3. Transport of energy: radiation

specific intensity, radiative flux optical depth absorption & emission equation of transfer, source function formal solution, limb darkening temperature distribution grey atmosphere, mean opacities 1 Energy flux conservation

No sinks and sources of energy in the atmosphere  all energy produced in stellar interior is transported through the atmosphere  at any given radius r in the atmosphere:

4().rFr2  const L

F is the energy flux per unit surface and per unit time. Dimensions: [erg/cm2/sec] The energy transport is sustained by the temperature gradient. The steepness of this gradient is dependent on the effectiveness of the energy transport through the different atmospheric layers. 2 Transport of energy

Mechanisms of energy transport a. radiation: Frad (most important) b. convection: F (important especially in cool stars) conv c. heat production: e.g. in the transition between solar

cromosphere and corona d. radial flow of matter : corona and stellar wind e. sound waves: cromosphere and corona

We will be mostly concerned with the first 2 mechanisms: F(r)=Frad(r) + Fconv(r). In the outer layers, we always have Frad >> Fconv

3 The specific intensity

Measures of energy flow: Specific Intensity and Flux

The amount of energy dEν transported through a surface area dA is proportional to dt (length of time), dν (frequency width), dω (solid angle) and the projected unit surface area cos θ dA.

The proportionality factor is the specific Intensity Iν(cosθ)

dEν = Iν (cos θ ) cos θ dA dω dν dt 2 1 1 1 ([Iν ]: erg cm− sr− Hz− s− )

c Iλ = λ2 Iν (from Iλdλ = Iν dν and ν = c/λ)

Intensity depends on location in space, direction and frequency 4 Invariance of the specific intensity

The area element dA emits radiation towards dA’. In the absence of any matter between emitter and receiver (no absorption and emission on the light paths between the surface elements) the amount of energy emitted and received through each surface elements is:

dEν = Iν (cos θ) cos θ dA dω dν dt dEν0 = Iν0 (cos θ0) cos θ0 dA0 dω0 dν dt

5 Invariance of the specific intensity

projected area dA0 cos θ0 dω = distance2 = r2 energy is conserved: dEν = dEν0 and dA cos θ dω0 = r2 dE = I (cos θ) cos θ dA dω dν dt and ν ν dEν0 = Iν0 (cos θ0) cos θ0 dA0 dω0 dν dt Specific intensity is constant along rays - as long as there is no absorption and emission Iν = Iν0 of matter between emitter and receiver

In TE: I = B

6 Spherical coordinate system and solid angle dω

dA solid angle : dω = r2 4πr2 Total solid angle = = 4π r2

dA = (r dθ)(r sin θ dφ)

dω = sin θ dθ dφ →

define μ = cos θ

dμ = sin θ dθ − dω = sin θ dθ dφ = dμ dφ − 7 Radiative flux

How much energy flows through surface element dA?

dE ~ I cosθ dω  integrate over the whole solid angle (4):

2π π πFν = Iν (cos θ) cos θ dω = Iν (cos θ) cos θ sin θdθdφ 0 0 “astrophysical flux” 4Zπ Z Z

Fν is the monochromatic radiative flux. The factor π in the definition is historical.

Fν can also be interpreted as the net rate of energy flow through a surface element. 8

Radiative flux

The monochromatic radiative flux at frequency  gives the net rate of energy flow through a surface element. dE ~ I cosθ dω  integrate over the whole solid angle (4): 2π π

π Fν = Iν (cos θ) cos θ dω = Iν (cos θ) cos θ sin θdθdφ “astrophysical flux” 0 0 4Zπ Z Z We distinguish between the outward direction (0 <  < /2) and the inward direction (/2 < so that the net flux is

+ πF = πF πF − = ν ν − ν 2π π/2 2π π = Iν (cos θ) cos θ sin θdθdφ + Iν (cos θ) cos θ sin θdθdφ Z0 Z0 Z0 Zπ/2 Note: for /2 < cosθ < 0  second term negative !! 9

Total radiative flux

Integral over frequencies 

∞ πF dν = ν Frad Z0

Frad is the total radiative flux.

It is the total net amount of energy going through the surface element per unit time and unit surface.

10 Stellar luminosity

At the outer boundary of atmosphere (r = Ro) there is no incident radiation  Integral interval over θ reduces from [0,π] to [0,π/2].

