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Log N -- Log S distribution as a new test for
cooling curves of neutron stars
S.B. Popov # , H. Grigorian +,## , R. Turolla # and D. Blaschke §,¶
# Universitetski 13, 119992 Moscow, Russia
+ D­18051 Rostock, Germany
## Alex Manoogian 1, 375025 Yerevan, Armenia
# Marzolo 8, 35131, Padova, Italy
§ D­33615 Bielefeld, Germany
¶ 141980, Dubna, Russia
Abstract. We propose to use population synthesis studies as an independent approach to test the
physics governing the star cooling. Theoretical LogN­LogS distributions depend on the assumed
neutron star thermal evolution. We have computed distributions for several different cooling scenar­
ios and found that comparison with the observed LogN­LogS of isolated neutron stars is effective
in discriminating among cooling models.
Keywords: Neutron stars, Thermal evolution, Population synthesis
PACS: 97.60.Jd
INTRODUCTION
The determination of the equation of state at or above nuclear densities is a long sought
goal in high energy astrophysics. In this respect direct observations of neutron stars
(NSs) are bound to provide invaluable insight into several key issues of fundamental
physics. For the time being X­ray observations of NSs are, and will still be for some time,
central in fostering our understanding of the star interior (see [1] for a review). Isolated
neutron stars which emit at X­ray energies as they cool are particularly promising in this
respect. Their thermal radiation, in fact, directly comes from the star surface, carrying
information on the physical conditions of the emitting matter, in particular on the star
surface temperature (see [2] for a review).
A customary way of testing predictions of cooling calculations is to construct a
temperature vs. age (T--t for short) plot for the largest sample of sources. Despite its wide
application and undisputed usefulness, this test has a number of limitations. In this paper
we suggest to use the LogN--LogS distribution of close­by NSs as an additional probe
for NS cooling models. The idea is based on the comparison of present observational
data with NS population synthesis calculations in which cooling curves are one of the
ingredients. Our approach extends the actual calculation of observational properties of
the population of close­by young NSs.

LOG N -- LOG S TEST
Here we briefly describe the LogN--LogS test of cooling curves of isolated NSs. Details
can be found in [3].
The LogN--LogS distribution is a widely used tool in many branches of astronomy.
One immediate advantage of the LogN--LogS test is that, at variance with the T--
t test, no degree of arbitrariness is introduced when observational data are analyzed:
both the fluxes and (of course) the number of sources are well measured. In addition,
this approach is a ``global'' one. In our scenario it would not be possible to explain
some particular sources by invoking slight changes in the cooling physics. Once the
parameters of the model other than those related to the cooling process are fixed, a
particular cooling curve either fits the population as a whole or not. Furthermore, the
LogN--LogS sample is a uniform one, i.e. objects are flux (and probably volume)
limited, and no strong selection criteria are introduced.
We run a population synthesis model (see [4] for a brief review of population synthesis
in astrophysics) of close­by young isolated NSs. A calculated LogN--LogS distribution
is compared with the data observed by ROSAT.
For illustrating the capabilities of the proposed approach, we applied it to nine sets of
cooling curves from [5]. The application of the LogN­LogS test ruled out at least six
out of nine investigated cooling models, resulting in just three models which were able
to pass both tests.
One of the most challenging questions for the application of the test suggested in this
paper is to use it next for a possible discrimination between purely hadronic compact star
cooling scenarios and hybrid ones for stars having a color superconducting quark core
which have already successfully passed the T--t test. Our conclusion is that the LogN--
LogS may therefore become a powerful strategy in uncovering the properties of dense
nuclear matter under the extreme conditions in neutron star interiors. The LogN--LogS
test is mostly effective when used together with the standard T--t test.
ACKNOWLEDGMENTS
S.P. thanks the Organizers for support and hospitality. The work was supported by the
RFBR grant 04­02­16720 and by the ``Dynasty'' Foundation (Russia).
REFERENCES
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R. Maiolino, F. Mannucci, World Scientific Publishing, Singapore, p. 319 (astro­ph/0305435)
2. D. Page et al., ApJS, 155, 623­650, 2004
3. S.B. Popov et al., A&A, 2005 (in press, astro­ph/0411618)
4. S.B. Popov and M.E. Prokhorov, ``Population synthesis in astrophsics'', in Hot points in Astrophysics
and Cosmology, edited by V. Belyaev et al., Joint Institute for Nuclear Research, Dubna, 2005 (in
press, astro­ph/0411792)
5. D. Blaschke et al., A&A, 424, 979­992, 2004