Coefficients of heat and electric conductivity inmagnetized neutron star
Glushikhina М.V., Bisnovatyi-Kogan G.S.

Space Research InstituteIntroduction

It is widely accepted that NSs are born at the final stage of evolution of normalstars of mass M ≥ 8 ∗ M in gravitational collapse of their cores. During
collapse, the matter of central layers is compressed to nuclear densities andenriched by neutrons. As a result, a compact NS is created of mass M ∼ M
and radius R ∼ 10km.

Structure

4 internal regions:ρ0 = 2.8 ∗ 1014 g /cm3

The cooling

I The cooling of a NS is accompanied by the loss of its thermalenergy which is mainly stored in the stellar core. The energy iscarried away through two channels:1) by neutrino emission from the stellar body,2) by heat conduction through the internal stellar layers
towards the surface, and further, by thermal emission ofphotons from the surface.
I The nature of the cooling depends on many parameters of thestar: the equation of state of the inner layers, NS mass,
magnetic field, chemical composition of the surface layers, andsuperfluidity of nucleons in the core, etc. Comparison of thecooling theory with observations can impose restrictions onthese parameters.

Observations

I It has been understood since Greenstein & Hartke (1983) thatin presence of a sufficiently strong magnetic field, ≥ 1010 G ,the surface temperature of a neutron star (NS) will not be
uniform as is expected in the unmagnetized case.
I Page (1995) and Page & Sarmiento (1996) applied theGreenstein & Hartke formula to explain the lightcurves of theisolated thermally emitting NSs PSR 0833-45 (Vela), PSR0656+14, PSR 0630+178 (Geminga) and PSR 1055-52,
considering dipolar field configurations.

Observations
I This result, that the geometry of the magnetic field in theinterior of the NS leaves an observable imprint at the surface,potentially allows us to study the internal structure of themagnetic field through modelling of the spectra and pulse
profile of thermally emitting NSs.

Formulation of the problem

I To analyse temperature distribution on the surface of NS wehave to obtain temperature distribution in the crust of NSfrom the border region to the insulated surface.
∂T
I the heat flux density:qi = −λik∂rk
I the current density ji = ene hvi i
I It is necessary to consider the tensor properties of the thermal

conductivity and electrical conductivity:
5 k 2 Tn 1((a0 − a11 )δik − ikn Bn (b01 − b11 ) + Bi Bk (c01 −λik =2 mc11 ), i, k = x, y , zkTn
(x0 δik − ikn Bn b0 + Bi Bk z0 ), i, k = x, y , z
σik =mAnalysis of previous work

I In previous works devoted to the kinetic coefficients in amagnetized neutron star, the following approximation was used
for the coefficients of thermal and electric conductivity along
and across the magnetic field lines: (Flowers, Itoh 1975 ),(Yakovlev, Urpin 1980)
1λ⊥
I=.
λk
1 + (ωτ )2σ⊥1I=.
σk
1 + (ωτ )23h3 neeBI ω =, τ=me c32π 2 Z 2 e 4 nN me ΛIsotherms
I Simulation of the isotherms in an isolated magnetized neutronstar by Perez-Azorin, Miralles, Pons 2005:

I The ratio for the coefficients along and across the magneticfield lines:

λ⊥1=.
λk
1 + (ωτ )2Analysis of previous work
There are a lot of papers, devoted to kinetic coefficients ofmagnetized plasma in Maxwell’s approximation, performed bya number of authors: W. Marshall (1961), Bisnovatyi-Kogan
G.S.(1964) и R. Lansdorf(1959). The approximation obtainedin these papers for the coefficients along and across themagnetic field lines for nondegenerate plasma is:
1λ⊥=2λk17(ωτ ) + 2.76(ωτ )41+2.58 + 0.39(ωτ )2σ⊥1I=,σk11.93(ωτ )2 + 2.07(ωτ )41+1.93 + 1.06(ωτ )2√
3 me (kT )3/2
I τ = √
4 2πZ 2 e 4 ne ΛNew approximation

I We assume that for the degenerate case, we can obtain a more

complex ratio between the coefficients along and across themagnetic field lines, that it was used in previous papers. For
large values of the magnetic field the value of the coefficient infront of (ωτ ) can be substantial.
I So we need to calculate coefficients of heat and electricalconductivity by solving Boltzmann equation.
I Obtained result can be used for calculation of isotherm

geometry on the surface and in the crust of neutron star.

