Constraints on neutrino natal kicks from black-hole binary VFTS 243

The recently reported observation of VFTS 243 is the first example of a massive black-hole binary system with negligible binary interaction following black-hole formation. The black-hole mass ($\approx 10\ M_{\odot}$) and near-circular orbit ($e\approx 0.02$) of VFTS 243 suggest that the progenitor star experienced complete collapse, with energy-momentum being lost predominantly through neutrinos. VFTS 243 enables us to constrain the natal kick and neutrino-emission asymmetry during black-hole formation. At 68% C.L., the natal kick velocity (mass decrement) is $\lesssim 10$ km/s ($\lesssim 1.0\ M_{\odot}$), with a full probability distribution that peaks when $\approx 0.3\ M_{\odot}$ were ejected, presumably in neutrinos, and the black hole experienced a natal kick of $4$ km/s. The neutrino-emission asymmetry is $\lesssim 4$%, with best fit values of $\sim$0-0.2%. Such a small neutrino natal kick accompanying black-hole formation is in agreement with theoretical predictions.

Introduction.-Starsseveral times more massive than the Sun end their lives with the collapse of their iron cores.The explosion mechanism and the physics driving the formation of the compact object are still a matter of intense research [1][2][3][4][5][6][7][8].In the delayed neutrinodriven mechanism, neutrinos revive the stalled shock wave, eventually leading to a successful explosion.In this case, the stellar mantle is successfully ejected and the compact-object remnant is a neutron star (NS) in most cases.However, if the explosion mechanism fails, continuous accretion of matter onto the transiently stable proto-NS pushes the latter over its mass limit and a black hole (BH) forms.
In the extreme scenario of complete collapse into a BH, the ejecta mass and natal kicks are thought to be very low (∼1-10 km/s) [29][30][31].In this case, mass-energy is lost via neutrinos and, to a lesser extent, gravitational waves [1,21].This differs from the archetypical scenario in which anisotropic baryonic ejecta are the main carriers of momentum [32].
Observations of stellar-mass BHs in high-mass X-ray binaries (HMXB) have been employed to constrain the impact of natal kicks of collapsing stars on the orbital configuration of massive binary systems [33][34][35].HMXBs are binary systems comprised of OB-type stars that transfer matter via stellar winds [36] to their compactobject companion [37].To date about 150 HMXBs have been discovered in the Milky Way, but only a handful of these are associated with BH companions [38,39].
Recently, spectroscopy [40][41][42][43] and astrometry [44,45] have allowed the first detection of inert (i.e., X-ray quiescent or dormant) stellar-mass BH binaries [46] in the field [47].This population of BH binaries is especially intriguing.These objects have wider orbital periods than HMXBs and the stellar companions are well within their Roche lobes.This configuration, which implies little interaction following BH formation (see Supplemental Material), makes inert BH binaries better probes of natal Constraints on the natal kick from VFTS 243.This figure shows the probability distribution of matching the current orbital period, eccentricity and systemic velocity of VFTS 243, given a certain mass loss (dM ) and natal-kick magnitude (v kick ) following BH formation.In the heatmap, darker (lighter) regions indicate a higher (lower) probability for a certain dM -v kick pair to reproduce the orbital configuration of VFTS 243.The blue shaded region, delimited by the solid blue vertical lines, shows the estimates for neutrino mass-loss decrements from stellar models of BH progenitors [48].The side panels display the marginalized probability distribution (PDF, gray histograms) and cumulative distribution functions (CDF, red curves).With CDF(dM = 1.0 M⊙) = CDF(v kick = 10 km/s) = 0.68, we conclude that solutions with low mass loss and small natal kicks are preferred.The marginalized distributions peak at around dM = 0.3 M⊙ and v kick = 4 km/s, which is consistent with a solution where the mass decrement comes exclusively from neutrino emission.kicks than HMXBs.In this Letter, we present direct inference on neutrino natal kicks for the most massive inert BH detected to date: VFTS 243 [41].Our findings are summarized in Fig. 1.
