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Normal Modes of Radial Pulsation of Stars at the End Point of Thermonuclear Evolution
... The idea that radial pulsations might occur out of chemical equilibrium, i.e. that weak interactions responsible for adjusting the stellar composition to density changes may proceed more slowly than or on timescales comparable to the pulsation period, is not a new concept. Soon after the relativistic radial oscillation equations were formulated by Chandrasekhar (1964), early works considered the possibility that the adiabatic index could differ significantly when matter is not in chemical equilibrium (Meltzer & Thorne 1966;Chanmugam 1977). Although these pioneering studies identified considerable differences between equilibrium and non-equilibrium adiabatic indices, such differences did not produce any evident modification to the standard stability criterion. ...
... and measures how pressure changes with local baryon number density under the set of physical conditions {C} that characterize the perturbation. To understand which conditions {C} should be applied, it is necessary to consider the typical timescales τ nuc of the various nuclear reactions that occur within the NS and compare them with the pulsation period T (Meltzer & Thorne 1966), which is on the order of 0.1 to 1 millisecond. As emphasized by Haensel et al. (2002), equilibration processes in NSs occur on vastly different timescales relative to T (also see the recent work of Rau & Salaben 2023, where this is extended to hybrid stars). ...
... Chandrasekhar (1964) derivation of relativistic stellar oscillation equations is based on the assumption that perturbations occur while maintaining chemical equilibrium (catalyzed matter). Subsequent investigations into pulsations out of chemical equilibrium (Meltzer & Thorne 1966;Chanmugam 1977;Gourgoulhon et al. 1995;Gondek et al. 1997) adopted these same equations, but simply substituted the equilibrium adiabatic index Γ EQ with the non-equilibrium index Γ 1 , making no other adjustments. Rau & Salaben (2023) recently attempted to re-derive the pulsation equations in a similar setting but unfortunately introduced certain incorrect terms. ...
We investigate the radial stability of neutron stars under conditions where their composition may or may not remain in chemical equilibrium during oscillations. Using different equations of state that include nucleons, hyperons, and/or resonances, we compute stellar configurations and examine their fundamental mode frequencies in two limiting scenarios. In one limit, nuclear reactions are fast enough to maintain chemical equilibrium throughout the pulsation, resulting in a lower effective adiabatic index, , and softer stellar responses. In the opposite limit, nuclear reactions are too slow to adjust particle abundances during oscillations, yielding a higher index, , and stiffer stellar responses. We find that the equilibrium scenario triggers dynamic instability at the maximum mass configuration, whereas the frozen composition scenario allows stable solutions to persist beyond this mass, extending the stable branch. This effect is modest for simpler equations of state, but becomes increasingly pronounced for more complex compositions, where the emergence of new particle species at high densities leads to a significant disparity between and . Realistic conditions, in which different nuclear reactions have distinct timescales, will place the effective between these two extreme values. Short-timescale reactions push the star toward the equilibrium limit, potentially restricting the length of the stable branch. Conversely, slow reactions preserve a frozen composition, allowing the stable branch to grow. Thus, the actual extent of the stable configuration range depends critically on the interplay between nuclear-reaction timescales and the star's fundamental oscillation period.
... This is reflected in a non-adiabatic response of matter. For example, several studies on NS oscillations (Meltzer & Thorne 1966;Chanmugam 1977;Gourgoulhon et al. 1995;Haensel et al. 2002) and thermal fluctuation in accreting neutron stars (Ushomirsky et al. 2000) pointed out that that the stellar crust matter can be not in full thermodynamical equilibrium during the build-up of internal stresses. In particular, beta equilibrium is known to have an important role in the damping of NSs oscillations (Haensel et al. 2002;Yakovlev et al. 2018;Andersson & Pnigouras 2019). ...
... For a given equation of state (EoS), it is possible to define a related adiabatic index that determines the changes of pressure associated with variations of the local baryon density (Shapiro & Teukolsky 1983). This fundamental quantity appears into the equations governing small-amplitude NSs pulsations (Thorne & Campolattaro 1967) and, in a relativistic context, provides a criteria of stability for cold stars (Meltzer & Thorne 1966;Chanmugam 1977;Gourgoulhon et al. 1995). When the star is perturbed around the hydrostatic equilibrium configuration, the adia-batic index value must be calculated by taking into account for the possible slowness of the equilibration channels, typically mediated by the weak interaction (Haensel et al. 2007). ...
... Clearly, in this case the adiabatic index is not a function of n b , but takes the constant value γeq = (n+1)/n = 2. On the other hand, we have little clues about the actual value of γ f . It is expected that γ f > γeq, but the actual relation between them strongly depends on the microscopic model underlying the specific EoS (Meltzer & Thorne 1966;Chanmugam 1977;Haensel et al. 2002;Ushomirsky et al. 2000). However, from the practical point of view, the uncertain value of γ f is not a strong limitation in the present work: we will assume different values and study how the estimated stress and strain change, starting from values that differ by only some percent from γeq, up to the incompressible limit γ f γeq. ...
We introduce a Newtonian model for the deformations of a compressible neutron star that goes beyond the widely used Cowling approximation. We employ this model to rigorously study the role played by the adiabatic index in the calculation of rotation-induced deformations: we assume a polytropic equation of state for the matter at chemical equilibrium but, since the equilibration reactions may be slow, the perturbations with respect to the unstressed configuration are modeled by using an equation of state with a different polytropic index. Hence, we quantify the impact of this departure of the adiabatic index on the calculated stresses and strains. We obtain that a small variation in the adiabatic index which regulates the perturbation can cause large variations in the calculated displacements and strains, the effect being larger for lighter stars. As a first practical application of our model, we calculate the strain developed between consecutive glitches in the Vela pulsar, confirming the difficulty that arises when trying to explain the trigger of pulsar glitches with starquakes: in order for the quake to be a possible trigger, the solid crust must never fully relax after a glitch, making the sequence of starquakes in a neutron star an history-dependent process.
... There is, however, a difference in the sense that the Chandrasekhar mass is larger than the Oppenheimer-Volkoff maximum mass while, in the present context, the Newtonian maximum mass of dilute axion stars is smaller than the general relativistic maximum mass of dense axion stars. In this sense, the mass-radius relation of axion stars is more similar to the mass-radius relation of compact stars obtained with the Skyrme-Cameron-Saakyan (SCS) equation of state (see Fig. 3 of [149]). Indeed, in more realistic models of neutron stars, the maximum mass is substantially larger than the ideal Oppenheimer-Volkov limit, being in the range 2 − 2.4 M , and overcomes the Chandrasekhar limiting mass of white dwarf stars. ...
... 6.3) in which the maximum mass of neutron stars is larger than the maximum mass of white dwarf stars. This figure, which can be seen as a synthesis of Figs. 2 and 3 of [149] is similar to Figs. 34 and 35 of axion stars. ...
... We finally note that we can determine the stability of the different configurations of fermion stars represented in Fig. 7 of [144], in Figs. 2 and 3 of [149], in Fig. 11.2 of [150] and in Fig. 6.3 of [151] by using the Poincaré theory of linear series of equilibria, as shown here in the case of axion stars. The Poincaré theory provides an alternative, and a generalization, of the M (R) theorem devised by Wheeler [144] in the physics of compact objects (white dwarfs and neutron stars) and used in [149][150][151]. ...
We study the nature of phase transitions between dilute and dense axion stars interpreted as self-gravitating Bose-Einstein condensates. We develop a Newtonian model based on the Gross-Pitaevskii-Poisson equations for a complex scalar field with a self-interaction potential involving an attractive term and a repulsive term. Using a Gaussian ansatz for the wave function, we analytically obtain the mass-radius relation of dilute and dense axion stars for arbitrary values of the self-interaction parameter . We show the existence of a critical point above which a first order phase transition takes place. We qualitatively estimate general relativistic corrections on the mass-radius relation of axion stars. For weak self-interactions , a system of self-gravitating axions forms a stable dilute axion star below a general relativistic maximum mass and collapses into a black hole above that mass. For strong self-interactions , a system of self-gravitating axions forms a stable dilute axion star below a Newtonian maximum mass , collapses into a dense axion star above that mass, and collapses into a black hole above a general relativistic maximum mass . Dense axion stars explode below a Newtonian minimum mass and form dilute axion stars of large size or disperse away. We determine the phase diagram of self-gravitating axions and show the existence of a triple point separating dilute axion stars, dense axion stars, and black holes. We make numerical applications for QCD axions and ultralight axions.
