NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Effects of the superfluid neutrons on the dynamics of the crust Lars Samuelsson,

The Complex Physics of Compact Stars

Ladek Zdrój 28 February 2008

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Effects of the superfluid neutrons on the dynamics of the crust

Lars Samuelsson, Nordita (Stockholm)

Nils Andersson

Kostas Glampedakis

[Karlovini & LS, CQG 20 3613 (2003), Carter & LS CQG 23 5367 (2006) LS & Andersson,

MNRAS 374 256 (2007)]Umberto Boccioni: Elasticity, 1912



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Punchlines

—We may potentially constrain the high density Eos if the properties of the crust are accurately known.

—We need properties beyond the Eos in order to describe neutron star dynamics (shear moduli, entrainment parameters, transport properties,...).



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Outline― Motivation

― Equations of motion for continuous matter in GR

― Example: axial modes in non-magnetic stars

― Application: QPOs in the tails of giant flares and seismology

― Conclusions



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Neutron stars

Not perfect fluid



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A minimal model

—Solid outer crust—Solid inner crust with superfluid neutrons—Superfluids and superconductors coexisting in

the core—Huge magnetic fields – possibly bunched (Type

I) or in flux tubes (Type II)—Rotation – hence vortices

Here I will only consider the crust without magnetic fields



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Continuous matter in GR— Variational approach [Brandon Carter et al.]

— Amounts to specifying a Lagrangian masterfunction.

,...)( axn

— The ... represent “structural” fields describing eg. the relaxed geometry of the solid or the frozen in magnetic field.

— nxa is the four current. The conjugate variables

ax

xa n

are the four-momenta.



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EntrainmentFor multi-fluids it is convenient to consider the Lagrangian to be a function of the scalars that can be formed from the currents:

xn as well as (x≠y)

This leads a momentum given by

xy2xy

An

This illustrates the key fact that the current and the momentum for a given fluid need not be parallel. It is known as the entrainment effect, and is important for superfluid neutron stars.

xy

by

xybx

xab

xa nAnBg

by

axabxy nngn 2



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The currents and momenta

The quantities xa are both the canonically conjugate

and the physical (four) momenta.

Note that

),1( vnn xax

),( pxa

The four-currents describe the flow of particles and are related to the physical velocity. Due to entrainment the momenta are not parallel to the velocity. Warning: Landau’s superfluid velocities are vs = p/m and are not the physical velocities of the average motion of the particles.



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Equations of motion for Multifluids

(no summation over x)

Assuming that each particle species is conserved, we get

02 ][ xba

ax

xa nf

0 axan

Note: Tab is not the whole story

flowmatter pressure dgeneralise

nb

an

pb

apb

anc

cn

pc

cpb

a nnnnT



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Equations of motion with an elastic component

—Define ab given by the energy minimum under volume preserving deformations

—Define the strain tensor as:

ababab hs 2

1

The strain tensor measures volume preserving deformations

Simplest case: isotropic solid

abab sP 2ab

bSa Pf

0naf 0 S

aca ff



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Total Stress-energy tensor

bdaccdab

magnetic FFggB

16

1

8

2

The magnetic contribution is just:

elastic

ba

magnetic

ba

ba

ba

nb

an

pb

apb

anc

cn

pc

cpb

a

sBBBuuB

nnnnT

228

1 22

flowmatter pressure dgeneralise



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The Lagrangian densitymagneticsolidtentrainmenEOS

The EOS contribution is the contribution from the rest mass density and the part of the internal energy that does not depend on relative motion or the state of strain in the solid. :

Assuming small relative velocities the entrainment can be represented by

),( npEOS nn

20*2

1

2

1vmmnnnm

n nBA

ABn

tentrainmen

The solid contribution can similarly be expanded assuming small strain

2

2

1sssE ABAB

ABCDsolid



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Example: axial modes in the Cowling approximation

—Due to the static spherical background the neutron equation of motion become very simple. For non-static perturbations it amounts to

