stellar atmospheres

Basics processes & equations

*Stellar Atmospheres

Giovanni CatanzaroINAF – Catania Astrophysical

ObservatoryItaly

Spring School of Spectroscopic Data Analysis

2

*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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4

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.


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


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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=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


=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

stellar atmospheres

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