Lecture course in Asteroseismology Part I · Asteroseismology Observables: some kind of periodic...
Transcript of Lecture course in Asteroseismology Part I · Asteroseismology Observables: some kind of periodic...
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Lecture course in Asteroseismology
Part IPart I
Margarida S. Cunha
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Christensen-Dalsgaard
Stellar pulsatorsStellar pulsators
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Christensen-Dalsgaard
Classification
Intrinsically unstableClassical
Origin Intrinsically stable
Solar-like
Acoustic wavesp modes
Nature Internal Gravity waves
g modes
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Asteroseismology
Observables: some kind of periodic variation
intensity; Velocity
Properties of the oscillations
frequencies, angular form
Properties = f (interior)
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Hydrodynamics
Continuity
Motion++−∇=
=+
Fgpt
t
D
vD
0v div D
D
ρρ
ρρ
rrr
r
Energy
Energy
Eqs ofstate
Γ−
−Γ=
=+=
t
p
t
p
tp
t
E
t
q
D
D
D
D
)1(
1
D
)1(D
D
D
D
D
1
3
ρρρ
ρ
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Hydrodynamics
++−∇=
=+
Fgpt
t
D
vD
0v div D
D
ρρ
ρρ
rrr
r Continuity
Motion
Γ−
−Γ=
=+=
t
p
t
p
tp
t
E
t
q
D
D
D
D
)1(
1
D
)1(D
D
D
D
D
1
3
ρρρ
ρEnergy
Energy
Eqs ofstate
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Hydrodynamics
++−∇=
=+
Fgpt
t
D
vD
0v div D
D
ρρ
ρρ
rrr
r Continuity
Motion
Small perturbations about a background state which is:static and in thermal equilibrium
Γ−
−Γ=
=+=
t
p
t
p
tp
t
E
t
q
D
D
D
D
)1(
1
D
)1(D
D
D
D
D
1
3
ρρρ
ρEnergy
Energy
Eqs ofstate
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Hydrodynamics
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Small perturbations about a state:
• static
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
• static • thermal equilibrium
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Hydrodynamics
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Small perturbations about a state:
• static
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
• static • thermal equilibrium
fff ∇⋅+′= rr
δδ
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Hydrodynamics
Most common approximations
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
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Hydrodynamics
Most common approximations
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Adiabatic approximation
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
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Hydrodynamics
Most common approximations
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Adiabatic approximation
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
Why is the adiabatic approximation adequate?
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Hydrodynamics
Adiabatic approximationMost common approximations
Typical values in the deep interior: 107 years (~10-15 s-1)Typical values in solar convection zone: 103 years (~10-11 s-1)
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Hydrodynamics
Adiabatic approximationMost common approximations
Outer layers of a 2.2 Msun star
Typical values in the deep interior: 107 years (~10-15 s-1)Typical values in solar convection zone: 103 years (~10-11 s-1)
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Hydrodynamics
Most common approximations
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Adiabatic approximation
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
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Hydrodynamics
Most common approximation
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Cowling approximation
Adiabatic approximation
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
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Hydrodynamics
Most common approximation
′+′+′−∇=∂∂
=+∂
∂
ggpt
t
ρρρ
ρδρ
v
0v div
rrr
r
Cowling approximation
( )dV
rr
trG
g
V
∫ ′−
′′=Φ′
Φ′∇=′
rr
r
r
,ρ
Adiabatic approximation
∂
∂Γ−
∂∂
−Γ=
∂∂
∂
t
p
t
p
t
q
t
δρρ
δρ
δ 1
3 )1(
1
Note: Cowling approximation not used for radial modes
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Simple waves
gpt
rr
ρδ
ρ ′+′−∇=∂
∂2
2 r
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Simple waves
gpt
rr
ρδ
ρ ′+′−∇=∂
∂2
2 r
Acoustic wavesNeglect last term on rhs
( )[ ]tii ω−⋅∝ rkexpperturrr
high ω
222
222
k
k
r
r
c
c
=
′=′
ω
ρρω
( )[ ]tii ω−⋅∝ rkexpperturrr
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Simple waves
gpt
rr
ρδ
ρ ′+′−∇=∂
∂2
2 r
Acoustic waves Internal gravity waves
Motion cannot be purely radialNeglect last term on rhs
( )[ ]tii ω−⋅∝ rkexpperturrr ( )[ ]tii ω−⋅∝ rkexppertur
rr
low ω
222
222
k
k
r
r
c
c
=
′=′
ω
ρρω
hr
r
r
h
r
g
kk1
k
k1
22
2
2
+=
=
′=
+
ω
ξ
ρξρω
( )[ ]tii ω−⋅∝ rkexpperturrr ( )[ ]tii ω−⋅∝ rkexppertur
