It is my aim to revise the relevant features of the cusp structures arise on from the accretion objects of neutron stars. The theoretical characteristics of the known schemes and the pertinent observational techniques and evidences are described. The complete description of a neutron star is that of the celestial object surrounded of macroscopic matter (solution of the Einstein Field Equations), where the scheme of the dust can be takes as a simple instance, from which the cusps structures arise. The accretion object can be modellized as a thin disc, a thick disc, an advection-dominated flow (which is analysed as gaseous) , or an accretion ring. The presence of magnetic fields is included in the magneto-hydro-dynamics complete parametrization of the neutron star. The identification of the neutron-star accretion objects from those of other relativistic stars has been a relevant effort, i.e., the discrimination of the neutron star accretion objects from the blackhole accretion objects has been pursued both from a theoretical point of view, and experimentally, according also to the luminosity of the objects and from the ejecta. At the moment, the theoretical interpretation of the accretion objects of neutron stars is delineated, and the present limitations in the description of cusps are elucidated. The new consequences of the understandings of the further missing elements in the theoretical analysis of the analytical functional dependences among the description of the foundational components of the neutron stars (i.e., spin, mass, and radius) are exposed. The geometries of these spacetimes are described after the metric tensor; the use of cataclysmic variables is commented as a valid tool: nevertheless, new sets of variables are to be considered. The role of the gravitational pressure within the radiation-pressure-supported objects is revised to be applied; as a further development, the implementation of the new corresponding General-Relativistic-Magneto-Hydro-Dynamics is to be now looked for.
- Neutron stars
- accretion objects
- cusps
1. Introduction
Recent studies of accretion tori around rotating neutron stars (NS), examine radiation-pressure-supported, equilibrium fluid structures in General Relativistic Hartle-Thorne geometry. These models analyze how angular momentum and quadrupole moments affect torus structure, size, and shape, crucial for interpreting quasiperiodic oscillations (QPOs) in low-mass X-ray binaries (LMXBs) [1].
Structure and Shape: The studies utilize a full general relativistic approach to model ”cusp” tori (critically thick). The interplay between the NS’s frame-dragging (angular momentum) and quadrupole moment can lead to structures similar to the non-rotating case at specific radii. Impact of Rotation: The size of the tori is strongly influenced by spacetime parameters, particularly when comparing moderate rotation to non-rotating scenarios.
Applications: These models help interpret the twin-peak QPOs in LMXBs, with simulations suggesting that fluid flow precession matches observed frequencies better than geodesic models.
2. Characterization of the Neutron-Star Accretion Objects
Formation and Structure: These tori are formed from material with significant angular momentum, which, instead of falling directly onto the star, spins around it. The inner edge of the torus is shaped by a cusp, where the material flows onto the neutron star.
Relativistic Effects: The structure of these tori is heavily influenced by the General Relativity (GR) effects of the rapidly rotating NS, including its angular momentum and quadrupole moment.
Cusp Accretion: The torus is considered ’critically thick’ when its inner edge (the cusp) reaches the Roche-like limit, allowing material to flow onto the star due to pressure and gravity; it is my aim to stress that the role of the gravitational pressure had not been investigated yet.
QPO Explanation: Oscillations of these cusp-filling tori are a leading model for explaining the twin-peak High-Frequency QPOs (HFQPOs) observed in Low-Mass X-ray Binaries (LMXBs).
Modelling: Research uses both numerical simulations and analytical methods (such as those for Hartle-Thorne geometry) to analyze these tori. Studies have shown that the torus size is sensitive to the neutron star’s equation of state (EoS).
Key Findings from Recent Studies: The ’Polish Doughnuts’: These, often, described, as, ”Polish, doughnuts,” are, analyzed, using, relativistic, polytropic, models. Magnetic Field Impact: While the torus itself forms from the material, the NS’s magnetic field can disrupt this structure and direct the flow into hot spots.
Fundamental characterizations of observation: These models are used to match the X-ray, spectral, and, timing, data, from, instruments, like LAXPC on AstroSat.
3. Cusps of Accretion Objects of Neutron Stars
Accretion tori with cusps around rotating neutron stars (NS’s) are equilibrium fluid configurations where gravity and pressure gradients drive accretion without needing viscosity. In 2024 and 2025, research has focused on how the NS’s angular momentum and quadruple moment influence these tori.
3.1. Key Characteristics pf the NS cusps
The Cusp: This is the inner edge of the torus where the equipotential surface self-crosses. Matter ”overflows” through this point directly onto the neutron star surface.
