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