A&A 383, 524-532 (2002)

DOI: 10.1051/0004-6361:20011758

**Sudip Bhattacharyya**

Joint Astronomy Program, Indian Institute of Science,
Bangalore 560012, India

Indian Institute of Astrophysics,
Bangalore 560034, India

Received 9 July 2001 / Accepted 28 November 2001

**Abstract**

We calculate the accretion disc
temperature profiles, disc luminosities and
boundary layer luminosities for rapidly rotating neutron stars considering the
full effect of general relativity. We compare the theoretical values of
these quantities with their values inferred from *EXOSAT* data for
four low mass X-ray binary sources: XB 1820-30, GX 17+2, GX 9+1 and GX 349+2
and constrain the values of several properties of these sources. According
to our calculations, the neutron stars in GX 9+1 and GX 349+2 are rapidly
rotating and stiffer equations of state are unfavoured.

**Key words: **accretion, accretion discs - relativity -
stars: neutron - stars: rotation - X-rays: binaries

A low mass X-ray binary (LMXB) is believed to contain either a weakly magnetised
neutron star or a black hole as the central accretor. The X-ray emission
arises from the innermost region of the accretion disc around the compact
star. In the case of a neutron star, there is an additional X-ray component
coming from the boundary layer of the star. Mitsuda et al. (1984) showed
that the spectrum of a luminous LMXB can be fitted by the sum of a single
temperature blackbody spectrum (believed to come from the boundary layer)
and a multicolour blackbody spectrum (may originate from the accretion
disc). However these authors used Newtonian models to fit the observed
spectra. Near the surface of a neutron star,
the accretion flow is expected to be governed by the laws of
general relativity due to the presence of strong gravity. Therefore
general relativistic models should be used
to get the correct best-fit values of the parameters.
Besides, the principal motivation behind the study of the spectral and
temporal behaviours of neutron star LMXBs is to understand the properties
of very high 10^{15} g cm^{-3}) density matter at the neutron star
core (van der Klis 2000).
This is a fundamental problem of physics, which cannot be addressed
by any kind of laboratory experiment. The only way to answer this question
is to assume an equation of state (EOS) model for the neutron star core, to
calculate the structure parameters of the neutron star and hence to
calculate an appropriate spectral model. By fitting such models (for different
chosen EOSs) to the observed data, one can hope to constrain the existing
EOS models and hence to understand the properties of high density matter.
However, general relativistic calculation is essential to calculate the
structure parameters of a neutron star and therefore to constrain the EOS
models.

It is expected that the neutron stars in LMXBs are rapidly rotating due to accretion-induced angular momentum transfer. LMXBs are thought to be the progenitors of milli-second (ms) radio pulsars (Bhattacharya & van den Heuvel 1991) like PSR 1937+21 with ms (Backer et al. 1982). The recent discovery of ms ms) X-ray pulsations in XTE J1808-369 (Wijnands & van der Klis 1998) has strengthened this hypothesis. Therefore it is necessary to calculate the structure of a rotating neutron star considering the full effect of general relativity. This was done by Cook et al. (1994) and the same procedure was used by Thampan & Datta (1998), to calculate the luminosities of the disc and the boundary layer.

The disc temperature profile for a rapidly rotating neutron star was first
calculated by Bhattacharyya et al. (2000). These authors also compared their
theoretical results with the *EXOSAT* data (analysed by White et al. 1988) to constrain different properties of the LMXB source Cygnus X-2.
The present work is a continuation of theirs, in which we constrain several
properties of four LMXB sources: XB 1820-30, GX 17+2, GX 9+1 and GX 349+2,
using the same procedure.
These sources were also observed by *EXOSAT* and the data were analysed
by White et al. (1988).

