Transcript elodie 95

Solar-like Oscillations in other Stars
or
The only way to test directly stellar structure
theory
I.
Scaling Relations
II.
Results
Asteroseismology: A relatively new field
Kjeldsen & Bedding, 1995, A&A, vol. 293, 87
„Asteroseismology of solar-like stars has so far produced
disappointing results. Despite repeated attempts and everincreasing sensitivity, there have been no unambiguous
detection of solar-like oscillations on any star except the
Sun.“
Solar-like oscillations: Any oscillation that is excited by
convection in the outer part of the star analogous to the 5
minute oscillations.
Only low-degree modes can be detected
Unlike the sun for which we have
detected mode with l up to 400,
this is because the Sun is
resolved. For stars for which we
measure integrated light, there is
a high degree of cancellation. We
can only detect low degree
modes, l = 0-3
The Asymptotic Limit for p-mode Oscillations
For high order modes: n >> l
nn,l = Dn0 (n + l /2 + e) – l (l +1 )D0
Dn0 = large spacing
D0 is sensitive to the sound speed near the core
e is sensitive to the surface layers
The last term is related to the small frequency spacing
Note: l = 0 → radial pulsations. For pure radial modes the frequency
spacing is the large spacing
If there is no small frequency spacing nn+1,0 = nn,2 but the gradient
of the sound speed (small frequency spacing) lifts this degeneracy
Definition:
dn02 is the small separation as the frequency spacing of
adjacent modes with l = 0 and l=2
dn13 is the small separation as the frequency spacing of
adjacent modes with l = 1 and l =3
dn01 is the amount modes l = 1 are offset between the midpoint of l =0 modes on either side.
If the asymptotic relation holds exactly then:
D0 =
dn02
6
=
dn01
2
=
dn01
10
The Solar Power Spectrum showing the large and
small spacings:
dn01
A Diagnostic Tool: Echelle Diagrams
Dn
l=0
l=1
l=2
nn,l = n0 + kDn + n1
n0 = a reference
k = integer
n1 = 0 → Dn
Echelle diagrams take advantage of the fact that in the asymptotic
relationship p-modes are equally-spaced in frequency.
An echelle diagram basically cuts the frequency axis into chunks of
Dn and stacks them on each other
dn13
dn02
l=1
Radial
order n,
increasing
Frequency
l=0
l=3
l=2
0
Dn1(= Dn0/2)
Modulo Frequency
Dn0
The „Asteroseismic“ H-R Diagram
The goal of asteroseismology is to detect enough
modes to derive the large and small frequency
spacings. From these you get the mass and age of the
star from the „Asteroseismic“ H-R diagram
Estimating the Photometric Amplitude of
Oscillations (Kjeldsen & Bedding, 1995, 293, 87)
For adiabatic oscillations
To first order the density compression for
an adiabatic sound wave
dL
(L
(
The change in luminosity of the star is due
almost entirely to temperature changes
dT
T
∝
∝
bol
dr
=
r
dT
T
dr
r
v
cs
Estimating the Photometric Amplitude of
Oscillations
The adiabatic sound speed:
[
cs =
 ln P
[  ln r
P
r
ad
Ideal gas law give:
P∝ rT
2
cs ∝ T
 ln P
[  ln r
[
2
=
ad
5/3
Estimating the Photometric Amplitude of
Oscillations
(
dL
(L
~
bol
vosc
Assume T ≈ Teff
√Teff
This expresses the luminosity amplitude of stellar oscillations in
terms of the velocity amplitude. Note: this is for small amplitude
variations.
In comparing to observations, which are made at specific wavelengths
we have to take into account that the luminosity amplitude of an
oscillation depends on the wavelength that it is observed.
dL
(L
l
(l
bol
(
~
dL
(L
l
(l
bol
l
l
(
(
bol
(
=
(
dL
(L
lbol
Combining:
(
dL
(L
l
=
vosc
l
Teff
–1.5
=
623 nm
Teff /5777K
When comparing to real data a better fit is obtained using an exponent
of –2 instead of –1.5. This is not unsurprising since we used an
adiabatic approximation so there may be temperature corrections
when comparing to real data. The revised equation becomes:
(
=
(l/ 550 nm) (Teff / 5777 K)
l
dL
(L
(
dL
(L
vosc / m s–1
bol
2
20.1 ppm
vosc / m s–1
=
(Teff / 5777 K)
2
1 ppm = 1 part per million (10–6) = 1.086 mmag
17.7 ppm
This shows the predicted
luminosity variations versus
observed variations for a
variety of pulsating stars.
