The Complex Physics of Compact Stars

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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.

• 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

are the four-momenta.
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Entrainment
For multi-fluids it is convenient to consider the
Lagrangian to be a function of the scalars that
can be formed from the currents
as well as (x?y)
This leads a momentum given by
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.
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The currents and momenta
The quantities mxa are both the canonically
conjugate and the physical (four) momenta. Note
that
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 Landaus 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
Assuming that each particle species is conserved,
we get
(no summation over x)
Note Tab is not the whole story
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Equations of motion with an elastic component

• Define hab given by the energy minimum under
  volume preserving deformations
• Define the strain tensor as

The strain tensor measures volume preserving
deformations
Simplest case isotropic solid
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Total Stress-energy tensor
The magnetic contribution is just
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The Lagrangian density
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
The solid contribution can similarly be expanded
assuming small strain
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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

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

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Dynamical equation
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Approximate frequencies
Fundamental
Overtones
Crust thickness
Leads to
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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 gtgt Ekin
• Three giant flares to date
• March 5, 1979 SGR 0526-66
• August 27, 1998 SGR 190014
• 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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Observations
Newtonian limit, homogeneous stars, no dripped
neutrons
Fundamental mode (n 0)
Overtones (n gt 0)
crust thickness 0.1 R
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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

Where is the Alfvén velocity and G

• Generic conclusion
• If the crust is set to oscillate the magnetars
  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?

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Toy model I Glampedakis, LS, Andersson, MNRAS
371, L74 (2006)

• Assume uniform density, shear modulus and
  magnetic field, ideal MHD
• Correct MHD conditions at interface Rc couple
  crust and core provided

Key effect crust-core resonance at the crustal
frequencies
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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

Consistent with QPO data

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

• Low frequency QPOs
• Example SGR 1806-20. Identify
  Hz
• Then

Hz
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Modelling the QPOs Input data
Eos by Haensel Pichon, Douchin Haensel
Shear modulus (bcc) by Ogata Ichimaru
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Seismology exemplified by SGR 1806-20
2 0
? 1
T. Strohmayer A. Watts, astro-ph/0608463
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Seismology exemplified by SGR 1806-20
2 0
? 1
T. Strohmayer A. Watts, astro-ph/0608463
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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

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Commercial
NORDITA (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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Corrections

• Magnetic field Sotani et al astro-ph/0608626,
  0611666
• Main effect
• a effect of EOS in inner core - Douchin
  Haensel AA 380, 151 (2001), Baym, Bethe
  Pethick, Nucl Phys A 175, 225 (1971), Negele
  Vautherin, Nucl Phys A 207, 298 (1973)
• Main dependence p/r at interface
• limits
• Anisotropy and reduction of the shear moduli in
  the pasta phases. Pethick Potekhin Phys Lett
  B 427 (1998). (Eagerly avaiting results of N.
  Chamel W. G. Newton J. R. Stone)