XB 1820-30 is an atoll source which shows type I X-ray bursts. GX 17+2 and
GX 349+2 are Z sources, of which the former shows X-ray bursts. GX 9+1 is an
atoll source. As all of them are LMXBs (van Paradijs 1995), the magnetic
field of the neutron stars are believed to be decayed to lower values
10^{8} G; see Bhattacharya & Datta 1996 and Bhattacharya &
van den Heuvel 1991). Therefore, we ignore the effect of the
magnetic field on the accretion disc structure in our calculations.

In Sect. 2, we give the formalism of the work. We present the results and discussion in Sect. 3 and give a summary in Sect. 4.

In order to calculate the disc temperature profile, the disc luminosity
and the boundary layer luminosity for a rapidly rotating neutron star
considering the full effect of general relativity, we need to compute the
structure of the rotating star. To do this, following Cook et al. (1994),
we choose a stationary, axisymmetric, asymptotically flat and
reflection-symmetric (about the equatorial plane) metric,
given by

where the metric coefficients and the angular speed () of zero-angular-momentum-observer (ZAMO) with respect to infinity, are all functions of the quasi-isotropic radial coordinate () and polar angle (). The quantity is related to the Schwarzschild-like radial coordinate (

With the
assumption that the star is rigidly rotating and a perfect
fluid, we solve Einstein's field equations and the equation of
hydrostatic equilibrium (the equations are given in the
appendix) self-consistently and numerically
from the centre of the star upto infinity
to obtain the metric coefficients and
(angular speed of neutron star with respect to infinity)
as functions of
and .
The inputs of this
calculation are a chosen EOS and assumed values of the central density and
the ratio of polar to equatorial radii. The outputs are bulk structure
parameters, such as gravitational mass (*M*),
equatorial radius (*R*), angular momentum (*J*), moment of inertia
(*I*) etc. of the neutron star. We can also calculate
the specific disc luminosity (),
the specific boundary layer luminosity (
),
the radius (
)
of the innermost stable circular orbit (ISCO) and specific
energy (), specific angular momentum ()
& angular
speed (
)
of a test particle in a Keplerian orbit (see
Thampan & Datta 1998
for a description of the method of calculation).

Then we calculate the effective temperature of a geometrically
thin blackbody disc, which is given by

where is the Stefan-Boltzmann constant and

(3) |

where

Here is the disc inner edge radius and a comma followed by a variable as subscript to a quantity represents a derivative of the quantity with respect to the variable. The values for and are given by the two conditions (circularity and extremum) for orbits (see Thampan & Datta 1998; Bhattacharyya et al. 2000):

= | (5) | ||

= | (6) |

where is the physical velocity of the matter. The equation of motion in the azimuthal direction and that in the time direction yield the Keplerian angular speed as

(7) |

Equation (4), that is exactly valid for a black hole accretor, is also valid for a neutron star accretor, if the disc does not touch the star. But, for a disc that extends up to the surface of the neutron star, the torque at the disc inner edge will not vanish, and Eq. (4) will not be strictly valid for such a case. However, for a very rapidly rotating neutron star, the angular speed of a particle at the disc inner edge will be close to the spin rate of the star, and hence the torque is expected to be negligible. According to Bhattacharyya et al. (2000), even when the spin rate is not large, Eq. (4) should give approximately correct results. This is because, Page & Thorne (1974) argued that the error in the calculation of will not be substantial outside a radial distance . In our calculation, we find that almost whole of the disc X-ray flux comes from well-outside this radial distance.

Equation (2) gives the effective disc temperature
with respect to an observer comoving with the disc. For our purpose,
this expression of the temperature must be changed to that in the observer's
frame, taking into account the
gravitational redshift and the rotational Doppler effect.
In order to keep our analysis tractable, we use
the expression given in Hanawa (1989) for this modification:

(8) |

With this correction for (1+

where the superscript "max'' denotes the maximum temperature value in the profile. is related to the colour temperature by , where

The structure of a neutron star for a given EOS is described
uniquely by two parameters: the gravitational mass (*M*) and
the angular speed (
). For each adopted EOS, we construct
constant *M* equilibrium sequences with
varying from the
non-rotating case (static limit) up to the centrifugal mass shed limit (rotation
rate at which inwardly directed gravitational forces are balanced
by outwardly directed centrifugal forces). So we are able to calculate
,
and
as functions of
*M* and
for a chosen EOS model.