The previous expression is
a good approximation even
when the oscillations are
nonlinear or non-adiabatic
Comparison of the predictions for this simple expression to
model predictions from Christensen-Daalsgard & Frandsen
(1983, Sol. Physics, 82, 469)
Estimating the Velocity Amplitude
Stellar models suggest that
the velocity amplitude
scales:
Vosc ~ L/M
We can thus take the
solar amplitudes and
scale these
according the values
to other stars
Scaling the Velocity Amplitude to other stars
Amplitude Scaling Laws
vosc
(
dL
(L
l
L/L‫סּ‬
(23.4 ± 1.4) cm/sec
=
M/M‫סּ‬
L/L‫( סּ‬4.7 ± 0.3) ppm
=
(l/ 550 nm) (Teff / 5777 K)2 (M/M‫)סּ‬
These equations scale to values observed for the sun
The Solar Power Spectrum
nmax ≈ 3000 mHz
Dn0 ≈ 135 mHz
How do we scale these to other stars?
nmax
nmax +1
l =0
Dn0
l =1
Dn1
Each peak in the frequency spectrum corresponds to a harmonic mode
characterized by a radial order n, and an angular degree l. For stars
observed in integrated light we most likely detect only l = 0,1
nnl ≈ Dn0 (n + l/2 + e)
R
–1
Dn0 ≈ [2∫ dr/cs]
e ~ 1.6 for the sun
cs is the sound speed
0
In other words the large spacing is the inverse travel time of a
sound wave passing directly through the star. It is 134.92 mHz
for the Sun (i.e. about 2 hours travel time)
This sound travel time is related to the global dynamical Timescale
of the star:
d2R
dt2
GM
= –
R2
Msun = 2 ×1033 gm
=GrR
Rsun
r = mean density
= 7 ×1010 cm
t = (G r )–½
r = 1.4 gm/cm3
The dynamical time for the sun is about 1 hour. This would
be the period of radial pulsations, if they were present. Or, if
you were to hit the sun, this is the fastest it could respond
But recall that the adiabatic sound speed satisfies:
2
cs ∝ T
R
–1
Dn0 ≈ [2∫ dr/cs]
0
≈
cs
R
∝
√T
R
Where T is now the average internal Temperature
Footnote: Equation for Hydrostatic Equilibrium
1. Hydrostatic Equilibrium
dA
P +dP
P + dP
P
r + dr
A
dr
r
dm
P
M(r)
Gravity
The gravity in a thin shell should be balanced by the outward gas
pressure in the cell
Fp = PdA –(P + dP)dA = –dP dA
dM
FG = –GM(r) r2
Gravitational Force
r
M(r) = ∫ r(r) 4pr2 dr
0
dM = r dA dr
Both forces must balance:
FP + FG = 0
dP
dr
Pressure Force
r(r)M(r)
= –G
r2
Some more approximations:
Stellar structure equation for
hydrostatic equilibrium
dP
GM
r
= –
2
dr
r
dP
M
GM
= –
R3
dr
R2
P
R
GM2
= –
R5
M2
P∝
R4
Ideal gas law:
P∝ rKT
M2
R4
∝ M
R3
M
T ∝
R
T
Dn0
Dn0
∝
∝
M
T∝
R
√T
R
½
M
(R)
3
∝
½
r
The frequency splitting is directly proportional to the square
root of the mean density of the star!
(M/M‫)סּ‬1/2
Dn0 ≈
(R/R‫)סּ‬3/2
134.9 mHz
The power has an envelope whose maximum is at nmax ≈ 3mHz For
the sun. The shape of the envelope and value of nmax is determined by
excitation and damping. The acoustic cutoff, nac, defines a dynamical
timescale for the atmosphere, so we expect nmax to scale as nac
For frequencies above the acoustic cutoff the energy of the mode
decreases exponentially with height in the atmosphere
nac ∝ cs/Hp
And what is the scale height of the atmosphere?
