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

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. 2017, 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 ultrastripped supernovae (USSNe; Tauris et al. 2013, 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 the collapse of a carbon oxygen (CO) core is responsible for a successful explosion with an explosion energy of E_SN ∼ 10⁵⁰ 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 ultrastripped progenitors (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 that coalesce within a cosmological timescale?" In fact, the estimated USSN rate, ${{ \mathcal R }}_{\mathrm{USSN}}\gtrsim {\rm{a}}\,\mathrm{few}\,\times 1000\,{\mathrm{yr}}^{-1}\,{\mathrm{Gpc}}^{-3}$ (e.g., Hijikawa et al. 2019), is an order of magnitude higher than the observed merger rates of BNSs and NS–BHs, ${{ \mathcal R }}_{\mathrm{BNS}}=10\mbox{--}1700\,{\mathrm{yr}}^{-1}\,{\mathrm{Gpc}}^{-3}$ and ${{ \mathcal R }}_{\mathrm{NS}-\mathrm{BH}}$ = 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 ⁵⁶Ni masses given by the same theoretical calculations are at most ∼0.01 M_⊙ 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). ⁷

In order to answer the above questions, one needs to have direct evidence of the formation of a 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 Figure 1). Following 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.

Zoom In Zoom Out Reset image size

This paper is constructed as follows. In Section 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 Section 3, we theoretically model fallback accretion onto nascent BNSs and NS–BHs. We calculate the resultant X-ray emission in Section 4 and the detectability in Section 5. Section 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_⊙.

Let us first review the basic properties of a USSN accompanying formation of a CB that coalesces within a cosmological time.

2.1. Progenitor Binary System

The GW inspiral time of the CB is calculated as

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{GW}}\sim 0.16\,\mathrm{Gyr}\,{a}_{11}^{4}{\left(\displaystyle \frac{\mu }{0.7\,{M}_{\odot }}\right)}^{-1}\\ \times \,{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{-2}{(1-{e}^{2})}^{7/2},\end{array}\end{eqnarray} \tag{ 1 }$$

where m = m₁ + m₂ is the total mass, μ = m₁ m₂/(m₁ + m₂) is the reduced mass, a is the semimajor axis, and e is the eccentricity of the binary. The corresponding orbital period is

$$\begin{eqnarray}&&{t}_{\mathrm{orb}}\sim 0.1\,\mathrm{day}\,{a}_{11}^{3/2}{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{-1/2}.\end{eqnarray} \tag{ 2 }$$

Hence, to merge within a cosmological time, say t_GW ≲ 10 Gyr, the CB orbit at its birth needs to satisfy the following criteria,

$$\begin{eqnarray}&&\begin{array}{l}a\lesssim 3\,\times {10}^{11}\,\mathrm{cm}{\left(\displaystyle \frac{{t}_{\mathrm{GW}}}{10\,\mathrm{Gyr}}\right)}^{1/4}\\ \times \,{\left(\displaystyle \frac{\mu }{0.7\,{M}_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{1/2}{(1-{e}^{2})}^{-7/8},\end{array}\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{orb}}\lesssim 0.5\,\mathrm{day}{\left(\displaystyle \frac{{t}_{\mathrm{GW}}}{10\,\mathrm{Gyr}}\right)}^{3/8}\\ \times \,{\left(\displaystyle \frac{\mu }{0.7\,{M}_{\odot }}\right)}^{3/8}{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{1/4}{(1-{e}^{2})}^{-21/16}.\end{array}\end{eqnarray} \tag{ 4 }$$

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 a (near) contact binary of the NS/BH and a helium star. In this case, the case BB Roche lobe 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).

2.2. Ultrastripped Supernovae

Hydrodynamic simulations of the core collapse of ultrastripped progenitors have been performed (Suwa et al. 2015; Moriya et al. 2017; Yoshida 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 ∼ 10⁵⁰ erg, an ejecta mass of M_ej ∼ 0.1 M_⊙, and an ejected ⁵⁶Ni 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 ultrastripped 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 ⁵⁶Ni mass, which is mainly due to the fallback (see Sawada et al. 2022, for the details).

