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4. Transport of energy: convection
convection in stars, solar granulation
Schwarzschild criterion for convective instability
mixing-length theory
numerical simulation of convection
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Convection as a means of energy transport
Mechanisms of energy transport
a. radiation: Frad (most important)
b. convection: Fconv (important especially in cool stars)
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
The sum F(r) = Frad(r) + Fconv(r) must be used to satisfy the condition of energy equilibrium, i.e. no sources and sinks of energy in the atmosphere
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View MK à Haleakala
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Solar granulation
resolved size ~ 500 km (0.7 arcsec) ∆ T ' 100 K v » few km/s
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Solar granulation
warm gas is pulled to the surface
cool dense gas falls back into the deeper atmosphere
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Convection as a means of energy transport
Hot stars (type OBA) Radiative transport more efficient in atmospheres CONVECTIVE CORES
Cool stars (F and later) Convective transport more important in atmospheres (dominant in coolest stars) OUTER CONVECTIVE ZONE
When does convection occur? à strategy
Start with an atmosphere in radiative equilibrium.
Displace a mass element via small perturbation. If such displacement grows larger and larger through bouyancy forces à instability (convection)
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The Schwarzschild criterion for instability
Assume a mass element in the photosphere slowly (v < vsound) moves upwards (perturbation) adiabatically (no energy exchanged)
Ambient density and pressure decreases ρ!ρa
mass element (‘cell’) adjusts and expands, changes ρ, T inside
2 cases:
a. ρi > ρa: the cell falls back à stable
b. ρi < ρa: cell rises à unstable
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The Schwarzschild criterion for instability
denoting with subscript ‘ad’ the adiabatic quantities and with ‘rad’ the radiative ones
we can substitute ρi ! ρad, Ti ! Tad and ρa ! ρrad, Ta ! Trad
the change in density over the radial distance ∆x:
at the end of the adiabatic expansion ρ and T of the cell : ρ ! ρi, T ! Ti
we obtain stability if: |∆ρad | < |∆ρrad |
outwards: ρ decreases! à density of cell is larger than surroundings,
inwards: ρ increases! à density of cell smaller than surroundings
2 cases:
a. ρad > ρrad: the cell falls back à stable
b. ρad < ρrad: cell rises à unstable
stability !!
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The Schwarzschild criterion for instability
à equivalently stability if:
We assume that the motion is slow, i.e. with v << vsound
à pressure equilibrium !!!
à pressure inside cell equal to outside: Prad = Pad
from the equation of state: P ~ ρ T à ρrad Trad = ρad Tad
equivalent to old criterium
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The Schwarzschild criterion for instability
equation of hydrostatic equilibrium: dPgas = -g ρ(x) dx
Pgas = (k/mH µ) ρT
à stability if: Schwarzschild criterion
instability: when the temperature gradient in the stellar atmosphere is larger than the adiabatic gradient à convection !!!
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The Schwarzschild criterion for instability
to evaluate
Adiabatic process: P ~ ργ γ = Cp/Cv
perfect monoatomic gas which is completely ionized or completely neutral: γ = 5/3
from the equation of state: P = ρkT/µ à Pγ-1 ~ Tγ$
specific heat at constant pressure, volume
C = dQ/dT
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The Schwarzschild criterion for instability
“simple” plasma with constant ionization à γ = 5/3 à
changing degree of ionization à plasma undergoes phase transition
à specific heat capacities dQ = C dT not constant!!!
à adiabatic exponent γ changes
H p + e- γ ≠ 5/3
¯̄̄¯d ln Td ln P
¯̄̄¯ad
= 0 .4
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The Schwarzschild criterion for instability
the general case:
for Xion = 0 or 1:
for Xion = 1/2 a minimum is reached
(example: inner boundary of solar photosphere at T = 9,000 K) Isocontours of adiabatic gradient in (logP, logT)-plane
note that gradient is small, where H, HeI, HeII change ionization
Unsoeld, 68
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Conditions for convection
instability when
can depend on: 1. small 2. large
1. HOT STARS: Teff > 10,000 K H fully ionized à γ = 5/3
convection NOT important
1. COOL STARS: Teff < 10,000 K contain zones with Xion = 0.5
convective zones
γ−effect
¯̄̄¯d ln Td ln P
¯̄̄¯ad
= 0 .4
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Conditions for convection
2.
large (convective instability) when κ large (less important: flux F or scale height H large) κ−effect
Example: in the Sun this occurs around T ' 9,000 K because of strong Balmer absorption from H 2nd level
κ−effect
γ−effect
hydrogen convection zone in the sun
Pressure scale height
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Conditions for convection
instability when 1. small 2. large
• The previous slides considered only the simple case of atomic hydrogen ionization and hydrogen opacities. This is good enough to explain the hydrogen convection zone of the sun.
• However, for stars substantially cooler than the sun molecular phase transitions and molecular opacities become important as well. This is, why convection becomes more and more important for all cool stars in their entire atmospheres. The effects of scale heights also contribute for red giants and supergiants.
