Observational constraints of the compactness of isolated neutron stars

We report on our observational attempt to constrain the compactness of the isolated neutron stars via X-ray spin phase-resolved spectroscopy. There are seven thermally emitting neutron stars known from X-ray and optical observations, which are young (up to few Myrs), nearby (hundreds of pc), and radio-quiet with blackbody-like X-ray spectra. A model with a condensed iron surface and partially ionized hydrogen-thin atmosphere allows us to fit simultaneously the observed general spectral shape and the broad absorption feature (observed at 0.3 keV) in different spin phases. We constrain a number of physical properties of the X-ray emitting areas, including their temperatures, magnetic field strengths at the poles, and their distribution parameters. In addition, we place some constraints on the geometry of the emerging X-ray emission and the gravitational redshift of three isolated neutron stars.


Introduction: Thermally Emitting Isolated Neutron Stars (INSs)
The studies of INSs can provide an important input in our understanding of neutron stars and can put stringent constraints on the possible form of the equation of state of superdense matter. Observations and modeling of the thermal emission by INSs allow us to infer their surface temperatures and their total fluxes measured by a distant observer, as well as to estimate their real parameters, such as their apparent radii, provided the distances to the INSs are known accurately. On the other hand, the detection and identification of any absorption or emission feature in the spectrum or a rotational phase-resolved spectroscopy of INSs allow us to determine gravitational redshift, which in turn permits us to determine directly their mass-to-radius M/R ratio (see, e.g. [5,6]).
The ROSAT X-ray observatory has discovered a small group (so far 7) nearby, thermally emitting and radio-quiet INSs which have the following common characteristics: (a) soft spectra, well described by blackbody radiation with the temperature in the range 60-120 eV, (b) absence of other spectral features, (c) no association with a known supernova remnant (SNR), (d) absence of radio emission or X-ray pulsations and, finally, (e) large ratio of the X-ray to optical emission f x /f opt . It has been suggested that these objects are either old INSs whose surface is reheated by accretion from the interstellar medium or young cooling stars which radiate away their thermal energy acquired at birth. Meanwhile, intensive X-ray observations using XMM-Newton and Chandra telescopes, as well as observations of these objects in optical and ultraviolet (UV), demand a revision of this origin of the thermal emission and provide further intriguing physical insight. Indeed, X-ray pulsations were found in six objects (with periods clustered in the range from 3 to 11 sec) and the period derivatives were determined for some of them. In the classical P −Ṗ diagram for the radio-pulsars these INSs occupy the region which is intermediate between radio pulsars and magnetars. Their inferred characteristic magnetic field strengths are above 10 13 G.
Furthermore, imaging CCD-spectroscopy with XMM-Newton uncovered absorption features in at least three of INSs. It is likely that another three INSs feature similar absorption lines. At the current resolution (E/dE ∼ 60) these lines are well described by Gaussian absorption lines. Their interpretation is not unique and they have been attributed to magnetically shifted atomic transitions, cyclotron resonances or an absorption feature from condensed surface (atmosphere). The inferred magnetic field strengths are again above 10 13 G.
Another surprise came with the optical and ultraviolet observations of the fluxes from INSs, which were found to lie generally a factor of ∼ 5−10 above the extrapolation of the X-ray spectrum (the so-called "optical excess", see, e.g. [9]).
Here we present the results of a rotational phase-resolved X-ray spectroscopy of the three brightest INSs (RBS 1223, RX J0720.4−3125 and RX J1856.5−3754) based on the multi-epoch observations conducted by XMM-Newton.

Data analysis and results
High quality rotation phase resolved spectra are needed in order to fit using magnetized atmosphere models of neutron stars and to constrain their gravitational redshift [13,6].  We used the data collected with XMM-Newton EPIC pn from the 12 publicly available (similar instrumental setup, i.e. Full Frame, Thin1 Filter) observations, in total comprising about 175 ks of effective exposure time. We extracted spin-phase resolved spectra with high S/N ratio (see Figs. 1 and 2) and fitted simultaneously with highly magnetized INS surface/atmosphere models [13]. These models are based on various local models and compute rotational phase dependent integral emergent spectra of INS using analytical approximations. The basic model includes temperature/magnetic field distributions over isolated neutron star surface 1 , viewing geometry and gravitational redshift. Three local radiating surface models are also considered, namely, a naked condensed iron surface [1] and partially ionized hydrogen model atmospheres, semi-infinite or finite atop of the iron condensed surface. The observed phase resolved spectra (i.e. energy-spin phase image, Fig. 2) of the INS RBS 1223 are satisfactorily fitted with two models, which have slightly different physical and geometrical characteristics of emitting areas. The fits show that the INS is an orthogonal rotator. The first model is parameterized by a Gaussian absorption line which is superimposed on a blackbody spectrum. The second model assumes a condensed iron surface above which there is a partially ionized, optically thin hydrogen atmosphere. Note, that the latter model is more physically motivated. We have additionally performed Markov Chain Monte Carlo (MCMC) fitting and estimated parameters and their uncertainties from posterior probability density functions.

RX J1856.5−3754
In contrary to the INSs RBS 1223 and RX J0720.4−3125, the X-ray spectrum of RX J1856.5−3754 does not show any significant absorption feature and the pulsed fraction is quite low (∼ 1.5%). However, in this case there is a constraint on a viewing geometry from the observed bow-shock at this INS [10]. In addition, RX J1856.5−3754 is the nearest INS and the distance (d = 123 +11 −15 pc) is known with relatively good accuracy [16]. As in the previous cases of the INSs, the spin phase-resolved spectra were fitted in the same manner. The results of the estimation of gravitational redshift and therefore the compactness of this object, together with other studied INSs, are shown in Fig. 6. It is worthwhile to note that we obtained statistically acceptable fit also assuming a non-vanishing toroidal component of magnetic field of INS. This corresponds to the values of the parameter a 1 which imply very small and hot emitting areas around the magnetic poles and yield small pulsed fraction, see footnote 1.
Nevertheless, our targeted parameter, the estimated gravitational redshift was approximately the same (z = 0.22 ± 0.07, see also [7]) independent of the model. Note, also the relatively broad   posterior probability density function of the gravitational redshiht in comparison to the cases of RBS1223 and RX J0720.4−3125 (see Fig. 5).

Conclusions and outlook
To summarize, X-ray spin phase-resolved spectroscopic study of three thermally emitting INSs and the fit of highly magnetized atmospheric models allowed us to estimate their compactness, which are suggesting on a stiff equation of state (see, Fig. 6). More work for detailed spectral model computation will be certainly worth to do in the near future and application to the phaseresolved spectra of other INSs. In particular, analysis of the high resolution spectra observed by XMM-Newton and Chandra with possibly other absorption features [5,11].