Table 1. Properties of an Ultrastripped Supernova Explosion Given by Sawada et al. (2022)

Model	MCO a	ESN b	MNS c	Mej d	MNi e	$\widetilde{{M}_{\mathrm{fb}}}$ f	$\widetilde{{t}_{\mathrm{fb}}}$ g	XC h	XO i	XNe j	XSi k	XFe l
	(M⊙)	(1051 erg)	(M⊙)	(M⊙)	(M⊙)	(10−2 M⊙)	(s)
CO145	1.45	0.17	1.35	0.097	1.63 × 10−2	0.18	103	0.028	0.33	0.055	0.22	0.0057
CO20	2.0	0.12	1.64	0.286	7.78 × 10−5	1.7	103	0.064	0.51	0.36	0.073	0.0020

Notes.

^a Progenitor CO core mass. ^b Explosion energy. ^c NS mass. ^d Ejecta mass. ^{e 56}Ni mass. ^{f ,g}Fitting parameters for the late-phase fallback (Equation (13)). ^{h ,i,j,k,l} Fractional abundance of carbon, oxygen, neon, silicon, and iron in the ejecta.

Download table as:  ASCIITypeset image

On the basis of the results of the hydrodynamic simulations, the basic properties of a 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_ej ², or

$$\begin{eqnarray}&&{\tau }_{\mathrm{sc}}\sim 13\,{\kappa }_{\mathrm{sc},\ -0.7}{M}_{\mathrm{ej},-1}{v}_{\mathrm{ej},\ 9}^{-2}{t}_{5.9}^{-2},\end{eqnarray} \tag{ 5 }$$

where κ_sc = 0.2 cm² g⁻¹ κ_sc,−0.7 is the electron scattering opacity and v_ej ∼ 10⁹ cm sec⁻¹ E_sn,50 ^1/2 M_{ej, −1} ^−1/2 is the velocity of the ejecta. The photon diffusion time through the ejecta is given by t_dif ≈ τ_sc r_ej/c, or

$$\begin{eqnarray}&&{t}_{\mathrm{dif}}\sim 4.3\,\mathrm{days}\,{\kappa }_{\mathrm{sc},\ -0.7}{M}_{\mathrm{ej},-1}{v}_{\mathrm{ej},\ 9}^{-1}{t}_{5.9}^{-1}.\end{eqnarray} \tag{ 6 }$$

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

$$\begin{eqnarray}&&{t}_{\mathrm{opt},\mathrm{peak}}\sim 6.5\,\mathrm{days}\,{\kappa }_{\mathrm{sc},\ -0.7}^{1/2}{M}_{\mathrm{ej},-1}^{1/2}{v}_{\mathrm{ej},\ 9}^{-1/2}.\end{eqnarray} \tag{ 7 }$$

The energy injection rate by the ⁵⁶Ni decay is

$$\begin{eqnarray}&&{\dot{Q}}_{\mathrm{Ni}}(t)={M}_{\mathrm{Ni}}{\epsilon }_{\mathrm{Ni}}\exp \left(-\displaystyle \frac{t}{{t}_{\mathrm{Ni}}}\right),\end{eqnarray} \tag{ 8 }$$

where _Ni = 3.9 × 10¹⁰ erg s⁻¹ g⁻¹ and t_Ni = 8.8 days. In the case of the USSN, t_opt,peak ≲ t_Ni, and if the ⁵⁶Ni decay is the main energy source, the peak luminosity can be estimated as

$$\begin{eqnarray}&&{L}_{\mathrm{opt},\mathrm{peak}}\approx {\dot{Q}}_{\mathrm{Ni}}({t}_{\mathrm{opt},\mathrm{peak}})\sim 7.8\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{M}_{\mathrm{Ni},-2}.\end{eqnarray} \tag{ 9 }$$

High-cadence photometric surveys are detecting rapidly evolving optical transients broadly consistent with Equations (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 postshock cooling emission from a dense circumstellar matter (Ho et al. 2021), some cases are spectroscopically confirmed to be compatible with ultrastripped progenitors, e.g., iPTF 14gqr (De et al. 2018) and SN 2019dge (Yao et al. 2020). Sawada et al. (2022) demonstrated that SN 2019dge can be consistently explained by the neutrino-driven explosion of a more stripped progenitor as the CO145 model and the light curve, powered by ⁵⁶Ni decay. On the other hand, if iPTF 14gqr is also powered by ⁵⁶Nidecay, the required amount of ejected ⁵⁶Ni mass is M_Ni ∼ 0.05 M_⊙, which is shown to be difficult to synthesize by a neutrino-driven explosion of ultrastripped progenitors. This apparent ⁵⁶Ni 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.