Hartle-Thorne Geometry: Researchers utilize this general relativistic solution to model the spacetime around rotating, oblate neutron stars.
Size and Mass: A rotating, oblate NS can host a torus with central density and mass several orders of magnitude higher than one around a non-rotating star.
Balancing Effects: There is an interplay where the NS’s angular momentum and its quadrupole moment (caused by rotation-induced oblateness) almost counterbalance each other’s impact on the torus shape at specific radii.
3.2. Observational Validation
QPO Origins: Oscillations in these tori-specifically the radial epicyclic mode-are a leading explanation for high-frequency quasi-periodic oscillations (QPOs) observed in X-ray binaries.
Lense-Thirring Precession: Recent 2025 studies have identified vertical (Lense-Thirring) precession in these flows as a sensitive indicator of the NS’s rotational properties and its equation of state (EoS).
Accretion Rate: The mass accretion rate from the torus carries a ’modulation imprint’ of the torus oscillations, which directly impacts the observed boundary layer luminosity.
4. Evolutions of the Study of the Neutron-Stars Cusps
In the work [2], the analysis of the relative number of low-luminosityX-ray sources and that of the high-luminosity ones in Galactic globular clusters is juxtaposed as far as the theoretical schematizations [3][4][5]. of which the experimental evidence is gathered.
In the work [6], the extended galactic populations of low-luminosity X-ray sources was compared with the diffuse X-ray background; as a result, the ’flattened’ galactic-halo distribution could be described as a thick disc with typical requirements about the luminosity which overlaps the analyses of the fluctuations of the unresolved XRB.
The possible nature of the sources was compared with subdwarfs, with LMXB’s. and with ’old’ neutron stars as far as the optical luminosity of the sources is concerned, for which observational test are proposed ibidem., where the possibility to match the cluster emission with the ROSAT observation from 1993 [7]- indeed, in the work of Hasinger, the background is resolved in discrete sources according to the flux after faint X-ray sources counts according to both the typical observations and to spectroscopic identification [8], where the latter technique has been undergoing improvements up to recent times, i.e., with NIR imaging.
A particular possible new class of neutron stars was eventually proposed in [9]. It is my aim to stress the the potentialities of the use of cataclysmic variables has not been fully investigated yet.
5. Cusps on Neutron Star Accretion Tori
The torus orbiting a neutron star can be studied analytically, as well as its cusp shapes. In the work of Matszukova´a et al. [10], the torus is characterized as 'radiation-pressure-supported. The Hartle Thorne description is utilized ibidem i.e., such that the NS angular momentum and the NS quadrupole moment are included within the analysis. It is my aim to stress that there arise diverse momenta in the description of NS structures, which have not been investigated yet.
The torus is characterized after its shape, its size, its structure and according to the presence of cusps, differently form the blackhole accretion objects, the instance of NS tori is affected after the role of the torus superdense macroscopic matter.
The observation if iron lines is adopted as a description parameter; the analysis of the quasi-periodic oscillations (QPO’s) is prescribed.
Further analyses of more complicated realizations are proposed in [11].
6. The Torus Equation of State
The reverence of the detection of gravitational waves in NS mergers is stressed from the work of Raithel [12] on the GW are scrutinized, the pertinent constraints on the NS EOS are reviewed ibidem; it this respects, attention is paid to the interplay between the tidal deformability of the merger and the mass of the NS, as discussed in [13]: the presence of the torus is discussed in Fig. 5 ibidem.
In [13], the importance of the mass of the remnant and directly that of the torus within the study of the ejecta, and in the radiation hydrodynamical properties, where the latter are modified after neutrino emission, are examined ibidem.
The rest-mass density of a merger remnant is investigated ibidem as well.
The electron fraction within the same context is examined as well, for remnant whose torus is orbiting is examined: the components of the ejecta are analyzed in Fig. 1 ibidem as being determined also after the mass.
The torus Equations of State (EOS) are constrained accordingly.
6.1. The Properties of the Dust Ring
In the work [14], the FR ratio galaxies are analyzed, and particular attention is devoted to the mechanisms leasing to the accretions rates of the different regions of the dust ring, after using the thermometric photometry- jet relationship, the passage from the thin disc model to the ADAF scheme is postulated. The Bondi accretion model is utilised for the energy spectrum to be fitted.
Ibidem, the FR I radio galaxies and the FR II ones are recalled to be discriminated after [15] according to the characterization of the optical magnitude and those of the total radio magnitude. In particular, from [15], one learns how to distinguish between the two types of galaxies.
