Post on 26-Jul-2020
Asteroseismology
Steve KawalerIowa State University
observed pulsations• operate on the dynamical time scale
• accessible on convenient time scale
• probe global and local structure
• periods change on ‘evolutionary’ time scale (thermal or nuclear) - depend on global properties
• amplitudes change on ~ ‘local’ thermal time scale
2
dynamical stability• “stable” configuration represents a stable
mean configuration
• on short time scale, oscillations occur, but the mean value is fixed on longer time scales
• simple example: a pendulum (single mode)• most likely position - extrema• mean position is at zero displacement• with no damping would oscillate forever
• more complex example: a vibrating string• multiple modes with different frequencies• enumerated by number of nodes
3
a more complex example: a star
• multiple oscillation modes
• radial modes - enumerated by number of nodes between center and surface
• non-radial modes - nodes also across surface of constant radius
• modes frequencies determined by solution of the appropriate wave equation
4
stability, damping, and driving
• zero energy change: constant amplitude oscillation
• energy loss via pulsation: oscillation amplitude drops with time
• if net energy input: amplitude increases with time (if properly phased)
5
Okay, start your engines...•PG 1159: light curve
• what kind of star might this be?
• what kind of star can this not possibly be?
• what about the amplitude over the run?
•PG 1336 light curve
• huh? what time scale(s) are involved
• what kind of star (or stars)?
• tell us everything you can about this!
6
Multimode pulsation
• Oscillations at “normal mode” frequencies
• mode = specific eigensolution of equations of motion within the confines of a stellar structure
• normal mode frequencies parallel structural properties
• simple example: radial fundamental is one mode, 1st overtone (a node within) is another mode
7
towards the wave equation I • continuity equation:
• equation of motion (HSE when RHS=0):
• perturb r, P, and ρ:
• and assume δx << x so we can linearize
8
@2r
@t2= �GMr
r2� 4⇡r2 @P
@Mr
@Mr
@r= 4⇡r2⇢
x(t, Mr) = xo(Mr)1 +
�x(t, Mr)xo(Mr)
�
• replace x with x+δx in the two equations, subtract off the equilibrium equations, and keep only 1st-order terms to find:
• linearized continuity equation
• linearized equation of motion
9
�⇢
⇢o= �3
�r
ro� ro
@(�r/ro)@ro
⇢orod2�r/ro
dt2= �
✓4�r
ro+
�P
Po
◆@Po
@ro� Po
@(�P/Po)@ro
towards the wave equation II • assume adiabatic relationship between P and ρ
• combine continuity and equation of motion:
• assume exponential (complex) time dependence
10
towards the wave equation III
�r(t, ro)ro
=�r(ro)
roei�t = ⌘(ro)ei�t
�1 =✓
@ lnP
@ ln ⇢
◆
ad
so�P
Po= �1
�⇢
⇢o
@2⌘
@t2=
1⇢r4
@
@r
✓�1Pr4 @⌘
@r
◆+ ⌘
1r⇢
⇢@
@r[(3�1 � 4) P ]
�
• substitute to yield the Linear Adiabatic Wave Equation (LAWE):
• This is a wave equation: L(η) = σ2η in the displacement η.
• the eigenvalue σ2 corresponds to the oscillation frequency
11
towards the wave equation IV
L(⌘) = � 1⇢r4
@
@r
✓�1Pr4 @⌘
@r
◆� ⌘
1r⇢
⇢@
@r[(3�1 � 4) P ]
�= �2⌘
the LAWE: a simple case• assume Γ1 and η both constant throughout
the star (homologous motion)
• LAWE becomes
• now, assume a constant density, and use HSE to replace the pressure derivative to find
• look familiar?!
