A&A 367, 582-587 (2001)
K. Strobel - M. K. Weigel
Sektion Physik, Ludwig-Maximilians Universität München, Am Coulombwall 1, 85748 Garching, Germany
Received 23 September 2000 / Accepted 28 November 2000
The minimum and maximum mass of protoneutron stars and neutron stars are investigated. The hot dense matter is described by relativistic (including hyperons) and non-relativistic equations of state. We show that the minimum mass (0.88-1.28) of a neutron star is determined by the earliest stage of its evolution and is nearly unaffected by the presence of hyperons. The maximum mass of a neutron star is limited by the protoneutron star or hot neutron star stage. Further we find that the delayed collapse of a neutron star into a black hole during deleptonization is not only possible for equations of state with softening components, as for instance, hyperons, meson condensates etc., but also for neutron stars with a pure nucleonic-leptonic equation of state.
Key words: stars: evolution - stars: neutron - dense matter - equation of state
It is expected that a neutron star (NS) is born as a result of the gravitational collapse of the iron core of a massive ( ) evolved star (e.g. Bethe 1990). Shortly after core bounce (some 10 ms) a hot, lepton rich NS, called protoneutron star (PNS), is formed. This PNS evolves in some minutes into a cold, lepton pure NS due to the loss of neutrinos (e.g. Burrows & Lattimer 1986; Keil & Janka 1995; Pons et al. 1999). During these first minutes the PNS can collapse delayed to a black hole (BH) if its initial mass is high enough, either due to the loss of neutrinos or due to post bounce accretion.
The aim of this work is to study the minimum and the maximum mass of NSs and the possibility of BH formation during the deleptonization period. Recently Gondek et al. (1997,1998), Goussard et al. (1998) and Strobel et al. (1999a) have calculated limits on the minimum mass of NSs, using equations of state (EOSs) including nucleons (n, p) and leptons. In this paper refinements to these approaches are given by calculating PNSs for a large sample of EOSs using a non-relativistic Thomas-Fermi model (Myers & Swiatecki 1996; Strobel et al. 1999b,a) and a relativistic Hartree model (Serot & Walecka 1986; Schäfer 1997) including hyperons. Limits on the maximum mass of NSs were studied by numerous authors (e.g. Takatsuka 1995; Bombaci 1996; Ellis et al. 1996; Prakash et al. 1997; Gondek et al. 1998) using different kinds of EOSs. All these investigations start with models of PNSs about 1-3 s after core bounce (see Sect. 2 for more details). In this paper the earliest stage of a PNS is taken additionally into account, which is reached about 100 ms after core bounce, to calculate the maximum mass of a NS (see Sect. 2 for more details).
As pointed out in the Introduction a special topic of this investigation is the more detailed incorporation of the earliest stage of the PNS, about 0.1 - 1 s after core bounce (Burrows & Lattimer 1986; Burrows et al. 1995; Keil et al. 1996; Pons et al. 1999). This early type PNS is characterized by a hot shocked envelope with an entropy per baryon of (in units of the Boltzmann constant, ) for baryon number densities , an unshocked core with 1-2 for densities , and a transition region between these densities (Burrows et al. 1995; Keil et al. 1996). For the description of this early stage of the PNS we use, in accordance with the investigations of Burrows & Lattimer (1988), Burrows et al. (1995) and Keil et al. (1996), baryon densities in the range of 0.002-0.02 fm-3 for and 0.1 fm-3 for . The entropy per baryon is chosen between 5-6 in the envelope and between 1-2 in the core, respectively. The investigation is performed for the early type PNS models with trapped neutrinos using a constant lepton number ( ) for densities above fm-3.
We assume that post bounce accretion onto the protoneutron star stops within the first 500ms after core bounce. The amount of accreted matter, and the time at which the accretion through the shock front stops are still open questions. The amount of accreted matter after core bounce ranges from during the first 10 ms or so to 0.001 to during the following 100 ms or so, but in most calculations significant accretion stops about 500 ms after core bounce (for a further discussion of this topic see e.g. Burrows & Lattimer 1988; Chevalier 1989; Herant et al. 1992; Burrows et al. 1995; Janka & Müller 1996; Mezzacappa et al. 1998; Fryer & Heger 2000).