The possible origin of NS has been investigated in [14].
7. Emission Phenomenology of the NS Accretiontori Cusp
7.1. r-Ray-Burst Phenomenology
The phenomenology of r-ray bursts (GRB’s) from the NS’s is appreciated as well for remnants surrounded of a torus in [16][17][18].
From [16], the role of the torus within the phenomenology of GRB170817A is to power the mass jet in the off-axis emission: the jet is found to interact with the torus giving raise to a ’cocoon’ emission; the process is referred to as ’trigger-ing’ (i.e., as from [17]) and was first detected after the Fermi-Gamma-Ra-Burst Monitor (GBM) on 17 August 2017. From ibidem, the gravitational radiation and the EM radiation are defined unambiguously from the same Astrophysical source: the start of GBM multi-messenger Astronomy was this way inaugurated.
7.2. Gravitational-Wave Phenomenology
The role of the torus in the GW emission is this way outlined within the frame-work of the evolution of the remnant, the post-merger being eventually signaled after the collapse of the torus.
8. Characterization of the Experimental Validation
In the work [10], the torus orbiting a rotating NS is studied analytically, The torus is characterized as ’radiation pressure’ supported.
The Harthle-Throne description is utilized ibidem such that the NS angular momentum and the NS quadrupole momentum are included within the analysis.
The torus is characterized via its shape, its (irr) size and its structure and the presence of the cusp; differently from the case of accretion objects of blackholes, the instance of the NS tori is affected after the role of the torus ’superdense’macroscopic matter'.
The observation of iron lines is outlined. The analysis of the quasi-periodic oscillations is prescribed.
The role of the torus in the GW emission is the way outlined within the frame-work of the evolution of the remnant, the post-merger being eventually signaled after the collapse of the torus, as analyzed in more detail in [19], in [20], and in [21]: the ejected mass is determined to cause a low X-ray emission in the merger, which is characterized after geometrical asymmetry of the two merge NS’s: the GW emission is described during the inspiralling, during the merger and during the formation of the disc.
8.1. Items of Description from the Experimental Data
In the work of Shi et al. [22], the kHz QPO’s of neutron stars re studied in the case of compressed magnetosphere comprehended in MHD: a functional dependence between the complete range of the kHz QPO frequencies and the accretion rates id ibidem newly evidentiated; as a resulting description, the parallel tracks which are explained as after slow variation of the magnetic field are found to be shifted.
Moreover, the spin, the mass and the radius of the NS are included within the picture, after the analysis of diverse sources.
Within the model, the usual relations between X-ray emissions and accretion process are assumed, the classifications of the several kHz QPO’s ar scrutinized ibidem as in charges to outline how the characteristic frequencies are related with the geometry, and a new role of the effective magnetic field is proposed. In the work of Shi et al. [23], the high-frequency QPO’s are studied in low-mass X-ray binaries; ibidem, the discrimination is provided with to identify NS’s from blackholes in LMXB’s.
The changes in the metric tensor of a gravitationally-collapsing mass are described in [24].
In [25], the environment of the accreting NS’s is further characterized Physically.
References
- M. Matuszkova’, G. Toeroek1, D. Lanc’ova’ et al., Accretion tori aroundrotating neutron stars I. Structure, shape, and size, A&A 691, Article Num-ber A167 (2024).
- J.E. Grindlay, Origin and Evolution of X-Ray Binaries in Globular Clusters.In: Truemper, J., Lewin, W.H.G., Brinkmann, W. (eds) The Evolutionof Galactic X-Ray Binaries, NATO ASI Series 167, Springer, Dordrecht(1986).
- J.E. Grindlay, Globular cluster origin of X-ray bursters, Advances in SpaceResearch 3(10-12), 19-28 (1984).
- X-raying the dynamics of globular clusters, IAU Symp 113, held in Prince-ton, New Jersey (USA), 29 May - 1 June 1984 J. Goodman P., Hut ed.’s,Dynamics of star clusters, 43-61 (1985).
- J.E. Grindlay, P. Hertz, Origin and Evolution of X-Ray Binaries in GlobularClusters, in Cataclysmic Variables and Low-Mass X-Ray Binaries, D.Q.Lamb, J. Patterson ed.’s., Dordrecht: D. Reidel, 79-91 (1985).
