X-raying the Birth of Binary Neutron Stars and Neutron Star-Black Hole Binaries

We consider fallback accretion after an ultra-stripped supernova (USSN) that accompanies formation of a binary neutron star (BNS) or a neutron star-black hole binary (NS-BH). The fallback matter initially accretes directly to the nascent NS, while it starts to accrete to the circumbinary disk, typically $0.1\mbox{-}1\, \mathrm{day}$ after the onset of the USSN explosion. The circumbinary disk mass further accretes, forming mini disks around each compact object, with a super-Eddington rate up to a few years. We show that such a system constitutes a binary ultraluminous X-ray source (ULX), and a fraction of the X rays can emerge through the USSN ejecta. We encourage follow-up observations of USSNe within $\lesssim 100\,\rm Mpc$ and $\sim 100\mbox{-}1,000\,\mathrm{day}$ after the explosion using Chandra, XMM Newton and NuSTAR, which could detect the X-ray counterpart with time variations representing the properties of the nascent compact binary, e.g., the orbital motion of the binary, the spin of the NS, and/or the quasiperiodic oscillation of the mini disks.


INTRODUCTION
The LIGO-Virgo detector network has detected gravitational waves (GWs) from O(100) of coalescing compact binaries (The LIGO Scientific Collaboration et al. 2021a).Most of them are confirmed to be binary black holes (BBHs) except for two binary neutron stars (BNSs; Abbott et al. 2017Abbott et al. , 2020) ) and two neutron star-black hole binaries (NS-BHs; Abbott et al. 2021).Their stellar progenitors are of great astrophysical interest.
In the isolated binary evolution scenario, the births of the BNSs and NS-BHs are likely accompanied by ultra-stripped supernovae (USSNe; Tauris et al. 2013Tauris et al. , 2015)); in order for the compact binary (CB) to lose its orbital energy via GW emission and merge within a cosmological time, the orbital separation at the birth needs to be comparable to the size of a massive stellar core, which means that the progenitor of the second-born NS inevitably experiences a significant envelope stripping via the binary interaction.
Properties of such USSNe have been inferred both by theory and observation.Neutrino-radiation hydrodynamic simulations show that a collapse of a carbon oxygen (CO) core is responsible for a successful explosion with an explosion energy of E SN ∼ 10 50 erg and an ejecta mass of M ej ∼ 0.1 M (e.g., Suwa et al. 2015;Müller et al. 2018).See also Mor et al. (2022) for the electron-capture SN explosion.Either way, the weak explosion induces a weak natal kick, preventing the disruption of the binary.Besides, high-cadence transient surveys are identifying more and more USSN candidates, i.e., those with faster light curves than ordinary core-collapse SNe and spectroscopic signatures of ultra-stripped progenitor (e.g., De et al. 2018;Yao et al. 2020).
With the increasing USSN samples at hand, the question is "Which USSNe accompany what type of BNS/NS-BH formation, in particular, those coalesce within a cosmological timescale?".In fact, the estimated USSN rate, R USSN a few × 1, 000 yr −1 Gpc −3 (e.g., Hijikawa et al. 2019), is an order of magnitude higher than the observed merger rates of BNS and NS-BH, R BNS = 10-1, 700 yr −1 Gpc −3 and R NS−BH = 7.4-320 yr −1 Gpc −3 (The LIGO Scientific Collaboration et al. 2021b).On the other hand, "What is the energy source of USSNe?" is another important question regarding the SN mechanism; although the theoretically calculated explosion energy and ejecta mass are consistent with those inferred from observations, the 56 Ni masses given by the same theoretical calculations are at most ∼ 0.01M and insufficient to explain some USSNe, for example, iPTF14gqr (Sawada et al. 2022).This may indicate that additional energy injection from the nascent NS and/or CB is necessary (Sawada et al. 2022). 1  In order to answer the above questions, one needs to have direct evidence of the formation of CB in a USSN and probe their properties, e.g., orbital separation and eccentricity of the binary, magnetic field strength and spin period of the NSs.To this end, we here consider fallback accretion occurring after the USSN and propose to search for the X-ray counterpart (See Fig. 1).An orbital timescale after the explosion, the fallback matter should start to accrete to the circumbinary disk and mini disks are also formed around each compact object.We show that the accretion rate well exceeds the Eddington limit for a few years, and the nascent CB can be a binary ultraluminous X-ray source (ULX).The physical properties of the nascent CB are imprinted in the X-ray emission, in particular in its time variations.