For BH binaries, the plausibility of the complete collapse scenario can be assessed because the impact on the orbital evolution of mass ejection (dM ) and a natal kick (v kick ) during compact object formation on binary star systems are well understood [49][50][51].For example, nearinstantaneous (quasi-)spherically symmetric mass ejection during the stellar collapse results in a recoil (the Blaauw effect [52]) that leads to a systemic velocity of the binary as a whole, widens the orbit, makes it more eccentric and can even disrupt the binary [53].On the other hand, asymmetries in the ejected mass lead to a wider variety of configurations, potentially modifying the sepa- ration (increase or decrease), eccentricity and inclination of the orbit.Such asymmetries could play an important role in the formation of BH-BH mergers [54], BH HMXBs [55], and inert BH binaries [56,57].
Properties of VFTS 243.-VFTS 243 belongs to the family of inert BH binaries recently discovered in the Milky Way [43] and the Large Magellanic Cloud [41] via spectroscopy (Fig. 2).It is comprised of a mainsequence O star with inferred mass of M * = 25.0±2.3M ⊙ and a BH companion with M BH = 10.1 ± 2.0 M ⊙ , orbital period of P = 10.4031 ± 0.0004 d, and eccentricity e = 0.017±0.012,where the errors are the 1σ uncertainty intervals [41].The relatively high mass of the BH and the nearly circular orbit suggest that the system experienced complete collapse.With a stellar radius (R) well within the binary Roche lobe (f RL = R/R RL ≈ 0.33) [41], VFTS 243 is a relatively wide BH binary (for example, in contrast, Cygnus-X1 is close to filling its Roche lobe with f RL > ∼ 0.99 [58,59]).Moreover, the super-synchronously rotating massive star of VFTS 243 suggests that the effects of tides in synchronizing and circularizing the orbit are negligible [41].
Constraints on natal kicks.-Inorder to analyze the formation of VFTS 243, we use a semi-analytic approach [34] to calculate the probability that a circular pre-collapse binary that received a natal kick during BH formation could reproduce the orbital configuration of the system as observed today.Models of the pre-collapse binary suggest it initially had a short (∼1 d) orbital period and experienced mass transfer [41].For such configurations, the tidal circularization timescale (∼ 10 4 yr [60]) is significantly shorter than the time the stars spend on the main sequence (∼ 10 6 yr [61]) prior to BH formation.Therefore, we assume the binary was circularized prior to collapse.Given that the current configuration is barely eccentric, it is reasonable to assume that the orbit was circular and the marginal eccentricity was induced during BH formation.Alternatively, the current eccentricity could be a small residual from incomplete circularization via tides prior to collapse.For the precollapse orbital configurations we assume independent, uniform wide prior distributions on the orbital period between 5 ≤ P i /d ≤ 15, on the natal-kick magnitudes between 0 ≤ v kick /(km/s) ≤ 200 and on the amount of mass ejected between 0 ≤ dM/M ⊙ ≤ 5. We also assume that the direction of the imparted natal kick is isotropic.We use the post-collapse orbital period, eccentricity and systemic radial velocity as independent constraints (see Supplemental Material), which we justify by the lack of correlation in the observationally inferred orbital parameters [41].
Figure 1 shows the results of our analysis.The heatmap presents two distinct hotspots.In the limit where v kick → 0, we have the solution for pure mass loss (Blaauw effect [52]), where the amount of ejected mass is linearly proportional to the newly acquired eccentricity of the binary.The other spot has dM → 0, i.e., a very small amount of mass is ejected asymmetrically in the frame of reference of the exploding star, and a natal kick makes the orbit eccentric.The marginalized one-dimensional probability distributions have modes at dM ≈ 0.3 M ⊙ and v kick ≈ 4 km/s.Explosions with moderate mass loss ( > ∼ 1 M ⊙ ) and natal kicks ( > ∼ 10 km/s) can occur (see yellow in the heatmap from Fig. 1), but require natal kick directions and magnitudes fine-tuned to balance the mass loss in order to form a low-eccentricity BH binary [56], as demonstrated by the diagonal structure of Fig. 1.Under the assumption that the binary was circular prior to BH formation and has a small but non-zero eccentricity at present, there is no support for a solution with dM = v kick = 0.
