Transcript Numerical Model Atmospheres (Gray 9) Equations Hydrostatic Equilibrium
Numerical Model Atmospheres (Gray 9)
Equations Hydrostatic Equilibrium Temperature Correction Schemes 1
Summary: Basic Equations
Equation Radiative transfer Corresponding State Parameter Mean intensities,
J ν
Radiative equilibrium Hydrostatic equilibrium Statistical equilibrium Charge conservation Temperature,
T
Total particle density,
N
Populations,
n i
Electron density,
n e
2
Physical State
• Recall rate equations that link the populations in each ionization/excitation state • Based primarily upon temperature and electron density • Given abundances,
n e
,
T
we can find
N
,
P g
, and
ρ
• With these state variables, we can calculate the gas opacity as a function of frequency 3
Hydrostatic Equilibrium
• • Gravitational force inward is balanced by the pressure gradient outwards,
P
g
• Pressure may have several components: gas, radiation, turbulence, magnetic
P
P g
P R
P t
P m
NkT
4
c
m N H
1 2 2
turb
B
2 4
μ
= # atomic mass units / free particle in gas 4
Column Density
• Rewrite H.E. using column mass inwards (measured in g/cm 2 ), “RHOX” in ATLAS
z dm
dz
0
dP
gm
dm
• Solution for constant
T
,
μ P
(scale height):
dP dz
g P
m g H
exp
kT
kT P m g H
d
z
ln
P
m g H P
e
kT dz
5
Gas Pressure Gradient
• Ignoring turbulence and magnetic fields:
g
dP dm
dP g dm
dP R dm
dP g dm
1
dP R dz
dP g dm
4
c
dK
d
d
dP g dm
4
c
dP g dm
4
c
T eff
4
c
• Radiation pressure acts against gravity (important in O-stars, supergiants) 6
Temperature Relations
• If we knew
T(m)
get
ρ(m)
and
P(m)
then we could (gas law) and then find
χ ν
and
η ν
• Then solve the transfer equation for the radiative field (
S ν
=
η ν / χ ν
) • But normally we start with
T( τ)
not
T(m)
• Since
dm
=
ρ dz
=
d τ ν
/
κ ν
we can transform results to an optical depth scale by considering the opacity 7
ATLAS Approach (Kurucz)
• H.E.
dP dz
g
dP d
g
• Start at top and estimate opacity κ from adopted gas pressure and temperature • At next optical depth step down,
P g
P g
0 1 0 • Recalculate κ for mean between optical depth steps, then iterate to convergence • Move down to next depth point and repeat 8
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
R T eff
4 / • Use temperature correction schemes based upon radiative equilibrium 9
Solar Temperature Relation
• From Eddington-Barbier (limb darkening)
τ 0
=
τ
(5000 Å) 10
Rescaling for Other Stars
T
T eff T eff
(
sun
)
T sun
Reasonable starting approximation 11
Temperature Relations for Supergiants • Differences small despite very different length scales 12
Other Effects on
T( τ)
Convection Including line opacity or line blanketing 13
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) 14
Some
T
Correction Methods
• Λ iteration scheme 0
T
T
B
T T
0
T
B
T d
T
(
J
B
T B
)
d
d
• Not too good at depth (cf. gray case) 15
Some
T
Correction Methods
• Unsöld-Lucy method similar 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) 16
Some
T
Correction Methods
• Auer & Mihalas (1969, ApJ, 158, 641) linearization method: build in
ΔT
correction in Feautrier method
I
I
B
I
B
B
T
T
I
B
B
T
(
J
B
T B
)
d
d
• Matrices more complicated • Solve for intensities then update
ΔT
17