... In General Relativity these are a set of radial normal modes. These represent undamped modes, that turn into unstable modes when for a given equation of state the maximum of the neutron star mass is passed [2,[40][41][42][43][44][45][46][47]. The usual nomenclature for these modes is the fundamental F-mode and the excited H-modes. ...
... When considering radial perturbations in General Relativity, stars are seen to possess a family of normal modes with ω 2 ∈ R, that become unstable beyond the maximum mass of the equation of state. These modes are well known in the literature [2,[41][42][43][44][45][46][47]. However, in General Relativity radial perturbations cannot propagate gravitational radiation outside the neutron star. ...
We study polar quasinormal modes of relativistic stars in scalar-tensor theories, where we include a massive gravitational scalar field and employ the standard Brans-Dicke coupling function. For the potential of the scalar field we consider a simple mass term as well as a potential associated with R ² gravity. The presence of the scalar field makes the spectrum of quasinormal modes much richer than the spectrum in General Relativity. We here investigate radial modes ( l = 0) and quadrupole modes ( l = 2). The general relativistic l = 0 normal modes turn into quasinormal modes in scalar-tensor theories, that are able to propagate outside of the stars. In addition to the pressure-led modes new scalar-led ϕ -modes arise. We analyze the dependence of the quasinormal mode frequencies and decay times on the scalar field mass.
... In General Relativity these are a set of radial normal modes. These represent undamped modes, that turn into unstable modes when for a given equation of state the maximum of the neutron star mass is passed (Chandrasekhar, 1964b,a;Bardeen et al., 1966;Meltzer and Thorne, 1966;Chanmugam, 1977;Glass and Lindblom, 1983;Vaeth and Chanmugam, 1992;Datta et al., 1998). The usual nomenclature for these modes is the fundamental F-mode and the excited H-modes. ...
... When considering radial perturbations in General Relativity, stars are seen to possess a family of normal modes with ω 2 ∈ R, that become unstable beyond the maximum mass of the equation of state. These modes are well known in the literature (Chandrasekhar, 1964a,a;Bardeen et al., 1966;Meltzer and Thorne, 1966;Chanmugam, 1977;Glass and Lindblom, 1983;Vaeth and Chanmugam, 1992;Datta et al., 1998). However, in General Relativity radial perturbations cannot propagate gravitational radiation outside the neutron star. ...
We study polar quasinormal modes of relativistic stars in scalar-tensor theories, where we include a massive gravitational scalar field and employ the standard Brans-Dicke coupling function. For the potential of the scalar field we consider a simple mass term as well as a potential associated with gravity. The presence of the scalar field makes the spectrum of quasinormal modes much richer than the spectrum in General Relativity. We here investigate radial modes (l=0) and quadrupole modes (l=2). The general relativistic l=0 normal modes turn into quasinormal modes in scalar-tensor theories, that are able to propagate outside of the stars. In addition to the pressure-led modes new scalar-led -modes arise. We analyze the dependence of the quasinormal mode frequencies and decay times on the scalar field mass.
... The radiation fluid equation of state P = ρ/3 plays a critical role in physics and astrophysics. By using this equation of state we can describe the physical properties of the cores of neutron stars, assumed to consist of cold, non-interacting and degenerate fermions [55,56]. The interesting possibility of the existence of stars described by the radiation equation of state, and therefore consisting of a radiation fluid, was also analyzed from different perspectives [57][58][59][60]. ...
... The first of these equations, the stiff fluid equation of state [52], gives the limiting case of the causality condition, and guarantees that the speed of sound cannot exceed the speed of light in the baryonic matter. The radiation fluid equation of state plays an important role in astrophysics and physics, and it may be used to model the dense cores of neutron stars [55,56]. Of particular interest are the quark matter [65,66] and Bose-Einstein Condensate matter [71,72] equations of state. ...
We consider interior static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert–Einstein action with a Lagrangian constructed from a three-form field A α β γ , which generates, via the field strength and a potential term, a new component in the total energy-momentum tensor of the gravitational system. We formulate the field equations in Schwarzschild coordinates and investigate their solutions numerically for different equations of state of neutron and quark matter, by assuming that the three-field potential is either a constant or possesses a Higgs-like form. Moreover, stellar models, described by the stiff-fluid, radiation-like, bag model and the Bose–Einstein condensate equations of state are explicitly obtained in both general relativity and three-form gravity, thus allowing an in-depth comparison between the astrophysical predictions of these two gravitational theories. As a general result we find that, for all the considered equations of state, three-form field stars are more massive than their general relativistic counterparts. As a possible astrophysical application, we suggest that the 2.5 M ⊙ mass compact object, associated with the GW190814 gravitational wave event, could be in fact a neutron or a quark star described by the three-form gravity theory.
... The radiation fluid equation of state P = ρ/3 plays a critical role in physics and astrophysics. By using this equation of state we can describe the physical properties of the cores of neutron stars, assumed to consist of cold, non-interacting and degenerate fermions [53,54]. The interesting possibility of the existence of stars described by the radiation equation of state, and therefore consisting of a radiation fluid, was also analyzed from different perspectives [55][56][57][58]. ...
... The first of these equations, the stiff fluid equation of state [50], gives the limiting case of the causality condition, and guarantees that the speed of sound cannot exceed the speed of light in the baryonic matter. The radiation fluid equation of state plays an important role in astrophysics and physics, and it may be used to model the dense cores of neutron stars [53,54]. Of particular interest are the quark matter [63,64] and Bose-Einstein Condensate matter [69,70] equations of state. ...
We consider interior static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert-Einstein action with a Lagrangian constructed from a three-form field , which generates, via the field strength and a potential term, a new component in the total energy-momentum tensor of the gravitational system. We formulate the field equations in Schwarzschild coordinates and investigate their solutions numerically for different equations of state of neutron and quark matter, by assuming that the three field potential is either a constant or possesses a Higgs-like form. Moreover, stellar models, described by the stiff fluid, radiation-like, bag model and the Bose-Einstein condensate equations of state are explicitly obtained in both general relativity and three-form gravity, thus allowing an in-depth comparison between the astrophysical predictions of these two gravitational theories. As a general result we find that for all the considered equations of state, three-form field stars are more massive than their general relativistic counterparts. As a possible astrophysical application of the obtained results, we suggest that the 2.5 mass compact object, associated with the GW190814 gravitational wave event, could be in fact a neutron or a quark star described by the three-form field gravity theory.
... In General Relativity these are a set of radial normal modes. These represent undamped modes, that turn into unstable modes when for a given equation of state the maximum of the neutron star mass is passed (Chandrasekhar, 1964b,a;Bardeen et al., 1966;Meltzer and Thorne, 1966;Chanmugam, 1977;Glass and Lindblom, 1983;Vaeth and Chanmugam, 1992;Datta et al., 1998). The usual nomenclature for these modes is the fundamental F-mode and the excited H-modes. ...
... When considering radial perturbations in General Relativity, stars are seen to possess a family of normal modes with ω 2 ∈ R, that become unstable beyond the maximum mass of the equation of state. These modes are well known in the literature (Chandrasekhar, 1964a,a;Bardeen et al., 1966;Meltzer and Thorne, 1966;Chanmugam, 1977;Glass and Lindblom, 1983;Vaeth and Chanmugam, 1992;Datta et al., 1998). However, in General Relativity radial perturbations cannot propagate gravitational radiation outside the neutron star. ...
Besides scalarized black holes and wormholes, Einstein-scalar-Gauss-Bonnet theories allow also for particlelike solutions. The scalar field of these particlelike solutions diverges at the origin, akin to the divergence of the Coulomb potential at the location of a charged particle. However, these particlelike solutions possess a globally regular metric, and their effective stress-energy tensor is free from pathologies, as well. We determine the domain of existence for particlelike solutions in a number of Einstein-scalar-Gauss-Bonnet theories, considering dilatonic and power-law coupling functions, and we analyze the physical properties of the solutions. Interestingly, the solutions may possess pairs of lightrings, and thus represent ultracompact objects. We determine the location of these lightrings, and study the effective potential for the occurrence of echoes in the gravitational-wave spectrum. We also address the relation of these particlelike solutions to the respective wormhole and black-hole solutions, and clarify the limiting procedure to recover the Fisher solution (also known as the Janis-Newman-Winicour-Wyman solution).