Footnote #2 :Scale Height of Atmosphere
Pressure
dP = –gr dh
F = GMm/R2
A
r = Pm/kT
h
mg
dP = –
P dh
kT
F =grAh
F/A = g r h
r
gravity
mgh
( kT ) = P
r = r exp (– h )
H
P = Po exp –
o
Scale height H = kT/mg
o
(
exp –
h
H
)
Hp is the pressure scale height where the pressure decreases
exponentially: P = Po e–h/Hp
Hp = kT/mg
nmax ∝ nac ∝ cs/Hp
cs ∝ √T
M
nmax ∝ g/√T ∝ 2
R √T
nmax =
M/M‫סּ‬
(R/R‫)סּ‬2√Teff/5777K
3.05 mHz
The maximum power in the sun is seen for modes with n ≈ 21
nmax =
M/M‫סּ‬
((R/R ) √T
‫סּ‬
2
eff/5777K
½
)
X 22.6 – 1.6
Summary of Scaling Relationships
vosc
(23.4 ± 1.4) cm/sec
M/M‫סּ‬
dL
L
(
(
L/L‫סּ‬
=
l
(R/R‫)סּ‬
=
nmax =
(4.7 ± 0.3) ppm
(l/ 550 nm) (Teff / 5777 K)2 (M/M‫)סּ‬
(M/M‫)סּ‬1/2
Dn0 ≈
nmax
=
L/L‫סּ‬
134.9
3/2
mHz
M/M‫סּ‬
3.05 mHz
(R/R‫)סּ‬2√Teff/5777K
M/M‫סּ‬
((R/R ) √T /5777K
‫סּ‬
2
eff
½
)
X 22.6 – 1.6
Scaling between Dn and nmax
Stello et al. 2009, MNRAS, 400, L80
Dn0 = (0.263 ± 0.009) mHz (nmax/mHz)0.772±0.005
The previous expression is useful when you do not have enough
data to derive the large spacing. Fit a Gaussian to the envelope
of excess power, find nmax and compute Dn0
nmax ≈ 950 mHz →Dn0 ≈ 52 mHz. This is data for Procyon,
and as we shall see this is near the correct value
Stellar Oscillations (or not) of Procyon
Procyon is a bright (mv = 0.36) star in the winter sky that is
slightly evolved off the main sequence
Spectral Type: F5 IV
Vosc =
0.8 m/s
Teff = 6450 K
dL/L ≈ 2 x 10–5
Mass = 1.42 M‫סּ‬
Dn0 ≈ 54 mHz
Radius = 2.07 R‫סּ‬
nmax ≈ 1 mHz
L = 7.03 L‫סּ‬
P ≈ 17 min
nmax ≈ 11
First possible detections of p-mode
oscillations with radial velocity
measurements were made with a
fiber fed spectropgraph: FOE
More convincing evidence…
Martic et al. 1999 found
convincing evidence for p-mode
oscillations in Procyon using
ELODIE and the simultaneous
Th-Ar for radial velocity
measurements
Excess power is in the same
frequency range as found by
Brown et al.
Most probable large frequency
spacing ≈ 55 mHz
Even more convincing evidence…
2-site campaign using
ELODIE and AFOE
but…
Microvariability and Oscillations of STars1
MOST is a 15cm telescope (Canada‘s First Space Telescope)
designed to study stellar oscillations. It can continuosly observe stars
for up to 60 days. In 2004 MOST observed Procyon for 32 days.
1
PI: Jaymie Matthews, also known as Matthew‘s Own Space Telescope
MOST found no evidence of solar-like oscillations in the
photometry of Procyon casting doubt on the radial velocity results.
Expected power
of oscillations at
1 mHz
Top panels: simulated power spectra of oscillations for Procyon and with 3
time scales for the mode lifetimes. In the lower panel nose has been added to
the simulated data to reach the noise level of MOST. Conclusion #1: MOST
could not have detected the pulsations even if they were present
Bedding et al. A&A, 432, L43, 2005
The „power density“ of the MOST observations is significantly
higher than for the EW measurements of Kjeldsen et al. 1999, and
for the Sun. Conclusion #2, the noise level of MOST is too high.
