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Numerical Model Atmospheres (Gray 9)

EquationsHydrostatic Equilibrium

Temperature Correction Schemes

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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

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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

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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

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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

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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

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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

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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

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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

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Solar Temperature Relation

• From Eddington-Barbier (limb darkening)

τ0 = τ(5000 Å)

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Rescaling for Other Stars

TT

T sunTeff

e ffsun

( )

Reasonable starting approximation

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Temperature Relations for Supergiants

• Differences smalldespite very different length scales

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Other Effects on T(τ)

ConvectionIncluding line opacity

or line blanketing

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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)

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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

( )

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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)

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Some T Correction Methods

• Auer & Mihalas (1969, ApJ, 158, 641) linearization method: build in ΔT correction in Feautrier method

• Matrices more complicated• Solve for intensities then update ΔT

II B

I BB

TT

I BB

T

J B d

B

Td

( )