Fallback Rate Research Articles

A simple and accurate prescription for the tidal disruption radius of a star and the peak accretion rate in tidal disruption events

ABSTRACT A star destroyed by a supermassive black hole (SMBH) in a tidal disruption event (TDE) enables the study of SMBHs. We propose that the distance within which a star is completely destroyed by an SMBH, defined rt,c, is accurately estimated by equating the SMBH tidal field (including numerical factors) to the maximum gravitational field in the star. We demonstrate that this definition accurately reproduces the critical βc = rt/rt,c, where rt = R⋆(M•/M⋆)1/3 is the standard tidal radius with R⋆ and M⋆ the stellar radius and mass, and M• the SMBH mass, for multiple stellar progenitors at various ages, and can be reasonably approximated by βc ≃ [ρc/(4ρ⋆)]1/3, where ρc (ρ⋆) is the central (average) stellar density. We also calculate the peak fallback rate and time at which the fallback rate peaks, finding excellent agreement with hydrodynamical simulations, and also suggest that the partial disruption radius – the distance at which any mass is successfully liberated from the star – is βpartial ≃ 4−1/3 ≃ 0.6. For given stellar and SMBH populations, this model yields, e.g. the fraction of partial TDEs, the peak luminosity distribution of TDEs, and the number of directly captured stars.

Cooling Envelope Model for Tidal Disruption Events

We present a toy model for the thermal optical/UV/X-ray emission from tidal disruption events (TDEs). Motivated by recent hydrodynamical simulations, we assume that the debris streams promptly and rapidly circularize (on the orbital period of the most tightly bound debris), generating a hot quasi-spherical pressure-supported envelope of radius R v ∼ 1014 cm (photosphere radius ∼1015 cm) surrounding the supermassive black hole (SMBH). As the envelope cools radiatively, it undergoes Kelvin–Helmholtz contraction R v ∝ t −1, its temperature rising T eff ∝ t 1/2 while its total luminosity remains roughly constant; the optical luminosity decays as . Despite this similarity to the mass fallback rate , envelope heating from fallback accretion is subdominant compared to the envelope cooling luminosity except near optical peak (where they are comparable). Envelope contraction can be delayed by energy injection from accretion from the inner envelope onto the SMBH in a regulated manner, leading to a late-time flattening of the optical/X-ray light curves, similar to those observed in some TDEs. Eventually, as the envelope contracts to near the circularization radius, the SMBH accretion rate rises to its maximum, in tandem with the decreasing optical luminosity. This cooling-induced (rather than circularization-induced) delay of up to several hundred days may account for the delayed onset of thermal X-rays, late-time radio flares, and high-energy neutrino generation, observed in some TDEs. We compare the model predictions to recent TDE light-curve correlation studies, finding both agreement and points of tension.

Revisit the Rate of Tidal Disruption Events: The Role of the Partial Tidal Disruption Event

Tidal disruption of stars in dense nuclear star clusters containing supermassive central black holes (SMBH) is modeled by high-accuracy direct N-body simulation. Stars getting too close to the SMBH are tidally disrupted, and a tidal disruption event (TDE) happens. The TDEs probe the properties of SMBHs, their accretion disks, and the surrounding nuclear stellar cluster. In this paper, we compare the rates of full tidal disruption events (FTDEs) with partial tidal disruption events (PTDEs). Since a PTDE does not destroy the star, a leftover object emerges; we use the term “leftover star” for it. Two novel effects occur in the simulation: (1) variation of the leftover star’s mass and radius and (2) variation of the leftover star’s orbital energy. After switching on these two effects in our simulation, the number of FTDEs is reduced by roughly 28%, and the reduction is mostly due to the ejection of the leftover stars from PTDEs originally coming from a relatively large distance. The number of PTDEs is about 75% higher than the simple estimation given by Stone et al., and the enhancement is mainly due to the multiple PTDEs produced by the leftover stars residing in the diffusive regime. We compute the peak mass fallback rate for the PTDEs and FTDEs recorded in the simulation and find that 58% of the PTDEs have a peak mass fallback rate exceeding the Eddington limit, and the number of super-Eddington PTDEs is 2.3 times the number of super-Eddington FTDEs.

