numerical model atmospheres (gray 9)
DESCRIPTION
Numerical Model Atmospheres (Gray 9). Equations Hydrostatic Equilibrium Temperature Correction Schemes. Summary: Basic Equations. Physical State. Recall rate equations that link the populations in each ionization/excitation state Based primarily upon temperature and electron density - PowerPoint PPT PresentationTRANSCRIPT
1
Numerical Model Atmospheres (Gray 9)
EquationsHydrostatic Equilibrium
Temperature Correction Schemes
2
Summary: Basic EquationsEquation Corresponding
State Parameter
Radiative transfer Mean intensities, Jν
Radiative equilibrium Temperature, T
Hydrostatic equilibrium Total particle density, N
Statistical equilibrium Populations, ni
Charge conservation Electron density, ne
3
Physical State
• Recall rate equations that link the populations in each ionization/excitation state
• Based primarily upon temperature and electron density
• Given abundances, ne, T we can find N, Pg, and ρ
• With these state variables, we can calculate the gas opacity as a function of frequency
4
Hydrostatic Equilibrium
• Gravitational force inward is balanced by the pressure gradient outwards,
• Pressure may have several components: gas, radiation, turbulence, magnetic
• μ = # atomic mass units / free particle in gas
P g
P P P P P
NkTc
K dB
m N
g R t m
turb
H
4 1
2 42
2
5
Column Density
• Rewrite H.E. using column mass inwards (measured in g/cm2), “RHOX” in ATLAS
• Solution for constant T, μ (scale height):
dm dz m x dx
dP
dmg P m gm P
z
0
0
dP
dzg
m g
kTP d P
m g
kTdz
P z Pm g
kTz P e
H H
H z H
ln
exp /0 0
6
Gas Pressure Gradient
• Ignoring turbulence and magnetic fields:
• Radiation pressure acts against gravity (important in O-stars, supergiants)
gdP
dm
dP
dm
dP
dm
dP
dm
dP
dz
dP
dm c
dK
dd
dP
dm cH d
dP
dmg
cH d g
T
c
g R g R
g g
g eff
1
4 4
44
7
Temperature Relations
• If we knew T(m) and P(m) then we could get ρ(m) (gas law) and then find χν and ην
• Then solve the transfer equation for the radiative field (Sν = ην / χν )
• But normally we start with T(τ) not T(m)
• Since dm = -ρ dz = dτν / κν we can transform results to an optical depth scale by considering the opacity
8
ATLAS Approach (Kurucz)
• H.E.
• Start at top and estimate opacity κ from adopted gas pressure and temperature
• At next optical depth step down,
• Recalculate κ for mean between optical depth steps, then iterate to convergence
• Move down to next depth point and repeat
dP
dzg
dP
d
g
P Pg
g g
1 00
1 0
9
Temperature Distributions
• If we have a good T(τ) relation, then model is complete: T(τ) → P(τ) → ρ(τ) → radiation field
• However, usually first guess for T(τ) will not satisfy flux conservation at every depth point
• Use temperature correction schemes based upon radiative equilibrium
F d TR eff 4 /
B d J d
14
Temperature Correction Schemes
• “The temperature correction need not be very accurate, because successive iterations of the model remove small errors. It should be emphasized that the criterion for judging the effectiveness of a temperature correction scheme is the total amount of computer time needed to calculate a model. Mathematical rigor is irrelevant. Any empirically derived tricks for speeding convergence are completely justified.”(R. L. Kurucz)
15
Some T Correction Methods
• Λ iteration scheme
• Not too good at depth (cf. gray case)
B T T B T TB
T
B d TB
Td J d
TJ B d
B
Td
T
0 0
0
( )
16
Some T Correction Methods
• Unsöld-Lucy methodsimilar to gray case: find corrections to the source function = Planck function that keep flux conserved (good for LTE, not non-LTE)
• Avrett and Krook method (ATLAS)develop perturbation equations for both T and τ at discrete points (important for upper and lower depths, respectively); interpolate back to standard τ grid at end (useful even when convection carries a significant fraction of flux)