talk start talk files all talks SUMMARY OF SOLAR SPECTRUM FORMATION

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SUMMARY OF SOLAR SPECTRUM FORMATION
Rob Rutten, Utrecht University & Oslo University
Jun 21 2007, Oslo
Rob Rutten: “Radiative Transfer in Stellar Atmospheres” (RTSA)
Michael Stix: “The Sun”, 2nd edition, Springer 2004
examples: local – nonlocal – converted photons
white light corona
EUV corona
EUV bright/dark
[Zanstra & Bowen PN lines]
radiative transfer basics
basic quantities
LTE solar radiation escape
Schadeenium in LTE
simple emission line
solar spectrum formation
constant Sν
Planck
H-minus in LTE
LTE lines
solar ultraviolet spectrum
Ca II H versus Halpha
radiative transfer in stellar atmospheres: math
radiative transfer in stellar atmospheres: scattering
thick transfer
Boltzmann-Saha
simple absorption line
VALIIIC model
[RTSA 3.4; 4.1 – 4.2]
[RTSA 4.3]
NLTE solar radiation escape
bb processes
bb rates
solar radiation processes
VAL3C continuum formation
VAL3C radiation budget
realistic absorption line
Na D1
Ca II H from RADYN chromosphere
summary
RTSA rap
coronium lines
bb equilibria
radiative cooling
Na D1 blends
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WHITE LIGHT CORONA
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Stix section 9.1.3
Grotrian (1931): Thomson scattering 8000 Z
km s−1 electrons
(S 9.2–9.3)
Z ∞
∞
ρ j(ρ)
p
ρ2 = x 2 + y 2
I(x) = 2
j(ρ) dy = 2
dρ
ρ2 − x 2
0
x
Ne from inverse Abel transform Z= isotropically scattered irradiation
(S 9.4–9.5)
Z
1 ∞ dI/dx
1
p
j(ρ) = −
dx = σT Ne
I (θ) dΩ
π ρ
4π
x 2 − ρ2
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“CORONIUM” LINES
Stix section 9.1.3
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http://laserstars.org/spectra/Coronium.html
Grotrian, Edlén 1942: forbidden lines high ionization stages (Stix Table 9.2 p. 398)
name
wavelength identification ∆λD
v
Aul
previous ion
green line 530.29 nm
yellow line 569.45
red line
637.45
Coronal sky at Dome C
[Fe XIV]
[Ca XV]
[Fe XI]
0.051 nm
0.087
0.049
29 km/s 60 s−1
46
95
23
69
Fe XIII
Ca XIV
Fe IX
χion
355 eV
820
235
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EUV CORONA
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Stix section 9.1.3
bf equilibria: collisional ionization = radiative recombination ⇒ only f (T ), not f (Ne )
bb equilibria: collisional excitation = spontaneous deexcitation ⇒ f (T, Ne ) (S 9.9–9.10)
−1
Z ∞
Z
Z
X
dT
2
dT ≡ EM
nl Clu = nl Ne
σlu f (v) v dv ≈ nu Aul
hν ∝ nion Ne dz = Ne
dz
v0
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BRIGHT AND DARK IN EUV IMAGES
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• iron lines
– Fe IX/X 171 Å: about 1.0 MK
– Fe XII 195 Å: about 1.5 MK
– Fe XIV 284 Å: about 2 MK
• bright
– collision up, radiation down
– thermal photon creation, NLTE equilibrium
– 171 Å: selected loops = special trees in forest
• dark
– radiation up, re-radiation at bound-free edge
– matter containing He+ , He, H: 104 – 105 K
– large opacity
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BASIC QUANTITIES
RTSA 2.1–2.2, Stix 4.2.5
Monochromatic emissivity (Stix uses ε)
dEν ≡ jν dV dt dν dΩ
units jν : erg cm−3 s−1 Hz−1 ster−1
dIν (s) = jν (s) ds
Iν : erg cm−2 s−1 Hz−1 ster−1
Monochromatic extinction coefficient
dIν ≡ −σν n Iν ds
dIν ≡ −αν Iν ds
units: per particle (physics)
per cm (RJR lectures)
dIν ≡ −κν ρ Iν ds
per gram (astronomy)
Monochromatic source function
Sν ≡ jν /αν = jν /κν ρ
Sνtot
P
jν
=P
αν
Sνtot =
thick: (κν , Sν ) more independent than (κν , jν )
jνc + jνl
Sνc + ην Sνl
=
ανc + ανl
1 + ην
stimulated emission negatively into αν , κν
Transport equation along the beam with τν as optical thickness
dIν
dIν
= jν − αν Iν
= Sν − Iν