A Model for the Possible Connection Between a Tidal Disruption Event and Quasi-periodic Eruption in GSN 069

Quasi-periodic eruptions (QPEs) are found in the center of five galaxies, where a tidal disruption event (TDE)-like event has been reported in GSN 069, which occurred a couple of years before the QPEs. We explain the connection of these phenomena based on a model of a highly eccentric white dwarf (WD) 104−6 M ⊙ massive black hole (MBH) binary formed by the Hill mechanism. In this system, the tidally induced internal oscillation of a WD can heat the WD envelope thereby inducing tidal nova and inflating the WD envelope, which can be captured by the MBH and form a TDE. The tidal stripping of the surviving WD in the eccentric orbit can produce QPEs. We also apply this model to the other four QPE sources. Based on the estimated fallback rate, we find that the remaining time after the QPE-observed time for these QPEs is only around 1–2 yr based on our simple model estimation, after which the WD will be fully disrupted. We also show that the accretion rate can be much higher than the Eddington accretion rate in the final stage of these QPE sources. The peak frequency of the spectral energy distribution of the disk stays in the soft X-ray band (∼0.1–1 keV), which is consistent with observational results.

Spectropolarimetry of the tidal disruption event AT 2019qiz: a quasi-spherical reprocessing layer

ABSTRACT We present optical spectropolarimetry of the tidal disruption event (TDE) AT 2019qiz on days +0 and +29 relative to maximum brightness. Continuum polarization, which informs the shape of the electron-scattering surface, was found to be consistent with 0 per cent at peak brightness. On day +29, the continuum polarization rose to ∼1 per cent, making this the first reported spectropolarimetric evolution of a TDE. These findings are incompatible with a naked eccentric disc that lacks significant mass outflow. Instead, the spectropolarimetry paints a picture wherein, at maximum brightness, high-frequency emission from the accretion disc is reprocessed into the optical band by a nearly spherical, optically thick, electron-scattering photosphere located far away from the black hole. We estimate the radius of the scattering photosphere to be $\sim 100\rm \, au$ at maximum brightness – significantly larger than the tidal radius ($\sim 1\rm \, au$) and the thermalization radius ($\sim 30\rm \, au$) where the optical continuum is formed. A month later, as the fallback rate drops and the scattering photosphere recedes, the continuum polarization increases, revealing a moderately aspherical interior. We also see evidence for smaller scale density variations in the scattering photosphere, inferred from the scatter of the data in the Stokes q−u plane. On day +29, the H α emission-line peak is depolarized to ∼0.3 per cent (compared to ∼1 per cent continuum polarization), and displays a gradual rise towards the line’s redder wavelengths. This observation indicates the H α line formed near the electron-scattering radius.

From Pericenter and Back: Full Debris Stream Evolution in Tidal Disruption Events

When a star passes too close to a supermassive black hole, it gets disrupted by strong tidal forces. The stellar debris then evolves into an elongated stream of gas that partly falls back toward the black hole. We present an analytical model describing for the first time the full stream evolution during such a tidal disruption event (TDE). Our framework consists of dividing the stream into different sections of elliptical geometry, whose properties are independently evolved in their comoving frame under the tidal, pressure, and self-gravity forces. Through an explicit treatment of the tidal force and the inclusion of the gas angular momentum, we can accurately follow the stream evolution near pericenter. Our model evolves the longitudinal stream stretching and both transverse widths simultaneously. For the latter, we identify two regimes depending on whether the dynamics is entirely dominated by the tidal force (ballistic regime) or additionally influenced by pressure and self-gravity (hydrostatic regime). We find that the stream undergoes transverse collapses both shortly after the stellar disruption and upon its return near the black hole, at specific locations determined by the regime of evolution considered. The stream evolution predicted by our model can be used to determine the subsequent interactions experienced by this gas that are at the origin of most of the electromagnetic emission from TDEs. Our results suggest that the accretion disk may be fed at a rate that differs from the standard fallback rate, which would provide novel observational signatures dependent on black hole spin.

