Ultra Compact Objects (UCOs) with radius ( mass of the star) are interesting entities. In principle, trapping of mass-less particles in UCOs potential well is possible or the object can oscillate in its quasinormal modes (Kembhavi & Vishveshwara 1980; Chandrasekhar & Ferrari 1991). Van Paradijs (1979) had pointed out the peculiar behavior of redshift for . Existence of UCO was speculated by Iyer & Vishveshwara (1985), Iyer et al. (1985), and Lattimer et al. (1990). The first calculations, showing the existence of trapped photon or neutrino orbits inside such a UCO were made by Kuchowicz (1965) and de Felice (1969). Recently, Negi & Durgapal (1996) have obtained various types of trajectories of such particles (photons or neutrinos), for different initial conditions, inside a UCO characterized by parabolic density variation.
Furthermore, the rotation period and mass of dense matter objects with are important regarding sub-millisecond pulsars (smps). Haensel & Zdunik (1989) discussed the uniform rotation for a static mass of , and found that nearly all the existing realistic equations of state (EOSs) fail to provide a suitable model for smps. They found that in order to have a successful model for smps, the equation of state (EOS) should correspond to the matter of maximum stiffness [which corresponds to the condition that the speed of sound, = speed of light (= 1 in geometrized units) where P and E are, respectively, the pressure and the energy-density], that is,
and is the value of E at the surface of the configuration.
Characteristics of the super-dense objects like neutron stars are based on the calculations of the EOS for the matter at very high densities. However, the nuclear interactions beyond the density of g cm-3 are empirically not well known (Dolan 1992) and all the known EOS are only extrapolations of the empirical results far beyond this density range. In this regard, various EOS based on theoretical manipulations are available in the literature (Arnett & Bowers 1977).
In order to obtain an upper limit for the neutron star masses various theoretical models have been proposed. As the status of EOS of nuclear matter cannot be established empirically, one can apply physical constraints to obtain an upper bound of neutron star mass. Brecher & Caporaso (1976) assumed that the speed of sound in the nuclear matter equals the speed of light and obtained a value of as an upper limit of the neutron star masses by using EOS given by Eq. (1) and a surface density, g cm-3. However, the matter described by this equation has a super-dense self-bound state at , which represent the `abnormal state of matter' (Lee 1975; Haensel & Zdunik 1989). This `abnormality' may be specified as the pressure vanishes at the order of nuclear densities, or in other words, the matter represent a super-dense self-bound state of matter even at vanishing small pressure, and the speed of sound remains equal to that of light in these conditions. This `abnormality' can be removed if we ensure continuity of all the metric parameters and their derivatives at the boundary of the structure.
Earlier, Rhoades & Ruffini (1974), without going into the details of the nuclear interactions, assumed that beyond a certain density g cm-3 [the range of densities where no extrapolated EOS is known], the EOS in the core is given by the criterion that the speed of sound attains the speed of light, that is, , and matched the core to an envelope with the BPS (Baym et al. 1971) EOS and obtained an upper limit for the neutron star mass as .
Hartle (1978) emphasized that the maximum masses of neutron stars obtained in this manner involve a scale factor, say, , such that, the matching density, , plays a sensitive role to obtain an upper bound on neutron star masses. Usually, is taken to be equal to or greater than one in all the conventional models. For the densities less than , the matter composing the object is assumed to be known and unique. That is, the EOS of the envelope of these stars are chosen so that the `abnormalities' in the sense mentioned above are removed. Friedman & Ipser (1987) calculated the masses of the neutron stars for different values of the matching densities by using EOS's in the envelope given by BPS and NV (Negele & Vautherin 1973), respectively, and concluded that for each case the mass in the envelope is negligible compared to that in the core containing the most stiff material. Furthermore, in all these cases the EOS chosen for the envelope are also uncertain and have little empirical support.
