Turolla, Zane, and Drake, 2004 Zane, Turolla and Drake, 2004

1. Neutron Star Crustal Emission: a basic, unanswered question Silvia Zane, MSSL, UCLMagnetic fields & Neutron Star Surface Cocoyoc, Mexico, February 12-14 2007 • Turolla, Zane, and Drake, 2004 • Zane, Turolla and Drake, 2004 • Perez Azorin, Miralles and Pons 2005, 2006 • Van Adelsberg et al. 2005 • Perez Azorin, et al 2006 • Ho et al. 2007

2. WHY? • 1970s: commonly accepted that radiation by NSs came from their solid surface (Brinkmann 1980 and refs) • Later, the role of the thin gaseous layer which covers the star crust in shaping the emergent radiation spectrum was appreciated and model atmospheres became the standard tool for interpreting the observed emission from isolated NSs. • However, highly magnetized NSs may be left without atmosphere if they are cool enough. Onset of a phase transition that turns the gaseous atmosphere into a solid when the surface T drops below Tcrit (B) (for a given chemical composition). • Moreover, crustal emission should be the “external boundary condition” for any cooling model, and the “internal boundary condition” (at the basis of the gaseous layer) for any atmospheric model !

3. X-ray Pulsating Dim Isolated Neutron Star: the largest class of thermally emitting Neutron Stars(Treves et al, 2000, Haberl et al. 2004) • Soft X-ray sources in ROSAT survey; no radio emission • BB-like X-ray spectra, no non thermal hard emission • Low absorption, nearby (NH~1019-1020 cm-2) • Constant X-ray flux over ~years: BUT 0720! • No obvious association with SNR or binary companion • Optically faint BUT shows a definite optical excess! • THEIR SPECTRUM CANNOT BE REPRODUCED BY SINGLE T/SINGLE B ATMOSPHERIC MODELS!

4. The striking case of RX J1856.5-3754 • 500 ks DDT Chandra exposure (i) RX J1856.5-3754 has a featureless X-ray continuum (ii) better fit with a simple bb than with more sophisticated atmospheric models (Burwitz et al, 2001, Drake et al, 2002) • Optical excess of ~6 over the Rayleigh-Jeans tail of the X-ray best fitting bb (Walter & Lattimer, 2002) • Lowest measured pulsed fraction: 0.015 P ~ 7s (Tiengo and Mereghetti 2006) • d ~170 pc (Kaplan et al, 2006)  radiation radius of only 6-7 km!(Drake et al, 2002)

5. 1) Is RX J1856.5-3754 the first quark/strange star discovered ? (Drake et al, 2002; Xu, 2002) Bare quark stars not covered by an atmosphere would presumably emit a pure blackbody spectrum (2th component for the optical emission) 2) Other options : NS models based on a two-T surface distribution (Pons et al, 2002; Walter & Lattimer, 2002 ) May account for X-ray to optical emission  Give acceptable values for the star radius But how to produce a featureless spectrum from a NS covered with an optically thick atmosphere ?? How the spectrum of a quark star looks like ??

6. An alternative explanation: BARE NSs • Lai & Salpeter (1997), Lai (2001): NSs may be left without an atmosphere if they are cool enough. Onset of a phase transition • A gaseous atmosphere turns into a solid when T < Tcrit(B) If B >> mee3c/h32.35 x109 G atoms and condensed matter change: Strong magnetic confinement on e-; atoms have cylindrical shape elongated atoms may form molecular chains by covalent bonding along B Interactions between linear chains can then led to the formation of 3-D condensates 4  level deviations from a bb (Burwitz et al 2003): F ~ E B = 1.28  0.30 Mimic XB , with X= absorption factor X = 1 - xx = reflection factor

7. VALUE OF Tcrit ? Lai & Salpeter (1997), Lai (2001): still very crude expressions, although Representing the most recent available estimetes For H, the infinite linear chains (and metallic hydrogen) are certainly bound, favoring the possibility of condensation for low T and/or high B Tcrit for phase separation between condensed H and vapor: Situation more uncertain for heavy elements (as Fe). It is likely that, unless B12>>100, the linear chains are unbound for Z>6. Lai 2001: The cohesive energy is only a tiny fraction of the atomic binding energy. But, even such weak cohesion of the Fe condensate can give rise to a phase transition at sufficiently low T. By equating ion density of the condensed phase near zero pressure to the gas density in the vapor Cohesive energy of the 3D condensate

8. The coolest thermally emitting NSs with available B + RX J1856.5-3754 [1] Burwitz et al, 2001; [2] Drake et al, 2002; [3] Paerels et al, 2001; [4]Zane et al, 2002; [5} Kaplan et al; [6] Pavlov et al, 2001; [7] Taylor et al, 1993; [8] Halpern & Wang, 1997; [9] Bignami & Caraveo, 1996; [10] Marshall & Schulz 2002; [11] Greivendilger et al, 1996. Critical T for H and Fe. Condensation is possible in the shaded region for Fe and in the cross-hatched region for H. Filled circles are the NSs listed in the table. The horizontal line is the color temperature of RX J1856.5-3754 Most Isolated Neutron Stars have T well in excess of THcrit : if surface layers are H-dominated an atmosphere is unavoidable. But: if some objects have not accreted much gas:  we may detect thermal emission directly from the iron surface layers  depending on B the outer layers of RX J1856.5-3754 might be in form of condensed matter ! SPECTRUM?

9. Turolla, SZ & Drake, 2004 (see also Brinkmann 1980) dA = 2 R2sin  d = surface element at magnetic co-latitude   i,  = angle of incidence and azimuth of the incident wave vector k  = total surface reflectivity for incident unpolarized radiation  = (1 - ) =absorption coefficient j =B (T) = emissivity (Kirchoff’s law) Total flux: Anisotropy of the medium response properties   strongly depends on the direction of the refracted ray • Pure vacuum outside the star (neglect vacuum birefringence) • EM wave incident at the surface with (E,k) is partly reflected (E’,k’) and partly refracted • Birefringence of the medium: the refracted wave is sum on an ordinary (E’’1,k’’1) and an extraordinary (E’’2,k’’2) mode.

10. First: write down the dielectric tensor ij  ij = k’i k’j - |k|2ij +(2/c2)ij = Maxwell tensor |  ij | = 0 = dispersion relation  refractive index nm , m=1,2 Once nm, m=1,2 are known: solve the wave equation for the two refracted modes:  ij (nm)E’m,j =0 obtain the ratios E’m,x /E’m,z andE’m,y /E’m,z put these ratios into the BCs at the interface between the two media (Fresnel equations) obtain the E-field of the reflected wave in terms of the E-field of the incident wave E||” = f1 E + f2 E|| E” = f3 E + f4 E||  = ||, + , = reflected/incident wave amplitude || = |f2|2 + |f4 |2 , = |f1|2 + |f3| 2

11. ħ p,0 plasma frequency ħ p 0.7 Z1/5 B123/5 (/s) keV s  560 AZ-3/5 B12 6/5 g/cm3, ion density near zero pressure (Lai, 2001) • dielectric tensor ij for a cold electron plasma • (unrealistic but instructive!) - No contribution from the ion lattice, - No damping of free electrons due to collisions

12. The Spectrum by a Bare NSs is not necessarily a bb • Strong (angle-dependent) absorption for photons with energy comparable or lower than the plasma frequency. • Strong absorption around the e- cyclotron frequency. • Below the plasma freq (and close to the resonant frequency -, one of the two modes may be non-propagating: a whistler. Whistlers have very large, divergent refractive index (Melrose, 1986). • Appearance of cut-off energies (n0) and evanescent modes which can not propagate into the medium. • If the refractive index has large imaginary part : highly damped modes. They can not penetrate much below the surface (Jackson, 1975).

13. OSS: the existence of these damped waves is not in contradiction with having neglected collisional damping. • The “conductivity” sigma plij of a cold plasma can be computed from the dielectric tensor as: ij =  ij + i (4 /)plij and is therefore purely imaginary (see Jackson ’75, Meszaros ‘92). • Quotation marks are placed on “conductivity” because there is no resistive loss of energy in this case. • Yet, depending on their frequency and grazing angle, electromagnetic waves may be exponentially damped even in the cold electron plasma. RESULTS: absorption features may or may not appear in the X-ray spectrum, depending on the model parameters (mainly on B).

14. Monochromatic absorption coefficient vs energy for a given surface element. Plots are for B=1012 G and different values of the magnetic field angle. From top to bottom: 2/=0.05, 0.2, 0.4, 0.6, 0.8, 0.9, 0.95. Total absorption coefficient averaged over the star surface, for different values of B. Bp=5e12 G (full), 1e13 G (dotted), 2e13G (dashed), 5e13 G (dash-dotted). The “reference” plasma frequency ħp0 has been used.

15. Spectra and a few numbers • Teff = 75 eV , B = 3e13 G • Left : T=cost • Right : T() as given by Greenstein & Hartke ‘83 • Dashed line: bb at Teff • Dashed-dotted line: best fitting bb in the 0.1-2 keV range • Solid lines: spectra • Upper/lower curves: po and 2.5 po For B  5 x 1013 G: No features whatsoever in the 0.1-2 keV band The Spectrum is within  4% from the best-fitting bb, no hardening The total power radiated by the surface in the 0.1-2 keV band is  30-50% of the bb power, slightly larger for the meridional temperature variation models

16. Correcting the Angular Size RX J1856.5-3754 R /(d/100 pc) = 4.12  0.68 km (Drake et al, 2002) Ratio of the emitted to the bb power in the 0.1-2 keV range for different values of the plasma frequency. Circles: B =3 x1013 G. Diamonds: B =5x1013 G Filled: T=constant. Open: T(). SINCE d~ 170 pc, IF THE CORRECTION IS AS LOW AS 0.3-0.5, THEN R  ~10-12 Km COMPATIBLE WITH THE PREDICTIONS OF EOS of NSs (Lattimer & Prakash, 2001)

17. B) A (slightly) more realistic case: accounting for e-/phonon interactions • In the solid crust e- are strongly degenerate. If ~s quantum effects due to the B-field are not negligible. • In the degenerate surface layers charge (and heat) is transported primarily by electrons.  needs an accurate determination of the electrical conductivity (Potekhin ’99 and refs therein) • Above the crystallization temperature of ions  is dominated by scattering off ions, but below the crystal melting point the dominant process is scattering by crystal lattice vibrations (phonons) through Umklapp processes. Eventually, if T decreases further, impurities in the crystal structure and scattering off lattice defects are important. • B-fields in the NS crust complicate e-transport making it anisotropic. e-move freely parallel to B, but their motion perpendicular to the field is quantized.

18. THE B-FIELD STRENGHT AFFECTS THE CHARGE TRANSPORT (Potekhin et a. 2003) 1)T>>TB =ħe B / (me c): Non-quantizing (classical) regime. A non-quantizing B-field essentially does not affect the thermodynamic properties of matter, BUT hampers transverse motion and produces Hall currents. weakly quantizing. When e- populate several Landau levels 2)T << TB strongly quantizing regime. When the B-field confines most e- in the ground Landau level. Transition to strongly quantizing regime: < B7x103(A/Z) B123/2 g/cm3 Note that the zero pressure density S is << B for B >> 1010 G.

19. relaxation times Fermi energy. The electrical conductivity tensor is computed from the transporttensor (Hernquist ’84, Potekhin ’99) For strongly degenerate e- and not too close to the Landau thresholds: • 3 independent components: parallel to the field ( zz = ||), transverse ( xx =  yy =  ), and one off-diagonal (Hall) component (xy = H), which is non-dissipative. • In rotating coordinates only  || and  are needed

20. Potekhin ’99: most complete expressions for  Accounts for correlation effects in the strongly coupled Coulomb liquid and multi-phonon scattering in the Coulomb crystal. • in a weakly quantizing field: the conductivity in the longitudinal and transverse direction oscillates around their classical values. • for strong quantization: oscillations are quite prominent and may reach several orders of magnitudes transport properties of matter much different from the classical regime. Electrical and thermal conductivities for two values of log T. Dashed lines: classical limit. From P99.

22. Damping frequencies In the two directions • We write down the “conductivity” of the cold electron plasma and that related to e-/phonon scattering • We define the corresponding relaxations times • We sum the effective frequencies of the two processes

23. e-/phonon damping seriously affects the emission properties of the surface. Below ~1keV the emissivity declines quite rapidly with decreasing photon energy. Suppression mainly due e-phonon damping in the transverse plane, more pronounced at lower B. No damping Total absorption coefficient averaged over the star surface, for different values of B. Bp=5e12 G (full), 1e13 G (dotted), 2e13G (dashed), 5e13 G (dash-dotted). The “reference” plasma frequency ħp0 has been used. Damping

24. Spectra and a few numbers • Teff = 75 eV • B = 3e13 G and 5e13 G • Left : T=cost • Right : T() as given by Greenstein & Hartke ‘83 • Dashed line: bb at Teff • Dashed-dotted line: best fitting bb in the 0.1-2 keV range • Solid lines: spectra Fit with a bb in the 0.1-2 keV range still acceptable: max deviations below 20% (4% in the cold e- case) Larger suppression: only 35% of the bb power radiated for B=2e13G  solve the “small radius problem”? (but feature at 300 eV) At low fields B<<5e12 G virtually no emission is expected below ~0.5keV. For B~5e13G the surface can radiate down to a few tens of eV's, but for B<<5e12G the star surface behaves as a perfect reflector below ~100 eV.

25. The basic, unanswered question:How the crust emits at low energies?Can crustal emission be responsible for the enhanced optical excess (possible with some reprocessing from a low density gaseous layer)?What is the role of ions?

26. Zane et al. (Cospar symp., 2002) e- and p (no damping) e- only Our original “mistake”: In a preliminary version of our work, we attempted to account for the ion contribution by including free protons in the dielectric tensor. • By including protons as free particles gives a “flat” enhanced emissivity at low E (and a proton cycl. feature in absorption). • However, this treatment of protons is obviously incorrect. Ions form a lattice and cannot be considered as free particles (as well as those e- which are bounds to ions). •  The simplistic treatment of the dielectric tensor as an e-/ion plasma is unrealistic and not applicable.

27. More recent works and applications • Turolla, Zane, and Drake, 2004: e- and phonon damping • Zane, Turolla and Drake, 2004 : e- and phonon damping + gaseous layer (see also Motch et al. 2003) • Perez Azorin, Miralles and Pons 2005, A&A, 433, 275: similar to TZD 2002 + models with free ions • Van Adelsberg et al. 2005 : include free ions ! • Perez Azorin, Miralles and Pons, 2006, A&A 451, 1009 2006: • effects of toroidal fields on heat transport, crustal mission as boundary condition • Perez Azorin et al. 2006, A&A, 459, 175: explain the excess in RXJ0720. Need free ions ? (no clear) • Ho, et al. 2007, MNRAS: explain the excess in RXJ1856 with free ions 