2.3. 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 a neutrino-driven explosion, the fallback starts when the neutrino luminosity from the proto-NS significantly decreases, typically ∼10–100 s 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, 2016b; Sukhbold et al. 2016; Janka et al. 2022). In the later phase, the fallback rate comes to follow the asymptotic relation;

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{fb}}(t)=\displaystyle \frac{2\widetilde{{M}_{\mathrm{fb}}}}{3\widetilde{{t}_{\mathrm{fb}}}}{\left(\displaystyle \frac{t}{\widetilde{{t}_{\mathrm{fb}}}}\right)}^{-5/3}.\end{eqnarray} \tag{ 10 }$$

Table 1 includes the parameters characterizing the late-phase fallback ($\widetilde{{M}_{\mathrm{fb}}}$, $\widetilde{{t}_{\mathrm{fb}}}$) 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 a USSN accompanying CB formation, the fallback accretion mode should change with time (see Figure 1). The important parameter is the fallback radius ${r}_{\mathrm{fb}}\approx {({{GM}}_{\mathrm{NS}}{t}^{2})}^{1/3}$ 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 accretes 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., ${\dot{M}}_{\mathrm{NS}}(t)\approx {\dot{M}}_{\mathrm{fb}}(t)\propto {t}^{-5/3}$ 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 can 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),

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{vis}}\approx \displaystyle \frac{1}{3\pi \alpha {(h/r)}^{2}}{t}_{\mathrm{orb}}\sim 10\,\mathrm{days}{\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1}\\ \times \,{\left(\displaystyle \frac{h/r}{0.1}\right)}^{-2}{a}_{11}^{3/2}{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{-1/2},\end{array}\end{eqnarray} \tag{ 11 }$$

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, ${\dot{M}}_{\mathrm{CB}}(t)$, 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.,

$$\begin{eqnarray}&&\begin{array}{l}{\dot{M}}_{\mathrm{CB}}(t)\approx {\dot{M}}_{\mathrm{fb}}(t)\displaystyle \frac{t}{{t}_{\mathrm{vis}}}=\displaystyle \frac{2\widetilde{{M}_{\mathrm{fb}}}}{3{t}_{\mathrm{vis}}}{\left(\displaystyle \frac{t}{\widetilde{{t}_{\mathrm{fb}}}}\right)}^{-2/3}\\ \qquad \propto \,{t}^{-2/3}\,({t}_{\mathrm{orb}}\lesssim t\lesssim {t}_{\mathrm{vis}}).\end{array}\end{eqnarray} \tag{ 12 }$$

For t ≳ t_vis, the accretion rate onto the CB is determined by the mass supply rate to the circumbinary disk, i.e.,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{CB}}(t)\approx {\dot{M}}_{\mathrm{fb}}(t)\propto {t}^{-5/3}\ \ \ (t\gtrsim {t}_{\mathrm{vis}}).\end{eqnarray} \tag{ 13 }$$

The accreted mass from the circumbinary disk forms mini disks around each compact object and finally accretes onto them. A 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 = m₂/m₁ > 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 the hardening of a 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 ${\dot{M}}_{\mathrm{CB}}$.

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 × 10¹¹ cm, 3 × 10¹¹ cm, and 9 × 10¹¹ cm. We set m = 2.8 M_⊙, and assume α = 0.1 and h/r = 0.1 for the circumbinary disk.

Zoom In Zoom Out Reset image size

Next let us consider emission from the accreting nascent CB. From Equations (12) and (13), the average accretion rate normalized by the Eddington rate, ${\dot{M}}_{\mathrm{Edd}}$ ∼ 4.0 × 10⁻¹⁵ M_⊙ s⁻¹ η₋₁ ⁻¹ κ_{sc, − 0.7} ⁻¹ (m/2.8 M_⊙), is described as