... The constant n must be positive to obey the boundary conditions, but it is otherwise arbitrary. These types of trial functions were considered in Ref. [66], where they were used with more elaborate techniques to estimate the first few eigenvalues of the Chandrasekhar pulsation equation. In scenarios in which the star is very inhomogeneous these trial functions can lead to misleading estimates. ...
A long-standing problem in physics is why observed masses are always positive. While energy conditions in quantum field theory can partly answer this problem, in this paper we find evidence that classical general relativity abhors negative masses, without the need for quantum theory or energy conditions. This is done by considering many different models of negative-mass ``stars'' and showing they are dynamically unstable. \emph{A fortiori}, we show that any barotropic negative-mass star whose pressure is not a monotonically increasing function of the energy density must be dynamically unstable. Furthermore, we argue that all acceptable barotropic models of negative-mass stars must be in this class, and are thus unstable.
... In the mid-1960s, researchers were interested in the radial oscillation modes of neutron stars presumed to arise in their initial formation in supernovae. Curiosity about the duration of these oscillations after the birth of the neutron star, as well as the possibility of the conversion of vibrational energy into heat, leading to enhanced radiation from the neutron star surface (see the contemporary review articles [21,22]), lead to preliminary calculations (Finzi [23], Meltzer & Thorne [24], and Hansen & Tsuruta [25]) of vibrational energy dissipation arising from chemical equilibration. In these early works, which considered neutron stars built of neutron-proton-electron (npe) matter (see Sec. II), modified Urca processes (Sec. ...
In this chapter, I describe bulk viscosity as a general concept, and then focus on bulk viscosity in the dense matter present in compact objects. While this review is focused on bulk viscosity in the conditions present in neutron star mergers, I present a history of bulk viscosity research in dense matter, from its role in damping radial oscillations in neutron stars through its current applications in neutron star mergers. The majority of the chapter consists of calculations of the bulk viscosity from Urca processes in generic neutron-proton-electron (npe) matter, and then in dense matter containing muons ( matter) as well. I make several approximations in these calculations to keep the focus on the concepts. More precise calculations exist in the literature, to which I refer the reader. One concept I attempt to elucidate is the thermodynamic behavior of a fluid element throughout an oscillation and how that leads to bulk-viscous dissipation. I conclude with a discussion of the recent research into the role of weak interactions and bulk viscosity in neutron star mergers.
... The radial spectrum of sound speed o↵ers an insight into the dynamic interplay between gravitational forces and nuclear interactions [67]. The speed of small perturbations in form of acoustic waves propagating through the radial and tangential directions in a compact star encapsulates the response of dense stellar matter to the gravitational compression [49]. ...
A charged compact star is modelled within the framework of general relativity and electromagnetism to investigate the intricate complexities arising from charge accumulation through the accretion of surrounding charged baryonic matter, charged dark matter, or both. The Einstein–Maxwell field equations are solved for the compact star PSRJ0740+6620 in anisotropic regime by employing a Gaussian type density profile over a coherent background. A radially modulated exponential function is used as a seed ansatz for coherently connecting the class-one type metric. The structural stability and feasibility are then probed through physical bounds on stellar parameters at equilibrium. The key findings associated with charge accretion emphasised: (i). the existence of a transition zone indicating the formation of a core–shell type stellar structure (ii). the plane shifting of intrinsic force fields and (iii) the spin retardation, both suggest a non-vanishing spin–charge coupling between the stellar spin and the accreted charge, whether it be from baryonic matter, dark matter, or both.
... The stellar core's intense density causes the sound velocity to move into the ultra-relativistic region [111]. In this context, to assess the viability of the stellar model [112,113], we can check the distribution of maximum sound velocity in the inner region. The upper bounds for radial sound velocity, v 2 r (r ) < 1, and tangential sound velocity, v 2 t (r ) < 1, are established using the causality limit. ...
In this paper, we introduce an anisotropic model using a dark matter (DM) density profile in Einstein–Gauss–Bonnet (EGB) gravity using a gravitational decoupling method introduced by Ovalle (Phys Rev D 95:104019, 2017), which has provided an innovative approach for obtaining solutions to the EGB field equations for the spherically symmetric structure of stellar bodies. The Tolman and Finch–Skea (TFS) solutions of two metric potentials, g tt and g rr , have been used to construct the seed solution. Additionally, the presence of DM in DM halos distorts spacetime, causing perturbations in the g rr metric potential, where the quantity of DM is determined by the decoupling parameter β . The physical validity of the solution, along with stability and equilibrium analysis, has also been performed. Along with stability and equilibrium analysis, the solution’s physical validity has also been examined. Additionally, we have shown how both constants affect the physical characteristics of the solution. Using a M - R diagram, it has been described how the DM component and the GB constant affect the maximum permissible masses and their corresponding radii for various compact objects. Our model predicts the masses beyond the 2 M ⊙ and maximum radii 11 . 92 - 0.01 + 0.02 and 12 . 83 - 0.02 + 0.01 for larger value of α under density order 10 15 g / cm 3 and 10 14 g / cm 3 , respectively, while the radii become 11 . 96 - 0.01 + 0.01 and 12 . 81 - 0.02 + 0.01 for larger value of β .
... The extremely high density within the stellar core results in an ultra-relativistic sound speed [63][64][65] . To assess the viability of the stellar model, we examine the maximum speed at which sound waves propagate within the stellar interior. ...
We present a singularity free and smoothly connected complete solution for a three-layered compact star with deconfined quark matter core surrounded by coherent quantum fluid of a Bose-Einstein condensate, all enclosed under a thin envelope of a repulsive neutron-Coulomb fluid. A modified form of Buchdahl-I type spatial metric is used as a seed to obtain the explicit temporal metric potential for each layer using Einstein field equation. The process allows us to obtain insights into the pressure-density profile and associated thermodynamic properties within stellar interior, shedding light on its structural stability, mass-radius relation and physical plausibility. One of the key finding of our work is the close similarity between the M-R curve of our three-phase model star Vela X-1 and that of a strange quark star model.
... The relativistic adiabatic exponent, defined as 1 = ∂ log P ∂ log n ad , where n is the baryonic number density, can be expressed as (Meltzer & Thorne 1966 ;Thorne 1967 ) ...
Ultra-massive white dwarf stars are currently being discovered at a considerable rate, thanks to surveys such as the Gaia space mission. These dense and compact stellar remnants likely play a major role in type Ia supernova explosions. It is possible to probe the interiors of ultra-massive white dwarfs through asteroseismology. In the case of the most massive white dwarfs, General Relativity could affect their structure and pulsations substantially. In this work, we present results of relativistic pulsation calculations employing relativistic ultra-massive ONe-core white dwarf models with hydrogen-rich atmospheres and masses ranging from 1.29 to 1.369M⊙ with the aim of assessing the impact of General Relativity on the adiabatic gravity (g)-mode period spectrum of very-high mass ZZ Ceti stars. Employing the relativistic Cowling approximation for the pulsation analysis, we find that the critical buoyancy (Brunt-Väisälä) and acoustic (Lamb) frequencies are larger for the relativistic case, compared to the Newtonian case, due to the relativistic white dwarf models having smaller radii and higher gravities for a fixed stellar mass. In addition, the g-mode periods are shorter in the relativistic case than in the Newtonian computations, with relative differences of up to ∼50 % for the highest-mass models (1.369M⊙) and for effective temperatures typical of the ZZ Ceti instability strip. Hence, the effects of General Relativity on the structure, evolution, and pulsations of white dwarfs with masses larger than ∼1.29M⊙ cannot be ignored in the asteroseismological analysis of ultra-massive ZZ Ceti stars.
... The relativistic adiabatic exponent, defined as Γ 1 = log log ad , where is the baryon number density, can be expressed as (Thorne 1967;Meltzer & Thorne 1966): . The relativistic sound speed, , is given by (Curtis 1950): ...