This paper is constructed as follows.In Sec. 2, we review the progenitor system of BNSs and NS-BHs that merge within a cosmological timescale and the USSN explosion associated with the second-born NS formation.In Sec. 3, we theoretically model fallback accretion onto nascent BNSs and NS-BHs.We calculate the resultant X-ray emission in Sec. 4 and the detectability in Sec. 5. Sec.6 is for summary and discussion.We use the notation Q = 10 x Q x in CGS units, except for some mass parameters in units of M = 10 y M y M .

ULTRA-STRIPPED SUPERNOVA ACCOMPANYING COMPACT BINARY FORMATION
Let us first review the basic properties of a USSN accompanying formation of a CB that coalesces within a cosmological time.

Progenitor binary system
The GW insprial time of the CB is calculated as where m = m 1 + m 2 is the total mass, µ = m 1 m 2 /(m 1 + m 2 ) is the reduced mass, a is the semi-major axis, and e is the eccentricity of the binary.The corresponding orbital period is (2) Hence, to merge within a cosmological time, say t GW 10 Gyr, the CB orbit at its birth needs to satisfy the following criteria, and In the isolated binary evolution scenario, such a CB can be formed from an OB star binary with an initial orbital separation of a 1 AU (see e.g., Postnov & Yungelson 2014, for a review): In the post-main-sequence phase of the primary, the first common envelope may develop.After spiraling in to some extent, the primary stellar core explodes and forms an NS or a BH.If the natal kick of the first-born compact object is sufficiently small not to significantly enlarge the binary separation, the second common envelope phase may occur.The system evolves into (near) contact

Circumbinary disk
Fallback radius BNS or NS-BH binary with mini disks Ultra-stripped supernova

Time
Accretion rate e N e y 2 o i W q 6 d q i V j T h y + h j X 8 A r q S U Q w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U K a j w e N e y 2 o i W q 6 d q i V j T h y + h j X 8 A r q S U Q w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U K a j w e N e y 2 o i W q 6 d q i V j T h y + h j X 8 A r q S U Q w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U K a j w binary of the NS/BH and a helium star.In this case, the Case BB Roche robe overflow occurs; the envelope of the helium star is further stripped, and its mass becomes close to the Chandrasekhar mass.The fate of such an ultrastripped star may be an electron-capture SN or a core-collapse SN driven by the neutrino mechanism (Tauris et al. 2015).In this paper, we focus on the latter case (see, e.g., Mor et al. (2022) for the former case).

Ultra-stripped supernovae
Hydrodynamic simulations of core collapse of ultra-stripped progenitors have been performed (Suwa et al. 2015;Yoshida et al. 2017;Moriya et al. 2017;Müller et al. 2018;Suwa et al. 2018;Sawada et al. 2022).It typically leads to a successful explosion by the neutrino mechanism with an explosion energy of E SN ∼ 10 50 erg, an ejecta mass of M ej ∼ 0.1 M , and an ejected 56 Ni mass of M Ni 0.01 M .Here we refer to two representative cases obtained by Sawada et al. (2022) (see Table 1).The CO145 and CO20 models are for ultra-stripped progenitors with CO core masses of 1.45 M and 2.0 M , respectively.The former (latter) represents a more (less) stripped progenitor and can be regarded to be in a binary system with a relatively small (large) orbital separation.The former (latter) shows a relatively strong (weak) explosion with a larger (smaller) ejecta mass.Noticeably, the latter has a significantly small 56 Ni mass, which is mainly due to the fallback (see Sawada et al. 2022, for the details).