In Fig. 1, the vertical blue band shows the estimated mass decrements associated with the total radiated neutrino energies of collapsing stripped stars [48].Numerical simulations of the collapse of massive stars without concomitant supernova explosion in 2D [31] and 3D [62] by the Garching group yield final neutrino-induced BH kick velocities of magnitudes similar to those favored by our analysis of VFTS 243 (0-4 km/s), in particular also for a pre-collapse stellar model with similar mass.Contesting 3D simulations by the Princeton group found that the final neutrino-induced natal kick velocities of BHs formed via failed supernovae could be larger (7-8 km/s [63]), but still within our 1-σ credible intervals.These quantitative differences may be connected to different life times of the NS prior to BH formation or to intrinsic differences of the neutrino asymmetries due to transport.
Under the hypothesis of complete collapse, we perform a back-of-the-envelope estimate on long-term (≫ 1 s) neutrino asymmetries during BH formation.We follow Refs.[14,29] and parameterize the linear momentum transferred to the BH as a result of anisotropic neutrino emission as where α ν is the neutrino emission asymmetry factor, v ν,kick is the neutrino natal kick, dM ν is the mass decrement from neutrino emission and c is the speed of light in vacuum.To estimate α ν and v ν,kick we assume the following (see Supplemental Material for more details).We choose M BH = 10 M ⊙ , similar to VFTS 243.We obtain the interval E tot ν ≈ 1-11 × 10 53 erg from the strippedstar BH progenitors from Ref. [48].Finally, the value of v ν,kick is determined by obtaining the subset of the complete natal kick probability distribution that is consistent with complete collapse.These assumptions result in an upper limit on the neutrino asymmetry of ≈ 4%, where 0% implies spherically symmetric neutrino emission (in the rest frame of the NS), and 33% implies that the dipole component of the neutrino luminosity equals the monopole amplitude [62].However, the asymmetries are between ∼ 0-0.2% for the most likely solution of v ν,kick ≈ 3 km/s (see Supplemental Material), though a solution with no natal kick is allowed.These values are in overall agreement with the estimates of the anisotropy parameter of the total neutrino emission in recent 3D core-collapse simulations with detailed neutrino transport, namely 0.45-0.76%for successful supernova explosions [62] and 0.05-0.15% in simulations of non-exploding models [62] (see, however, Ref. [21], which reports considerably higher values, probably because of differences in the analysis).
Discussion.-Our results provide evidence of neutrinoinduced natal kicks and add strong support to the complete collapse formation scenario [41].In general, the total natal kick of a compact object has a baryonic, neutrino and gravitational-wave component.If matter ejection in a successful explosion takes place, it generally dominates the total kick for higher-mass progenitors [32,62,64,65], and gravitational waves contribute to the natal kick negligibly, since they carry orders of magnitude less energy with respect to baryons and neutrinos (e.g., [31,65]).Our calculations provide constraints on the total natal kick and mass loss, which we find to be largely in agreement with mass loss exclusively through neutrino emission and an associated natal kick, rather than baryonic mass ejecta.Depending on the magnitude and direction, a natal kick from baryonic ejecta [57] could further reduce our limit on neutrino natal kicks.Alternatively, there could be a yet-to-be-determined physical mechanism that results in baryonic and neutrino kicks of a similar magnitude that are anti-aligned and almost completely cancel.
