Superfluid instabilities and neutron star dynamics

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Superfluid instabilities and neutron star dynamics
Superfluid instabilities and neutron star
dynamics
Kostas Glampedakis
University of Tuebingen
Superfluid instabilities and neutron star dynamics
1/16
Setting the scene
We observe aspects of neutron star dynamics, like glitches and
precession, that we do not really understand.
A key point: the observed dynamics is very sensitive to the
presence of superfluids in neutron star interiors → could be used
to probe the properties of neutron star matter.
Existing work on glitches/precession is based on
phenomenological “rigid-body” models, and/or studying the local
vortex dynamics (eg. vortex pinning/creep ...). This is an obvious
restriction.
What can we learn if we address these problems using global
superfluid hydrodynamics ?
Superfluid instabilities and neutron star dynamics
2/16
Why superfluidity?
• Neutron stars cool down
below Tc soon after birth.
Critical temperatures for neutron and proton superfluidity
core
crust
10
10
1
10
1
10
• At least two distinct fluids
in the interior (likely more in
the presence of ’exotica’ in
the inner core).
Tc (K)
3
10
10
10
• A “realistic” neutron star
model should be based on
multifluid hydrodynamics.
9
10
8
1
14
3
ρ (10 g/cm )
Superfluid instabilities and neutron star dynamics
S0 neutron
P2 neutron
9
10
7
S0 proton
10 10
8
7
1
14
3
ρ (10 g/cm )
3/16
10
The two-fluid model
• “Minimal” model for a superfluid neutron star:
neutron fluid = superfluid neutrons, ’proton’ fluid = superconducting
protons and electrons.
• Hydrodynamics:
(∂t +
(∂t +
vnj ∇j )
vpj ∇j )
vpi
vni
+ ǫn w
− ǫp w
i
i
i
+ ǫn wj ∇i vnj + ∇i (µ̃n + Φ) = fmf
i
− ǫp wj ∇i vpj + ∇i (µ̃p + Φ) = −(ρn /ρp )fmf
i
∂t ρn,p + ∇i (ρn,p vn,p
)=0
where wi = vpi − vni .
• Couplings:
i
, and
Entrainment parameters {ǫn , ǫp }, mutual friction force fmf
chemical potentials µ̃n,p (through the EoS)
Superfluid instabilities and neutron star dynamics
4/16
Mutual friction
• Mutual friction is a vortex-mediated force:
f~mf
R2
R
ω̂n × (~ωn × w)
~ +
(~
ωn × w)
~ ,
=
2
2
1+R
1+R
ωn = ∇ × ~vn
~
.
• The drag R makes contact with the dynamics of a single vortex:
drag force
R∼
Magnus force
R → 0 zero coupling limit, R → ∞ perfect pinning limit.
• Diversity in drag forces:
– Weak drag R ≪ 1 (electron-vortex scattering).
– Strong drag R ≫ 1 (interaction with fluxtubes (core) or lattice (crust)).
Superfluid instabilities and neutron star dynamics
5/16
Exploring some new dynamics
• Lets consider linear oscillations ( vni → vni + δvni etc) including two
key properties:
(1) In the background the fluids are in rigid-body rotation with different
spin frequencies:
~ n,p × ~r,
~vn,p = Ω
~ n 6= Ω
~ p.
Ω
(2) Mutual friction is present.
• For simplicity we ignore entrainment and assume a uniform
j
background (⇒ ∇j δvn,p
= 0).
• Main result: We find two new instabilities associated with inertial
modes.
Superfluid instabilities and neutron star dynamics
6/16
A new r-mode instability
• Global mode analysis: we obtain axial r-mode solutions (for ℓ = m).
• One of the r-mode solutions becomes unstable provided:
R≫1
Ωn − Ωp
6 0
=
and ∆ =
Ωp
• The instability appears for:
m > mc ≈ 320
0.05
xp
1/2 10
∆
−4
1/2
• The instability is dynamical. Growth timescale:
τgrow
P
x 1/2 10−4 1/2
p
≈ 2.5
0.05
∆
Superfluid instabilities and neutron star dynamics
7/16
Pulsar glitches
• Crust & proton fluid: spin-down
together (Ωp = Ωc )
spin frequency
• Neutron fluid: unable to follow
spin down if vortices are pinned
(Ωn > Ωp ).
pinned superfluid
Ωs
Ωc
crus
t+p
spin
dow
n
lasm
a
relaxation
time
• At a critical spin-lag vortex
unpinning is forced, and angular
momentum is transferred to the
crust:
Ic ∆Ωc
Ωs − Ωc
≈
≈ 5 × 10−4
∆g =
Ωc
Is Ωc
• Data for Vela-like glitches:
∆Ωc /Ωc ∼ 10−6 ,
Superfluid instabilities and neutron star dynamics
Is /Ic . 0.02
8/16
A glitch-trigger mechanism
• The r-mode instability needs to overcome viscous dissipation,
(dominated by electron shear viscosity). Combining τgrow < τsv and
m > mc leads to a critical ∆ at which the instability first appears:
∆c ≈ 6 × 10−5
P
0.1 s
2/3 T
108 K
−4/3
• Estimate glitch size:
∆g = ∆ ≥ ∆c
⇒
∆Ωc
& 10−6
Ωc
Is /Ic
0.02
P
0.1 s
2/3 T
108 K
−4/3
• Given an estimated (“observed”) core temperature, and the moment
of inertia ratio Is /Ic we can confront observations.
