Disk Instability Research Articles

The outcome of collisions between gaseous clumps formed by disk instability

The disk instability model is a promising pathway for giant planet formation in various conditions. At the moment, population synthesis models are used to investigate the outcomes of this theory, where a key ingredient of the disk population evolution are collisions of self-gravitating clumps formed by the disk instabilities. In this study, we explored the wide range of dynamics between the colliding clumps by performing state-of-the-art smoothed particle hydrodynamics simulations with a hydrogen-helium mixture equation of state and investigated the parameter space of collisions between clumps of different ages, masses (1--10 Jupiter mass), various impact conditions (head-on to oblique collisions) and a range of relative velocities. We find that the perfect merger assumption used in population synthesis models is rarely satisfied and that the outcomes of most of the collisions lead to erosion, disruption or a hit-and-run. We also show that in some cases collisions can initiate the dynamical collapse of the clump. We conclude that population synthesis models should abandon the simplifying assumption of perfect merging. Relaxing this assumption will significantly affect the inferred population of planets resulting from the disk instability model.

The Impact of Positive AGN Feedback on the Properties of Galaxies in a Semianalytic Model of Galaxy Formation

We introduce the state-of-the-art semianalytic model Formation and Evolution of GAlaxies (FEGA), which incorporates updated prescriptions for key physical processes in galaxy formation. Notably, FEGA features an unprecedented semianalytic modeling of positive active galactic nucleus (AGN) feedback. The model combines the latest prescriptions for gas infall and cooling, a revised star formation recipe that incorporates the extended Kennicutt–Schmidt relation, disk instability, updated supernova feedback, reincorporation of ejected gas, hot gas stripping from satellite galaxies, and the formation of diffuse light. A novel description of AGN feedback is introduced, describing the positive mode as a burst of star formation from a cooling gas fraction. FEGA is rigorously calibrated using a Markov Chain Monte Carlo procedure to match the evolution of the stellar mass function from high redshift to the present. Subsequently, the model is tested against several observed and predicted scaling relations, including the star formation rate (SFR)–mass, black hole–bulge and stellar mass, stellar-to-halo mass, and red fraction–mass relations. Additionally, we test FEGA against other galaxy properties, such as the distribution of specific SFRs, stellar metallicity, and morphology. Our results demonstrate that the inclusion of positive AGN feedback can coexist with its negative counterpart without drastic alterations to other prescriptions. Importantly, this inclusion improves the ability of the model to describe the primary scaling relations observed in galaxies.

Multi-kilogauss magnetic field driving the magnetospheric accretion process in EX Lupi

EX Lupi is the prototype of EX Lup-type stars, which are classical T Tauri stars (cTTSs) with luminosity bursts and outbursts of 1 to 5 magnitudes that last for a few months to a few years. These events are ascribed to an episodic accretion that can occur repeatedly, but whose physical mechanism is still debated. We aim to investigate the magnetically driven accretion of EX Lup in quiescence. We include for the first time a study of the small- and large-scale magnetic field. This allows us to characterise the magnetospheric accretion process of the system completely. We used spectropolarimetric times series acquired in 2016 and 2019 with the Echelle SpectroPolarimetric Device for the Observation of Stars and in 2019 with the SpectroPolarimètre InfraRouge at the Canada-France-Hawaii telescope during a quiescence phase of EX Lup. We were thus able to perform a variability analysis of the radial velocity, the emission lines, and the surface-averaged longitudinal magnetic field in different epochs and wavelength domains. We also provide a small-scale magnetic field analysis using Zeeman intensification of photospheric lines and a large-scale magnetic topology reconstruction using Zeeman-Doppler imaging. Our study reveals that typical magnetospheric accretion is ongoing on EX Lup. A main accretion funnel flow connects the inner disc to the star in a stable fashion and produces an accretion shock on the stellar surface close to the pole of the magnetic dipole component. We also measure one of the strongest fields ever observed on cTTSs. This strong field indicates that the disc is truncated by the magnetic field close to but beyond the corotation radius, where the angular velocity of the disc equals the angular velocity of the star. This configuration is suitable for a magnetically induced disc instability that yields episodic accretion onto the star.

