Neutron Stars 4: Magnetism
Departamento de Astronomía y Astrofísica
Pontificia Universidad Católica de Chile
• General books:
– Russell M. Kulsrud, Plasma Physics for Astrophysics
– Leon Mestel, Stellar Magnetism
– Alice Harding & Dong Lai, Physics of strongly magnetized neutron stars,
Rep. Prog. Phys., 69, 2631 (2006): includes interesting physics (QED,
etc.) that occurs in magnetar-strength fields - not covered in this
– A. Reisenegger, conference reviews:
• Origin & evolution of neutron star magnetic fields, astro-ph/0307133: General
• Magnetic fields in neutron stars: a theoretical perspective, astro-ph/0503047:
• Magnetic field evolution in neutron stars, arXiv:0710.2839: Theoretical, short
– Goldreich & Reisenegger 1992, ApJ
– Hoyos, Reisenegger, & Valdivia 2008, A&A
– Reisenegger 2009, A&A
• Classes of NSs, evidence for B
• Magnetohydrodynamics (MHD) & flux freezing
• Comparison to other, related stars, origin of B in NSs
• Magnetic equilibria
• Observational evidence for B evolution
• Physical mechanisms for B evolution
– External: Accretion
– Internal: Ambipolar diffusion, Hall drift, resistive decay
Caution: Little is known for sure – many speculations!
Spin-down I 2 d B 2 4
(magnetic dipole model) 3 2
Kaspi et al. 1999
Objects Emission B determination log B [G] log age [yr]
Classical pulsars Radio to Spin-down 11-13 3-8
Millisecond Radio to Spin-down 8-9 8-10
Magnetars Gamma, X, Spin-down, LX 14-15 (-16?) 3-5
(SGRs & AXPs) IR
RRATs Radio, X Spin-down 12-14 5-7
Isolated thermal X, optical Spin-down, 13-14 4-6
“Magnificent 7” cyclotron lines
Thermal CCOs X Spin-down 12.5??? 2.5-4.5
HMXBs X Cyclotron lines 12 young
LMXBs X Absence of 8-9? old
Note large range of Bs, but few if any non-magnetic NSs
Neutron star magnetic fields
• Strongest B in the Universe, up to at least ~1015G.
• Cause rotational energy loss: accounts for bolometric luminosity of pulsars
• Soft gamma-ray repeaters (SGRs) & Anomalous X-ray Pulsars (AXPs):
X/gamma-ray luminosity >> rotational energy loss or cooling
Magnetically powered neutron stars or “Magnetars”
(Thompson & Duncan 1993, 1995, 1996)
Quasi-periodic oscillations (QPOs) may be probing magnetic structure inside the
star (Levin 2007)
• (Slight) deformation of NS due to B might cause:
– Precession (observed?)
– Gravitational waves (hope!)
From R. Duncan’s “magnetar” web page, http://solomon.as.utexas.edu/~duncan/magnetar.html
L RI 0 I e L
r 1 L r 2
L~ 2 R~ tdecay ~ 2
c r R c
• tdecay is long in astrophysical contexts (r large),
>> Hubble time in NSs (Baym et al. 1969)
• Alternative: deform the “circuit” in order to move
the magnetic field MHD
dv j B
Assume 1 fluid moving with P
Electrons have small mass: neglect their inertia, gravity, etc.:
E vB 0
Induction equation B
c E (v B)
(advection of field lines)
Current density is secondary, calculated by j B
Magnetic field origin?
• Fossil: flux conservation during core collapse:
– Woltjer (1964) predicted NSs with B up to ~1015G.
• Dynamo in convective, rapidly (differentially)
rotating proto-neutron star (~ minutes)
– Scaling from solar dynamo led to prediction of “magnetars”
with B~1016G (Thompson & Duncan 1993)
• Both?: Some memory of initial conditions, but
strongly modified by differential rotation, etc.?
Highly disordered field:
(random component~kG) >>
Inversion every 11 yrs
Probably due to convection
+ differential rotation
A&A, 358, 929 (2000)
Upper main Only small fraction detectably magnetic
(Ap, Bp or CP=“Chemically Peculiar”)
sequence Ordered field: low-order multipoles ~ kG
(Ap, Bp stars) Convective core + stable, radiative envelope
A&A, 358, 929 (2000)
Small fraction of all
massive than non-
Ordered field, low
multipoles ~ MG
Stars with long-lived, ordered B-fields
Radius Bmax [G] Flux
[solar units] R2Bmax
Upper main 3 3104 (“Ap” stars) 106
White dwarfs 10-2 109 3105
Neutron stars 10-5 1015 (magnetars) 3105
In all cases, (magnetic pressure) < 10-6 (fluid pressure).
All are stably stratified.