The neutron star structure parameters are quite sensitive to the chosen EOS models. For the purpose of a general study, we have used the same four EOSs as considered in Bhattacharyya et al. (2000), namely, (A) Pandharipande (1971), (B) Baldo et al. (1997), (C) Walecka (1974) and (D) Sahu et al. (1993). EOS model (A) is for hyperonic matter. It is assumed that hyperonic potentials are similar to the nucleon-nucleon potentials, but altered suitably to represent the different isospin states. Model (B) is a microscopic EOS for asymmetric nuclear matter, derived from the Brueckner-Bethe-Goldstone many-body theory with explicit three-body terms. The three-body force parameters are adjusted to give a reasonable saturation point for nuclear matter. EOS model (C) corresponds to pure neutron matter and is based on a mean-field theory with exchange of scaler and (isoscalar) vector mesons representing the nuclear interaction. Model (D) is a field theoretical EOS for neutron-rich matter in beta equilibrium based on the chiral sigma model. The model includes an isoscalar vector field generated dynamically and reproduces the empirical values of the nuclear matter saturation density and binding energy and also the isospin symmetry coefficient for asymmetric nuclear matter. Of these, model (A) is soft, (B) is intermediate in stiffness and (C) & (D) are stiff EOSs, with (D) as the stiffest.

We choose four LMXB sources observed by *EXOSAT*
(data analysed by White et al. 1988).
For each source, we take the best-fit values of the parameters
,
and
,
where *L* denotes
the luminosity. On the other hand, for the chosen values of *f*,
the accretion rate ()
and *M*, we theoretically
calculate the values of
,
and
as functions of
,
for an
assumed EOS model.
Then comparing these theoretical values with the observed ones,
we constrain different parameters of the chosen source
(see Bhattacharyya et al. 2000 for detailed description).
However, to take into account
the uncertainties in the fitting procedure and in the value of *z*, and
also those arising due to the simplicity of the model, we consider a range of
acceptable
values for
,
and
.
We take two combinations of deviations around the best-fit values, namely,
(,
)
and (,
), where the first number in parentheses
corresponds to the error in
and the second to
the error in the best-fit luminosities.

In this paper, we calculate gravitational mass sequences for different
EOS models and constrain several properties of four LMXB sources.
For the neutron star in each of the sources, we assume
*M* = 1.4
(i.e., the canonical mass value for neutron stars).
We take two values for
(*i* is the inclination angle of the source)
for each source, namely, 0.2 and 0.8. These two widely
different values ensure the sufficient generality of our results.
For the four sources, the best-fit values (White et al. 1988) of the parameters
,
and
are
given in Table 1.

Source name | ||||

(10^{7} K) |
(10^{38} ergs s^{-1}) |
(10^{38} ergs s^{-1}) |
||

XB 1820-30 | 1.59 | 1.49 | 0.37 | 0.26 |

GX 17+2 | 1.76 | 6.49 | 1.62 | 0.71 |

GX 9+1 | 2.25 | 6.01 | 1.50 | 0.25 |

GX 349+2 | 2.07 | 8.54 | 2.14 | 0.48 |

We take the distance (*D*) of the source XB 1820-30 as 6.4 kpc
(Bloser et al. 2000). We assume *D* = 8 kpc for both GX 17+2 and
GX 9+1, as their locations are believed to be near the galactic centre
(Deutsch et al. 1999; Hertz et al. 1990) and distance of the galactic
centre is
kpc, as concluded by McNamara et al. (2000).
For GX 349+2, we take *D* = 9 kpc (Deutsch et al. 1999).

