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Astron. Astrophys. 350, 497-512 (1999)

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4. Structure of rotating and non-rotating protoneutron stars

The structure of rotating PNS's and NS's is governed by the Einstein equations in stationary, axisymmetric, and asymptotic flat space-time. Under these special conditions to the space-time symmetry the ten Einstein equations reduce to four non-trivial equations which are elliptic in quasi-isotropic coordinates (Bonazzola et al. 1993). The four non trivial Einstein equations together with the energy-momentum conservation are solved via a finite difference scheme (Schaab 1998). We follow Bonazzola et al. (1993) in compactifying the outer space to a finite region by using the transformation [FORMULA]. The boundary condition of approximate flatness can then be exactly fulfilled. The neutron star model is uniquely determined by fixing one of the parameters: central density, gravitational mass, or baryon number, as well as one of the parameters: angular velocity, angular momentum, or stability parameter [FORMULA]. The models of maximum mass and/or maximum rotation velocity can also be calculated.

4.1. Protoneutron star and neutron star sequences

Fig. 10 shows the gravitational mass as function of the central energy-density for the non-rotating PNS and NS models. Only models whose gravitational masses increase with central energy-density are stable against axisymmetric perturbations. Whereas the maximum mass differs only by [FORMULA] between the various star models, the minimum mass of the EPNS models, [FORMULA], is much larger than the minimum mass of the CNS models, [FORMULA].

[FIGURE] Fig. 10. The gravitational mass versus central energy-density of non-rotating NS's and PNS's. The lower long dashed line corresponds to the [FORMULA] EOS and the upper long dashed line to the [FORMULA] EOS. The lower solid line corresponds to the [FORMULA] EOS and the upper solid line to the [FORMULA] EOS.

Fig. 11 shows the Kepler frequency, i.e. the frequency at which mass shedding sets in, as function of the gravitational mass. Also shown is the rotational frequency, [FORMULA] s-1, of the fastest pulsar known, PSR 1937+214 (Backer et al. 1982). The gravitational mass of this pulsar is unfortunately unknown but is typically assumed to be in the range 1.0-2.0 [FORMULA]. Since the Kepler frequency of the CNS models is larger than the rotational frequency of PSR 1937+214 for this mass range, this model is consistent with the observation.

[FIGURE] Fig. 11. Kepler frequency versus gravitational mass of PNS's and NS's for different EOS's. The upper long dashed line corresponds to the [FORMULA] EOS and the lower long dashed line to the [FORMULA] EOS. The upper solid line corresponds to the [FORMULA] EOS and the lower solid line to the [FORMULA] EOS.

Fig. 12 shows the sequences of stars rotating with Kepler frequency. As expected the masses are now larger at fixed central energy density, but the maximum mass is reached for lower central energy densities (see also Table 5). The mass increase due to rotation is smaller for PNS's than for CNS's, since the radii of PNS's are larger and thus the Kepler frequencies are smaller.

[FIGURE] Fig. 12. The gravitational mass versus central energy-density of NS's and PNS's rotating at their Kepler frequency. The lower long dashed line corresponds to the [FORMULA] EOS and the upper long dashed line to the [FORMULA] EOS. The lower solid line corresponds to the [FORMULA] EOS and the upper solid line to the [FORMULA] EOS.

We also show the mass-radius relation for PNS's in Figs. 13 and 14 8. In comparison with CNS's, where the radius only slightly changes in the relevant mass region around [FORMULA], one obtains for PNS's a much stronger increase of the radius with decreasing mass, which is caused by the stiffer EOS's for the PNS's.

[FIGURE] Fig. 13. The gravitational mass versus stellar radius (as measured by an observer located at infinity) of non-rotating NS's and PNS's. The abbreviations for the different EOS's are described in Table 1. The lower long dashed line corresponds to the [FORMULA] EOS and the upper long dashed line to the [FORMULA] EOS. The lower solid line corresponds to the [FORMULA] EOS and the upper solid line to the [FORMULA] EOS.

[FIGURE] Fig. 14. The gravitational mass versus equatorial radius (as measured by an observer located at infinity) of NS's and PNS's rotating at their Kepler frequency. The lower long dashed line corresponds to the [FORMULA] EOS and the upper long dashed line to the [FORMULA] EOS. The lower solid line corresponds to the [FORMULA] EOS and the upper solid line to the [FORMULA] EOS.