2π π/2 + πFν (Ro) = πFν (Ro) = Iν (cos θ) cos θ sin θdθdφ Z0 Z0 This is the monochromatic energy that each surface element of the star radiates in all directions

2 If we multiply by the total stellar surface 4πR0 2 4πRo πFν (Ro)= Lν  monochromatic stellar luminosity at frequency  ·

and integrating over   total stellar luminosity 2 ∞ + 4πRo πFν (Ro)dν = L (Luminosity) · 11 Z0 Observed flux

What radiative flux is measured by an observer at distance d?

 integrate specific intensity Iν towards observer over all surface elements note that only half sphere contributes

Eν = dE = ∆ω ∆ν ∆ t Iν(cos θ) cos θ dA 1/2 sZphere 1/2 sZphere

2 in spherical symmetry: dA = Ro sin θ dθ dφ

2π π/2 E = ∆ω ∆ν ∆t R2 I (cos θ) cos θ sin θ dθ dφ → ν o ν Zo Z0

+ πFν

 because of spherical symmetry the integral of intensity towards + the observer over the stellar surface is proportional to F , the

flux emitted into all directions by one surface element !! 12 Observed flux

Solid angle of telescope at distance d:

∆ω = ∆A/d2

+

2 + Eν = ∆ω ∆ν ∆t Ro πFν (Ro)

flux received = flux emitted x (R/r)2 radiative energy R2 unlike I , decreases with obs = = o πF +(R )  F Fν area frequency time d2 ν o increasing distance · ·

This, and not I, is the quantity generally measured for stars. For the Sun, whose disk is resolved, we can also measure I  ∞ obs 2 0 ν, dν = 1.36 KW/m F ¯ (the variation of I over the solar disk is called the limb R darkening) 13

Mean intensity, energy density & radiation pressure

Integrating over the solid angle and dividing by 4:

1 J = I dω mean intensity ν 4π ν 4Zπ

radiation energy 1 4π u = = I dω = J energy density ν volume c ν c ν 4Zπ

1 p = I cos2 θ dω radiation pressure ν c ν 4Zπ (important in hot stars)

force d momentum(= E/c) 1 pressure = = area dt area 14 Moments of the specific intensity

1 1 2π 1 1 1 0th moment Jν = Iν dω = dφ Iν dμ = Iν dμ 4π 4π 0 1 2 1 Z Z Z− Z− for azimuthal symmetry

1 1 1 Fν st Hν = Iν cos θ dω = Iν μ dμ = 1 moment 4π 2 1 4 Z Z− (Eddington flux)

1 1 2 1 2 c Kν = Iν cos θ dω = Iν μ dμ = pν 2nd moment 4π 2 1 4π Z Z−

15 InteractionsInteractions betweenbetween photonsphotons andand mattermatter

absorption of radiation

ds dI = κ I ds ν − ν ν

κν : absorption coefficient I 1 [κν ] = cm−

microscopical view: n  loss of intensity in the beam (true absorption/scattering)

Over a distance s: s κν ds s o − 0 o Iν (s) = Iν e R I I(s) s optical depth τν := κν ds Convention:  = 0 at the outer edge (dimensionless) of the atmosphere, increasing Z0 16 inwards or: d =  ds

optical depth

The optical thickness of a layer o τν determines the fraction of the Iν (s) = I e− ν intensity passing through the layer

o Iν o We can see through if τν = 1 Iν = 0.37 Iν → e ' atmosphere until  ~ 1

optically thick (thin) medium:  > (<) 1  The quantity  has a geometrical interpretation in terms of mean  free path of photons s¯ :

s¯

τν = 1 = κν ds Zo photons travel on average for a length s¯ before absorption 17 photon mean free path

What is the average distance over which photons travel?

∞ expectation value < τν >= τν p(τν ) dτν Z0

probability of absorption in interval [+d]

= probability of non-absorption between 0 and  and absorption in d

∆I(τ) Io I(τν ) I(τν ) - probability that photon is absorbed: p(0, τν ) = = − = 1 Io Io − Io I(τ ) ν τν - probability that photon is not absorbed: 1 p(0, τν ) = = e− − Io

dIν - probability that photon is absorbed in [τν , τν + dτν ] : p(τν , τν + dτν ) = = dτν I(τν )