Vocal-friend: internet of social-things framework to aid verbal communication

Purpose Deficits in social verbal communication in individuals with Social Communication Disorder (SCD) is of concern and SCD in the human community is prevalent in large population throughout the globe. Deficits in verbal social communication are prevalent in a large population. This paper aimed to propose internet connected multi-system architecture which is capable to support verbal communication in a social environment for individuals with social communication deficits. Material and methods Implementation methodology was included with corpus collection for specific communication, deep learning based machine training for intelligent communication, and implementation of the trained algorithm on internet connected electronic multiple social communication devices. The implemented system is smart enough to initiate and maintain two types of communication; the first type includes communication between multiple individuals on the remote location and the second type includes communication with the individual present in the physical listening range. Results The system was investigated in terms of its algorithmic parameters and found 97% to 100% in terms of training and testing accuracy with negligible mean squared error. Vocal-Friend analysed results based on audio-bot simulative conditions provide more than 91% accuracy, interaction rate and fallback rate. On the basis of the satisfaction analysis, above average results were noticed. Conclusion In terms of technical implementations and satisfaction analysis, results found acceptable with above average score. IMPLICATION FOR REHABILITATION Proposed framework is easy to use by caregivers with even having little knowledge. Support individual with deficit to learn social verbal communication skill to survive in society. Aiding parents, caregivers and professionals to understand the communication needs of individuals with communication deficits. Since technology is also grooming in the domain of rehabilitation, so this system could be used in various future applications such as social robots, social virtual assistants etc.

The nozzle shock in tidal disruption events

ABSTRACT Tidal disruption events (TDEs) occur when a star gets torn apart by the strong tidal forces of a supermassive black hole, which results in the formation of a debris stream that partly falls back towards the compact object. This gas moves along inclined orbital planes that intersect near pericentre, resulting in a so-called ‘nozzle shock’. We perform the first dedicated study of this interaction, making use of a two-dimensional simulation that follows the transverse gas evolution inside a given section of stream. This numerical approach circumvents the lack of resolution encountered near pericentre passage in global three-dimensional simulations using particle-based methods. As it moves inward, we find that the gas motion is purely ballistic, which near pericentre causes strong vertical compression that squeezes the stream into a thin sheet. Dissipation takes place at the resulting nozzle shock, inducing a rise in pressure that causes the collapsing gas to bounce back, although without imparting significant net expansion. As it recedes to larger distances, this matter continues to expand while remaining thin despite the influence of pressure forces. This gas evolution specifies the strength of the subsequent self-crossing shock, which we find to be more affected by black hole spin than previously estimated. We also evaluate the impact of general relativistic effects, viscous dissipation, magnetic fields, and radiative processes on the nozzle shock. This study represents an important step forward in the theoretical understanding of TDEs, bridging the gap between our robust knowledge of the fallback rate and the more complex following stages, during which most of the emission occurs.

Piezoelectric-Thermal Coupling Driven Biomimetic Stick-Slip Bidirectional Rotary Actuator for Nanomanipulation.

Substantial improvement of rotation driving accuracy is urgently needed and facing challenges. Miniature bidirectional rotary actuators with high-precision and controllable fallback rate require novel driving principles. Here, on the basis of a proposed biomimetic stick-slip motion principle, a novel piezoelectric-thermal coupling bidirectional rotary actuator was developed. The integrated mantis grasping leglike biomimetic claws and heating rods could realize the clockwise macroscopic rotation and anticlockwise macroscopic fallback of a cylindrical rotator, generated by piezoelectric stick-slip and thermal expansion, respectively. The rotation fallback was effectively inhibited at relatively lower frequencies and higher voltages, as a slight fallback rate of 0.095 was confirmed in term of 0.5 Hz and 80 V. An extraordinary piezoelectric-driven macroscopic rotation resolution of 0.2 μrad and thermal-induced microscopic resolution of 0.00073°/°C were experimentally revealed with the aid of real-time observation of the clockwise slow sticking and anticlockwise instantaneous slipping processes by using three-dimensional optical imaging.

Fallback Accretion Halted by R-process Heating in Neutron Star Mergers and Gamma-Ray Bursts

Abstract The gravitational wave event GW170817 with a macronova/kilonova shows that a merger of two neutron stars ejects matter with radioactivity including r-process nucleosynthesis. A part of the ejecta inevitably falls back to the central object, possibly powering long-lasting activities of a short gamma-ray burst (sGRB), such as extended and plateau emissions. We investigate fallback accretion with r-process heating by performing one-dimensional hydrodynamic simulations and developing a semi-analytical model. We show that the usual fallback rate dM/dt ∝ t −5/3 is halted by the heating because pressure gradients accelerate ejecta beyond an escape velocity. The suppression is steeper than Chevalier’s power-law model through Bondi accretion within a turn-around radius. The characteristic halting timescale is ∼104–108 s for the GW170817-like r-process heating, which is longer than the typical timescale of the long-lasting emission of sGRBs. The halting timescale is sensitive to the uncertainty of the r-process. Future observations of fallback halting could constrain the r-process heating on the scale of a year.