Astrometric microlensing has been suggested as a method to constrain BH natal kicks [66,67]; as long as the observed velocity dispersion is much larger than the presumed neutrino kicks, the latter are unconstrained.The entire BH kick distribution cannot be completely explored with bound binaries [53], but systems similar to or more massive than VFTS 243 are ideal to explore and constrain low natal kicks.The broad implications of the natal kick that VFTS 243 received during BH formation has been recently investigated in the literature [41], showing support for small [57] and moderate natal kicks [56].The solution of very low natal kicks for VFTS 243 is in agreement with neutrino kick estimations from hydrodynamical simulations [31,68,69].In particular, recent hydrodynamical simulations of corecollapse events yield a range of model-dependent neutrino kicks between ≈0-4 km/s in 2D [31] and between ≈0-3 km/s in 3D [62,69].However, other simulations of BH-forming collapses produce neutrino natal kicks between 30 km/s < ∼ v ν,kick < ∼ 100 km/s [21], which would be partially compatible with the tail of our distribution in Fig. 1 but may be difficult to reconcile with other existing theoretical constraints, if confirmed.
For BH progenitors that have been stripped during a mass transfer episode, the complete collapse scenario suggests almost no baryonic ejecta and a mass decrement exclusively from neutrinos.A very massive proto-NS can lead to a total neutrino energy emission of E tot ν ≈ 10 54 erg [48], which roughly corresponds to a neutrino mass decrement of dM max ν ≈ 0.6 M ⊙ .Therefore dM < 0.6 M ⊙ , and more realistically 0.1 < ∼ dM/M ⊙ < ∼ 0.4 (see Supplemental Material), is consistent with natal kicks induced from neutrino emission exclusively [48].Alternatively, dM > ∼ 0.6 M ⊙ implies at least some baryonic-mass ejecta.
The sample of single line spectroscopic (SB1) inert BH binaries is diverse (Fig. 2).VFTS 243 is the most massive and least eccentric of SB1 BH binaries.HD 130298 is slightly less massive than VFTS 243 (with M BH ≈ 9 M ⊙ and M * ≈ 24 M ⊙ ), has a similarly small Roche-lobe filling factor (f RL = 0.26) and a similar orbital period (P orb ≈ 15 d), but it is quite eccentric (e ≈ 0.5).VFTS 514 is even less massive (with M BH ≈ 5 M ⊙ and M * ≈ 19 M ⊙ ) and is also quite eccentric (e ≈ 0.4); with a long orbital period (P orb ≈ 185 d), it is likely far from Roche-lobe overflow.VFTS 779 is the least massive of the SB1 BH binaries (with M BH ≈ 4 M ⊙ and M * ≈ 14 M ⊙ ), has a near circular orbit (e ≈ 0.02) and a long orbital period (≈ 60 d).The spectral type of the star of VFTS 779 indicates it has evolved past the main sequence [42]; at this stage, it is developing a convective envelope which makes tidal circularization orders of magnitude more efficient than for main-sequence stars with radiative envelopes (see also discussion in Supplemental Material).
The sample of BH HMXBs [39,75] is broader in the mass parameter space, but otherwise is more homogeneous (Fig. 2).Most BH HMXBs have short orbital periods (P orb < 6 d), small eccentricities (e < ∼ 0.1) and large Roche-filling factors (f RL > ∼ 0.9).Large Rochefilling factors in HXMBs are in agreement with the theory of wind-accreting X-ray binaries, which predicts a threshold of f RL > ∼ 0.8-0.9 as a condition for observability [76].Cygnus X-1 is the archetype of BH HMXBs, with M BH ≈ 21 M ⊙ and M * ≈ 41 M ⊙ , a short-day orbital period (P orb ≈ 6 d), a large Roche-lobe filling factor (f RL > 0.99), and a near-circular orbit (e ≈ 0.02).While the eccentricities of Cygnus X-1, LMC X-1 and M33 X-1 are similar to that of VFTS 243, both wind accretion onto the compact object [35,60] and tides could have played a role in circularizing these BH HMXBs.For massive, main-sequence stars with radiative envelopes, the dynamical tide is the dominant tidal dissipation mechanism [77].Dynamical tides are not an efficient dissipation mechanism for VFTS 243 [42], the most compact inert BH binary detected to date.The circularization timescale via dynamical tides is a strong function of the separation (τ circ ∝ (a/R) 21/2 [78], where a is the semimajor axis of the binary), which implies that tides are significantly less efficient for the SB1 sample than for the observed BH HMXBs.For example, doubling the orbital period of Cygnus X-1 would result in a similar period to VFTS 243 and would reduce the impact of tides by ∼ 2 orders of magnitude.Moreover, HD 130298 has rather similar properties (masses, periods and Roche-filling factors) to VFTS 243, but is rather eccentric; this demonstrates that dynamical tides are not efficient at circularizing the orbit at such large separations.
