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Astron. Astrophys. 353, 641-645 (2000)

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3. Methodology and the core envelope model

For spherically symmetric and static configurations we make use of the metric

[EQUATION]

where [FORMULA] and [FORMULA] are functions of r alone. The Oppenheimer-Volkoff (O-V) equations, resulting from Einstein's field equations, for systems with isotropic pressure P and energy-density E can be written as

[EQUATION]

where [FORMULA] is the mass, contained within the radius r, and the prime denotes radial derivative.

The core-envelope model consists of a core with most stiff EOS in the region [FORMULA] and an envelope with a polytropic EOS in the region [FORMULA] as given below

(i) The core: [FORMULA]

For the models of neutron stars considered here, we have chosen the core of most stiff material (Eq. (1)) as

[EQUATION]

where [FORMULA] is the value of density at the surface of the configuration, where pressure vanishes.

(ii) The envelope: [FORMULA]

The envelope of this model is given by the equation of state

[EQUATION]

or

[EQUATION]

where K is a constant to be worked out by the matching of various variables at the core-envelope boundary.

At the boundary, [FORMULA], the continuity of [FORMULA], and [FORMULA] require

[EQUATION]

where [FORMULA] is given by (see, e.g., Tooper 1965)

[EQUATION]

Thus, the continuity of [FORMULA], at the boundary gives

[EQUATION]

Thus, the continuity of [FORMULA] and [FORMULA] at the core-envelope boundary is ensured, for the static and spherically symmetric configuration.

The coupled Eqs. (3), (4), (5), are solved along with Eqs. (1) and (6) for the boundary conditions (7) and (8) [at the core-envelope boundary, [FORMULA]], and the boundary conditions, [FORMULA], [FORMULA], [FORMULA] at the external boundary, [FORMULA].

For the sake of numerical simplification, we assign the central density, [FORMULA]. It is seen that the degree of softness of the envelope is restricted by the inequality, [FORMULA]. For the minimum value of [FORMULA], we obtain various quantities, such as, core mass, [FORMULA], core radius, [FORMULA], density at the core-envelope boundary, [FORMULA], total mass, M, and the corresponding radius, R, of the configuration in dimensionless form as shown in Table 1 for various assigned values of the central pressure to density ratio, [FORMULA].


[TABLE]

Table 1. Properties of the causal core-envelope models, with a core given by the most stiff EOS, [FORMULA], and the envelope is characterized by the polytropic EOS, (d[FORMULA]d[FORMULA], such that, all the parameters, [FORMULA] and the speed of sound, [FORMULA], are continuous at the core-envelope boundary, [FORMULA]. The maximum value of [FORMULA] for the structure is obtained (Fig. 1), when the minimum value of the ratio of pressure to density at the core-envelope boundary, [FORMULA], reaches about 0.014. The maximum value of the binding-energy per baryon, [FORMULA] where [FORMULA] is the rest mass of the configuration] [FORMULA] also occurs for the maximum stable value of u. The subscript `0' and `b' represent, the values of respective quantities at the centre, and at the core-envelope boundary. [FORMULA] stands for the surface redshift. The calculations are performed for an assigned value of the central energy-density, [FORMULA]. The slanted values represent the limiting case upto which the structure remains dynamically stable. Dimensionless values of various quantities for [FORMULA] 0.014


To determine the stability of the models given in Table 1, we need to draw the mass-radius diagram for the structures. For this purpose, we have normalized the boundary density, [FORMULA] g cm-3, and obtained the mass-radius diagram as shown in Fig. 1 [Notice that the value of [FORMULA] chosen in this way (and hence also the mass and the radius obtained in conventional units as shown in Fig. 1) is purely arbitrary. These values have nothing to do with the actual maximum mass and the corresponding radius of the stable neutron star obtained in the present paper. The choice of [FORMULA] does not affect the maximum value of u upto which the structure remains dynamically stable]. The maximum stable value of u of the whole configuration is obtained as 0.3574, which would determine the maximum mass of the stable neutron star from the knowledge of the rotation period of PSR 1937 + 214, or PSR 1957 + 20. It is interesting to note that for this value of u, the binding energy per baryon, [FORMULA], where [FORMULA] is the rest-mass (Zeldovich & Novikov 1978) of the configuration] also approaches maximum ([FORMULA]) as shown in Table 1.

[FIGURE] Fig. 1. Mass-Radius diagram of the model for an assigned value of [FORMULA] g cm-3 at the core-envelope boundary [FORMULA], such that the ratio of pressure to density [FORMULA] at [FORMULA] reaches about 0.014. The pressure, energy-density, [FORMULA], and the speed of sound, [FORMULA] are continuous at the core-envelope boundary.

Koranda et al.(1997) have obtained the following empirical formula for the class of minimum period EOSs which is fairly insensitive to the matching density

[EQUATION]

where [FORMULA] is the (minimum) rotation period of the maximal (uniformly) rotating configuration, and M and R are maximum mass and the corresponding size of the non-rotating configuration. Rewriting Eq. (9) in terms of compaction parameter, [FORMULA], and angular velocity [FORMULA] we obtain

[EQUATION]

where [FORMULA] is the maximum value of u of the non-rotating configuration, such that the configuration becomes dynamically unstable when u exceeds [FORMULA], and R represents the corresponding radius of the configuration.

Let us denote the average density of the configuration by [FORMULA] [because [FORMULA], or [FORMULA]] by using Eq. (10) we obtain

[EQUATION]

and,

[EQUATION]

Thus, the average density of the configuration depends only upon the rotation period, and not upon the compaction parameter u. Therefore, it is clear from Eqs. (11) - (12) that for a given value of the rotation period, [FORMULA], the maximum mass of the stable configuration depends only upon the maximum value of u. For [FORMULA] ms, Eq. (11) gives the average density, [FORMULA], of the configuration as [FORMULA] g cm-3, the substitution of [FORMULA] in Eq. (12) gives the maximum mass of the configuration, [FORMULA] and the corresponding radius, [FORMULA] km.

The ratio, M(envelope)/M(star), is about [FORMULA], which may be relevant to explain rotational irregularities in pulsars known as timing noise and glitches (see, e.g., D' Allessandro 1997, and references therein).

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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