low energy theory of the neutron star crust and its ... · star crust and its observable...

Low Energy Theory of the Neutron Star Crust and its Observable

Implications. Sanjay Reddy

Los Alamos National Laboratory

Collaborators: Aguilera, Cirigliano, Chamel, Cumming, Page, Pethick, Pons &

Sharma

INT @ 20. The future of nuclear physics and its intersections. July 1-2, 2010Tuesday, July 6, 2010

nuclei

neutronsuperfluid

Neutron Star Crust:

Tuesday, July 6, 2010

New Phenomena in Neutron Stars

• Crustal heating and subsequent thermal relaxation in accreting neutron stars.

• Possible excitation of shear modes in the solid crusts of magnetars during giant flares.

- a window into the thermal and mechanical properties of the crust.


Simulations by Dany Page

Transient Accretion• Nuclear reactions heat the crust during accretion.

• Crust relaxes during quiescence. Haensel & Zdunik 1990, Brown, Bildsten, Rutledge (1998)

Shternin & Yakovlev (2007), Brown & Cumming (2009)

crust


Simulations by Dany Page

Transient Accretion• Nuclear reactions heat the crust during accretion.

• Crust relaxes during quiescence. Haensel & Zdunik 1990, Brown, Bildsten, Rutledge (1998)

Shternin & Yakovlev (2007), Brown & Cumming (2009)


More than one source !Cackett et al. 2006 Cackett et al. 2008

4.4 yr6.6 yrMXB 1659-29 KS 1731-260

!Cool = 305± 50 days!Cool = 465± 25 days


Connecting to Crust Microphysics

!Cool !CV

"(!R)2



!Cool !CV

"(!R)2

Crustal Specific Heat



!Cool !CV

"(!R)2

Crustal Specific Heat

Thermal Conductivity



!Cool !CV

"(!R)2

Crust ThicknessCrustal Specific Heat



Explosions on Magnetars: Giant Flares

SGRs exhibit powerful outburst ~ 1046 ergs/s

http://www.physics.mcgill.ca/~pulsar/magnetar/main.html

Anomalous X-Ray Pulsars (10)Soft Gamma Repeaters (8)

Inferred to have surface fields of the order of 1015 Gauss.

SGR 0525-66 : (1979)SGR 1806-20 (1979/1986/2004)* SGR 1900+14 (1979/1986/1998)SGR 1627-41 (1998)

SGR 1806-20: 2004 Flare

Hurley et al. (2005)




QPOs are likely to be shear modes in the solid crustDuncan (1998), Strohmayer, Watts (2006)

!n=0,l=2 ! 2 ctR

!n=1 ! " ctR

!R

R

Similar frequencies observed in 2 sources.

SGR 1806 2004 Giant Flare


Shear mode

velocity

QPOs are likely to be shear modes in the solid crustDuncan (1998), Strohmayer, Watts (2006)

!n=0,l=2 ! 2 ctR

!n=1 ! " ctR

!R

R

Similar frequencies observed in 2 sources.

SGR 1806 2004 Giant Flare


Microscopic Structure of the Crust

Negele & Vautherin (1973)

2 aneutrons

protons

Out

er C

rust

Inner Crustdripped

superfluid neutrons

Baym Pethick & Sutherland (1971)Tuesday, July 6, 2010

Microscopic Structure of the Crust

Negele & Vautherin (1973)

2 aneutrons

protons

Out

er C

rust

Inner Crustdripped

superfluid neutrons

Baym Pethick & Sutherland (1971)

Theory of electr

ons and phonon

s


Separation of Scales•Protons cluster (pairing + shell gaps) •Proton clusters form a Coulomb lattice. •Neutrons pair to form a superfluid.

!Debye !c

a! 0.45 !plasma

Ener

gy (

Tem

pera

ture

)

Longitudinal and Transverse Lattice

Phonons

!plasma =

!4"# Z2 nI

A mn

Nuclei (protons)

Superfluid Phonons

! ! EFn exp!

"1N(0) Vnn

"

Neutrons

Sing

le p

artic

le

exci

tatio

nsC

olle

ctiv

e ex

cita

tions


Superfluid Critical Temperature

Pairing gap is difficult to calculate in strong coupling.No small expansion parameter kF a > 1.

Cold atom experiments at kF a =∞ validate QMC. Polarization measurements in imbalanced systems are exponentially sensitive to the gap.

Tc ! 0.57 !

At kF a =∞: ! =!EF

= 0.45± 0.05

Gezerlis & Carlson (2009)

Carlson & Reddy (2010)

(?)