To demonstrate the differences of the internal structure of an EPNS and a CNS, we show in Figs. 15 and 16 the iso-energy-density surfaces for models with [FORMULA]. It turns out that EPNS's contain mainly matter of densities below nuclear matter density (Fig. 15). Contrary to this, matter in a CNS is dominated by matter of densities around and above nuclear matter density (Fig. 16).

[FIGURE] Fig. 15. Iso-energy-density surfaces for a [FORMULA] model with a baryonic mass of [FORMULA] rotating at its Kepler frequency. The surfaces correspond to energy-densities (beginning at the centre), [FORMULA], where [FORMULA] g cm-3 denotes the central energy-density.

[FIGURE] Fig. 16. Iso-energy-density surfaces for a CNS model with a baryonic mass of [FORMULA] rotating at its Kepler frequency. The surfaces correspond to energy-densities (beginning at the centre), [FORMULA], where [FORMULA] g cm-3 denotes the central energy-density.

Another important point concerns the stability of the star models against secular or dynamical instabilities. A certain configuration is dynamically stable against axisymmetric perturbations if the gravitational mass is minimum with respect to variations at fixed baryon number and angular momentum. Along a star sequence with fixed angular momentum, this is the case if the gravitational mass increases with the central energy density (see Shapiro & Teukolsky 1983, p. 151). Please note that the angular momentum is not fixed in the sequences of stars rotating with Kepler frequency. The maximum mass configuration differs therefore slightly from the marginally stable configuration. A fully general relativistic analysis of dynamical and secular stability against non-axisymmetric perturbations is extremely difficult and has been performed only by means of approximations and/or special assumptions in literature. It turned out however that the configurations can be classified by a stability parameter defined by the ratio of the kinetical energy and the absolute value of the gravitational energy, [FORMULA]. For values, [FORMULA], the models are probably dynamically unstable against the bar mode ([FORMULA]), whereas they become already secularly unstable for [FORMULA] (see, e.g., Durisen 1975; Managan 1985; Imamura et al. 1985; Bonazzola et al. 1996). Higher modes, [FORMULA], of perturbations are suppressed by the finite viscosity of neutron star matter. It is however uncertain whether modes with [FORMULA] or 4 are also suppressed. If this is not the case, secular instability sets already in above a critical value, [FORMULA] (Friedman et al. 1986).

Fig. 17 shows the value of the stability parameter for stars rotating with their Kepler frequency. Since the timescale for the growing of the secular instabilities is larger than the evolution timescale of PNS's and HNS's, only the dynamical instability is important for PNS's and HNS's. The critical value for the dynamical instability, [FORMULA], is reached by none of the models. Depending on the internal structure of the CNS's, the CNS's models rotating with Kepler frequency may become secular unstable against modes with [FORMULA] or [FORMULA] for star masses [FORMULA].

[FIGURE] Fig. 17. Stability parameter [FORMULA] versus gravitational mass of PNS's and NS's rotating at their Kepler frequency for different EOS's. The upper long dashed line corresponds to the [FORMULA] EOS and the lower long dashed line to the [FORMULA] EOS. The upper solid line corresponds to the [FORMULA] EOS and the lower solid line to the [FORMULA] EOS.

4.2. Evolution of a non-rotating protoneutron star

The evolution of a PNS to a CNS can be followed by means of several "snapshots" taken at different times after core bounce (see Sect. 2). The evolution path is determined by fixing the baryonic mass and the angular momentum if accretion of matter and loss of angular momentum is neglected. First, we study the evolution of a non-rotating PNS with [FORMULA] (see Table 3 and Fig. 18).

[FIGURE] Fig. 18. Gravitational mass versus baryonic mass for the non-rotating [FORMULA] EOS and the non-rotating CNS EOS. The [FORMULA] EOS lies exactly on the [FORMULA] line and is not shown by that reason. The arrows show the evolution (from left to right) for the minimum mass [FORMULA], the minimum mass [FORMULA], the [FORMULA], and the maximum mass configurations.


