New Test for Cooling Curves Population Synthesis of Close-by Neutron Stars S.B. Popov 1, H. Grigorian 2, R. Turolla 3, D. Blaschke 4 1 Sternberg Astronomical Institute 2 Yerevan State University; Rostock University 3 Padova University 4 Bielefeld University; BLTP JINR Dubna

Abstract We propose to use population synthesis studies as an independent approach to test the physics governing the star cooling. Theoretical Log N – Log S distributions depend on the assumed neutron star thermal evolution. We have computed distributions for several different cooling scenarios and found that comparison with the observed Log N – Log S of isolated neutron stars is effective in discriminating among cooling models. The Log N - Log S test appears capable to ideally complement the standard temperature vs. age test used up to now. Details can be found in astro-ph/

Introduction Significant progress in the understanding of NS thermal evolution has been made in recent years and cooling curves have been computed by several groups (see e.g. Kaminker et al. 2002; Tsuruta et al. 2002; Page et al. 2004; Blaschke et al and references therein). The present work is based on NS cooling calculations performed in the Nuclear medium cooling scenario (Blaschke et al. 2004) which differs from the other above mentioned approaches by a consistent inclusion of medium effects. Processes of internal heating (Tsuruta et al. 2002) are not included. Since the cooling history crucially depends on the assumed physical conditions inside the star, comparison with observations may rule out some models in favor of others. In order to exemplify this we shall vary assumptions on the nuclear pairing gaps, the relation between crust and surface temperature as well as presence or absence of pion condensation. 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. Here we suggest to use the Log N - Log S distribution of close-by 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 NSs thermal history that has already been developed by Popov et al. (2003, 2004) (hereafter Paper I and II) by including nuclear medium effects in the cooling code and by investigating the contribution of massive progenitors in the Gould Belt to the local NS population.

The T – t test The most natural plot. Perfect illustration. Does not care about uncertainties of the “third” parameters. Only internal uncertainties of the model are important. Uncertainties in t. Very often an age is determined by or as a SNR age. Uncertainties in T. Atmospheres etc. Comparison of calculations with observations. Temperature is not measured directly, it is always obtained as a fit. Non-uniform data set. Data samples of coolers are non- uniform. There are sources of different types. The sample is nor flux, neither volume limited. We want to underline one important advantage of this test in comparison with the Log N - Log S proposal: there are no additional theoretical uncertainties except the ones of the model itself. I.e. theoretical cooling curves do not depend on unknown (or poorly known) astrophysical parameters which are not directly connected with the model of cooling. Discussing contras we stress the reader's attention on the last one. The sample of “test sources” is a strongly selected one. The sample is not uniform in any sense. So these sources do not form any real population of NSs. That is why in our opinion it is necessary to apply in addition the Log N - Log S test. Obviously T-t test is the most natural one if it is necessary to compare results of thermal evolution calculations with observations. Still there are some well known drawbacks of this probe. Below we summarize them. Pro Contra

The Log N – Log S test There are many uncertainties of the scenario. - Initial distribution of progenitors. - Mass spectrum. - Emission properties There can be unknown correlations between parameters of synthesis. (For example mass-magnetic field due to fall- back; velocity-internal structure due to deconfinement). Finite sample. It is difficult to take into account statistical fluctuations if a data sample is small. “Hidden” population (unobserved objects). Uses uniform sample of objects. Takes into account “populational effects”. No unsertainties in properties of observed objects! Log N - Log S distribution is well known and widely used in astronomy. Here in our proposal the idea is to compute the distribution with a population synthesis model, and then compare it with the ROSAT data on close-by NSs. The most inevitable one is related to a low statistics: around us there are not too much bright coolers. Other problems can be solved at least in principle. It is possible to take into account detailed picture of the progenitors distribution. It is (hopefully) possible to know the mass spectrum and to apply different atmospheric models. Anyway at the moment there are serious uncertainties in the scenario and one has to have it in mind. However simultaneous usage of the two methods can help to avoid bias due to different internal uncertainties of each test. ProContra

The Population Synthesis Model the initial NS spatial distribution; the kick velocity distribution; the NS mass spectrum; the cooling curves; the surface emission; the interstellar absorption. The main physical ingredients which enter our population synthesis model are: The calculation of the NS spatial evolution as they move in the Galactic gravitational potential follows that presented in Papers I and II. The same treatment of the interstellar absorption is retained and the kick distribution is that proposed by Arzumanian et al. (2002). We do not account for atmospheric reprocessing of thermal radiation, and assume that the emitted spectrum is a pure blackbody. Although this is clearly an oversimplification, it is a reasonable starting assumption and will serve for our, mainly illustrative, purposes. A more detailed description of surface emission may be easily accommodated in our model later on. For the time being, we perform our calculations for eight different sets of cooling curves among those discussed in Blaschke et al.(2004) to which we refer for all details. This issue, together with the initial spatial distribution of NSs and their mass spectrum, is further discussed below.

Model I Cooling curves for eight masses Full line: R belt =300 pc, non-truncated mass spectrum. Dotted line: R belt =500 pc and truncated mass spectrum.

Model II Cooling curves for eight masses. Full line: R belt =300 pc, non-truncated mass spectrum. Dotted line: R belt =500 pc and truncated mass spectrum.

Model III Cooling curves for eight masses The dashed line refers to a calculation in which the full (non-truncated) mass spectrum was used and R belt was assumed to be 500 pc.

Mass spectrum We use eight mass bins centered at M/M sun = 1.1, 1.25, 1.32, 1.4, 1.48, 1.6, 1.7, According to the mass spectrum, each curve has a statistical weight of 31.75%, 25.75%, 11%, %, 0.875%, 1.125%, 0.75%, and 0.625%. For the truncated one the weights of the first two bins are replaced by 0 and 57.5% respectively. Details of calculations of the mass spectrum can be found in Paper I and II and in astro-ph/

List of models I. Cooling of NS configuration with superfluid nuclear matter with medium effects and with pion condensation. The gaps are taken from (Takatsuka, Tamagaki 2004), the neutron gap 3 P 2 is additionally suppressed. The T s -T in relation is given by a fit (see Blaschke et al. 2004). II. Cooling of NS configuration with superfluid nuclear matter with medium effects but without pion condensation. The gaps are taken from (Yakovlev et al. 2004), the neutron gap 3 P 2 is additionally suppressed. The T s -T in relation is given by the Tsuruta law. III. Cooling of NS configuration with superfluid nuclear matter with medium effects and with pion condensation. The gaps are taken from (Yakovlev et al. 2004), the neutron gap 3 P 2 is additionally suppressed. The T s -T in relation is given by a fit. We calculated the Log N – Log S distribution for 9 sets of cooling curves from Blaschke et al. (2004), which successfully passed the T – t test. Below we present just three models. Other plots can be found in astro-ph/

Conclusions Here we propose to use the Log N – Log S distribution of close-by isolated cooling neutron stars as an additional test for cooling curves. Starting with 16 sets calculated by Blaschke et al. (2004) we take 9 of them that successfully passed the T – t test and apply the Log N – Log S. We show (see astro-ph/ ) that among 9 sets only 3 passed the second test (two of them just marginally). We conclude that the Log N – Log S distribution can be used as an additional test for cooling curves of neutron stars.