We display the constrained values with the help of four tables.
It is to be noted that here
is presented
in unit of
g s^{-1}. The
Eddington rate is
,
with
.
Therefore, as the actual value of
is much lesser than
1.0 (generally not greater than 0.3 and for rapidly rotating neutron star,
typically less than 0.2), the value of Eddington accretion rate is
much higher than
.
For all the sources, as the stiffness of the EOS models increases,
the absolute values of the allowed spin frequencies
and rotational
frequencies in the ISCO
decreases. This is because, for a
stiffer EOS model, neutron star radius is higher and it can support lesser
amount of rotation. The energy conversion efficiency is also lesser
for a stiffer EOS model (as the neutron star for this case is less compact)
and therefore higher accretion rate is needed to generate the observed
luminosity (as seen from the tables). In the following, we describe the
results for four sources in four subsections and give a general discussion
in the last subsection.

We display the allowed ranges of different parameters for the source
XB 1820-30 in Table 2.

EOS | f |
R |
||||||

kHz | kHz | km | km | |||||

(A) | 0.2 | L | 1.31[1.10] | 1.751[1.726] | 1.756[1.756] | 11.2[10.2] | 18.6[18.1] | 7.7[3.8] |

U | 1.91[2.41] | 1.755[1.755] | 1.819[2.078] | 11.4[11.4] | 18.7[18.7] | 16.1[20.3] | ||

(B) | 0.2 | L | 1.45[1.30] | 1.103[1.059] | 1.137[1.126] | 15.0[13.7] | 23.0[21.6] | 9.7[4.5] |

U | 2.07[2.61] | 1.112[1.113] | 1.197[1.372] | 15.5[15.6] | 23.5[23.6] | 19.4[24.4] | ||

(C) | 0.2 | L | 1.49[1.30] | 0.961[0.913] | 0.979[0.973] | 16.5[15.0] | 24.8[23.1] | 10.4[4.9] |

U | 2.12[2.71] | 0.968[0.968] | 1.042[1.206] | 17.2[17.2] | 25.5[25.6] | 21.2[26.7] | ||

(D) | 0.2 | L | 1.59[1.40] | 0.735[0.687] | 0.748[0.743] | 19.9[17.7] | 29.1[26.5] | 12.2[5.6] |

U | 2.25[2.81] | 0.740[0.740] | 0.795[0.941] | 20.6[20.7] | 30.0[30.1] | 25.0[31.4] | ||

(A) | 0.8 | L | 1.79[1.50] | 1.463[0.000] | 1.822[1.571] | 9.9[7.5] | 18.1[18.1] | 1.2[0.5] |

U | 3.06[4.70] | 1.751[1.754] | 2.165[2.165] | 11.2[11.4] | 20.4[22.3] | 4.5[6.4] | ||

(B) | 0.8 | L | 1.94[1.71] | 0.498[0.000] | 1.207[1.152] | 11.3[11.0] | 20.2[20.2] | 1.4[0.9] |

U | 3.22[4.20] | 1.102[1.110] | 1.782[1.782] | 14.9[15.4] | 22.9[23.4] | 5.6[7.9] | ||

(C) | 0.8 | L | 1.99[1.70] | 0.175[0.000] | 1.046[0.991] | 12.3[12.3] | 21.5[21.0] | 1.5[1.0] |

U | 3.36[4.02] | 0.960[0.966] | 1.573[1.573] | 16.5[17.0] | 24.7[25.4] | 6.1[8.5] | ||

(D) | 0.8 | L | 2.11[1.80] | 0.000[0.000] | 0.806[0.758] | 14.7[14.7] | 23.1[23.1] | 1.8[1.2] |

U | 3.30[3.90] | 0.733[0.739] | 1.212[1.212] | 19.7[20.5] | 28.9[29.8] | 7.2[9.9] |

It is seen that for , the spin frequency () of the neutron star comes out to be very high. But in the case of , it is not possible to constrain for (20%, 50%) uncertainty set (for all EOS models) and for both the uncertainty sets (for EOS model D). The ranges of the colour factor are in general consistent with the results of Shimura & Takahara and Borozdin et al. (1999) (

Table 3 shows the results for the source GX 17+2.