[TABLE]

Table 3. Properties of non-rotating and with Kepler frequency rotating PNS's and NS's, for a fixed baryonic mass [FORMULA]. The EOS's are summarized in Table 1. The entries are: gravitational mass, [FORMULA]; baryonic mass, [FORMULA]; circumferential radius (as measured in infinity), [FORMULA]; central baryon number density, [FORMULA]; central temperature, [FORMULA]; Kepler frequency, [FORMULA]; angular momentum, J ([FORMULA] km [FORMULA] g cm2 s-1).


The first snapshot corresponds to approximately 50-100 ms when the envelope is characterized by a high entropy per baryon and high lepton number (see curves labeled [FORMULA] and [FORMULA]). After 0.5-1 seconds the PNS reached our LPNS stage, which is characterized by an approximately constant entropy per baryon, [FORMULA], throughout the star (model [FORMULA]). Due to the higher entropy per baryon in the core of the PNS, the central density and thus the gravitational binding energy decreases. On the other hand the entropy per baryon decreases in the envelope and therefore the radius decreases, too. For a lower value of the entropy per baryon and/or a lower value for the lepton number, the gravitational binding energy increases compared to the EPNS stage (models [FORMULA], [FORMULA], [FORMULA]). After about 10-30 seconds the neutrinos escape from the star and its EOS softens (HNS models). The gravitational binding energy thus increases by roughly 3 %. Finally, the CNS model is even more compressed.

Due to a smaller increase of the pressure with increasing temperature, Burrows & Lattimer (1986) and Pons et al. (1999) obtain a monotonous increase of the central density during the evolution of the PNS to the CNS. In contrast, our results, and also those of Keil & Janka (1995), show a more complex behaviour of the central density (see discussion in the paper of Pons et al. 1999). This difference has mainly two reasons: Firstly, Pons et al. (1999) use an entropy profile obtained in a supernova collapse simulation of a 1.08 [FORMULA] NS, which leads to a higher central entropy per baryon, [FORMULA], at the EPNS stage and an increase to [FORMULA] at the LPNS stage. We use the calculations of Burrows et al. (1995) who simulated the supernova collapse of a NS with [FORMULA]. They obtained a smaller starting central entropy per baryon, [FORMULA], and an increase to [FORMULA] at the LPNS stage. The second reason is the fact, that they use an approximation for the temperature influence on the pressure (derived by Prakash et al. 1997) of the nucleons:

[EQUATION]

[EQUATION]

where [FORMULA] denotes the Fermi temperature of the quasi particles ([FORMULA]), [FORMULA], the thermal pressure and [FORMULA], the pressure at zero temperature as sum of kinetic and potential pressure. This approximation holds under the assumption that in the dense parts of the PNS the temperature is small in comparison with the Fermi temperature, i.e. [FORMULA]. However in our case these ratio reaches [FORMULA] ([FORMULA]) at [FORMULA] and [FORMULA] ([FORMULA]) at [FORMULA] fm-3 for neutrons (protons) in the [FORMULA] model (see also the Fermi-Dirac distribution function in Fig. A1 of Appendix A and Takatsuka et al. (1994) for this purpose, in which it can be seen that more than [FORMULA] of the matter is non-degenerate). Therefore this approximation underestimates the pressure increase due to thermal effects, e.g. the thermal pressure [FORMULA] is exact (approximative) 2.37 MeVfm-3 (1.26 MeVfm-3) at [FORMULA] and 54.62 MeVfm-3 (28.83 MeVfm-3) at [FORMULA] fm-3 for the nucleons of the [FORMULA] case of our model. Hence in our opinion, higher order terms should be included in the treatment. Since the temperature dependence of the GM3 model of Pons et al. (1999) is smaller than in our case, the deviations between the exact solution and the approximation may be small.

4.3. Maximum rotational frequency of a neutron star

We follow now the evolution of an EPNS with [FORMULA], which rotates with Kepler frequency at its EPNS state. During its evolution, the angular momentum is assumed to be conserved (see Table 4). As in the non-rotating case, the star becomes more and more compressed during its evolution. With only a few exceptions, the central baryon density increases, whereas the gravitational mass and the circumferential radius decreases. It is obvious that this trend has to be counterbalanced by an increasing angular velocity in order to keep the angular momentum constant. Compared to the EPNS state, the angular velocity in the CNS state is increased by 51 % or 32 % for the sequences starting with the model [FORMULA] or [FORMULA], respectively. Nevertheless, the angular velocity in the CNS state reaches at most 60 % of the Kepler frequency.


