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

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

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

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

Equation of state of degenerate electron gas

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

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

Density profile in the envelope of a canonical neutron star

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)

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 ) ?

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

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

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

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!

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)

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

THERMAL CONDUCTIVITY OF DEGENERATE ELECTRONS

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 )

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