On the basis of the results of the hydrodynamic simulations, the basic properties of USSN light curve can be inferred as follows.The optical depth of the ejecta evolves with time as τ sc ≈ 3κ sc M ej /4πr 2 ej , or f,g Fitting parameters for the late-phase fallback (Eq.13); h,i,j,k,l Fractional abundance of carbon, oxygen, neon, silicon, and iron in the ejecta.
where κ sc = 0.2 cm 2 g −1 κ sc,−0.7 is the electron scattering opacity and v ej ∼ 10 9 cm sec −1 E 1/2 sn,50 M −1/2 ej,−1 is the velocity of the ejecta.The photon diffusion time through the ejecta is given by t dif ≈ τ sc r ej /c, or The SN light curve takes its peak when the diffusion time becomes comparable to the dynamical time of the ejecta, t dif (t) ≈ r ej (t)/v ej .Solving this equation for time, the peak time can be estimated as t opt,peak ≈ (3κ sc M ej /4πcv ej ) 1/2 , or t opt,peak ∼ 6.5 day κ The energy injection rate by the 56 Ni decay is where Ni = 3.9 × 10 10 erg sec −1 g −1 and t Ni = 8.8 day.In the case of the USSN, t opt,peak t Ni , and if the 56 Ni decay is the main energy source, the peak luminosity can be estimated as High-cadence photometric surveys are detecting rapidly evolving optical transients broadly consistent with Eqs. ( 7) and ( 9) (e.g., Drout et al. 2014;Arcavi et al. 2016;Tanaka et al. 2016).Although a good fraction of them are likely explained by shock breakout or post-shock cooling emission from a dense circumstellar matter (Ho et al. 2021), some cases are spectroscopically confirmed to be compatible with ultra-stripped progenitors, e.g., iPTF14gqr (De et al. 2018) and SN2019dge (Yao et al. 2020).Sawada et al. (2022) demonstrated that SN2019dge can be consistently explained by the neutrino-driven explosion of a more stripped progenitor as the CO145 model and the light curve powered by 56 Ni decay.On the other hand, if iPTF14gqr is also powered by 56 Ni decay, the required amount of ejected 56 Ni mass is M Ni ∼ 0.05 M , which were shown to be difficult to synthesize by neutrino-driven explosion of ultra-stripped progenitors.This apparent 56 Ni problem can be resolved by incorporating an additional energy injection into the USSN ejecta from the nascent NS (Arcavi et al. 2016;Sawada et al. 2022) or CB.

Nascent compact binary
As a remnant of a USSN explosion in a close binary system, a BNS or a NS-BH is formed.Here, we consider the physical parameters of the nascent CB.
If the orbital parameters satisfy Eq. ( 3), the binary merges within a cosmological time and can be detected by the GW detector network.For GW events, the masses of compact objects are determined; M NS ∼ 1.5 and 1.3 M for GW170817 (Abbott et al. 2017), M NS ∼ 2.0 and 1.4 M for GW190425 (Abbott et al. 2020), M BH = 8.9 +1.2 −1.5 M and M NS = 1.9 +0.3 −0.2 M for GW200105, and M BH = 5.7 +1.8 −2.1 M and M NS = 1.5 +0.7 −0.3 M for GW200115 (Abbott et al. 2021).Although the sample is limited, we assume that these masses are typical for NSs and BHs in compact binaries that coalesce within a cosmological timescale.In principle, spins of the compact objects can be simultaneously determined from the GWs, but so far the uncertainties are relatively large.