rr
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Summary
Acoustic waves Internal gravity waves
Simple waves
• maintained by gradient of pressure fluctuation;
• maintained by gravity acting on density fluctuation;
• Radial or non-radial;
• Propagate in convectively stable or non-stable
regions
on density fluctuation;
• Always non-radial;
• Propagate in convectively stable regions only
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p modes g modes
Summary
Simple waves
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Waves in a star
])er(~
Re[),r( tiftf ωδδ −=rr
Separate solutions in time. Any perturbation δ f admits solutions :
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])er(~
Re[),r( tiftf ωδδ −=rr
Waves in a star
Separate solutions in time. Any perturbation δf admits solutions :
δf (r)~Substituting in perturbed equations to get Eqs forf (r)
)~
(~2 ff δδω L=
Nontrivial solutions satisfying the Eqs + boundary
conditions exist only for particular values of ω
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])er(~
Re[),r( tiftf ωδδ −=rr
Waves in a star
Separate solutions in time. Any perturbation δf admits solutions :
Separate solutions in angular space. Consider (r,θ,φ) . If the
equilibrium state has spherical symmetry, any scalar perturbation δfequilibrium state has spherical symmetry, any scalar perturbation δfadmits solutions:
]e),()(Re[),r( tim
lYrftf ωφθδδ −=(r
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])er(~
Re[),r( tiftf ωδδ −=rr
Waves in a star
Separate solutions in time. Any perturbation δf admits solutions :
Separate solutions in angular space. Consider (r,θ,φ) . If the
equilibrium state has spherical symmetry, any scalar perturbation δfequilibrium state has spherical symmetry, any scalar perturbation δfadmits solutions:
]e),()(Re[),r( tim
lYrftf ωφθδδ −=(r
and the displacement δ r admits solutions :
∂∂
+∂
∂+= − ti
m
l
m
lh
m
lr
YYrYrt ω
φθ φθθξξδ e a
sin
1a)(a)(Re),r(r r
rr
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Degree l : related to the horizontal wavenumber
Spherical Harmonics Ylm
R
L
R
llh =
+=
)1(k
Azimuthal order m : number of nodes along the equator
=> orientation on the sphere
l=0 l=1
m=0
l=1
m=1
Note: |m| ≤ l
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]e),()(Re[),r( tim
lYrftf ωφθδδ −=(r
Waves in a star
Assume perturbation has the form
Substitute in perturbed equations to get Eqs for δf (r)
centre surfaceradius
ω=ω(n,l,m)
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Spherical symmetry
Waves in a star
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Spherical symmetry
Waves in a star
)~
(~2 ff δδω L=
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Spherical symmetry
Waves in a star
)~
(~2 ff δδω L=
Eigenfrequencies ω of modes with the same n and l are degenerate.
∑=m
m
lmYaf~
δOne eigenmode any combination
ω=ω(n,l,m)
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Waves in a star
Radial dependence
Under the adiabatic and Cowling approximations, the perturbed Eqs can be combined to give:
0d 2
2
=Ψ+Ψ
K )()( where rpur δ=Ψ0d
d 2
2=Ψ+
ΨrK
r)()( where rpur δ=Ψ
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Waves in a star
Radial dependence
Under the adiabatic and Cowling approximations, the perturbed Eqs can be combined to give:
0d 2
2
=Ψ+Ψ
K )()( where rpur δ=Ψ0d
d 2
2=Ψ+
ΨrK
r)()( where rpur δ=Ψ
Kr2>0 => oscillatory behaviour => propagation
Kr2<0 => exponential behaviour => evanescent
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Waves in a star
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
Kh2
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Waves in a star
p modes
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
Kh2K2
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Waves in a star
p modes
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
Kh2K2 2
22 Kr
c=ω2
222 Kr
cc =− ωω
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Waves in a star
p modes
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
Kh2K2 2
22 Kr
c=ω
Near the surface Kr2 < 0 if ω2 < ωc
2 : ~independent of L
Deep in the interior Kr2 < 0 if ω2 < c2 Kh
2 : dependent of
2222 Kr
cc =− ωω
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Waves in a star
p modes
Different colours
Different l
Near the surface Kr2 < 0 if ω2 < ωc
2 : ~independent of L
Deep in the interior Kr2 < 0 if ω2 < c2 Kh
2 : dependent of
Different l
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Waves in a star
g modes
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
Kh2
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Waves in a star
g modes
−−
−=
2
2
2
2
2
222
1ω
ωω N
r
L
cK c
r
2N 2
Kh2
hr KK1
22
+=
Nω
hr
KK1
22
+=ω
Kr2 < 0 if ω2 < N2
g modes can propagate only in convectively stable regions.