• Remark: Expect a glitch to occur after the object has “evolved” into
the instability region (i.e. τgrow minimum): m & 1.3mc ⇒ ∆ & 1.7∆c
Superfluid instabilities and neutron star dynamics
9/16
Confronting glitch observations
10
-4
2.5mc
B2334+61
mc
-5
∆Ωc/Ωc
10
1.3 mc
B1706-44
0.01
Vela
10
0.005
-6
Crab
-7
10 0
10
5
2/3
8
(P/0.1s) (T/10 K)
-4/3
• For systems without T data we use a standard URCA cooling law.
• Curves for Is /Ic = 0.02, 0.01, 0.005
Superfluid instabilities and neutron star dynamics
10/16
The “Glaberson-Donnelly” instability
• We carry out a local plane-wave analysis:
i
δvn,p
=
i
iσt+ikj xj
An,p e
,
Ain,p = constant
• We find that inertial waves could go unstable when there is sufficient
~ n, Ω
~ p.
misalignment between Ω
• Instability similar to the Glaberson-Donnelly instability in superfluid
Helium (where it leads to superfluid turbulence).
• The instability naturally applies to precessing neutron stars where
the precession modes are derived by assuming rigid body rotation and
~ n, Ω
~ p.
misaligned Ω
Superfluid instabilities and neutron star dynamics
11/16
Precession of superfluid neutron stars
• Nature of precession modes
decided by mutual friction R.
(Sedrakian et al, ’99)
Ppr ≈ 3
P
1s
−8
10
yr
ǫ
fast mode
2
13
10
10
12
10
timescales / P
• Weak drag precession:
slow mode
τd
10
10
1
Ppr
11
10
10
Ppr
τd
0
10
9
10
-1
8
10
10
or
7
5
Ppr ≈ 10
In /Ip
10
P
1s
−4
10
s
R
10
weakly damped precession
weakly damped precession
-2
10
6
10 -4
10
-2
10
0
10
2
10
4
10
-4
10
-2
10
0
10
2
10
mutual friction drag R
• Strong drag precession:
Ppr ≈ 0.1
10
In /Ip
P
1s
s
Superfluid instabilities and neutron star dynamics
12/16
4
10
Is slow precession unstable?
• Slow precession is only possible for weak mutual friction drag R ≪ 1.
• The instability is active above a critical wobble angle:
θw
> 1900
1◦
P
1s
1/2 10
ǫ
−8
10 km
R
• Conclusion:
– Long period precession is typically stable (e.g for a precessor like
PSR B1828 − 11).
– A notable exception: the instability could be relevant for
millisecond-period neutron stars.
Superfluid instabilities and neutron star dynamics
13/16
Is fast precession unstable?
• Strong drag leads to fast “Shaham”
precession.
3
P = 1 s, R = 10
10
2
10
1
5
• Wobble angle constraint:
P
1s
1/2 R
103
τgrow (s)
θw
> 0.014
◦
1
10
10
10
Ppr
7
10 K
I
-3
9
-4
10
• Growth timescale:
-1
-2
10
10
10 K
0
10 K
-5
-6
10 0.1
τgrow ≈ 140
θw
1◦
−1 P
1s
R
103
λ
R
1
10
100
1000 10000 1e+05
wavelength λ (cm)
s
• Conclusion:
Fast rigid-body precession is generically unstable.
Superfluid instabilities and neutron star dynamics
14/16
Yet another twist: the magnetic field
• Our model does not account for hydromagnetic stresses.
– This simplification should be obviously relaxed, since the magnetic
restoring force may dominate small-wavelength perturbations.
• In general, the magnetic field will play a stabilising (?) role.
• Local analysis: can be easily modified to include the magnetic field
(van Hoven & Levin ’08). The constraint on fast precession becomes
weaker, but the instability is still astrophysically relevant:
θw
>3
1◦
P
1s
B
1012 G
xp 0.1
• Global analysis: not so easy to modify ... but there is no reason why
our proposed glitch mechanism should not work in magnetised neutron
stars.
Superfluid instabilities and neutron star dynamics
15/16
Outlook
• The two fluid model for superfluid neutron stars predicts new
dynamical instabilities provided there is relative motion in the
background and mutual friction coupling.
• These instabilities may well be relevant for “real life” neutron star
dynamics (glitches, precession ...).
• Obviously, the model is far from being “realistic”. In order to make
more reliable predictions, one should worry about things like:
– Additional restoring forces (magnetic and crustal stresses)
– The phenomenology assumed in the mutual friction force
For example, vortex pinning in the core and the crust is poorly
understood → need for a consistent description of vortex dynamics.
Superfluid instabilities and neutron star dynamics
16/16

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