primary forms a BH as a result of the KL dynamics, which we have not modeled here. To estimate ζ, we conservatively consider that all stellar triples whose initial KL timescale is less than the lifetime of the primary star (∼ 7 Myr;Iben 1991;Hurley et al. 2000;Maeder 2009) in the inner binary merge as MS stars (Rodriguez & Antonini 2018). We find that the fraction of these triples is ζ ∼ 0.60 on average, except for Model A3 and Model A4 where we find ζ ∼ 0.35. Furthermore, we check the fraction of systems that produce a stellar mean sequence TDE instead of a WD TDE, i.e. before the secondary star in the inner binary forms a WD. We estimate this fraction to be κ ∼ 0.15 from the results of Fragione et al. (2019). In our calculations, we adopt for the triple fraction f3 = 0.25 and fWD−TDE ∼ 0.21 on average (see Tab. 1). The fraction of stable systems after the SNe depends on the value of σBH and σWD, and on the details of the distributions of initial parameters. We report f stable for all our models in Tab. 1. Using the minimum and maximum values of f stable in Tab. 1, our final estimated WD-TDE rate is in the range, For a log-uniform distribution of mass ratios, we estimate a rate ∼ 1.5 times larger. Considering the signal up to z = 0.1, the WD-TDE rate becomes, ΓWD−TDE(z 0.1) = 1.2 × 10 −3 − 1.4 yr −1 .
Finally, we note that we are not taking into consider- ation fallback in our calculations, whose effect would be to increase the WD-TDE rates for large σBH's since it would give smaller natal kick velocities.

ELECTROMAGNETIC COUNTERPARTS OF BLACK HOLE-WHITE DWARF TIDAL DISRUPTION EVENTS
Previous works have considered possible electromagnetic signatures of the tidal disruption of a WD by a stellar-mass compact object, such as a NS or a BH (Fryer et al. 1999;King et al. 2007;Metzger 2012;Fernández & Metzger 2013;Margalit & Metzger 2016;Zenati et al. 2019;Fernández et al. 2019). As discussed, the WD is tidally disrupted once its orbital pericenter radius, Rp, decreases below the tidal radius RT (see Eq. 6). Tidal pinching of the WD and/or tidal tail intersection can in principle result in thermonuclear burning during the WD disruption process (Luminet & Pichon 1989;Rosswog et al. 2009;MacLeod et al. 2016;Kawana et al. 2018), particularly for high penetration factors However, unlike the focus of the present paper, most of these works consider massive 100 M black holes, for which high β 1 and thus strong tidal compression is possible. 2 We do not generally expect significant nuclear burning during the disruption by lower-mass black holes.
The tidal disruption imparts a specific energy spread to the WD debris (Rees 1988 This greatly exceeds the initial orbital binding energy of the WD, E orb ∼ GMBH/a, for initial WD semi-major axes obeying The condition ∆Et E orb is easily satisfied by the WD-TDEs in our population. In this case, the half of the disrupted WD furthest from the BH at the time of disruption receives positive energy and is ejected promptly from the system. The other half of the WD is tightly bound to the BH and returns to the tidal radius over a characteristic fallback time corresponding to the orbital period of matter with binding energy ∆Et = GMBH/at (e.g. Stone et al. 2013) In the top panel of Figure 6, we illustrate t fb for Models A1-A4.
Also note that we are justified in neglecting the influence of a binary companion on the dynamics of the mass fallback (Coughlin et al. 2017;Liu & Lai 2019). This is because the apocenter radii ∼ a of the bound debris a 0.41(M/10M ) 1/3 (T /1 month) 2/3 AU, where T is the elapsed time since the disruption, is much smaller than the separations of the outer companion of the systems considered here (Fig. 3).
For the bound fallback material to circularize and hence form an accretion disk, it must lose a significant amount of energy. Circularization is believed to be aided by relativistic effects, since apsidal precession causes highly eccentric debris streams to self-intersect (e.g. Hayasaki et al. 2016;Sadowski et al. 2016;Stone et al. 2019). However, whether circularization can be fully realized before the end of the actual TDE still remains an issue of discussion (see e.g. Piran et al. 2015); in the case of stellar mass BHs, it is aided by the fact that the bound debris are not highly eccentric ). Additionally, a large fraction of the tidally disrupted material is expected to be flung out and become unbound as a result of heating associated with inter-stream shocks (Ayal et al. 2000).