$$\begin{eqnarray}&&\begin{array}{c}\dot{m}\approx \frac{{\dot{M}}_{\mathrm{CB}}}{{\dot{M}}_{\mathrm{Edd}}}\\ \sim \left\{\begin{array}{c}1900\,{\eta }_{-1}{\tilde{M}}_{\mathrm{fb},-3}{\tilde{t}}_{\mathrm{fb},3}^{2/3}\left(\frac{\alpha }{0.1}\right){\left(\frac{h/r}{0.1}\right)}^{2}{a}_{11}^{-3/2}{\left(\frac{m}{2.8\,{M}_{\odot }}\right)}^{-1/2}{\kappa }_{\mathrm{sc},-0.7}{t}_{6}^{-2/3}\,({t}_{\mathrm{orb}}\lesssim t\lesssim {t}_{\mathrm{vis}}),\\ 36\,{\eta }_{-1}{\tilde{M}}_{\mathrm{fb},-3}{\tilde{t}}_{\mathrm{fb},3}^{2/3}{\left(\frac{m}{2.8\,{M}_{\odot }}\right)}^{-1}{\kappa }_{\mathrm{sc},-0.7}{t}_{7}^{-5/3}\,(t\gtrsim {t}_{\mathrm{vis}}),\end{array}\right.\end{array}\end{eqnarray} \tag{ 14 }$$

where η is the bolometric radiation efficiency. The accretion rate is higher than the Eddington rate up to

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{Edd}}\sim 990\,\mathrm{days}\,{\eta }_{-1}^{3/5}{\widetilde{M}}_{\mathrm{fb},-3}^{3/5}{\widetilde{t}}_{\mathrm{fb},3}^{2/5}\\ \times \,{\left(\displaystyle \frac{m}{2.8\,{M}_{\odot }}\right)}^{-3/5}{\kappa }_{\mathrm{sc},-0.7}^{3/5}.\end{array}\end{eqnarray} \tag{ 15 }$$

Figure 3 shows the time evolution of the bolometric luminosity of the accreting CB, ${L}_{\mathrm{bol}}=\eta {\dot{M}}_{\mathrm{CO}}{c}^{2}$, 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 review). 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

$$\begin{eqnarray}&&\begin{array}{l}\overline{{L}_{{\rm{X}}}}={\eta }_{{\rm{X}}}{\dot{M}}_{\mathrm{CB}}{c}^{2}\sim 2.6\times {10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\\ \times \,{\eta }_{{\rm{X}},\ -1}{\widetilde{M}}_{\mathrm{fb},\ -3}{\widetilde{t}}_{\mathrm{fb},\ 3}^{2/3}{t}_{7}^{-5/3},\end{array}\end{eqnarray} \tag{ 16 }$$

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.

Zoom In Zoom Out Reset image size

As for the NSs, the observed properties of the ULX pulsars can be used as a reference (Bachetti et al. 2014; Fürst et al. 2016; Israel et al. 2017a, 2017b; Carpano et al. 2018; Wilson-Hodge et al. 2018; Sathyaprakash et al. 2019; Chandra et al. 2020; Rodríguez Castillo et al. 2020). 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 an accretion column near the NS surface and an accretion disk beyond the Alfvén 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 Alfvén radius R_M where the magnetic pressure and the ram pressure of the accreting material balance, the corotation 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 et al. 2018). They are estimated as

$$\begin{eqnarray}&&\begin{array}{l}{R}_{{\rm{M}}}\approx {\left(\displaystyle \frac{{\mu }_{{\rm{M}}}^{4}}{2{{GM}}_{\mathrm{NS}}{\dot{M}}^{2}}\right)}^{1/7}\\ \sim \left\{\begin{array}{l}1.4\times {10}^{6}\,\mathrm{cm}\,{B}_{10}^{4/7}{\dot{m}}_{2}^{-2/7}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right)}^{-1/7},\\ 7.3\times {10}^{7}\,\mathrm{cm}\,{B}_{13}^{4/7}{\dot{m}}_{2}^{-2/7}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right)}^{-1/7},\end{array}\right.\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\begin{array}{l}{R}_{\mathrm{co}}={\left(\displaystyle \frac{{{GM}}_{\mathrm{NS}}{P}_{{\rm{s}}}^{2}}{4{\pi }^{2}}\right)}^{1/3}\\ \sim \left\{\begin{array}{l}1.6\times {10}^{6}\,\mathrm{cm}\,{P}_{{\rm{s}},-3}^{2/3}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right)}^{1/3},\\ 3.6\times {10}^{7}\,\mathrm{cm}\,{P}_{{\rm{s}},-1}^{2/3}{\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right)}^{1/3}\\ \end{array}\right.\end{array}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&\begin{array}{l}{R}_{\mathrm{sph},\ \mathrm{NS}}\approx \displaystyle \frac{27}{4}\displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}\displaystyle \frac{{{GM}}_{\mathrm{NS}}}{{c}^{2}}\\ \sim \,1.4\times {10}^{8}\,\mathrm{cm}\,{\dot{m}}_{2}\left(\displaystyle \frac{{M}_{\mathrm{NS}}}{1.4\,{M}_{\odot }}\right),\end{array}\end{eqnarray} \tag{ 19 }$$

