Excitations collectives dans la croˆute interne des ´etoiles `a neutrons Luc Di Gallo

S´eminaire interdisciplinaire, ENSICAEN, jeudi 15 mars 2012

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Neutron star formation Neutron star structure Observables Neutron star cooling Matter signature

Neutron stars are formed after a gravitational supernova type II.

Figure: Crab nebula with a Pulsar in the center.

2/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Neutron star formation Neutron star structure Observables Neutron star cooling Matter signature

Neutron star characteristics are: A radius: R ' 10 − 15 km A mass: M ' 1 − 3 M¯ Compacity: Ξ =

GM ' 0.2 Rc 2

Average density: ρ ' 2.1014 g.cm−3 Temperature: T ' 106 − 1010 K Period of rotation: P ' 0.001 − 10 s Magnetic field: B ' 107 − 1015 G =⇒ Test for fundamental physics Figure: Neutron star structure, D. Page

3/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Several observations: Radio + high energy emission Thermal emission Accretion outburst Neutrino (Not yet...) Gravitational waves (Near future...)

Neutron star formation Neutron star structure Observables Neutron star cooling Matter signature

0.08 normalized counts s−1 keV−1

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

0.06 0.04 0.02

ratio

0 1.2 1.1 1 0.9 0.5

1

2 Energy (keV)

5

Figure: Chandra observations of Cassiopeia A Neutron Star at several moments between 2000 and 2009 (Heinke et al. 2010) Figure: Magnetic field of a Pulsar

4/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Neutron star formation Neutron star structure Observables Neutron star cooling Matter signature

Thermal emission + estimated age =⇒ constraint on thermal evolution models

TC = 1

0 9K

TC = 5

.5x10 8 K

Figure: Cooling curve and observational data (Gusakov et al. 2004)

TC = 0

Figure: Cooling in Cassiopeia A (Page et al. 2011)

→ Accurate thermal evolution model to interpret these constraints.

5/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Neutron star formation Neutron star structure Observables Neutron star cooling Matter signature

Specific heat + thermal conductivity + ν emissivity =⇒ Thermal identity of the matter

→ Investigation of a new contribution to the specific heat from the collective excitations.

9

T=10 K Ions Electrons Protons Non-superfluid neutrons Weakly paired neutrons Strongly paired neutrons

22 log10 (CV [erg cm-3 K-1])

Specific heat is a sum over the different contributions from the different excitations (nuclei, phonons, electrons,...) The crust is important for thermal evolution models (Gnedin et al. 2001, Brown and Cumming 2009) Nucleonic contribution in the inner crust is strongly suppressed

21 20 19 18 17 16 10

11

12

13

14

15

log10 (ρ [g cm-3])

Figure: Specific heat contribution as a function of the density at T = 109 K (Fortin et al. 2010)

6/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

MOTIVATIONS At low temperature collective modes are present Pairing energy of the order of 1010 K =⇒ No excitations coming from pair breaking of nucleons at T < 1010 K Due to pairing =⇒ matter is superfluid Superfluidity =⇒ Collective excitation at low energy Cover a large range of excitation wavelengths

Figure: Collective excitations regime VS single particle excitation regime function of the temperature (Page and Reddy 2012)

→ Hydrodynamic approximation to model collective behaviour of nucleons

7/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

MOTIVATIONS 10 Neutron 3 P2 HGRR AO T

Tc [10 9 K]

8

6

4

Neutron 1S0 AWP II AWP III SCLBL Proton 1S0 T

2

0

x 0.4

At low temperature collective modes are present Pairing energy of the order of 1010 K =⇒ No excitations coming from pair breaking of nucleons at T < 1010 K Due to pairing =⇒ matter is superfluid Superfluidity =⇒ Collective excitation at low energy Cover a large range of excitation wavelengths

1012

1013 1014 Log ρ [g/cm 3 ]

1015

Figure: Critical temperature for several models of pairing (Page 1997)

→ Hydrodynamic approximation to model collective behaviour of nucleons

7/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

Two basic equations to derive non-relativistic hydrodynamics of uncharged superfluids (Prix 2004): Conservation of particle number: ∂t na + ∇. ja = 0 with a = n,p Euler equation: ∂t Pa = ∇π a with a = n,p 1 −π a = µa − ma va2 + va · pa 2

(1)

Characteristics of superfluids come from quantum properties: No viscosity Locally irrotational No entropy transport Entrainment between the two fluids (n,p): non dissipative interaction which misalign velocities and momenta =⇒ coupling between fluids → Microscopic input from nuclear interaction

