Uncertainty regarding the behaviour of nuclear matter in the deep neutron star interior has compromised a complete description of the dense matter equation of state (EOS). Various theoretical models in circulation predict a range of macroscopic observables, such as masses, radii, temperatures and maximum rotation rates, and as such are open to scrutiny with empirical data. Relativistic effects complicate such observations, with the true radius, R, related to the apparent radius at a distance (d ) as
with M being the mass of the compact object, and z the associated gravitational redshift. Theory to date indicates that 7km 20km, dependent on the stiffness of the EOS (e.g. Lindblom 1992). Binary studies suggest a common value of neutron star mass approximately that of the canonical estimate of 1.4 (Van Kerwijk et al. 1995). Typically, the model neutron star is assumed to have M 1.4 and 13km (R 10 km). Based on observations of the neutron star's thermal emission in the extreme UV (EUV) and soft X-ray bands, it should in principle be feasible to compute the ideal blackbody spectral energy distribution (SED) as a function of (, & ), and for a known distance d, an estimate of the apparent neutron star radius. It is generally agreed that such a measurement would have a profound impact in constraining EOS models. However, strong galactic HI absorption at these wavelengths restricts observations to the closest neutron stars, and uncertainty in X-ray detector sensitivities at these low ( 0.2 keV) energies (e.g. Walters & An 1998) has compromised attempts to accurately determine estimates for and . Furthermore, such an analysis may be complicated by neutron star phenomenology, such as atmospheric opacity effects, an active magnetosphere, hot polar cap regions and accretion processes. It is critical to separate the various contributions, so as to estimate the genuine total surface component. For the older, isolated neutron stars (INS), phase-resolved studies in the soft X-ray, EUV and optical wavebands can provide such a SED, and thus a real possibility of determining . The Rayleigh-Jeans tail in the optical regime provides stringent constraints to any SED model-fit, and in this waveband atmospheric opacity effects are expected to have the most noticeable impact. In the X-ray regime where the blackbody continuum peaks, low Z atmospheres preferentially transmit radiation from the lower, hotter regions of the photosphere, producing a X-ray spectrum suggestive of a `hotter' source (Romani 1987). The Rayleigh-Jeans tail is unaffected by this deviation, and discrepancies between optical spectrophotometry and higher energy extrapolations may be rigorously tested (Pavlov et al. 1998). However the intrinsic faintness of these astrophysical objects in the optical regime coupled with the limitations of current technology have restricted previous optical studies to deep integrated photometry. These observations have in some ways aided such thermal continuum studies (e.g. Walters & Matthews 1997), but the differentiation of pulsed (predominantly nonthermal) and unpulsed (thermal) components would be ideal in a more rigorous treatment. Recently, the TRIFFID high speed optical photometer detected pulsations in the B band from the optical counterparts of the middle aged pulsars Geminga and PSR B0656+14 (Shearer et al. 1997, Shearer et al. 1998). Both light curves are highly pulsed and suggest a dominant nonthermal mode of emission optically. Despite their extreme faintness, it was possible in both cases to determine upper limits to each pulsar's thermal component of emission. In this letter, we combine these unpulsed limits with the results of recently published ground-based and HST photometric analysis on these two pulsars (Pavlov et al. 1997, Martin et al. 1998 hereafter & ), and derive radius/distance estimates for both. In particular, for the case of Geminga with a known parallax distance of 160 pc, we provide for the first time an upper limit for based upon such phase-resolved photometry.
We approach the analysis of the optical photometry by assuming a model fit incorporating both nonthermal power-law and thermal components of emission in the UBVRI regime, as originally adopted by . This two-component model fit is defined as:
with = 8.766 Hz an arbitrary reference frequency, and the interstellar extinction determined following Savage & Mathis (1979). and are taken to be the values of the nonthermal and thermal fluxes at . The latter can be expressed, for the chosen value of , as
where K is the apparent neutron star brightness temperature, km the radius and pc the distance to the blackbody. Model fits for both pulsars have suggested the presence of both emission modes, yet uncertainty in the optimum (,, ) solution based on EUV/soft X-ray datasets restricts accurate differentiation. By providing upper bounds for the unpulsed flux, and indirectly via the estimated towards the pulsar, we can derive an independent estimate of G and in this way, constrain (, ) space. The use of the lower bounds to for a given neutron star would then yield upper bounds to the parameter.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998