dτν ≡ αν ds
ds
αν ds
Plane-parallel transport equation with τν as radial
Z optical depth
z0
dτν ≡ −κν ρ dz
ην ≡ ανl /ανc
τν (z0 ) = −
κν ρ dz
∞
µ
dIν
= Sν − Iν
dτν
dIν
= Iν − Sν
dτν
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CONSTANT SOURCE FUNCTION
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Transport equation along the beam (τν = optical thickness)Z
τν
dIν
= Sν − Iν
Iν (τν ) = Iν (0) e−τν +
Sν (tν ) e−(τν −tν ) dtν
dτν
0
Invariant Sν
Iν (D) = Iν (0) e−τν (D) + Sν 1 − e−τν (D)
example: Sν = Bν for all continuum and line processes in an isothermal cloud
Thick object
Iν (D) ≈ Sν
Thin object
Iν (D) ≈ Iν (0) + [Sν − Iν (0)] τν (D)
Iν (0) = 0 : Iν (D) ≈ τν (D) Sν = jν D
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THICK TRANSFER
RTSA 2.2.2; Stix 4.1.1
Radial optical depth
dτν = −κν ρ dr
r radial
αν cm−1 = cm2 /cm3
κν cm2 /gram
Hubený τνµ
Transport equation
µ
dIν
= Iν − Sν
dτν
Integral form
−(τ0−τν )/µ
Iν (τν , µ) = Iν (τ0 , µ) e
Z
τ0
+ (1/µ)
0
Sν (τν0 ) e−(τν−τν )/µ dτν0
τν
“formal solution”
both directions
Emergent intensity
Z
Iν (0, µ) = (1/µ)
pm: Doppler anisotropy Sν
∞
Sν (τν ) e−τν /µ dτν
0
Eddington-Barbier approximation
Iν (0, µ) ≈ Sν (τν = µ)
exact for linear Sν (τν )
σν cm2 /particle
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PLANCK
RTSA Chapt. 2; SSA 3; Stix 4.5
2hν 3
1
Planck: Bν (T ) = 2 hν/kT
c e
−1
2hν 3 −hν/kT
Wien: exp(hν/kT ) 1 : Bν (T ) ≈ 2 e
c
2ν 2 kT
Rayleigh-Jeans: exp(hν/kT ) 1 : Bν (T ) ≈
c2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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SOLAR VISIBLE AND INFRARED
RTSA Chapt. 8; SSB 2
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Saha–Boltzmann level populations
Boltzmann distribution per ionization stage:
partition function: Ur ≡
X
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nr,s
gr,s −χr,s /kT
=
e
Nr
Ur
gr,s e−χr,s /kT
s
Saha distribution over ionization stages:
3/2
1 2 Ur+1 2πme kT
Nr+1
=
e−χr /kT
2
Nr
Ne Ur
h
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Saha–Boltzmann Schadeenium
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LTE LINES
RTSA Chapt. 2; Stix 4.1.3
continuum optical depth scale
ην ≡ κl /κC
dτν = dτC + dτl = (1 + ην ) dτC
emergent intensity
Z at disk center in LTE Z
∞
Iν (0, 1) =
Bν exp(−τν ) dτν =
0
∞
τC
Z
(1 + ην ) Bν exp −
0
(1 + ην )
dτC0
dτC
0
Eddington-Barbier: Iν (0, 1) ≈ Bν [τν = 1] = Bν [τC = 1/(1+ην )]
extinction coefficient
shape = Maxwell Gauss ⊗ damping Lorentz
r
Z +∞
γ
exp(−(ν −ν 0 )2 /∆νD2 ) 0
1
ν0 2RT
2
H(a, v)
dν = √
+ ξt
φ(ν) = √
∆νD ≡
0
2
2
c
A
π∆νD
π∆νD −∞ [2π(ν −ν0 )] + γ /4
Voigt function
ν −ν0
v≡
∆νD
γ
a≡
4π∆νD
a
H(a, v) ≡
π
Z
+∞
−∞
line extinction coefficient
hν
πe2
e2 fl
√
σl =
Blu φ(ν) =
fl φ(ν) =
H(a, v)
4π
me c
40 me c π∆νD
κl = σl nLTE
ρ (1−e−hν/kT ) =
l
2
e−y
dy
(v − y)2 + a2
σl
n nij nijk
Pi
(1−e−hν/kT )
µmH
ni ni nij
Aul = 6.67×1013
area
√
π
gl flu −1
s (λ nm)
gu λ2
i, j, k : species, stage, lower level
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SIMPLE ABSORPTION LINE
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• extinction: bb process gives peak in κtotal = κC + κl = (1 + ην ) κC
• optical depth: height-invariant κtotal ⇒ linear (1 + ην ) τC
• source function: same for line (bb) and continous (bf, ff, electron scattering) processes
• intensity: Eddington-Barbier (nearly) exact
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SIMPLE EMISSION LINE
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• extinction: bb process gives peak in κtotal = κC + κl = (1 + ην ) κC
• optical depth: height-invariant κtotal ⇒ linear (1 + ην ) τC
• source function: same for line (bb) and continous (bf, ff, electron scattering) processes