Boltzmann’s equation∂fe1∂f∂f+ ci+ (Ei + ikl ck Bl )+ J = 0. (1)∂t∂rimc∂ci
I For collision integral we take into account only the binary
collisions: J = Jee + JeN ,R 0 00I Jee = 2m3 /(2π~)3 [f f (1 − f )(1 − f1 ) − ff1 (1 − f )(1 −10f1 )] × gee Wee (θ, gee )dΩdc1i ,R 0 00I JeN = [f f (1 − f ) − ffN (1 − f )] × geN WeN (θ, geN )dΩdcNi .
NII gee =| c1i − c2i | is relative velocity of colliding electrons.

geN is relative velocity of colliding electron and nuclei.
I Wee , WeN are differential cross sections for collisions ofparticles with relative speed gee or geN , inclined at an angle θand lying after a collision in the solid angle dΩ.

solution

I The zero approximation for the distribution function:v 2 −2µ −1
f0 = [1 + exp me 2kT] .
I The first approximation for the electrons: f = f0 [1 + χ(1 − f0 )]I The first approximation for the nuclei:fN = fN0 (1 + χN ).
I χ and χN are linear and admits representation of the solution

in the form:GI χ = −Ai ∂ ln T − ne Di di 5/2 ,∂ri
G3/2
GI χN = −ANi ∂ ln T − nN DNi di 5/2
∂ri
G3/2

solution

I Functions AI , ANi и DI , DNi determine heat conductivity anddiffusion.
I It was shown by Bisnovatyi-Kogan (1964) that in the presenceof the magnetic field, we can seek Bi vectors Ai and ANi inthe form :I Ai = A1 vi + A2 ijk vj Bk + A3 Bi (vj Bj ),I ANi = A1 vi + A2 ijk vj Bk + A3 Bi (vj Bj ),NN
N
I Let’s introduce:
I ξ = A1 + iBA2 ,I ξN = A1 + iBA2
NNsolution

I Generalized equation for ξ and ξN can be written in the form:

Z5G5/2i eme B
2dci −)ui = −ui (fN0 ξN fN0 uN
f0 (1 − f0 )(u −2G3/2
3 ρe kTc
Zeξ− f0 (1 − f0 ) ξf0 (1 − f0 )ui2 dci ) + iBf0 (1 − f0 )ui +me c+ Iee (ξui ) + IeN (ξNi uNi )I Solving this equation by polynomial expansion, analogous toSonine expansion, we obtain the following results2(5)Results: coefficientsof tensor of heat conductivity andelectro conductivity
I For degenerate case:kTne −τ (1.06 + 1.1(ωτ )2 ),m 1 + 6.5(ωτ )2 + 4.1(ωτ )4kT −τ (0.2(ωτ ) + 1.2(ωτ )3 )σxy = −σyx =,m 1 + 6.5(ωτ )2 + 4.1(ωτ )4kT
σzz = −1.06τ
mI For degenerate case:5 k 2 Tne π 2 τ (0.14 + 0.6(ωτ )2 )I λxx = λyy =,2 m 1 + 6.5(ωτ )2 + 4.1(ωτ )4
5 k 2 Tne −π 2 τ (0.2(ωτ ) + 1.2(ωτ )3 )λxy = −λyx =,2 m
1 + 6.5(ωτ )2 + 4.1(ωτ )4k 2 ne T
λzz = 0.33π 2 τmI σxx = σyy =Relation of coefficients of heat- and electocunductivities
across and along magnetic fields lines for degenerate plasma.

λ⊥=λk1.
0.31(ωτ )2 + 0.55(ωτ )4
1+0.14 + 0.6(ωτ )2σ⊥1I=.
σk5.6(ωτ )2 + 4.3(ωτ )41+1.06 + 1.1(ωτ )2I

Conclusion

I In our work we have reconsidered the thermal conductivity ofelectrons in magnetic field produced by electron - ion
scattering. The new coefficients can be used for calculation oftemperature distribution on the surface and in the crust ofmagnetized neutron star. The temperature distribution can bevery useful in understanding of geometry of magnetic field thethe dense regions of the star.
I There are some ways of elaboration of our results such asconsideration of the quantum effects of particle interaction athigh magnetic fields.

Thank you for attention!