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Waves in a star
Propagation diagram
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Lecture course in Asteroseismology
Part IIPart II
Margarida S. Cunha
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Asteroseismology
Observables: some kind of periodic variation
intensity; Velocity
Properties of the oscillations
frequencies, angular form
Properties = f (interior)
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Asteroseismology
Observables: some kind of periodic variation
intensity; Velocity
Properties of the oscillations
frequencies, angular form
Properties = f (interior)
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counts
ω=ω(n,l,m)
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Different colours
Different l
ω=ω(n,l,m)
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Asymptotic analysis
How is the oscillation spectrum supposed to look like?
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Asymptotic analysis
Adiabatic oscillations in the Cowling approximation.
High n, low l, acoustic oscillations:
1
0
0
0
2 where
2
−
=∆
+∆
++≈
∫R
nl
c
dr
termsorderhigherl
n
ν
ναν
0 c
• ∆ν0 prop (M/R3)1/2
• α function of ν and is due to surface effects
• Note: ν=ω/2π
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Asymptotic analysis
Adiabatic oscillations in the Cowling approximation.
High n, low l, acoustic oscillations:
∆ν0 prop (M/R3)1/2
...2
0 +∆
++≈ νανl
nnl
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Asymptotic analysis
Large separations ∆νnl
termsorderhigherl
nnl 2
0 +∆
++≈ ναν
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Asymptotic analysis
Large separations ∆νnl
termsorderhigherl
nnl 2
0 +∆
++≈ ναν
0,,1∆ ννν ∆≈−= + lnlnnlν α (M/R3)1/2
νννν
∆νnlSchematicPowerSpectrum
n-1,0 n-1,1 n,0 n,1 n+1,0
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( )[ ]
−
∆=
+∆
−+−∆
++≈
∫R
nl
nl
r
dc
R
RcA
lAll
n
00
2
00
)(
4
1 where
...12
νπ
νν
δναν
Asymptotic analysis
Adiabatic oscillations in the Cowling approximation.
High n, low l, acoustic oscillations:
∆ rR004 νπ
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( )[ ]
−
∆=
+∆
−+−∆
++≈
∫R
nl
nl
r
dc
R
RcA
lAll
n
00
2
00
)(
4
1 where
...12
νπ
νν
δναν
Asymptotic analysis
small separations δνnl
∆ rR004 νπ
( ) ∫∆
+−≈−= +−
R
ln
lnlnnlr
dcl
0,
2
02,1,
464
νπν
ννδν
νννν
∆νnlSchematicPowerSpectrum
n-1,2
δνnl
n-1,0 n-1,1 n,0 n,1
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Asymptotic analysis
Sun as a star
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Asymptotic analysis
Sun as a star
Harder for classical pulsators
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Asymptotic analysis
( ) ( ) ∫∆
+−≈+−= ++−
R
ln
lnlnlnnlr
dcl
0,
2
01,1,1,
)1(
422
2
1
νπν
ννννδ
Alternative small separations δ(1)νnl
νννν
∆νnlSchematicPowerSpectrum
δ(1)νnl
n-1,0 n-1,1 n,0 n,1
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Asymptotic analysis
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MsunZAMS
Asymptotic analysis
JWKB breaks down
Msun8 Gyr
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Signatures of sharp transitions
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e.g: boundaries of convective regions;regions of different chemical composition;Ionization regions, etc
...2
0 +∆
++≈ νανl
ns
nlDeviations from smooth ν
Signatures of sharp transitions
20
nl
nl
s
nlnl νδνν ~+=
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Modes of different frequency ‘feel’ the transition region differently.
e.g: boundaries of convective regions;regions of different chemical composition;Ionization regions, etc
Signatures of sharp transitions
Transition region
differently.
( )[ ]ϕτπννδ +dnlnl A 22cos~~
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Signatures of sharp transitions
( )[ ]ϕτπννδ +dnlnl A 22cos~~
∫==R
ddc
drr )(ττ ∫==
r
dd
dc
r )(ττ
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Signatures of sharp transitions
( )[ ]ϕτπννδ +dnlnl A 22cos~~
∫==R
ddc
drr )(ττ ∫==
r
dd
dc
r )(ττ
How to isolate the perturbation due to the sharp transition?
