Neutron Star Magnetic Mountains: Realistic Massflux and Equation of State

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Neutron Star Magnetic Mountains Realistic
Mass-flux and Equation of State
Maxim Priymak Supervisor Dr. A. Melatos
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Accreting Neutron Stars
Binary evolution through HMXB stage

• Accreting Neutron Stars (NS)
• X-ray sources
• LMXB (M lt M?)
• IMXB (1.5 M? M 10 M?)
• HMXB (M 10 M?)
• Angular momentum transfer
• NS spin up
• NS spins measured via
• X-ray pulsations
• Burst oscillations
• QPO ?

Binary evolution through LMXB stage
Rosetta stones

Tauris van den Heuvel 2003
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Neutron Star Spins

• Spin distribution cut off gt 700 Hz
• None at Obreak up (1500-3000 Hz)
• NOT due to selection effects
• Contradicts evolution theory
• Two mechanisms explain this
• Gravitational Wave (GW) emission
• Propeller effect
• Inconclusive evidence to support only one
  mechanism
• Both are thought to contribute

NS spin versus propeller effect spin equilibrium
? XTE J1739-285 ?
From tabulated data of Watts et al. 2008
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Gravitational Wave Torque

• Gravitational Wave (GW) torque Accretion torque
  
• Max spin (Bildsten 1998)
• Accretion torque
• GW torque
• Equilibrium spin
• Insensitive to M and Q
• Spin clustering
• Need mass quadrupole

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Magnetic Mountain Gravitational Waves

• Accretion driven (LMXB / IMXB / HMXB)
• B confines matter at the magnetic poles
• Hydrostatic P exceeds magnetic P
• Accreted matter spreads
• Magnetic field distorted
• NS asphericity (biaxial in 2D / triaxial 3D )
• Magnetic axis misaligned with spin axis
• Q ? 0
• CW gravitational radiation
• Advantages (cf. inspiralling binaries GW
  chirp signal)
• X-ray / Optical / Radio observations
• Known position / signal frequency
• Persistent signal

2D axisymmetric magnetic mountain
3D non-axisymmetric magnetic mountain
Distance (m)
Distance (m)
All pictures courtesy of M. Vigelius
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Current Magnetic Mountain Models
Time evolution of 3D perturbed magnetic mountain

• 2D (Payne Melatos 2004)
• Ideal MHD
• Grad-Shafranov equilibrium
• Axisymmetric structure
• Stable
• 3D (Vigelius Melatos 2008)
• Non-ideal MHD
• ZEUS-MP 2D equilibrium
• Realistic accreted masses
• Stable
• Current model deficiencies
• Rigid crust no sinking
• Irrotational NS no Coriolis force
• Axisymmetric accretion no inclination
• Constant BCs no crustal freezing
• Isothermal no variable resistivity
• Ideal isothermal EoS (P cs2?)

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Ideal Magnetohydrostatics (MHS)
The equations of non-ideal magnetohydrodynamics
(MHD) (in SI) are
mass conservation
Pressure gradient
momentum equation
Gravitational force
Net Force
Lorentz force (pressure tension)
induction equation
In the ideal magnetostatic limit (ie.
, , ), the continuity
and induction equations are satisfied. The
momentum equation becomes
This must be supplemented by the condition
and an equation of state (EoS)
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2D MHS Mass - Flux Conservation
To preserve the information encoded in the
continuity and induction eqns impose an
auxiliary constraint mass-fluxinitial state
mass-fluxfinal state mass-fluxaccreted
material
Mathematically, this mass-flux can be expressed
as
Initial State
Final State
?1
?2
?1
?3
?2
?3
?4
?4
?5
?5
?6
?6
?7
?7
?8
?9
?8
?10
?9
?10
?1 lt ?2 lt...lt ?9 lt ?10 and
as mass
cannot cross magnetic flux surfaces in the ideal
limit
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Grad - Shafranov Equation Ideal Isothermal EoS
In the case of ideal isothermal EoS (
), and an axisymmetric B field
( ), the MHS momentum
equation can be solved to yield
1)
2)
3)
Grad-Shafranov operator is NB Eqn 3) is
derived by substituting Eqn 1) into the relation
for dM(?)/d?. Eqns 2) and 3) are solved
iteratively, as F(?)initial ? F(?)final.even
though dM(?)/d?initial dM(?)/d?final
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Grad - Shafranov Equation Adiabatic EoS
In the case of an adiabatic EoS (
), the MHS equilibrium equations are
1)
2)
3)
NB F(?) on both sides of the equation
where ,
, ,