12
�⌘1⇢r
(3�1 � 4)@P
@r= �2⌘
⇧ =2⇡
�=
p⇡q
G⇢̄ (�1 � 43 )
the LAWE: standing wave solutions
• boundary conditions:
• center: zero displacement (η = 0)
• surface: perfect wave reflection [d(δP/P)/dr = 0]
• asymptotic analysis:
• clever change of variables renders LAWE as:
• recognizing the sound speed when we see it:
13
d2w(r)dr2
+
�2⇢
�1P� �(r)
�w(r) = 0
d2w(r)dr2
+�2
c2s
� �(r)�
w(r) = 0
the LAWE: asymptotic solution
• represent the eigenfunction as: where kr is the (local) radial wavenumber and varies slowly with radius so, locally:
• for a standing wave, we need an integral number of half-wavelengths between inner and outer reflection points:
14
d2w(r)dr2
+�2
c2s
� �(r)�
w(r) = 0
w(r) / eikrr
k2r =
�2
c2s(r)
� �(r)
Z b
akr dr = (n + 1)⇡
the LAWE: asymptotic solution
• if then
• i.e. high-frequency (high overtone, n) radial modes are equally spaced in frequency, with
σo2 ≈ G<ρ>
15
Z b
akr dr = (n + 1)⇡
�2
c2s
� �
� = (n + 1)⇡
"Z b
a
dr
cs
#�1
= (n + 1) �o
k2r =
�2
c2s(r)
� �(r)where
Nonradial oscillations• preserve angular derivatives in LAWE
• similar operator structure for radial part (as before), now along with angular part
where the quantity A is:
A < 0 when radiative A > 0 when convective
(the ‘Schwarzschild A’)
16
A =d ln ⇢
dr� 1
�1
d lnP
dr=
1�P
�T
�⇢[r�rad]
d2�rdt2
= �r✓
P 0
⇢+ 0
◆+ A
�1P
⇢r · �r
Decompose into Spherical Harmonics
• position perturbation decomposition
• produces (after some work):
where the operator L2 (the Legendrian) is:
• which has eigenstates Ylm such that:
17
�r = �r er + r�✓ e✓ + r sin ✓�� e�
r · �r =1r2
@(r2�r)@r
� 1�2
r2L2
✓P 0
⇢+ 0
◆
L2 = � 1sin ✓
@
@✓
✓sin ✓
@
@✓
◆� 1
sin2 ✓
@2
@�2
L2Y ml (✓, �) = l(l + 1)Y m
l (✓, �)
l=1, m=0 l=1, m=1
i=70
l=3, m=0 l=3, m=1 l=3, m=3
Spherical Harmonics.... courtesy asteroseismology.org (Travis Metcafe)
• now we have
• expanding into components:
using Spherical Harmonics19
r · �r =1r2
@(r2�r)@r
� l(l + 1)�2r2
✓P 0
⇢+ 0
◆
r2 d⌘t
dr=
1 +
Ag
�2
�r⌘r +
(�Ag)
r
g� 1
�r⌘t
r2 d⌘r
dr=
gr
c2s
� 2�
r⌘r + r2 l(l + 1)r2
1� �2r2
c2s
1l(l + 1)
�r⌘t
frequency-2
another frequency2
• so:
• where we’ve defined 2 “structural” frequencies
• the acoustic (Lamb) frequency Sl :
• the Brunt-Väisälä (buoyancy) frequency N:
the two characteristic frequencies20
r2 d⌘r
dr=
gl(l + 1)
S2l
� 2r
�⌘r + l(l + 1)
1� �2
S2l
�r⌘t
r2 d⌘t
dr=
1� N2
�2
�r⌘r +
N2 r
g� 1
�r⌘t
S2l =
l(l + 1)r2
c2s
N2 = �Ag = �g
d ln ⇢
dr� 1
�1
d lnP
dr
�
page
Propagation diagram, ZAMS solar model21
Sl2, l=1
N2
CZbase
NRP dispersion relation• identify the horizontal wave number(s)
• allows the wave equation(s) to reduce to a local dispersion relation, as with the radial case, to provide relationship between kr and σ:
22
k2t =
l(l + 1)r2
=S2
l
c2s
k2r =
1�2c2
s
(�2 �N2)(�2 � S2l )
asymptotic analysis
• kr2 > 0 (kr real) when
• σ2 > N2, Sl2 - or - σ2 < N2, Sl2
• kr real means oscillatory eigenfunctions
• kr2 < 0 (kr imaginary) when
• Sl2 > σ2 > N2 or Sl2 < σ2 < N2
• kr real means evanescent (exponentially decreasing or increasing) eigenfunctions
23
k2r =
1�2c2
s
(�2 �N2)(�2 � S2l )
page
Propagation diagram, ZAMS solar model24
Sl2, l=1
N2
CZbase
l=1 n=1l=1 n=2l=1 n=3l=1 n=4
page 25
Sl2, l=1
N2
CZbase
l=1 n=1l=1 n=2l=1 n=3l=1 n=4
Propagation diagram, ZAMS solar model
evanescentzonesin gray
page 26
Sl2, l=1
N2
CZbase
l=1 n=1l=1 n=2l=1 n=3l=1 n=4
evanescentzonesin gray
Propagation diagram, ZAMS solar model