1. Fit a smooth function to the frequencies as function of n and remove that function from the frequencies => residual
2. Calculate the second differences:lnlnlnnl ,1,,12 2 −+ +−=∆ νννν
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Signatures of sharp transitions
( )[ ]ϕτπννδ +dnlnl A 22cos~~
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Direct fitting
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Direct fitting
[ ] ref
nl
ref
surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=
Set of non-seismic and seismic observables, yi errors σi
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Direct fitting
[ ] ref
nl
ref
surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=
Set of non-seismic and seismic observables, yi errors σi
Set of model parameters aj
ageYXMa MLj ,,,, 00 α=
Set of model parameters aj
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Direct fitting
[ ] ref
nl
ref
surfeffi rRHFeLTy 020 ,,,,/,, νν ∆=
Set of non-seismic and seismic observables, yi errors σi
Set of model parameters aj
ageYXMa MLj ,,,, 00 α=
Set of model parameters aj
From models construct yimod and minimize
( )[ ]∑ −=i
iii yy2mod2 σχ
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Inversion
Combine different seismic data (e.g. frequencies) to infer localised information.
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Inversion
Combine different seismic data (e.g. frequencies) to infer localised information.
∫ Ω=V
ii rdrrkdrrr
)()(
We want localized information about Ω at position r0
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Inversion
Combine different seismic data (e.g. frequencies) to infer localised information.
∫ Ω=V
ii rdrrkdrrr
)()(
[ ]∫∑∑ Ω=V i
ii
i
ii rdrrkadarrr
)( )(
We want localized information about Ω at position r0
),( 0 rrrr
K
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Seismic inference
• Asymptotic properties to get information on mean density and about the core
• Deviation from asymptotic behaviour to characterize sharp transitions characterize sharp transitions
• Direct fitting to determine model parameters
• Inverse methods to derive localize information
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Deviations from spherical symmetry
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Deviations from spherical symmetry
Fgpt
rrr
++−∇= ρρD
vDe.g. rotation; magnetic field
ω=ω(n,l,m)
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Deviations from spherical symmetry
Fgpt
rrr
++−∇= ρρD
vDe.g. rotation; magnetic field
ω=ω(n,l,m)
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Deviations from spherical symmetry
Fgpt
rrr
++−∇= ρρD
vDe.g. rotation; magnetic field
ω=ω(n,l,m)
l=1
m=-1
l=1
m=1
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Deviations from spherical symmetry
Fgpt
rrr
++−∇= ρρD
vDe.g. rotation; magnetic field
ω=ω(n,l,m)
l=1
m=-1
l=1
m=1)cos(
]e),(Re[),r(
tm
Ytf tim
l
ωφ
φθδ ω
−∝
∝ −r
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Axisymmetric Star
)~
(~2 ff δδω A=
AAAA
Reference frame aligned with axis of symmetry
one eigenmode one single m state: Ylm
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Axisymmetric Star
)~
(~2 ff δδω A=
AAAA
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Axisymmetric Star
)~
(~2 ff δδω A=
AAAA
1 – perturbation invariant under reflection about the equator:- degeneracy partially lifted: ω=ω(n,l,|m|)
2 – perturbation not invariant under reflection about the equator:- degeneracy totally lifted: ω=ω(n,l,m)
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Axisymmetric Star
Slowly rigid rotating star
‘Geometrical’ effect
Ω
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Axisymmetric Star
Slowly rigid rotating star
‘Geometrical’ effect
Ω
mΩobs
m += 0ωω
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Axisymmetric Star
Slowly rigid rotating star
Coriolis effect
In a reference frame rotating with the star, to first order in Ω/ω
rirrrrrr
δωδδω ×Ω−= 2)(2L rirr δωδδω ×Ω−= 2)(L
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Axisymmetric Star
Slowly rigid rotating star
Coriolis effect
In a reference frame rotating with the star, to first order in Ω/ω
rirrrrrr
δωδδω ×Ω−= 2)(2L rirr δωδδω ×Ω−= 2)(L
ΩmCnl
rot
m −= 0ωω
Where Cnl is the Ledoux constant: typically << 1 and ->0 for high n
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Axisymmetric Star
Slowly rotating star
Coriolis effect
In a reference frame of the observer, to first order in Ω/ω
ΩCm nl
obs
m )1(0 −+= ωω ΩCm nlm )1(0 −+= ωω
If Ω = Ω(r)
nlnl
obs
m ΩCm )1(0 −+= ωω
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Axisymmetric Star
Slowly rotating star