τν total probability: e− dτν 18 photon mean free path

∞ ∞ x x τν xe− dx = (1 + x) e− < τν >= τν p(τν ) dτν = τν e− dτν = 1 − Z Z0 Z0

mean free path corresponds to <>=1

1 if κν (s) = const : ∆τν = κν ∆s ∆s = s¯ = → κν (homogeneous material)

19 Principle of line formation

observer sees through the

atmospheric layers up to   1

In the continuum  is smaller than in the line  see deeper into the atmosphere

T(cont) > T(line) 20 radiative acceleration

In the absorption process photons release momentum E/c to the atoms, and the corresponding force is:

momentum(=E/c) force =dfphot = dt

The infinitesimal energy absorbed is:

abs dEν = dIν cos θ dA dω dt dν = κν Iν cosθ dA dω dt dν ds

The total energy absorbed is (assuming that κν does not depend on ω):

∞ ∞ abs E = κν Iν cos θ dω dν dA dt ds = π κν Fν dν dA dt ds Z0 4Zπ Z0

 F 21 radiative acceleration

∞ κν Fν dν π df = 0 dA dt ds = g dm (dm = ρ dA ds) phot c R dt rad

π ∞ g = κ F dν rad cρ ν ν Z0

22 emission of radiation

ds

d dA

dV=dA ds

energy added by emis sion processes within dV

em dEν = ²ν dV dω dν dt

²ν : emission coefficient 3 1 1 1 [²ν ] = erg cm− sr− Hz− s−

23 TheThe equationequation ofof radiativeradiative tranfertranfer

If we combine absorption and emission together: dEabs = dIabs dA cos θ dω dν dt = κ I dA cos θ dω dt dν ds ν ν − ν ν em em dEν = dIν dA cos θ dω dν dt = ²ν dA cos θ dω dν dt ds

dEabs+dEem = (dIabs+dI em) dA cos θ dω dν dt = ( κ I +² ) dA cos θ dω dν dt ds ν ν ν ν − ν ν ν

dI = dIabs + dIem = ( κ I + ² ) ds ν ν ν − ν ν ν differential equation dIν = κ I + ² describing the flow of ds − ν ν ν radiation through matter 24 TheThe equationequation ofof radiativeradiative tranfertranfer

Plane-parallel symmetry

dx = cos θ ds = μ ds

d d = μ ds dx

μ dIν (μ,x) = κ I (μ, x) + ² dx − ν ν ν

25 TheThe equationequation ofof radiativeradiative tranfertranfer

Spherical symmetry

d dr ∂ dθ ∂ = + ds ds ∂r ds ∂θ

dr dr = ds cos θ = cos θ (as in plane parallel) → ds − dθ sin θ r dθ = sin θ ds (dθ < 0) = − → ds − r

∂ ∂μ ∂ ∂ = = sin θ ∂θ ∂θ ∂μ − ∂μ

d ∂ sin2 θ ∂ ∂ 1 μ2 ∂ = = μ + = μ + − ⇒ ds ∂r r ∂μ ∂r r ∂μ

∂ 1 μ2 ∂ μ ∂r Iν (μ, r) + −r ∂μ Iν (μ, r) = κν Iν (μ, r) + ²ν angle  between ray and radial direction − is not constant 26 TheThe equationequation ofof radiativeradiative tranfertranfer

Optical depth and source function

optical depth increasing In plane-parallel symmetry: towards interior:

κν dx = dτν dI (μ,x) − μ ν = κ (x) I (μ, x) + ² (x) x dx − ν ν ν τ = κ dx ν − ν RZo

dIν (μ,τν ) μ = Iν (μ, τν ) Sν (τν ) dτν −

S = ²ν ν κν source function Observed emerging intensity I(cos ,= 0) depends on = cos  ,  (R ) and S dim [S] = [I]  i 

dτν ∆τν 1 = 1 corresponds to free mean path of κν =  The physics of S is crucial for radiative transfer ds ≈ ∆s ≈ s¯ photons  ²ν S = ² s¯ source function S corresponds to intensity ν κν ν  ≈ · emitted over the free mean path of photons 27

TheThe equationequation ofof radiativeradiative tranfertranfer

Source function: simple cases

a. LTE (thermal absorption/emission)

S = ²ν = B (T ) Kirchhoff’s law ν κν ν photons are absorbed and re-emitted at the local independent of radiation field temperature T

Knowledge of T stratification T=T(x) or T()

 solution of transfer equation I(,)