For the debris which remains bound, at times t t fb , the fall-back rate obeyṡ wherė is the peak fall-back rate. Once in a circular disk at Rout ∼ 2Rt, matter is fed onto the BH on the viscous timescale, where H/Rout ∼ 1 is the aspect ratio of the disk and α its effective viscosity. To the extent that (MBH/MWD) 1/2 10 1 10 2 t fb (s) α −1 , the viscous timescale is generally longer than the fallback time. However, for simplicity we adopt Eq. (18) for the BH accretion rate hereafter (though note that the true accretion rate could be smaller if tvisc t fb ). As matter accretes deeper into the potential well approaching the BH, the increasingly high densities and temperatures of the accretion flow will burn the WD material into increasingly heavy elements at sequentially smaller radii, generating an onion-skin like radial structure to the disk composition (Metzger 2012).
Given the very high accretion rates, photons are trapped in the accretion flow and radiative cooling is inefficient. Under these conditions, powerful disk winds driven by the released accretion energy are likely to carry away most of the accreting material before it reaches the central BH (e.g. Narayan & Yi 1995;Blandford & Begelman 1999 where p ≈ 0.7. The ∼ 99% of the matter not accreted by the BH is ejected in a wind (e.g. Margalit & Metzger 2016;Fernández et al. 2019). In the bottom panel of Figure 6, we illustrateṀBH|t fb for Models A1-A4. The inner parts of the accretion flow (near the central BH) could generate a relativistic jet similar to those which give rise to gammaray bursts (e.g., Fryer et al. 1999;King et al. 2007). From Fig. 6, we predict peak accretion rates of ∼ 10 −3 − 10 −2 M s −1 , with a typical timescale varying between a few tens and a few hundreds of seconds (models A1 and A2 have especially long timescales). Assuming a jet launching efficiency of j 0.1, the peak jet power could therefore be jṀ c 2 10 50 −10 51 erg s −1 . Internal dissipation (e.g. shocks or magnetic reconnection) within such a jet could plausibly give rise to a short-lived (∼ several tens to several hundreds of seconds) gamma-ray or X-ray transient with a luminosity 10 49 − 10 50 erg s −1 .
Have such events already been observed? The distribution of accretion rates and durations largely falls within the range observed for the canonical long GRBs, with the tail of the distribution approaching the timescales of a sub-class with similar characteristics, but longer duration, known as the 'ultra-long' GRBs (ULGRBs) (Levan et al. 2014). It remains debated whether ULGRBs are simply the longest lasting members of a single, continuous LGRB population (Zhang et al. 2014) or whether they represent a distinct class with potentially different progenitors (Levan et al. 2014). Either way, an important observational feature of the TD events discussed here would be the lack of signatures of a core-collapse supernova in association with a GRB-like type event. Wolf-Rayet stars are believed to be the progenitor stars of the canonical LGRBs (Woosley & Bloom 2006), while a working model for the ULGRBs is the collapse of a blue supergiant star (Perna et al. 2018; though competing models exist involving millisecond magnetar engines, e.g. Greiner et al. 2015;Metzger et al. 2015). Among the events observed so far, an interesting one is GRB060614, with a duration of 102 s. At a redshift of z = 0.125, its associated core-collapse SN should have been detected, but it was not, calling for the possibility of a new γ-ray burst classification (Gehrels et al. 2006), which King et al. (2007) suggested might be indicative of a WD-NS merger. Future events of this kind will hence deserve special attention.