Ultra-massive white dwarf stars are currently being discovered at a considerable rate, thanks to surveys such as the {\it Gaia} space mission. These dense and compact stellar remnants likely play a major role in type Ia supernova explosions. It is possible to probe the interiors of ultra-massive white dwarfs through asteroseismology. In the case of the most massive white dwarfs, General Relativity could affect their structure and pulsations substantially. In this work, we present results of relativistic pulsation calculations employing relativistic ultra-massive ONe-core white dwarf models with hydrogen-rich atmospheres and masses ranging from 1.29 to with the aim of assessing the impact of General Relativity on the adiabatic gravity (g)-mode period spectrum of very-high mass ZZ Ceti stars. Employing the relativistic Cowling approximation for the pulsation analysis, we find that the critical buoyancy (Brunt-V\"ais\"al\"a) and acoustic (Lamb) frequencies are larger for the relativistic case, compared to the Newtonian case, due to the relativistic white dwarf models having smaller radii and higher gravities for a fixed stellar mass. In addition, the g-mode periods are shorter in the relativistic case than in the Newtonian computations, with relative differences of up to \% for the highest-mass models () and for effective temperatures typical of the ZZ Ceti instability strip. Hence, the effects of General Relativity on the structure, evolution, and pulsations of white dwarfs with masses larger than cannot be ignored in the asteroseismological analysis of ultra-massive ZZ Ceti stars.
... The radiation fluid equation of state plays a major role in astrophysics. Such an equation of state can describe the dense core of neutron stars, assumed to consist of coldᾱ degenerate (non-interacting) fermions [84][85][86]. Moreover, self-gravitating high density photon stars, obeying the radiation fluid equation of state, could also exist [87][88][89][90]. ...
We consider the structure and physical properties of specific classes of neutron, quark, and Bose-Einstein condensate stars in the conformally invariant Weyl geometric gravity theory. The basic theory is derived from the simplest conformally invariant action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To keep the theory conformally invariant the trace condition is imposed on the matter energy-momentum tensor. The field equations are derived by varying the action with respect to the metric tensor, Weyl vector field and scalar field. By adopting a static spherically symmetric interior geometry, we obtain the field equations, describing the structure and properties of stellar objects in Weyl geometric gravity. The solutions of the field equations are obtained numerically, for different equations of state of the neutron and quark matter. More specifically, constant density stellar models, and models described by the stiff fluid, radiation fluid, quark bag model, and Bose-Einstein condensate equations of state are explicitly constructed numerically in both general relativity and Weyl geometric gravity, thus allowing an in depth comparison between the predictions of these two gravitational theories. As a general result it turns out that for all the considered equations of state, Weyl geometric gravity stars are more massive than their general relativistic counterparts. As a possible astrophysical application of the obtained results we suggest that the recently observed neutron stars, with masses in the range of 2M⊙ and 3M⊙, respectively, could be in fact conformally invariant Weyl geometric neutron or quark stars.
... The extreme density in the stellar core pushes the sound speed to ultra relativistic regime [44] . Thus checking the distribution of maximum possible sound speed in the interior region, r < R provides a measure for testing the feasibility of the stellar model [45,46] . We use the causality limit to set the upper bound for radial sound speed, v r (r) < 1 and tangential sound speed, v t (r) < 1. ...
We present a singularity free and smoothly connected complete solution for a three-layered compact star with deconfined quark matter core surrounded by coherent quantum fluid of a Bose-Einstein condensate, all enclosed under a thin envelope of a repulsive neutron-Coulomb fluid. A modified form of Buchdahl-I type spatial metric is used as a seed to obtain the explicit temporal metric potential for each layer using Einstein field equation. The process allows us to obtain insights into the pressure-density profile and associated thermodynamic properties within stellar interior, shedding light on its structural stability, mass-radius relation and physical plausibility. One of the key finding of our work is the close similarity between the M--R curve of our three-phase model star \nn and that of a strange quark star model.
... However, the problem is that the branch that corresponds to the holographic quark matter in the mass-radius relation is unstable according to the Bardeen-Thorne-Meltzer (BTM) stable criterion stated in Refs. [70,71]. The BTM stable criterion is stated in the following. ...
The equation of state (EoS) for cold dense matter inside neutron stars is investigated by using holographic QCD models in the framework of the Einstein-Maxwell-dilaton (EMD) system and the improved Karch-Katz-Son-Stephanov (KKSS) action for matter part. This method of describing holographic nuclear matter in the EMD+KKSS framework is different from that by using the Dirac-Born-Infeld (DBI) action and the Chern-Simons (CS) terms. Combining with the Hebeler-Lattimer-Pethick-Schwenk (HLPS) intermediate equation of state (EoS), the hybrid EoS inside the neutron stars is constructed. The obtained hybrid EoS is located in the range that is defined by the low-density chiral effective theory, the high-density perturbative QCD, and the polytropic interpolations between them, and is constrained by the astrophysics observations. The square of the sound velocity reaches a maximum value larger than 0.8 in the region of times the saturation baryon number density and approaches the conformal limit at the high baryon density range. The mass-radius relation and the tidal deformability of the neutron stars are in agreement with astrophysical measurements. The possible maximum mass for the neutron star is about and the radius is about then. It is noticed that the holographic quark matter branch in the mass-radius relation is always unstable and the holographic nuclear matter can produce a stable branch. These results indicate that even in the core of the NS, the matter is still in the confinement phase and the quark matter is not favored.
... We refer to it as the frozen adiabatic index γ f . This index rules the star's response when crust matter is not in full thermodynamical equilibrium during the buildup of stresses, such as in the case of NS oscillations (Meltzer & Thorne 1966 ;Chanmugam 1977 ;Gourgoulhon, Haensel & Gondek 1995 ;Haensel, Levenfish & Yakovlev 2002 ) or thermal fluctuation in accreting NSs (Ushomirsky et al. 2000 ). Generally speaking γ f > γ eq , but the peculiar relation between the two indices strongly depends on the microscopic model underlying the specific EoS. ...
So far, only transient Gravitational waves (GWs) produced by catastrophic events of extra-galactic origin have been detected. However, it is generally believed that there should be also continuous sources of GWs within our galaxy, such as accreting neutron stars (NSs), that could in principle be detected in the next near future. In fact, in these objects, centrifugal forces can be so strong to break the neutron star crust (causing a starquake), thus producing a quadrupole moment responsible for the continuous emission of GWs. At equilibrium, the angular momentum gained by accretion and the one lost via GWs emission should balance each other, stopping the stellar spin-up. We hereinafter investigate the above physical picture within the framework of a Newtonian model describing compressible, non-magnetized and self-gravitating NSs. In particular, we calculate the rotational frequency need to break the stellar crust of an accreting pulsar and we estimate the upper limit for the ellipticity due to this event. We find that the maximum starquake-induced ellipticity ranges from 10−9 to 10−5, depending on the stellar mass and its equation of state. The corresponding equilibrium frequency that we calculate is in good agreement with observations and, for all the scenarios, it is below the higher NS frequency observed of 716.36 Hz. Finally, we also discuss possible observational constraints on the ellipticity upper limit of accreting pulsars.
... As a last remark of our results concerning the massradius diagrams, we observe that for each parametrization constructed, namely, sets 1 to 3 of the DDHδ-PNJL0 hadron-quark model, one identifies linear branches in which M decreases as R decreases. From the investigation of the static condition for the star's stability, this kind of branch would indicate unstable configurations [88,89]. ...
We propose a three-flavor version of the Polyakov-Nambu-Jona-Lasino (PNJL) model at zero temperature regime, by implementing a traced Polyakov loop (Φ) dependence in the scalar, vector and ’t Hooft channel strengths. We study the thermodynamics of this model, named as PNJL0, with special attention for the first-order confinement/deconfinement phase transition for which Φ is the order parameter. For the symmetric quark matter case, an interesting feature observed is a strong reduction of the constituent strange quark mass (Ms) at the chemical potential related to point where deconfinement takes place. The emergence of Φ favors the restoration of chiral symmetry even for the strange quark. We also investigate the charge neutral system of quarks and leptons in weak equilibrium. As an application, we construct a hadron-quark phase transition with a density dependent hadronic model coupled to the SU(3) PNJL0 model. In this case, the quark side is composed by deconfined particles. This approach is used to determine mass-radius profiles compatible with recent data from the Neutron Star Interior Composition Explorer mission.