A multi-site campaign from 9 observatories was conducted on
Procyon in 2008
Above: the radial velocity measurements
from the various sites (black: TLS). Left:
preliminary power spectrum
Stellar Oscillations h Boo
Spectral Type: G0 IV
Vosc =
mv´= 2.68
dL/L ≈ 2.5 x 10–5 (25 ppm)
Teff = 6050 K
Dn0 ≈ 36 mHz
Mass = 1.6 M‫סּ‬
nmax ≈ 0.61 mHz
Radius = 2.8 R‫סּ‬
P ≈ 27 min
L = 9.5 L‫סּ‬
nmax ≈ 8
1.4 m/s
Looking for oscillations through temperature changes
L ∝ T4 (assuming constant radius). dL/L ∝ 4 dT/T. The strength of the
Balmer lines of hydrogen are sensitive to temperature and the variations
are expected to be 6 ppm. Advantage: you do not need a high resolution
spectrograph to measure the strength of the hydrogen lines!
Power Spectrum of Temperature Variations of h Boo
Echelle diagram based on
equivalent width (temperature)
variations of Hydrogen lines
Echelle diagram combining
temperature and radial velocity
measurements
Stellar Oscillations b Hya
Spectral Type: G2 IV
Teff = 5774 K
Mass = 1.1 M‫סּ‬
Radius = 1.87 R‫סּ‬
L = 3.53 L‫סּ‬
Vosc ≈ 0.8 m/s
dL/L ≈ 15 ppm
Dn0 ≈ 55 mHz
nmax ≈ 1 mHz
P ≈ 17 min
nmax ≈ 11
Radial velocity
measurements from a
multi-site campaign
(HARPS + UCLES).
Black are HARPS,
red are UCLES
Vosc ≈ 0.8 cm/s
Stellar Oscillations a Cen A
Spectral Type: G2 IV
Teff = 5810 K
Mass = 1.1 M‫סּ‬
Radius = 1.22 R‫סּ‬
L = 1.5 L‫סּ‬
Vosc =
0.3 m/s
dL/L ≈ 6.3 ppm
Dn0 ≈ 105 mHz
nmax ≈ 2.2 mHz
P ≈ 7.5 min
nmax ≈ 11
Radial velocity measurements taken with an iodine cell:
Power spectrum
l=2
l=0
l=1
Bouchy & Carrier 2003
Dn0 ≈ 105.5 mHz
Power spectrum of data taken with
a different instrument shows power
at the right frequency range
Bazot et al. 2007, A&A, 470, 295
Radial Velocity Measurements of
a Cen A with HARPS
Power Spectrum
Bazot et al. 2007, A&A, 470, 295
Large separation as a
function of frequency for
l=0,1,2 modes
Small separation as a for l=0
And a questionable claim of rotational splitting:
nnlm ≈ nm,l ± mW
Bazot et al. 2007, A&A, 470, 295
For l=2
This gives W ≈ 1 mHz → P = 11.5 days, but estimated rotational
period is 28.8 days
Stellar Oscillations a Cen B
Spectral Type: K1 V
Vosc =
0.13 m/s
mv = 1.33
dL/L ≈ 3 ppm
Teff = 5260 K
Dn0 ≈ 160 mHz
Mass = 0.90 M‫סּ‬
nmax ≈ 3.9 mHz
Radius = 0.86 R‫סּ‬
P ≈ 4.3 min
L = 0.5 L‫סּ‬
nmax ≈ 20
The Radial Velocity and Power
Spectrum of a Cen B
Eggenberger et al. 2004, A&A, 417, 235
Stellar Oscillations m Ara
Spectral Type: G3 IV-V
Teff = 5813 K
Mass = 1.1 M‫סּ‬
Radius = 1.3 R‫סּ‬
L = 1.91 L‫סּ‬
[Fe/H] = +0.3
Vosc =
0.4 m/s
dL/L ≈ 8 ppm
Dn0 ≈ 95 mHz
nmax ≈ 2 mHz
P ≈ 8.4 min
nmax ≈ 17
2x metals as the sun
A 14 Mearth planet around m Ara
was discovered as part of an
asteroseismic run.