The active role of co-evolving haloes in stellar bar formation

ABSTRACT We use idealized N-body simulations of equilibrium discs in live and static haloes to study how dark matter co-evolution impacts the assembly of stellar particles into a bar and the halo response. Initial conditions correspond to a marginally unstable disc according to commonly used disc stability criteria, and are evolved for the equivalent of about 150 disc dynamical times (10 Gyr). An extensive convergence study ensures accurate modelling of the bar formation process. Live haloes lead to the formation of a strong bar, but the same disc remains unbarred when evolved in a static halo. Neither seeded disc instabilities nor longer (60 Gyr) simulations result in the formation of a bar when the halo is static. When the live halo is replaced with a static analogue at later times, the previously robust bar slowly dissipates, suggesting that (1) the co-evolution of the disc and halo is critical for the assembly and long-term survival of bars in marginally unstable discs and (2) global disc stability criteria must be modified for discs in the presence of live haloes. In our live halo runs, a ‘dark bar’ grows synchronously with the stellar bar. Processes that inhibit the transfer of angular momentum between the halo and disc may stabilize a galaxy against bar formation, and can lead to the dissolution of the bar itself. This raises further questions about the puzzling stability of observed discs that are marginally unstable, but unbarred.

Local gravitational instability of two-component thick discs in three dimensions

Aims. The local gravitational instability of rotating discs is believed to be an important mechanism in different astrophysical processes, including the formation of gas and stellar clumps in galaxies. We aim to study the local gravitational instability of two-component thick discs in three dimensions. Methods. We use as a starting point a recently proposed analytic three-dimensional (3D) instability criterion for discs with non-negligible thickness that takes the form Q3D < 1, where Q3D is a 3D version of the classical 2D Toomre Q parameter for razor-thin discs. Here, we extend the 3D stability analysis to two-component discs, considering first the influence on Q3D of a second unresponsive component, and then the case in which both components are responsive. We present the application to two-component discs with isothermal vertical distributions, which can represent, for instance, galactic discs with both stellar and gaseous components. Finally, we relax the assumption of vertical isothermal distribution, by studying one-component self-gravitating discs with polytropic vertical distributions for a range of values of the polytropic index corresponding to convectively stable configurations. Results. We find that Q3D < 1, where Q3D can be computed from observationally inferred quantities, is a robust indicator of local gravitational instability, depending only weakly on the presence of a second component and on the vertical gradient of temperature or velocity dispersion. We derive a sufficient condition for local gravitational instability in the midplane of two-component discs, which can be employed when both components have Q3D > 1.

𝒫𝒯 and anti-𝒫𝒯 symmetries for astrophysical waves

Context. Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics. Aims. We aim to show how 𝒫𝒯 and anti-𝒫𝒯 symmetries determine the behaviour of linear perturbations in a wide class of astrophysical problems. They set the location of ‘exceptional points’ in the parameter space and the associated transitions to instability, and are associated with the conservation of quadratic quantities that can be determined explicitly. Methods. We study several classical local problems: the gravitational instability of isothermal spheres and thin discs, the Schwarzschild instability, the Rayleigh-Bénard instability and acoustic waves in dust–gas mixtures. We calculate the locations and the order of the exceptional points using the resultant of two univariate polynomials, as well as the conserved quantities in the different regions of the parameter space using Krein theory. Results. All problems studied here exhibit discrete symmetries, even though Hermiticity is broken by different physical processes (self-gravity, buoyancy, diffusion, and drag). This analysis provides genuine explanations for certain instabilities, and for the existence of regions in the parameter space where waves do not propagate. Those two aspects correspond to regions where 𝒫𝒯 and anti-𝒫𝒯 symmetries are broken respectively. Not all instabilities are associated to symmetry breaking (e.g. the Rayleigh-Benard instability).

Rossby wave instability and substructure formation in 3D non-ideal MHD wind-launching discs