... A few of them are described as follows. At zero temperatures: In Meltzer & Thorne (1966) the periods and e-folding of the lowest three normal radial pulsations of white dwarf equilibrium configurations at the end point of thermonuclear evolution were investigated. The authors found that at mass densities of 2.5 × 10 8 < ρ < 1.3 × 10 9 [g cm −3 ], there are metastable white dwarfs with e-folding time 10 10 yr, which corresponds to Hubble time. ...
We investigate the structure and stability against radial oscillations, pycnonuclear reactions, and inverse β -decay of hot white dwarfs. We consider the fluid matter to be made up of nucleons and electrons confined in a Wigner–Seitz cell surrounded by free photons. It is considered that the temperature depends on the mass density considering the presence of an isothermal core. We find that the temperature produces remarkable effects on the equilibrium and radial stability of white dwarfs. The stable equilibrium configuration results are compared with those for white dwarfs estimated from the Extreme Ultraviolet Explorer survey and the Sloan Digital Sky Survey. We derive masses, radii, and central temperatures for the most massive white dwarfs according to the surface gravity and effective temperature reported by the surveys. We note that these massive stars are in the mass region where general relativity effects are important. These stars are near the threshold of instabilities due to radial oscillations, pycnonuclear reactions, and inverse β -decay. Regarding the radial stability of these stars as a function of the temperature, we find that it decreases with the increment of central temperature. We also find that the maximum-mass point and the zero eigenfrequencies of the fundamental mode are determined at the same central energy density. Regarding low-temperature stars, pycnonuclear reactions occur in similar central energy densities, and the central energy density threshold for inverse β -decay is not modified. For T c ≥ 1.0 × 10 ⁸ [K], the onset of radial instability is attained before pycnonuclear reaction and inverse β -decay.
... Since the analysis of the stability of compact objects against small radial perturbations developed in Chandrasekhar's seminal works (Chandrasekhar 1964a,b), several articles investigated the radial stability of white dwarfs at zero and nonzero temperature, without taking into account the Wigner-Seitz cell. To name a few of them, at zero temperature, in (Meltzer & Thorne 1966) the periods and e-folding of the lowest three normal radial pulsation of white dwarf equilibrium configurations at the end point of thermonuclear evolution are investigated. The authors found that between the mass densities 2.5 × 10 8 < ρ < 1.3 × 10 9 [g cm −3 ], there are metastable white dwarfs with e-folding time 10 10 years, which corresponds to the Hubble time. ...
We investigate the structure and stability against radial oscillations, pycnonuclear reactions, and inverse -decay of hot white dwarfs. We regard that the fluid matter is made up for nucleons and electrons confined in a Wigner-Seitz cell surrounded by free photons. It is considered that the temperature depends on the mass density considering the presence of an isothermal core. We find that the temperature produces remarkable effects on the equilibrium and radial stability of white dwarfs. The stable equilibrium configuration results are compared with white dwarfs estimated from the Extreme Ultraviolet Explorer Survey and Sloan Digital Sky Survey. We derive masses, radii, and central temperatures for the most massive white dwarfs according to surface gravity and effective temperature reported by the survey. We note that these massive stars are in the mass region where the general relativity effects are important. These stars are near the threshold of instabilities due to radial oscillations, pycnonuclear reaction, and inverse -decay. Regarding the radial stability of these stars as a function of the temperature, we obtain that the radial stability decreases with the increment of central temperature. We also obtain that the maximum mass point and the zero eigenfrequencies of the fundamental mode are determined at the same central energy density. Regarding low-temperature stars, the pycnonuclear reactions occur in almost similar central energy densities, and the central energy density threshold for inverse -decay is not modified. For , the onset of the radial instability is attained before the pycnonuclear reaction and the inverse -decay.
... As is well know, in GR radial oscillations of relativistic stars do not source gravitational wave (GW) emissions (although in principle they can couple to non-radial modes [77][78][79] and potentially be observable during the post-merger phase [80][81][82]). For this reason, they are typically studied only for assessing the stability of NS solutions [83][84][85][86][87]. However, in ST theories a new family of modes typically appears in association with the additional degree of freedom [88,89]. ...
Scalar-tensor theories whose phenomenology differs significantly from general relativity on large (e.g. cosmological) scales do not typically pass local experimental tests (e.g. in the solar system) unless they present a suitable "screening mechanism". An example is provided by chameleon screening, whereby the local general relativistic behavior is recovered in high density environments, at least in weak-field and quasi-static configurations. Here, we test the validity of chameleon screening in strong-field and highly relativistic/dynamical conditions, by performing fully non-linear simulations of neutron stars subjected to initial perturbations that cause them to oscillate or even collapse to a black hole. We confirm that screened chameleon stars are stable to sufficiently small radial oscillations, but that the frequency spectrum of the latter shows deviations from the general relativistic predictions. We also calculate the scalar fluxes produced during collapse to a black hole, and comment on their detectability with future gravitational-wave interferometers.
... The case of pathological amplitudes near the surface, i.e., within the outer layers of NS, which occurs in Cases 1 and 2, has already been reported in Ref. [44] and is known to be harmless due to its dilute nature. Our calculations show that the same happens for the EoSs of Kurkela et al. [17] and the first set of Annala et al. [22] with amplitudes around ξ n ≲ 30. ...
We study the stability against infinitesimal radial oscillations of neutron stars generated by a set of equations of state obtained from first-principle calculations in cold and dense QCD and constrained by observational data. We consider mild and large violations of the conformal bound, cs=1/3, in stars that can possibly contain a quark matter core. Some neutron star families in the mass-radius diagram become dynamically unstable due to large oscillation amplitudes near the core.
... In the context of relativistic stars, Meltzer and Thorne [53] discuss the stability properties of very dense relativistic stars. They predict that perturbations localized to the core of a very dense star will result in gravitational collapse, which is consistent with our methodology, (see Sects. ...
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift . We prove their linear instability when is sufficiently large, i.e., when they are strongly relativistic, and prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel’dovich & Podurets (for the Einstein–Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein–Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein–Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.
... This means that after performing the perturbations, the particle compositions are not modified and remain fixed 12 (or frozen, as it is usually known), i.e. constant values for the electron, Y e , and other particle fractions, Y i . In contrast, the case τ reac ≈ τ dyn would produce nuclear transformations (reactions) that could affect strongly the EoS (and Γ), thus complicating considerably the dynamical analysis of stability [18,159,160]. Then, after pointing out the above considerations, one is allowed to write the Γ-index for cold, adiabatic and isentropic (in this case zero entropy) compact star matter as (only in this equation we write explicitly constant particle fractions "Y i ") ...
In this work, we study the effects that interacting quark matter has on the stellar structure of strange and charm quark stars. Additionally, their stability against radial pulsations is analyzed using a first-order formalism for adiabatic general relativistic oscillations. Besides, the early stage of stellar evolution of neutron stars after the supernovae explosion, i.e. protoneutron stars, is investigated by considering the possibility of a first-order phase transition to quark matter rich in leptons, where the dynamics of conversion between phases is studied within a thermal nucleation model. For each kind of compact star mentioned we use for the quark phase the equation of state calculated within cold perturbative QCD (pQCD), parametrized only by its renormalization scale. We note that the original pQCD framework is manipulated appropriately to include neutrinos and extended to add heavy quarks to the original system composed only by up, down and strange quark flavors.
... However, these are not all the modes the stars possess. It is well known that NSs in GR are characterized by a set of radial normal modes, i.e., not damped modes, that become unstable modes beyond the maximum mass NS for a given equation of state (EOS) [26][27][28][29][30][31][32][33]. These modes consist of a fundamental F -mode, and the excited H Nmodes. Clearly, in GR, radial perturbations cannot propagate gravitational radiation outside of the NS, and thus these modes play no role during the ringdown phase. ...
The spectrum of frequencies and characteristic times that compose the ringdown phase of gravitational waves emitted by neutron stars carries information about the matter content (the equation of state) and the underlying theory of gravity. Typically, modified theories of gravity introduce additional degrees of freedom/fields, such as scalars, which result in new families of modes composing the ringdown spectrum. Simple but physically promising candidates are scalar-tensor theories, which effectively introduce an additional massive scalar field (i.e., an ultra-light boson) that couples non-minimally to gravity, resulting in scalarized neutron stars. Here we present the first calculation of the full ringdown spectrum in such theories. We show that the ringdown spectrum of neutron stars with ultra-light bosons is much richer and fundamentally different from the spectrum in general relativity and that it possesses propagating ultra-long-lived modes.
