A&A 420, 987-991 (2004)
DOI: 10.1051/0004-6361:20034538
M. Bejger^{1} - P. Haensel^{1,2}
1 - N. Copernicus Astronomical
Center, Polish
Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
2 - LUTH du CNRS, Observatoire de Paris, 92195 Meudon Cedex, France
Received 20 October 2003 / Accepted 28 February 2004
Abstract
The upper bound on the value of the surface gravity,
,
for neutron stars with equations of state respecting
,
is derived. This bound is inversely
proportional to the maximum allowable mass
,
and it
reads
.
It implies an absolute
upper bound
if one uses the
lower bound on the neutron mass measured recently in 4U1700-37,
.
A correlation between
and
the compactness parameter 2GM/Rc^{2} for baryonic stars is
analyzed. The properties of
of strange quark stars
and its upper bounds are discussed using the scaling properties of
the strange-star models.
Key words: dense matter - equation of state - stars: neutron
The surface gravity of neutron stars is by many orders of magnitude larger than for other stars; it is 10^{5} times larger than for white dwarfs, and 10^{8} times stronger than the gravity at the solar surface. As we show in the present paper, at a given neutron-star mass the value of depends very strongly on the largely unknown equation of state (EOS) of dense matter at supra-nuclear densities. It is therefore of interest to derive upper bounds on . In Sect. 2 we derive such upper bounds on resulting from the condition of subluminality of the EOS (speed of sound not exceeding c). We show how an upper bound on can be obtained from measured neutron star masses. In Sect. 3 we analyze the EOS-dependence of for a set of 31 models of dense matter and compare the maximal surface gravity, reached at the maximum allowable mass, with subluminal upper bounds. In Sect. 4 we consider surface gravity of strange quark stars. We derive a scaling formula for of strange stars and use it to relate the maximum surface gravity for various models of such stars. Concluding remarks are presented in Sect. 5.
It is well known that (see, e.g., Shapiro & Teukolsky 1983)
Let us introduce the dimensionless compactness parameter
where
is the Schwarzschild
radius. Then
For subluminal EOSs of dense matter, a strict upper bound on x is
(see, e.g., Haensel et al. 1999,
and references therein). Using this value we get, from Eq. (2),
an upper bound for a surface gravity of non-rotating neutron star of mass M,
The expression on the right-hand-side of (4) deserves an additional comment. Apart from the numerical constant, it is identical to the upper bound on the frequency of stable rigid rotation of neutron stars with subluminal EOS (Koranda et al. 1997). An alternative derivation of an approximate (but very precise) upper bound on rotation frequency, relating it to the maximum stellar compactness (surface redshift) for subluminal EOS, was given by Haensel et al. (1999). In the latter work, the starting point was the "empirical formula'' for the maximum rotation frequency. Its upper bound was also obtained by maximizing a function of .
We have calculated for neutron-star models based on a set of 31 baryonic EOSs of dense matter. The EOSs were obtained under different assumptions regarding the composition of the matter at , where is the normal nuclear density.
The EOSs can be subdivided into several groups. Within one group
of models, the matter consists of
nucleons and leptons (Baldo et al. 1997;
Bombaci 1995; Balberg & Gal 1997; Balberg et al. 1999; model I
of Bethe & Johnson 1974 (usually called BJI); Pandharipande 1971;
Pandharipande & Ravenhall 1989; Douchin & Haensel 2001; Wiringa
et al. 1988; Walecka 1974; Haensel et al. 1980;
Akmal et al. 1998). Within the second group, the matter is assumed
to consist of nucleons, hyperons and leptons (Glendenning 1985;
Balberg & Gal 1997; Weber et al. 1991). The third group involves
an exotic high-density phase: de-confined quark matter mixed
with baryonic matter (Glendenning 1997), pion condensate
(Muto & Tatsumi 1990) and kaon condensate (Kubis 2001). Finally,
one EOS is the so called "maximally-stiff-core'' (MSC) EOS. It consists
of the BJI EOS below the baryon density
,
matched continuously to the EOS with
at higher density.
Figure 1: Surface gravity in the units of against neutron-star mass for 31 EOSs of dense baryonic matter. Only stable configurations are shown, so that the curves terminate at the maximum allowable mass. Thick dashed line: maximally-stiff-core EOS. Remaining EOSs: thin lines. For further explanation see the text. | |
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In Fig. 1 we show the surface gravity versus stellar (gravitational) mass for all 31 baryonic EOSs. The values of for a given mass strongly depend on the EOS. For , ranges from 1.1 to 2.8, while for the predicted values are between 1.4 and 4.7.