ABSTRACT Rings and gaps are routinely observed in the dust continuum emission of protoplanetary discs (PPDs). How they form and evolve remains debated. Previous studies have demonstrated the possibility of spontaneous gas rings and gaps formation in wind-launching discs. Here, we show that such gas substructures are unstable to the Rossby wave instability (RWI) through numerical simulations. Specifically, shorter wavelength azimuthal modes develop earlier, and longer wavelength ones dominate later, forming elongated (arc-like) anticyclonic vortices in the rings and (strongly magnetized) cyclonic vortices in the gaps that persist until the end of the simulation. Highly elongated vortices with aspect ratios of 10 or more are found to decay with time in our non-ideal magnetohydrodynamic (MHD) simulation, in contrast with the hydro case. This difference could be caused by magnetically induced motions, particularly strong meridional circulations with large values of the azimuthal component of the vorticity, which may be incompatible with the columnar structure preferred by vortices. The cyclonic and anticyclonic RWI vortices saturate at moderate levels, modifying but not destroying the rings and gaps in the radial gas distribution of the disc. In particular, they do not shut-off the poloidal magnetic flux accumulation in low-density regions and the characteristic meridional flow patterns that are crucial to the ring and gap formation in wind-launching discs. Nevertheless, the RWI and their associated vortices open up the possibility of producing non-axisymmetric dust features observed in a small fraction of PPDs through non-ideal MHD, although detailed dust treatment is needed to explore this possibility.

Instability of Circumnuclear Gas Supply as an Origin of the “Changing-look” Phenomenon of Supermassive Black Holes

The origin of the “changing-look” (CL) phenomenon in supermassive black holes (SMBHs) remains an open issue. This study aims to shed light on this phenomenon by focusing on a sample that encompasses all known repeating CL active galactic nuclei (AGNs). Through the identification of a characteristic timescale for the CL phenomenon, it was observed that larger SMBHs possess shorter characteristic timescales, while smaller SMBHs exhibit longer timescales. These findings reveal a significant contrast to the traditional AGN variability that has been adequately explained by the AGN’s disk instability model. This stark discrepancy highlights a distinct origin of the CL phenomenon, distinguishing it from traditional AGN variability. By properly predicting the characteristic timescale and its dependence on SMBH mass, we propose that the CL phenomenon is likely a result of a variation in accretion rate caused by a sudden change in the supply of circumnuclear gas during the transition between active and passive SMBH fueling stages.

X-Ray Quasi-periodic Eruptions and Tidal Disruption Events Prefer Similar Host Galaxies

In the past 5 yr, six X-ray quasi-periodic eruption (QPE) sources have been discovered in the nuclei of nearby galaxies. Their origin remains an open question. We present Multi Unit Spectroscopic Explorer integral field spectroscopy of five QPE host galaxies to characterize their properties. We find that 3/5 galaxies host extended emission-line regions (EELRs) up to 10 kpc in size. The EELRs are photoionized by a nonstellar continuum, but the current nuclear luminosity is insufficient to power the observed emission lines. The EELRs are decoupled from the stars both kinematically and in projected sky position, and the low velocities and velocity dispersions (<100 km s−1 and ≲75 km s−1, respectively) are inconsistent with being driven by active galactic nuclei (AGNs) or shocks. The origin of the EELRs is likely a previous phase of nuclear activity. QPE host galaxies share several similarities with tidal disruption event (TDE) hosts, including an overrepresentation of galaxies with strong Balmer absorption and little ongoing star formation, as well as a preference for a short-lived (the typical EELR lifetime is ∼15,000 yr), gas-rich phase where the nucleus has recently faded significantly. This suggests that QPEs and TDEs may share a common formation channel, disfavoring AGN accretion disk instabilities as the origin of QPEs. If QPEs are related to extreme mass ratio inspiral systems (EMRIs), e.g., stellar-mass objects on bound orbits about massive black holes, the high incidence of EELRs and recently faded nuclei could be used to localize the hosts of EMRIs discovered by low-frequency gravitational-wave observatories.

Fresh view of the hot brown dwarf HD 984 B through high-resolution spectroscopy

Context. High-resolution spectroscopy has the potential to drive a better understanding of the atmospheric composition, physics, and dynamics of young exoplanets and brown dwarfs, bringing clear insights into the formation channel of individual objects. Aims. Using the Keck Planet Imager and Characterizer (KPIC; R « 35 000), we aim to characterize a young brown dwarf HD 984 B. By measuring its C/O and 12CO/13CO ratios, we expect to gain new knowledge about its origin by confirming the difference in the formation pathways between brown dwarfs and super-Jupiters. Methods. We analysed the KPIC high-resolution spectrum (2.29–2.49 μm) of HD 984 B using an atmospheric retrieval framework based on nested sampling and petitRADTRANS, using both clear and cloudy models. Results. Using our best-fit model, we find C/O = 0.50 ± 0.01 (0.01 is the statistical error) for HD 984 B which agrees with that of its host star within 1σ (0.40 ± 0.20). We also retrieve an isotopolog 12CO/13CO ratio of 98-25+20 in its atmosphere, which is similar to that of the Sun. In addition, HD 984 B has a substellar metallicity with [Fe/H] =-0.62-0.02+0.02. Finally, we find that most of the retrieved parameters are independent of our choice of retrieval model. Conclusions. From our measured C/O and 12CO/13CO, the favored formation mechanism of HD 984 B seems to be via gravitational collapse or disk instability and not core accretion, which is a favored formation mechanism for giant exoplanets with m &lt; 13 MJup and semimajor axis between 10 and 100 au. However, with only a few brown dwarfs with a measured 12CO/13CO ratio, similar analyses using high-resolution spectroscopy will become essential in order to determine planet formation processes more precisely.