However, while not accompanied by canonical corecollapse SNe, WD-BH mergers may be accompanied by fast-evolving supernova-like transients. As mentioned above, half of the white dwarf is unbound promptly during the tidal disruption process at a characteristic velocity vt ∼ (GMBH 1/3 M 2/3 WD /RWD) 1/2 ≈ 4.4 × 10 3 km s −1 . Due to outflows from the accretion disk, the majority of the bound half of the WD will also be ejected, with a range of velocities ∼ 10 4 − 10 5 km s −1 (Metzger 2012; Margalit & Metzger 2016; Fernández et al. 2019). Due to the low ejecta mass ∼ MWD 1 M , any thermal transient would be ex-pected to peak much faster than normal supernovae, e.g. on a timescale of tsn ∼ days instead of weeks. What source of luminosity would power the supernovalike emission? Although little radioactive 56 Ni is likely to be produced during the tidal compression, a small quantity of 56 Ni is produced by the inner regions of the accretion flow (Metzger 2012). Fernández et al. (2019) predict that the accretion flows generated by the merger of quasi-circular BH-WD systems will generate ∼ 10 −3 −10 −2 M in ejected 56 Ni (see their Table 2), in which case the resulting thermonuclear supernovae would peak at a luminosity ∼ 10 40 − 10 41 erg s −1 , i.e. 10-100 times less luminous than normal Type Ia supernovae.
The luminosity of the thermonuclear supernova could be substantially boosted if the ejecta is heated by ongoing outflows from the central engine (e.g. Dexter & Kasen 2013). In particular, if 10% of the accretion power reaching the BH goes into powering the supernova luminosity through accretion-disk winds, then from equations (18) and (20), we see that the peak luminosity could reach L pk ≈ 0.1ṀBH|t fb c 2 tsn t fb −5/3 ∼ 10 43 − 10 44 erg s −1 (21) for characteristic parameters, e.g. tsn ∼ 1 d. Consistent with this, numerical simulations of the radiation-hydrodynamic evolution of disk winds by Kremer et al. (2019) showed that it can produce optical transients with luminosities ∼ 10 41 − 10 44 erg s −1 , on timescales varying from about a day to about a month. A number of fast-evolving blue supernovae with luminosities in this range have been discovered in recent years (Drout et al. 2014). However, the total estimated rate of this (highly heterogeneous) population of transients ≈ 4800−8000 Gpc −3 yr −1 exceeds the rate of BH-WD mergers found in this paper, hence suggesting another progenitor origin for the bulk of this population. Furthermore, the closest example yet discovered (AT2018cow; e.g. Prentice et al. 2018) showed evidence for hydrogen in the spectrum, ruling out a BH-WD origin.

GRAVITATION WAVE COUNTERPARTS
It is well known that the extreme mass ratio inspiral of a WD into an IMBH is a target for multimessenger observations including GW detections with LISA (e.g. Hils & Bender 1995;Kobayashi et al. 2004;Dai & Blandford 2013;Eracleous et al. 2019). Stellar-mass BH-WD and NS-WD inspirals could also be observed coincidentally in GWs with In Fig. 7 we show the distributions of the total characteristic GW strain for stellar-mass BH-WD systems that lead to a merger in Models A1-A4, at a distance of D = 10 Mpc. Note that the characteristic noise amplitude for LISA is 8 × 10 −21 at 0.09 Hz (see Fig. 6 in Robson et al. 2019). We follow Robson et al. (2019) to calculate the signal to noise ratio (SNR) of detecting the GWs for a WD-BH inspiral using the current design of LISA with arm length of 2.5 × 10 6 km. For a 4 year observation, we find that the binary orientation averaged SNR is higher than 8 up to 10 Mpc for MBH = 40 M , MWD = 1.0 M . We conclude that the GW observations of WD mergers with stellar-mass BHs or NSs will be limited to the local Universe typically within ∼ 10 Mpc.
The detection volume is VGW ∝ h 3 c /[f Sn(f )] 1/2 , where Sn is the noise power spectral density, which scales as The GW-observed TDE rate is strongly biased towards higher BH mass and higher WD masses.
Note that BH-WD and NS-WD binaries typically emit GWs in the LISA band for thousands of years before merger. Furthermore, KL oscillations induced by a triple companion leaves a time-dependent imprint on the GW spectrum of the inner binary (Randall & Xianyu 2019; Hoang et al. 2019). Thus, LISA observations of GWs emitted by binaries in the Galaxy may be used directly to constrain the expected number of BH-WD and NS-WD TDE rate in the Universe.