RV variation with long term variation due
to planet
Full data set
Short time segment showing oscillations
l= 1
l= 2
l= 0
l= 3
Power spectrum
Dn0 = 90 mHz
The Planet-Metallicity Connection
These are stars with metallicity [Fe/H] ~ +0.3 – +0.5
Valenti & Fischer
There is believed to be a
connection between metallicity
and planet formation. Stars with
higher metalicity tend to have a
higher frequency of planets.
The Planet-Metallicity Connection
Two scenarios have been proposed to explain the high
metallicity of planet hosting
Scenario 1: The high metal content is primordial and reflects the
abundance in the star and thus the proto-planetary disk. A higher
metal content implies that the planets are easier to form (core
accretion theory) → the high metal abundance forms more planets
Scenario 2: The high metal content is only on the surface layers of
the star and result from the accretion of planetary bodies onto the
star → the planets cause the high metalic abundance
The two different scenarios should produce different asteroseismic
(acoustic) signals.
Over-metallic star
Accretion model
A slightly better fit is provided with the accretion model, however the main
difference is the cross-over of the l =0,2 modes at 2.5 mHz. Unfortunately,
this is beyond the highest frequency of the observations. One day
asteroseismology may provide the answer to the planet-metallicity effect.
Stellar Oscillations n Ind
Spectral Type: G0III
Teff = 5300 K
Mass = 0.9 M‫סּ‬
Radius = 3 R
L = 5.5 L‫סּ‬
Vosc =
1.4 m/s
dL/L ≈ 34 ppm
Dn0 ≈ 25 mHz
nmax ≈ 0.3 mHz
P ≈ 55 min
nmax ≈ 6
Stellar Oscillations n Ind
Dno = 24.25 mHz
Best fits to the
Teff, Radius,
Mass, and Age of
n Ind from model
fitting to the
observed
frequencies
What about Main Sequence stars?
Stellar Oscillations t Cet
Spectral Type: G8 V
Teff = ~5400 K
Mass = 0.9 M‫סּ‬
Radius = 0.79 R‫סּ‬
(interferometry)
L = 0.49 L‫סּ‬
Vosc =
0.13 m/s
dL/L ≈ 3 ppm
Dn0 ≈ 178 mHz
nmax ≈ 4.4 mHz
P ≈ 3.8 min
nmax ≈ 31
Note: t Cet is often used as a radial velocity
standard star by planet search programs
HARPS data
Instrumental effects!
Always have a
control star!
Power
spectra after
correcting
instrumental
effects
Echelle Diagram
for t Cet
Interferometry gives a radius of 0.79 R‫סּ‬. From the large spacing
one gets a mass of 0.783 ± 0.012 M‫ סּ‬good to 1.6%
Comparison of observed nmax to those predicted
by the scaling relationships:
The „Asteroseismic“ H-R Diagram
t Cet
a Cen B
m Ara
Procyon
bHya
a Cen A
hBoo
The models of course need refinement but one can say that most of these
stars are evolved.
Asteroseismic Targets
Most asteroseismic targets have been evolved stars because these
produce the highest amplitudes.
Also showing solar-like (p-mode) oscillations, but will be
discussed later:
• rapidly oscillating Ap stars
• K giant stars
• solar-like oscillations from Space Missions: expected
amplitudes are 5-20 ppm. Such precision can only be
obtained from space. Space also offers the possibility for
continuous coverage
Stellar Oscillations Network Group
SONG plans a series of 1-m telescopes equipped with
spectrographs + iodine absorption cells and spread across
the globe.
Summary
1. Scaling relationships work remarkably well for predicting the
amplitude and frequencies of solar-like oscillations in other
stars over a wide range of amplitudes, periods, spectral types,
etc.
2. About a dozen solar-type stars have been studied with stellar
oscillations using ground based observations. These have
exclusively used the radial velocity method. Photometric
amplitudes are expected to be ≈ 10–5 → you need to go into
space
3. From the ground multi-site campaigns are the most effective
means of studying stellar oscillations
4. Asteroseismology it the best means of deriving the mass,
radius, effective temperature, helium and heavy element
fraction and internal structure of the star.