- E. Maoz, J.E. Grindlay, An Extended Galactic Population of Low-Luminosity X-Ray Sources (CVs?) and the Diffuse X-Ray Background,Astrophys.J. 444, 183 (1995).
- G. Hasinger, R. Burg, R. Giacconi et al., A Deep X-Ray Survey in the Lockman-Hole and the Soft X-Ray log N-log S, A&A 275, 1 (1993).
- K. Morihana, M. Tsujimoto, K. Ebisawa et al., Deep Near-infrared ImagingObservation of the Faint X-ray Point Sources Constituting the GalacticBulge X-ray Emission, Publications of the Astronomical Society of Japan74(2), 283-297 (2022).
- J.I. Katz, A FRB in a Globular Cluster: Why Is This Neutron Star DifferentFrom (Almost) All Other Neutron Stars? Monthly Notices of the RoyalAstronomical Society: Letters 508(1), L12-L16 (2021).
- M. Matuszkov´a, G. Toeroek, D. Lanc’ov´a et al., Accretion tori aroundrotating neutron stars, A&A 691, A167 (2024).
- M. Matuszkov´a, G. Toeroek, K. Klimovic’ov´a et al., Accretion tori aroundrotating neutron stars II: Oscillations and precessions, A&A 691, A168(2024).
- C.A. Raithel, Constraints on the neutron star equation of state fromGW170817, The European Physical Journal A 55(5), article id. 80 (2019).
- M. Shibata, S. Fujibayashi, K. Hotokezaka et al., Modeling GW170817based on numerical relativity and its implications, Physical Review D 96,123012 (2017).
- Y.-J. Zhao, J,-f. Hu, FR I Radio Galaxy 3C 348 Accumulation Patternsand Dust Ring Occlusion Effects, e-Print: 2503.05327 (2025).
- S. Buttiglione, A. Capetti, A. Celotti et al., An optical spectroscopic surveyof the 3cr sample of radio galaxies with z 0.3-ii. spectroscopic classes andaccretion modes in radio-loud agn, Astronomy & Astrophysics 509, A6(2010).
- LIGO Scientific Collaboration and Virgo Collaboration, Fermi Gamma-rayBurst Monitor, and INTEGRAL, Astrophys. J. Lett. 848, L13 (2017).
- A. Goldstein et al., Astrophys. J. Lett. 848, L14 (2017).A. Goldstein, P. Verese, E. Burns et al.., An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A, Astrophys. J. Lett. 848, L14 (2017).
- V. Savchenko et al., Astrophys. J. Lett. 848, L15 (2017).V. Savchenko, C. Ferrigno, E. Kuulkers et al., INTEGRAL Detection of the First Prompt Gamma-Ray Signal Coincident with the Gravitational-wave Event GW170817, Astrophys. J. Lett. 848, L15 (2017).
- R. Aguilera-Miret, J.-E. Christian, S. Rosswog et al., e-printarXiv:2504.10604 (2025).R. Aguilera-Miret, J.-E. Christian, S. Rosswog et al., Robustness of Magnetic Field Amplification in Neutron Star Mergers, e-print arXiv:2504.10604 (2025).
- J. Bamber, A. Tsokaros, M. Ruiz et al., Phys. Rev. D 111, 044038 (2025).J. Bamber, A. Tsokaros, M. Ruiz et al., Postmerger multimessenger analysis of binary neutron stars: Effect of the magnetic field strength and topology, Phys. Rev. D 111, 044038 (2025).
- L. Brandes, W. Weise, Phys. Rev. D 111, 034005 (2025).L. Brandes, W. Weise, Implications of latest NICER data for the neutron star equation of state, Phys. Rev. D 111, 034005 (2025).
- A possible explanation of the parallel tracks in kilohertz quasi-periodicoscillations from low-mass X-ray binaries, Monthly Notices of the RoyalAstronomical Society 479(4), 5049-5059 (2018).
- C.S. Shi, S.N. Zhang, X.D. Li, The possible origin of high frequency quasi-periodic oscillations in low mass X-ray binaries, in: High-mass X-ray Bina-ries: Illuminating the Passage from Massive Binaries to Merging CompactObjects, Proceedings of the International Astronomical Union 346, 277-280(2019).
- I.D. Novikiv, Ya.B. Zel’dovich, Gravitational collapse of nonsymmetric androtating masses, J. Exp.. Theoret. Phys. (U.S.S.R.) 49, 170-181 (1965).
- J. Wang, Physical Environment of Accreting Neutron Stars, Advances inAstronomy 2016, Article ID 3424565 (2016).