Transcript Convection in Neutron Star

Convection
in Neutron Stars
Convection in the surface layers of neutron stars
Juan A. Miralles , V. Urpin , K. Van Riper
ApJ , 480:358-363, 1997
Department of Physics
National Tsing Hua University
G.T. Chen
2004/5/20
Outline
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Ideas
Assumptions
Basic Equations
Perturbation  I , II
Results
Problems and Future Work
Ideas
Convective transport
T may exceed
(T )ad
T
Superadiabatic gradient is
necessary condition for convective instability
use theoretical equations to
examine the condition in neutron
star surface layer
Assumptions
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Pure 56Fe atmosphere
Gravitational mass=1.4 M0
Radius=16.4 km
Superadiabatic zones are sensitive to surface
temperature
The thickness of the zone is almost equal to
the scale height of atm. (~cm)
Plane-parallel approximation with gravity g
perpendicular to the layer
Assumptions in Eq.s
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Use Boussinesq approximation
Neglect the viscous term in eq. of motion
Consider incompressible fluid
Magnetic permeability is not departure
Variations of pressure are small and their
contribution to thermal balance is
negligible
Next>
Due to smallness of coefficient of volume
expansion 
the variation of the density  in equations
can be ignored
but in external force term should not be
neglected. Because the acceleration resulting
from  ~ T can be quite large
Treat ρ as a constant in eq. of motion
except the one in external force
<back
Hydrodynamic and
Hydromagnetic stability ,
Chandrasekhar
Basic Equations
A
B
C
D
E
Basic Equations
v= fluid velocity
 j = c ▽ × B / 4π= electric current
 △▽ T = ▽ T - ▽ Tad
 χ=κ/ρcp η=c2 R/4π
χ=thermal conductivity
R =electric resisitivity
cp=specific heat at const. P
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Next>
A = eq. of motion
dv

F
dt
4 J
 B 
c
v
P
1
 (v )v  
g
J B
t

c
Equation (1)
B=incompressible fluid
d

 0   v 
dt
t
Continuity eq.
Eq. (2)

   (v )  0
t
C=Ohm’s law taken curl
Introduction to Plasma
Theory , Nicholson
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At low frequency
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Current is small J  B
1
E   vB
 c
J
t
J
Take curl
can be ignored
can be ignored
Eq. (3)
D
………………XD
(大家都知道吧)
E=energy conservation
dQ
dT
T
   q   c p
 cp (
 v T )
dt
dt
t
T
 v  (T )    ( ˆ T )  J  E
t
and
E 
J

Eq. (5)
T  (T )  (T )ad
Basic Equations
ˆ T  11T    T    b  T
ˆ B  1 (  B)1   (  B)    b  (  B)
b
B
B

is the so-called Hall component
b1
T
b  T
b
Perturbation
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Linearize eq.(1)~(5) by perturbation X  X 0  X1
Assumption:
v0  0
 ln 
  (
)p
T
j0  0
g   gez
q1    B1
B0
is uniform
1  T1
g
z
x
Perturbation
Perturbation
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Assume perturbed terms ~
e
( rt  ikx )
The dependence on vertical coordinate z is
given by eq. (6)~(10)
deduce to one eq.of higher order
Q 2  2  k 2
CA 
B
4
is the Alfen velocity
Boundary Condition
Assume the component of fluid velocity
vanishes at both bounding surfaces
v (z=0) = v (z=a) = 0
q1z  A sin(
z
a
)
2
2  k a
2
( 2  k 2 a 2 )
Frequency of
oscillations in
Hall current
Inverse timescales
of dissipation
Frequency of
Alfven wave
Frequency
of buoyant
wave
Derivation I
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Assume r=real  dynamical unstable
Set r=0  g 2   2 A2 k
 
The min. is approached for k infinity
Next page
Derivation I

4 

 2
 c  c p R
The ratio of the thermal and magnetic diffusivities
perpendicular to the magnetic field
(T )cr1
The convection will occur when the value of
superadiabatic adiabatic smaller than critical
value
Derivation II
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Assume r=imaginary  oscillating modes
Set r= i
Derivation II
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Assume the frequency of Alfven mode is
higher than  for the most unstable
perturbations
Consider the solutions at k  A
Derivation II
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Because
(19)
(20)
2
0
for an oscillating convection
Derivation II
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The value for superadiabatic tend to infinity
both at k  0 and k  infinity

There is a flat min. between k ~ a and k~ k0
( A   )
1/ 2
( A   )
1/ 2


a
a
k0  (A  )
1/ 2
k0  (aA  )
1/ 2
The convection will occur
when the value of
superadiabatic adiabatic
smaller than critical value
Results
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Dynamically unstable convection tend to occur in
region with  1 ,whereas a oscillating
convection seems to be more appropriate for
region with  1
The value  and the type of convection in the
surface layers of neutron star are strongly
dependent on the surface temperature
Results
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Take g~3*1014 cm s-2
Density ~ 1 g/cm3
a~ H (scale height) ~ 0.1-1 cm
The critical field stabilizing convections is of
the order of 107 ~ 109 G for ξ~104 ~10-4
These fields are small in comparison with the standard
field of neutron stars, and therefore convection can
probably arise only in very weakly magnetized neutron
stars
Problems & Future Work
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Read books to understand the properties
of convection in fluid mechanics and
plasma physics
Work out the detail in this paper
Check the assumptions in basic eq.s
consider r= real part + imaginary part ??
Use another temperature profile
To be continued…