6 Alternative models

The absence of periodic variations and spectral features in the observed radiation imposes very stringent constraints on any model. The simplest way to produce a time constant flux would be to assume a uniform temperature distribution across the stellar surface. For a temperature of $kT^{\infty}_{\rm bb}=63$ eV the measured optical spectrum requires a blackbody radius $R_{\rm bb}^{\infty}\simeq12.3$ km. Consequently, the X-ray emissivity has to be below that of a blackbody by a substantial factor. Using the parameters derived from the Chandra data (cf. Fig. 3 right) we find this factor to be about 0.15 (it can be about 0.45 if one adopts the parameters given by the ROSAT data). This would mean that the radiating surface should have a high reflectivity as may be expected for a condensed matter surface (Lenzen & Trümper 1978; Brinkmann 1980). In this case the spectrum may be represented by a $\alpha_{\rm X}~\times$ $B_\nu$ dependence ($B_\nu$ is the Planck function), where $\alpha_{\rm X}$ is the absorption factor ( $\sim[1~-\rho_{\rm X}]$, with $\rho_{\rm X}$ being the reflection factor), which in the general case will be energy-dependent (Brinkmann 1980). We have tested this hypothesis by fitting the Chandra LETGS spectrum with a Planckian $B_\nu$ multiplied by an energy dependent absorption factor $\alpha_{\rm X}$ = $E^\beta$ where E is the photon energy. It turns out that the best fit yields $\beta=1.28\pm 0.30$, $kT^{\infty}_{\rm bb}=54\pm 2$ eV, and $n_H=(5.1\pm0.3)\times 10^{19}$ cm^-2. This indicates that at a 4 $\sigma$ level we find find deviations from a Planckian spectrum which may result from an energy dependent absorption factor. In this case the radius required from the optical spectrum is $R_{\rm bb}^{\infty}\simeq13.3~\alpha_{\rm opt}^{-1/2}~ (d/120~{\rm pc})$ km, where $\alpha_{\rm opt}\le1$is the absorption factor of the surface in the optical domain.

In conclusion, the two versions of our one-component model yield radii of $R^{\infty}=12.3$ km and 13.3 km, respectively. These are lower limits as the absorption factor $\alpha$ may be smaller than unity in the optical band. We note that the observed radii correspond to true NS radii of R>9.1 km and R>10.3 km (for a NS mass of $1.4~ M_\odot$), respectively, which are consistent with a soft equation of state. But, of course, the possible range of parameters allows stiff equations of state as well.

Two main conditions have to be fulfilled to make this model work. Firstly, the NS has to have a condensed matter surface, which requires a low temperature and a strong magnetic field. The former seems to be fulfilled in this case: with a temperature of kT=54-63 eV RX J1856 is the coldest one of all detected isolated NSs. But its magnetic field is still unknown. The second condition is that the condensed matter surface really exhibits the required high reflectivity ($\sim$0.55-0.85) in the X-ray domain. It remains to be seen whether a detailed analysis of the optical properties of magnetically condensed matter substantiates this hypothesis.

Note added in proof: Turolla et al. (2002) have submitted a paper to ApJ in which they treat the emissivity of a condensed matter surface from a theoretical point of view, using the method of Brinkman (1980).

Acknowledgements

The XMM-Newton and the Chandra LETG projects are supported by the Bundesministerium für Bildung und Forschung/Deutsches Zentrum für Luft- und Raumfahrt (BMBF/DLR) and the Max-Planck Society. We would also like to thank the anonymous referee for contructive comments.