Parametric Survey of Nonaxisymmetric Accretion Disk Instabilities: Magnetorotational Instability to Super-Alfvénic Rotational Instability

Accretion disks are highly unstable to magnetic instabilities driven by shear flow, where classically, the axisymmetric, weak-field magnetorotational instability (MRI) has received much attention through local WKB approximations. In contrast, discrete nonaxisymmetric counterparts require a more involved analysis through a full global approach to deal with the influence of the nearby magnetohydrodynamic (MHD) continua. Recently, rigorous MHD spectroscopy identified a new type of ultralocalized, nonaxisymmetric instability in global disks with super-Alfvénic flow. These super-Alfvénic rotational instabilities (SARIs) fill vast unstable regions in the complex eigenfrequency plane with (near eigen)modes that corotate at the local Doppler velocity and are radially localized between Alfvénic resonances. Unlike discrete modes, they are utterly insensitive to the radial disk boundaries. In this work, we independently confirm the existence of these unprecedented modes using our novel spectral MHD code Legolas, reproducing and extending our earlier study with detailed eigenspectra and eigenfunctions. We calculate the growth rates of SARIs and MRI in a variety of disk equilibria, highlighting the impact of field strength and orientation, and find correspondence with analytical predictions for thin, weakly magnetized disks. We show that nonaxisymmetric modes can significantly extend instability regimes at high mode numbers, with maximal growth rates comparable to those of the MRI. Furthermore, we explicitly show a region filled with quasi-modes whose eigenfunctions are extremely localized in all directions. These modes must be ubiquitous in accretion disks, and play a role in local shearing box simulations. Finally, we revisit recent dispersion relations in the appendix, highlighting their relation to our global framework.

Thermal Structure Determines Kinematics: Vertical Shear Instability in Stellar Irradiated Protoplanetary Disks

Turbulence is crucial for protoplanetary disk dynamics, and vertical shear instability (VSI) is a promising mechanism in outer disk regions to generate turbulence. We use the Athena++ radiation module to study VSI in full and transition disks, accounting for radiation transport and stellar irradiation. We find that the thermal structure and cooling timescale significantly influence VSI behavior. The inner rim location and radial optical depth affect disk kinematics. Compared with previous vertically isothermal simulations, our full disk and transition disks with small cavities have a superheated atmosphere and cool midplane with long cooling timescales, which suppresses the corrugation mode and the associated meridional circulation. This temperature structure also produces a strong vertical shear at τ * = 1, producing an outgoing flow layer at τ * < 1 on top of an ingoing flow layer at τ * ∼ 1. The midplane becomes less turbulent, while the surface becomes more turbulent with effective α reaching ∼10−2 at τ * ≲ 1. This large surface stress drives significant surface accretion, producing substructures. Using temperature and cooling time measured/estimated from radiation-hydro simulations, we demonstrate that less computationally intensive simulations incorporating simple orbital cooling can almost reproduce radiation-hydro results. By generating synthetic images, we find that substructures are more pronounced in disks with larger cavities. The higher velocity dispersion at the gap edge could also slow particle settling. Both properties are consistent with recent near-IR and Atacama Large Millimeter/submillimeter Array (ALMA) observations. Our simulations predict that regions with significant temperature changes are accompanied by significant velocity changes, which can be tested by ALMA kinematics/chemistry observations.

