Convection in Neutron Stars

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Oct 24, 2013 (4 years and 15 days ago)

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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