Another possibility which leads to an entirely different property of compact objects is that, in the absence of gravity high density baryonic matter is bound by purely strong forces. It can be shown that non-gravitationally bound bulk hadronic matter is consistent with nuclear physics data (Bahcall et al. 1989) suggesting that bulk hadronic matter is just as likely to be the correct description of matter at high densities as the conventional unbound hadronic matter. In general, the high-density non-gravitationally bound states of baryons are called "baryon matter" (Bahcall et al. 1990) and the terms "hadronic matter" or "quark Matter" are used for baryon matter described by theories of hadrons or quarks. Baryon matter refers to the saturating, large fermion number limit of these states in theories of either quarks or hadrons. Bahcall et al. (1990) gave an EOS, based upon effective field theory, which is consistent with all nuclear physics data, and low energy interaction data (Lynn et al. 1990), and they argued that possibly all the neutron stars are Q-stars with mass much larger than those obtained by conventional models. The term `Q-star' is used for the objects whose self gravity is important, and also to distinguish these models from conventional models in which large numbers of baryons are not bound without gravity. In the Q-star model, baryon matter is a perfect fluid, and so the Oppenheimer-Volkoff equations can be integrated using the equation of state derived from a particular effective field theory. The Q-star boundary conditions define a stellar surface where the total hadronic pressure vanishes. At this point the energy density has not yet vanished, since the zero-hadronic-pressure state is just baryonic matter, but it drops exponentially to zero on a scale of fermis. Because there are no experimental data available for an EOS of many baryon system with densities close to nuclear density, may take a value less than g cm-3, and the upper limit on the maximum mass of compact objects which are not black holes (and also not neutron stars) could be arbitrarily large. Even if one is willing to dismiss the particular object resulting from the new EOS as being currently undiscovered in nature, the possibility that some EOS other than those previously extrapolated to nuclear densities may contain the correct physics at these densities indicated that the densities at which we know the form of EOS to be unique is lower than g cm-3. In any case, the important point is that the density for which an EOS is known to be unique is lower than g cm-3. Revealing the fact that the density range, 1010 g cm-3 g cm-3, remains valid for Q-star equation of state, Bahcall et al. (1990) have obtained the stable Q-star masses as large as [for the matching density, g cm-3]. But, Lynn (priv. comm.) showed that to represent a physically viable model of Q-star, the upper mass limit would be significantly less than .
Thus, we can summarize the whole scenario of EOS for the super-dense objects as follows:
(a) If we consider a hadronic matter we expect the density of the matter to vanish with the vanishing pressure, that is, near the surface of the star we must have an equation of state pertaining to a vanishing density. The model of the super-dense object may have a high density core represented by some stiff EOS surrounded by matter represented by EOS derived from the known nuclear interactions and extrapolated to densities at which these EOS are matched with the stiff EOS in the core. The matter represented by EOS corresponding to hadronic matter is surrounded by empirically known EOS [one or more in sequence] such that we obtain a vanishing small density at the surface of the star where the pressure vanishes.
The examples for such models are due to Rhoades & Ruffini (1974), Hartle (1978), and Friedman & Ipser (1987). Recently, Kalogera & Baym (1996) considered a model in which the most stiff core is matched to the WFF (Wiringa et al. 1988) EOS in the envelope, which is then matched to a crust with the EOS given by BPS. Glendenning (1992) considered a rotating structure with a core of most stiff EOS up to a density of g cm-3 surrounded by a constant pressure region which is then surrounded by BPS matter. Koranda et al. (1997) also considered a rotating structure with a core of most stiff EOS surrounded either by (i) an envelope of zero pressure or by (ii) a region of constant pressure which is covered by FPS (Lorenz et al. 1993) EOS.
(b) We may choose a baryonic matter surrounding the stiff core or an entire structure consisting of baryonic matter. In this case we are free to choose a finite density at the surface of the structure where the pressure vanishes. The baryonic matter represents a perfect fluid and can have densities lower than g cm-3. One can get a large mass [but not arbitrary large (Lynn priv. comm.)] for the superdense objects [Q-stars] by introducing a lower value of in the equation containing the scale factor .
Thus, to obtain a physically viable, upper mass limit for superdense objects, one may introduce certain constraint on the matching density, , from the observational evidences known at present. The important observational evidences are:
(A) The minimum rotation period of the fastest rotating pulsars, PSR 1937 + 214, or PSR 1957 + 20 known to date is 1.558 ms [see, e.g., Müther et al. 1991, and references therein].
Assuming this pulsar as a Q-star, Hochron et al. (1993) used the observational fact (A) to obtain a constrained value of [instead of assuming ], and then obtained a strict upper bound on Q-star masses as . However, based upon the observational data, Dolan (1992) had already shown that the mass of an unseen X-Ray binary (Cyg XR-1), , may not necessarily represent a black-hole.
To avoid the discrepancy in the theoretical results of Hochron et al. (1993), and the observational data put forward by Dolan (1992), one may use values of less than g cm-3, and obtain a maximum mass larger than for self-bound (Q) stars [by using a physically viable and causally consistent self-bound EOS (Negi & Durgapal 1999)]. Alternatively, it is also justified to construct a model of gravitationally bound star (neutron star) by choosing a core of the most stiff material [i.e., ], matched to the envelope given by any physically viable and causally consistent EOS [not necessarily those given by BPS, NV, or FPS].
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999