Other physical parameters of the NSs can be also inferred from observations of Galactic NSs.In Galactic BNSs with t GW < 1 Gyr, the first-born NSs have relatively weak magnetic fields (B ∼ 10 9-10 G) and show fast rotation (P s < 200 ms) (Beniamini & Piran 2016, and references therein); the first-born NS has experienced a common envelope phase and a Case BB Roche robe overflow, it tends to evolve into a millisecond pulsar.The parameters of the secondborn NSs at their birth are relatively uncertain; if they are similar to those of young NSs in the Galaxy, B = 10 12-14 G and P s = 10-100 ms (e.g., Enoto et al. 2019).The weak natal kick of the second-born NS can still induce a non-negligible eccentricity of the binary orbit, as inferred by binary population synthesis calculations explaining the detected GW events (e.g., Kinugawa et al. 2022).Observationally, the Galactic BNSs with t GW < 1 Gyr have eccentricities ranging from 0.085 < e < 0.4 (Beniamini & Piran 2016, and references therein).Note that half of them have a relatively small value of e < 0.2, which is more compatible with the weak natal kick expected for USSNe.

SUPERNOVA FALLBACK ONTO NASCENT COMPACT BINARY
In an SN explosion, a fraction of the ejecta falls back onto the nascent compact object (Colgate 1971;Zel'dovich et al. 1972;Michel 1988).In the case of neutrino-driven explosion, the fallback starts when the neutrino luminosity from the proto-NS significantly decreases, typically ∼ 10-100 sec after the onset of the core collapse, and is the most significant in the early phase; the total fallback mass is sensitive to the progenitor structure and the SN explosion dynamics (e.g., Ertl et al. 2016a,b;Sukhbold et al. 2016;Janka et al. 2022).In the later phase, the fallback rate becomes to follow the asymptotic relation; Table 1 includes the parameters characterizing the late-phase fallback ( M fb , t fb ) obtained for the CO145 and CO20 models by Sawada et al. (2022).The CO20 model shows a more intense fallback since it has a more dense core structure and a smaller explosion energy.
In the case of an USSN accompanying CB formation, the fallback accretion mode should change with time (see Fig. 1).The important parameter is the fallback radius r fb ≈ (GM NS t 2 ) 1/3 where the fallback material arriving at the central region at time t starts falling toward the central region.For r fb a, or t t orb , where the fallback radius is smaller than the orbital separation of the nascent CB, the fallback is mainly on the second-born NS.The fallback material would not have sufficient angular momentum to form a disk around the second-born NS, thus directly accrete onto the NS magnetosphere or the surface (e.g., Zhong et al. 2021).The mass accretion rate onto the NS should be comparable to the fallback rate, i.e., ṀNS (t) ≈ Ṁfb (t) ∝ t −5/3 for t t orb .On the other hand, for r fb a, or t t orb , the fallback is on the binary system.The fallback material can be scattered by the binary orbit and on-average gain an angular momentum to form a circumbinary disk.The circumbinary disk mass further accretes with a viscous timescale (e.g., Farris et al. 2014), where α is the viscous parameter and h/r is the scale height of the disk.The time-averaged accretion rate from the circumbinary disk to the CB, ṀCB (t), behaves differently before and after t ≈ t vis : For t orb t t vis , the accretion rate is regulated by the viscous timescale of the circumbinary disk, i.e., For t t vis , the accretion rate onto the CB is determined by the mass supply rate to the circumbinary disk, i.e., The accreted mass from the circumbinary disk forms mini disks around each compact object and finally accrete onto them.Similar situation, in particular accretion on a BBH, has been numerically investigated (e.g., Farris et al. 2014;D'Orazio et al. 2016;Tagawa et al. 2018).For cases with a sufficiently large mass ratio q = m 2 /m 1 > 0.04, the accretion rate on each compact object is on average comparable, but fluctuates with orbital motion (Farris et al. 2014;D'Orazio et al. 2016).We note that hardening of BNS / NS-BH by interacting with the fallback material is negligible since M fb,tot M NS (Tagawa et al. 2018).We also note that the viscous timescales of the mini-disks are much smaller than those of both t orb and t vis and can be neglected for the estimate of ṀCB .