Relevant Temperature Scales in the Crust

Electro

ns TF

Ion TPlasma

Ions TDeBroglie

Ion TMelt (Γ=200)

TUmklapp

InnerCrust

OuterCrust



Electro

ns TF

Ion TPlasma

Ions TDeBroglie

Ion TMelt (Γ=200)

TUmklapp

InnerCrust

OuterCrust

Acc

retin

g an

d M

agne

tized

Neu

tron

Sta

rs



Electro

ns TF

Ion TPlasma

Ions TDeBroglie

Ion TMelt (Γ=200)

TUmklapp

InnerCrust

OuterCrust

Tc

Acc

retin

g an

d M

agne

tized

Neu

tron

Sta

rs


1011 1012 1013 1014

ρ (g/cm3)

0.01

0.1

Spee

d (in

uni

ts o

f c=3

x1010

cm

/s)

Longitudinal Lattice PhononsTransverse Lattice PhononsSuperfluid Phonons

Low Energy Excitations

1011 1012 1013 1014

ρ (g/cm3)

0.01

0.1

Spee

d (in

uni

ts o

f c=3

x1010

cm

/s)

Longitudinal Lattice PhononsTransverse Lattice PhononsSuperfluid Phonons

vs

cl

ct

!lPh(q) = cl q

!tPh(q) = ct q

!sPh(q) = vs q

!electron = q

How are these low energy modes coupled ? Tuesday, July 6, 2010

Low Energy Effective Field Theory

Neutron superfluid: Goldstone excitation is the phase of the condensate.

Proton (clusters) move collectively on lattice sites. Displacement is a good coordinate.

neutrons

protons

neutrons

protons



!i(x, y, z)



neutrons

protons

neutrons

protons



!i(x, y, z)



neutrons

protons

neutrons

protons

“coarse-grain”

Collective coordinates:

Vector Field: Scalar Field:

!i(r, t)!(r, t)

!!!(r)!"(r)" = |!| exp (#2i ")


Inner Crust EFT

Lp =12npm !t"i !t"i !

12

K !i"i!i"i !14

µs "ij "ij + · · ·

Compressibility

K ! !2E

!np!np

Shear Modulus

µs !Z2 e2

a4

!ij = "i!j + "j!i !23#ij "k!k

Protons:

Neutrons:

Son & Wingate (2006)

!!!(r)!"(r)" = |!| exp (#2i ")

! = µn t

Ground-state

Ln = P (µn)Tuesday, July 6, 2010

Inner Crust EFT

Lp =12npm !t"i !t"i !

12

K !i"i!i"i !14

µs "ij "ij + · · ·

Compressibility

K ! !2E

!np!np

Shear Modulus

µs !Z2 e2

a4

!ij = "i!j + "j!i !23#ij "k!k

Protons:

Neutrons:

Son & Wingate (2006)

!!!(r)!"(r)" = |!| exp (#2i ")

! = µn t

Ground-state

! !

Fluctuations (Superfluid Phonons)

Ln = P (µn)Tuesday, July 6, 2010

Coupling Neutrons and Protons. (or the superfluid and the lattice)

Gibbs-Duhem Relation: !µn = Enn !nn + Enp!np

!µn = !"t#! Enpnp"i$i

Velocities and current-current coupling :

Ln = P (µn) +!P

!µn"µn +

12

!2P

!µn!µn"µ2

n + · · ·

!µn = ! ("i#)2

2m+

12$ m (%vn ! %vp)2

!vn ="i#

m!vp = m "t$i

density-density interaction

current-current interaction


Coupling Neutrons and Protons. (or the superfluid and the lattice)

Gibbs-Duhem Relation: !µn = Enn !nn + Enp!np

!µn = !"t#! Enpnp"i$i

Velocities and current-current coupling :

Ln = P (µn) +!P

!µn"µn +

12

!2P

!µn!µn"µ2

n + · · ·

!µn = ! ("i#)2

2m+

12$ m (%vn ! %vp)2

!vn ="i#

m!vp = m "t$i

density-density interaction

current-current interaction

Ln = P (µn) + nn !µn +12"n! µ2

n + · · ·


The Coupled SystemLn+p =

12(!t")2 ! 1

2v2

s (!i")2 +12(!t#i)2 !

12(c2

l ! g2) (!i#i)2

v2s =

nf

m!nc2l =

K + 4µs/3m(np + nb)

nb = ! nnBound neutrons:Free neutrons: nf = nn (1! !){Entrainment: protons

drag neutrons.

Velocities :



12(!t")2 ! 1

2v2

s (!i")2 +12(!t#i)2 !

12(c2

l ! g2) (!i#i)2

v2s =

nf

m!nc2l =



drag neutrons.

Velocities :

+ g !t" !i#i + $̃ !i" !t#i

Longitudinal lattice phonons and superfluid phonons are coupled:

g = np Enp

!!n

m(np + nb)"̃ =

!nb vs"(np + nb)nf



12(!t")2 ! 1

2v2

s (!i")2 +12(!t#i)2 !

12(c2

l ! g2) (!i#i)2

v2s =

nf

m!nc2l =



drag neutrons.

Velocities :

+ g !t" !i#i + $̃ !i" !t#i

Longitudinal lattice phonons and superfluid phonons are coupled:

g = np Enp

!!n

m(np + nb)"̃ =

!nb vs"(np + nb)nf

Lt =12(!t"i)2 !

