1. Superfluid turbulence and neutron star dynamics

2. The theory challenge
Neutron stars are cosmic laboratories of extreme physics, and we want to use observations to probe matter at supranuclear densities.
A “minimal” model requires:
supranuclear equation of state (hyperons, quarks etc.)
temperature profiles (exotic cooling mechanisms)
superfluids/superconductors (vortices, flux tubes)
elastic crust
magnetic fields (configuration, currents?)
rotation (various instabilities)

3. Superfluidity
Since mature neutron stars are cold (108K<< TFermi=1012K) they should be either solid or superfluid!
Theory:
Since late 1950’s, nuclear physics calculations indicate neutron and proton “BCS-like” pairing gap energies.
Exotica:
Deep core may contain superfluid hyperons and/or colour superconducting deconfined quarks.

4. Pulsar glitches
Many young neutron stars undergo spin-up events known as “glitches”.
The standard model for large glitches is based on transfer of angular momentum from a superfluid component to the crust.
the crust slows down due to magnetic braking
the superfluid can only spin down if vortices move outwards
if the vortices are pinned to the crust, the superfluid lags behind
at some critical level, a large number of vortices are released. As a result the crust is spun up.
Observations suggest unpinning of vortices at relative rotation
Our understanding is, however, far from satisfactory. In particular, we do not know what mechanism triggers a glitch.

5. The r-mode instability
The r-modes belong to a large class of “inertial” modes, which are driven unstable by the emission of gravitational waves at all rates of rotation!
The l=m=2 r-mode grows on a timescale of a few tens of seconds.
Instability window depends on uncertain core-physics.
Need to account for “exotic” states of matter :
hyperon and/or quark bulk viscosity
superfluid mutual friction
Suppressed by superfluidity
Stabilises the f-modes completely?

6. Superfluid hydrodynamics
Assume that:
Electrons/muons in the core are coupled to the protons on very short timescales.
Vortices and fluxtubes are sufficiently dense that a smooth-averaging can be performed.
The system is reduced to a two-fluid model.
One fluid is the superfluid neutrons in the inner crust and core, and the other fluid is a conglomerate of all charged constituents.
The equations of motion can be derived from a variational principle, and are analogous to the Landau model for superfluid Helium.

7. Equations of motion
The equations describing a Newtonian (“inviscid”) superfluid neutron star can be written (x & y are n or p):
Here we have defined the relative velocity and the momentum is given by
This encodes the “entrainment effect”, due to which the velocity of each fluid does not have to be parallel to its momentum.
Can be thought of in terms of an “effective mass”

8. Helium? It is useful to compare and contrast to the Helium case:
For He4 the two fluids represent the condensate and excitations.
In the (simplest) neutron star case we are at zero temperature and have a mixture of
superfluid neutrons
superconducting proton (+electrons)
At finite temperature we must also account for excitations and the associated entropy. This is likely to be very difficult…
Good news:
The basic multifluid aspects and the vortex dynamics are similar in the two problems.

9. Mutual friction
The presence of vortices leads to “mutual friction” between interpenetrating superfluids (e.g. the neutrons and protons):
Momentum of the neutrons will induce a flow in part of the protons (entrainment).
Magnetic fields are formed on the vortices.
Electrons scatter dissipatively off the magnetic fields.
Standard form for mutual friction considered is analogous to Hall-Vinen model for Helium (assumes a straight vortex array).

10. “Glitch relaxation” Predict that the system evolves according to
following a glitch event. This corresponds to a typical coupling timescale of
“Since this is much faster than the observed relaxation time in, for example, the Vela pulsar, glitches are unlikely to be associated with the stars core.”

11. Turbulence
The model may, however, be wrong… If there is a large scale flow along the vortex array, then short wavelength inertial modes become unstable.
The system is turbulent (overwhelming evidence for Helium), and the mutual friction may have a different form
Note: This leads to non-exponential relaxation (locally).

12. Free precession Ppr = P/e with e =10-8
Recent evidence for pulsar free precession.
Best candidate PSR B , with rotation period 0.4 s and precession period about 1000 d.
Data fits simple crust deformation model.
Ppr = P/e with e =10-8

13. Free precession Ppr = P/e with e =10-8
Recent evidence for pulsar free precession.
Best candidate PSR B , with rotation period 0.4 s and precession period about 1000 d.
Data fits simple crust deformation model.
Ppr = P/e with e =10-8
However, if the protons in the neutron star core form a type II superconductor, then the neutron vortices may be effectively pinned.
As a result, the star should precess rapidly…
New perspective: The misalignment between the two rotation axes facilitates an instability and leads to turbulence.

14. Two-stream instability
Instability sets in via inertial modes;
Short wavelength modes are not strongly damped by shear viscosity.
Similar behaviour both for weak and strong superfluid “drag”.
Severe constraint on fast precession:

15. Unstable r-modes (again!)
Now studying global inertial modes in a superfluid neutron star with relative rotation (as required in pulsar glitch models)
Preliminary results:
The mutual friction coupling renders the r-modes unstable in much of the relevant parameter space
for weak drag the instability is “secular”
for strong drag there is a dynamical instability with very fast growth (for high order modes)
Phenomenology very similar to precession problem!
Can we perhaps “explain” pulsar glitches?

16. Summary
Neutron star superfluidity impacts on key astrophysics, like pulsar glitches, free precession and oscillations (gravitational waves).
Ongoing effort to understand the dynamics of multifluid systems in general and neutron stars in particular. A number of interesting results.
understand entrainment and the “standard” mutual friction damping
know that this picture may be wrong; need to worry about turbulence
may prevent fast precession
may lead to unstable oscillations (and perhaps explain glitches?)

17. A few questions
Can a single model account for the rich glitch phenomenology?
How relevant is superfluid turbulence? (possible Helium experiments?)
What about proton superconductivity and fluxtubes?
Colour superconductivity?