“low” σoscillatory
zonesin green
“high” σoscillatory
zonesin pink
the NRP LAWE: asymptotic solutions
• again, integrate dispersion relation over propagation regions:
• two classes of solutions:
• σ2 > N2, Sl2:
• “p-modes”; pressure as the restoring force
• σ2 < N2, Sl2:
• “g-modes”: buoyancy as the restoring force
27
k2r =
1�2c2
s
(�2 �N2)(�2 � S2l )
Z b
akr dr = (n + 1)⇡ where
�nl = (n + l/2)�o ; �o =Z b
a
dr
cs
⇧nl = n⇧op
l(l + 1); ⇧o = 2⇡2
"Z b
a
N
rdr
#�1
page 28
Sl2, l=1
N2
CZbase
l=1 n=1l=1 n=2l=1 n=3l=1 n=4
evanescentzonesin gray
Propagation diagram, ZAMS solar model
“low” σoscillatory
zonesin green
“high” σoscillatory
zonesin pink
l=1 n=1l=1 n=2l=1 n=3l=1 n=4
page 29 page
Pulsation PeriodsPeriod of ‘radial fundamental’ ~ tff
g-modes p-modes
Periods Π > tff Π < tffrestoring force buoyancy pressure
asymptotic behavior
Π ∝ Πo x n σ ∝ σo x n
examples white dwarfs Cepheids,the Sun
p-modes: ~ equally spaced in frequency31
n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2
......frequency
�nl = (n + l/2)�o ; �o =Z b
a
dr
cs
l l l l l l l l l
n-2l+1
n-1l+1
nl+1
n+1l+1
n+2l+1
n+3l+1
n+4l+1
n+5l+1
• if modes of different l present, observed spacing ~ σo / 2
p. 32
20 parts per million!
Solar Oscillations: Full-disk photometry Frohlich et al. 1997: SOHO/VIRGO
1 hr 10 min
5 min
p.Solar Oscillations: Full-disk photometry Frohlich et al. 1997: SOHO/VIRGO
33
20 parts per million!
1 hr 10 min
5 min
p. 34
How to observe all these low amplitude modes?
have your friends buy a $600,000,000 photometer!
p.
16 Cyg A and B (Metcalfe et al. 2012)35 p.
asteroseismic radius determination (i.e. Chaplin et al. 2011)
36
• νmax scales with acoustic cutoff frequency ~ gTe-1/2
• Δν measures mean density:�
��
���
�2
��
M
M�
� �R
R�
��3
��max
�max,�
��
�M
M�
� �R
R�
��2 �Te�
Te�,�
��0.5
p.
Kepler 93b (Ballard et al. 2014)
37 p.
Kepler 93b (Ballard et al. 2014)
38
p. 39
Kepler 93b (Ballard et al. 2014)
g-modes: ~ equally spaced in period40
n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2
......Period
n n+1 n+2 n+3 n+4 n+5 n+6n-1n-2
.........
...Period
l=1
l=2
⇧nl = n⇧op
l(l + 1); ⇧o = 2⇡2
"Z b
a
N
rdr
#�1
• spacing depends on l
PG 1159-035: a g-mode pulsator
P P-390
390 0
424 34 ?
451 61 3x20.3
495 105 5x21.0
516 126 6x21.0
539 149 7x21.3
645 255 12x21.3
832 442 21x21.0
an example: hot white dwarf PG 1159-035 (Corsico et al. [WET] 2008)
in white dwarfs: Πodepends on total stellar mass
PG 1159 as an example(Winget et al. 1991)
Rotational splitting of nonradial oscillations (uniform, slow rotation)
equal frequency spacing: triplets (l=1), quintuplets (l=2) etc.
page
Pulsating stars in the HR diagram
from J. Christensen-Dalsgaard
“Solar-like” oscillations• globally stable (damped) but constantly excited
• damping time τ generally ~ days
• continuously re-excited by turbulence
• frequencies locked to normal modes of star
• excited mode periods ~ minutes
• broad mode selection, low amplitude
• (integrated) velocity amplitude < meters / second
• photometric amplitude ~ parts per million
Consequences of stochasitic excitation
• lots of modes present . . . but
• coherence time of (only) days lowers peak amplitudes• reduces detectability
• phase instability broadens FT peaks• Lorentzian envelope
• reduces frequency accuracy
• confuses mode identification and rotational splitting effects
from J. C.-D.
