Phase transition effects on the dynamical stability of hybrid neutron stars
We study radial oscillations of hybrid non-rotating neutron stars composed by a quark matter core and hadronic external layers. At first, we deduce the junction conditions that should be imposed between two any phases in these systems when perturbations take place. Then we compute the spectrum exhibited by the radial oscillations of hybrid neutron stars focusing on the effects of slow and rapid phase conversions at the quark-hadron interface. For the quark matter core, we use a generic MIT bag model that allows the inclusion of the effects of finite quark masses, strong interactions and color superconductivity. For the hadronic phase, we use a relativistic mean field theory widely used to describe hadronic matter in neutron stars.
We find that the general relativistic version of the reaction mode in classical stars (first excited mode), uniquely due to rapid transitions, could actually be either the fundamental mode or the first excited one, depending on the equation of state parametrizations. Thus, the speed of the phase conversion at the hybrid interface could play a more fundamental role on the stability of stars when general relativistic effects are taken into account given the hierarchy of the eigenvalues in a Sturm-Liouville problem. We also show that the usual static stability condition , where is the central density of a star whose total mass is , remains always true for rapid phase transitions but breaks down in general for slow phase transitions. In fact, 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 viscous dissipation and nonlinear damping mechanisms suppress secular instability’s exponential growth, which is expected to be the case below some critical rotation rate in cold catalyzed NSs, it would be possible the existence of twin or even triplet stars with the same gravitational mass but different radii which, if observed, could reveal the nature of reactions in stars and by consequence their hybrid nature and aspects of their innermost phases.
Subject headings:stars: neutron – stars: oscillations – dense matter – gravitation
It is known since long ago that the stability of one-phase stars can be assessed by means of families of static solutions with different central densities , such that unstable ones have (Harrison et al., 1965; Shapiro & Teukolsky, 1986). However, it has been shown in several different scenarios (charged strange quark stars (Arbañil & Malheiro, 2015), hybrid stars with a quark and hadronic phases (Vásquez Flores et al., 2012), color superconducting quark stars (Vásquez Flores & Lugones, 2010), etc.) that this simple condition may not hold in general. Studies on multi-phase neutron stars are pertinent because they could unveil characteristics uniquely pertaining to these stars and thus lead to potential observables for their probe. Of special interest, due to stability assessments and motivated by the advent of gravitational wave astronomy (Abbott et al., 2016b, a, 2017a, 2017b), is the eigenfrequency spectrum of these systems. This is very important because it carries information of the stars’ structures, even their hybrid nature (Vásquez Flores & Lugones, 2014). The most natural approach thereof is by means of radial perturbations, the route we take in this work.
As regards hybrid stars’ perturbations, in general one cannot integrate the system of equations as is done in one-phase stars, but must instead take due care of additional boundary conditions at the phase-splitting interfaces, which in essence capture the main physics taking place in their vicinities. There are basically two kinds of physical behavior around such surfaces in the presence of perturbations: either volume elements are converted from one phase to the other or they keep their nature and are only stretched or compressed. The first kind of reactions is related to rapid phase transitions, while the other regards slow phase transitions (Haensel et al., 1989). Suggestions of possible physics behind them include quantum and thermal nucleation processes of one phase into the other that result in characteristic conversion times that are very sensitive to several microphysical details of both phases (see e.g. Lugones & Grunfeld (2011); Bombaci et al. (2016); Lugones (2016); Kroff & Fraga (2015) and references therein). As a consequence, the transition might be either rapid or slow, which motivates the investigations performed in this paper. More details about the aforesaid reactions will be given in the subsequent sections.
There are many studies related to radial oscillations of hybrid stars, focusing for instance on the influence of the mixed phase (Gupta et al., 2002; Sahu et al., 2002), electric charge (Brillante & Mishustin, 2014), the observational consequences of a possible quark core formation (mixed phase is taken into account) (Mishustin et al., 2003), as well as pion condensation (Migdal et al., 1979; Haensel & Proszynski, 1982). Puzzling enough, it seems that rapid phase transitions and its consequences for hybrid stars have not been investigated much in the literature. To the best of our knowledge it has been pioneered in the context of Newtonian stars by Haensel and collaborators (Haensel et al., 1989), who found the so called reaction mode, but general relativistic analyses seem yet pending. Additionally, a systematic analysis of stellar stability in the case of slow phase transitions has not yet been performed in the literature. We try to partially fill this gap here.