8/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

Hydrodynamic parameters are calculated within a nuclear interaction: Chemical potential from energy density Entrainment from the translation of the Fermi sphere q = qn − qp . In the neutron star inner crust these differences are small compare with the Fermi momenta → Landau-Fermi liquid model

Figure: Fermi sphere translation

Two interactions are employed: Relativistic Mean Field interaction DDHδ with σ − ω − ρ − δ mesons and density dependent parameters defined in (Avancini et al. 2009) Skyrme type interaction SLy4 (Chabanat et al. 1995) → Hydrodynamic mode equations

9/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

Few hypothesis: Zero temperature β equilibrium =⇒ proportion of n,p Hydrostatic equilibrium Non relativistic hydrodynamics Two basic equations for superfluid hydrodynamics: Conservation of particle number: ∂t na + ∇. ja = 0 with a = n,p Euler equation: ∂t Pa = ∇π a with a = n,p

10/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Motivations Superfluid hydrodynamics Hydrodynamic parameters Hydrodynamic modes

Few hypothesis: Zero temperature β equilibrium =⇒ proportion of n,p Hydrostatic equilibrium Non relativistic hydrodynamics Two linearised equations:

vitesse du son u (c)

0.3

±

=⇒ Two eigenvectors (U ) with associated sound velocity (u± )

0.2

0.1

0

Conservation of particle number: ∂t δna +na ∇. δva = 0 with a = n,p Euler equation: ∂t δPa = −∇δµa with a = n,p

u− u+ un

0

0.05

0.1 nB (fm -3)

0.15

Figure: Sound velocities as a function of the density

→ Hydrodynamic modes in the inner crust

10/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

The inner crust structures:

Figure: Neutron star inner crust, Newton et al 2011

Inner crust = transition from homogeneous matter to a lattice of atomic nuclei Inner crust = lattice of nuclei immersed in a neutron fluid Pasta phase = very deformed nuclei Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

11/29

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

The inner crust structures:

Surface tension

Electrostatic force

Figure: Neutron star inner crust, Newton et al 2011

strong

weak

strong

pasta

phase separation

weak

Wigner crystal like

amorphous

Figure: Effect of Coulomb force and nuclear surface tension, Watanabe and Maruyama 2011

Inner crust = transition from homogeneous matter to a lattice of atomic nuclei Inner crust = lattice of nuclei immersed in a neutron fluid Pasta phase = very deformed nuclei Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

11/29

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

The inner crust structures:

Figure: Pasta structures, G. Watanabe

Figure: Proton distribution in cylinder at nb = 0.033 fm−3 from a numerical simulation (Watanabe et al. 2003)

Inner crust = transition from homogeneous matter to the lattice of atomic nuclei Inner crust = lattice of nuclei immersed in a neutron fluid Pasta phase = very deformed nuclei Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

12/29

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: The neutron (upper) and proton (lower, shaded) distributions along the straight lines joining the centers of the nearest spherical nuclei at nb = 0.055 fm−3 (Oyamatsu 1993) Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

13/29

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: The neutron (upper) and proton (lower, shaded) distributions along the straight lines joining the centers of the nearest spherical nuclei at nb = 0.055 fm−3 (Oyamatsu 1993)

D = Thickness of the skin → Simplification of the structures D=0

14/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: Representation of ”1D structure”

Description of the model: Model with 1D geometry Structures correspond to a periodic alternance of two slabs (”gaseous” and ”liquid”) with different proton and neutron densities Superfluid hydrodynamics approximation in each slab

15/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: Representation of ”1D structure”

Description of the model: Model with 1D geometry Structures correspond to a periodic alternance of two slabs (”gaseous” and ”liquid”) with different proton and neutron densities Superfluid hydrodynamics approximation in each slab → Plane waves propagating in each slab

15/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: Representation of ”1D structure”

Description of the model: Model with 1D geometry Structures correspond to a periodic alternance of two slabs (”gaseous” and ”liquid”) with different proton and neutron densities Superfluid hydrodynamics approximation in each slab → Plane waves propagating in each slab

=⇒ 4 coefficients of amplitude

15/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Figure: Representation of ”1D structure”

Description of the model: Model with 1D geometry Structures correspond to a periodic alternance of two slabs (”gaseous” and ”liquid”) with different proton and neutron densities Superfluid hydrodynamics approximation in each slab → Plane waves propagating in each slab =⇒ 4 coefficients of amplitude =⇒ 4 boundary conditions are needed in order to describe the transmission/reflection of hydrodynamic modes at the interfaces