Partial, Zombie, and Full Tidal Disruption of Stars by Supermassive Black Holes

Abstract We present long-duration numerical simulations of the tidal disruption of stars modeled with accurate stellar structures and spanning a range of pericenter distances, corresponding to cases where the stars are partially and completely disrupted. We substantiate the prediction that the late-time power-law index of the fallback rate n ∞ ≃ −5/3 for full disruptions, while for partial disruptions—in which the central part of the star survives the tidal encounter intact—we show that n ∞ ≃ −9/4. For the subset of simulations where the pericenter distance is close to that which delineates full from partial disruption, we find that a stellar core can reform after the star has been completely destroyed; for these events the energy of the zombie core is slightly positive, which results in late-time evolution from n ≃ −9/4 to n ≃ −5/3. We find that self-gravity can generate an n(t) that deviates from n ∞ by a small but significant amount for several years post-disruption. In one specific case with the stellar pericenter near the critical value, we find that self-gravity also drives the recollapse of the central regions of the debris stream into a collection of several cores while the rest of the stream remains relatively smooth. We also show that it is possible for the surviving stellar core in a partial disruption to acquire a circumstellar disk that is shed from the rapidly rotating core. Finally, we provide a novel analytical fitting function for the fallback rates that may also be useful in a range of contexts beyond tidal disruption events.

Two regimes of tidal-stream circularization by supermassive black holes

Stars that approach a supermassive black hole (SMBH) too closely can be disrupted by the tidal gravitational field of the SMBH. The resulting debris forms a tidal stream orbiting the SMBH which can collide with itself due to relativistic apsidal precession. These self-collisions dissipate energy, causing the stream to circularize. We perform kinematic simulations of these stream self-collisions to estimate the efficiency of this circularization as a function of SMBH mass $M_\bullet$ and penetration factor $\beta$, the ratio of the tidal radius to the pericenter distance. We uncover two distinct regimes depending on whether the time $t_c$ at which the most tightly bound debris circularizes is greater or less than the time $t_{\rm fb}$ at which the mass fallback rate peaks. The bolometric light curve of energy dissipated in the stream self-collisions has a single peak at $t > t_{\rm fb}$ in the slow circularization regime ($t_c > t_{\rm fb}$), but two peaks (one at $t < t_{\rm fb}$ and a second at $t_{\rm fb}$) in the fast circularization regime ($t_c < t_{\rm fb}$). Tidal streams will circularize in the slow (fast) regime for apsidal precession angles less (greater) than 0.2 radians which occur for $\beta \lesssim (\gtrsim) (M_\bullet/10^6M_\odot)^{-2/3}$. The observation of prominent double peaks in bolometric TDE light curves near the transition between these two regimes would strongly support our model of tidal-stream kinematics.

On the Origin of Late-time X-Ray Flares in UV/optically Selected Tidal Disruption Events

Abstract We propose a model to explain the time delay between the peak of the optical and X-ray luminosity, Δt O−X hereafter, in UV/optically selected tidal disruption events (TDEs). The following picture explains the observed Δt O−X in several TDEs as a consequence of the circularization and disk accretion processes as long as the sub-Eddington accretion. At the beginning of the circularization, the fallback debris is thermalized by the self-crossing shock caused by relativistic precession, providing the peak optical emission. During the circularization process, the mass fallback rate decreases with time to form a ring around the supermassive black hole (SMBH). The formation timescale corresponds to the circularization timescale of the most tightly bound debris, which is less than a year to several decades, depending mostly on the penetration factor, the circularization efficiency, and the black hole mass. The ring will subsequently evolve viscously over the viscous diffusion time. We find that it accretes onto the SMBH on a fraction of the viscous timescale, which is 2 years for the typical given parameters, leading to X-ray emission at late times. The resultant Δt O−X is given by the sum of the circularization timescale and the accretion timescale and significantly decreases with increasing penetration factor to several to ∼10 yr typically. Since the X-ray luminosity substantially decreases as the viewing angle between the normal to the disk plane and line of sight increases from 0° to 90°, a low late-time X-ray luminosity can be explained by an edge-on view. We also discuss the super-Eddington accretion scenario, where Δt O−X is dominated by the circularization timescale.