• intensity: Eddington-Barbier (nearly) exact
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SOLAR ULTRAVIOLET SPECTRUM
Scheffler & Elsässer, courtesy Karin Muglach
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VALIIIC MODEL
Vernazza, Avrett, Loeser 1981ApJS...45..635V
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SOLAR SPECTRUM FORMATION
Avrett 1990IAUS..138....3A
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CaÍI H & Hα in LTE
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Leenaarts et al. 2006A&A...449.1209L
http://dot.astro.uu.nl
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BOUND-BOUND PROCESSES AND EINSTEIN COEFFICIENTS
Spontaneous deexcitation
Aul ≡ transition probability for spontaneous deexcitation
from state u to state l per sec per particle in state u
Radiative excitation
ϕ
Blu J ν0 ≡ number of radiative excitations from state l to state u
per sec per particle in state l
Induced deexcitation
χ
Bul J ν0 ≡ number of induced radiative deexcitations from state u
to state l per sec per particle in state u
Collisional excitation and deexcitation
Clu ≡ number of collisional excitations from state l to state u
per sec per particle in state l
Cul ≡ number of collisional deexcitations from state u to
state l per sec per particle in state u
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BOUND-BOUND RATES
RTSA 2.3.1, 2.3.2, 2.6.1; Stix 4.2
Monochromatic bb rates expressed in Einstein coefficients (intensity units)
nu Aul χ(ν)/4π
nu Bul Iν ψ(ν)/4π
nl Blu Iν φ(ν)/4π
nu Cul
spontaneous emission
Einstein relations
gu Bul = gl Blu
stimulated emission
radiative excitation
(gu /gl )Aul = (2hν 3 /c2 ) Blu
nl Clu
collisional (de-)excitation
Cul /Clu = (gl /gu ) exp(Eul /kT )
required for TE detailed balancing with Iν = Bν , but hold universally
General line source function
hν
hν
nu Aul χ(ν)
αν =
[nl Blu φ(ν) − nu Bul ψ(ν)]
jν =
4π
4π
Sl =
nu Aul χ(ν)
nl Blu φ(ν) − nu Bul ψ(ν)
Simplified line source function
CRD: χ(ν) = ψ(ν) = φ(ν)
Sl =
nu Aul
2hν 3
1
= 2 gu nl
nl Blu − nu Bul
c
−1
gl nu
Statistical equilibrium equations for level j
N
N
X
X
nj
Rji =
nj Rij
Rji = Aji + Bji Jji + Cji
j6=i
j6=i
time-independent population
bb rates per particle in j
1
Jji ≡
4π
Boltzmann: Sl = Bν (T )
Z
4πZ ∞
Iν φ(ν) dν dΩ
0
0
total (= mean) mean intensity
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BOUND-BOUND EQUILIBRIA
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RTSA missing; Stix 4.1.2
• LTE = large collision frequency – interior, low photosphere
– up: mostly collisional = thermal creation (d + e)
– down: mostly collisional = large destruction probability (a)
– photon travel: “honorary gas particles” or negligible leak
• NLTE = statistical equilibrium or time-dependent – chromosphere, TR
– photon travel: non-local impinging (pumping), loss (suction)
– two-level scattering: coherent/complete/partial redistribution
– multi-level: photon conversion, sensitivity transcription
• coronal equilibrium = hot tenuous – coronal EUV
– up: only collisional = thermal creation (only d)
– down: only spontaneous (only d)
– photon travel: escape / drown / scatter bf H I, He I, He II
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SOLAR ATMOSPHERE RADIATIVE PROCESSES
RTSA Chapt. 8; Stix Fig. 4.5
• bound-bound – Sν , κν NLTE? PRD?
– neutral atom transitions
– ion transitions
– molecule transitions
• bound-free – Sν , κν NLTE? always CRD
– H− optical, near-infrared
– H I Balmer, Lyman; He I, He II
– Fe I, Si I, Mg I, Al I electron donors
• free-free – Sν always LTE, κν NLTE
– H− infrared, sub-mm
– H I radio
• electron scattering – always NLTE, Doppler?
– Thomson scattering
– Rayleigh scattering
• collective – p.m.
