Basics processes & equations * Stellar Atmospheres Giovanni Catanzaro INAF – Catania Astrophysical Observatory Italy
Feb 22, 2016
Basics processes & equations
*Stellar Atmospheres
Giovanni CatanzaroINAF – Catania Astrophysical
ObservatoryItaly
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*Working definitions
*We call stellar atmosphere the external layers of a star*These are the layers where radiation
created in the stellar core can escape freely into interstellar medium*In practice, the atmosphere is the only
part of a star from which we receive photons
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Theory of stellar atmospheres
How radiation produced in the stellar core propagates and interacts with the external layers
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*Basic definitions𝐼𝜈=𝑙𝑖𝑚 Δ 𝐸𝜈
𝑐𝑜𝑠 𝜃 Δ 𝐴 Δ𝜔 Δ𝑡 Δ𝜈
𝐼𝜈=𝑑𝐸𝜈
𝑐𝑜𝑠𝜃 𝑑𝐴𝑑𝜔𝑑𝑡 𝑑𝜈Specific intensity = the energy that flow through an element of area dA in the unit of solid angle, time and frequency
𝐽𝜈=14 𝜋∮ 𝐼𝜈𝑑𝜔
Integrating the specific intensity over all the directions we obtain the so-called mean intensity
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𝐹 𝜈=𝑙𝑖𝑚∑ Δ 𝐸𝜈
Δ 𝐴 Δ𝑡 Δ𝜈Flux = net energy flow across element of area over the unit of time and frequency 𝐹 𝜈=
∮𝑑 𝐸𝜈
𝑑𝐴 𝑑𝑡 𝑑𝜈
Looking at a point on the physical boundary of a radiating sphere, we can write:
𝐹 𝜈=∫0
2 𝜋
𝑑𝜙 ∫0
𝜋 /2
𝐼𝜈 sin θ𝑐𝑜𝑠 𝜃𝑑𝜃=2𝜋 ∫0
𝜋 /2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 𝑑𝜃=𝜋 𝐼𝜈
Flux could be related with the specific intensity 𝐹 𝜈=∮ 𝐼𝜈𝑐𝑜𝑠𝜃𝑑𝜔
Κ𝜈=14𝜋∮ 𝐼𝜈𝑐𝑜𝑠
2𝜃𝑑𝜔 yields→
𝑃𝑅=4𝜋𝑐 ∫
0
∞
Κ𝜈𝑑𝜈K integral physically related with radiation pressure
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*Absorption coefficient and optical depth
Processes contribuiting: true absorption and scattering.No emission.
Since the absorption, radiation interacts with plasma. We can say that it sees neither nor dx alone, rather their combination, so we define:
𝜏𝜈=∫0
𝐿
𝜅𝜈𝜌 𝑑𝑥𝑑𝜏𝜈=𝜅𝜈𝜌 𝑑𝑥 Optical depth
𝐼𝜈 (0 ) 𝐼𝜈+𝑑 𝐼𝜈
𝑑𝑥𝜌
𝑑 𝐼𝜈=−𝜅𝜈𝜌 𝐼𝜈𝑑𝑥 [𝜅𝜈 ]=𝑐𝑚2𝑔− 1
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𝑑 𝐼𝜈=− 𝐼𝜈𝑑𝜏𝜈 𝐼𝜈=𝐼❑0 𝜈𝑒− 𝜏𝜈
The optical depth =1 corresponds, for a given frequency and absorption coefficient, to the distance at which the intesity is reduced by 1/e
>> 1 plasma optically thick
<< 1 plasma optically thin
It measures a characteristic of matter and radiation coupled together and for a given frequency.
Plasma could be optically thin for radiation of frequency n1 and optically thick for another frequency n2
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*Emission coefficient and Source function
Like we did for absorption, let define the increment of the radiation if there is emission:
Processes contribuiting: true emission and scattering into direction.No absorption.
Source function
𝑆𝜈=𝑗𝜈𝜅𝜈
It could be considered as the specific intensity emitted at some point in a hot gas.
