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Unfortunately, the non-thermal radiation from magnetospheres of neutron stars (§ 1.4.4) is usually so bright that the thermal radiation from their surfaces is hardly observable. In particular, this is so for young active pulsars like the Crab pulsar (age ~ 103 yr). The thermal radiation of middle-aged (age ~ 104-106 yr) and sufficiently warm neutron stars, with effective temperatures (0.3 — 1) x 106 K, can dominate in soft X-ray and UV ranges. Surfaces of old neutron stars (age > 106 yr) are too cold to emit observable thermal radiation, but polar caps of old active pulsars can be hot enough to be detected in X-rays.

That is why the thermal radiation from neutron star surface was detected only after the launch of X-ray satellites Einstein (1978-1981) and EXOSAT (1983-1986). First reliable spectra of the thermal radiation from several pulsars were obtained even later, with the sensitive X-ray observatory ROSAT (19901998) as reviewed, e.g., by Ogelman (1995) and Becker (1999). A new era in observing the thermal emission of neutron stars has started since 1999 with the launch of X-ray observatories of outstanding capabilities, Chandra and XMM-Newton (e.g., Pavlov et al. 2002).

Extracting neutron star parameters from these observations is a complicated task. It is not sufficient to obtain a good high-quality spectrum; its interpretation meets several serious difficulties. First, as a rule, a spectrum includes both thermal and non-thermal components, so that one has to carefully separate the thermal one. Second, theoretical interpretation of observations depends on the composition of a neutron star atmosphere, which is unknown a priori, so that one should try several possibilities. Third, even if the atmosphere composition is fixed, the model of the outgoing spectral flux depends on many parameters such as M, R, and the effective surface temperature Ts. For a strongly magnetized star, one should also take into account the effects of the magnetic field B and its geometry and an associated non-uniform temperature distribution over the stellar surface. For instance, the polar cap of a pulsar can be heated by a stream of electrons or positrons moving along open field lines from the magnetosphere. This is what happens in the so called inner gap model of pulsar emission, which stems from the seminal paper by Sturrock (1971). The temperature of "hot spots" produced by this heating on the stellar surface may be much higher than outside them. Fourth, the detected spectrum depends on the distance d to the star and the column density of interstellar hydrogen, which are usually poorly known. Therefore, it is not surprising that numerous attempts to determine neutron star radii from observations of thermal radiation have been largely inconclusive.

Generally, while fitting an observed spectrum, one infers possible values of Ro within some confidence interval. An assumed EOS of dense matter can be ruled out, if the theoretical curve Ro (M) does not intersect this interval. As a rule, these results should be taken with a grain of salt, because of difficulties in estimating the errorbars of Ro . The emission of active pulsars may be mainly produced by hot polar caps, rather than by entire colder surfaces. In these cases one will infer apolar cap radius (much smaller than R^) from observations. For instance, Zavlin etal. (2002), using a hydrogen atmosphere model, estimated the radius of the polar cap of the millisecond pulsar J0437—4715 as Rpc = 2.Ü+0 3 km. It can be taken as the observational lower bound to the stellar radius. Obviously, it is too low to be useful for constraining the EOS in neutron star cores.

However, there are several cooling (isolated) middle-aged neutron stars, which apparently show a thermal-like emission and whose distance has been determined from parallax measurements. The distance determination eliminates the largest uncertainty in constraining R^. These neutron stars are Geminga (PSR B0633+17), PSR B0656+14, Vela (PSR B0833—45), and RX J1856—3754, which we consider in the rest of this section.

Golden & Shearer (1999) used HST and BTA optical photometry of Geminga and PSR B0656+14 to estimate their radii. Previous X-ray observations had indicated that a significant part of soft X-ray flux from these neutron stars is of thermal origin. Golden & Shearer (1999) compared the sum of the Rayleigh-Jeans tail of the thermal blackbody flux (a v2, where v is the photon frequency) and the synchrotron-emission flux (a v-a with a = 1.9 for Geminga and 1.4 for PSR B0656+14) with the pulsed optical flux detected from these pulsars and with the upper limit to the non-pulsed optical flux. Assuming that the thermal emission powered by neutron star cooling is non-pulsed, they obtained the upper bound on the Rayleigh-Jeans parameter G = T^R^/d5oo)2, where Rf0 = R^/10 km, = T™/106 K is the effective surface temperature (in MK) as detected by a distant observer, and d50o = d/5ÜÜ pc. For Geminga, the restriction reads G < 3.9 (at the 3a level), while d = 159—pc from the parallax measurement (Caraveo et al., 1996). If one fixes d & 160 pc, then the blackbody fit of the X-ray spectrum gives T6° & 0.4-0.6 (Halpern et al., 1996), so that R^ < 10 km, below the absolute lower limits R^ > 12- 14 km for most realistic EOSs (§ 6.6.6). However, taking into account rather large errors in d, one obtains R^ < 18 km (Haensel, 2001), which is satisfied for any reasonable EOS and, therefore, is not useful for the EOS selection. This may indicate that d is actually larger than the assumed 160 pc. On the other hand, one cannot exclude that other, non-blackbody models (e.g., magnetized atmosphere models) yield noticeably larger values of R^.