0na

—The remaining equation is nearly identical to the purely elastic case. The only difference is that the frequency is multiplied by a factor

2

*

2

*

22 1)(

)(1~

m

mX

mmnp

pXn

n



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Dynamical equation

0)2)(1(~ln

2

222

2

24

Fr

llve

v

eFerF



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Application: Flares in Soft Gamma-Ray repeaters

— SGRs: persistent X-ray sources envisaged as magnetars– B ~ 1015 G

– P ~ 1-10 s

— Key property: Emag >> Ekin

— Three giant flares to date– March 5, 1979: SGR 0526-66

– August 27, 1998: SGR 1900+14

– December 27 2004: SGR 1806-20

— Flares are associated with large scale magnetic activity and crust fracturing

— Quasi-periodic oscillations discovered in the data

T. Strohmayer & A. Watts, ApJ. 653 (2006) p.593



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ObservationsSGR fQPO (Hz) fmode l n

1806-20 18 ? - -

26 ? - -

29 29 2 0

93 92 6 0

150 151 10 0

626 308 ? 1 ?

1840 ? ? ?

1900+14

28 28 2 0

54 59 4 0

84 88 6 0

155 159 11 0

Newtonian limit, homogeneous stars, no dripped neutrons:

Fundamental mode (n = 0):

Overtones (n > 0):

= crust thickness ~ 0.1 R

cl RR

llvf

)2)(1(22

0

2

nvfnl



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Magnetic crust-core coupling— The strong magnetic field threads both the crust and the fluid

core (assuming non-type-I superconductor...)— The coupling timescale is the Alfvén crossing timescale

— Generic conclusion:– If the crust is set to oscillate the magnetar’s core gets

involved in less than one oscillation period– Pure crustal modes replaced by global MHD modes

— Puzzle: Why do we observe the seismic frequencies?

Where is the Alfvén velocity and G



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Mode excitation— Modes in the vicinity of a crustal mode

frequency are preferable for excitation by a “crustquake” as they communicate minimum energy to the core:

— Our model naturally predicts the presence of excitable modes below the fundamental crustal frequency

— Low frequency QPOs:

Example: SGR 1806-20. Identify Hz

Then:

Consistent with QPO data

Hz



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Shear modulus (bcc) by Ogata & Ichimaru:

Modelling the QPOs: Input data

Eos by Haensel & Pichon, Douchin & Haensel



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6 0 10 0

? 1

2 0

Seismology – exemplified by SGR 1806-20

frequency l n

18

26

29

93

150

626

(720)

1837

(2387)

T. Strohmayer & A. Watts, astro-ph/0608463

M=0.96Mo

R=11.4 km



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6 0 10 0

? 1

2 0

Seismology – exemplified by SGR 1806-20

frequency l n

18

26

29

93

150

626

(720)

1837

(2387)

T. Strohmayer & A. Watts, astro-ph/0608463

M=1.05Mo

R=12.5 km



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Conclusions— From a theoretical point of view we have come a long

way towards a description of neutron star dynamics— Need better understanding of

– Dissipation in GR– Superconductor fluid dynamics– Magnetic field dynamics

— We need microscopic calculations providing better understanding on matter properties beyond the equation of state: eg Superfluid parameters, shear modulus, pinning, vortex/fluxtube interactions, dissipation, ...

— The potential return is a “point” in the mass radius diagram implying constraints for the high density equation of state but...



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Conclusions continued

— We need to understand the dynamics and structure of the magnetic field.

— We need accurate Eos of the crust including shear moduli/us and effective neutron mass

— In particular the seismology is sensitive to

cc

ccp

1



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CommercialNORDITA (recently moved to Stockholm) provide the opportunity to organizing programmes of 1-2 month duration.

Applications for funding are open to the whole theoretical physics community.

See http://www.nordita.org/ for details.

There will be a 2 week mini-programme next year on the physics of the crust and beyond, tentatively in the spring.



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Documents

NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Effects of the superfluid neutrons on the dynamics of the crust Lars Samuelsson,