About 1-3 s after core bounce the PNS has a nearly constant entropy per baryon ( ) and an approximately constant lepton number ( ) for densities n>6 10-4fm-3. The next stage in the lifetime of a NS is reached after about 10-30 s, where the hot NS ( ) is now transparent to neutrinos. During the following minutes this hot NS evolves into a cold NS. For a more detailed description of the evolution of PNSs see, for instance, Prakash et al. (1997) and Strobel et al. (1999a).
The EOSs used in this paper for the description of PNSs are: (i) A non-relativistic Thomas-Fermi model (TF) for finite temperatures developed by Strobel et al. (1999a,b). (ii) A relativistic Hartree model for finite temperatures developed by Schäfer (1997). The Hartree model includes the following hyperons and nucleonic isobars: (the Delta resonances are not included in the NL1 model).
The main properties of cold symmetric nuclear matter
of the different models are listed in
Table 1 (see Strobel et al. 1999a; Glendenning & Moszkowski 1991; Reinhard 1989).
The coupling constants for the Hartree
models (GM1, GM3, NL1) are listed in Table 2.
|Figure 1: Pressure versus baryon number density in the high density region for different EOSs. The solid curves correspond (from the bottom to the top) to the cold EOS, the EOS and the EOS of the TF model. The long dashed curves correspond to the cold EOS (lower) and the EOS (upper) of the NL1 model without hyperons (dot-dashed lines, hyperons included)|
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|Figure 2: Pressure versus baryon number density in the high density region for different EOSs. The dotted curves correspond to the cold EOS (lower) and the EOS (upper) of the GM1 model without hyperons (long dashed lines, hyperons included). The dot-dashed curves correspond to the cold EOS (lower) and the EOS (upper) of the GM3 model without hyperons (short dashed lines, hyperons included)|
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The NL1 coupling constants lead to the stiffest EOSs at high densities, since the high density behaviour is dominated by the strength of the -coupling. However the incompressibility is the smallest. In this respect one has to keep in mind that the incompressibility determines the stiffness only near saturation and for higher densities the second and third derivatives of the incompressibility are important for the stiffness (see for instance Ramschütz et al. 1990).
The minimum and maximum mass of non-rotating NSs for the
different EOSs and different
choices of the transition region between the hot shocked envelope
and the unshocked core of the PNS are given in Table 3.
It turned out that the minimum mass of a NS is determined by the
earliest stage of the lifetime of a PNS. The values for the
minimum gravitational (baryonic) mass,
lie in the range of
for the different models (see Table 3).
|TF||0.1 - 1||6.0||2.0||0.002||0.1||1.338||1.284||2.366||-||yes|
|0.1 - 1||6.0||1.0||0.002||0.1||0.949||0.878||2.371||-||yes|
|0.1 - 1||6.0||1.0||0.005||0.1||1.117||1.021||2.372||-||yes|
|0.1 - 1||5.0||1.0||0.01||0.1||1.020||0.940||2.372||-||yes|
|0.1 - 1||5.0||1.0||0.02||0.1||1.229||1.114||2.374||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||2.364||1.969||-|
|10 - 30||2.0||2.0||-||-||-||-||2.449||-||-|
|GM1||0.1 - 1||6.0||1.0||0.005||0.1||1.255||1.206||2.629||-||yes|
|0.1 - 1||5.0||1.0||0.005||0.1||1.019||0.988||2.628||-||yes|
|0.1 - 1||5.0||1.0||0.01||0.1||1.140||1.100||2.629||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||2.613||2.251||-|
|10 - 30||2.0||2.0||-||-||-||-||2.670||-||-|