𝐼𝜈 (0 ) 𝐼𝜈+𝑑 𝐼𝜈
𝑑𝑥𝜌
𝑑 𝐼𝜈= 𝑗𝜈𝜌 𝑑𝑥 [ 𝑗𝜈 ]=𝑒𝑟𝑔 𝑠−1𝑟𝑎𝑑−1𝐻𝑧− 1𝑔−1
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*Source function: physical meaning
Let write the number of photons emitted in an element of volume dV over all directions, for frequency n and unit time dt
𝑁 𝑒𝑚=4 𝜋h𝜈 𝑗𝜈𝜌 𝑑𝑥𝑑𝐴𝑑𝜈 𝑑𝑡
dV
Energy emitted in the volume dV
Transform energy in number of photons
Integration over solid angle
𝑗𝜈 𝜌 𝑑𝑥=𝑗𝜈𝜅𝜈
𝜅𝜈 𝜌𝑑𝑥
𝑁 𝑒𝑚=𝑆𝜈𝑑𝜏𝜈4𝜋h𝜈 𝑑𝐴𝑑𝜈 𝑑𝑡
𝑆𝜈∝𝑁𝑒𝑚 /𝑑𝜏𝜈
𝑗𝜈 𝜌 𝑑𝑥=𝑗𝜈𝜅𝜈
𝜅𝜈 𝜌𝑑𝑥=𝑆𝜈𝑗𝜈 𝜌 𝑑𝑥=
𝑗𝜈𝜅𝜈
𝜅𝜈 𝜌𝑑𝑥=𝑆𝜈𝑑𝜏𝜈
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*Source functions: 2 simple cases Pure isotropic
scattering
𝑑 𝑗𝜈=14𝜋 𝜅𝜈 𝐼𝜈𝑑𝜔
‘absorbed’ energyIntegrating over w
𝑗𝜈=14 𝜋∮𝜅𝜈 𝐼𝜈𝑑𝜔=
𝜅𝜈
4𝜋∮ 𝐼𝜈𝑑𝜔=𝜅𝜈 𝐽𝜈𝑆𝜈=
𝑗𝜈𝜅𝜈
= 𝐽𝜈
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Pure absorption
All the absorbed photons are destroyed and all the emitted photons are newly created with a distribution governed by the physical state of the material.
Thermodinamic equilibrium
𝑆𝜈 (𝑇 )=2h𝜈3
𝑐21
𝑒h𝜈𝑘𝑇 −1
The source function is equal to the Planck function, depends on frequency and temperature of the material
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*The transfer equationAlong a
line𝑑 𝐼𝜈=−𝜅𝜈𝜌 𝐼𝜈𝑑𝑠+ 𝑗𝜈 𝜌 𝑑𝑠
𝑑 𝐼𝜈𝑑𝜏𝜈
=− 𝐼𝜈+𝑆𝜈
𝑑 𝐼𝜈𝜅𝜈𝜌 𝑑𝑠
=− 𝐼𝜈+𝑗𝜈𝜅𝜈
𝐼 𝜈 (𝜏𝜈 )=∫0
𝜏𝜈
𝑆𝜈 (𝑡𝜈 )𝑒− (𝜏𝜈− 𝑡𝜈) 𝑑𝑡𝜈+ 𝐼𝜈 (0 )𝑒−𝜏𝜈
𝑡𝜈 𝜏𝜈𝑡𝜈−𝜏𝜈
𝐼𝜈 (0 ) 𝐼𝜈+𝑑 𝐼𝜈
𝑑𝑠 ,𝜌
Differential form
Integral form
𝑑𝜏𝜈
𝑆𝜈
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*The transfer equationDifferent
geometries• Let consider polar coordinates with z axis along the line of sight. In this case a projection factor, cos q, should be considered.
• Atmosphere is thin with respect the radius, so a plane parallel approximation could be used.
• That means that cos q does not depend upon z.