Soybean oil supplementation for sows in the first three days after farrowing

ABSTRACT One of the main causes of death in newborn piglets is the low level of energy reserves to maintain their body temperature, which can lead to hypothermia and, subsequently, death. Thus, the objective of this study was to evaluate the effect of soybean oil supplementation to sows in the first three days after farrowing to save piglet fat reserves through the higher nutritional intake of sow milk. In total, 604.5, 750, 1000, 1250 and 1500 g of soybean oil were provided for each sow during the three days of supplementation. A total of 60 sows were evaluated per treatment, distributed in a random block design, supplemented in the first three days after farrowing with soybean oil added on top of the feed at the time of feeding. Performance and reproductive data and milk samples were collected from the sows to determine fat levels. Piglets were evaluated for fall-back rate and survival. There was no significant effect of soybean oil supplementation on any of the parameters evaluated for both sows and their milk. Therefore, soybean oil supplementation for sows in the first three days after farrowing does not influence performance parameters, reproduction and milk fat of the sows and mortality and fall-back rate of the piglets.

Global simulations of tidal disruption event disc formation via stream injection in GRRMHD

ABSTRACT We use the general relativistic radiation magnetohydrodynamics code KORAL to simulate the accretion disc formation resulting from the tidal disruption of a solar mass star around a supermassive black hole (BH) of mass 106 M⊙. We simulate the disruption of artificially more bound stars with orbital eccentricity e ≤ 0.99 (compared to the more realistic case of parabolic orbits with e = 1) on close orbits with impact parameter β ≥ 3. We use a novel method of injecting the tidal stream into the domain, and we begin the stream injection at the peak fallback rate in this study. For two simulations, we choose e = 0.99 and inject mass at a rate that is similar to parabolic TDEs. We find that the disc only becomes mildly circularized with eccentricity e ≈ 0.6 within the 3.5 d that we simulate. The rate of circularization is faster for pericenter radii that come closer to the BH. The emitted radiation is mildly super-Eddington with $L_{\rm {bol}}\approx 3{-}5\, L_{\rm {Edd}}$ and the photosphere is highly asymmetric with the photosphere being significantly closer to the inner accretion disc for viewing angles near pericenter. We find that soft X-ray radiation with Trad ≈ 3–5 × 105 K may be visible for chance viewing angles. Our simulations suggest that TDEs should be radiatively inefficient with η ≈ 0.009–0.014.

A Necessary Condition for Supernova Fallback Invading Newborn Neutron-star Magnetosphere

Abstract We numerically investigate the dynamics of a supernova fallback accretion confronting with a relativistic wind from a newborn neutron star (NS). The time evolution of the accretion shock in the radial direction is basically characterized by the encounter radius of the flow r enc and a dimensionless parameter ζ ≡ L / M ̇ fb c 2 , where L is the NS wind luminosity and M ̇ fb is the fallback mass accretion rate. We find that the critical condition for the fallback matter to reach near the NS surface can be simply described as ζ &lt; ζ min ≡ GM * / c 2 r enc or r enc L / GM * M ̇ fb &lt; 1 independent of the wind Lorentz factor, where M * is the NS mass. With combining the condition for the fallback matter to bury the surface magnetic field under the NS crust, we discuss the possibility that the trifurcation of NSs into rotation-powered pulsars, central compact objects, and magnetars can be induced by supernova fallback.

From Neutrino- to Photon-cooled in Three Years: Can Fallback Accretion Explain the X-Ray Excess in GW170817?

Abstract Excess X-ray emission from the neutron star merger GW170817 above the predicted afterglow was recently detected t ≈ 3.4 yr post-merger. One possible origin is accretion onto the newly unshrouded black hole (BH) remnant. While fallback of bound dynamical ejecta is insufficient to generate the excess luminosity, L X ∼ 5 × 1038 erg s−1, fallback from the disk wind ejecta—due to their larger mass and lower velocity—remains a possibility. We present hydrodynamic α-viscosity simulations of the post-merger disk evolution that extend to timescales t ≈ 35 s post-merger, necessary to capture the asymptotic evolution into the radiatively inefficient regime. Due to inefficient neutrino cooling, the BH accretion rate decays rapidly at late times ( M ̇ bh ∝ t − β bh , where β bh ≈ 2.4–2.8), which is incompatible with the late-time excess. However, matter falls back to the inner disk from the equatorial region more gradually, M ̇ fb ∝ t − β fb with β fb ≈ 1.43 in our α ≈ 0.03 simulations. By the present epoch t ≈ 3.4 yr, the fallback rate has become sub-Eddington and the disk can again accrete efficiently, i.e., M ̇ bh ≈ M ̇ fb , this time due to photon instead of neutrino cooling. The predicted present-day X-ray accretion luminosity, L X ≈ 0.1 M ̇ bh c 2 ≈ ( 2 – 70 ) × 10 38 erg s−1 for β fb ≈ 1.43–1.66, thus supports (with caveats) an accretion-powered origin for the X-ray excess in GW170817. The suppressed BH accretion rate prior to the sub-Eddington transition, weeks to months after the merger, is key to avoid overproducing the kilonova luminosity via reprocessing.