Finally, we highlight that for BH binaries with M * > M BH there seems to be a sharp drop in eccentricity for systems with M BH > ∼ 10 M ⊙ .We speculate that this drop could denote the transition between luminous, massshedding explosions and complete collapse [79][80][81][82].The early theory of stellar collapse suggested that less massive cores eject some mass during BH formation, while complete collapse could only occur for cores of mass > ∼ 11 M ⊙ [83].However, a growing body of theoretical work shows evidence that the compact object remnant mass function is a non-monotonic function of mass [1,6,[84][85][86][87][88][89][90][91][92][93] that could even be stochastic [94], dependent on initial rotation and composition [95] and that might be affected by binary interactions [96][97][98].This means that more massive stars do not always result in more massive compact objects; some stellar cores above a certain threshold can result in mass-shedding explosions and form NSs [99].Based on the sharp eccentricity transition between HD 130298 and VFTS 243 (both SB1 BH binaries with similar masses and orbital periods) and the results presented in this Letter, we consider that helium cores with a mass of ≈ 10 M ⊙ could undergo complete collapse.However, this does not rule out the possibility that complete collapse occurs for less massive stars, or that incomplete collapse returns at higher masses.Future observations should provide more precise values on the actual mass threshold for complete collapse.
Conclusions.-Weexplore the effect of mass loss and natal kicks during BH formation in VFTS 243, the heaviest inert stellar-mass BH binary detected to date.Our results show that the marginalized distributions peak around dM = 0.3 M ⊙ and v kick = 4 km/s, respectively; these values are consistent with recent simulations of the stellar collapse predicting mass loss via neutrino emission exclusively [48], fully in agreement with the complete collapse scenario [84].Under the hypothesis of complete collapse, we estimate the asymmetry in the neutrino emission to be ∼ 0-0.2% and provide an upper limit of ≈ 4%.We find that mass loss < ∼ 1.0 M ⊙ and v kick < ∼ 10 km/s are preferred at a 68% C.L., in agreement with other studies in the literature [41,57].Moreover, our results also accommodate solutions with no natal kick and no asymmetries in neutrino emission.The progenitor of the BH component of VFTS 243 is likely a massive stripped star [42] which experiences negligible mass loss during BH formation.Following recent observations of inert BH binaries in the Local Group (Fig. 2), we suggest that complete collapse can occur for stars that end their lives with cores of > ∼ 10 M ⊙ .

Supplemental Material Observational evidence for neutrino natal kicks from black-hole binary VFTS 243
In this Supplemental Material, we first describe the modeling of the natal kicks, discuss the caveats in our model, and compare with other studies in the literature.Then, we describe our analysis on the estimates of the neutrino asymmetries in the context of the black hole (BH) of VFTS 243.Finally, we present a more detailed description on the binary evolution leading to the formation and fate of VFTS 243, as well as justify the assumptions made in our analysis.

Modeling of natal kicks
Evidence for large neutron star (NS) natal kicks comes from observations of young, isolated pulsars with speeds of the order of a few 100 km/s, which are assumed to reflect their speeds at birth [11,100,101].If the natal kick is mainly due to asymmetric baryonic mass loss, any fallback leading to BH formation is expected to suppress the natal kick.For modeling purposes, the natal-kick reduction is often assumed for simplicity to scale proportionally with the fraction of the envelope material that falls back onto the nascent BH [29,83].Ref. [102] finds that BHs in X-ray binary systems could have natal kicks up to a few 100 km/s, following a possibly bimodal distribution.