[TABLE]

Table 4. Evolution of 1.5 [FORMULA] star at constant angular momentum, [FORMULA] [FORMULA] km and [FORMULA] [FORMULA] km for the sequences starting with the model [FORMULA] and [FORMULA], respectively.


In this respect, the minimum rotational period is determined by the Kepler rotating EPNS model. Fig. 19 shows the the general evolution of the rotational period for the two EPNS models. For a CNS with a typical baryonic mass of [FORMULA] one obtains a minimum rotational period, P, between 1.56 and 2.22 ms. Recently, Goussard et al. (1997, 1998) found similar results for the minimum rotational period, which confirms our calculations. The period of the fastest known pulsar PSR 1937+214, [FORMULA] ms (Backer et al. 1982) is at the lower limit of this range. Pulsars that rotate even faster cannot be born with such small periods but have to be accelerated after their formation, as long as a typical baryonic mass, [FORMULA] is assumed. Andersson (1998) has recently shown that not too cold NS's rotating with [FORMULA] are unstable against r-modes. This means that the minimum rotational period for a young NS is even higher, [FORMULA] ms. Both the results by Andersson (1998) and the results presented here strengthen the view of millisecond pulsars as being recycled by accretion (Lorimer 1996).

[FIGURE] Fig. 19. Minimum rotational period P versus baryonic mass of the EPNS's and CNS's. The figure shows the spin up during the evolution of the EPNS's to CNS's. The dot-dashed (short dashed) line shows the [FORMULA] ([FORMULA]) model rotating at its Kepler frequency. The solid (long dashed) line shows the corresponding CNS model obtained by cooling at constant angular momentum. The dotted line shows CNS's rotating at their Kepler frequency for comparison. The arrows represent the evolution from the EPNS stage to the CNS stage.

Fig. 20 shows the behaviour of [FORMULA] during the evolution from the EPNS to the CNS state. The stability parameter increases during the evolution. Nevertheless, its maximum value, [FORMULA], reached for the most massive star of the [FORMULA] EOS, is only a little larger than the critical value [FORMULA] for the onset of the secular instabilities with [FORMULA] (see Sect. 4.1).

[FIGURE] Fig. 20. Variation of the stability parameter [FORMULA] during the evolution from EPNS's to CNS's at constant angular momentum. The arrows represent the evolution from the EPNS stage to the CNS stage.

4.4. Maximum and minimum mass of a neutron star

Table 5 shows, that the maximum baryonic mass of a non-rotating (rotating) EPNS is [FORMULA] ([FORMULA]), whereas the maximum baryonic mass of a CNS is 2.41 [FORMULA] (2.7 [FORMULA]) (see also Fig. 18). Since the maximum baryonic mass of a LPNS and HNS is larger than the maximum baryonic mass of a EPNS, too, it is obvious that a PNS based on our nuclear EOS cannot collapse to a black hole during its Kelvin-Helmholtz cooling phase if any further accretion is neglected (see Baumgarte et al. 1996). The situation changes if hyperons, meson condensation, or a quark-hadron phase transition are included (see, e.g., Brown & Bethe 1994; Pandharipande et al. 1995; Prakash et al. 1995; Glendenning 1995). Another important point concerning the maximum mass of a CNS is due to the different maximum masses of EPNS's and CNS's rotating at their Kepler frequencies. The EPNS can support a baryonic mass of [FORMULA], which is only slightly higher than the maximum baryonic mass of the non-rotating CNS ([FORMULA]), but [FORMULA] less than the maximum baryonic mass of the CNS rotating at its Kepler frequency. Such supramassive CNS's enter unstable regions during their spin down evolution due to unstability against axisymmetric perturbations and may finally collapse to a black hole. For a discussion of these supramassive CNS's, see Cook et al. (1994) and Salgado et al. (1994).

In first approximation, the gravitational mass of non-rotating PNS's and HNS's increases quadratically with the entropy per baryon, which is taken constant throughout the star (see Prakash et al. 1997):

[EQUATION]


[TABLE]

Table 5. Properties of the maximum gravitational mass configurations of non-rotating and with Kepler frequency rotating PNS's and NS's.