EOS | f |
R |
||||||

kHz | kHz | km | km | |||||

(A) | 0.2 | L | 1.01[1.00] | 1.754[1.748] | 1.756[1.756] | 11.4[11.0] | 18.7[18.5] | 36.1[19.8] |

U | 1.43[1.82] | 1.755[1.755] | 1.773[1.869] | 11.4[11.4] | 18.7[18.7] | 68.7[82.6] | ||

(B) | 0.2 | L | 1.12[1.00] | 1.108[1.097] | 1.128[1.122] | 15.3[14.7] | 23.3[22.6] | 44.4[24.4] |

U | 1.57[1.96] | 1.113[1.113] | 1.163[1.236] | 15.6[15.7] | 23.6[23.6] | 82.6[101.7] | ||

(C) | 0.2 | L | 1.15[1.00] | 0.966[0.954] | 0.974[0.971] | 16.9[16.0] | 25.2[24.2] | 47.5[26.1] |

U | 1.62[2.01] | 0.968[0.968] | 1.000[1.091] | 17.2[17.2] | 25.5[25.6] | 88.5[111.5] | ||

(D) | 0.2 | L | 1.22[1.03] | 0.738[0.728] | 0.744[0.742] | 20.3[19.2] | 29.7[28.3] | 55.9[30.7] |

U | 1.72[2.13] | 0.740[0.740] | 0.766[0.849] | 20.7[20.7] | 30.1[30.1] | 106.4[131.0] | ||

(A) | 0.8 | L | 1.39[1.20] | 1.702[0.000] | 1.782[1.571] | 9.9[7.5] | 18.1[18.1] | 6.6[2.0] |

U | 2.20[3.72] | 1.754[1.755] | 2.166[2.166] | 11.3[11.4] | 18.6[22.3] | 18.5[25.5] | ||

(B) | 0.8 | L | 1.53[1.31] | 1.009[0.000] | 1.172[1.141] | 13.1[11.0] | 21.1[20.2] | 7.7[3.2] |

U | 2.31[3.35] | 1.107[1.111] | 1.463[1.782] | 15.2[15.5] | 23.2[23.5] | 22.8[30.7] | ||

(C) | 0.8 | L | 1.61[1.30] | 0.858[0.000] | 1.010[0.983] | 14.3[12.3] | 22.4[21.0] | 8.3[3.6] |

U | 2.33[3.20] | 0.965[0.967] | 1.289[1.568] | 16.8[17.1] | 25.1[25.4] | 25.0[32.9] | ||

(D) | 0.8 | L | 1.66[1.41] | 0.631[0.000] | 0.775[0.750] | 16.8[14.7] | 25.4[23.1] | 9.5[4.4] |

U | 2.47[3.10] | 0.737[0.740] | 1.011[1.212] | 20.2[20.6] | 29.5[30.0] | 29.3[38.6] |

Here the ranges of are similar to those for XB 1820-30. But for GX 17+2, the value of

The results for the source GX 9+1 are given in Table 4.