We modeled the effects of a supernova in a binary system with arbitrary natal kick and mass loss based on Ref [34].We do not distinguish baryonic (dM bar ) from neutrino (dM ν ) mass losses, so our resulting upper limits apply to dM = dM bar + dM ν .Similarly, upper limits are placed on the overall natal kick v kick , which is the magnitude of the vector sum of any hydrodynamic and neutrino natal kicks.The momentum radiated in gravitational waves is negligible [103] and we therefore ignore it in the natal-kick analysis.We assume the orbit was circular prior to the supernova, and that there is no change to the mass or instantaneous velocity of the companion star.
Fixing the present-day masses for a given binary, we sample from a set of independent initial distributions on dM , v kick , and the pre-collapse orbital period P i .In our analysis, we assume independent uniform initial distributions on all three parameters, with ranges between 0 ≤ dM/M ⊙ ≤ 5, 0 ≤ v kick /(km/s) ≤ 200, and 5 ≤ P i /d ≤ 15.The final joint distribution is marginalized over P i .The kick direction is taken to be isotropic in the frame of the collapsing star.We construct a probability for each triplet (dM, v kick , P i ) from the fraction of the sampled kick directions which result in a final orbital period P f = 10.4031 ± 0.0004 d and eccentricity e f = 0.017 ± 0.012, where the errors are the 1σ uncertainty intervals provided in the detection paper of VFTS 243 [42].We use the systemic radial velocity as an additional constraint.The systemic radial velocity of VFTS 243 is 260.2 ± 0.9 km/s, as reported by the discovery paper [42].The 1σ of the measured mean of the Tarantula region of 271.6 ± 12.2 km/s suggests a low (∼ 1-10 km/s) systemic radial velocity.We only consider systems with systemic velocity of less than 17 km/s, which correspond to the 68% percentile on the absolute value of the difference of the two normal distributions for VFTS 243 and the Tarantula region.From our analysis, the natal kicks that result in a VFTS 243-like binary have a systemic velocity distribution that peaks around 3 km/s (Fig. S1).
In Fig. S1, we show the initial and final distributions leading to 10 6 VFTS 243-like systems.We find that the choice of the range of the initial distributions completely span the pre-supernova parameter-space of interest leading to the configuration of VFTS 243 as observed today, i.e., our analysis is not constrained by the chosen prior boundaries.The impact on the period and eccentricity from significant mass loss can be offset by a large and fortuitously directed natal kick; this fortuitousness is quantified by the solid angle on the kick direction unit sphere within which the final orbital properties are consistent with observations, which is precisely the probability we estimate.For very large dM and v kick , the probability of producing a low eccentricity orbit is vanishingly small, providing a fairly tight constraint on both dM and v kick , as shown in Fig. 1.
In the context of near-circular BH binaries, the scenario of an eccentric pre-collapse binary is unlikely, but cannot be ruled out.From observations of short-period (< 10 d) massive O-type binaries on the main sequence, all of them have e < 0.4, and approximately half have e < 0.1 (cf.Fig. 3 of Ref. [104]).Moreover, VFTS 243 is likely to have experienced a mass transfer episode where the binary could have been further circularized [42], either via tidal interactions, gas drag or both.In the context of Cygnus X-1, the plausibility of an eccentric pre-collapse orbit was explored and concluded to be unlikely [35]; we expect something similar for VFTS 243.Moreover, an eccentric pre-collapse orbit leading to a circular post-collapse orbit requires a fortuitous kick.This rare fortuitous kick would likely be non-negligible in magnitude, broadening the natal kick distributions presented in Fig. 1.We consider the scenario of a near circular pre-supernova orbit as the more likely one, given that it requires less fine tuning.On the other hand, it is possible that some of the observed eccentricity is due to a small residual from the pre-collapse orbit rather than a neutrino kick, which would further reduce our neutrino mass loss and kick estimates.We now briefly contrast the impact of our assumptions on the natal-kick magnitude distribution with the ones in previous work on VFTS 243 [41,56,57].Ref. [57] explores the impact of a uniform prior-consistent with our approach-with that of a Maxwellian prior for the natal kick magnitude.They find that the Maxwellian distribution leads to a slightly larger posterior in the natal-kick parameter space.This is likely because the Maxwellian distribution rarely allows for very low natal kicks, which are the preferred solution in the VFTS 243 configuration.Moreover, the Maxwellian natal-kick distribution is motivated by observations of isolated pulsars [100] and therefore might not be adequate for black holes formed via fallback or complete collapse; in fact, the apparent Maxwellian distribution of NS velocities may be an accidental consequence of the integration of a physical distribution of individual kicks over all progenitor systems [105].We consider the uniform natal-kick magnitude prior to be an agnostic and adequate choice.Ref. [56] perform a full Bayesian analysis, using wide priors for all parameters and comparing uniform and Maxwellian velocity priors.Their results are in agreement with our work and the results from Ref. [57], once the use of different (90%) confidence intervals and large (> 5 M ⊙ ) mass ejecta is accounted for.Finally, they show that the eccentricity induced by large ejecta can be canceled by a strong, fortuitously aligned natal kick; however, this outcome is rather unlikely.