Figure 2 shows the evolution of the time-averaged accretion rate onto the central compact objects based on the CO145 and CO20 models of Sawada et al. (2022) with orbital separations of a = 1 × 10 11 cm, 3 × 10 11 cm, and 9 × 10 11 cm.We set m = 2.8 M , and assume α = 0.1 and h/r = 0.1 for the circumbinary disk.Accretion rate [M sec −1 ] CO20 a = 1 ×10 11 cm a = 3 ×10 11 cm a = 9 ×10 11 cm Figure 2. Fallback mass accretion rate onto a nascent compact binary system after a ultra-stripped supernova (USSN) at the birth of the second-born neutron star with an orbital separation of a = (1, 3, 9) × 10 11 cm.The total mass of the binary is set to be m = 2.8 M .The vertical lines indicate the orbital timescale and the viscous timescale of the circumbinary disk for the case with a = 1 × 10 11 cm.The USSN progenitor models are the CO145 (left) and CO20 (right) models in Table 1.

BINARY ULTRALUMINOUS X-RAY SOURCE
Next let us consider emission from the accreting nascent CB.From Eqs. ( 12) and ( 13), the average accretion rate normalized by the Eddington rate, ṀEdd ∼ 4.0 where η is the bolometric radiation efficiency.The accretion rate is higher than the Eddington rate up to t Edd ∼ 990 day η (15) Fig. 3 shows the time evolution of the bolomotric luminosity of the accreting CB, L bol = η ṀCO c 2 , for the CO145 and CO20 models with η = 0.1.Note that L bol includes contributions other than radiation, for example, the kinetic luminosity of the outflow.For t vis t t Edd , the accretion rate is comparable to the observed ULXs (see Kaaret et al. 2017, for a recent raview).In this case, a dominant fraction of the accretion luminosity can be converted to X-rays, i.e. η X η, and the luminosity can be described as which is the sum of the contributions from the two ULXs.The detail value of η X and the spectral shape in the X-ray bands should depend on the physical properties of the accreting nascent CB, e.g., accretion rate, surface magnetic field strength and spin of the NS, and mass and spin of the BH.
As for the NSs, the observed properties of the ULX pulsars can be used as reference (Bachetti et al. 2014;Fürst et al. 2016;Israel et al. 2017a,b).The observed spectra of ULX pulsars are typically fitted by a double blackbody spectrum or a blackbody plus cutoff power law in an energy range of ∼ 1-10 keV (e.g., Koliopanos et al. 2017;Walton et al. 2018;Tao et al. 2019).Although still under debate, the hard and soft components are considered to be coming from accretion column near the NS surface and accretion disk beyond the Alfvèn radius, respectively (see e.g., Mushtukov et al. 2019, and references therein).The relative importance of each component is determined by the spin and magnetic field of the accretor.The key parameters are the Alfvèn radius R M where the magnetic pressure and the ram pressure of the accreting material balance, the co-rotation radius R co where the rotation angular velocity of the NS and accretion disk becomes equal, and the spherization radius R sph where the accretion luminosity reaches the Eddington limit (Walton respectively.The upper and lower rows in Eqs. ( 17) and ( 18) correspond to the first and second NS, respectively.For the first-born NS, we expect R NS ∼ R co ∼ R M R sph,NS ; due to the relatively weak magnetic field, R M becomes close to the surface and ∼ 100 ṁ2 times smaller than R sph .In such case, the optically thick outflow should be relevant.Radiation efficiency may be suppressed to η 1/ ṁ, that is, the total luminosity of X-rays becomes L Edd (Shakura & Sunyaev 1973).However, even in this case, the apparent isotropic luminosity can be super-Eddington in a beamed direction (e.g., King & Lasota 2016;King et al. 2017).For the second-born NS, we expect R co R M R sph,NS .In this case, the impacts of the optically thick outflow are relatively minor.The strong magnetic field may enable to maintain an accretion column near the surface and the radiation efficiency can be as high as η X ∼ 0.1 (e.g., Kawashima et al. 2016).Since R co R M , the second-born NS is likely spinning up.The parameters of the second-born NS are broadly consistent with those inferred for the observed ULX pulsars (although our fiducial spin period is slightly smaller than the observed values).
