! ""# " "!! $" % $ & ' %! % () % ) *&

+ # # # % &, &#&# # # #%, # &

% # #! %"! + & %!% ( % ))% ) * +"- &

. % + %

/ , "

"# % % %& ! ' + % # , -! % # 0

topgas rad

bottomgas rad

( )

( )

F g V P P

P P

ρ σ

σ

↓ ↓

↑ ↑

= − ∆ − + ⋅ +

+ + ⋅

0F =

, , gT P kνρ

ρ∆

σtop

… in stars with very low surface gravity

/% , % % $ 1 &#&%!% , # ! $

#+ % " #% , # 2 # $ , 34, " ! # # !&

1 1 nP K ρ +=( )gas rad

22 0

1 1gas

( )

( ) 4r

n

d P Pg r

drG

g r s dsr

P K

ρ

ρ π

ρ +

+= −

=

=

gas rad gas(1 )P P Pβ+ ≡ + ⋅

/%

2 # #+ % " , #+ !

-" 1# #,! !" 5 5 !%

/) %+ " , # )+ $ &

6+ 7

+ !+ ! , % "! &! & "! #+!)#% + % &8 #! ' "! % & # # % "! # + # &

Example of the Solar atmosphere /

≈96: 1&/ !

,2 % ;6411&/

≈ 644 # #+!

( #+!*3&96<= # "! &4>, % &/ #+! ?% ## &#

3<&<<

% # # !(@ ?*# % &2 + % σ <3

/ % # )% ) ! % + , A

!%%

6* / ! * #+!4* B! $"<* #! % "! :* ! $"* C 5% % !! # !9* ?% + D* B# % E* , ) ) # 6* (% !1 *66*#

$ / " + 1 % #( 7' *&% + % % !) &#&

+ !" ?,

1 #!%

7' " #! + %%

" , #

P kT N= ⋅k

e eP kT N= ⋅

2 2 3 2m kT=

part total( ) / exp( 2 )N N m kT→ ∆ ∝ − ∆

exp( )i iN E kT∝ −

.% ,

θI+

I-

θτ

Stellar surface

Stellar interiors

s - geometrical depth

τ - optical depth, pointsdown

θ- angle between thebeam and the normalto the surface. Wedefine µ≡cosθ, so thatthe path through thephotosphere is µ∆sor µ∆τ

I± - are the intensities of the two radiation streams

Atmosphere

C 5% +&%

! % % !

!"

#

!$

%(. * ! F #56 5 B'56 56GF #56 5 H56 56G&

$

!% F #56 54 B'56 56G&

?% % % %

" & !F#54G ? " F #56G), + ? F56G&

2E I I I I cλ ν λ ν

νλ ν∆ ≡ ∆ = ∆ =

dI k I ds j dsν ν ν νρ= − +Jν

k dsν ρ

/ ?

!&

I %% %

, $ " ,

d k dsντ ρ= −

dII S

dν

ν νµτ

= −

S j kν ν ν ρ=

+ #%

%!1=

+ $

)()0()( seIsI ντ−νν =

0 0

e d e dν ντ τν ν ντ τ τ

∞ ∞− −

ν

νντ −τ−

ντ−

ννν +=τ0

)()()0()( dtetSeII t

0 ,! , , "+ # $ J

/ #

1

$ 5 ,)%%

, , ,

( ) (1 )S B T I

I I d

ν ν ν

ν ν

ε ε= + −

= Ω

&

& #

$ ! $"( ,#+! *

+ , 1 , " , #% ! ++ % , #

, # % , ++ $ #! + $ &

gas rad

2rad

1cos sin

4

dP dPg

ds ds

P I dπ

νπ

ρ

θ θ θπ

+

−

+ = −

= Iνρ

gas kP Tmρ=

/ !

/ 1 #% "% + ?% ! % ( + ),+ ), *

(C)) ! ! * " + # #!? ! !" # $

%" & '

(" ) * '

+" &

," %

! 2$" / % + % % ( &#& )#* ' "!% ! % &" + $1!" , # , % &-' .-/

'()% % # # ! % %&% %, !K#L&

, !%$1!, ! !& % " "! # # ("% % *& *)% &

2 " C 1 % #" &

( )B Tν

?

(,*6* !

"