EOS | f |
R |
||||||

kHz | kHz | km | km | |||||

(A) | 0.2 | L | 1.33[1.13] | 1.755[1.755] | 1.756[1.756] | 11.4[11.4] | 18.7[18.7] | 36.9[22.8] |

U | 1.85[2.25] | 1.755[1.755] | 1.756[1.761] | 11.4[11.4] | 18.7[18.7] | 59.9[72.0] | ||

(B) | 0.2 | L | 1.47[1.24] | 1.112[1.110] | 1.120[1.117] | 15.6[15.4] | 23.6[23.4] | 43.4[27.4] |

U | 2.04[2.49] | 1.114[1.114] | 1.130[1.147] | 15.7[15.7] | 23.6[23.7] | 73.6[90.6] | ||

(C) | 0.2 | L | 1.51[1.28] | 0.968[0.967] | 0.970[0.970] | 17.2[17.1] | 25.5[25.4] | 47.5[29.3] |

U | 2.09[2.56] | 0.968[0.968] | 0.975[0.986] | 17.3[17.3] | 25.6[25.6] | 80.7[99.3] | ||

(D) | 0.2 | L | 1.61[1.36] | 0.740[0.739] | 0.742[0.741] | 20.7[20.5] | 30.1[29.9] | 55.9[34.4] |

U | 2.24[2.74] | 0.740[0.740] | 0.745[0.754] | 20.7[20.7] | 30.1[30.1] | 94.9[116.7] | ||

(A) | 0.8 | L | 1.84[1.61] | 1.752[1.728] | 1.756[1.756] | 11.2[10.3] | 18.6[18.1] | 7.9[3.8] |

U | 2.69[3.52] | 1.755[1.755] | 1.815[2.064] | 11.4[11.4] | 18.7[18.7] | 16.5[20.3] | ||

(B) | 0.8 | L | 2.05[1.80] | 1.103[1.064] | 1.136[1.126] | 15.0[13.8] | 22.9[21.7] | 9.7[4.6] |

U | 2.92[3.72] | 1.112[1.113] | 1.200[1.361] | 15.5[15.6] | 23.5[23.6] | 19.4[24.4] | ||

(C) | 0.8 | L | 2.10[1.80] | 0.961[0.919] | 0.978[0.972] | 16.5[15.0] | 24.8[23.2] | 10.4[5.0] |

U | 3.00[3.81] | 0.968[0.968] | 1.041[1.195] | 17.2[17.2] | 25.5[25.6] | 21.2[26.7] | ||

(D) | 0.8 | L | 2.24[1.92] | 0.734[0.692] | 0.748[0.743] | 19.8[17.8] | 29.0[26.7] | 12.2[5.7] |

U | 3.19[4.00] | 0.740[0.740] | 0.802[0.932] | 20.6[20.7] | 30.0[30.1] | 25.0[31.4] |

For this source, -value comes out to be very high for all the EOS models and for both the -values. Here, the allowed values for

The allowed ranges of different parameters for the source GX 349+2 are
given in Table 5.