|GM1||0.1 - 1||6.0||1.0||0.005||0.1||1.255||1.206||2.462||-||yes|
|0.1 - 1||5.0||1.0||0.005||0.1||1.019||0.987||2.461||-||yes|
|0.1 - 1||5.0||1.0||0.01||0.1||1.140||1.100||2.462||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||2.401||-||yes|
|10 - 30||2.0||2.0||-||-||-||-||2.304||2.015||-|
|GM3||0.1 - 1||6.0||1.0||0.005||0.1||1.255||1.205||2.204||1.939||no|
|0.1 - 1||5.0||1.0||0.005||0.1||1.039||1.005||2.203||1.938||no|
|0.1 - 1||5.0||1.0||0.01||0.1||1.145||1.104||2.204||1.939||no|
|1 - 3||2.0||2.0||-||-||-||-||2.212||-||-|
|10 - 30||2.0||2.0||-||-||-||-||2.249||-||-|
|GM3||0.1 - 1||6.0||1.0||0.005||0.1||1.255||1.205||2.047||-||yes|
|0.1 - 1||5.0||1.0||0.005||0.1||1.039||1.005||2.046||-||yes|
|0.1 - 1||5.0||1.0||0.01||0.1||1.145||1.104||2.047||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||2.015||-||yes|
|10 - 30||2.0||2.0||-||-||-||-||1.898||1.699||-|
|NL1||0.1 - 1||5.0||1.0||0.005||0.1||1.175||1.111||3.218||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||3.162||2.663||(yes)|
|10 - 30||2.0||2.0||-||-||-||-||3.202||(2.689)||-|
|NL1||0.1 - 1||5.0||1.0||0.005||0.1||1.169||1.104||3.017||-||yes|
|1 - 3||2.0||2.0||-||-||-||-||2.942||-||yes|
|10 - 30||2.0||2.0||-||-||-||-||2.871||2.399||-|
Let us now turn to the question, why there is a minimum mass in this early stage of the evolution. A star becomes dynamically unstable if its mean adiabatic index, , lies below 4/3. In the case of a cold NS this is caused by the neutron drip at a density of fm-3. For the earliest stage of the PNS the drop below is caused by the entropy drop between the hot shocked envelope and the unshocked core of the PNS. For a complete description of the stability criteria of neutron stars see Shapiro & Teukolsky (1983).
Quite interesting is the result for the maximum gravitational mass for
NSs, for which we obtain values in the range between
(see Table 3) for the different
EOSs used in this paper, with the possibility
of the formation of a BH during deleptonization for EOSs with a pure
nucleonic-leptonic composition and EOSs which include hyperons.
For instance, for the NL1 EOS, the maximum baryonic mass for the first
larger than the allowed maximum baryonic mass of 3.162
1-3 s, which implies the possibility of a BH formation
(see Table 3 and Fig. 3).
|Figure 3: Maximum gravitatinal versus maximum baryonic mass for different evolutionary stages of the NL1 EOSs. The two lines denote the mass window for the delayed collapse|
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Similar behaviour is found for the other EOSs with exception of the
nucleonic-leptonic EOS GM3, where the allowed maximum mass after
1-3 s exeeds the maximum masses of the first period.
The reason why this possibility of a BH formation was dismissed so far
for a pure nucleonic-leptonic composition (e.g. Takatsuka 1995; Bombaci 1996; Ellis et al. 1996; Prakash et al. 1997; Gondek et al. 1998) was that these authors start their calculations
about 1-3 s after core bounce, since the following stages
allow only larger baryonic masses.
One should mention in this context that for the nucleonic-leptonic
EOSs the differences in the maximum masses in the first seconds are more than
three times smaller (<0.056
than for the softer EOSs with
hyperons (0.146-0.158 ). Due to this,
for the EOSs including hyperons the possibility of a BH
formation shows up even by starting the calculation
1-3 s after core bounce (see Table 3).
It turned out, that the mass window for the delayed collapse
of the pure nucleonic-leptonic EOSs is an increasing function of the maximum
possible mass of the different models (see Fig. 4).