Stochastically excited modeheat-engine mode
1/(observing time)
1/lifetime
Observational Differences
1980 1990 2000 2010 2020
0.00
0.20
0.40
0.60
0.80
1.00T = 5d
1980 1990 2000 2010 2020
T = 10d
1980 1990 2000 2010 2020
T = 30d
coherent pulsators: frequency precision ~ 1/T
1980 1990 2000 2010 2020
0.00
0.20
0.40
0.60
0.80
1.00T = 5d
1980 1990 2000 2010 2020
T = 10d
1980 1990 2000 2010 2020
T = 30d
0.00
0.20
0.40
0.60
0.80
1.00T = 5d T = 10d T = 30d
0.00
0.20
0.40
0.60
0.80
1.00T = 5d T = 10d T = 30d
stochastic pulsators - freq. precision poorer than 1/T
Asymptotics of low-degree p modes
Large frequency separation:
Frequency separations:
Small frequency separations
Asteroseismic HR diagram
from J. Christensen-Dalsgaard
A Search for Habitable Planets
Borucki et al. 2009 -
Kepler’s Optical Phase Curve of the Exoplanet HAT-P-7b
HAT-P7 asteroseismology HAT-P7 asteroseismology
1.40 Mo
1.45 Mo
Asteroseismic HR diagram
from J. Christensen-DalsgaardSunα Cen A
HAT-P-7
HAT-P7 asteroseismology
spectroscopyPal et al. 2008
M=1.40±0.02
t = 1.6±0.4 Gyr
r = 1.94±0.05 Rsun
Xc=0.19
quick seismic fitISUEVO
Solar-like pulsators: KIC 6603624 Chaplin et al. 2010
Solar-like pulsators: KIC 3656476 Chaplin et al. 2010
Solar-like pulsators: KIC 11026764 Chaplin et al. 2010
Solar-like - FTs
Sunα Cen A
HAT-P-7
Asteroseismic HR diagram
KIC 11026764 is post-MS!
p.oscillations beyond the MS w/ Kepler(Chaplin & Miglio 2013)
64
p.Solar-like oscillations w/ Kepler(Chaplin & Miglio 2013)
65
MS
MS (16 Cyg A)
Subgiant
Subgiant
Base of RGB
500 1000 2000 3000 4000 uHz
p.Solar-like oscillations w/ Kepler(Chaplin & Miglio 2013)
66
MS
MS (16 Cyg A)
Subgiant
Subgiant
Base of RGB
500 1000 2000 3000 4000 uHz
p. 67
First-ascent RGB
First-ascent RGB
First-ascent RGB
Red Clump
20 40 60 80 100 uHz
oscillations beyond the MS w/ Kepler(Chaplin & Miglio 2013)
p.
scaling relations MS to RGB68
R/Rsun log g Δν [uHz] νmax
MS 1 4.44 135 3300
RGB base 5 3.04 12.1 140
RGB top 20 1.84 1.5 9 < 1/d1/wk
p. 69
multiplel=1 modesper order?!
Bedding et al. (2010) Ensemble asteroseismology of solar-type stars with Kepler
Chaplin et al. 2011 (Science, today!)
• determination of Δν and νmax for ~ 500 solar-type stars
• νmax scales with acoustic cutoff frequency ~ gTe-1/2
• Δν measures mean density:�
��
���
�2
��
M
M�
� �R
R�
��3
��max
�max,�
��
�M
M�
� �R
R�
��2 �Te�
Te�,�
��0.5
Chaplin et al., Science, 7 April 2011 Chaplin et al., Science, 7 April 2011
• Δν measures mean density:
• νmax scales with acoustic cutoff frequency ~ gTe-1/2
• then one can determine M and R:
• and, with Teff from multicolor photometry, one gets L
���
���
�2
��
M
M�
� �R
R�
��3
��max
�max,�
��
�M
M�
� �R
R�
��2 �Te�
Te�,�
��0.5
Chaplin et al., Science, 7 April 2011
R
R��
��max
�max,�
� ���
���
��2 �Te�
Te�,�
�0.5
M
M��
��max
�max,�
�3 ���
���
��4 �Te�
Te�,�
�1.5
Chaplin et al., Science, 7 April 2011