The strategy of this work is as follows. In the next three sections we summarize the equations of state and the relevant equations concerning radial stability analyses. In Sec. 5 we deduce the relevant boundary conditions for matching the phases of hybrid stars when perturbations take place. Section 6 is devoted to a brief explanation of the numerical approaches we have made use of in order to obtain our results, present in Sec. 7. Finally, in Sec. 8 we discuss the main issues raised in our analyses. We work with geometric units unless otherwise stated.
2. Equations of state
In this work we investigate the imprint of different phase transitions into the stability of hybrid stars having quark and hadronic phases. Given the unknown constitution of matter at very high densities, the equation of state (EOS) is usually derived from phenomenological descriptions trying to take into account the expected properties of matter at each density regime.
For the hadronic phase we use a non-linear Walecka model (Walecka, 1974; Glendenning & Moszkowski, 1991) including the whole baryon octet, electrons and the corresponding antiparticles. The Lagrangian is given by
where the indices , and refer to baryons, mesons and leptons respectively. The only baryons considered here are nucleons (neutrons and protons), thus reads:
The contribution of the mesons , and is given by
Electrons are included as a free Fermi gas, , in chemical equilibrium under weak interactions with all other particles.
The constants in the model are determined by the properties of nuclear matter and hyperon potential depths known from hypernuclear experiments. In the present work we use the NL3 parametrization for which we have fm, fm, fm, and (Lalazissis et al., 1997; Glendenning & Moszkowski, 1991). At low densities we use the Baym, Pethick and Sutherland (BPS) model (Baym et al., 1971). For details on the explicit form of the equation of state derived from this Lagrangian the reader is referred to Lugones et al. (2010); do Carmo et al. (2013) and references therein.
For the quark matter, bag-like models are appealing due to their simplicity and ability to effectively capture some key aspects of QCD. Here, we use the bag-like model presented in Alford et al. (2005), with free parameters , and , which could encompass several physical effects, as explained below. The model is defined by the following grand thermodynamic potential
where is the quark chemical potential, and is the grand thermodynamic potential for electrons .
For hybrid systems, though, it happens that the contribution to the thermodynamic quantities coming from electrons is negligible due to the following reason. Local charge neutrality implies that , where refers to the number density of the ith particles involved. From the chemical equilibrium relations and , it follows that at zero temperature (Alford et al., 2005), and therefore, . The effect of electrons in the quark matter EOS can be estimated through the ratio . Since we are interested in hybrid systems, the quark matter EOS is used essentially in the high density regime where is significantly larger than (typically, MeV and MeV), which means a very small contribution of the electrons to the quark EOS and hence to the star’s macroscopic parameters. This is in agreement with other estimates in the literature (see e.g. page 368 of Haensel et al. (2007) and references therein). Therefore, we neglect the contribution of electrons in Eq. (4), which has the clear advantage of leading to analytic thermodynamic expressions.
As explained in Alford et al. (2005), the above phenomenological model is convenient because, besides the standard MIT-bag model (), it also allows exploring the effect of strong interactions (by modifying ), as well as the effects of color superconductivity (, being the energy gap associated with the quark pairing).
From Eq. (4), one can obtain the pressure , the baryon number density
and the energy density
Nevertheless, it is not for all values of and the effective bag constant that hybrid stars exist. They only do if the phase transition pressure is larger than zero. In this work we assume that the interface between hadrons and quarks is a sharp discontinuity (justifications for it are given in Sec. 8) at which the Gibbs conditions and are satisfied, being the Gibbs free energy per baryon (Shapiro & Teukolsky, 1986). Thus, the quark-hadron interface will be located at only if , with
where is the Gibbs free energy per baryon of quark matter evaluated at null pressure. By the Gibbs condition, turns out to be exactly the energy per baryon of pressureless hadronic matter, taken here to be that of the iron, of approximately MeV. The above condition defines the 3-flavor line in Fig. 1.
One should also guarantee that the hadronic part of a hybrid star is not in metastable equilibrium. This is done by imposing that the energy per baryon of quark matter is larger than the iron binding energy. For two-flavor quark matter in the absence of electrons and in the massless limit one has that
where (due to local charge neutrality). From and , one has that the aforesaid 2-flavor stability condition leads to , where
which defines the 2-flavor line in Fig. 1.
Fig. 1 depicts the regions in the plane (for a specific value of ) associated with hybrid neutron stars and absolutely stable strange quark stars. For model parameters above the 3-flavor line, stars containing quark matter must be hybrid. The region between the 3-flavor and 2-flavor lines defines the range of parameters related to absolutely stable strange quark stars, made up of quark matter from the center all the way up to the surface. The unshaded region of parameters (below the 2-flavor line) is excluded due to the known existence and stability of nuclei.