28 TheThe equationequation ofof radiativeradiative tranfertranfer

Source function: simple cases

b. coherent isotropic scattering (e.g. Thomson scattering) the absorption process is characterized by the  = ’ scattering coefficient , analogous to  incident = scattered dI = σ I ds em sc ν ν ν dEν = ²ν dω − 4Zπ em abs abs and at each frequency  dEν = dEν dEν = σν Iνdω 4Zπ sc ²ν dω = σν Iν dω 4Zπ 4Zπ sc sc ²ν 1 ²ν dω = σν Iν dω = Iν dω σν 4π 4Zπ 4Zπ 4Zπ completely dependent on radiation field 29 Sν = Jν not dependent on temperature T TheThe equationequation ofof radiativeradiative tranfertranfer

Source function: simple cases

c. mixed case

sc sc ²ν + ²ν κν ²ν σν ²ν Sν = = + κν + σν κν + σν κν κν + σν σν

sc ²ν + ²ν κν σν Sν = = Bν + Jν κν + σν κν + σν κν + σν

30 Formal solution of the equation of radiative tranfer

linear 1st order differential equation we want to solve the equation of RT dIν with a known source function and in μ = Iν Sν plane-parallel geometry dτν −

- multiply by e and integrate between 1 (outside) and 2 (> 1, inside)

d S e τν /μ τν /μ ν − check, whether this really yields transfer (Iν e− ) = equation above dτν − μ

τ2 τ τν 2 τν dtν Iν e− μ = Sν e− μ τ1 − μ τZ h i 1 31 τ2 τ τν 2 τν dtν Iν e− μ = Sν e− μ τ1 − μ h i τZ1

Formal solution of the equation of radiative tranfer

integral form of equation of radiation transfer τ2 τ τ t τ 2− 1 − 1 dt I (τ , μ) = I (τ , μ) e− μ + S (t) e− μ ν 1 ν 2 ν μ τZ1

intensity originating at 2 decreased by exponential factor to  1 contribution to the intensity by

emission along the path from 2 to 1 (at each point decreased by the exponential factor)

Formal solution! actual solution

can be complex, since S can depend on I 32 τ2 τ τ t τ 2− 1 − 1 dt I (τ , μ) = I (τ , μ) e− μ + S (t) e− μ ν 1 ν 2 ν μ τZ1

Boundary conditions

solution of RT equation requires boundary conditions, which are different for incoming and outgoing radiation a. incoming radiation:  < 0 at   0

usually we can neglect irradiation from outside: I(  0,  < 0) = 0

0 t τ in − ν dt I (τ , μ) = S (t) e− μ ν ν ν μ τZν

33 τ2 τ τ t τ 2− 1 − 1 dt I (τ , μ) = I (τ , μ) e− μ + S (t) e− μ ν 1 ν 2 ν μ τZ1

Boundary conditions

b. outgoing radiation:  > 0 at  = max  ∞

+ finite slab or We have either I (τ , μ) = I (μ ) ν max ν shell

τ/μ or lim Iν (τ, μ) e− = 0 semi-infinite τ case (planar →∞ or spherical) I increases less rapidly than the exponential

∞ t τ out − ν dt I (τν , μ) = Sν (t) e− μ ν and at a given position  in μ the atmosphere: τZν out in Iν(τν ) = Iν (τν ) + Iν (τν ) 34 Emergent intensity

from the latter  emergent intensity

 = 0,  > 0

∞ t dt I (0, μ) = S (t) e− μ ν ν μ Z0

intensity observed is a weighted average of the source function along the line of sight. The contribution to the emerging intensity comes mostly from each depths with  < 1.

35 Emergent intensity

suppose that S is linear in  (Taylor expansion around  = 0):

Sν (τν ) = S0ν + S1ν τν

x x xe− dx = (1 + x) e− − Z emergent intensity ∞ t dt I (0, μ) = (S + S t)e− μ = S + S μ ν 0ν 1ν μ 0ν 1ν Z0

Iν (0, μ) = Sν (τν = μ) Eddington-Barbier relation

we see the source function the emergent intensity corresponds to at location   the source function at  = 1 along the line of sight 36

Emergent intensity

 = 1 (normal direction):

Iν (0, 1) = Sν (τν = 1)

 = 0.5 (slanted direction):