– cyclotron, synchrotron radiation
– plasma radiation
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VAL3C CONTINUUM FORMATION
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VAL3C CONTINUUM FORMATION
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VAL3C CONTINUUM FORMATION
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VAL3C CONTINUUM FORMATION
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VAL3C CONTINUUM FORMATION
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VAL3C CONTINUUM FORMATION
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RADIATIVE COOLING
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RTSA 7.3.2
Radiative equilibrium condition
Φtot (z) ≡
dFrad (z)
=0
Zdz
∞
αν (z) [Sν (z) − Jν (z)] dν
Z ∞Z +1
= 2π
[jνµ (z) − ανµ (z) Iνµ (z)] dµ dν
= 4π
0
0
−1
Net radiative cooling in a two-level atom gas
Φul = 4πανl 0 (Sνl 0 − J ν0 )
= 4πjνl 0 − 4πανl 0 J ν0
= hν0 nu (Aul + Bul J ν0 ) − nl Blu J ν0
= hν0 [nu Rul − nl Rlu ]
Net radiative cooling in a one-level-plus-continuum gas
Z ∞
bi
bc −hν/kT
LTE
−hν/kT
Φci = 4π ni bc
σic (ν) Bν 1 − e
− Jν 1 − e
dν
bc
bi
ν0
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VAL3C RADIATION BUDGET
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VAL3C RADIATION BUDGET
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REALISTIC ABSORPTION LINE
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• extinction: bb peak lower and narrower at larger height
• optical depth: near-log-linear inward increases
• source function: split for line (bb) and continous (bf, ff, electron scattering) processes
• intensity: Eddington-Barbier for Sνtotal = (κC SC + κl Sl )/(κC + κl ) = (SC + ην Sl )/(1 + ην )
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SOLAR NaI D2
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Uitenbroek & Bruls 1992A&A...265..268U
Na I D2 is a good example of two-level scattering with complete redistribution: very dark
Eddington-Barbier approximation: line-center τ = 1 at h ≈ 600 km
chromospheric velocity response but photospheric brightness response
Can you use the Eddington-Barbier approximation to estimate the formation height of the
blend lines?
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SOLAR NaI D2
Eddington-Barbier for the blends? Moore, Minnaert, Houtgast 1966sst..book.....M:
5888.703
5889.637
5889.756 *
5889.973M *
5890.203 *
5890.495
5891.178
5891.178
10.
14.
752.
752.
752.
5.
17.
17.
2.
2.
4.
120.SS
3.
1.S"
3.S
3.S
ATM H2O
ATM H2O
ATM H2O
NA 1(D2)
ATM H2O
FE 1P
ATM H2O
FE 1P
R4
R4
R3
0.00
R4
5.06
R3
4.65
302
401
401
1
302
1313
401
1179
26
26
26
26
17,26
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Ca II H2V GRAIN SIMULATION
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• observation
Lites, Rutten & Kalkofen 1993
– sawtooth line-center shift
– triangular whiskers
– H2V grains
• simulation
Carlsson & Stein 1997
– 1D radiation hydrodynamics
– subsurface piston from Fe I blend
– observer’s diagnostics
• analysis
(RR radiative transfer course)
– source function breakdown
– dynamical chromosphere
– H2V grains = acoustic shocks
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DYNAMIC LOWER CHROMOSPHERE
Carlsson & Stein, ApJ 440, L29, 1995
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SHOCK GRAIN DIAGNOSIS
Carlsson & Stein, ApJ 481, 500, 1997
Z
Iν (0) =
∞
Sν e
0
−τν
Z
dτν =
0
∞
Sν τν e−τν
d ln τν
dz
dz
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BASIC RADIATIVE TRANSFER EQUATIONS
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RTSA last page
specific intensity
Iν (~r, ~l, t) erg cm−2 s−1 Hz−1 ster−1
emissivity
jν erg cm−3 s−1 Hz−1 ster−1
extinction coefficient
radial optical depth
αν cm−1 σν cm2 part−1
P
P
Sν = jν / αν
R∞
τν (z0 ) = z0 αν dz
plane-parallel transport
µ dIν /dτν = Iν − Sν
source function
κν cm2 g−1
Iν = I0 + (Sν − I0 ) τν
R∞
thick emergent intensity Iν+ (0, µ) = 0 Sν (τν ) e−τν /µ dτν /µ
thin cloud
mean mean intensity
Iν+ (0, µ) ≈ Sν (τν = µ)
R ∞ R +1
ϕ
J ν0 = 12 0 −1 Iν ϕ(ν −ν0 ) dµ dν
photon destruction
εν = ανa /(ανa + ανs ) ≈ Cul /(Aul + Cul )
complete redistribution
Sνl 0 = (1 − εν0 ) J ν0 + εν0 Bν0
√
Sν0 (0) = εν0 Bν0
Eddington-Barbier
isothermal atmosphere
ϕ
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