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José Antonio Fontwww.uv.es/jofontro
Magneto-elastic oscillations of neutron stars
Outline of the talk1. Astrophysical motivation2. Magneto-elastic oscillations with crust3. Different magnetic field configurations4. Superfluid core5. Conclusions
Collaborators
Michael Gabler, Pablo Cerdá-Durán (Valencia) Nikolaos Stergioulas (Thessaloniki), Ewald Müller (MPA)
ReferencesCerdá-Durán, Stergioulas & Font, MNRAS, 397, 1607 (2009)Gabler, Cerdá-Durán, Font, Stergioulas & Müller, MNRAS, 410, L37 (2011)Gabler, Cerdá-Durán, Stergioulas, Font & Müller, MNRAS, 421, 2054 (2012) Gabler, Cerdá-Durán, Font, Müller & Stergioulas, MNRAS, 430, 1811 (2013)Gabler, Cerdá-Durán, Stergioulas, Font & Müller, arXiv:1304.3566 (2013)
In quiescence, persistent X-ray emission at ~1035 erg/sspin period (s)
period
der
ivat
ive
(s/s
)Erot = −4π2I
P
P 3Spinning down NS
magnetars
Magnetar properties
• Intense magnetic fields with surface strengths > 1014-1015 G.
• Young age (<104 years)
• Distance of 15 kpc (Galaxy) - 55 kpc (LMC)
SGRs very active in ϒ-rays
• Frequent weak bursts with L~1041 erg/s, duration < 1s• Intermediate bursts with L~1041-1043 erg/s• Less frequent giant flares with L~1044-1046 erg/s
A giant flare has been observed in 3 out of 4 SGRs.
Giant flares have a strong initial peak in ϒ-rays with duration of ~0.2-0.5s, followed by an X-ray tail, lasting for tens to hundreds of seconds.
Magnetars and giant flares in SGRs
Three giant flares have been detected so far:
• SGR 0526-66 on March 5, 1979• SGR 1900+14 on August 27, 1998• SGR 1806-20 on December 27, 2004
Magnetar bursts are magnetic-field-driven quakes in the crust of neutron stars (Duncan & Thompson 1992).
R. Mallozzi, UAH/NASA MSFC
Magnetic field evolves:- Reconnection in magnetosphere- e-e+ pairs created and trapped by ultra strong magnetic field
Light curves for giant flares in SGRs
SGR 1900+14August 27, 1998
SGR 1806-20December 27, 2004
SGR 0526-66March 5, 1979
QPOs in the decaying X-ray tail
High frequency variations (QPOs) discovered in the tail of SGR 1806-20 (Israel et al. 05, Watts & Strohmayer 06).
Similar QPOs discovered in the tail of SGR 1900+14 (Strohmayer & Watts 05).
Where do QPOs come from?
Possible origin of observed frequencies:
Discrete shear modes (crust)
Alfvén oscillations at a turning point of a continuum (crust+core)
Magnetospheric oscillations.
Coupled crust-core (magneto-elastic) oscillations
Glampedakis et al 2006, Levin 2007, Van Hoven & Levin 2011, 2012, Colaiuda et al 2010, 2011, 2012, Gabler et al 2011, 2012, 2013.
Very little is known from observations about internal B-field configuration of magnetars. Strong outer dipole field responsible for observed spin down.
Most studies of magneto-elastic oscillations restricted to limited set of magnetic field configurations (dipole field).
Purely toroidal and purely poloidal fields unstable in stars (Tayler 1973; Markey & Tayler 1973), confirmed by non-linear simulations (Braithwaite & Spruit 2006; Kiuchi et al 2011; Ciolfi et al 2011; Lasky et al 2011; Lander & Jones 2011). Twisted torus configuration (mixed poloidal and toroidal field) expected.
Exist attempts to model equilibrium axisymmetric configurations with such mixed fields (Colaiuda et al 2008; Kiuchi & Kotake 2008; Ciolfi et al 2009; Lander & Jones 2009). No stable configuration found for barotropic stars (Lander & Jones 2012).
Possibilities to stabilize magnetic fields in NS: 3D B-field structure, stratification, presence of solid crust. Issue not yet settled.