𝑑𝜏𝜈=− 𝜅𝜈 𝜌𝑑𝑟𝑑𝑠=−𝑑𝑟𝑑𝜏𝜈=𝜅𝜈𝜌 𝑑𝑠
𝑐𝑜𝑠𝜃𝑑 𝐼𝜈𝑑𝜏𝜈
=𝐼𝜈−𝑆𝜈
1𝜅𝜈𝜌
𝑑 𝐼𝜈𝑑𝑧 =− 𝐼𝜈+𝑆𝜈
1𝜅𝜈𝜌 ( 𝜕 𝐼𝜈𝜕𝑟
𝑑𝑟𝑑𝑧 +
𝜕 𝐼𝜈𝜕 𝜃
𝑑𝜃𝑑𝑧 )=− 𝐼𝜈+𝑆𝜈
1𝜅𝜈𝜌 ( 𝑑 𝐼𝜈𝑑𝑟 𝑐𝑜𝑠𝜃)=− 𝐼𝜈+𝑆𝜈
Polar coordinates 𝑑𝜃
𝑑𝑧 =0
𝑑𝑟=𝑐𝑜𝑠𝜃 𝑑𝑧
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𝑐𝑜𝑠𝜃𝑑 𝐼𝜈𝑑𝜏𝜈
=𝐼 𝜈−𝑆𝜈*Elementary solutions
1) No absorption (kn = 0), no emission (jn = 0)
𝐼𝜈=𝑐𝑜𝑠𝑡Trivial solution: in absence of any interaction with the medium the radiation intensity remains constant
2) No absorption (kn = 0), only emission (jn > 0)
𝐼𝜈 (𝑥 )=∫0
𝑥
𝑗𝜈 (𝑥 ′ ) sec𝜃𝑑𝑥 ′
Outcoming radiation from an optically thin radiating slab
3) No emission (jn = 0), only absorption (kn = 0)
𝐼𝜈 (0 ,𝑐𝑜𝑠𝜃 )=𝐼𝜈 (𝜏𝜈 ,𝑐𝑜𝑠𝜃 )𝑒− 𝜏𝜈𝑐𝑜𝑠 𝜃
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4) General case: emission (jn > 0), absorption (kn > 0)
𝜏𝜈
𝜏𝜈=0
𝜏𝜈=∞
𝐼𝜈❑𝑜𝑢𝑡𝐼𝜈❑𝑖𝑛
𝑧→
𝐼 𝜈 (𝜏𝜈 )= 𝐼𝜈❑𝑜𝑢𝑡 (𝜏𝜈 )+ 𝐼𝜈❑𝑖𝑛 (𝜏𝜈 )=¿
¿∫𝜏𝜈
0
𝑆𝜈𝑒− (𝑡 𝜈−𝜏𝜈) sec 𝜃 sec𝜃 𝑑𝑡𝜈−∫
∞
𝜏𝜈
𝑆𝜈𝑒−( 𝑡𝜈−𝜏𝜈 ) sec 𝜃 sec𝜃𝑑 𝑡𝜈
At the surface, where tn=0: 𝐼 𝜈 (0 )=∫0
∞
𝑆𝜈𝑒−𝑡 𝜈 sec 𝜃 sec 𝜃𝑑𝑡𝜈
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5) Special case: linear source function
𝑆𝜈 (𝜏𝜈 )=𝑎+𝑏𝜏𝜈
𝐼𝜈 (0 )=∫0
∞
(𝑎+𝑏𝜏𝜈 )𝑒−𝑡 𝜈 sec 𝜃 sec 𝜃𝑑𝑡𝜈=𝑎+𝑏𝑐𝑜𝑠 𝜃
The values of emergent intensity for all angle p/2 < q < 0 then map the values of the source function between optical depths 0 and 1.