Iν (0, 0. 5) = Sν (τν = 0.5)

in both cases:   1

spectral lines: compared to

continuum  = 1 is reached at higher layer in the atmosphere

line cont S < S 37  a dip is created in the spectrum Line formation

simplify:  = 1, 1=0 (emergent intensity), 2 =  S independent of location 

τν τ t τ τ I (0) = I (τ ) e− ν + S e− dt = I (τ ) e− ν + S (1 e− ν ) ν ν ν ν ν ν ν − Z0

38 Line formation

τ τ Optically thick object: I (0) = I (τ ) e− ν + S (1 e− ν ) = S ν ν ν ν − ν   ∞

Optically thin object: I (0) = I (τ ) + [S I (τ )] τ ν ν ν ν − ν ν ν

exp(-)  1 - 

39 I =  S =  ds S e.g. HII region, solar corona

enhanced 

independent of , no line (e.g. black body B)

I (0) = I (τ ) + [S I (τ )] τ ν ν ν ν − ν ν ν

e.g. stellar absorption spectrum (temperature decreasing outwards)

e.g. stellar spectrum with temperature increasing outwards (e.g. Sun in the UV)

40 From Rutten’s web notes

Line formation example: solar corona

I =  S

41 The diffusion approximation

At large optical depth in stellar atmosphere photons are local: S  B

Expand S B as a power-series:

∞ n d Bν n Sν (t) = n (t τν ) /n! dτν − nX=0

In the diffusion approximation ( >>1) we retain only first order terms:

dBν Bν (t) = Bν (τν ) + (t τν ) dτν −

∞ out dBν (t τν )/μ dt Iν (τν , μ) = [Bν (τν ) + (t τν )]e− − dτν − μ 42 τZν ∞ out dBν (t τν )/μ dt Iν (τν , μ) = [Bν (τν ) + (t τν )]e− − dτν − μ τZν

The diffusion approximation

t τν Substituting: t u = − dt = μ du → μ → ∞ k u u e− du = k! Z0

∞ out dBν u dBν Iν (τν , μ) = [Bν (τν ) + μu]e− du = Bν (τν ) + μ dτν dτν Z0

τν /μ in dBν u Iν (τν , μ) = [Bν (τν ) + μu]e− du − dτν Z0

At  = 0 we obtain the Eddington-Barbier relation for the observed emergent intensity.

It is given by the Planck-function and its gradient at  = 0.

It depends linearly on μ = cos θ.

43 diffusion approximation:

Solar limb darkening

center-to-limb variation of B (0)+ dBν μ intensity Iν (0,μ) ν dτν = dB Iν (0,1) B (0)+ ν ν dτν

from the intensity measurements

 B, dB/d

3 dBν 2hν 1 B (t) = B (0) + t = a + b t = 2 ν ν dτν c ehν/kT (t) 1 · −

T(t): empirical temperature stratification of solar photosphere 44 Solar limb darkening

...and also giant planets 45 Solar limb darkening: temperature stratification

∞ t dt I (0, μ) = S (t) e− μ ν ν μ Z0

exponential extinction varies as - /cos

From S = a + b

Iν(0, cos θ) = aν + bν cos θ

I (0, μ) = S (τ = μ) ν ν ν S

R. Rutten, web notes

Unsoeld, 68 46 we want to obtain an approximation for the radiation field – both inward and outward radiation - at large optical depth  stellar interior, inner boundary of Eddington approximation atmosphere

In the diffusion approximation we had:

dBν Bν (t) = Bν (τν ) + (t τν ) dτν −

out dBν Iν (τν, μ) = Bν(τν) + μ 0 <  < 1 dτν

τν /μ in dBν u Iν (τν , μ) = [Bν (τν ) + μu]e− du -1 <  < 0 − dτν Z0

 >> 1 dBν Iν−(τν, μ) = Bν(τν ) + μ dτν

47 With this approximation for Iν we can calculate the angle averaged momenta of the intensity

simple approximation for photon flux and a

relationship between mean intensity Jν and Kν Eddington approximation  very important for analytical estimates

1 1 Jν = Iν dμ = Bν (τν ) 2 1 Z−

1 Fν 1 1 dBν 1 1 dBν 1 dBν dT Hν = = μ Iν dμ = = = −3 κν dx −3κν dT dx 4 2 1 3 dτν Z− flux F ~ dT/dx diffusion: flux ~ gradient (e.g. 1 heat conduction) 1 2 1 Kν = μ Iν dμ = Bν (τν ) 2 1 3 Z−