3. Equilibrium Equations
Before the study of radial oscillations, the compact star’s equilibrium structure has to be taken into account. We consider the unperturbed (hybrid) star to be composed of layers of effective perfect fluids, whose stress-energy tensors read as
At the same time, we set a static spherically symmetric background spacetime for each phase given by the following line element
After solving Einstein’s equations with the definition we come up with the following set of stellar structure equations (Tolman-Oppenheimer-Volkoff equations (Oppenheimer & Volkoff, 1939))
where , and , are the pressure, the energy density and the gravitational mass respectively, as measured in a proper frame of reference, and is the fluid’s four-velocity. All the quantities have well defined values at the radius and they will be used as inputs in the numerical integration of the oscillation equations for each phase of the star.
The metric function has the boundary condition
where is the radius of the star and its total mass (as measured by faraway observers). With this condition the metric function will match smoothly the Schwarzschild metric outside the star and will have a definite value at its origin. Another boundary condition to the system naturally is , which expresses the regularity of the metric at the center of the star. Finally, we must have at the stellar surface.
4. Radial Oscillation Equations
The first form of the radial oscillation equations can be found in Chandrasekhar’s pioneering article (Chandrasekhar, 1964), in which the main goal was the study of the dynamical stability of relativistic compact stars. After his work, there was a lot of interest in the understanding of the behavior of compact stars under radial disturbances. Since our objective here is to study phase transition’s effects on the dynamical stability of compact hybrid stars, we will not scrutinize the homogeneous (one-phase) case. For aspects of the stellar stability under radial perturbations without phase transitions, see e.g. Glendenning (2000); Haensel et al. (2007); Harrison et al. (1965).
Chandrasekhar’s work showed that the equations governing each phase (quark or hadronic) can be obtained by making perturbations in fluid and spacetime variables and inserting them into Einstein’s and baryon number conservation equations. When first order terms are retained a second order differential equation is obtained. For reasons pertaining to advantages in numerical treatments, the oscillation equation can be split into two first order equations. This has been done by Vaeth & Chanmugam (1992), who derived a set of first order equations for the quantities and (see Eqs. (9) and (10) of Vaeth & Chanmugam (1992)). More recently, Gondek et al. (1997) obtained a set of equations for the relative radial displacement and the Lagrangian perturbation of the pressure (see Eqs. (11) and (12) of Gondek et al. (1997)). These sets of equations are equivalent. In this work we adopt the equations of Gondek et al. (1997), because they are particularly suitable for numerical applications and boundary conditions in terms of them have a simpler physical interpretation and a more natural derivation. Another important advantage of this system of oscillation equations is that they do not involve any derivatives of the adiabatic index . Adopting we have
with the coefficients given by
where is the eigenfrequency and the quantities and are assumed to have a harmonic time dependence (. To solve equations (17)–(22) in each phase of the star one needs two boundary conditions there. The condition of regularity at requires that the coefficient associated with the term in Eq. (17) must vanish (Vaeth & Chanmugam, 1992; Gondek et al., 1997; Gondek & Zdunik, 1999). Thus, we have
Note that the eigenfunctions can be normalized in order to have . The surface of the star is determined by the condition that for , one has at all times. This implies that the Lagrangian perturbation in the pressure at the stellar surface is zero. Therefore another boundary condition is
Boundary conditions for possible interfaces splitting different phases of a star are still in order, which we turn our attention to in the next section.
5. Junction conditions at the interface
We now discuss the junction conditions needed for the numerical integration of the oscillations’ equations when a sharp interface due to a first order phase transition takes place inside a hybrid compact star. Such conditions are intrinsically related to the rate of reactions near the surface splitting any two phases (see Haensel et al. (1989) for further details). We discuss them in what follows.
5.1. Slow Transitions
When the characteristic timescale of reactions transforming one phase into another is much larger than those of the perturbations, we are in the scenario of slow phase transitions. In this case, volume elements near a surface splitting two phases do not change their nature due to the perturbations but they just co-move with the splitting surface, stretching and squashing (due to pressure changes), and there is no mass transfer from one phase to another. This implies that the jump of across the interface, , should always be null because one can always track down such near surface elements:
The instantaneous constancy of the pressure (different from the equilibrium value, though) on both sides of the splitting-phase surface (directly related to the absence of thin shell surface quantities, such as thin shell surface tensions and energy densities; see, e.g., Pereira et al. (2014)) also guarantees that
These results can be rigorously derived by making use of generalized distributions (Pereira & Rueda, 2015).