The EOSs including hyperons did not show a similar behaviour,
the mass window is nearly constant for all used EOSs.
|Figure 4: Baryonic mass difference between the early type PNS stage (0.1-1 s after core bounce) and the late type PNS stage (1-3 s after core bounce) versus resulting maximum baryonic mass for the different pure nucleonic-leptonic EOSs|
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It is an open question how large the maximum mass of a NS really is. It is clear that an EOS should allow NS masses larger than since measurements of pulsars in binary systems show up values around this mass (e.g. Thorsett & Chakrabarty 1999). If quasi periodic oscillations (e.g. van der Klis 2000) are taken into account, the possible maximum mass of an EOS has to be larger than (e.g. Schaab & Weigel 1999). This whould lead to the conclusion that only the GM1 and the NL1 EOSs (with and without hyperons) are stiff enough to allow NSs with a larger mass.
In this paper we presented a calculation of the minimum and maximum mass of a NS taking the early evolution, shortly after core bounce, into account. We found that the minimum mass of a NS is limited by the earliest stage of its evolution (0.1-1 s after core bounce). Therefore the minimum gravitational mass has values between 0.878 and 1.284 . These values lie in the same range as those found in recent works by Goussard et al. (1998) and Strobel et al. (1999a). Additionally we have examined the influence of hyperons on the minimum mass. As expected it turned out that the minimum mass is nearly unaffected by the inclusion of hyperons. Another possibility of NS production is the accretion induced collapse of a white dwarf (Canal & Schatzman 1976), but the numerical simulation of Woosley & Baron (1992) showed that the resulting NS will have a mass of 1.4 since the white dwarf collapses without significant mass loss after reaching the maximum mass of a white dwarf. From the results reported it is obvious that NSs, which are born in type-II supernovae or due to accretion induced collapse of a white dwarf, have a minimum mass about ten times larger than the possible minimum mass of a cold NS (0.1 ).
With respect to the maximum mass of a NS calculated by solving the Tolman-Oppenheimer-Volkoff equation for NS matter the outcome is rather uncertain, since the EOS in the high density region rests strongly on theoretical extrapolations. The underlying model, for instance, different many-body approximations and/or inclusion of more massive baryons, quarks, condensates etc., can render the results for the maximum masses significantly (e.g. Prakash et al. 1997; Huber et al. 1998). However, as a result of our work it turned out, that the maximum mass of a NS is limited by the first 30 s of its evolution. The maximum baryonic mass of a NS, constructed with pure nucleonic-leptonic EOSs and EOSs including hyperons, is always reached during this early epoch of the evolution.
Finally we found that BH formation during deleptonization is also possible for NSs with pure nucleonic-leptonic EOSs if the initial mass is high enough. This property of NSs was expected till now to be possible only for EOSs including softening ingredients, like hyperons, meson condensates or a quark-hadron phase transition (e.g. Brown & Bethe 1994; Glendenning 1995). There are numerous papers on this topic (e.g. Takatsuka 1995; Bombaci 1996; Ellis et al. 1996; Prakash et al. 1997; Gondek et al. 1998), but all these investigations started their calculations about 1-3 s after core bounce. In this paper we included the earliest stage and found that the maximum baryonic mass of the early type PNSs (0.1-1 s after core bounce) could be up to 0.056 larger than that of the late type PNS (1-3 s after core bounce). Due to this result it seems possible that a NS with a pure nucleonic-leptonic EOS could also undergo delayed collapse into a BH during the first seconds of its evolution. This question could only be finally decided in time evolution calculations like this including softening ingredients (e.g. Baumgarte et al. 1996), since we have used approximate values for the entropy per baryon and the lepton number at fixed times. Post bounce accretion (e.g. Chevalier 1989; Herant et al. 1992) could lead also to a delayed collapse, but it is uncertain till now how much matter falls through the shock front within the first second after core bounce, so that if a cut off of a neutrino signal after a few seconds is detected in the future, the two ways of BH formation could possibly be indistinguishable.
We want to thank A. Schäfer for providing us with his program. We want to thank H.-Th. Janka and Ch. Schaab for many helpful discussions. One of us, K. S., gratefully acknowledges the Bavarian State for financial support.