15/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Several boundary conditions: Kinematic boundary conditions Neutrons can pass through slab Fluids are impenetrable since ωτ < 1 where τ is the relaxation time =⇒ Contact is maintained =⇒ Continuity of perpendicular fluid velocities A common surface for protons and neutrons Dynamical boundary conditions Continuity of the pressure =⇒ Conservation of the momentum density Continuity of chemical potentials =⇒ Conservation of the momentum per particle

16/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Inner crust structures Characteristics of the model Boundary conditions Hydrodynamic mode propagation

Several compatible sets of four boundary conditions are possible. The most probable is : Two continuity of perpendicular fluid velocities (neutrons and protons) A common surface for protons and neutrons Continuity of the pressure To complete the set of equations: Invariance along r|| =⇒ Snell-Descartes laws for angles ³ ´ ± ± ± k||1 = k||2 = k||3 = q|| = q cos(θ) We use the Floquet-Bloch theorem to take into account the periodicity U(r + L) = U(r)e iq.L where L is the periodicity → The problem is formulated with a 6 × 6 matrix. Solutions are the zeros of the matrix determinant

17/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Table: Lasagne characteristics for the average baryonic density ρ0 from the DDHδ nB = 0.08 fm−3 ∼ 2 model of Avancini et al. 2009 Parameter L (fm) nB (fm−3 ) Yp = np /nB u or u+ (c) u− (c)

Slab 1 9.4 0.070 0 0.064

Slab 2 7.4 0.093 0.045 0.038 0.15

Total 16.8 0.080 0.023

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

-1

h ω = 4.5 MeV et qz= 0.13 fm −

1.5 1 δ P/δ P1(z=0)

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

0.5 0 -0.5 δP1/δP1(0) δP2/δP1(0) δP3/δP1(0)

-1 -1.5 0

5

10

15

20

25

z (fm)

=⇒ Solutions are non planar waves

Figure: Real part of the pressure fluctuation along z axis for lasagna at nB = 0.08 fm−3 , ω = 4.5 MeV and q = (0, 0, qz = 0.13 fm−1 )

18/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

Excitations spectrum at zero angle of propagation: A basic model of lasagne leads to: 6

L2 L L1 = 2 + n u2 nB us2 nB1 us1 B2 s2

5

Collectives excitations at θ =0 Linear dispersion ω = us.q

with usi2 =

3

−

h ω (MeV)

4

2

¯ 1 ∂Pi ¯¯ m ∂nBi ¯Ypi

(2)

(3)

This model gives us = 0.073 c. → The acoustic branch has a small enough slope to contribute significantly to the specific heat

1

0 0

0.05

0.1

0.15

|q| (fm -1)

Figure: Excitation spectrum at nB = 0.08 fm−3 and θ = 0

19/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

Solutions at a given energy are the intersection between zeros of the matrix determinant and the line defined by q|| = qz tan(θ): θ =π/4 −

h ω = 1 MeV

6

0.15

0.1

q|| (fm -1)

0.05

5 Mode 1 Mode 2 Droite q|| = tan(π/4) qz

4

0

3

-0.05

2

-0.1

1

-0.15 -0.2

0 -0.1

0 qz (fm -1)

0.1

0.2

0

0.05

0.1

0.15

0.2

0.25

|q| (fm -1)

Figure: Solutions for ~ω = 1 MeV (left) and excitation spectrum at nB = 0.08 fm−3 and θ = π/4 (right).

→ The second mode depends weakly on qz

20/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

θ =π/4

θ =π/2

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

−

h

ω (MeV)

θ=0

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

0

0

0

0.05

0.1

|q| (fm -1)

0.15

0 0

0.05

0.1

0.15

|q| (fm -1)

0.2

0.25

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 |q| (fm -1)

Figure: Excitation spectrum at nB = 0.08 fm−3 for θ = 0 (left), θ = π/4 (middle), θ = π/2 (right)

→ The second mode is close to the dispersion relation ω ' us0 q|| with a slope varying from us0 = 0.041 c to us0 = 0.051 c. Both acoustic mode can be associated to a Goldstone mode

21/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

θ=0

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

θ =π/4

θ =π/2

7

7

6

6

5

5

4

4

3

3

2

2

2

1

1

1

6

4

Linear dispersion ω = us.q

3

−

h

ω (MeV)

5

0

0 0

0.05

0.1 -1

|q| (fm )

0.15

0 0

0.05

0.1

0.15 -1

|q| (fm )

0.2

0.25

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 |q| (fm -1)

Figure: Excitation spectrum at nB = 0.08 fm−3 for θ = 0 (left), θ = π/4 (middle), θ = π/2 (right) with SLy4 interaction.

