Transcript m/s

Stellar Oscillations in Giant Stars
1. K giants
2. Mira
3. RV Tau stars
Giant stars are A-K starts that have evolved off the main
sequence and on to the giant branch
K giants occupy a „messy“
region of the H-R diagram
Progenitors are higher mass
stars than the sun
Giant stars are of particular interest to planet hunters. Why?
Because they have masses in the range of 1-3 M‫סּ‬
A 2 M‫ סּ‬star on the main
sequence
A 2 M‫ סּ‬star on the giant
branch
Stars of higher mass than the Sun are ill-suited for RV searches.
However the problem with this is getting a good estimate for the mass
of the star
The story of variability in K giant stars began in 1989:
Smith et al. 1989 found a 1.89 d period variation in the
radial velocity of Arcturus:
1989 Walker et al. Found that RV variations are common
among K giant stars
These are all IAU radial
velocity standard stars !!!
Inspired by the Walker et al.
Paper, Hatzes & Cochran
began a radial velocity
survey of a small sample of
K giant stars.
The Long Period Variability: Planets?
1990-1993 Hatzes & Cochran surveyed 12 K giants with
precise radial velocity measurements and found significant
period
The „3 Muskateers“
Many showed RV
variations with periods of
200-600 days
The nature of the long period variations in K giants
Three possible hypothesis:
1. Pulsations (radial or non-radial)
2. Spots (rotational modulation)
3. Sub-stellar companions
What about radial pulsations?
Pulsation Constant for radial pulsations:
Q=
M 0.5
P(
)
M‫סּ‬
R –1.5
(R ) =
‫סּ‬
P
(
r 0.5
)
r‫סּ‬
For the sun:
Period of Fundamental (F) = 63 minutes = 0.033 days (using
extrapolated formula for Cepheids)
Q = 0.033
What about radial pulsations?
K Giant: M ~ 2 M‫ סּ‬, R ~ 20 R‫סּ‬
Period of Fundamental (F) = 2.5 days
Q = 0.039
Period of first harmonic (1H) = 1.8 day
→ Observed periods too long
What about radial pulsations?
Alternatively, let‘s calculate the change in radius
V = Vo sin (2pt/P),
p/2
DR =2 ∫ Vo sin (2pt/P) =
VoP
p
0
b Gem: P = 590 days, Vo = 40 m/s, R = 9 R‫סּ‬
DR ≈ 0.9 R‫סּ‬
Brightness ~ R2
Dm = 0.2 mag, not supported by Hipparcos photometry
What about non-radial pulsations?
p-mode oscillations, Period < Fundamental mode
Periods should be a few days → not p-modes
g-mode oscillations, Period > Fundamental mode
So why can‘ t these be g-modes?
Hint: Giant stars have a very large, and deep convection zone
Rotation (and pulsations) should be
accompanied by other forms of variability
Planets on the other hand:
1. Have long lived and coherent RV variations
2.
No chromospheric activity variations with RV period
3.
No photometric variations with the RV period
4.
No spectral line shape variations with the RV period
The Planet around Pollux (b Gem)
McDonald 2.1m
CFHT
McDonald 2.7m
TLS
The RV variations of b Gem taken with 4 telescopes over a time span of 26 years. The
solid line represents an orbital solution with Period = 590 days, m sin i = 2.3 MJup.
Ca II H & K core emission is a measure of magnetic activity:
Active star
Inactive star
Ca II emission variations for b Gem
If there are no Ca II variations with the RV period, it probably is
not activity
Hipparcos Photometry
If there are no photometric variations with the RV period,
spots on the surface are not causing the variations.
Test 2: Bisector velocity
Spectral Line Bisectors
From Gray (homepage)
For most phenomena like spots, surface structure, or stellar pulsations,
the radial velocity variations are all accompained by changes in the
shape of the spectral lines. Planets on the other hand cause an overall
Doppler shift of the line without an accompanying change in the lines.
Spectral line bisectors are a common way to measure line shapes
The Spectral line shape variations of b Gem.