For the first-born BH in a NS-BH, the accretion disk should be truncated at the innermost stable circular orbit (ISCO), R ISCO ∼ 4.5 × 10 6 cm (M BH /5 M ).Other than the innermost structure, the basic properties can be similar to super-Eddington accretion to a first-born NS in a BNS; an outflow would be prominent within the spherization radius R sph,BH ≈ (27/4) × ( Ṁ / ṀEdd ) × (GM BH /c 2 ) ∼ 5.0 × 10 8 cm ṁ2 (M BH /5 M ), and the total X-ray luminosity may not be significantly larger than the Eddington limit, while the apparent isotropic X-ray luminosity can be super-Eddington in a beamed direction.The X-ray spectrum of the BH accretion disk in general consists of a multi-temperature disk component and a power-law component produced in the coronal region.In the cases with a high accretion rate we are interested in, the latter component would be minor, as observed in ULXs.
The X-ray emission from the accreting nascent CB intrinsically can have time variability.First, the emission may show modulation due to the orbital motion of the binary.Such a modulation has been detected from an accreting supermassive black hole binary candidate (Graham et al. 2015).Based on the numerical simulations (e.g., Farris et al. 2014), the accretion rate on each compact object periodically fluctuates by more than a few 10%, which can be decomposed into evenly spaced frequencies, ω i ≈ 0.2 × (2π/t orb ) × i, where i is positive integer.Second, the emission from near-surface regions, i.e., the accretion column and the inner disk, may show a (quasi)periodic variability associated with the secondary NS spin.In the case of the observed ULX pulsars, the light curves are sinusoidal and the pulse fraction is ∼ 10% in the soft X-ray band and slightly larger for the hard X-ray band (Bachetti et al. 2014).The spin period may decrease with time as the NS spins up due to the accretion.Third, the accretion dynamics of the mini disks can be imprinted in high-frequency quasi-periodic oscillations (QPOs) and low-frequency variability of the X-ray light curve (e.g., Remillard & McClintock 2006;Pasham et al. 2014).In the BH case, the former would be correlated to the position of the ISCO and can be used to measure the BH mass.

X-RAYING NASCENT COMPACT BINARIES?
As we showed in the previous section, an accreting nascent compact binary in a USSN remnant can be a binary ULX, and the physical properties of the binary are imprinted in the X-ray emission.However, whether we can observe the signal is not straightforward.Just after the explosion, the SN ejecta is optically thick for electromagnetic waves.For X-rays, inelastic Compton scattering and bound-free absorption are the main obstacles (e.g., Metzger et al. 2014).
The energy loss by inelastic Compton scattering is predominantly determined by the scattering optical depth; X-rays with an energy larger than 5.9 (20) will be lost in the ejecta.From Eq. ( 20), a good fraction of the X-ray luminosity is deposited to the USSN ejecta via inelastic scattering for t 10 day, and can contribute to power the SN light curve.In Fig. 3, we compare the bolometric luminosity with the decay rate of 56 Ni synthesized and ejected in the explosion.In the case of a more stripped progenitor (CO145), the deposition of the X-rays and / or the kinetic luminosity of the outflow can give a comparable contribution to the 56 Ni decay, while in the case of a less stripped progenitor (CO20), the deposition can be the main energy source of the USSN.In particular, the peak luminosity of the USSN can be even much higher than that of the canonical core-collapse SNe when the orbital separation is relatively small.This case may be applicable to the bright end of rapidly evolving optical transients (e.g., Coppejans et al. 2020).Detailed calculations of the optical light curve powered by the energy injection from the nascent CB and comparison with the observed transients will be presented elsewhere (Sawada et al., in prep).