EOS | f |
R |
||||||

kHz | kHz | km | km | |||||

(A) | 0.2 | L | 1.12[1.00] | 1.755[1.754] | 1.756[1.756] | 11.4[11.4] | 18.7[18.7] | 50.9[30.7] |

U | 1.55[1.92] | 1.755[1.755] | 1.756[1.771] | 11.4[11.4] | 18.7[18.7] | 86.4[103.9] | ||

(B) | 0.2 | L | 1.24[1.04] | 1.112[1.109] | 1.122[1.119] | 15.5[15.3] | 23.5[23.3] | 61.2[37.7] |

U | 1.72[2.11] | 1.113[1.114] | 1.137[1.159] | 15.7[15.7] | 23.6[23.7] | 106.3[130.8] | ||

(C) | 0.2 | L | 1.27[1.08] | 0.968[0.966] | 0.971[0.970] | 17.2[17.0] | 25.5[25.3] | 67.1[40.4] |

U | 1.76[2.17] | 0.968[0.968] | 0.979[1.000] | 17.2[17.3] | 25.6[25.6] | 116.6[140.1] | ||

(D) | 0.2 | L | 1.35[1.14] | 0.740[0.738] | 0.742[0.741] | 20.6[20.4] | 30.0[29.7] | 78.8[47.5] |

U | 1.88[2.31] | 0.740[0.740] | 0.748[0.765] | 20.7[20.7] | 30.1[30.1] | 136.9[168.5] | ||

(A) | 0.8 | L | 1.55[1.34] | 1.748[1.678] | 1.760[1.756] | 11.0[9.7] | 18.4[18.1] | 10.6[4.5] |

U | 2.29[3.10] | 1.755[1.755] | 1.879[2.147] | 11.4[11.4] | 18.7[18.7] | 23.3[30.0] | ||

(B) | 0.8 | L | 1.71[1.50] | 1.096[0.955] | 1.148[1.129] | 14.6[12.7] | 22.6[20.7] | 13.1[5.2] |

U | 2.47[3.22] | 1.110[1.112] | 1.244[1.532] | 15.4[15.6] | 23.4[23.6] | 28.0[36.0] | ||

(C) | 0.8 | L | 1.76[1.53] | 0.953[0.798] | 0.984[0.974] | 16.0[13.8] | 24.2[21.9] | 14.0[5.6] |

U | 2.54[3.30] | 0.967[0.968] | 1.093[1.350] | 17.1[17.2] | 25.4[25.5] | 30.7[38.6] | ||

(D) | 0.8 | L | 1.87[1.61] | 0.727[0.557] | 0.753[0.744] | 19.1[16.2] | 28.2[24.6] | 16.5[6.3] |

U | 2.69[3.40] | 0.739[0.740] | 0.845[1.069] | 20.6[20.7] | 29.9[30.1] | 36.0[46.4] |

As is the case for GX 9+1, here also the value of comes out to be very high for all the chosen EOS models and -values. The allowed values for

Here we have constrained the values of several properties of four LMXB sources. For all of them, the accretion rates come out to be very high (always 0.5 . This is in accord with the fact that these are very luminous sources.

The rotation rate of neutron star in each of the sources is very high (close to the mass shed limit) for . This is because, the values of are very low for these cases (see Thampan & Datta 1998; Bhattacharyya et al. 2000). But, for , rotation rate cannot be constrained effectively for the sources XB 1820-30 and GX 17+2. Therefore, for these two sources, no general conclusion (about the values of can be drawn. However, the allowed ranges (combined for all the cases considered in a table) of are 0.93-1.00 and 0.75-1.00 for the other two sources GX 9+1 and GX 349+2 respectively (here is the at the mass shed limit; see Bhattacharyya et al. 2000 for the mass shed limit values). Therefore the neutron stars in these two sources can be concluded to be rapidly rotating in general.

Our calculated allowed ranges for *f* are in accord with the results
obtained by Shimura & Takahara (1995). However,
if we take the value *f* = 2.6 (obtained by Borozdin et al. 1999),
one would require a very stiff EOS model or a mass greater than
*M* = 1.4
for most of the cases with
.

High frequency quasi-periodic-oscillations (kHz QPO) have been observed for three (XB 1820-30, GX 17+2 and GX 349+2) of the chosen sources. The observed maximum kHz QPO frequencies are 1.100 kHz (XB 1820-30), 1.080 kHz (GX 17+2) and 1.020 kHz (GX 349+2) (van der Klis 2000). Now, as pointed out in Bhattacharyya et al. (2000), the maximum possible frequency (i.e., the shortest time scale) of the system should be given by the rotational frequency in ISCO ; Col. 5 of the tables). Therefore, the stiffest EOS model D is unfavoured for for the source XB 1820-30, as the maximum value of (=0.941 kHz, Table 2) is less than the observed maximum kHz QPO frequency. For the same reason, EOS model D is unfavoured for for the source GX 17+2. It can also be seen from Table 3 that if we use only the narrower limits on the luminosities and colour temperature, EOS model D (for and EOS model C (for are unfavoured for the same source. Same is true for EOS model D for the source GX 349+2. As we also see from Table 5, EOS model C is unfavoured for for this source. Therefore, we may conclude that the stiffer EOS models are unfavoured by our results.