respectively. The upper and lower rows in Equations (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 comes close to the surface and is ∼100 $\,{\dot{m}}_{2}$ times smaller than R_sph. In such case, the optically thick outflow should be relevant. Radiation efficiency may be suppressed to $\eta \gtrsim 1/\dot{m}$, 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) with a weak or negligible beaming (e.g., Rodríguez Castillo et al. 2020; Bachetti et al. 2021). 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 × 10⁶ 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 ${R}_{\mathrm{sph},\ \mathrm{BH}}\,\approx (27/4)\times (\dot{M}/{\dot{M}}_{\mathrm{Edd}})\times ({{GM}}_{\mathrm{BH}}/{c}^{2})$ $\sim 5.0\,\times \,{10}^{8}\mathrm{cm}\,{\dot{m}}_{2}({M}_{\mathrm{BH}}/5\,{M}_{\odot })$, 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 multitemperature 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 the 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 nearly sinusoidal. The pulsed fractions are typically ∼10%, but can be ≲100% in some cases (e.g., Carpano et al. 2018; Wilson-Hodge et al. 2018), and are larger in the hard X-ray band than the soft X-ray band. The spin period may decrease significantly with time as the NS spins up due to the accretion (e.g., Vasilopoulos et al. 2018). Third, the accretion dynamics of the mini disks can be imprinted in high-frequency quasiperiodic 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.

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

$$\begin{eqnarray}&&\begin{array}{l}{(h\nu )}_{\mathrm{sc}}\approx \displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{{\tau }_{\mathrm{ej}}^{2}}\\ \,\sim \,3.1\,\mathrm{keV}\,{\kappa }_{\mathrm{sc},\ -0.7}^{-2}{M}_{\mathrm{ej},-1}^{-2}{v}_{\mathrm{ej},\ 9}^{4}{t}_{5.9}^{4}\end{array}\end{eqnarray} \tag{ 20 }$$

will be lost in the ejecta. From Equation (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 Figure 3, we compare the bolometric luminosity with the decay rate of ⁵⁶Ni 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 ⁵⁶Ni 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 even be 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 (R. Sawada et al. 2022, in preparation).

Hereafter, we focus on the transmitted X-ray emission through the USSN ejecta at t ≳ 10 days. We pick up two cases, the CO145 model with a = 1 × 10¹¹ cm and the CO20 model with a = 9 × 10¹¹ cm, for which the optical light curve will be broadly consistent with a typical USSN. For t ≳ 10 days, bound–free absorption (e.g., Osterbrock & Ferland 2006) may still be important. The threshold energy for the absorption is given by

$$\begin{eqnarray}&&{(h\nu )}_{\mathrm{bf}}\approx 870\,\mathrm{eV}\,{Z}_{8}^{2},\end{eqnarray} \tag{ 21 }$$

where Z = 8 Z₈ is the average degree of ionization of the ejecta. The ionization rate is given by Γ_ion ≈ n_γ σ_γ c, or

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{ion}}\sim 0.76\,{\sec }^{-1}\,{Z}_{8}^{-4}{\overline{{L}_{{\rm{x}}}}}_{,41}{v}_{\mathrm{ej},9}^{-2}{t}_{5.9}^{-2},\end{eqnarray} \tag{ 22 }$$

where ${n}_{\gamma }=\overline{{L}_{{\rm{x}}}}/[4\pi {{cr}}_{\mathrm{ej}}^{2}{(h\nu )}_{\mathrm{bf}}]$ is the ionizing photon density and σ_γ ∼ 1 × 10⁻¹⁹ cm² Z₈ ⁻² is the cross section. On the other hand, the recombination rate is given by Γ_rec ≈ n_e α_rec, or