In a more general case when the interface has thin shell surface degrees of freedom, the dynamics of radial oscillations could be modified. As one expects from the lack of transmutation of underneath surface volume elements as regards a phase-splitting surface into above ones (and vice-versa), even when thin shell surface effects take place we still have . Nevertheless, due to the presence of a nontrivial surface dynamics, ceases to be continuous (Pereira & Rueda, 2015). Although interesting, in the present work we limit ourselves to the situations in which thin shell surface degrees of freedom are absent when the system is in equilibrium.
5.2. Rapid Transitions
Rapid phase transitions are characterized by reaction rates transforming one phase into another whose timescales are much smaller than those of the perturbations (Haensel et al., 1989). For practical purposes this means instantaneous change of nature of volume elements (from one state of matter to another) near a phase-splitting boundary due to perturbations. This would imply mass transfers between the two phases. Since the reaction rates are very fast, the surface that splits two any phases can be seen in thermodynamic equilibrium at all times, which means one should characterize it by a constant pressure, equal to that in the absence of perturbations, implying thus , which in turn would lead to have a null jump across the surface :
Given the practically instantaneous change of nature of volume elements near , one would imagine it would be hard to keep trace of them. Thus, one would expect the Lagrangian displacements of volume elements immediately above and below to be discontinuous, which would be the key difference between rapid and slow phase transitions regarding boundary conditions.
We now deduce the boundary condition for in the case of rapid phase transitions by means of physical considerations alone. This will be obtained by demanding that be well-localized, i.e., , where is the radial position of the phase-splitting surface with respect to the radial coordinates above and below it, respectively. Let us consider that in equilibrium is at the position . When perturbations take place, we should generically have , where so far are unknowns and obviously are of the order of . From the definition of the Lagrangian displacement of the pressure, it follows that
where we have defined the prime operation as the radial derivative and stands for the pressure at in the absence of perturbations. At , due to the nature of rapid phase transitions, it follows that . We have . Thus, from the above and Eq. (28), it follows that
From the condition we have
We point out that the conditions we have obtained for and are identical to the ones found in Karlovini et al. (2004), though different reasonings have been used.
For numerical purposes, as we have seen previously, it is more convenient to work with . Since is always continuous across we can split both sides of Eq. (30) by it, resulting thus in
5.3. Classical limit of rapid phase transition’s boundary condition
It is instructive to calculate the classical limit of Eq. (31) in order to check if it agrees with the one calculated in Haensel et al. (1989). This can be easily done by assuming that and from Eq. (13), which leads to
where we restored the units only for future convenience and is the mass density of the system. It is also known in the classical case that (Shapiro & Teukolsky, 1986)
with , the squared adiabatic speed of the sound.
6. Numerical procedure
Let us now succinctly describe the numerical procedure that will be used subsequently. The oscillation equations are solved numerically by means of a shooting method. At first, for a given central pressure and for each set of parameters of the EOS, we integrated the Tolman-Oppenheimer-Volkoff stellar structure equations in order to obtain the coefficients of the oscillation equations. Then, we start at the core with the numerical integration of Eqs. (17)–(22) for a trial value of and a given set of values of and such that the boundary condition at the center is fulfilled. The equations are integrated outwards until the quark-hadron interface is reached, where we use the junction conditions (Eqs. (25) and (26) for the slow case and Eqs. (27) and (31) for the rapid case) to obtain the correct values of and at the other side of the interface. Thereafter, the integration proceeds outwards trying to match the boundary condition at the stellar surface. After each integration, the trial value of is corrected until the desired precision on the boundary condition is achieved. The discrete values of for which the oscillations equations are satisfied are the eigenfrequencies of the star (for more details see Vásquez Flores & Lugones (2010)).
We have also implemented a shooting to a fitting point method in which two “shots” are made, one from the center and the other from the surface of the star, trying to match the junction conditions at the quark-hadron interface. Both methods produced identical results.