Tidal Disruption Events

The concept of stars being tidally ripped apart and consumed by a massive black hole (MBH) lurking in the center of a galaxy first captivated theorists in the late 1970s. The observational evidence for these rare but illuminating phenomena for probing otherwise dormant MBHs first emerged in archival searches of the soft X-ray ROSAT All-Sky Survey in the 1990s, but has recently accelerated with the increasing survey power in the optical time domain, with tidal disruption events (TDEs) now regarded as a class of optical nuclear transients with distinct spectroscopic features. Multiwavelength observations of TDEs have revealed panchromatic emission, probing a wide range of scales, from the innermost regions of the accretion flow to the surrounding circumnuclear medium. I review the current census of 56 TDEs reported in the literature, and their observed properties can be summarized as follows: ▪ The optical light curves follow a power-law decline from peak that scales with the inferred central black hole mass as expected for the fallback rate of the stellar debris, but the rise time does not. ▪ The UV-optical and soft X-ray thermal emission come from different spatial scales, and their intensity ratio has a large dynamic range and is highly variable, providing important clues as to what is powering the two components. ▪ They can be grouped into three spectral classes, and those with Bowen fluorescence line emission show a preference for a hotter and more compact line-emitting region, whereas those with only Heii emission lines are the rarest class.

Partial tidal disruption events by stellar mass black holes: Gravitational instability of stream and impact from remnant core

ABSTRACT In dense star clusters, such as globular and open clusters, dynamical interactions between stars and black holes (BHs) can be extremely frequent, leading to various astrophysical transients. Close encounters between a star and a stellar mass BH make it possible for the star to be tidally disrupted by the BH. Due to the relative low mass of the BH and the small cross-section of the tidal disruption event (TDE) for cases with high penetration, disruptions caused by close encounters are usually partial disruptions. The existence of the remnant stellar core and its non-negligible mass compared to the stellar mass BH alters the accretion process significantly. We study this problem with SPH simulations using the code Phantom, with the inclusion of radiation pressure, which is important for small mass BHs. Additionally, we develop a new, more general method of computing the fallback rate which does not rely on any approximation. Our study shows that the powerlaw slope of the fallback rate has a strong dependence on the mass of the BH in the stellar mass BH regime. Furthermore, in this regime, self-gravity of the fallback stream and local instabilities become more significant, and cause the disrupted material to collapse into small clumps before returning to the BH. This results in an abrupt increase of the fallback rate, which can significantly deviate from a powerlaw. Our results will help in the identification of TDEs by stellar mass BHs in dense clusters.

Giant X-Ray and Optical Bump in GRBs: Evidence for Fallback Accretion Model

Abstract The successful operation of dedicated detectors has brought us valuable information for understanding the central engine and the progenitor of gamma-ray bursts (GRBs). For instance, the giant X-ray and optical bumps found in some long-duration GRBs (e.g., GRBs 121027A and 111209A) imply that some extended central engine activities, such as the late X-ray flares, are likely due to the fall-back of progenitor envelope materials. Here we systemically search for long GRBs that consist of a giant X-ray or optical bump from the Swift GRB sample, and eventually we find 19 new possible candidates. The fall-back accretion model could interpret the X-ray and optical bump for all candidates within a reasonable parameter space. Six candidates showing simultaneous bump signatures in both X-ray and optical observations, which could be well fitted at the same time when scaling down the X-ray flux into optical by one order of magnitude, are consistent with the standard F ν ∝ ν 1/3 synchrotron spectrum. The typical fall-back radius is distributed around 1010–1012 cm, which is consistent with the typical radius of a Wolf–Rayet star. The peak fall-back accretion rate is in the range of ∼10−11–10−4 M ⊙ s−1 at time ∼102–105 s, which is relatively easy to fulfill as long as the progenitor’s metallicity is not too high. Combined with the sample we found, future studies of the mass supply rate for the progenitors with different mass, metallicity, and angular momentum distribution would help us to better constrain the progenitor properties of long GRBs.