We considered also three EOSs of self-bound absolutely stable strange quark matter, forming hypothetical strange stars (Zdunik et al. 2000; Dey et al. 1998) - they will be studied separately in Sect. 4.
Contrary to the significant scatter of the
plots for
baryonic stars,
is much less EOS-dependent: it can
be rather well (within better than 20%) reproduced by the
approximate formula
The maximum value of for stable stars with a given EOS is reached at the maximum allowable mass. The values of for the selected EOSs are shown in Fig. 3. For comparison, we show also the absolute upper bounds, Eq. (3), at . The dense matter models were divided into groups denoted by specific symbols. Subluminal models involving only nucleons () give . The stiffer the EOS, the closer to the subluminal upper bound .
Subluminal hyperonic EOSs () give which is typically lower than . For these EOSs, can be as small as one-fifth of the upper bound .
The subluminal EOSs with an exotic high-density phase () have relatively low . A phase transition softens the EOS, lowering the radius. Simultaneously, however, the softening leads to a decrease of . The latter effect dominates over the former one. This strongly pushes up the upper bound . If the stellar interior consists mostly of a mixed quark-baryon phase (EOSs G6, G7, G8 in Fig. 3), then , only one fifth of .
The MSC EOS () yields which is quite close to . A similar situation occurs for the EOSs that give models with superluminal cores, labeled . Their values of range from to . Note that even for those EOSs . This is because the compactness parameter is always smaller than the upper bound for subluminal EOSs, , as is shown in Fig. 2.
Figure 2: Plots of versus compactness parameter 2GM/Rc^{2}. Notation as in Fig. 1. Thick solid line represents approximate formula, Eq. (5). | |
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Figure 3: Maximum values of surface gravity in units of (centers of the symbols at the bottom end of the vertical segments) for 34 baryonic and quark EOSs of dense matter, and the upper-bounds at the maximum allowable mass for these EOSs obtained from Eq. (3) (dashes at the upper end of the vertical segments). Nucleonic EOS: B1, B2 - Baldo et al. (1997); D - Bombaci (1995); E1, E3 Balberg & Gal (1997), Balberg et al. (1999); C - model I of Bethe & Johnson (1974); J - Pandharipande (1971); F - Pandharipande & Ravenhall (1989); H - Douchin & Haensel (2001); L1, L2, L3 - Wiringa et al. (1988); 0 - Walecka (1974); I - Haensel et al. (1980); A - Akmal et al. (1998). Hyperonic EOSs: G1-G5 - Glendenning (1985); E2, E4 - Balberg & Gal (1997), Balberg et al. (1999); M1, M2 - Weber et al. (1991). EOSs with exotic high-density phase: G6-G8 - Glendenning (1997); K - Kubis (2001); P1, P2 - Muto & Tatsumi (1990). Strange quark matter: S1, S2 - Zdunik et al. (2000); S3 - Dey et al. (1998). N labels the MSC (maximally-stiff-core) EOS. For further explanations see the text. | |
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Figure 4: Surface gravity in units of versus strange star mass ( upper panel) and compactness parameter ( lower panel), for three EOSs of strange quark matter. Dashed line in the lower panel is obtained by transforming (see text for details) the S2 EOS curve into S3 EOS curve using Eq. (7). | |
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EOSs of self-bound quark matter (strange
matter) are derived from various different models of the quark
structure of hadrons. Despite differences between the
underlying models, the EOSs relevant for stable models of strange
stars can be well represented (fitted) by a linear relation
between the pressure P and the mass density
(Zdunik
2000),
We have studied for a set of 31 EOSs of baryonic matter. The dependence of on the stellar mass M is very sensitive to the EOS. On the contrary, the dependence of on the stellar compactness has a generic character for baryonic EOSs. The maximum surface gravity is sensitive to the EOS of dense matter and ranges from about to for subluminal baryonic EOSs.
The dependence of on the mass and compactness of strange stars is very different from that of baryonic stars. However, the range of is quite similar to baryonic stars.
We hope that the EOS-sensitive features of will be useful in extracting information about the EOS of dense matter, for instance, by combining the values of obtained fitting the thermal component of the observed photon spectra with atmospheric models and the surface redshift measured for the identified spectral lines.
Acknowledgements
This work was motivated by a question of A. Majczyna concerning the possible range of surface gravity in neutron star atmospheres, which is gratefully acknowledged. We are grateful to D. G. Yakovlev for the reading of the manuscript and for helpful remarks. This work was partially supported by the KBN grants No. 2P03D.019.24 and 5P03D.020.20.