energy transportastro.phys.au.dk/jcd/astrofysik2/lecture04_convection_handouts.pdf06/09/2013 1...
TRANSCRIPT
06/09/2013
1
Energy transport
• Radiation (photons)• Convection (gas
cells)• Heat transport
(between atoms and electrons)
• Particle radiation (neutrinos)
Convection
ConvectionSolar convection Rayleigh-Bénard convection
The instability condition
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2
ggfbuoy*
2rGmg
*
(6.1)The instability condition
0buoyf
2rGmg
*
Acceleration: Instable
0
0
gfbuoy
The instability condition
*1
*1 P
*2
*2 P
11 P
22 P
r
The instability condition
*1
*1 P
*2
*2 P
11 P
22 P
r
1*
1 PP 1
*1
The instability condition
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3
The motion is so slow that there is pressure balancebetween the element and the surroundings
How to determine Δρ?
min 3012/13
dyn
GMRt (6.2)
The motion is so slow that there is pressure balancebetween the element and the surroundings
How to determine Δρ?
min 3012/13
dyn
GMRt
How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings
The motion is so fast that there is no heat lossto the surroundings
min 3012/13
dyn
GMRt
yr mill. 30S
tot
S
2
KH L
URL
GMt (6.3)
How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings
The motion is so fast that there is no heat loss to the surroundings
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How to determine Δρ?
ggfbuoy*
How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings
2*2
but:
2*
2 PP
How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings
The motion is so fastthat there is no heat loss to the surroundings
2*
2 PP 2*
2 PP
Adiabatic
How to determine Δρ?The motion is so slow that there is pressure balancebetween the element and the surroundings
The motion is so fastthat there is no heat loss to the surroundings
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S
S
S
T
TP
P
ln ln1
ln ln
1
ln ln
3
2
2
1
Adiabatic exponents
entropy
S
S
S
T
TP
P
ln ln1
ln ln
1
ln ln
3
2
2
1
Adiabatic exponents
entropy
SS
PP
P
ln
ln 1
rrP
P
PP
PP
ddd11d
d1d1d
1
**
1*
*
1*
*
(6.4)
entropy
inside element outside
How to determine Δρ?
2*2
How to determine Δρ?
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2*2 12
*1
*2
(6.5)How to determine Δρ?
2*2
rr
rrP
P
dd
dd11
111
12*1
*2
rrP
P
PP
PP
ddd11d
d1d1d
1
**
1*
*
1*
*
(6.5)How to determine Δρ?
Taylor expansionof 2 about 1=1
*
2*2
rr
rrP
P
dd
dd11
111
12*1
*2
rrr
rrr
PP
dd
dd:
dd
dd1
ad
11
1
S
P
ln
ln 1
How to determine Δρ?
rrr
dd
dd
ad
How to determine Δρ?
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rrr
dd
dd
ad
0Instability if:
How to determine Δρ?
rrr
dd
dd
ad
0
rr dd
dd
ad
(6.7)
How to determine Δρ?
Instability if:
rr dd
dd
ad
rr dd
dd
ad
1
dd
rPP
1
dd
rPP
Negative
rr dd
dd
ad
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PP lndln d
lndln d1
ad1
If this condition is satisfied convection
(6.8)
rr dd
dd
ad
1
dd
rPP
1
dd
rPP
rr dd
dd
ad
PP lndln d
lndln d1
ad1
Convection when
”light”
”heavy”
rr dd
dd
ad
PP lndln d
lndln d1
ad1
”light”
”heavy”
53
Convection when Express instability condition in terms of T gradient
addd
dd
rT
rT
PP lndln d
lndln d
ad
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TP
km
B
u
rT
TrP
Pr dd1
dd1
dd1
(6.10)
Express instability condition in terms of T gradient
ideal EOS:TP
km
B
u
rT
TrP
Pr dd
dd
dd
rr dd
dd
ad
rT
TrP
PrP
P dd
dd
dd1
1
”definition”
Express instability condition in terms of T gradient
ideal EOS:
TP
km
B
u
rT
TrP
Pr dd
dd
dd
rr dd
dd
ad
rT
TrP
PrP
P dd
dd
dd1
1
”definition”: i.e. (d/dr)ad is density gradient resulting from adiabatic motion in the given pressure gradient.