... The mass-radius stability criterion was proposed in [196] and further developed in [208,209,207]. Spherical stars of arbitrarily low central density are just terrestrial matter, and thus are stable to radial perturbations -they have zero unstable modes. ...
After an introduction to the QCD phase diagram, the nuclear equations of state, and neutron star mergers, I discuss three projects related to transport and nuclear matter in neutron star mergers. The first is the nature of beta equilibrium in the portion of a merger that is transparent to neutrinos. We calculate the weak interaction (Urca) rates and find that the beta equilibrium condition needs to be modified by adding an additional chemical potential, which changes slightly the particle content in neutrino-transparent beta equilibrium. Secondly, we calculate the bulk viscosity in neutrino-transparent nuclear matter in conditions encountered in neutron star mergers. Bulk viscosity arises from a phase lag between the pressure and density in the nuclear matter, which is due to the finite rate of beta equilibration. When bulk viscosity is sufficiently strong, which happens when the equilibration rate nearly matches the frequency of the density oscillation, it can noticeably dampen the oscillation. We find that in certain thermodynamic conditions likely encountered in mergers, oscillations in nuclear matter can be damped on timescales on the order of 10 milliseconds, so we conclude that bulk viscosity should be included in merger simulations. Finally, we study thermal transport due to axions in neutron star mergers. We conclude that axions are never trapped in mergers, but instead escape, carrying energy away from the merger. We calculate the cooling time due to the energy carried away by axions and find that within current constraints on the axion-nucleon coupling, axions could cool fluid elements in mergers on timescales which could affect the dynamics of the merger.
... To determine the eigenfrequency of the fundamental mode we assume a trial function ψ(r ) ∝ r 3 which corresponds to a homologous vibration. It can be shown [42] that this trial function is a sufficiently close approximation to the true eigenfunction of the fundamental mode. ...
Newtonian gravity predicts the existence of white dwarfs with masses far exceeding the Chandrasekhar limit when the equation of state of the degenerate electron gas incorporates the effect of quantum spacetime fluctuations (via a modified dispersion relation) even when the strength of the fluctuations is taken to be very small. In this paper, we show that this Newtonian “super-stability” does not hold true when the gravity is treated in the general relativistic framework. Employing dynamical instability analysis, we find that the Chandrasekhar limit can be reassured even for a range of high strengths of quantum spacetime fluctuations with the onset density for gravitational collapse practically remaining unaffected.
... However, these are not all the modes the stars possess. It is well known that NSs in GR are characterized by a set of radial normal modes, i.e., not damped modes, that become unstable modes beyond the maximum mass NS for a given equation of state (EOS) [26][27][28][29][30][31][32]. Clearly in GR, radial perturbations cannot propagate gravitational radiation outside of the NS, and thus these modes play no role during the ringdown phase. ...
The spectrum of frequencies and characteristic times that compose the ringdown phase of gravitational waves emitted by neutron stars carry information about the matter content (the equation of state) and the underlying theory of gravity. Typically, modified theories of gravity introduce additional degrees of freedom/fields, such as scalars, which result in new families of modes composing the ringdown spectrum. One simple but physically promising candidate is gravity, which effectively introduces an additional massive scalar field that couples non-minimally to gravity, resulting in scalarized neutron stars. Here we show that the ringdown spectrum of neutron stars is much richer and fundamentally different from the spectrum in GR, possessing for instance ultra long lived modes that can propagate away from the star in the form of scalar gravitational radiation.
... These are called Magnetars with magnetic fields about 10 14 − 10 15 Gauss [Duncan and Thompson, 1992]. The rapidly rotating magnetic field of a neutron star accelerates particles to relativistic energies and produces synchrotron radiation emitting broadband electromagnetic radiation from Radio to X-rays [Meltzer and Thorne, 1966], [Bisnovatyi-Kogan and Fridman, 1969]. If a neutron star or a black hole is accompanied by a normal star in a gravitationally bound orbit, the gas from the companion star is accreted by the compact object and forms an accretion disc of gas around it. ...
The recent progresses in the development of multilayer mirrors have revolutionized the field of astronomical X-rays optics. A variety of multilayer mirrors are now being developed for several unique applications such as hard X-ray imaging telescopes and soft X-ray polarimeters. Technology development to fabricate good quality multilayer mirrors carries a significant importance for realization of next generation X-ray instruments. In this thesis, we have presented our progress in fabricating and characterizing high quality W/B4C multilayer mirrors for various applications. We have also discussed the design and development of two X-ray instruments using the combination of grazing incidence X-ray concentrator and multilayer mirrors.
We fabricated W/B4C multilayer mirrors with varied design parameters using magnetron sputtering technique. We studied the performance and structural stability of these mirrors over time and by subjecting these mirror to the temperature variation analogous to the satellite in low earth orbit using soft X-ray, hard X-ray reflectivity as well as scanning electron microscopic studies for estimating the contamination and surface quality. We observed that multilayers with small thickness are more stable than the large thickness multilayers.
We designed a multilayer mirror based soft X-ray polarimeter to operate at energies less than 1 keV. We proposed this design coupled with a hard X-ray polarimeter as a simultaneous back-end instrument to a hard X-ray telescope. For this application, to make multilayer mirrors transparent to hard X-rays, we etched the Silicon substrate of the mirrors to reduce the absorption. We observed that the etching process significantly degraded the performance of large thickness multilayers (> 5 nm) while the process did not affect the performance of short thickness multilayers (< 3 nm).
... In this paper we assume adiabatic oscillations for which the fluid element does not exchange heat with its surrounding, then ∆s = 0. Also we need to compare the time scale of the nuclear reaction rate of the fluid element t react with the oscillation time t osc ≈ 10 −3 s. If t react ≫ t osc the fluid element composition does not change during the oscillation, and ∆x i = 0, otherwise it is necessary to consider the composition profile see [42]. ...
We perform a detailed analysis of the fundamental f-mode frequencies and damping times of nonrotating boson stars in general relativity by solving the nonradial perturbation equations. Two parameters which govern the microscopic properties of the bosonic condensates, namely the self-coupling strength and the mass of scalar particle, are explored. These two quantities characterize oscillations of boson star. Specifically, we reexamine some empirical relations that describe the f-mode parameters in terms of mass and radius of the boson stars. We found it is possible to constrain the equation of state if the fundamental oscillation mode is observed.
... To determine the eigenfrequency of the fundamental mode we assume a trial function ψ(r) ∝ r 3 which corresponds to a homologous vibration. It can be shown [41] that this trial function is a sufficiently close approximation to the true eigenfunction of the fundamental mode. ...
Newtonian gravity predicts the existence of white dwarfs with masses far exceeding the Chandrasekhar limit when the equation of state of the degenerate electron gas incorporates the effect of quantum spacetime fluctuations (via a non-commutative formulation) even when the strength of the fluctuations is taken to be very small. In this paper, we show that this Newtonian super-stability of non-commutative white dwarfs does not hold true when the gravity is treated in the general relativistic framework. Besides, we find that the Chandrasekhar limit can be reassured even for high strengths of quantum spacetime fluctuations with the onset density for gravitational collapse practically remaining unaffected.
... In the context of relativistic stars, Meltzer and Thorne [47] discuss the stability properties of very dense relativistic stars. They predict that perturbations localized to the core of a very dense star will result in gravitational collapse, which is consistent with our methodology, see Sections 4.2 and 5.6. ...
... In the context of relativistic stars, Meltzer and Thorne [53] discuss the stability properties of very dense relativistic stars. They predict that perturbations localized to the core of a very dense star will result in gravitational collapse, which is consistent with our methodology, see Sections 4.2 and 5.6. ...
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parametrized by their central redshift . We prove their linear instability when is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich \& Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, \& Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.
... In the case of hadronic matter, there are many relaxation channels associated with strong and Coulombic elastic collisions, strong-interaction processes conserving strangeness but changing baryon fractions, modified and direct Urca processes, and nonleptonic processes (Haensel et al. 2002). In general, the relaxation time in hadronic matter is significantly larger that the dynamical timescale of the perturbation (Haensel et al. 2002), and the value of the adiabatic index relevant for oscillations is that with fixed (frozen) composition (Γ FR ) and not that calculated from the EOS of catalyzed (i.e. in full thermodynamic equilibrium) matter (Meltzer & Thorne 1966;Chanmugam 1977;Gourgoulhon et al. 1995;Haensel et al. 2002). Nonetheless, in the present work we focus on hybrid stars where hadronic matter is present mainly in a not-too-high density regime. ...