When talking about the stability of hybrid systems, independent of whether slow or rapid phase transitions take place, the fundamental mode with frequency is crucial. It is defined as the nodeless solution to Eqs. (17)–(22), supplemented by Eqs. (13), (14) and (15). When it is purely imaginary, the star is unstable because perturbations could grow without limits. We have previously seen that hybrid stars only exist for certain values of the effective bag constant larger than a minimum one, for given and , defined by Eq. (8) (see Fig. 1). In our forthcoming analysis this fact is taken into account. Besides, for the hadronic phase of the star, we make use of the relativistic mean field EOS with the NL3 parametrization already presented in Sec. 2. We assume that the phase transition is sharp (no mixed phase), and determine the interface by imposing the Gibbs equilibrium conditions (equality of the pressure and the Gibbs free energy per baryon of the quark and hadronic phases). Figure 2 shows some examples of equations of state we will work with.
In Fig. 3 we calculate the maximum masses associated with some , and . As clearly seen from this figure, several different configurations result in maximum masses larger than , in agreement with the recent discovery of two very high mass pulsars (Demorest et al., 2010; Antoniadis et al., 2013). Naturally, these are of main physical interest but other less massive configurations are also interesting for having insights into the macroscopic fingerprints and conspicuous aspects of different phase transitions. In our analysis we take into consideration both above-mentioned characteristics.
7.1. Rapid versus slow modes
In Fig. 4 we calculate the fundamental and some excited modes assuming rapid and slow transitions for a sequence of stars with the quark EOS parametrization MeV, and MeV. Hybrid stars in this case have maximum mass around and the pressure where the phase transition occurs is around 31 MeV fm. Notice that the fundamental modes of rapid and slow phase transitions tend to the same value in the limit the system becomes one-phase (purely hadronic, occurring when the radius of the innermost phase goes to zero). Additionally, for , the th excited rapid mode matches continuously the th slow one when the quark core shrinks.
Note that some rapid transition frequencies could overlap slow transition ones. The fundamental frequencies associated with rapid phase transitions are smaller than their slow counterparts for all possible masses of the system. As indeed expected from the densities involved, frequencies for the modes are ordinarily around some kHz. We do not elaborate on aspects of one-phase stars because they are already very well known (Glendenning, 2000; Haensel et al., 2007; Harrison et al., 1965).
7.2. The reaction mode
Regarding rapid phase transitions, a new mode appears, which is the general relativistic generalization of the reaction mode discovered by Haensel et al. (1989). We find that in general the reaction mode can either be the first excited or the fundamental one, depending upon the EOS parametrizations. For example, if the NL3 parametrization of the hadronic equation of state is used and we take MeV), then the reaction mode is the fundamental mode when MeV. Indeed, as one can see for the case MeV and of Fig. 4, the reaction mode is the first excited eigenfrequency for each configuration. It does not converge to any slow or one-phase mode when the radius of the core goes to zero, being thus the fingerprint of rapid reactions. The main relevance of the reaction mode to the stability of the star is naturally when it is the fundamental mode.
Figure 5 exemplifies a situation contrasting with Fig. 4 where is increased beyond MeV and the reaction mode is now the fundamental one. As one further increases for fixed and , a maximum value is reached above which the frequency of the fundamental mode of rapid reactions becomes purely imaginary. We will come back to this issue later when we give their estimates and associated star aspects for some selected .
Since when one increases for a fixed the minimum value of such that hybrid stars exist also increases (see Fig. 1), the reaction mode will eventually only be the fundamental one. This is exemplified by Fig. 6 for the case MeV and . Observe in this case that there is a region around the critical mass marking the onset of hybrid stars where the reaction eigenfrequencies do not exist, while they do for slow phase transitions. We discuss this issue in the next subsection.
7.3. Zero-frequency mass, maximum mass and stellar stability
As depicted in Fig. 4, the stellar background mass at which the frequency of the fundamental mode is zero, , depends on the nature of the phase transition. The difference between and for the parameters of Fig. 4 is very small, around 0.05%, while for the case of Fig. 5 it is conspicuous. Taking the same parameters of Figs. 4 and 5, Fig. 7 shows that coincide with the maximum masses of their plots, and that the reality of the fundamental mode is intrinsically associated with the condition . However, for slow phase transitions the zero-frequency mass corresponds to a model with a central density , being the central density of the model with . We argue now by means of other examples that this is a generic aspect of rapid and slow reactions in hybrid stars and hence a potential way to differentiate them.
Consider for instance the case associated with Fig. 6. As clear from Fig. 8 for rapid reactions, differently from slow phase transitions, the system is unstable (nonexistent/imaginary fundamental modes) for the situations where . Figs. 9 and 10 confirm that the stability criterion for rapid phase transitions can also be associated with the critical points of the mass as a function of both the quark phase’s radius and star’s radius, and , respectively. Note, however, that this is not the case when one works with the mass of the inner (quark) phase, , as a function of or (see Figs. 9 and 8, respectively).