Estimating neutrino asymmetries
We have explored the natal-kick parameter space of VFTS 243 and found that low-mass loss (dM ≈ 0.3 M ⊙ ) and low natal-kick magnitudes (v kick ≈ 4 km/s) are the preferred parameters to match VFTS 243 as observed today (Fig. 1).However, there are also less likely, but plausible, dM -v kick combinations with larger mass loss and natal kicks that could explain the current orbital configuration of VFTS 243.Here we consider exclusively the dM -v kick pairs that are in agreement with models where there are no baryonic mass ejecta and the mass decrement and natal kick come exclusively from neutrinos, which implies v ν,kick = v kick .In practice, this means that we select a subset of the complete natal kick distribution that is within the blue shaded region from Fig. 1.Under this assumption, we calculate the asymmetry in the neutrino emission in the BH of VFTS 243.
We use the results from Ref. [48], which investigates successful and failed supernova explosions with parameterized several uncertainties in the evolutionary channel just introduced.Many initial conditions could result in the current configuration of VFTS 243.The eccentricity distribution of close ( < ∼ 10 d) OB-type binaries suggest that they are never extremely eccentric (e < 0.4) and that they have moderate eccentricities (e ≈ 0.1 − 0.4) or are effectively circular [104,109].The initial orbital period determines whether the first mass transfer episode will occur while the primary is on the main sequence (case A) or just after it has left the main sequence (case B).Case A mass transfer is expected for orbital periods < ∼ 10 d; alternatively case B mass transfer is expected for longer orbital periods [46].This first mass transfer episode can significantly alter the secondary star, which becomes spun-up, rejuvenated and polluted [110].Moreover, the fraction of the donated mass that is accreted, i.e. the conservativeness of the mass transfer episode, directly impacts the orbital evolution; this conservativeness is one of the most poorly constrained parameters in massive binary evolution, along with the specific angular momentum lost during non-conservative mass transfer [111].After the primary has been stripped, the time before it collapses into a BH is relatively short with respect to the duration of the main sequence.If the BH forms via complete collapse, the orbit is not significantly affected, as described in this Letter.Now that the primary has formed a BH, the evolution of the secondary determines the evolution of the system.There are three main physical processes that govern the evolution of detached binaries, or in this case BH binaries: gravitational waves, stellar winds and tides.Gravitational waves dissipate energy and result in the inspiral and circularization of the binary [112].For systems as wide as VFTS 243, gravitational waves are an extremely inefficient dissipation mechanism and their effect is negligible.Stellar winds can generally widen the binary, but hydrodynamical gas drag can play a role if the wind velocity is comparable to the orbital velocity [113].In order to determine the role of gas drag we compare the terminal wind speed of v wind > ∼ 2000 km/s [41] to the relative orbital velocity of VFTS 243 v rel = G(M * + M BH )/a ≈ G(25 + 10 M ⊙ )/(0.45AU) ≈ 260 km/s, where G is the gravitational constant, M * is the mass of the star, M BH is the mass of the BH and a is the semi-major axis.We conclude that for VFTS 243 the binary will likely widen due to rapid ("Jeans mode") mass loss but the eccentricity will remain unchanged [113].Finally, tidal dissipation drives the binary toward synchronization and circularization.The dynamical tide is the dominant tidal dissipation mechanism for main sequence O-type stars [77].For VFTS 243, the dynamical tide is not an efficient dissipation mechanism.The orbital period is too long (∼ 10 d), the cirularization timescale is too long ( > ∼ 10 4 Myr) and the lifetime of the system is too short ( < ∼ 10 Myr).Moreover, VFTS 243 is not yet synchronously rotating, which strongly suggest that tides have not played a significant role in the evolution of the system [41].