→ Sound velocities change but the structure of excitation spectrum is conserved

22/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

-3

-3

nB= 0.08 fm « lasagne »

-3

nB= 0.07 fm « spaghetti »

nB= 0.03 fm « gnocchi »

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

−

h

ω (MeV)

Non planar waves Excitation spectrum for θ = 0 Angle dependence Using SLy4 nuclear interaction Extrapolation to other densities

0

0 0

0.05

0.1 -1

|q| (fm )

0.15

0 0

0.02 0.04 0.06 0.08 0.1 0.12 -1

|q| (fm )

0

0.02

0.04

0.06

0.08

|q| (fm -1)

Figure: Excitation spectrum for θ = 0 at nB = 0.08 fm−3 (left), at nB = 0.07 fm−3 (middle), at nB = 0.03 fm−3 (right).

→ Slope of acoustic mode decrease with the decreasing density

23/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Collective excitations contribution to specific heat The specific heat as a function of the density The specific heat as a function of the temperature

Collective modes have a Bose distribution with zero chemical potential. We can integrate over all momenta in order to obtain energy density: Z π/L Z 2 d q|| dqz 1 ~ω(q) ~ω(q)/k T E(T ) = (4) 2 B (2π) e −1 −π/L 2π Temperature derivation leads to the specific heat: ¯ ∂E ¯¯ Cv (T ) = ∂T ¯

(5)

V

For a linear dependence ω = us q of the energy: 2π 2 kB4 T 3 ≡ bT 3 15~3 us3

(6)

3ζ(3)kB3 T 2 ≡ aT 2 π~2 us0 2 L

(7)

Cvs = For a linear dependence ω = us0 q|| : 0

Cvs = with ζ(3) = 1.202

24/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Collective excitations contribution to specific heat The specific heat as a function of the density The specific heat as a function of the temperature

Figure: Specific heat at T = 109 K as a function of the density from Fortin et al. 2010

25/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Collective excitations contribution to specific heat The specific heat as a function of the density The specific heat as a function of the temperature

CV (1018 erg cm-3 K-1)

T=109 K

1

0.1

Electrons Weakly paired neutrons Collective excitations Homogeneous matter

0.01 0.077 0.078 0.079

0.08 0.081 0.082 0.083 0.084 nB(fm-3)

Figure: Specific heat at T = 109 K as a function of the density compare with other contributions from Fortin et al. 2010

→ Collective excitations contribute significantly to the specific heat

26/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Collective excitations contribution to specific heat The specific heat as a function of the density The specific heat as a function of the temperature

The electronic contribution is: 10

Cv (10

18

-1

-3

erg.K .cm )

9 8 7

Cvel. =

Collective excitations Electrons Interpolation: a T2 + b T3

6

kB2 µ2e T 3(~c)3

(8)

Collective excitations contribution is interpolated by

5 4 3

Cv = aT 2 + bT 3

2

(9)

1 0 0.5

1

1.5

2

2.5

3

9

T (10 K)

Figure: Specific heat contribution as a function of the temperature at nB = 0.08 fm−3

with us = 0.079 c and us0 = 0.046 c. → Collective excitations contribution can dominate electrons contribution. Contribution from acoustic mode and good agreement between exact solutions and interpolation.

27/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Conclusion Formalism for collective excitation in 1D structures within superfluid hydrodynamics approach has been considered. The dispersion relations show two interesting acoustic modes and several optic branches One of the acoustic branch results of the considered geometry Collective excitations contribution to the specific heat is important compare with other contributions.

28/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons

Introduction Hydrodynamic modes The inner crust Excitation spectrum Application to specific heat Conclusion and perspective

Perspectives Extension of the model for other geometrical structures Consider electrons and Coulomb interaction Estimate the impact of this specific heat on the thermal evolution of the neutron stars Estimate the impact of nuclear parameters on this specific heat Calculate heat conductivity related to this collective mode

Figure: Electron and phonon thermal conductivity in the presence of large magnetic fields, Page and Reddy 2012

29/29 Luc Di Gallo

Excitations collectives dans la croˆ ute interne des ´ etoiles ` a neutrons