𝐼𝜈 (0 ,𝑐𝑜𝑠𝜃 )=𝑆𝜈 (𝜏𝜈=𝑐𝑜𝑠 𝜃 )
Eddington-Barbier relation
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*The flux integral
𝐹 𝜈=2𝜋∫0
𝜋
𝐼 𝜈𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑑𝜃=2𝜋∫0
𝜋2
𝐼𝜈𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃𝑑𝜃+2𝜋∫𝜋2
𝜋
𝐼𝜈𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃𝑑𝜃
𝐹 𝜈 (0 )=2𝜋∫0
∞
𝑆𝜈 (𝑡𝜈 )𝐸2 (𝑡𝜈 )𝑑𝑡𝜈
The theoretical stellar spectrum is for tn = 0
𝐸𝑛 (𝑥 )=∫1
∞ 𝑒−𝑥𝑤
𝑤𝑛 𝑑𝑤
𝐹 𝜈 (0 )=2𝜋∫𝜏𝜈
∞
𝑆𝜈 (𝑡𝜈 )𝐸2 (𝑡𝜈−𝜏𝜈 )𝑑𝑡𝜈−2𝜋∫0
𝜏𝜈
𝑆𝜈 ( 𝑡𝜈 )𝐸2 (𝑡𝜈−𝜏𝜈 )𝑑 𝑡𝜈
Considering Sn isotropic, independent on q, we obtain:
Extinction factor
𝐹 𝜈=∮ 𝐼𝜈𝑐𝑜𝑠𝜃𝑑𝜔 We can write this expression in polar coordinates
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𝑆𝜈=𝑗𝜈𝜅𝜈
depend on physical properties of the layer
To compute Sn we must know the distributions of these quantities with tn
Computing Model Atmosphere
T, P, ni, ne...
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*Stellar atmosphere: basic equationsHypothesis: horizontally homogeneous, plane-parallel,
static
Hydrostatic equilibrium equation
Radiative equilibrium equation
Statistical equilibrium equation
Charge equilibrium equation
Radiative transfer equation
𝑐𝑜𝑠𝜃𝑑 𝐼𝜈𝑑𝜏𝜈
=𝐼𝜈−𝑆𝜈
Mean intensities, Jn
Pressure, total particle density N
Temperature, T
Populations, ni
Electron density, ne
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*Hydrostatic equilibrium equation
𝑑𝑃𝑑𝑧 =𝜌 𝑔 𝑑𝑃
𝑑𝜏𝜈=𝑔𝜅𝜈
𝑃=𝑃𝑔𝑎𝑠+𝑃𝑟𝑎𝑑+𝑃𝑡𝑢𝑟𝑏+𝑃 𝐵=𝑁𝑘𝑇+ 4𝜋𝑐 ∫0
∞
𝐾𝜈 𝑑𝜈+12𝜌𝑣 2𝑡𝑢𝑟𝑏
❑ + 𝐵2
8𝜋
Teff(K)
Pgas(dyn cm-
2)
Prad(dyn cm-
2)
B(G)
Vturb
(km s-
1)
4000 1x105 0.6 1121
7.5
8000 1x104 10 354 10.612000 3x103 52 194 13.016000 3x103 165 194 15.020000 5x103 403 251 16.7
𝑑𝑃 𝑔𝑎𝑠
𝑑𝜏𝜈= 𝑔𝜅𝜈−𝑑𝑃𝑟𝑎𝑑
𝑑𝜏𝜈
Effective gravity acceleration
𝑑𝜏𝜈=𝜅𝜈𝜌 𝑑𝑧 [𝑔𝑐𝑚−2 ]
Introducing:
From Gray D. (2005), chapter 9
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*Radiative equilibrium
08/04/2013
Ensure radiative equilibrium means ensure conservation of energy
𝑑𝐹 (𝑥)𝑑𝑥 =0
→𝐹 (𝑥 )=𝐹 0First radiative equilibrium condition is then
∫0
∞
𝐹 𝜈 (𝜏𝜈 )𝑑𝜈=𝐹0The other 2 consitions come from transfer equation, write as:
𝑐𝑜𝑠𝜃𝑑 𝐼𝜈𝑑𝑥 =𝜅𝜈𝜌 𝐼𝜈−𝜅𝜈 𝜌𝑆𝜈
Integrating over solid angle and over frequencies𝑑𝑑𝑥∫0
∞
𝐹 𝜈𝑑𝜈=4𝜋𝜌∫0
∞
𝜅𝜈 𝐽𝜈𝑑𝜈−4𝜋𝜌∫0
∞
𝜅𝜈𝑆𝜈 𝑑𝜈 ∫0
∞
𝜅𝜈 𝐽𝜈𝑑𝜈=∫0
∞
𝜅𝜈𝑆𝜈𝑑𝜈
Multiply by cosq and integrating over solid angle and frequencies