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{rec}}\sim 0.44\,{\sec }^{-1}\,{Z}_{8}^{2}{T}_{4}^{-0.8}{M}_{\mathrm{ej},-1}^{-1}{v}_{\mathrm{ej},9}^{-3}{t}_{5.9}^{-3},\end{eqnarray} \tag{ 23 }$$

where n_e = 3M_ej/8πm_u r_ej ³ is the electron density and α_rec ∼ 2 × 10⁻¹¹ cm³ sec⁻¹ Z₈ ² T_e,4 ^−0.8 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 Equations (16), (22), and (23), the condition Γ_ion > Γ_rec is realized for

$$\begin{eqnarray}&&\begin{array}{l}Z\lt 11\,{T}_{{\rm{e}},4}^{2/15}{M}_{\mathrm{ej},-1}^{1/6}{v}_{\mathrm{ej},9}^{1/6}{\eta }_{-1}^{1/6}{\widetilde{M}}_{\mathrm{fb},\ -3}^{1/6}{\widetilde{t}}_{\mathrm{fb},\ 3}^{1/9}{t}_{5.9}^{-1/9}.\end{array}\end{eqnarray} \tag{ 24 }$$

Equation (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.

Here we use the spectral synthesis code CLOUDY (Ferland et al. 1998, 2017). For simplicity, we fix the shape of the injection X-ray spectrum as a multitemperature blackbody with a temperature range of T_bb ∼ (1–3) × 10⁷ K, mimicking the observed ULX spectra. The total flux is determined from Equation (16). We assume a homologously expanding ejecta; the density profile and composition are set to be consistent with those obtained by hydrodynamic simulations (Sawada et al. 2022). The fractional abundances of some relevant elements are shown in Table 1. We neglect the interstellar absorption both in the host galaxy and in the Milky way.

Figure 4 shows the resultant X-ray spectra of the CO145 model with a = 1 × 10¹¹ cm and the CO20 model with a = 9 × 10¹¹ cm at 25, 50, 100, 200, 400, 800 days after the explosion. In both cases, the emitted X-rays from the accreting nascent CB are absorbed significantly at an early stage, but gradually escape through the ejecta from ∼1 month after the explosion. The ejecta becomes transparent first for hard X-rays and then for intermediate to soft X-rays. We note that the recombination lines of the iron group elements are less prominent in the CO20 model, since the abundance is lower (see Table 1). ⁸

Zoom In Zoom Out Reset image size

Figure 5 shows the corresponding light curves in the soft (0.4–6 keV), intermediate (6–10 rmkeV), and hard X-ray (10–30 keV) bands. For comparison, we indicate the sensitivities of Chandra ACIS and NuSTAR with an integration time of 10⁴ s 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 (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 days after the explosion in the intermediate and hard X-ray bands and within ≲1000 days in the soft X-ray band.

Zoom In Zoom Out Reset image size

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 Δt, an S/N ≳10(100) needs to be obtained with an integration time of ∼Δt, while if the variation is periodic, the S/N for the detection can be reduced by a factor of ∼(Δt/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.

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 follow-up observation of USSNe within ≲100 Mpc and ∼100–1000 days after the explosion using Chandra, XMM Newton, NuSTAR, and future X-ray satellites, which provides direct evidence of CB formation in a 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 above conclusions are based on the simplified calculations of the fallback accretion on and the X-ray emission from the nascent CB, which need to be justified by multidimensional radiation hydrodynamic simulations. As for the fallback accretion rate, we referred the two representative cases obtained by the numerical simulation of a neutrino-driven explosion of ultrastripped progenitors by Suwa et al. (2015) and Sawada et al. (2022). Given that we still lack a complete understanding of the supernova explosion mechanism, the detailed values of the fallback accretion parameters may also need to be refined by future more elaborate simulations to give more accurate numbers.

For t ≲ 100 days, 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 ⁵⁶Ni problem raised for, e.g., iPTF 14gqr (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 magnetized and rapidly rotating, the spindown 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, multiwavelength follow-up observations also in the radio, submillimeter, and gamma-ray bands would be useful.

K.K. would like to thank Vasilopoulos Georgios for the comments. K.K. also thanks Aya Bamba, Daichi Tsuna, Toshikazu Shigeyama, and Shigeo S. Kimura for discussion. This work is financially supported by JSPS KAKENHI grant 18H04573 and 17K14248 (K.K.), 18H05437, 20H00174, 20H01904, and 22H04571 (Y.S.), 21J00825 and 21K13964 (R.S.).