Even when , stars can also be stable when only slow reactions take place. Figure 11 exemplifies that for a case where the maximum mass of the system exceeds . One clearly sees that the violation range of the condition is considerable: is positive up to around four times the central density related to the maximum mass. This evidences the nontriviality of slow phase transitions and how perilous extrapolations of aspects of one-phase stars can be for hybrid systems, besides the ensuing uncertainty of astrophysical interpretations of objects whose only masses can be measured.
One may wonder whether the above analysis remains valid if one considers that NSs may rotate. In fact, for rotating compact stars the existence of a stable configuration can be threatened by nonaxisymmetric dynamical and secular instabilities.
The most studied type of rotational dynamical instability is the so called bar-mode instability. The classical bar-mode instability is excited in Newtonian stars when the ratio of the rotational kinetic energy to the gravitational binding energy is larger than , but general relativistic effects lower the critical value to (Shibata et al., 2000; Saijo et al., 2001). Differential rotation may change the scenario significantly. NSs with a high degree of differential rotation may be dynamically unstable for (Shibata et al., 2002, 2003). Additionally, an one-armed spiral instability may become unstable if differential rotation is sufficiently strong (Centrella et al., 2001; Saijo et al., 2003). However, differential rotation affects only very young protoneutron stars. In fact, a few minutes after a NS is born in a core collapse supernova, rigid rotation sets in due to the presence of viscosity or a sufficiently strong magnetic field (Haensel et al., 2016). Thus, these dynamical instabilities are not expected to have an impact in most cold catalyzed NSs.
Another class of nonaxisymmetric instabilities are secular instabilities, which require the presence of dissipation due, for example, to viscosity and/or gravitational radiation. A particularly interesting class of oscillation modes are modes, which are large-scale currents in NSs that couple to gravitational radiation and remove energy and angular momentum from the star in the form of gravitational waves (Andersson, 1998; Friedman & Morsink, 1998; Friedman & Schutz, 1978; Lindblom et al., 1998). In the absence of viscous dissipation, they are unstable at all rotation frequencies leading to an exponential rise of the mode amplitude. However, when viscous damping is taken into account the star is stable at low frequencies but there may remain instability regions at high frequencies (Lindblom et al., 1998; Alford et al., 2012b). In addition, the fact that fast-spinning compact stars are observed suggests that nonlinear damping mechanisms are present and limit the exponential grow of modes that would destroy the NS (Alford et al., 2012a). If this instability is stopped at a large amplitude, modes may be a strong and continuous source of gravitational waves in some objects (Haensel et al., 2016). Another, secular bar-mode instability, sets in at (Andersson & Comer, 2007).
In summary, it is reasonable to expect that the here-presented stability analysis based on radial oscillations should be valid for hybrid stars rotating below a yet unknown but not too small critical frequency.
The left panel of Figure 12 shows the eigenfunctions (for modes ranging from 0 to 3) using the same EOS parameters of Fig. 4 for a hybrid star with mass . For this particular choice there are rapid and slow representatives for each mode depicted. Their order is defined by the number of nodes they present, even in the case where jumps take place. The fact that rapid reactions differ from slow ones only by a boundary condition does not mean that slow and rapid eigenmodes differ only after the phase transition radius. Rather, they should generally be different throughout the whole star because the description of perturbations constitutes a Sturm-Liouville problem, whose solutions are extremely dependent upon boundary conditions. However, broadly speaking, one sees from the figure that the higher the mode the smaller the jump for fast reactions, which make them approach the slow eigenmodes. The right panel of Figure 12 shows the eigenfunctions using the same EOS parameters of Fig. 5 for the hybrid star with mass . Note from it that for such a mass the rapid fundamental mode is nonexistent. The qualitative behavior of the eigenmodes in this case is similar to that of the left panel of Fig. 12.
7.5. Twin and triplet stars
From the above analysis a very important conclusion ensues: slow phase transitions lead to the existence of twin hybrid stars (i.e. couples of stable stars with the same gravitational mass but different central densities), while rapid phase transitions do not. At the same time, it explicitly shows that the usual condition for stability, , should in general be superseded because it is reaction dependent. The general expression and the associated analysis from first principles will be presented in a forthcoming publication.