Quasi-periodic oscillation analysis for a sample of blazars at the optical band

Quasi-periodic behavior in the light curves of blazars can help us understand the physics. The identification of the supermassive black hole binary (SMBHB) is an active topic, and the periodicity analysis of light curves is a powerful method for searching for these sources. In this work, we aim to identify quasi-periodic oscillations (QPOs) in the optical band light curves in our sample and discuss the possible physical origins behind these targets. In this study, we collected 155 optical band light curves from three different monitoring programs. We searched for QPOs in our sample using the generalised Lomb-Scargle (GLS) and weighted wavelet Z-transform (WWZ) methods. We simulated $10^ $ artificial light curves and evaluated the significance of the results using the Monte Carlo method. Our work reveals that 18 targets show QPOs with timescales ranging from 200 days to 1400 days. These QPOs could be explained by three scenarios, including the SMBHB, instability of thick disks and jet precession. Since the frequencies corresponding to QPOs are in the nHz regime, our work provides candidates of SMBHBs for further verification.

A self-consistent model for dust settling and the vertical shear instability in protoplanetary disks

Abstract The spatial distribution of dust particles in protoplanetary disks affects dust evolution and planetesimal formation processes. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk region and affect dust diffusion. Turbulence driven by the VSI has a predominant vertical motion that can prevent dust settling. On the other hand, the dust distribution controls the spatial distribution of the gas cooling rate, thereby affecting the strength of VSI-driven turbulence. Here, we present a semi-analytic model that determines the vertical dust distribution and the strength of VSI-driven turbulence in a self-consistent manner. The model uses an empirical formula for the vertical diffusion coefficient in VSI-driven turbulence obtained from our recent hydrodynamical simulations. The formula returns the vertical diffusion coefficient as a function of the vertical profile of the cooling rate, which is determined by the vertical dust distribution. We use this model to search for an equilibrium vertical dust profile where settling balances with turbulent diffusion for a given maximum grain size. We find that if the grains are sufficiently small, there exists a stable equilibrium dust distribution where VSI-driven turbulence is sustained at a level of αz ∼ 10−3, where αz is the dimensionless vertical diffusion coefficient. However, as the maximum grain size increases, the equilibrium solution vanishes because the VSI can no longer stop the settling of the grains. This runaway settling may explain highly settled dust rings found in the outer part of some protoplanetary disks.

The evolution of supermassive black hole mass–bulge mass relation by a semi-analytical model, ν2GC

ABSTRACT We have investigated the redshift evolution of the relationship between supermassive black hole (SMBH) mass and host bulge mass using a semi-analytical galaxy formation model ν2GC. Our model reproduces the relation in the local universe well. We find that, at high redshift (z ≳ 3), two sequences appear in the SMBH mass–bulge mass plane. The emergence of these two sequences can be attributed to the primary triggers of the growth of the SMBHs and bulges: galaxy mergers and disc instabilities. The growth of SMBHs and bulges as a result of galaxy mergers is responsible for giving rise to the high-mass sequence, in which SMBHs are more massive for a given host bulge mass than in the low-mas sequence. Conversely, disc instabilities are accountable for the emergence of the low-mass sequence. At lower redshifts, galaxy mergers tend to become increasingly deficient in gas, resulting in a preferential increase of bulge mass without a corresponding growth in SMBH mass. This has the effect of causing galaxies in the upper sequence to shift towards the lower one on the SMBH mass–bulge mass plane. The galaxies that undergo dry mergers serve to bridge the gap between the two sequences, eventually leading to convergence into a single relation known in the local universe. Our results suggest that the observations of the SMBH mass–bulge mass relation in high redshifts can provide insight into their growth mechanisms.

Study of type II migration under the framework of the disk instability model for giant planet formation

Context. Hydrodynamic simulations of the migration of planets formed by gravitational instability suggest that after an initial phase of fast migration, planets can open gaps and continue to migrate on a type II migration timescale. The simulation time length is typically on the order of 104 yr. Aims. We study the effects of the subsequent type II migration during the disk lifetime on the final orbital radii of planets. Methods. We used a numerical disk model that follows the disk formation and evolution. The disk acquires mass through the mass influx from the collapse of its parent molecular cloud core. The model reflects the influence of the properties of the parent core on the disk. Considering clumps forming at different times in a disk and also in different disks with different parent core properties, we used the type II migration rate to follow the clump migration from the formation location. We studied the dependence of the clump migration on the properties of the parent core. Results. The mass influx drag enhances the migration process. The duration and viscosity of gravitational instability, viscosity in the dead zone, and the collapse time of the parent core play important roles in planet migration. As the angular momentum and mass of the parent core increase, migration is enhanced. The final radius is sensitive to the initial radius. Clumps forming at large radii might migrate outward with the disk expansion. Conclusions. Even though type II migration is slow, clumps can migrate over significant distances. A considerable proportion of clumps migrate to the central protostar via type II migration. Our calculations support the idea that the observed pile-up of planets at &lt;0.3 AU is explained by a scenario where planets might form at large radii, then migrate to orbits of &lt;0.3 AU, and halt by a stopping mechanism at this location.