Express instability condition in terms of T gradient
ideal EOS:
rr dd
dd
ad
rT
TrP
PrP
P dd
dd
dd1
1
rP
PrT
T dd1
dd
1
1
(6.11)
Correct thermodynamical treatment, which includes partial ionizationand departure from ideal gas law, shows that 1 in above equation mustbe replaced by 2.
Express instability condition in terms of T gradient
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rr dd
dd
ad
rT
TrP
PrP
P dd
dd
dd1
1
rP
PrT
T dd1
dd
2
2
Correct thermodynamical treatment, which includes partial ionizationand departure from ideal gas law, shows that 1 in above equation mustbe replaced by 2.
Express instability condition in terms of T gradient
rr dd
dd
ad
rP
PrT
T dd1
dd
2
2
ad
2
2
dd
dd
dd1
dd
rT
rT
T
rP
PT
rT
T
ad
2
2 : ln
ln1
SS PT
TP
PT
Adiabatic
Express instability condition in terms of T gradient
rr dd
dd
ad
rP
PrT
T dd1
dd
2
2
ad
2
2
dd
dd
dd1
dd
rT
rT
T
rP
PT
rT
T
0
0
addd
dd
rT
rT
Instability when:
(6.12)
Express instability condition in terms of T gradient
rr dd
dd
ad
rP
PrT
T dd1
dd
2
2
0
addd
dd
rT
rT
rr dd
dd
ad
Instability when:
Express instability condition in terms of T gradient
ad
2
2
dd
dd
dd1
dd
rT
rT
T
rP
PT
rT
T
0
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rr dd
dd
ad
rP
PrT
T dd1
dd
2
2
0
addd
dd
rT
rT
Instability when:
Express instability condition in terms of T gradient
ad
2
2
dd
dd
dd1
dd
rT
rT
T
rP
PT
rT
T
0
rr dd
dd
ad
rP
PrT
T dd1
dd
2
2
ad
2
2
dd
dd
dd1
dd
rT
rT
T
rP
PT
rT
T
0
0
addd
dd
rT
rT 1
dd
rP
TP
1
dd
rP
TP
Instability when:
Instability condition in terms of dimensionless T gradient
addd
dd
rT
rT 1
dd
rP
TP
1
dd
rP
TP
2
2 1 ln dln d
PT
(6.14)
Instability when:Instability condition in terms of dimensionless T gradient
This equation shows that instability sets in if the temperaturedecreases too rapidly outwards through the star.
addd
dd
rT
rT 1
dd
rP
TP
1
dd
rP
TP
2
2 1 ln dln d
PT
adadln
ln
PT
ad52
(6.16)
Instability when:
Convection sets in if
for fully ionized ideal gas
Instability condition in terms of dimensionless T gradient
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Where does convection occur?
Radiative energy transport
Energy transport by convection
32π16)( 3
dd
TrcarL
rT
Energy transport by radiation(dimensionless temperature gradient )
(6.17)
(5.8)
R
32π16)( 3
dd
TrcarL
rT
2
)( dd
rrGm
rP
Energy transport by radiation(dimensionless temperature gradient )R
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)( π16)( 3
dd
dd
2
32
1
rGmr
TrcarL
rT
rP
Energy transport by radiation(dimensionless temperature gradient )R
2
)( dd
rrGm
rP
)( π16)( 3
dd
dd
2
32
1
rGmr
TrcarL
rT
rP
XX X
X
Energy transport by radiation(dimensionless temperature gradient )R
2
)( dd
rrGm
rP
)(1
π16)( 3
dd
dd 3
1
rGmTcarL
rT
rP
umk
TP
B
Energy transport by radiation(dimensionless temperature gradient )R
umk
rGmTcarL
rT
rP
TP
)(1
π16)( 3
dd
dd B
3
1
Energy transport by radiation(dimensionless temperature gradient )R
umk
TP
B
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3B
)()(
π16 3
lndlnd
TrmrL
mGcak
PT
u
Energy transport by radiation(dimensionless temperature gradient )R
3B
R
)()(
π16 3
TrmrL
mGcak
u
(6.18)
Energy transport by radiation(dimensionless temperature gradient )R
3B
R
)()(
π16 3
TrmrL
mGcak
u
adR Instability when
Stability when adR
(6.19)
Criteria after Karl Schwarzschild (1905)
Energy transport by radiation(dimensionless temperature gradient )R
Where does convection occur?