We study radial oscillations of hybrid non-rotating neutron stars composed by a quark matter core and hadronic external layers. At first, we physically deduce the junction conditions that should be imposed between two any phases in these systems when perturbations take place. Then we compute the oscillation spectrum focusing on the effects of slow and rapid phase transitions at the quark-hadron interface. We use a generic MIT bag model for quark matter and a relativistic mean field theory for hadronic matter. In the case of rapid transitions at the interface we find a general relativistic version of the reaction mode which has similar properties as its classical counterpart. We also show that the usual static stability condition , where is the central density of a star whose total mass is M, remains always true for rapid transitions but breaks down in general for slow transitions. In fact, for slow transitions we find that the frequency of the fundamental mode can be a real number (indicating stability) even for some branches of stellar models that verify . Thus, when secular instabilities are suppressed, as expected below some critical stellar rotation rate, it would be possible the existence of twin or even triplet stars with the same gravitational mass but different radii, with one of the counterparts having . We explore some astrophysical consequences of these results.
... The radiation fluid stars have crucial importance in astrophysics. They can describe the core of neutron stars which is a collection of cold degenerate (non-interacting) fermions [1][2][3][4] and self-gravitating photon stars [4][5][6][7][8]. Such radiative stars which called Radiation Pressure Supported Stars (RPSS) can be possible even in Newtonian Gravity [9] and their relativistic extension which called Relativistic Radiation Pressure Supported Stars (RRPSS) [10] can describe the gravitational collapse of massive matter clouds to very high density fluid. ...
We propose a non-minimally coupled gravity model in form to describe the radiation fluid stars which have the radiative equation of state between the energy density and the pressure p given by Here is the Maxwell invariant and Y(R) is a function of the Ricci scalar R. We give the gravitational and electromagnetic field equations in differential form notation taking the infinitesimal variations of the model. We look for electrically charged star solutions to the field equations under the constraint eliminating complexity of the higher order terms in the field equations. We determine the non-minimally coupled function Y(R) and the corresponding model which admits new exact solutions in the interior of the star and the Reissner–Nordstrom solution at the exterior region. Using the vanishing pressure condition at the boundary together with the continuity conditions of the metric functions and the electric charge, we find the mass–radius ratio, charge–radius ratio, and the gravitational surface redshift depending on the parameter of the model for the radiation fluid star. We derive general restrictions for the ratios and redshift of the charged compact stars. We obtain a slightly smaller upper mass–radius ratio limit than the Buchdahl bound 4 / 9 and a smaller upper redshift limit than the bound of the standard general relativistic stars.
The Nemeth-Sprung equation of state is modified and used to obtain neutron star models. Contrary to the results of some authors it is found that neutron stars with central densities ≲ 10 ¹⁴ g cm ⁻³ are dynamically stable. It is suggested that some pulsars may belong to this category of stars.
The stability of relativistic systems is reviewed against the background of what is known in the corresponding contexts of the Newtonian theory. In particular, the importance of determining whether Dedekind-like points of bifurcation occur along given stationary axisymmetric sequences is emphasized: the occurrence of such points of bifurcation may signal the onset of secular instability induced by radiation-reaction. (At a Dedekind-like point of bifurcation, the system can be subject, quasistationarily, to a non-axisymmetric deformation with an e 2iϕ -dependence on the azimuthal angle ϕ.)
A formalism is described in terms of which the normal modes of axisymmetric oscillation of axisymmetric systems can be determined. Specialized to neutral modes of oscillation the formalism provides an alternative proof of Carter's theorem and clarifies the minimal requirements for its validity. A parallel formalism is described for ascertaining whether an axisymmetric system can be subject to a quasi-stationary non-axisymmetric deformation. The possibility of applying this latter formalism to determining whether a Dedekind-like point of bifurcation occurs along the Kerr sequence is considered.
Em fevereiro de 1968, a então estudante de pós-graduação Jocelyn Bell, seu orientador Antony Hewish e demais integrantes do grupo de radioastrônomos da Universidade de Cambridge publicaram um artigo na Revista Nature sobre incomuns sinais pulsados em ondas de rádio, identificados inicialmente em meados de agosto de 1967, gerando um amplo movimento da comunidade científica para entendimento deste suposto novo objeto astronômico: os pulsares. Neste artigo, objetivamos discutir aspectos históricos envolvidos no processo de compreensão conceitual dos pulsares, tendo como base artigos científicos publicados à época, comentários sobre o episódio elaborados por Jocelyn Bell Burnell e Antony Hewish, além de estudos secundários sobre a história dos pulsares. A partir de reflexões de Ludwik Fleck e Thomas Kuhn, as discussões de Natureza da Ciência evidenciadas consistem, por exemplo, na extensão temporal e na construção coletiva de uma descoberta científica e no processo complexo de circulação de teorias e de observações sobre um fenômeno entre astrônomos.
The equation of state (EOS) for cold dense matter inside neutron stars is investigated by using holographic QCD models that consist of the Einstein-Maxwell-dilaton system and the improved Karch-Katz-Son-Stephanov action for the flavor part. This method of describing holographic nuclear matter in the Einstein−Maxwell−dilaton+Karch−Katz−Son−Stephanov framework is different from that by using the Dirac-Born-Infeld action and the Chern-Simons terms. Combining with the Hebeler-Lattimer-Pethick-Schwenk intermediate EOS, the hybrid EOS inside the neutron stars is constructed. The obtained hybrid EOS is located in the range that is defined by the low-density chiral effective theory, the high-density perturbative QCD, and the polytropic interpolations between them, and is constrained by the astrophysical observations. The square of the sound velocity reaches a maximum value larger than 0.8 in the region of 2–5 times the saturation baryon number density and approaches the conformal limit at the high baryon density range. The mass-radius relation and the tidal deformability of the neutron stars are in agreement with astrophysical measurements. The possible maximum mass for the neutron star is about 2.5 M⊙ and the radius is about 12 km then. It is noticed that the holographic quark matter branch in the mass-radius relation is always unstable and the holographic nuclear matter can produce a stable branch. These results indicate that even in the core of the NS, the matter is still in the confinement phase and the quark matter is not favored.
After more than half a century since their unexpected discovery and identification as neutron stars, the observation and understanding of pulsars touches upon many areas of astronomy and astrophysics. The literature on pulsars is vast and the observational techniques used now cover the whole of the electromagnetic spectrum from radio to gamma-rays. Now in its fifth edition, this volume has been reorganised and features new material throughout. It provides an introduction in historical and physical terms to the many aspects of neutron stars, including condensed matter, physics of the magnetosphere, supernovae and the development of the pulsar population, propagation in the interstellar medium, binary stars, gravitation and general relativity. The current development of a new generation of powerful radio telescopes, designed with pulsar research in mind, makes this survey and guide essential reading for a growing body of students and astronomers.
This chapter focuses on the physics of neutron stars. It starts describing the compositions and equation of state (EoS) of dense matter in the neutron star interior at length. Various novel forms of the dense matter such as hyperon, Bose–Einstein condensate and quark are mentioned here. In this connection, models of the outer and inner crusts are presented. The EoSs of the dense matter in the core at the zero and finite temperatures are calculated using the relativistic field theoretical models. The theoretically calculated gross properties of neutron stars using the non-rotating, slowly rotating, and rapidly rotating neutron star models are compared with the observed masses, radii, and moments of inertia to constrain the compositions and EoS of the dense matter. The stable branch of compact stars beyond a neutron star branch is found to exist in the case of dense matter undergoing a phase transition from nuclear matter to any of novel forms of matter. The role of quantizing magnetic fields on the compositions and the EoS of neutron star matter is discussed. Finally, the EoS tables as a function of density, temperature, and positive charge fraction and their applications to supernova simulations are highlighted.