We obtain for [FORMULA] approximate values [FORMULA] ([FORMULA]) for the cases without (with) trapped lepton number (see Table 5). These values are in agreement with the values derived by Prakash et al. (1997). In the investigation of Goussard et al. (1997) the gravitational mass shows the opposite behaviour: The gravitational mass decreases slightly with increasing entropy per baryon. This is probably caused by the smaller temperature dependence of the EOS of Lattimer & Swesty (1991), as it was pointed out in Sect. 3. In the case of rotating stars the behaviour seems to be more complex since the value of [FORMULA] is negative for [FORMULA] and [FORMULA] and positive for [FORMULA] and [FORMULA]. Another interesting point is, that the maximum gravitational and baryonic mass of the non-rotating PNS's are determined by the EOS used in the core and do not depend on the EOS used in the envelope (compare the models [FORMULA], [FORMULA], and [FORMULA] in Table 5 and Figs. 10 and 13). In the case of rotating PNS's, the maximum gravitational mass of the EPNS is however smaller compared to the corresponding LPNS model (see Table 5 and Figs. 12 and 14). This behaviour is caused by the higher Kepler frequency which can be supported by the LPNS models (see Sect. 4.1).

The minimum baryonic mass of our non-rotating EPNS sequences is in the range [FORMULA]. The minimum mass is increased in the case of Kepler rotating stars to [FORMULA]. If accretion is neglected the baryonic mass is conserved during the evolution of the EPNS to the CNS. We follow the evolution of the minimum mass [FORMULA] and [FORMULA] model to a CNS via the LPNS and the HNS stage (see Table 6). The minimum mass of a CNS born in a supernova is therefore determined by the minimum mass model of the EPNS sequence. The above ranges of baryonic mass correspond to a lower limit of the gravitational CNS's mass in the range [FORMULA]. This is by a factor of ten larger than the minimum mass of the CNS sequence. This property of EPNS's was also recently found by Goussard et al. (1998).


[TABLE]

Table 6. Evolution of minimum mass star for the non-rotating EPNS[FORMULA] EOS ([FORMULA]) and the EPNS[FORMULA] EOS ([FORMULA]).


Variations of the location of the transition region between the envelope (high entropy per baryon) and the core (low entropy per baryon) of the EPNS's do not change the minimum mass considerably. Starting the transition region at lower densities will lead to smaller minimum masses of the EPNS's, but the lower the initial mass of the EPNS is, the higher is the entropy per baryon in the envelope and in the core (see e.g. Keil et al. 1996; Pons et al. 1999). This effect drives the minimum mass back to higher values, so that our results of the minimum mass range is a good approximation (see also Goussard et al. 1998).

Timmes et al. (1996) examine the most likely masses of NS's using the numerical data of Woosley & Weaver (1995) who simulated Type-II supernovae with progenitor stars in the mass range between 11 and 40 [FORMULA]. They obtain a lower limit of the NS mass, which depends on the mass and the composition of the progenitor, of [FORMULA] [FORMULA]. This lower limit is comparable to our results.

4.5. Sensitivity of the results

In Table 3, we compare the properties of our PNS and NS models for a fixed baryonic mass, [FORMULA]. This canonical value corresponds to the measured gravitational masses, [FORMULA], of neutron stars in binary systems (see Thorsett et al. 1993; Van Kerkwijk et al. 1995).

As can be inferred by comparing the LPNS models [FORMULA], [FORMULA], and [FORMULA] with the model [FORMULA] (see Sect. 2.2), the location of the neutrino sphere has nearly no effect on the gravitational mass and the central density. However, the circumferential radius and the Kepler frequency vary by up to [FORMULA] and [FORMULA], respectively.

The use of an isothermal, instead of an isentropic, EOS in the envelope of the HNS models [FORMULA] and [FORMULA] has only small effects on the properties of the HNS's (see Table 3). If thermal effects in the envelope are however neglected (models [FORMULA] and [FORMULA]), the circumferential radius is reduced by [FORMULA]. This yields to an increase of the Kepler frequency by [FORMULA]. Though the assumption of zero temperature in the envelope does not change the resulting mass and central densities, the error made in the circumferential radius and in the Kepler frequency might be rather large.

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Online publication: October 4, 1999
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