# Convection in Neutron Stars

Mechanics

Oct 24, 2013 (4 years and 6 months 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

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

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

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

(

)

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
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
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