IMPLEMENTED
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Grad-Shafranov Ideal Isothermal EoS Ma 10-5 M?
Density
Magnitude of the Magnetic Field
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
Poloidal Angle
Poloidal Angle
Magnetic Field Lines
Current Density
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
Poloidal Angle
e 1.0e-6
Poloidal Angle
GS with Adiabatic EoS being tested
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Realistic EoS Crustal Composition

• Realistic EoS includes
• Thermal degenerate (rel. non-rel.) electron
  pressures (Paczynski 1983)
• Degenerate (non-rel.) neutron pressure (Negele
  Vautherin 1973)
• Ionic lattice pressure (Farouki Hamaguchi 1993)
• These depend on composition (Z, A) of crust
• One Component Plasma (OCP) composition of Haensel
  Zdunik (1990) is assumed

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Realistic EoS

• Baryon density n, nuclear density nN,
  Wigner-Seitz cell radius a, plasma coupling
  parameter ? computed
• Degenerate e- lattice degenerate neutron
  pressure components calculated
• Pressure continuity across density jumps enforced
• Adiabatic index dlog(P)/dlog(?)

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Ideal/Realistic EoS Ma 10-5 M?
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Realistic EoS Ma 10-5 M? with/without
compositional variations
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LIGO/ALIGO Estimates

• (Axisymmetry) GW strain h is
• Ellipticity e is
• Relative ellipticities b/w 1D MHS equilibria
  suggest
• Realistic EoS decreases ellipticity ( 1 order
  of magnitude )
• GW strain is similarly modulated
• Compositional variations decrease ellipticity (
  5 )

LIGO locations
LIGO/ALIGO detectability curves
www.cs.unc.edu
Vigelius et al. 2008
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Realistic Accretion Rate/Mass-flux

• X-ray pulsations NS possesses an
  inclination angle between magnetic dipole axis
  and spin axis
• Need 3D accretion simulations to deduce realistic
  Mass-Flux Cornell University
• Have M(r, ?) (for GS) and M(r, ?, f) (for Zeus
  MP) on the NS surface for 4 NS magnetic dipole
  axis inclinations (ie. 5, 15, 30, 60 wrt the
  spin axis)

? 5
? 15
? 30
? 60
Default b 3
Default b 10
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GS equilibrium ? 5/Default, Ma1.0e-5 M?
Density
Magnitude of the Magnetic Field
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
Poloidal Angle
Poloidal Angle
Magnetic Field Lines
Current Density
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
e 1.2e-5
Poloidal Angle
Poloidal Angle
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GS equilibrium ? 15/Default, Ma1.0e-5 M?
Density
Magnitude of the Magnetic Field
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
Poloidal Angle
Poloidal Angle
Magnetic Field Lines
Current Density
Distance above NS (pressure scaleheights)
Distance above NS (pressure scaleheights)
e 8.5e-6
Poloidal Angle
Poloidal Angle
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Magnetospheric Feedback

• Cornell (M.Romanova, R.V.E.Lovelace,
  A.K.Kulkarni)
• 3D MHD simulations of accretion
  disk/magnetosphere interaction around a NS
  M(r, ?, f) at NS surface
• Melbourne (M.Priymak, M.Vigelius, A.Melatos
• 2D MHS (Grad-Shafranov) 3D MHD (Zeus MP)
  simulations of magnetic mountain distortion
  of B
• Cornell Melbourne (iterative process)
• Accretion generates a mountain distorts B
  affects the accretion disk change in
  accretion dynamics change in mountain
  dynamics (ie. dM/d?, M etc...)

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Current Work

• Obtain results for adiabatic GS
• Validate results of semi-phenomenological model
  
• Deduce effects on the accretion rate via
  magnetospheric feedback
• Generalize GS code for generic EoS Implement
  realistic EoS

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Future Work

• Implement in 3D non-ideal MHD ( Zeus-MP2, Flash
  etc )
• Gauge mountain stability
• Deduce time evolution of Q (mass quadrupole
  moment) and magnetic multipole moments
• Zeus-MP2 Time dependent Magnetospheric effects
• GS/Zeus-MP2 crust freezing/crust sinking
• Application to X-ray bursts
• Light curves cyclones
• Episodic decay of the mountain
• WHY?
• Quantify the effects on GW detectability by
  LIGO/ALIGO
• Construct search templates
• Infer NS properties (accreted mass, conductivity
  etc)

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The End
Thank you for your attention.
Any Questions?