12c2t (!i"j + !j"i)2

Transverse lattice phonons:

c2t =

µs

m(np + nb)⇒


List of low energy constants

Thermodynamic Derivatives: Enn =

!2E

!nn!nnEpp =

!2E

!np!npEnp =

!2E

!nn!np

cl ct vs g !̃

K µs Enn Enp nb

g ! 10!3 " 10!2

Coupling between superfluid and lattice: !̃ ! 10!3 " 10!2


Two Applications

• Superfluid heat conduction.

• Effects of mixing and entrainment on the sound speeds.


Mixing and DissipationMixing of sound modes

Dissipation of lPh leads to dissipation of sPh

1011 1012 1013 1014

ρ (g/cm3)

0.01

0.1

Spee

d (in

uni

ts o

f c=3

x1010

cm

/s)

Longitudinal Lattice PhononSuperfluid PhononTransverse Lattice Phonons




sPh mean free path

lPh mean free path


g̃mix = g + !̃ " =cl

vs

!lPh = cl "lPh



sPh mean free path

lPh mean free path


g̃mix = g + !̃ " =cl

vs

!lPh = cl "lPh



sPh mean free path

lPh mean free path

Away from resonance



107 K

108 K

109 K

neutron-drip

! =1

3Cv ! v ! "

Typically electrons dominate heat conduction.Processes:•Electron-phonon•Electron-impurity

Flowers & Itoh (1976)Uripin & Yakovlev (1980)



107 K

108 K

109 K

109 K

108 K

107 K

For T>108 K superfluid phonons play a role.

neutron-drip

! =1

3Cv ! v ! "

Typically electrons dominate heat conduction.Processes:•Electron-phonon•Electron-impurity

Flowers & Itoh (1976)Uripin & Yakovlev (1980)


Conductivity in Magnetized Neutron StarsMagnetic field suppresses transverse conduction

= Collision time

=Gyrofrequency

Canuto and Ventura (1977)Uripin & Yakovlev (1980)


Conductivity in Magnetized Neutron StarsMagnetic field suppresses transverse conduction

= Collision time

=Gyrofrequency

Canuto and Ventura (1977)Uripin & Yakovlev (1980)

Aguilera et al. (2009)

1013 G

1013 G1014 G

1014 G


Shear Mode Velocity:

Lowest energy modes dominate the specific heat.

c2t =

µs

m(np + nb)

Entrainment effects

µs !Z2e2

a4i

nb = ! nn

CV ! 4!2

15

T 3

c3t(T < TDebye)

Specific Heat

Chamel, Pethick, Reddy (in prep)Tuesday, July 6, 2010

Specific Heat at 1012 g/cm3

electronssh

ear m

ode

long

itudin

al m

ode

supe

rfluid

mod

e

107 108 109

T (K)

0.0001

0.001

0.01

0.1

1C

v (in

units

of n

i) Normal Neutrons !


Specific Heat at 1012 g/cm3

electronssh

ear m

ode

long

itudin

al m

ode

supe

rfluid

mod

e

107 108 109

T (K)

0.0001

0.001

0.01

0.1

1C

v (in

units

of n

i) Normal Neutrons !

CV ! CnormalV exp ("!

T)


Shear Mode Velocity & Magnetar QPOs

Duncan (1998)Watts & Strohmayer (2006)

!n=0,l=2 ! 2 ctR

!n=1 ! " ctR

!R

R


Conclusions

!Cool !CV

"(!R)2

Astronomy (Observations) Astrophysics

( Theory)

Nuclear Physics& Many-body

Theory

Insights about the nature of matter at extreme density


Back up slides


MixingMixing leads to oscillations

1011 1012 1013 1014

ρ (g/cm3)

0.01

0.1

Spee

d (in

uni

ts o

f c=3

x1010

cm

/s)

Longitudinal Lattice PhononSuperfluid PhononTransverse Lattice Phonons


Dissipative Processes

Electrons lPhs sPhs

Electron-phononprocesses

Impurity (Rayleigh)scattering

Multi electron and phonon processes


Coupling to Electrons - Landau Damping

Electron particle-hole excitations damp collective modesand vice-versa.

Lel!lph =1

fel!ph!i"i #†

e #e

Lel!sPh !g̃mix

v2s(1" !2)

1fel!ph

"t# $†e $e

Induced coupling between superfluid phonons and electrons:

g̃mix = g + !̃ " =cl

vswhere


Heating in an Accreting Crust

Haensel & Zdunik (2003,2008) Gupta et al. (2007 & 2009)

Non-equilibrium reactions:Electron capture: Neutron transfers:Pycno-nuclear fusion:

e! + [A,Z] ! [A,Z " 1] + !e + Heat

n + [A,Z] ! [A+ 1, Z] + Heat


low energy theory of the neutron star crust and its ... · star crust and its observable...

Documents