• EG ~ GM2/R ~ Nn ~ 1054 erg
• E = I2/2 ~ 1053 Pms-2 erg
• ET ~ N(kT)2/ n ~ 1046 T82 erg
• EB ~ (B2/8)(4R3/3) ~ 1048 B152 erg
Generally E , ET , EB << EG: small perturbations
Barotropic fluid: density = (P) [P = pressure]
Non-barotropic fluid: density = (P,Y),
where Y = another, independent variable:
• Specific entropy in radiative zones of stars (upper MS & WDs)
• Composition (e.g., proton fraction) in neutron stars
(Pethick 1992; Reisenegger & Goldreich 1992; Reisenegger 2009)
• Like water with non-uniform temperature or salinity:
– Colder or saltier water stays at the bottom
– Weak B can’t force substantial, non-radial motions
Equilibrium only in
• Force balance: P
• B as small perturbation:
– Background 0 P0 0 0
– Perturbation P 0
(fluid perturbation described by 2 independent scalars)
Stable magnetic field configurations
Braithwaite & Spruit 2004: simulation of ideal MHD in fluid, stably stratified star.
B quickly reaches an equilibrium configuration with poloidal & toroidal components.
Equilibria & stability
• Poloidal-toroidal decomposition:
– Pure poloidal & pure toroidal field are unstable
(Flowers & Ruderman 1977; Tayler 1973)
• Our current (semi-)analytic work
– Calculation of Flowers-Ruderman instability
– Construction of non-barotropic, poloidal +
toroidal equilibria (A. Mastrano, T. Akgün)
– Find unstable modes of toroidal fields, study
stabilizing effect of poloidal component (T.
Evidence for B-field evolution
• Magnetars: LX , | I |
B decay as main energy source?
requires internal field ~10x inferred dipole
• Young NSs have strong B (classical pulsars, HMXBs),
old NSs have weak B (MSPs, LMXBs).
Result of accretion?
• (Classical) Pulsar population statistics: no decay? -
contradictory claims (Narayan & Ostriker 1990; Bhattacharya
1992; Regimbau & de Freitas Pacheco 2001)
• “Braking index” in young pulsars n
progressive increase of inferred B
Material accreted in the LMXB stage is highly ionized
conducting magnetic flux is advected
Accreted material could screen the original B, which
remains inside the star, but is not detectable outside
(Bisnovatyi-Kogan & Komberg 1975, Romani 1993, Payne & Melatos
• Do instabilities prevent this?
• Why 108-9 G, but not 0?
Speculation: Magnetic accretion?
Can the field of MSPs have been transported onto
them by the accreted flow?
GM jB B2
Force balance: ~ ~
c 4 R
Mass transport: M ~ f 4 R 2 v ~ f ' 4 R 2 2GM
GMM 2 4
M M Edd 2
Combination: B~ ~ 108 G
2 5 f'
on magnetic accretion
The strongest magnetic field that might be forced onto a
neutron star by an LMXB accretion flow is close to that
observed in MSPs.
More serious exploration is required (S. Flores, PhD thesis in
– Hydrodynamic model: transport through “turbulent viscosity” or
– Is the magn. flux transported from the companion star?
– Is it generated in the disk (“magneto-rotational inst.”)?
– Is it coherent enough?
B evolution inside NS
Protons & electrons move through a fixed neutron background, colliding with each
other and with the background (Goldreich & Reisenegger 1992):
v A B
t ne e
• Ambipolar diffusion: Driven by magnetic stresses (Lorentz force), protons &
electrons move together, carrying the magnetic flux and dissipating magnetic
• Hall drift: Magnetic flux carried by the electric current; non-dissipative, may
cause “Hall turbulence” to smaller scales.
• Ohmic or resistive diffusion: very small on large scales; important for ending
“Hall cascade”. May be important in the crust (uncertain conductivity!).
Time scales depend on B (nonlinear!), lengthscales, microscopic interactions.
Cooper pairing (n superfluidity, p superconductivity) is not included (not well
understood, but see Ruderman, astro-ph/0410607).
• Spontaneous field decay is unlikely for parameters
characteristic of pulsars, unless the field is confined to a thin
surface layer (Goldreich & Reisenegger 1992)
• Spontaneous field decay could happen for magnetar
parameters (Thompson & Duncan 1996)
• Simulations (include moving neutrons):
– 1-d: Hoyos, Reisenegger, & Valdivia 2008
– 2-d: in progress
Magnetic fields have:
– Very small effect on structure of stars
– Strong effect on NS appearance & evolution (pulsar braking,
– Source currents due to moving p, e, or other charged particles
– Uncertain origin: fossil – dynamo – both ?
– (possibly) Stable equilibrium configurations with linked toroidal &
poloidal components, thanks to stable stratification
– Non-trivial evolution, even in the most “prosaic” NS models (no
need for ferromagnetism, quarks, Cooper pairs, etc. ...):
• Internal (ambipolar diffusion, weak interactions) in magnetars
• External (diamagnetic screening, flux accretion) in LMXBs MSPs