Hereafter, we focus on the transmitted X-ray emission through the USSN ejecta at t 10 day.We pick up two cases, the CO145 model with a = 1 × 10 11 cm and the CO20 model with a = 9 × 10 11 cm, for which the optical light curve will be broadly consistent with a typical USSN.For t 10 day, bound-free absorption (e.g., Osterbrock & Ferland 2006) may still be important.The threshold energy for the absorption is given by (hν) bf ≈ 870 eV Z 2 8 .
where Z = 8 Z 8 is the average degree of ionization of the ejecta.The ionization rate is given by Γ ion ≈ n γ σ γ c, or where is the cross section.On the other hand, the recombination rate is given by Γ rec ≈ n e α rec , or where n e = 3M ej /8πm u r 3 ej is the electron density and α rec ∼ 2×10 −11 cm 3 sec −1 Z 2 8 T −0.8 e,4 is the (case B) recombination coefficient.The electron temperature should be determined including the Compton heating effect, ranging from T e ∼ 10 4-5 K. From Eqs. ( 16 (24) Eq. ( 24) may infer that elements up to neon can be fully ionized by the X rays.To calculate the transmitted X-ray spectrum through the USSN ejecta, we need to solve the ionization state of heavier elements and the transfer of the radiation field consistently.
Fig. 5 shows the corresponding light curves in the soft (0.4-6 keV), intermediate (6-10 keV), and hard X-ray (10-30 keV) bands.For comparison, we indicate the sensitivities of Chandra ACIS and NuSTAR with an integration time of 10 4 sec for a USSN at a nominal distance of 40 Mpc.The X-ray counterpart in the soft and intermediate bands can be detectable up to 100 Mpc by these instruments (and XMM Newton with a sensitivity comparable to Chandra).The detection will also be promising for future X-ray satellites, e.g., XRISM (XRISM Science Team 2022) and Athena (Nandra et al. 2013).The expected event rate of such USSNe is a few yr −1 × f b where f b ∼ 0.1-1 is the beaming fraction of the X-ray emission.The hard X-ray counterpart can also be detected for cases with a larger fallback rate and/or a smaller distance to the source.To this end, follow-up observations of USSNe need to be done within a few 100 day after the explosion in the intermediate and hard X-ray bands and within 1, 000 day in the soft-X-ray band.
Time variability of the accreting nascent CB can be also detected if the signal-to-noise ratio (S/N) is sufficiently high: In order to detect an abrupt variation with an amplitude of 10(1)% and a duration of ∆t, an S/N 10(100) needs to be obtained with an integration time of ∼ ∆t, while if the variation is periodic, the S/N for the detection can be reduced by a factor of ∼ (∆t/T obs ) 1/2 , where T obs is the total observation time.The most promising target would be the orbital modulation with a period of t orb ∼ 0.1-1 day.The rotation period and its time derivatives of the second-born NS and the QPOs of the mini disks could also be detected when the signals are relatively persistent in time.

SUMMARY AND DISCUSSION
We have investigated the effect of USSN fallback occurring after the formation of the secondary NS in a BNS or a NS-BH that coalesces within a cosmological timescale.We showed that the nascent CB can be a binary ULX and the X-ray counterpart is detectable by a followup observation of USSNe within 100 Mpc and ∼ 100-1, 000 day after the explosion using Chandra, XMM Newton, NuSTAR, and future X-ray satellites, which provides a direct evidence of CB formation in USSN.Furthermore, information on the nascent CB, e.g. the orbital separation, eccentricity, and the NS spin, can be obtained from the time variability of the X-ray light curve.The conclusions are based on simplified calculations of the fallback accretion on and the X-ray emission from the nascent CB and need to be justified by multidimensional radiation hydrodynamic simulations.
For t 100 day, the X-ray counterpart could not be observed; though the accretion rate is even higher, the X-ray radiation efficiency may not be high and the X-rays, if any, are absorbed and converted into UV and optical photons.Instead, the injection of energy from the accreting nascent CB can power the USSN light curve, which may solve the apparent 56 Ni problem raised for, e.g., iPTF14gqr (Sawada et al. 2022).The detailed modeling and comparison with observed USSNe and rapidly evolving optical transients will be presented elsewhere.
