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Neutron Star Magnetic Fields: Genesis & Decay. Chris Thompson (CITA). Kramer & Stairs 2008. Spinning down Neutron stars (Not accreting). 10 10 T. 10 9 T. 10 8 T. Magnetars. B ~ 10 14 -10 15 G P ~ 5-10 s. 10 7 T. 10 6 T. L X ~ 10 35 -10 46 erg/s >> L spindown.

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Neutron Star Magnetic Fields: Genesis & Decay

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spinning down neutron stars not accreting

Kramer & Stairs 2008

Spinning downNeutron stars(Not accreting)

1010 T

109 T

108 T


B ~ 1014-1015 G

P ~ 5-10 s

107 T

106 T

LX ~ 1035-1046 erg/s

>> Lspindown

B ~ 108-109 G

P ~ 1-100 ms




QED Magnetic Field:

Radio pulsars generally B < BQED

Soft Gamma Repeaters

Anomalous X-ray Pulsars

Radio pulsars: 10-12 is small !

Why aren’t all neutron stars magnetars?

How strong is internal (e.g. toroidal) magnetic

field compared with the dipole -

Are some radio pulsars `very transient’ magnetars?


(threading one


Magnetic Flux



A-B-O stars






Helical equilibria of

a magnetic field

in a stably stratified star

Braithwaite 2009:

Braithwaite & Spruit (2004)

assumed random

initial magnetic field

Etoroidal > 30 Epoloidal

is possible numerically

with mainly toroidal

seed field

some magnetar formation channels
(Some) Magnetar Formation Channels

0. Maybe it’s just conserved `fossil’ magnetic flux

(any neutron star formation channel)

Wickramasinghe & Ferrario

Collapsed stellar core with rapid rotation

(+ seed magnetic field above threshold value)

T & Duncan, T & Murray, Blackman et al.

2. Merger of neutron star with dense stellar core

(and/or white dwarf)

Collapse of rapidly rotating white dwarf

AIC with moderately strong (106 G) B-field T & Duncan

or WD + WD merger King et al; Levan et al.;

Metzger et al.


Magnetars from Supernova Collapse

  • Violent convection extends close to -sphere:



Helical dynamo when

  • Accretion of angular momentum in outermost

from shock instabilities

Need to make

1016 G r.m.s.!



Buras et al. 2005


some magnetar formation channels1
(Some) Magnetar Formation Channels

0. Maybe it’s just conserved `fossil’ magnetic flux

(any neutron star formation channel)

Wickramasinghe & Ferrario

Collapsed stellar core with rapid rotation

(+ seed magnetic field above threshold value)

T & Duncan, T & Murray, Blackman et al.

2. Merger of neutron star with dense stellar core

(and/or white dwarf)

Collapse of rapidly rotating white dwarf

AIC with moderately strong (106 G) B-field T & Duncan

or WD + WD merger King et al; Levan et al.;

Metzger et al.


Some comments:

0. Most of the available energy in convection and

differential rotation is post-collapse

B > 107 G WD are a much smaller fraction of WD

pop (~ 0.01) than magnetars are of NS pop (> 0.1)

1./2./3. Dynamo within proto-neutron star,

vs. dynamo within accretion flow onto star

2. Mergers expected in some Be - NS binaries

but rate is too low (< 1/10 of magnetar formation)

3. AIC: WD rotates slowly if B > 106 G

WD-WD: Stable C burning(?) -> mini Wolf-Rayet star

white dwarf magnetic field distribution
White Dwarf Magnetic Field Distribution

Schmidt et al. (2003)

Isolated WD


~ 10% B > 106 G

B < 5x108 G


(synchronized CVs)

~25% CVs B < 4x107 G

Wickramasinghe & Ferrario (2001)

isolated magnetic white dwarf rotation
Isolated Magnetic White Dwarf Rotation

Non-magnetic white dwarfs: Prot ~ hours-days

Ultramagnetic white dwarfs (B > 100 MG)

with measured spin periods:

Prot , B = 130 min, 35 MG

725 s, ~300 MG

1.33 d, ~100 MG

98 min, ~120 MG

3.4 hr, ~200 MG



rotation of isolated white dwarfs
Rotation of Isolated White Dwarfs

Limiting spin period from AGB envelope collapse:

Disk forms if specific ang. mom.

Disk transfers ang. mom. outward,

so maximum spin ang. mom.

H-depleted core

Remnant AGB envelope

Mass (Spruit)


Dynamos in Massive Stars

Differential rotation driven by convection


Magnetic Energy  Gravitational Binding Energy

under an expansion or contraction


core carbon burning

• base of H-rich envelope

• convective H, He, C

burning in stellar core



• core collapse /

proto-neutron star

enforces nearly

solid-body rotation

pre main sequence convection
Pre-Main Sequence Convection

tKelvin ~ M/(dM/dt)

Building a massive star

by accretion

( )



Averaged magnetic flux

 ~ 1012 G x (10 km)2

Stronger B-fields if

Magnetic early-type stars

form by mergers?

Ferrario, Pringle et al. 2009

core collapse dynamos neutrino aided magnetic buoyancy
Core Collapse DynamosNeutrino-Aided Magnetic Buoyancy

Infall with



Toroidal B-field

amplified by

linear winding:

Strong shear


A magnetic flux tube is cooler than surroundings:

and is heated by neutrinos:


Competition between buoyancy and down-flow

neutrino heating

adiabatic cooling

Buoyancy speed:




outside -sphere


Magnetic field


emission of e/e

by decreasing

particle densities

gain region

(Q+ > Q-)


Fernandez & Thompson


Threshold seed magnetic field for a dynamo:

(shear concentrated

in surface layer)

Seed field is lower if angular momentum is

concentrated in outer layers of newborn neutron star

late fallback
Late Fallback

Neutrino heating is

too weak to drive


SASI helps to

trigger post-shock



origin of angular momentum fossil spin or shock instability
Origin of Angular Momentum:Fossil Spin or Shock Instability?

Spruit & Phinney (generalized random kicks)

Thompson 2000 (bernoulli fluctuations)

Blondin & Mezzacappa 2007 (m>0 SASI mode)


accretes all the angular momentum


Effective dynamo for


supernova accretion shock instability
Supernova Accretion Shock Instability

(2D, Full EOS with electron captures)

(Scheck et al. 2008)

Linear instability with neutrino flux suppressed at inner computational boundary but boundary moves rapidly inward


A linear instability is present in the ideal gas

accretion flow, especially for

What is its nature?

Does it persist when a significant fraction of

the flow kinetic energy is lost to nuclear dissociation?

Blondin &


Non-spherical shock

displacement 

Entropy + vortex perturbation 

Acoustic perturbation 

shock displacement

(SASI: Foglizzo & Tagger 2000)


Upstream of shock:

Downstream of shock:

Shock compression:


Shock displacements

much smaller at

finite dissociation


Expansion driven

mainly by turbulent

kinetic energy

Fernandez &

Thompson 2009ab

comparison of linear stability analysis and direct hydrodynamical simulation
Comparison of Linear Stability Analysisand Direct Hydrodynamical Simulation

Fernandez & T 2009



  • amplitudes grow
  • strongly just below
  • critical heating rate
  • for explosion
  • Asymmetries

driven by Bernoulli

fluctuations below

the shock

Fernandez & T 2009b


Neutron Stars:

Relativistic, degenerate

electron gas in outer, inner crust; core

Degenerate neutron gas

in core, inner crust;

proton gas in core

inner crust

and core


Time Evolution of the Magnetic Field

Very high electrical conductivity  

magnetic field lines `imprinted’ in fluid

on a MHD timescale ~ 10-1 sec

Induction equation:

fluid motions induced

by loss of stability

Drift of charged particle

+ field w.r.t neutrons

Hall drift: B-field advected

by the drift-motion of e-

Haensel et al. 1990;

Goldreich &

Reisenegger 1992


Ohmic Transport

White Dwarfs:

Electrical conductivity:

Ohmic decay time:

Neutron stars:

Ohmic decay in fluid core is very slow

Crust: scattering off impurities and lattice vibrations

hall drift
Hall Drift

Timescale depends on flux and charged particle

densities, not on transport coefficients

(White Dwarf)

(Neutron Star)

Density scale height ~ 0.3 km deep in NS crust

Hall drift is fast in outer NS crust,

or in presence of small-scale B-field:

twist ejection into magnetosphere

Mechanism of

Hall Drift

Twist Ejection into Magnetosphere

Turbulent cascade

Hall term in induction

equation is non-linear

Individual wave modes:

equivalent to whistlers

Goldreich &

Reisenegger 1992

T, Lyutikov &

Kulkarni 2002

ambipolar drift in neutron star cores
Ambipolar Drift in Neutron Star Cores

(Yakovlev et al. 1990; Goldreich & Reisenegger 1992; T & Duncan 1996)

~ 95% of mass is in neutral particles (neutrons)

Even for magnetars,

In degenerate Fermi fluid

Chemical equilibrium

chemical potential of species k


Low ion abundance

Adding gives


et al. 1990

Drag dominated:


Transport of charged particles pushes the

e-p-n fluid out of weak-interaction equilibrium


Transport then depends on rate of weak interactions,

which are very temperature sensitive:

Balancing gives

Pethick 1991; Goldreich & Reisenegger 1992

T & Duncan

an old question
An Old Question:

Do the neutrons in the core of a neutron

star form a superfluid (V.L. Ginzburg 1964)

Superfluidity results from binding of neutrons

into pairs near the Fermi surface

High neutron density -> pairing through P-wave

component of nuclear force

Pairing energy is very uncertain (medium effects….)

Forming and breaking of Cooper pairs leads

to enhanced cooling(Flowers, Ruderman & Sutherland)

an interesting observational clue
An interesting observational clue

Thermal X-ray output of Anomalous X-ray Pulsars

is clustered around 1x1035 erg/s (kTbb ~ 0.5 keV)

 buffered by neutrino cooling (Durant & Kerkwijk 2007)

Ambipolar heating of non-superfluid NS core

+ thermal conduction through 1015 G envelope

 Lbb = 0.5 x 1034 erg/s (T & Duncan 1996)

Lbb (Vela pulsar) = 10-2 Lbb (AXPs)

But: factor of 102 is hard to achieve just by

core magnetic heating

+ increased transmissivity of neutron star envelope

Neutrino emissivity of Vela needs to be higher


Increase thermal X-ray flux:

<Bsurface> is larger than spindown field (1x1015 G)

(3-5)x1015 G is needed to power SGR flares

Alternative: shallow magnetic heating

e.g. magnetospheric currents (T, Lyutikov & Kulkarni 2002)

field decay in upper crust (Kaminker et al. 2007)



Pairing temperature of core neutrons is 5-6 x 108 K

Magnetic field decay releases enough energy

to delay pairing transition from ~ 102 yrs to 104 yrs

(Arras, Cumming & Thompson 2004)


Delayed Core Superfluid Transition

(Tcn < 6x108 K)

 emissivities from

Yakovlev &


see Arras, Cumming, & T 2004

Page et al. 2009 for non-magnetic NS