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ad3B
)()(
π16 3
TrmrL
mGcak
u
Energy transport by convection(where does it occur?)
adR Instability when
Stability when adR
(6.19)
Criteria after Karl Schwarzschild (1905)
ad3B
)()(
π16 3
TrmrL
mGcak
u
If L/m is large. This condition is typically the case inthe interiors of massive stars.
Massive stars show convection in the core.
Energy transport by convection(where does it occur?)
ad3B
)()(
π16 3
TrmrL
mGcak
u
I the opacity is large. This is satisfied in the outer oartsof relatively light stars on the MS.
”cold” stars show convection in the outher part.
Energy transport by convection(where does it occur?)
ad3B
)()(
π16 3
TrmrL
mGcak
u
If this term is large… this is also satisfied in the outer parts of relatively ”cool” stars.
Low mass stars show convection in the outer part.
Energy transport by convection(where does it occur?)
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ad3B
)()(
π16 3
TrmrL
mGcak
u
If the adiabatic gradient is small. This is satisfied in the ionization zone of hydrogen
”cool” stars show convection in the outer part.
Energy transport by convection(where does it occur?)
sun M2
sun M,51
sun M1
sun M,50
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IonizationHe++, He+, H+
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3D - hydrodynamicalsimulations
3D - hydrodynamicalsimulations(M. Miesch)
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Trampedach (2009)
Star Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr
Bernd FreytagUppsala Universitet in Sweden
Star Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr
TemperatureStar Betelgeuse in Orion Mass 5 M(sun) Radius Ca. 600 R(sun) = 3 AELuminosity 41400 L(sun) Simulation 7.5 yr
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rrT
rTT
dd
dd
ad
(6.21)
Estimate of superadiabatic temperature gradient T
rrT
rTT
dd
dd
ad
Convective heat flux:
TcvF P con (6.22)
Estimate of superadiabatic temperature gradient T
surrounding receivesenergy u per unit volume
mean vertical velocity of convective element
rrT
rTT
dd
dd
ad
TcvF P con
Small: convection isvery efficient
Estimate of superadiabatic temperature gradient T
Convective heat flux:
Estimate of convective velocity v
Equate kinetic energy/volume to buoyancy work over distance r
Introduce dimensionless measure for superadiabatic temperature gradient
TcvF P con
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Estimate of superadiabatic temperature gradient T
With
Convective luminosity Lcon=4R2Fcon
With
and …. total internal energy of star
Estimate of superadiabatic temperature gradient T
Simple interpretation with~1
where is measure of the transported internal energy excess
and is the convective timescale tcon
tcon is a dynamical timescale but increased by , because onlydifference in density provides force!
Estimate of superadiabatic temperature gradient T
Similar to radiative transport, where a sufficiently large T-grad wasrequired, the dimensionless T-gradient has also to be sufficientlylarge for driving convection:
With and
For solar values of tdyn and tKH and for r/R=0.1 we obtain
Extremely small temperature gradient is sufficient to drive convection
addd
dd
rT
rT
If convection is present (~ 0):
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addd
dd
rT
rT
addd
dd
rT
rT
ad3B
)()(
π16 3
TrmrL
mGcak
u
Energy transport by efficient convection
addd
dd
rT
rT
Energy transport by inefficient convection
Needs to be estimated by means of a convection model, e.g. mixing-length model (e.g., Boehm-Vitensen 1956).
A convection model can in principle be interpreted as aninterpolation formulae for Fc between efficient (d~0) and inefficient (d>0) convection (Gough & Weiss 1976).
S … measure of convective efficacy
Fc
convective heat flux
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Convective timescale
Convection zones are chemically homogeneous !This has consequences on the evolution of stars.
<< tnuc
Convective momentum flux(turbulent pressure)
Additional to the convective heat flux Fc, convection also transportsthe flux of momentum pt, which has dimensions of pressure:
Convective momentum flux(turbulent pressure)
Additional to the convective heat flux Fc, convection also transportsthe flux of momentum pt, which has dimensions of pressure:
Sun
Convective momentum flux(turbulent pressure)
Sun
Momentum equation of stellar (envelope) structure: