1. HEAT BLANKETING ENVELOPES OF NEUTRON STARS D.G. Yakovlev Ioffe Physical Technical Institute, St.-Petersburg, Russia Ladek Zdroj, February 2008, Outer crust Density profile Thermal structure Main properties of heat blankets

2. OUTER CRUST Composition: electrons + ions (nuclei) Electrons (e): constitute a strongly degenerate, almost ideal gas, give the main contribution into the pressure Ions (A,Z): fully ionized by electron pressure, give the main contribution into the density Electron background

3. Equation of state of degenerate electron gas Frenkel (1928) Stoner (1932) Chandrasekhar (1935)

4. LIMITING CASES Non-relativistic electron gas Ultra-relativistic electron gas

5. Equation of state of degenerate electron gas

6. Universal Density Profile in a Neutron Star Envelope In a thin surface envelope locally flat space r r=R z=0 z surface gravity for a «canonical» neutron star

7. In the outer envelope: depth-scale the density profile in the envelope Limiting cases: Accumulated mass:

8. Density profile in the envelope of a canonical neutron star

9. THERMAL STRUCTURE OF HEAT BLANKETING ENVELOPES =F=const MAIN EQUATIONS heat transport in a thin envelope without energy sources and sinks = thermal conductivity (radiative+electron) = opacity hydrostatic equilibrium; g S =const – surface gravity (F) (H) Divide (F)/(H): The basic equation to be solved (TP)

10. Degenerate layer Electron thermal conductivity Non-degenerate layer Radiative thermal conductivity Atmosphere. Radiation transfer THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE Nearly isothermal interior Radiative surface T=T F = onset of electron degeneracy Heat blanket z Z=0 Heat flux F T=T S T=T b T S =T S (T b ) ?

11. NON-DEGENERATE RADIATIVE LAYER Assume: Kramers’s radiative thermal conductivity (free-free transitions): Pressure of nondegenerate matter (P=nk B T): Eq. (TP): (K) Insert Into (K): Constant thermal conductivity along thermal path Analytic model

12. Linear growth of T with z z CM =z/(1 cm) Density profile TEMPERATURE AND DENSITY PROFILES IN THE RADIATIVE LAYER ONSET OF ELECTRON DEGENERACY

13. ELECTRON CONDUCTION LAYER Analytic model Electron thermal conductivity of degenerate electrons (ei-scattering): Equation (TP) assuming P=P e (degenerate electrons): Integrate within degenerate layer with Temperature profile within degenerate layer

14. INTERNAL TEMPERATURE VERSUS SURFACE TEMPERATURE Typically,and T(z)  const=T b at z>>z d which is the temperature of isothermal interior The main thermal insulation is provided by degenerate electrons!

15. T S -T b RELATION FOR NEUTRON STARS Our semi-analytic approach: Exact numerical integration (Gudmundsson, Pethick and Epstein 1983) For estimates: “DETECTOR OF LIE” For iron heat blanketing envelopes (A=56, Z=26)

16. COMPUTER VERSUS ANALYTIC CALCULATIONS log T S [K] = 5.9 or 6.5 (Potekhin and Ventura 2001) s = radiative surface solid lines – computer d = electron degeneracy dashed lines – analytics t = transition between radiative and electron conduction

17. THERMAL CONDUCTIVITY OF DEGENERATE ELECTRONS

18. MAIN PROPERTIES OF HEAT BLANKETING ENVELOPES Self-similarity (regulated by g S ) Dependence on chemical composition (thermal conductivity becomes lower with increasing Z). Envelopes composed of light elements are more heat transparent (have higher T S for a given T b ) Dependence on surface magnetic fields (B-fields make thermal conductivity anisotropic). For a given T b magnetic poles can be much hotter than the magnetic equator – non-uniform surface temperature distribution Finite thermal relaxation (heat propagation) times: Actual heat blanket is typically thinner than the “computer one” (density <10 10 g/cc). When the star cools, the actual heat blanket becomes thinner (as well as degeneracy layer and the atmosphere) In very cold stars (T S <<10 4 K) the blanket disappears (T S  T b )

19. REFERENCES