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: , 1997

Outline Ideas Ideas Assumptions Assumptions Basic Equations Basic Equations Perturbation  I, II Perturbation  I, II Results Results Problems and Future Work Problems and Future Work

Ideas Convective transport Convective transport may exceed may exceed Superadiabatic gradient is 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 Pure 56 Fe atmosphere Gravitational mass=1.4 M 0 Gravitational mass=1.4 M 0 Radius=16.4 km Radius=16.4 km Superadiabatic zones are sensitive to surface temperature Superadiabatic zones are sensitive to surface temperature The thickness of the zone is almost equal to the scale height of atm. (~cm) The thickness of the zone is almost equal to the scale height of atm. (~cm) Plane-parallel approximation with gravity perpendicular to the layer Plane-parallel approximation with gravity perpendicular to the layer

Assumptions in Eq.s Use Boussinesq approximation Use Boussinesq approximationBoussinesq approximationBoussinesq approximation Neglect the viscous term in eq. of motion Neglect the viscous term in eq. of motion Consider incompressible fluid Consider incompressible fluid Magnetic permeability is not departure Magnetic permeability is not departure Variations of pressure are small and their contribution to thermal balance is negligible 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 the variation of the density in equations can be ignored can be ignored but in external force term should not be but in external force term should not be neglected. Because the acceleration resulting neglected. Because the acceleration resulting from ~ can be quite large from ~ can be quite large Treat ρ as a constant in eq. of motion Treat ρ as a constant in eq. of motion except the one in external force except the one in external force Hydrodynamic and Hydromagnetic stability, Chandrasekhar <back

Basic Equations A B C D E

v= fluid velocity v= fluid velocity j = c ▽ × B / 4π= electric current j = c ▽ × B / 4π= electric current △▽ T = ▽ T - ▽ T ad △▽ T = ▽ T - ▽ T ad χ=κ/ρc p η =c 2 R/4π χ=κ/ρc p η =c 2 R/4π χ=thermal conductivity χ=thermal conductivity R =electric resisitivity R =electric resisitivity c p =specific heat at const. P 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 At low frequency can be ignored Current is small 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 ) ( 大家都知道吧 ) 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 Linearize eq.(1)~(5) by perturbation Assumption: Assumption: is uniform g z x

Perturbation

Perturbation Assume perturbed terms ~ Assume perturbed terms ~ The dependence on vertical coordinate z is given by eq. (6)~(10) The dependence on vertical coordinate z is given by eq. (6)~(10) deduce to one eq.of higher order deduce to one eq.of higher order is the Alfen velocity

Boundary Condition Assume the component of fluid velocity vanishes at both bounding surfaces 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 Assume r=real  dynamical unstable Set r=0  Set r=0  The min. is approached for k  infinity 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 Assume r=imaginary  oscillating modes Set r= Set r=

Derivation II Assume the frequency of Alfven mode is higher than for the most unstable perturbations Assume the frequency of Alfven mode is higher than for the most unstable perturbations Consider the solutions at Consider the solutions at

Derivation II Because for an oscillating convection Because for an oscillating convection (19) (20)

Derivation II The value for superadiabatic tend to infinity both at k  0 and k  infinity The value for superadiabatic tend to infinity both at k  0 and k  infinity There is a flat min. between k ~ and k~ 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 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 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 Take g~3*10 14 cm s -2 Density ~ 1 g/cm 3 Density ~ 1 g/cm 3 a~ H (scale height) ~ cm a~ H (scale height) ~ 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 Read books to understand the properties of convection in fluid mechanics and plasma physics Work out the detail in this paper Work out the detail in this paper Check the assumptions in basic eq.s Check the assumptions in basic eq.s consider r= real part + imaginary part ?? consider r= real part + imaginary part ?? Use another temperature profile Use another temperature profile

To be continued …