1 Kν = 3 Jν Eddington approximation

48 After the previous approximations, we now want to calculate exact solutions for tha

radiative momenta Jν, Hν, Kν. Those are important to calculate spectra and Schwarzschild-Milne equations atmospheric structure

∞ t τ out − ν dt I (τ , μ) = S (t) e− μ ν ν ν μ 1 1 0 τZν 1 1 out 1 in 0 Jν = Iν dμ = Iν dμ + Iν dμ t τ 2 2 2 in − ν dt Z1 Z0 Z1 I (τν , μ) = Sν (t) e− μ − − ν μ τZν  > 0  < 0

1 0 τν ∞ 1 (t τ )/μ dt (t τ )/μ dt J = S (t)e− − ν dμ S (t)e− − ν dμ ν 2 ⎡ ν μ − ν μ ⎤ Z0 τZν Z1 Z0 ⎣ − ⎦ substitute w = 1 dw = 1 dμ w = 1 dw = 1 dμ μ ⇒ w − μ − μ ⇒ w − μ

τν ∞ ∞ ∞ 1 (t τ )w dw (τ t)w dw J = S (t)e− − ν dt + S (t)e− ν − dt ν 2 ⎡ ν w ν w ⎤ Z1 τZν Z1 Z0 ⎣ ⎦ 49 Schwarzschild-Milne equations

τν ∞ ∞ ∞ 1 (t τ )w dw (τ t)w dw J = S (t) e− − ν dt + S (t) e− ν − dt ν 2 ⎡ ν w ν w ⎤ τZν Z1 Z0 Z1 ⎣ ⎦ > 0 > 0

∞ ∞ ∞ 1 w t τ dw 1 J = S (t) e− | − ν| dt = S (t)E ( t τ )dt ν 2 ν w 2 ν 1 | − ν | Z0 Z1 Z0

Schwarzschild’s equation

50 Schwarzschild-Milne equations

where

∞ ∞ x tx dx e− E (t) = e− = dx 1 x x Z1 Zt is the first exponential integral (singularity at t=0)

Exponential integrals

∞ n 1 n x En(t) = t − x− e− dx Zt t E (0) = 1/(n 1), E (t ) = e− /t 0 n − n → ∞ → dEn = En 1, En(t) = En+1(t) dt − − − Z E (0) = E (0) = 1 E (0) = 1/2 E ( ) = 0 1 ∞ 2 3 n ∞

Gray, 92 51 Schwarzschild-Milne equations

1 ∞ Introducing the  operator: Λ [f(t)] = f(t)E ( t τ ) dt τν 2 1 | − ν | Z0

Jν (τν ) = Λτν [Sν (t)]

Similarly for the other 2 moments of Intensity:

Milne’s equations

J, H and K are all depth-weighted means of S 52

Gray, 92

Schwarzschild-Milne equations the 3 moments of Intensity:

J, H and K are all depth-weighted means of S  the strongest contribution comes from the depth, where

the argument of the exponential integrals is zero, i.e. t= 53 TheThe temperaturetemperature--opticaloptical depthdepth relationrelation

Radiative equilibrium

The condition of radiative equilibrium (expressing conservation of energy) requires that the flux at any given depth remains constant:

∞ ∞ ∞ (r) = πF = I cos θ dω dν = π F dν = 4π H dν F ν ν ν Z0 4Zπ Z0 Z0

∞ 4πr2 (r) = 4πr2 4π H dν = const = L F · ν Z0 ∞ In plane-parallel geometry r  R = const 4π Hν dν = const Z0

∞ and in analogy to the black body 4 radiation, from the Stefan-Boltzmann law 4π Hν dν = σTeff we define the effective temperature: Z0 54 TheThe effectiveeffective temperaturetemperature

∞ 4 The effective temperature is defined by: 4π Hν dν = σTeff Z It characterizes the total radiative flux 0 transported through the atmosphere.

It can be regarded as an average of the temperature over depth in the atmosphere. A blackbody radiating the same amount of total energy would have a

temperature T = Teff.