Another interesting feature that arises for some EOS parametrizations is the existence of some stable hybrid configurations with a gravitational mass smaller than the mass of the most massive one-phase (purely hadronic) object, i.e. the plateau mass in the plot. Such undercritical masses can be seen explicitly in Fig. 8 for and in Fig. 11 for where it is apparent that for some central densities above the one where the hybrid phase starts to exist, stars could have undercritical masses if the transition is slow.
When this is the case, one could even have “triplet stars” if the hadronic counterpart is also taken into account (see Fig. 8). As a thumb rule, when critical points do not take place in the plot of hybrid systems or when it only presents a local maximum, triplet stars would not be possible and twin stars would be related to hadronic stars turning into hybrid ones (therefore utterly different systems with the same mass).
Thus, only the existence of twin stars would already tell us about the kind of reactions taking place inside neutron stars, as well as their hybrid nature. Indeed, one-phase and hybrid stars presenting rapid reactions cannot have twin counterparts because they would violate the condition .
7.6. Density jumps for rapid reactions
For specificity, we focus here only on hybrid stars with rapid reactions. Naturally, for a given value of , different values of lead to different maximum masses for the associated stars; see Fig. 3. However, there exists a maximum value of above which the fundamental mode becomes unstable (imaginary eigenfrequency). The change of the effective bag constant also changes the density jump at the phase transition radius, . For small core radii, it is classically known that stable solutions should be related to (see Kämpfer (1981) and references therein). This could be violated when general relativistic corrections are taken into account and the above condition should be superseded by (Kämpfer, 1981; Seidov, 1971). By working numerically we obtain that when rapid reactions are considered indeed holds for small quark cores but it could be violated when cores are not small.
Given , , and a hadronic equation of state, one can find a unique transition pressure (related to the equality of the associated Gibbs functions of the quark and hadronic phases), which leads to
From the hadronic equation of state , one can simply obtain by inverting it and evaluating at .
Table 1 summarizes maximum values of and associated stellar parameters for configurations with when rapid reactions are considered. ( leads to for some range of central densities.) The horizontal line in the middle of Table 1 splits the (quark phase) parameters leading to stable stars with small quark cores (upper part) from those leading to stable stars with extended cores (lower part) [these parameters, however, lead to unstable stars when their cores are small, which happens when their central pressures are close to the associated s]. One indeed sees that the condition holds for stable stars with small cores while it could be violated for stable stars with extended cores (outside the scope of Seidov’s work (Seidov, 1971)). This issue is interesting on its own and we plan to investigate it more precisely elsewhere.
8. Discussion and Conclusions
It is known since long ago that in one-phase stars the stability analysis for radial oscillations is equivalent to the condition . This is very important because it gives a practical rule for determining whether or not a system lingers on in time when perturbed based solely on a sequence of its static solutions. However, when hybrid stars are taken into account, the aforementioned stability criterion should be taken cautiously. The reason is broadly due to the possibility of conversion reactions and the non-differentiability of some physical quantities at the phase-splitting surface, such as the pressure and the energy density, related to the total different natures of the fluids in each phase. However, one is always on the safe side when the perturbation equations are directly analyzed, because the stability of a nonrotating system is related to the reality of the fundamental eigenfrequencies. The price to pay is naturally the dealing with nonlinear equations in the presence of nontrivial boundary conditions. Fortunately, powerful numerical tools already exist, allowing us to solve them relatively easily, such as the one we have made use of in this work.
When extra boundary conditions are present, one expects the spectrum of eigenfrequencies and eigenmodes of the system to change, since the associated perturbation equations constitute a Sturm-Liouville problem. This, in turn, may in general lead the known conditions valid for one-phase systems to change. In particular, the practical rule that for a system to be stable should not be taken for granted.
In the present paper, we have showed that within a phenomenologically inspired bag-like model for the quark phase and a nonlinear Walecka model for the hadronic phase (NL3 parametrization), the spectrum of hybrid stars in general relativity is extremely dependent upon the nature of the reactions taking place near the surface splitting the two aforesaid phases. When rapid phase transitions are present a new mode appears, which, depending upon the EOS parameters, could be either the first excited mode (as classically known) or the fundamental one (to our knowledge something new). We have also found that for rapid reactions the frequency of the fundamental mode is a real number if and only if the condition is verified, coinciding with the practical rule for one-phase systems. Therefore, for rapid phase transitions, the above mentioned condition is necessary and sufficient for the stability of hybrid stars. When slow phase transitions are analyzed, though, is neither necessary nor sufficient in general, greatly contrasting with its one-phase counterpart. This shows that particular attention should be taken when analyzing the physical situations which lead to slow phase transitions.