If a NS-WD, BH-WD, or IMBH-WD TDE happens within the detectable LISA volume, the GW measurements can be used to determine the parameters fGW given by Eq. (22), the chirp mass of the binary M = M 3/5 BH M 3/5 WD /(MBH+MWD) 1/5 , and the distance to the source D independently of electromagnetic observations. Coincident GW detections may help to secure the identification of the electromagnetic counterpart. The joint multimessenger analysis of the merger of a WD with a stellar BH, NS, or IMBH offers to gain a more accurate understanding of these astrophysical sources.

CONCLUSIONS
The mergers of binaries comprised of two compact objects can produce diverse explosive transient events, such as GW chirps, Type Ia and GRBs. Though they have received comparatively less attention in the literature, the mergers of NS-WD and BH-WD binaries are expected to generate transients if the WD approaches the NS or BH close enough to be disrupted in a WD-TDE.
This paper explores a new triple channel for WD-BH mergers driven by the joint effect of GW emission and the LK mechanism. We explore the sensitivity of our results to different assumptions for the distributions of natal kick velocities imparted to the BH and the WD, the semi-major axes and eccentricities of the triple and the initial stellar masses. We estimate the rate of WD-TDEs in triples to be in the range 1.2 × 10 −3 − 1.4 Gpc −3 yr −1 for z 0.1, under the assumption of momentum-conserving natal kicks. Compared to stellar TDEs in triples, WD-TDEs are therefore a factor of ∼ 3-30 rarer.
In our simulations we check that the triple systems remain stable after each SN event. Systems that become unstable may still merge, but they are not taken into account in our results. Furthermore, we conservatively neglected the contribution of stable triple systems whose KL timescale is lower than the lifetime of the progenitor stars in the inner binary, since in these systems the merger may happen in the main sequence phase. Therefore, our inferred merger rates should be rather interpreted as lower limits. Moreover, we are assuming that the SN events take place instantaneously and do not simulate the systems during the main sequence lifetime of the progenitors. This and the details of the specific evolutionary paths, which depend on stellar winds, metallicity and rotation, of the stellar progenitors could reduce the available parameter space for BH-WD mergers (Shappee & Thompson 2013). The situation becomes even more complicated if mass loss during possible episodes of Roche-lobe overflows and common evolution phases in the triple are taken into account, which however are not modeled in a self-consistent way in triple systems because of the possible interplay with the KL cycles during the main sequence lifetime of the progenitors (Di Stefano 2019; Hamers & Dosopoulou 2019). Nevertheless, we tried to quantify these effects by accounting for the progenitors that could merge during their main sequence lifetime and that could lead to a stellar TDE, as in Fragione et al. (2019).
Accretion of the bound debris onto the BH following a BH-WD TDE could power a relativistic jet, generating a burst of high energy X-ray or gamma-ray emission with a duration similar to a long GRB. The heating of white dwarf debris (unbound during the tidal disruption event or in outflows from the accretion disk) by radioactivity or winds from the accretion disk, could generate a rapidly-evolving supernova-like optical transient. Such peculiar transients from BH-NS mergers might be observable by high energy satellites or upcoming time-domain optical surveys, such as LSST. The characterisation of WD-TDE events and their distributions is therefore a fundamental step in ultimately being able to identify them among the myriad of other cataclysmic events. Stellar mass BH-WD, NS-WD, and IMBH-WD, binaries may also be detected in GWs using LISA up to the point of TDE. LISA may also provide an accurate determination of the TDE rates from triples by observing systems thousands of years before merger in the Galaxy. Multimessenger studies of WD TDEs by stellar-mass BHs or NSs will be limited by the LISA detection range of ∼10 Mpc.
The future discovery of a population of WD-TDE could be used to study the demographics of BHs in nearby galaxies and to place constraints on the distributions of natal kicks at BH birth in a complementary way to what now probed by LIGO from BH-BH mergers.