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Neutron Stars 4 Magnetism

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					Neutron Stars 4: Magnetism

         Andreas Reisenegger
   Departamento de Astronomía y Astrofísica
    Pontificia Universidad Católica de Chile
                       Bibliography
• General books:
   – Russell M. Kulsrud, Plasma Physics for Astrophysics
   – Leon Mestel, Stellar Magnetism

• Reviews:
   – 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
     presentation
   – 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:
         Theoretical
       • Magnetic field evolution in neutron stars, arXiv:0710.2839: Theoretical, short

• Papers:
   – Goldreich & Reisenegger 1992, ApJ
   – Hoyos, Reisenegger, & Valdivia 2008, A&A
   – Reisenegger 2009, A&A
                               Outline
• 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!
                                                                          2
Spin-down  I  2 d   B 2 4
                                                                   2
                        
(magnetic dipole model)   3 2
                                                         3c       dt

                                                                   Magnetic field:
                                                                           
                                                                          ||
                                                                       B
                                                                            3


                                                                  Spin-down time
                                                                    (age?):
                                                                             
Lyne 2000,
                                                                       ts    
                                                                            2||
http://online.kitp.ucsb.edu/online/neustars_c00/lyne/oh/03.html
                                                   “Magnetars”




                                          Classical pulsars




                    Millisecond pulsars



Kaspi et al. 1999
Objects          Emission     B determination log B [G]      log age [yr]

Classical pulsars Radio to    Spin-down         11-13        3-8
                  gamma
Millisecond      Radio to     Spin-down         8-9          8-10
pulsars          gamma
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
in SNRs
HMXBs            X            Cyclotron lines   12           young
LMXBs            X            Absence of         8-9?        old
                              pulsations, others

     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.
• Persistent
• 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!)
 Magnetic
    field
 strengths




From R. Duncan’s “magnetar” web page, http://solomon.as.utexas.edu/~duncan/magnetar.html
  Flux freezing
                             R
 dI                          t
L  RI  0  I  e           L
 dt
   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) 
   “flux freezing”
 • Alternative: deform the “circuit” in order to move
   the magnetic field  MHD
          MagnetoHydroDynamics
                                            
                                         dv j  B
Assume 1 fluid moving with                       P  
                                         dt   c
Electrons have small mass: neglect their inertia, gravity, etc.:
                       1 
                      E  vB  0
                         c
                               
Induction equation            B                   
                                   c  E    (v  B)
(advection of field lines)
                              t
                                                      c      
Current density is secondary, calculated by          j     B
                                                        4
            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.?
                                        http://science.nasa.gov/ssl/pad/solar/maghstry.htm



                Sun
Highly disordered field:
  (random component~kG) >>
  (dipole component~50G)


Inversion every 11 yrs
Probably due to convection
  + differential rotation
  (dynamo effect)

http://solarscience.msfc.nasa.gov/3dfields.shtml
                                            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)
 Magnetic
white dwarfs

Small fraction of all
  WDs
(Statistically) more
  massive than non-
  magnetic WDs
Ordered field, low
  multipoles ~ MG
  Stars with long-lived, ordered B-fields
                Radius        Bmax [G]           Flux
                [solar units]                    R2Bmax
Upper main      3             3104 (“Ap” stars) 106
sequence
White dwarfs 10-2              109                   3105
Neutron stars 10-5             1015 (magnetars)      3105

In all cases, (magnetic pressure) < 10-6 (fluid pressure).
    Weak B!!
All are stably stratified.
                 NS energies
• EG ~ GM2/R ~ Nn ~ 1054 erg
• E = I2/2              ~ 1053 Pms-2 erg
• ET ~ N(kT)2/ n         ~ 1046 T82 erg
• EB ~ (B2/8)(4R3/3) ~ 1048 B152 erg


Generally E , ET , EB << EG: small perturbations
               Stable stratification
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
cross section




                Equilibrium only in
                non-barotropic fluid
           Magnetic equilibria
                         
                        jB
• Force balance:             P  
                         c



• B as small perturbation:
  – Background               0  P0   0 0
                          
                         jB
  – Perturbation                 P    0
                           c
  (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
     (P. Marchant)
   – Construction of non-barotropic, poloidal +
     toroidal equilibria (A. Mastrano, T. Akgün)
                                                        Braithwaite 2007
   – Find unstable modes of toroidal fields, study
     stabilizing effect of poloidal component (T.
     Akgün)
   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)
                                          2  3
• “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
  2004, 2007)

Questions:

• Do instabilities prevent this?

• Why 108-9 G, but not 0?
   Diamagnetic
    screening
   Speculation: Magnetic accretion?
Can the field of MSPs have been transported onto
  them by the accreted flow?

                    GM    jB    B2
Force balance:           ~     ~
                     R 2
                            c    4 R

Mass transport: M ~ f  4 R 2   v ~ f '  4 R 2  2GM
                
                                                          R
                                    1                         1
                            
                       GMM 2          4            
                                                   M M Edd  2
Combination:        B~                    ~ 108           G
                      2f' R
                          2   5                     f' 
                                                         
           Preliminary conclusions
            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
 progress):
   – Hydrodynamic model: transport through “turbulent viscosity” or
     wind
   – 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):
                                    
        B                         j 
                                              
                                                                
              v A  B     
                                 
                                                    c
                                          B                   j
        t                         ne e                       
Terms:
• Ambipolar diffusion: Driven by magnetic stresses (Lorentz force), protons &
   electrons move together, carrying the magnetic flux and dissipating magnetic
   energy.
• 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).
               Model conclusions

• 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
                        Conclusions
Magnetic fields have:
   – Very small effect on structure of stars
   – Strong effect on NS appearance & evolution (pulsar braking,
     magnetars)
   – 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

				
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