• Even for hot stars, helium and metal ionization can lead to convection (see Groth, Kudritzki & Heber, 1984, A&A 152, 107). However, here convection is only important to prevent gravitational settling of elements, the convective flux is small compared with the radiative flux and radiative equilibrium is still a valid condition.
γ−effect κ−effect
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If convection occurs, what changes?
• convective energy flux in addition to radiative flux Ftotal = Fconvective + Fradiative = σ T4
eff/π
• temperature gradient becomes flatter
• how to calculate convective flux and new T(x)??
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Calculation of convective flux: Mixing length theory
simple approach to a complicated phenomenon:
a. suppose atmosphere becomes unstable at r = r0 à mass element rises for a characteristic distance L (mixing length) to r0 + L
b. the cell excess energy is released into the ambient medium
c. cell cools, sinks back down, absorbs energy, rises again, ...
In this process the temperature gradient becomes shallower than in the purely radiative case
given the pressure scale height (see ch. 2)
we parameterize the mixing length by α = 0.5 – 1.5 (adjustable parameter)
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Mixing length theory: further assumptions
simple approach to a complicated phenomenon:
d. mixing length L equal for all cells
e. the velocity v of all cells is equal
Note: assumptions d., e. are made ad hoc for simplicity.
There is no real justification for them
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Mixing length theory
> 0 under the conditions for convection
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Mixing length theory: energy flux
for a cell travelling with speed v the flux of energy is:
Estimate of v:
1. buoyancy force: |fb| = g |∆ρ| ∆ρ: density difference cell – surroundings. We want to
express it in terms of ∆T, which is known à
2. equation of state:
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Mixing length theory: energy flux
in pressure equilibrium:
3. work done by buoyancy force: crude approximation: we assume
integrand to be constant over Integration interval
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Mixing length theory: energy flux
4. equate kinetic energy ρv2/ 2 to work
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Mixing length theory: energy flux
re-arranging the equation of state:
Note that Fconv ~ T1.5, whereas Frad ~ Teff4
à convection not effective for hot stars
Note that Fconv ~ α2 , α is a free parameter !!!!!!
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Mixing length theory: energy flux
We finally require that the total energy flux
Since the T stratification is first done on the assumption:
a correction ∆T(τ) must be applied iteratively to calculate the correct T stratification if the instability criterion for convection is found to be satisfied
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Mixing length theory: energy flux
comparison of numerical models with mixing-length theory results for solar convection
Abbett et al. 1997
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3D Hydrodynamical modeling Mixing length theory is simple and allows an analytical treatment More recent studies of stellar convection make use of radiative – hydrodynamical numerical simulations in 3D (account for time dependence), including radiation transfer. Need a lot of CPU power.
Nordlund 1999 Freitag & Steffen
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3D Hydrodynamical modeling
2009, Annual Meeting German Astronomical Society
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al. 2009
animation
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3D Hydrodynamical modeling
Steffen et al. 2009
animation
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
! Orionis (Freitag 2000)
Steffen et al., 2009
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3D Hydrodynamical modeling
! Orionis (Freitag 2000)
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Asplund, 2005
Gas temperature distribution in dynamic solar photosphere as a function of depth
Warm colors: abundance of gas
Cold colors: shortage of gas
- - - model assuming mixing length theory
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3D Hydrodynamical modeling
Temperature stratification as a function of optical depth
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3D Hydrodynamical modeling
Center to limb variation of intensity as a function of wavelength
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
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3D Hydrodynamical modeling
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3D Hydrodynamical modeling
Steffen et al., 2009
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3D Hydrodynamical modeling
Iron abundances from individual iron lines with different excitation potential
● Fe I
● Fe II
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Solar element abundances 1D (old) 3D (new)
He 10.98±0.01 10.98±0.01 C 8.56±0.06 8.47±0.05 N 7.96±0.06 7.87±0.05 O 8.87±0.06 8.73±0.05 Ne 8.12±0.06 7.97±0.10 Mg 7.62±0.05 7.64±0.04 Si 7.59±0.05 7.55±0.05 S 7.37±0.11 7.16±0.03 Ar 6.44±0.06 6.44±0.13 Fe 7.55±0.05 7.44±0.04
Asplund, 2009 X = log{N(X)/N(H)} + 12
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3D Hydrodynamical modeling low metallicity G dwarfs
Strong effects of metallicity
Asplund, 2005 - - - model assuming mixing length theory
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Spri
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3D convection simulation Betelgeuze
Large pressure scale height à huge convective cells
Chiavassa et al. 2011
2 AU
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Temperature structure: 3D vs. MARCS
Chiavassa et al.2011
MARCS
3D
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Spri
ng 2
013
55
3D effects on TiO and SED
1-D model which fits visual spectrum of 3D-model fails in the near-IR à consequences for Teff determination !!!
from Davies, Kudritzki et al., 2013, ApJ 767, 3