Scalar-tensor theories whose phenomenology differs significantly from general relativity on large (e.g., cosmological) scales do not typically pass local experimental tests (e.g., in the Solar System) unless they present a suitable “screening mechanism.” An example is provided by chameleon screening, whereby the local general relativistic behavior is recovered in high-density environments, at least in weak-field and quasistatic configurations. Here, we test the validity of chameleon screening in strong-field and highly relativistic/dynamical conditions by performing fully nonlinear simulations of neutron stars subjected to initial perturbations that cause them to oscillate or even collapse to a black hole. We confirm that screened chameleon stars are stable to sufficiently small radial oscillations, but that the frequency spectrum of the latter shows deviations from the general relativistic predictions. We also calculate the scalar fluxes produced during collapse to a black hole, and we comment on their detectability with future gravitational-wave interferometers.
We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi–Dirac entropy for fermions. The maximization of entropy at fixed mass energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman–Oppenheimer–Volkoff equations of hydrostatic equilibrium and the Tolman–Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of self-gravitating fermions in general relativity obtained in recent works (Roupas and Chavanis in Class Quant Grav 36:065001, 2019; Chavanis and Alberti in Phys Lett B 801:135155, 2020). We consider the nonrelativistic limit and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.
We study the general properties of compact objects made up of strange matter in the framework of a new equation of state in which the quark masses are parametrized as functions of the baryon density, so that they are heavy (light) at low (high) densities. This has been called the "quark mass-density-dependent model." In this approximation, the strange matter equation of state is rather similar to the corresponding to the MIT Bag Model, but it is significantly stiffer at low densities. Such a property modifies the structure of strange stars in a sizeable way. In this framework, we calculate the structure of strange stars (mass, radius, central density, gravitational redshift, moment of inertia, and total baryon number) finding that the resulting structures are rather similar to those obtained in the MIT Bag model, although some important differences appear. Comparing to the standard bagged case (with a bag constant in the range of B = 60 - 80 MeV fm-3), we find that these objects may be more massive and may show gravitational redshifts larger (up to ≈ 10%) than in the bag case. The moment of inertia and total baryon number may be larger than in the bagged case up to a factor of three. We also calculate the first three radial pulsation modes of these objects, finding that the relation of period vs. gravitational redshift is rather similar to the bag case. Also, we present an analytical treatment for such modes in the low-mass strange stars regime, which is in reasonable agreement with the numerical results.
Several authors have suggested that the recently discovered extraterrestrial sources of x rays may be hot neutron stars. The plausibility of this suggestion, and in fact the likelihood
that astronomers will ever be able to observe neutron stars by their x-ray emission, depend critically upon the cooling times of the hot stars.
The energy loss of a degenerate neutron gas by the "Urca" process is calculated in this paper; it is 7.36×104(T/109)8 erg g-1 for a density of 6×1014 g cm-3. It follows from this energy loss that the Urca process alone should have cooled the core of the neutron star created in the type I supernova of 1054 a.d. to a temperature around 5×108 °K. The emission power of the star should then be about one order of magnitude smaller than that of the x-ray source discovered recently in the Crab nebula; the source cannot be interpreted therefore simply as the thermal radiation of the star. This conclusion is consistent with the result of a recent experiment performed by the method of lunar occultation, indicating an angular size of the source comparable to that of the Crab nebula. A more refined experiment performed by the same method should, on the other hand, make it possible to decide whether a neutron star exists in the Crab nebula.
The behavior of superdense ("neutron") stars at absolute zero has been studied. It is shown that, with certain assumptions about the equation of state, the mass and radius of such a star approach constants in an oscillatory fashion as the star's central density increases. The assumptions about the equation of state are: dP/dρ>0 everywhere, and for sufficiently high-density ρ, the pressure divided by the density P/ρ approaches a constant. These assumptions are physically reasonable, especially if one assumes a real speed of sound which is finite, but always less than the speed of light. The results of the paper show that there exists for such stars an infinite series of ranges of the central density, in which ranges dM/dρ0 alternates in sign, where M is the total star mass and ρ0 is the central density. This indicates alternate local stability and instability; however, the total binding energy is positive for ρ0 greater than ∼1016 g/cm3, so that instability against large-scale deformation exists. A striking feature of the results of this paper is that their qualitative nature does not depend on whether or not the general relativistic form of the equations is used. The exact quantitative results do, of course, depend on the form of the equations, as well as on the exact equation of state used.
DOI:https://doi.org/10.1103/PhysRevLett.16.414
DOI:https://doi.org/10.1103/PhysRevLett.12.413
DOI:https://doi.org/10.1103/PhysRevLett.13.122
DOI:https://doi.org/10.1103/PhysRevLett.15.599
The reasons why nuclear electron-capture rates in stars depend on
temperature and density are discussed, and some astrophysical applications of
continuum electron-capture rates are reviewed. The modern theory of nuclear
BETA decay is then used to calculate stellar-continuum electroncapture rates for
transitions of an arbitrary degree of forbiddenness. The equations that are most
useful for astrophysical applications are discussed in detail; particular
emphasis is placed upon methods for predicting stellar rates that utilize,
whenever possible, terrestrial measurements. Three examples are discussed that
illustrate the use of the formulas given; the examples are: the electroncapture
lifetime of a proton, the stellar BETA decay of Mo, and the effect of forbidden
transitions on the abundances of elements in the iron peak. (auth)
The equation of state is considered for matter consisting of electrons
and nuclei of atomic weight A and charge Z, at zero temperature and at densities
much larger than that of the solid at zero pressure. Corrections are evaluated
to the energy and pressure of a degenerate Fermi gas of noninteracting electrons,
due to the following effectsn classical Coulomb energy of an ion lattice with
uniformly distributed electrons (this is the largest correction); Thomas-Fermi
deviations from uniform charge distribution of the electrons; and exehange energy
and spin-spin interactions between the electrons. The corrections increase with
decreasing density, and the approximations break down when the spacing between
nuclei is greater than the mean radius of the free Thomas-Fermi atom, and the
formulas are not applicable to the interior of planets. At very high densities,
where the electrons are extremely relativistic, the correction to the pressure is
a multiplying constant factor which is 0.994, 0.986, and 0.960, respectively, for
Z = 2, 6, and 26. It is shown that the nuclei form a lattice rather than a gas.
At very high densities, restrictions are found on possible values for A and Z due
to inverse beta decays and pycnonuclear reactions. For instance, C/sup 12/
changes to Ne/sup 24/ at densities above 6 x 10/sup 9/ gm/cc. (auth).
The semiempirical abundance table of Suess and Urey has been very helpful to nuclear physicists in determining the relative importance of various mechanisms of nucleogenesis in contributing to the natural abundances of the elements. The table was based primarily on abundance determinations in chondritic meteorites, together with the hypothesis that the abundances of odd mass-number nuclides should lie on a smooth curve. It has recently become apparent that certain elements are over- or underabundant in the chondritic meteorites without apparent chemical reason. In order to throw additional light oil such discrepancies and on certain less well-understood mechanisms of nucleogenesis, the abundance-curve has been re-examined, using some additional nucleogenesis-based criteria for the abundances of evcn mass-number nuclides, and a new table of abundances has been constructed. It was found necessary to change the Suess-Urey table substantially in the rare-enrth region and in the lead region and to make isolated changes in the abundances of certain additional elements. Some conclusions have been drawn regarding the process of neutron capture on a fast time scale and the methods of production of bypassed heavy nuclei. (auth)
WITH the discovery of X-ray sources in the sky1,2, speculation has arisen that they might be associated with neutron or hyperon stars formed during the internal collapse which triggers off supernova explosions (probably of type I). Rates of cooling of neutron star models have been calculated by Morton3, Chiu and Salpeter4,5, and Tsuruta6. It appears (J. Bahcall, personal communication) that the importance of the early cooling by emission of neutrinos from the ‘Urca’ process has been underestimated in the foregoing investigations. With rough allowance for this effect, the calculations of Miss Tsuruta indicate that a neutron star will rapidly cool to 3 or 4 × 106 °K, but that after 105 years its surface temperature will still be about 2 × 106 °K.
The analysis previously made of the average nuclear potential has been extended to consideration of the spin-orbit interactions. It has not been possible to find a satisfactory two-body interaction consistent with all the data; that suggested by the phase-shift analysis of nucleon-nucleon scattering is just within the region of possible forms.
Relativity and nuclear theory together predict the fate of a star which has burned all its nuclear fuel.