55 Radiative equilibrium

Let us now combine the condition of radiative equilibrium with the equation of radiative transfer in plane-parallel geometry:

dI μ ν = (κ + σ ) (I S ) dx − ν ν ν − ν

1 1 1 dI 1 μ ν dμ = (κ + σ ) (I S ) dμ 2 dx − 2 ν ν ν − ν Z1 Z1 − − 1 d 1 μ I dμ = (κ + σ ) (J S ) dx ⎡ 2 ν ⎤ − ν ν ν − ν Z1 ⎣ − ⎦

H

56 Radiative equilibrium

Integrate over frequency:

d ∞ ∞ H dν = (κ + σ ) (J S ) dν dx ν − ν ν ν − ν Z0 Z0

const

∞ κ ν σ ν s u b s t it u t e S ν = B ν + J ν (κν + σν ) (Jν Sν ) dν = 0 κ + σ κ + σ − ν ν ν ν Z0 ∞

⎛ κν Jν dν = absorbed energy⎞ Z ∞ ⎝ 0 ⎠ at each depth: κν [Jν Bν (T )] dν = 0 ∞ − ⎛ κν Bν dν = emitted energy⎞ Z0 Z ⎝ 0 ⎠ ∞ 4 in addition: 4π Hν dν = σTeff T(x) or T() Z0 57 Radiative equilibrium

∞ ∞ κ [J B (T )] dν = 0 4π H dν = σT 4 ν ν − ν ν eff Z 0 Z0

T(x) or T()

The temperature T(r) at every depth has to assume the value for which the left integral over all frequencies becomes zero.

 This determines the local temperature.

58 Iterative method for calculation of a stellar atmosphere:

the major parameters are Teff and g

a. hydrostatic equilibrium dP = gρ(x) dx − equation of transfer b. equation of radiation T(x), (x), B[T(x)],P(x), ρ(x) J(x), H(x) transfer

dIν μ = (κν + σν) (Iν Sν) dx − − c. radiative equilibrium

∞ ∞ κν (Jν Bν ) dν = 0 ? κν [Jν Bν (T )] dν = 0 0 − − Z0 ∞ 4 R4π 0 Hν dν = σ Teff ? d. flux conservation

∞ 4 R 4π Hν dν = σTeff Z0 e. equation of state T(x), (x), B[T(x)], ρ(x) ρ k T P = μ m H 59

Grey atmosphere - an approximation for the temperature structure

We derive a simple analytical approximation for the temperature structure. We assume that we can approximate the radiative equilibrium integral by using a frequency-averaged absorption coefficient, which we can put in front of the integral.

∞ ∞ κ [J B (T )] dν = 0 κ¯ [Jν Bν (T )] dν = 0 ν ν − ν − Z0 Z0

∞ ∞ ∞ ∞ σT 4 With: J = J dν H = H dν K = K dν B = B dν = ν ν ν ν π Z0 Z0 Z0 Z0

J = B 4 4πH = σTeff

60 Grey atmosphere

We then assume LTE: S = B. From 1 ∞ J (τ ) = Λ [S (t)] = S (t)E ( t τ )dt ν ν τν ν 2 ν 1 | − ν | Z0 and a similar expression for frequency-integrated quantities J(¯τ )=Λ¯τ [S(t)],d¯τ =¯κdx

and with the approximations S = B, B = J:

Milne’s equation 1 ∞ J (¯τ)=Λ [J(t)]= J (t)E ( t τ¯ )dt τ¯ 2 1 | − | !!! this is an integral Z0 equation for J)  61 The exact solution of the Hopf integral equation

Milne’s equation J() = [J(t)]  exact solution (see Mihalas, “Stellar Atmospheres”)

J() = const. [  + q()], with q monotonic

1 = 0.577 = q(0) q(τ¯) q( ) = 0.710 √3 ≤ ≤ ∞ Radiative equilibrium - grey approximation

J() = B() =  T4() = const. [  + q()]

with boundary conditions 

4 4 T () = ¾ T eff [ + q()] 62

A simple approximation for T

0th moment of equation of transfer (integrate both sides in d from -1 to 1)

dI dH σT 4 μ = κ¯(I B) = J B = 0 (J = B) H = const = eff dx − − dτ¯ − 4π

1st moment of equation of transfer (integrate both sides in d from -1 to 1)

dI dK σT 4 μ = κ¯(I B) = H = eff K(τ¯) = Hτ¯ + constant dx − − dτ¯ 4π

From Eddington’s approximation at large depth: K = 1/3 J

63 Grey atmosphere – temperature distribution

3πH σ T 4(τ¯) = (τ¯ + c) H = T 4 σ 4π eff

3 T 4(τ¯) = T 4 (τ¯ + c) T4 is linear in  4 eff

Estimation of c

1 ∞ ∞ Hν (τ¯ = 0) = 3H tE2(t) dt + c E2(t) dt ∞ 2 ⎡ ⎤ s s ! Z Z t E n (t) d t = 0 0 Z s + n ⎣ ⎦ 0 1/3 1/2 64 Grey atmosphere – Hopf function