Magnetic field configuration
Conservation of energy-momentum:
GRMHD equations for elastic bodiesds2 = −α2 dt2 + γij dx
i dxj
Tµν = (ρh+ b2)uµuν +
�p+
1
2b2�gµν − bµbν − 2µshearΣ
µν
∇µTµν = 0
Induction equation:
1√−g
�∂√γBj
∂t+
∂√−g(viBj − vjBi)
∂xi
�= 0
∂√γU
∂t+
∂√−gFi
∂xi= 0
GRMHD conservation law (flux-conservative hyperbolic system):
Magneto-elastic simulations in GR: equations
∂√γU
∂t+
∂√−gFi
∂xi= 0
U = (Sϕ, Bϕ)
Fi =
�−bϕBi
W− 2µsΣ
iϕ,−vϕBi
�
Σiϕ =1
2giiξϕ,i (i = r, θ) (ξϕ,i),t − (αvϕ),i = 0
Evolution eqs for displacement
Boundary conditions:• Surface: continuous traction and no surface currents• Crust-core interface: continuity of displacement & traction
Semi-analytic model (Cerdá-Durán+ 09): standing wave approach. Integration of a perturbation along B-field lines in the short wavelength limit. Extended in Gabler+ 12 to include elastic crust.
In linear regime and axisymmetry poloidal and toroidal perturbations decouple
Sϕ = (ρh+ b2)W 2vϕ − αbϕb0
Background magnetic field(Bocquet et al 1995; Lander & Jones 2012; Gabler et al. 2013a)
MAGSTAR routine of LORENE library (lorene.obspm.fr)
Extended to account for more general current distributions in Ampere’s law.
• Dipolar like configurations• Quadrupolar like and mixed quadrupolar-dipolar configurations• Mixed poloidal-toroidal configurations
MAGNETSTAR routine of LORENE library
Our works on magnetar QPOs: 1. dipole B-field + no crust (Cerdá-Duran et al 2009)2. dipole B-field + crust (Gabler et al 2011, 2012)3. various B-fields + crust (Gabler et al 2013a)4. superfluidity (Gabler et al 2013b)
Magneto-elastic simulations in GRMCoCoA code (CoCoNuT framework)- 2D-axisymmetric GRMHD code- Spherical coordinates- Finite-volume Riemann solvers + CT methods- Dynamical space-time (CFC)
Approximations- Torsional oscillations. Sound waves suppressed.- Low amplitude (linear)- Cowling (fixed spacetime)- Spherically symmetric background (non-rotating stars)- Ideal MHD
EOS- Core: APR (Akmal et al 1998) and L (Pandharipande & Smith 1975)- Crust: NV (Negele & Vautherin 1973) and DH (Douchin & Hansel 2001)
Other groups working in this field:Tuebingen, Sotani (linear simulation in GR)Y. Levin, A. Watts, Southampton (linear models in Newtonian limit)
Magneto-elastic model
0 2e+05 4e+05 6e+05 8e+05 1e+06X in [km]
1
1.5
2
2.5
3
frequ
ency
in [H
z]Turning Point
EdgeTurning Point
Turning Point Open Lines
Open Lines
Closed Lines
Continuum Gap
(Upper QPO)
(Upper QPO)
(Lower QPO)
(Edge QPO)
0
2
4
6
8
10
Y in
[km
]
open lineslast open lineclosed lines
The continuum Each f ie ld l ine has proper
eigenfrequency. Field l ines coupled through
boundary conditions at the surface or at the crust.
Calculate spectra with semi-analytic model (Cerdá-Durán et al 2009).
Long-lived QPOs exist at turning points and edges of the continuum.
Gaps between successive Alfvén overtones.
Crustal modes damped efficiently. For sufficiently strong fields Alfvén QPOs reach the surface (Gabler at el 2011, 2012); neglect crust for simplicity in some models.
Gabler et al (2012a)
Upper QPOs
Non-zero (maximum) amplitude at the surface
Number of nodes along magnet ic axis and location agrees with nodes computed with semi-analytic model (blue lines)
Alfvén oscillations: spatial pattern of effective amplitude
Amplitude only appreciable along magnetic axis
Cerdá-Durán et al (2009)
Symmetric (top) and antisymmetric (bottom)
Lower QPOs
QPO at the turning point
Amplitude only appreciable within the region of closed field lines
Cerdá-Durán+ 2009
Alfvén oscillations: spatial pattern of effective amplitude
Empirical relations- We find empirical relations that are independent of EOS.- Frequencies only depend on compactness M/R and B.- Numerical results reproduced to within a few % or better
Comparison to observed QPOs:
Upper limit on mean surface magnetic field 3-8 x 1015 G independent of EOS or mass of magnetar.
The integer ratios of 1:3:5 and the empirical relations were first pointed out in Sotani, Kokkotas & Stergioulas (2008).
A more realistic modelA number of additional effects must be included in order to arrive at a realistic model for magnetar QPOs:
- Elastic crust- Different current distributions that generate magnetic field- Toroidal magnetic field component- Type I superconductivity (B-field confined to the crust)- Higher multipoles in the magnetic field- Superfluidity
- Type II superconductivity- Coupling to poloidal oscillations- Nonaxisymmetric oscillations
} still missing
Effect of the crust on magneto-elastic oscillations
Kinetic+magnetic energy / field line
Without a crust, Alfvén wave packets travel roughly along B-field lines. No longer true when crust is present. A perturbation travelling along field lines from the star center to the surface (back and forth) will spread out past the crust-core interface.
The inclusion of the (scalar) shear smears the oscillations.
perturbation enters crust for the first time at ~70 ms.
Damping of crustal shear modesModel: APR+DH EoS, M=1.4Msun, R=12.26km, dipole B-field
• n=0 crustal shear modes efficiently damped by resonant absorption on timescales of ~0.2s for a lower limit on the dipole B-field strength of 5x1013G.
• Torsional shear oscillations of the NS crust excluded to explain low-frequency QPOs.
• After damping, only magneto-elastic oscillations remain.
Rapid Absorption of Crustal OscillationsCrustal oscillations are quickly damped and their energy absorbed by the Alfvén continuum of the core on timescales much shorter than the Alfvén timescale.
Upper QPOs in the continuum further away from the pole than in no crust model. Field lines get out of phase there due to the interaction through the extended crust because a significant fraction of the oscillation is refracted.
Upper QPO locations shifted towards equator
Lower QPOs at closed field lines (as in no crust case since closed lines unaffected by crust).
Edge QPOs at the edges of the continuum.
• B < 5x1013G crustal shear modes dominate evolution
• 5x1013G < B < 1015G Alfvén QPOs mainly confined to the core and crustal modes damped very efficiently
• B > 1015G magneto-elastic oscillations reach surface of star and approach behaviour of purely Alfvén QPOs
Need Strong B for Oscillations to Reach Surface
Magneto-elastic QPOs inside the magnetar
Different current distributionsY
[k
m]
X [km] X [km] X [km] X [km] X [km] X [km]
Y [
km
]
X [km] X [km] X [km] X [km] X [km] X [km]
Different currents (spherically symmetric, aligned with polar axis, non-spherical, two maxima) lead to similar magnetic field configurations.
current configurations
magnetic field configurations
spectra of Alfvén oscillations not too different from each other for the various models.
Alfvén QPOs - purely poloidal dipole-like fields
0 2 4 6Crossing radius with equator ! [km]
0
5
10F
req
uen
cy [
Hz]
A0C
0.1
C10
A1OF
U1
U2
• All models have a turning-point QPO (U1) near the pole
• Second turning points QPO (U2) only for some configurations (O, A1).
• Edge QPO at last open field line (weak)
spectra
Shear modes in gaps of Alfvén continuum?Colaiuda & Kokkotas 2011, van Hoven & Levin 2012
0 2 4 6 8Crossing radius with equator ! [km]
0
20
40
60
80
Fre
quen
cy [
Hz]
24.8
39.3
52.7
65.7
0 20 40 60 80 100Frequency [Hz]
10-19
10-18
10-17
10-16
10-15
10-14
Fo
uri
er a
mp
litu
de
B15
=3.7
B15
=1.85
B15
=0.93
Construct special model with very flat continuum with large gaps expected to produce very long-lasting QPOs (as almost all open field lines have similar frequency)
QPOs scale with B-field and have different frequencies
no crustal shear modes found in the continuum gaps
FFT of overlap integral with l = 2 crustal mode
Mixed poloidal-toroidal fieldsY
[k
m]
X [km] X [km] X [km]
Toroidal component limited to regions of closed field lines
Increasing toroidal field: closed field lines region shrinks and shifts towards surface (Lander & Jones 2012)
Moderate changes to oscillations in the region of open field lines, but qualitatively the same structure as for purely poloidal fields.
Mixed Dipole/Quadrupole fields
2 4 6 8 10X [km]
Q/D = 10
2 4 6 8 10X [km]
Q/D = 1
2 4 6 8 10X [km]
Q/D = 0.1
0 2 4 6 8 10X [km]
-10
-5
0
5
10
Y [
km
]pure Q
Magnetic field structure:Configuration with dipolar (D) and quadrupolar (Q) componentMay be realized during core-collapse supernovaNo equatorial symmetry (mixed fields)
Spectra of poloidal mixed Q/D fields
- Quite some more features in the spectra- Q/D=0.1 similar to the purely dipolar case- Q/D>1.0 complicated spectra shows different QPO families in both hemispheres- QPO frequencies matched by smaller (more realistic) fields
Example of QPOs for mixed Q/D fields
Numerical simulations. FFT at given frequencySimulations produce non-symmetric QPOs
Magnetic fields confined to the crust
0 2 4 6 8 10X [km]
0
2
4
6
8
10
Y [
km
]
0 2 4 6 8 100
2
4
6
8
10
0 2 4 6 8 10X [km]
0 2 4 6 8 10 0 2 4 6 8 10X [km]
0 2 4 6 8 10
D Q O
Magnetic field must be able to penetrate superconducting region in the core (Baym+ 1969).
Uncertainty of the supranuclear density EoS cannot rule out the possible presence of superconducting protons in the core which could expel the magnetic flux (Page+ 11; Shternin+ 11). Type I superconducting core.Axisym. configurations confined to crust (Aguilera+ 2008). Matched to exterior dipolar (D), quadrupolar (Q) & octupolar (O) field.
Magnetic fields confined to the crust
Not possible to explain lowest observed QPOs in SGR 1806-20 (18, 26 & 30 Hz)
QPO frequencies are incompatible with observations!
Accommodating low- and high-frequency QPOsLow frequencies < 150 Hz
18, 26, 30, 92, 150, 28, 53, 84, 155(SGR 1806-20, SGR 1900+14)
High frequencies < 500 Hz625, 1840
(SGR 1900+14)
Roughly match frequencies of crustal shear (torsional, n=0, 1) modes of unmagnetized stars. However, these modes quickly damped (resonant absorption) by magnetic field in the core.
Magneto-elastic QPOs explains observed low-frequency QPOs as excitations of fundamental turning-point QPO and of several overtones.
Observation of high-frequency QPOs poses a problem for magneto-elastic model: first overtone (n=1) crustal shear mode quickly absorbed into the Alfvén continuum (Gabler+ 12, vanHoven & Levi 12). Need for a new model that explains both low- and high-frequency QPOs.
Superfluid neutrons in the core (Baym+ 69). Favoured by pulsar glitches (Anderson & Itoh 75), and cooling curve of Cas A consistent with phase transition to superfluid neutrons (Shternin+ 11, Page+ 11).
Superfluid neutron star core Newtonian two fluid model
Neutrons:
Charged particles (protons):
∂tρn +∇ · (ρnvn) = 0
∂tρp +∇ · (ρpvp) = 0
(∂t + vn∇)(vn + εnwpn) +∇(Φ+ µn) + εnwpnk ∇vkn = 0
(∂t + vp∇)(vp + εpwnp) +∇(Φ+ µp) + εpwnpk ∇vkp =
(∇×B)×B
4πρp
wnp = −wpn = vn − vp ε: entrainment parameter
(see Passamonti & Andersson 2012 for details; perturbative approach)
(measure of interaction of different species)
Superfluid neutron star core One fluid approximation
Use an effective one fluid model (decoupling n from p). Only protons dynamically linked to magneto-elastic oscillations.
ρ → ρp ∼ 0.05ρ
Spϕ = (ρph+ b2)W 2vϕ − αbϕb
0
Fundamental QPOs exist as before but with:
fsf ∼1
tA∼ vA
R∼ B
R√ρp
∼ B
R√0.05ρ
∼ 5× fn
Less strong magnetic fields needed to match observed QPOs
2× 1014 ≤ B ≤ 1015 G
Andersson+ 00, Glampedakis+ 11, vanHoven & Levin 11 & 12, Passamonti & Lander 13
(broad agreement with spin down estimates)
B > 3.2× 1019(PP )1/2 G
0 10 20 30 40 50Time [ms]
-1
-0.5
0
0.5
1
Am
pli
tud
e o
f n
=1
cru
stal
mo
de
superfluidnormal fluid
10 20 30 40 50
0 200 400 600 800 1000Frequency [Hz]
10-6
10-4
10-2
100
Res
cale
d F
ouri
er a
mpli
tude
!=0.1 (s)!=1.5 (n)
High-frequency QPOs
Rapid initial damping
Long-lived QPOs at
f ∼ fn=1crust
B=1015 G
893 Hz (superfluid case)782, 806, 829 Hz (normal fluid)
Initial perturbation: crustal n=1 shear mode (f~760 Hz)
!2
!4
!6
!8
!10
8
10
6
4
2
0
1062 40 8 0 2 4 6 8
Y [
km
]
X [km] X [km]
Superfluid Normal fluid
Superfluid neutron star core - High-frequency QPOs
Normal fluid
Superfluid
n=1 radial shear mode structure localized close to equatorial plane
predominantly shear mode only in crust
B ⊥ r ⇒
n=1 radial shear mode structure localized close to poleResonance with Alfvén (~40th) overtone of core
• Approach to study magneto-elastic oscillations of magnetars presented
• n=0 crustal shear modes damped efficiently
• n=0 magneto-elastic modes can only explain observed low-frequency QPOs
• QPOs related to magnetic fields confined to the crust (type I superconducting core) cannot explain observed frequencies
• Inclusion of superfluid effects:- explains both low- and high-frequency QPOs- B-field values in agreement with spin down observations
Summary
For the first time in a realistic magnetar model both groups of frequencies can be explained. QPOs of SGRs are probably superfluid magneto-elastic QPOs.
Gabler+ 2013 arXiv:1304.3566
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