Thermal radiation from magnetic neutron star surfaces J.F. P ´ EREZ –A ZOR ´ IN , J.A. M IRALLES AND J.A. P ONS Departament de F´ ısica Aplicada, Universitat d’Alacant, 03690 Alacant,Spain ABSTRACT We investigate the thermal emission from magnetic neutron star surfaces in which the cohesive effects of the magnetic field have produced the condensation of the atmosphere and the external layers [7, 5, 11]. This may happen for suf- ficiently cool ( ) atmospheres with moderately in- tense magnetic fields (about G ). The thermal emission from an isothermal bare surface of a neutron star shows no remarkable spectral features, in agreement with recent ob- servations. However, the presence of the magnetic field is expected to produce a highly anisotropic temperature dis- tribution, resulting in an observed flux very similar to a BB spectrum, but depressed in a nearly constant factor at all energies. This may lead to a systematic underestimation of the area and size of the emitter by a factor 5-10. 1 Introduction In the last few years, the thermal emission of about 20 iso- lated neutron stars (NS) has been detected in the X-ray band. The observations can be summarized as follows: Most of them show a featureless thermal spectrum well fit by a blackbody (BB), with low temperatures (T eV). They seem to have high magnetic fields (B G). The optical counterparts have been detected in four cases, showing a systematic excess flux of about a fac- tor 5 10. Light element atmosphere models are ruled out by the multiwavelenght observations. Heavy element atmosphere models don’t explain the absence of spectral features. Single component models cannot account for the op- tical excess observed in 4 sources. Two component models (i.e. two BB) can reconcile the optical and X-ray observations. To explain the lack of spectral features and the 2–component spectrum we consider the EMISSION FROM A SOLID SURFACE with NON UNIFORM TEMPERATURE, but HOW CAN A NS BE LEFT WITHOUT AN ATMO- SPHERE? WHAT ROLE DOES THE MAGNETIC FIELD PLAY IN ALL OF THIS? 2 Effects of the magnetic field Highly magnetized NS can undergo a phase transi- tion that turns the gaseous atmosphere into a solid [5]. – Critical temperature (Fe): eV – Zero-pressure density: gr/cm Thermal conductivity is different in the directions parallel and perpendicular to B temperature varies over the surface. Need to solve the 2D diffusion equa- tion in the condensed envelope [8, 4]. 3 Solid surface model Suppose the solid surface emits according to Kirchoff’s law [2]: where emissivity; reflectivity 3.1 Dielectric tensor where the conductivity tensor( a ) is keV e plasma frequency effective relaxation times keV e cyclotron frequency keV ion cyclotron frequency keV ion plasma frequency damping frequencies a www.ioffe.rssi.ru/astro/conduct/condmag.html 3.2 Dispersion relation Introducing the Maxwell tensor, considering that and applying the complex Snell law , the dispersion relation leads to a quartic equation where 3.3 Boundary conditions and emissivity Imposing boundary conditions at the surface (con- tinuity of normal components of and , and tangential components of and ), we obtain the components of the reflected wave in terms of the components of the incident wave ( ): Parallel: Transverse: For each incident polarization we obtain the reflectiv- ity: – Parallel: – Transverse: Total reflectivity FINALLY THE EMISSIVITY 4 Emission properties(without ions) 4.1 Emissivity Normalized emissivity integrated in all possible incident angles as a function of energy for different orientations of the magnetic field (B G and T K): 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.001 0.01 0.1 1 10 α ν E (KeV) α=4.5 o α=18 o α=36 o α=45 o α=54 o α=63 o α=72 o α=85.5 o Emissivity is strongly reduced for low energies be- cause one mode developes a large imaginary part for The plasma resonance depends on the orientation of the magnetic field (neglecting damping): – POLE ( ): and – EQUATOR ( ): and For the emissivity approaches to BB ( ). 4.2 Spectral energy distribution Dipolar magnetic field: and con- stant temperature ( K) ( ) define the plane of incidence 1e-04 0.001 0.01 0.1 1 10 100 1000 0.001 0.01 0.1 1 F ν (arbitrary units) E (keV) BB B p =10 12 G B p =5 10 12 G B p =10 13 G B p =5 10 13 G Spectrum is essentially featureless At low energies, the behavior is similar for different magnetic field strenghts 4.3 Observed flux Assuming the following anisotropic temperature distribu- tion: with G and K. scale factor 1e-04 0.001 0.01 0.1 1 10 100 1000 0.001 0.01 0.1 1 F ν (arbitrary units) E(keV) BB θ o =0 o θ o =45 o θ o =90 o Red line is the observed flux for a single BB ( K). Optical band is not very much altered for different ob- servation angles ( ), but high energy tail is signifi- cantly depressed. Broadband spectrum mimics the BB, but with an over- all reduced flux. 5 ION’S EFFECT The effect of the ions is visible in the next figure. We com- pare the observed flux from two different NS models: a sin- gle temperature BB and the solid surface model described in 4.3 (with and without ions). In both cases, we take into ac- count interstellar medium absorption ( hydrogen col- umn density). Blackbody: K, . Model: K, , , G, 0.001 0.01 0.1 1 10 0.001 0.01 0.1 1 F ν (arbitrary units) E(keV) BB fit θ o =90 o θ o =90 o w/ ions X-ray spectrum is practically indistinguishable (notice that = 11.5). Optical flux including ion effects (green) is 4.5 times larger, similary to what has been observed in INS’s. The apparent estimated value of the radius ( ) is 3.4 times smaller for a single BB. 6 CONCLUSIONS Solid surface models show a significantly de- pressed spectrum (up to factor 10) compared to a single temperature BB with the same ef- fective temperature. Spectrum is almost featureless. Minor differ- ences only at energies where the interstellar medium absorption makes difficult to distin- guish between models. The presence of the magnetic field produces a large anisotropy in the surface temperature distribution. The size of the emitter can be understimated by a large factor if a simplified BB model is used. No need to appeal to strange stars to explain the apparent small radius of some NS [3, 9]. Depending on the magnetic field’s strength and including the effects of the ions, the op- tical flux is about 5 times larger than the cor- responding BB extrapolation of the best X- ray fit. This is similar to the observed spectra from INS. References [1] Burwitz, V. et al., 2001, A&A 379, L35 [2] Brinkmann, W. 1980, A&A 82, 352352 [3] Drake, J.J., et al., 2002, ApJ, 572, 996 [4] Geppert, U., et al., A&A, submitted, astro-ph/0403441. [5] Lai, D. 2001, Rev. of Mod. Phys. 73, 629 [6] Pavlov, G.G., et al., 1996, ApJ 472, L33 [7] P´ erez–Azor´ ın, J.F., Miralles J.A., & Pons J.A. 2004, submitted A&A. [8] P´ erez–Azor´ ın, J.F., Miralles J.A., & Pons J.A. 2004, in prepara- tion [9] Pons, J.A, et al., 2002, ApJ, 564, 981 [10] Potekhin, A.Y., 1999, A&A 351, 787 [11] Turolla, R., Zane, S. & Drake, J.J. 2004, ApJ, in press [12] Walter, F.M. & Lattimer, J.M.,2002, ApJL 575, L145 [13] Zavlin, V.E., et al., 1995, A&A, 297, 441