The First Quenched Galaxies: When and How?

Many quiescent galaxies discovered in the early Universe by JWST raise fundamental questions on when and how these galaxies became and stayed quenched. Making use of the latest version of the semianalytic model GAEA that provides good agreement with the observed quenched fractions up to z ∼ 3, we make predictions for the expected fractions of quiescent galaxies up to z ∼ 7 and analyze the main quenching mechanism. We find that in a simulated box of 685 Mpc on a side, the first quenched massive (M ⋆ ∼ 1011 M ⊙), Milky Way–mass, and low-mass (M ⋆ ∼ 109.5 M ⊙) galaxies appear at z ∼ 4.5, z ∼ 6.2, and before z = 7, respectively. Most quenched galaxies identified at early redshifts remain quenched for more than 1 Gyr. Independently of galaxy stellar mass, the dominant quenching mechanism at high redshift is accretion disk feedback (quasar winds) from a central massive black hole, which is triggered by mergers in massive and Milky Way–mass galaxies and by disk instabilities in low-mass galaxies. Environmental stripping becomes increasingly more important at lower redshift.

Discovery of a new IW And-type dwarf nova with both tilted disc and tidal instability

ABSTRACT IW And-type dwarf novae are anomalous Z Cam stars featured with outbursts happening during standstill states, which are not expected in the standard disc instability model. The physical mechanisms for these variations remain unclear. In this study, we report the discovery of a new candidate IW And-type dwarf nova J0652+2436, identified with its frequent outbursts from the slowly rising standstill states. Luckily, the Transiting Exoplanet Survey Satellite observations during a long standstill state and the earlier K2 observations give a chance to find the orbital and negative superhump period in the light curve of J0652+2436, allowing the measurement of its mass ratio of 0.366. This mass ratio is marginally possible for the tidal instability to set in according to previous smoothed particle hydrodynamics simulations. Thus, we propose that the outbursts in J0652+2436 are likely to be caused by the growing accretion disc during standstills, in favour of the previous hypothesis of the mechanisms lying in all IW And stars. We conclude that J0652+2436 might be the first IW And star with both a precessing tilted disc and tidal instability, which will be an important laboratory for studying the accretion disc dynamics and help understand the IW And phenomenon.

Instability and trajectories of buoyancy-driven annular disks: A numerical study

Instability and trajectories of buoyancy-driven annular disks: A numerical study

Formation of long-period radio pulsars

Abstract This study investigates the influence of different braking mechanisms on the formation of three long-period radio pulsars (PSRs J0250 + 5854, J2251-3711, and J0901-4046): plasma-filled magnetosphere in combination with magnetic field decay, fall-back disc, and r-mode instability. These braking mechanisms can also affect the thermal evolution of pulsars. By comparing the model-predicted values with observational data such as spin periods, period derivatives, and upper limits of the bolometric luminosity of these pulsars, we find that these three braking mechanisms can reasonably explain the spin period and the period derivative within a certain range of parameters for these sources. The model-predicted bolometric luminosity associated with magnetic field dissipation exceeds the upper limit for PSR J0901-4046 but falls below the upper limits for PSR J0250 + 5854 and PSR J2251-3711. The model-predicted bolometric luminosity within the fall-back disc model exceeds the observed results, whereas the luminosity within the r-mode instability falls below the observed upper limit for these three pulsars. However, a conflict arises when we consider the pulsar radio activity and the accretion phases within the fall-back disc model simultaneously. By combining data from radio and X-ray observations, along with precise measurements of surface thermal emissions, we can enhance our understanding of the braking mechanisms involved in the formation of long-period radio pulsars or constrain key model parameters. Finding more long-period pulsars in the future and conducting multi-band observations will enhance our understanding of the formation and nature of long-period radio pulsars.