The most comprehensive analysis of the data on PSR B0656+14 made in the pre-Chandra era was done by Edelstein et al. (2000). Chandra X-ray observations of PSR B0656+14 were analyzed by Marshall & Schulz (2002). Combined optical, X-ray, and UV data gave the best estimate T¡?° = 0.470.74 (associated with the soft spectrum component) and Rl^/dsoo = 1.4-4.0 at the 99% confidence level, with the most probable distance d & 200 pc (to be compared with the estimate d = 760 pc based on the dispersion measure,

Manchester et al. 2005). Later, Brisken et al. (2003) measured the parallax of this pulsar and obtained d = 288+2? pc. Analyzing previous R/d estimates, the latter authors concluded that, within the model uncertainties, any radius Ro is between ~ 13 and ~ 20 km, which is not useful for the EOS problem.

The parallax measurements ofthe Vela pulsar gave the distance d = 293+if pc (Caraveo et al., 2001; Dodson et al., 2003). The X-ray spectrum of this pulsar observed with Chandra (Pavlov et al., 2001) and XMM-Newton (Mori et al., 2004) can be fitted by a sum of non-thermal (power-law) and thermal (blackbody or hydrogen-atmosphere) components. The blackbody fit yields Tso & 1.5 x 106 K and Ro = 2.5 ± 0.2 km. It may describe the thermal radiation of a hot polar cap on the pulsar surface. On the other hand, a magnetized (B = 1012 G) hydrogen atmosphere model allows the neutron star to be "canonical" (see Eq. (1.6)), yielding the temperature To = 0.674-003? x 106 K and the distance dflt = 256+44 pc (Mori et al., 2004). Using the approximate scaling Ro a d and the parallax distance d, the canonical Ro = 13 km can be translated into

Another nearby neutron star with the thermal-like spectrum is RX J1856-3754, discovered in the ROSAT observations by Walter et al. (1996) and extensively studied in subsequent years. A faint optical counterpart of this star was found by Walter & Matthews (1997). The distance determined from the parallax measurements is d = 117 ± 12 pc (Walter & Lattimer, 2002). One-component blackbody and atmosphere models failed to explain the spectrum in both optical and X-ray ranges. The X-ray spectrum is well fitted by the blackbody spectrum with T6°° = 0.73 and R°§ = 0.44 (d/120 pc), whereas the optical data obey the Rayleigh-Jeans law with the flux ~ 7 times higher than that obtained by extrapolating the X-ray spectrum into the optical range (e.g., Burwitz et al. 2003 and references therein). This may indicate the presence of a softer thermal component, produced by the emission from the entire surface, while the "hard" component may be emitted from a hot polar cap. The constraints obtained by Burwitz et al. (2003) for the "soft" component read 4 eV < kBTo < 33.6 eV (i.e., 0.046 < T6°° < 0.39) and 1.63 < R°°0 < 4.6 (at d = 120 pc). However, the hot polar cap model implies pulsations of radiation from a rotating star. Many attempt to find these pulsations failed. Burwitz et al. (2003) proposed an alternative model, according to which the radiation comes from a solid or liquid surface (rather than a gaseous atmosphere), which emits the blackbody radiation but with reduced emissivity in the X-rays. This yields the lower bound Ro > 12.3 km at the 4a level consistent with the majority ofthe EOSs of dense matter (the upper bound is uncertain because of an unknown reflectivity in the optical).

The latter result stimulated extensive studies of the emissivity and spectrum of magnetized neutron stars with a condensed surface (van Adelsberg et al. 2005, and references therein). Simultaneously, the first self-consistent models of partially ionized hydrogen neutron star atmospheres with strong magnetic fields have been constructed (Ho et al., 2003; Potekhin et al., 2004).

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