Finally, we comment on the main differences between the population of inert BH binaries and BH HMXBs [39, 75] (Fig. S5).Most BH HMXBs are at orbital periods < ∼ 6 d; exceptionally, SS 433 has an orbital period of 13 d and a circumbinary disk [114,115], a unique feature with respect to the other BH binaries in the sample.Excluding HD 96670, for which the presence of a BH still requires validation, SS 433 is the only BH-HMXB with a massive companion for which an eccentric orbit has been derived.However, unlike for other HMXBs, the orbit of SS 433 relies on radial-velocity measurements of variable emission lines, deeming its orbital solution, and in particular the eccentricity, susceptible to systematic errors.By contrast, inert BH binaries have orbital period > ∼ 10 d.For BH HMXBs, their short orbital period implies that tidal dissipation is more efficient than for wide inert BH binaries.The formation of an accretion disk around the BH requires some mass and angular momentum transfer from the star on the compact object; this process is not particularly efficient at circularizing the orbit but can in principle also change the eccentricity of the system [35,60].Ref. [35] explored the natal kick of Cygnus X-1, as well as the orbital evolution of the system due to gravitational waves, stellar winds and tides.They find that stellar tides (winds) contribute the most to the change in the eccentricity (orbital period).
FIG. 1.Constraints on the natal kick from VFTS 243.This figure shows the probability distribution of matching the current orbital period, eccentricity and systemic velocity of VFTS 243, given a certain mass loss (dM ) and natal-kick magnitude (v kick ) following BH formation.In the heatmap, darker (lighter) regions indicate a higher (lower) probability for a certain dM -v kick pair to reproduce the orbital configuration of VFTS 243.The blue shaded region, delimited by the solid blue vertical lines, shows the estimates for neutrino mass-loss decrements from stellar models of BH progenitors[48].The side panels display the marginalized probability distribution (PDF, gray histograms) and cumulative distribution functions (CDF, red curves).With CDF(dM = 1.0 M⊙) = CDF(v kick = 10 km/s) = 0.68, we conclude that solutions with low mass loss and small natal kicks are preferred.The marginalized distributions peak at around dM = 0.3 M⊙ and v kick = 4 km/s, which is consistent with a solution where the mass decrement comes exclusively from neutrino emission.

FIG. 2 .
FIG. 2. Observed sample of field black-hole (BH) binaries with massive companions.For each binary, we show the BH mass (MBH) on the abscissa, the stellar companion mass (M * ) on the ordinate, and the eccentricity (e) through the colorbar.We use circles to indicate high-mass X-ray binaries (HMXBs) and triangles for inert BH binaries (Inert BHB).The symbol of VFTS 243 is slightly larger for clarity.

100 FIG
FIG. S1.Normalized distributions of the properties of binaries resulting in VFTS 243.We show the pre-collapse orbital period (Pi), mass decrement after collapse (dM ), and natal kick magnitude (v kick ) in the top row.The post-collapse orbital period (P f ), post-collapse eccentricity (e f ) and systemic velocity (vsys) plotted in the bottom row are constrained to match the reported values of VFTS 243.