Convection
in Neutron Stars
Department of Physics
National Tsing Hua University
G.T. Chen
2004/5/20
Convection in the surface layers of neutron stars
Juan A. Miralles , V. Urpin , K. Van Riper
ApJ , 480:358

363, 1997
Outline
Ideas
Assumptions
Basic Equations
Perturbation
I , II
Results
Problems and Future Work
Ideas
Convective transport
may exceed
Superadiabatic gradient is
necessary condition for convective instability
use theoretical equations to
examine the condition in neutron
star surface layer
Assumptions
Pure
56
Fe atmosphere
Gravitational mass=1.4 M
0
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
perpendicular to the layer
Assumptions in Eq.s
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 ~ can be quite large
Treat ρ as a constant in eq. of motion
except the one in external force
Hydrodynamic and
Hydromagnetic stability ,
Chandrasekhar
<back
Basic Equations
A
B
C
D
E
Basic Equations
v= fluid velocity
j = c
▽
×
B / 4π= electric current
△▽
T =
▽
T

▽
T
ad
χ=κ/ρc
p
η
=c
2
R/4π
χ=thermal conductivity
R =electric resisitivity
c
p
=specific heat at const. P
Next>
A = eq. of motion
Equation (1)
B=incompressible fluid
Continuity eq.
Eq. (2)
C=Ohm
’
s law taken curl
At low frequency can be ignored
Current is small can be ignored
Eq. (3)
Take curl
Introduction to Plasma
Theory , Nicholson
D
………………
XD
(
大家都知道吧
)
E=energy conservation
Eq. (5)
and
Basic Equations
is the so

called Hall component
Perturbation
Linearize eq.(1)~(5) by perturbation
Assumption:
is uniform
g
z x
Perturbation
Perturbation
Assume perturbed terms ~
The dependence on vertical coordinate z is
given by eq. (6)~(10)
deduce to one eq.of higher order
is the Alfen velocity
Boundary Condition
Assume the component of fluid velocity
vanishes at both bounding surfaces
v (z=0) = v (z=a) = 0
Inverse timescales
of dissipation
Frequency of
oscillations in
Hall current
Frequency of
Alfven wave
Frequency
of buoyant
wave
Derivation I
Assume r=real
dynamical unstable
Set r=0
The min. is approached for k
infinity
Next page
Derivation I
The ratio of the thermal and magnetic diffusivities
perpendicular to the magnetic field
The convection will occur when the value of
superadiabatic adiabatic smaller than critical
value
Derivation II
Assume r=imaginary
oscillating modes
Set r=
Derivation II
Assume the frequency of Alfven mode is
higher than for the most unstable
perturbations
Consider the solutions at
Derivation II
Because for an oscillating convection
(19)
(20)
Derivation II
The value for superadiabatic tend to infinity
both at k
0 and k
infinity
There is a flat min. between k ~ and k~
The convection will occur
when the value of
superadiabatic adiabatic
smaller than critical value
Results
Dynamically unstable convection tend to occur in
region with ,whereas a oscillating
convection seems to be more appropriate for
region with
The value and the type of convection in the
surface layers of neutron star are strongly
dependent on the surface temperature
Results
Take g~3*10
14
cm s

2
Density ~ 1 g/cm
3
a~ H (scale height) ~ 0.1

1 cm
The critical field stabilizing convections is of
the order of 10
7
~ 10
9
G for ξ~10
4
~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
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
…
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