We have ignored the magnetic fields of the neutron stars in our
calculations. Therefore, the necessary condition for the validity of
our results is that the Alfvén radius
be less than the
radius of the inner edge of the disc. This condition will always
be valid if
holds. Here
is given by
(Shapiro & Teukolsky 1983),

(10) |

where

It is also to be noted that our results are valid for a thin blackbody disc. However, as the spectra of the sources were well-fitted by a multicolour blackbody (plus a blackbody, presumably coming from the boundary layer; White et al. 1988), the assumption of thin blackbody disc may be correct.

In this paper, we have constrained the values of two neutron star parameters (spin frequency and equatorial radius) for the four LMXBs: XB 1820-30, GX 17+2, GX 9+1 and GX 349+2. We have also calculated the allowed ranges of the colour factor (for accretion disc), rotational frequency of a particle in the ISCO, the radius where the effective disc temperature is maximum and the accretion rate for these sources. These have been done for a chosen mass 1.4 (canonical mass) of the neutron stars and two values of inclination angles . The whole work has been repeated for four EOS models (from very soft to very stiff).

We have drawn the following main conclusions from our study. A comparison between the kHz QPO frequencies (observed from the sources) and our calculated values of has shown that the stiffer EOS models are unfavoured. By the constraining procedure, we have got very high accretion rates for all the sources, which is in accordance with their high luminosities. The neutron stars in the sources GX 9+1 and GX 349+2 have been found to be very rapidly rotating, and those in the other two sources may also be rapidly rotating (although we can not say decisively). This is in accordance with the belief that LMXBs are the progenitors of millisecond pulsars. It also shows that while calculating the spectral models, taking the rapid rotation of neutron stars into account is very important.

It is difficult to constrain EOS models effectively with the
poor quality *EXOSAT* data. However, the present generation
X-ray satellites *Chandra* and *XMM* have much better
resolving power. For example, the resolving power of *Chandra
HETGS* is 60-1000 in the energy range 0.5-10.0 keV.
The future generation X-ray satellite *Constellation-X* will
have even better resolving power (upto 3000).
With the spectral data of these X-ray observatories, it may be
possible to constrain EOS models and other parameters of LMXBs
effectively. Therefore we propose that it is essential to compute
EOS-dependent general relativistic spectral models to utilise
these good quality data in a fruitful way.

We acknowledge Arun Thampan for providing us with the neutron star structure calculation code and for discussions. We thank Dipankar Bhattacharya for reading the manuscript and giving valuable suggestions. We also thank Ranjeev Misra and late Bhaskar Datta for discussions and Pijush Bhattacharjee for encouragement.

Here we give Einstein's field equations and the equation of hydrostatic
equilibrium that were solved for the computation of the structure of the
rapidly rotating neutron star.
For an axisymmetric and equatorial plane symmetric configuration, the
computational domain in spherical polar coordinates covers
and
.
For numerical convenience,
we make a change of variables (
and
)
given by
and
,
where
is the quasi-isotropic radial
coordinate of the
equator. It is easy to see that *s* and
vary in the range
and
and at the equator *s* = 0.5.

The four Einstein's equations (Cook et al. 1994) to solve are given below:

= | |||

(11) |

= | |||

(12) |

= | |||

(13) |

= | |||

(14) |

where are the Legendre polynomials, are the associated Legendre polynomials and is a function of through . The effective sources 's are defined as (Cook et al. 1994)

= | |||

(15) |

= | |||

(16) |

= | |||

(17) |

where and . Here the tilde over a variable represents the corresponding dimensionless quantity and the variables and

The equation of hydrostatic equilibrium for a barytropic fluid is

= | (18) |

where is the dimensionless specific enthalpy as a function of pressure and and are the dimensionless values of pressure, t-component of the four-velocity and the specific enthalpy at the pole. The quantity is the (dimensionless) central value of the angular speed, which on the rotation axis is constant and equal to its value at the pole. The quantity is obtained from an integrability condition on the equation of hydrostatic equilibrium. Choosing the form of this function fixes the rotation law for the matter. Following Komatsu et al. (1989), we set it to . Here

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