We finally note that, if the second-born NS is strongly mangetized and rapidly rotating, the spindwon luminosity can be comparable to or larger than the total accretion luminosity of the nascent CB.It can both provide an enhanced USSN light curve (Hotokezaka et al. 2017;Sawada et al. 2022) and a late phase X-ray emission (Metzger et al. 2014;Kashiyama et al. 2016).To distinguish the energy sources, multi-wavelength follow-up observations also in the radio, submillimeter, and gamma-ray bands would be useful.
y 1 A R P S S 0 h s a m 5 p b W t m R 7 R 2 d X d 6 q n d z n 0 y o E l 8 p Z n e 8 G q a Y T C l q 7 I K 6 l s s e o H w n B M W 6 y Y p Z n a / k p F B K H 0 3 C W 1 6 4 t 1 x 9 h y 5 a a 0 D M V U M d W v 0 g X D 9 w N v J 6 2 K B c d Q 2 4 F T r c h w v 5 j K U I 6 i S P 8 E e g w y i G P O S 1 2 g g A 1 4 s F C G A w E X i r E N A y G 3 N e g g + M y t o 8 p c w E h G + w L 7 S L K 2 z F m C M w x m S z x u 8 W o t Z l 1 e 1 2 q G k d r i U 2 z u A S v T y N I 9 X d I z 3 d E V P d L b r 7 W q U Y 2 a l 1 2 e z b p W + M X u w / 7 F 1 3 9 V D s 8 K 2 5 + q P z 0 r b G I i 8 i r Z u x 8 x t V t Y d X 1 l 7 + h 5 c X I h W x 2 i U 3 p i / y f 0 Q L d 8 A 7 f y Y p 3 P i 4 V j J P k D 9 O / P / R M s j + Z 0 y u n z Y 5 m p 6 f g r W j G A Q Q z z e 4 9 j C r O Y Q 5 7 P P c A Z r n G T y 1 A R P S S 0 h s a m 5 p b W t m R 7 R 2 d X d 6 q n d z n 0 y o E l 8 p Z n e 8 G q a Y T C l q 7 I K 6 l s s e o H w n B M W 6 y Y p Z n a / k p F B K H 0 3 C W 1 6 4 t 1 x 9 h y 5 a a 0 D M V U M d W v 0 g X D 9 w N v J 6 2 K B c d Q 2 4 F T r c h w v 5 j K U I 6 i S P 8 E e g w y i G P O S 1 2 g g A 1 4 s F C G A w E X i r E N A y G 3 N e g g + M y t o 8 p c w E h G + w L 7 S L K 2 z F m C M w x m S z x u 8 W o t Z l 1 e 1 2 q G k d r i U 2 z u A S v T y N I 9 X d I z 3 d E V P d L b r 7 W q U Y 2 a l 1 2 e z b p W + M X u w / 7 F 1 3 9 V D s 8 K 2 5 + q P z 0 r b G I i 8 i r Z u x 8 x t V t Y d X 1 l 7 + h 5 c X I h W x 2 i U 3 p i / y f 0 Q L d 8 A 7 f y Y p 3 P i 4 V j J P k D 9 O / P / R M s j + Z 0 y u n z Y 5 m p 6 f g r W j G A Q Q z z e 4 9 j C r O Y Q 5 7 P P c A Z r n G T Figure 1.Schematic picture of ultra-stripped supernova fallback onto a nascent binary neutron star (BNS) or neutron star-black hole binary (NS-BH).

Figure 3 .
Figure 3. Time-averaged bolometric luminosity of the accreting compact binary system with an radiation efficiency of η = 0.1 for the same models as Fig. 2. The dotted line show the energy injection rate to the USSN by the radioactive decay of 56 Ni synthesised in the explosion.The vertical line indicates the calculated peak time of the optical light curve of the USSN.
a Progenitor CO core mass; b Explosion energy; c NS mass; d Ejecta mass; e 56 Ni mass;