1 3 2 H(0) = H = H(1 + c) c = 2 2 → 3

4 3 4 2 based on approximation K/J = 1/3 T (τ¯) = Teff (τ¯ + ) 4 3 T = Teff at  = 2/3, T(0) = 0.84 Teff

Remember: More in general J is obtained from J(¯τ )=Λ¯τ [S(t)]

3 T 4(τ¯) = T 4 [τ¯ + q(τ¯)] q(τ¯) : Hopf function 4 eff

Once Hopf function is specified  solution of the grey atmosphere (temperature distribution) 1 = 0.577 = q(0) q(τ¯) q( ) = 0.710 √3 ≤ ≤ ∞ 65 Selection of the appropriate κν ¯κ ⇒

In the grey case we define a ‘suitable’ mean opacity (absorption coefficient). ∞ ∞ κν ¯κ ⇒ I = Iν dν J = Jν dν ... 0Z 0Z non-grey grey

dIν dI Equation of transfer μ = κν(Iν Sν) μ = ˜κ (I S) dx − − dx − −

dH dH ν = κν(Jν Sν) = ˆκ (J S) 1st moment dx − − dx − −

dKν dK = ¯ nd = κνHν κ H 2 moment dx − dx − 66

Selection of the appropriate κν ¯κ ⇒

non-grey grey

dI dI μ ν = κν(Iν Sν) μ = ˜κ (I S) Equation of transfer dx − − dx − −

dH dH ν = κν(Jν Sν) = ˆκ (J S) 1st moment dx − − dx − −

dK dK ν = κνHν = ¯κ H 2nd moment dx − dx −

For each equation there is one opacity average that fits “grey equations”, however, all averages are different. Which one to select?

 For flux constant models with H() = const. 2nd moment equation is relevant  67

Mean opacities: flux-weighted

1st possibility: Flux-weighted mean

∞ κνHν dν κ¯ = 0 R H

allows the preservation of the K-integral (radiation pressure)

Problem: H not known a priory (requires iteration of model atmospheres)

68 dKν ∞ 1 dKν ∞ = κνHν κ dx dν = Hνdν dx − 0 ν − 0 R R Mean opacities: Rosseland

2nd possibility: Rosseland mean

to obtain correct integrated energy flux and use local T

∞ 1 dK 1 dK ν dν = H (grey) = H κν dx − ⇒ ⇒ κ¯ dx − Z0 1 Kν Jν , Jν Bν as τ ∞ 1 dKν dν → 3 → → ∞ 1 κν dx = 0 R dK dKν 1 dBν 1 dBν dT κ¯ = dx dx → 3 dx 3 dT dx

∞ 1 dBν (T ) dν κν dT 1 0 large weight for low-opacity (more = R κ¯Ross ∞ dBν (T ) transparent to radiation) regions dT dν 0 R 69 Mean opacities: Rosseland

at large  the T structure is accurately given by

4 3 4 T = Teff [τRoss + q(τRoss)] Rosseland opacities used 4 in stellar interiors

For stellar atmospheres Rosseland opacities allow us to obtain initial approximate values for the Temperature stratification (used for further iterations).

70 T4 vs. 

non-grey numerical

grey: q() = exact grey: q= 2/3

71 T vs. log()

non-grey numerical grey: q() = exact grey: q= 2/3

72 Iterative method for calculation of a stellar atmosphere:

the major parameters are Teff and g

a. hydrostatic equilibrium dP = gρ(x) dx − equation of transfer b. equation of radiation T(x), (x), B[T(x)],P(x), ρ(x) J(x), H(x) transfer

dIν μ = (κν + σν) (Iν Sν) dx − − c. radiative equilibrium

∞ ∞ κν (Jν Bν ) dν = 0 ? κν [Jν Bν (T )] dν = 0 0 − − Z0 ∞ 4 R4π 0 Hν dν = σ Teff ? d. flux conservation

∞ 4 R 4π Hν dν = σTeff Z0 e. equation of state T(x), (x), B[T(x)], ρ(x) ρ k T P = μ m H 73