∫0
∞ 𝑑𝐾 𝜈
𝑑𝜏𝜈𝑑𝜈= 𝐹0
4𝜋
Milne equations
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*Statistical and charge conservation equations
𝑛𝑖∑𝑗≠ 𝑖
(𝑅𝑖𝑗+𝐶𝑖𝑗 )=𝑛 𝑗∑𝑗≠ 𝑖
(𝑅 𝑗 𝑖+𝐶 𝑗 𝑖 ) R radiative rateC collisional rate
Total number of transitions out of level i
Total number of transitions into level i
∑𝑖𝑛𝑖𝑍 𝑖−𝑛𝑒=0 Zi is the charge associated with
level ini population of the level ine electron density
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*The grey atmosphere𝜅𝜈=𝜅∀𝜈
𝑐𝑜𝑠𝜃 𝑑𝐼𝑑𝜏=𝐼−𝑆 Milne
equations
𝐼=∫0
∞
𝐼𝜈𝑑𝜈
𝑆=∫0
∞
𝑆𝜈𝑑𝜈
𝑑𝜏=𝜅𝜌 𝑑𝑥
Eddington (1926) hemispheperically isotropic outward and inward specific
intensity
𝐼 (𝜏 )={𝐼𝑜𝑢𝑡 (𝜏 )0 ≤Θ≤ 𝜋2𝐼 𝑖𝑛 (𝜏 ) 𝜋
2≤Θ≤𝜋
𝐽 (𝜏 )=12 [ 𝐼𝑜𝑢𝑡 (𝜏 )+𝐼𝑖𝑛 (𝜏 ) ]
𝐹 (𝜏 )=𝜋 [ 𝐼𝑜𝑢𝑡 (𝜏 )− 𝐼 𝑖𝑛 (𝜏 ) ]𝐾 (𝜏 )=16 [ 𝐼𝑜𝑢𝑡 (𝜏 )+𝐼 𝑖𝑛 (𝜏 ) ]
𝑆 (𝜏 )= 𝐽 (𝜏 )=3𝐾 (𝜏) 𝐾 (𝜏 )=𝐹 0
4𝜋 𝜏+𝐹 0
4 𝜋𝑆 (𝜏 )=
3𝐹0
4𝜋 (𝜏+ 23 )𝑇 (𝜏 )=[(𝜏+ 23 ) ]
14 𝑇 𝑒𝑓𝑓
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*Conclusions• Modeling stellar spectrum means computing the flux
emerging at the stellar surface
• To accomplish this task we need to know the radiation specific intensity along the atmosphere
• The calculation of how the radiation propagates within a stellar atmospere requires knowledge of the source function
• Source function depends on emission and absorption coefficient
• Both jn and kn depend on the physical condition of the stellar material: T, P, electronic density and so on
• We need to resolve the equations of the model atmosphere
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*ReferencesI. I. Hubeny, «Stellar atmospheres theory: an
introduction» in: Stellar atmospheres: Theory and Observations, Lecture note in physics, J.P. De Greeve, R.Blomme, H. Hensberg (Eds.), Springer
II. D. Gray, «The observations and analysis of stellar photospheres»
III. D. Mihalas, «Stellar Atmospheres»
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*Thanks for your attention
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*The Einstein coefficients
Nl atoms per dVl
u Nu atoms per dVAul Bul Blu DEul = hn
𝑗𝜈 𝜌=𝒩𝓊 𝐴𝓊ℓ h𝜈Spontaneus emission
𝑝∝ 𝐴𝓊ℓ 𝑑𝑡𝑑𝜔 Probability that an atom will emit is quantum energy in dt and dw
Absorption
2 contributions
Stimulated emission
𝑝∝𝐵𝓊 ℓ 𝐼𝜈 𝑑𝑡𝑑𝜔
True absorption
𝑝∝𝐵ℓ𝓊 𝐼𝜈 𝑑𝑡𝑑𝜔𝜅𝜈 𝜌=(𝒩ℓ𝐵 ℓ𝓊−𝒩𝓊𝐵𝓊 ℓ )h𝜈