When rapid reactions take place, due to the condition , it would not be possible the occurrence of twin hybrid stars; see for instance the case related to Fig. 8 and Sec. VII-C. However, when slow phase transitions are taken into account, twin hybrid stars may emerge because some stars with may have real eigenmodes in this case (see Figs. 8 and 6). If one also takes into account the sequence of hadronic stars, then slow reactions could even lead to the existence of triplet stars. For rapid reactions, though, only twin stars could emerge. All of the above shows that there may be in principle macroscopic ways of assessing internal processes of neutron stars and therefore the possibility of learning about its constitution.
Our analyses have been performed in the context of non-rotating stars, where the issue of secular instabilities does not take place, but to date all known stars are known to rotate which in principle may induce such instabilities. Notwithstanding, our analyses are relevant when viscous dissipation and nonlinear damping mechanisms suppress secular instability’s exponential growth which is expected to be the case below some critical rotation rate in cold catalyzed NSs. In such circumstances dynamical stability determines NSs existence and stable configurations are characterized by (real eigenfrequencies) because it is only in this case that displacements of volume elements are bound (for radial displacements ). Hence this must be equivalent to the minimum of the total energy in the equilibrium solution (from the TOV equations) for fixed baryon number since it leads to the same conclusion regarding the dynamics of small volume displacements.
It has been speculated that hybrid stars may contain a mixed hadron-quark phase in their interiors. In such a phase it is assumed that the electric charge is zero globally but not locally, and therefore charged hadronic and quark matter may share a common lepton background, leading to a quark-hadron mixture extending over a wide density region of the star (Glendenning, 2001). The mixed phase entails a smooth variation of the energy density, leading in turn to a continuous density profile along the star. Whether the quark-hadron interface is actually a sharp discontinuity or a wide mixed region depends crucially on the amount of electrostatic and surface energy needed for the formation of the varying geometric structures of one phase embedded in the other all along the mixed phase (Voskresensky et al., 2003; Endo, 2011; Yasutake et al., 2014). If the energy cost of Coulomb and surface effects exceeds the gain in bulk energy, the scenario involving a sharp interface turns out to be favorable. However, there are still many points to be elucidated before arriving to an undisputed description of the quark-hadron coexistence. In particular, model calculations of the surface tension span a wide range of values (see Lugones et al. (2013); Lugones & Grunfeld (2017) and references therein) and as a consequence the very nature of the quark-hadron interface remains uncertain. Nonetheless, since in the presence of the mixed phase all physical quantities are differentiable everywhere, these stars behave essentially as one-phase objects whose stability is not affected by the junction conditions explored in the present work. This fully justifies our assumption of disregarding it in our analysis.
Besides twin stars, one could in principle unveil the nature of phase transitions and innermost phases in neutron stars by means of direct frequency observations. This would especially be so due to the fact the reaction mode is uniquely associated with rapid phase transitions, it is very sensitive to the quark EOS and in some cases, such as when it is the fundamental mode, it may differ significantly from its slow counterpart for some masses (see, e.g., Fig. 5). Such observations might be possible in the future through gravitational wave detectors because in the case of rotating objects we can expect some amount of gravitational radiation from even the lowest () quasiradial mode (Stergioulas, 2003; Passamonti et al., 2006). Additionally, purely radial oscillations could leave some electromagnetic imprint in the microstucture of pulsar emission or in magnetar flare lightcurves.
Summing up, in this work we have found the extra boundary conditions appropriate for slow and rapid reactions in hybrid stars. By using a bag-like model for the quark phase and a relativistic mean field theory model for the hadronic phase of a star, we have showed that rapid reactions lead the generalization of the classical reaction mode to be either the first excited or the fundamental eigenfrequency. The general relativistic stability jump condition for hybrid stars with small cores (Seidov’s condition) holds true for rapid reactions, while it could be violated for large (extended) cores. For hybrid stars experiencing rapid phase transitions the frequency of the fundamental mode is zero when , identical to what is known for one-phase stars. However, slow phase transitions lead to for the stellar configuration at which the frequency of the fundamental mode is zero. As a consequence, the range of central densities in a hybrid star where slow reactions take place at the interface is larger than in hybrid objects undergoing rapid reactions. Twin hybrid stars could exist for slow reactions, while this is not the case for rapid ones. When one-phase hadronic stars are also taken into account, even triplet stars could emerge for slow reactions. Therefore, internal aspects of stars could in principle be probed with these systems.
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