The Planet around b Gem
Period
RV Amplitude
e
a
Msin i
590.5 ± 0.9 d
40.1 ± 1.8 m/s
0.01 ± 0.064
1.9 AU
2.9 MJupiter
The Star
M = 1.9 Msun
[Fe/H] = –0.07
Planets have been found around ~ 30 Giant stars
The Planet around i Dra
Frink et al. 2002
P = 1.5 yrs
M = 9 MJ
From Michaela Döllinger‘s thesis
P = 517 d
m = 10.6 MJ
e = 0.09
M* = 1.84 M‫סּ‬
RV (m/s)
P = 272 d
m = 6.6 MJ
e = 0.53
M* = 1.2 M‫סּ‬
P = 657 d
m = 10.6 MJ
e = 0.60
M* = 1.2 M‫סּ‬
P = 159 d
m = 3 MJ
e = 0.03
M* = 1.15 M‫סּ‬
P = 1011 d
m = 9 MJ
e = 0.08
M* = 1.3 M‫סּ‬
P = 477 d
m = 3.8 MJ
e = 0.37
M* = 1.0 M‫סּ‬
JD - 2400000
M sin i = 3.5 – 10 MJupiter
a Tau
The Planet around a Tau
Period
RV Amplitude
e
a
Msin i
653.8 ± 1.1 d
133 ± 11 m/s
0.02 ± 0.08
2.0 AU
10.6 MJupiter
The Star
M = 2.5 Msun
[Fe/H] = –0.34
g Dra
The Planet around g Dra
Period
RV Amplitude
e
a
Msin i
712 ± 2.3 d
134 ± 9.9 m/s
0.27 ± 0.05
2.4
13 MJupiter
The Star
M = 2.9 Msun
[Fe/H] = –0.14
The evidence supports that the long period RV variations in many K giants are
due to planets…so what?
Setiawan et al. 2005
K giants can tell us about planet formation around stars more massive than the
sun. The problem is the getting the mass. This is where stellar oscillations can
help.
And now for the stellar oscillations…
Hatzes & Cochran 1994
Short period variations
in Arcturus consistent
with radial pulsations
n = 1 (1H)
n = 0 (F)
a Ari velocity variations:
Alias
n≈3 overtone radial mode
Photometry of a UMa with WIRE guide camera (Buzasi et al. 2000)
Equally spaced modes in frequency → pmodes. Observed Dn = 2.94 mHz
Buzasi et al get a mean spacing of 2.94 mHz and a lowest
frequency mode of 1.82 mHz (P = 6.35 d).
a UMa has an interferometric radius of 28 R‫סּ‬
The Fundamental radial mode is given by:
Q = P0 √r/r‫סּ‬
Where the pulsation constant Q = 0.038 – 0.116, so P = 2.8, to 8.6
days, if M ≈ 4 M‫ סּ‬, close to the first frequency. But…
M1/2
Dn0 ≈
135
mHz
R3/2
Based on the known radius and observed spacing, this gives M ≈
10 M‫סּ‬. So actual spacing may be one-half as a large and
one is not seeing all modes (odd or even radial order, n)
g Dra
g Dra : June 1992
g Dra : June 2005
g Dra
The short period variations of g Dra can also be explained by radial pulsations, but only
n order modes?
i Dra: A planet hosting K giant
P1 = 7 hrs A1= 5 m/s
n1 = 39.7 mHz
P2 = 6.4 hrs A2=6.35 m/s
n2 = 43.4 mHz
P3 = 5.9hrs d A3=4 m/s
n2 = 47.8 mHz
Mean Dn = 4.05 mHz
Recall our Scaling Relations
nmax
M/M‫סּ‬
=
(R/R‫)סּ‬2√Teff/5777K
3.05 mHz
Frequency spacing:
M1/2
Dn0 ≈ 135 3/2 mHz
R
nn,l = Dn0 (n + l/2 + d)
Thes can be solved for the radius of the star:
R ≈ ( n(mHz)max/3.05 )(135/Dn(mHz))2
We have 2 equations and 2 unknowns, these can be solved for M, R
nmax ≈ 40 mHz (max peak at P = 7 hrs) = 0.04 mHz
Mean Dn = 4.05 mHz
We have two cases:
1.
These are nonradial modes and the observed spacing is one-half the large
spacing
2.
These are radial modes and the observed spacing is the large spacing
Case 1:
Case 2:
R = 3.6 R‫סּ‬
R = 14.5 R‫סּ‬
M = 0.17 M‫סּ‬
M = 2.9 M‫סּ‬
Case 1 is in disagreement with evolutionary tracks (they cannot
be that wrong!) and Hipparcos distance. Conclusion: this is a giant
star and we are detecting radial modes.
Stellar Oscillations in b Gem
Nine nights of RV measurements of b Gem. The solid line represents a 17 sine
component fit. The false alarm probability of these modes is < 1% and most have FAP <
10–5. The rms scatter about the final fit is 1.9 m s–1
Amplitude (m/s)
DFT Velocities
Window
Observed RV Frequencies in b Gem
Amplitude in m/s
DFT Fit
The Oscillation Spectrum of Pollux
The p-mode oscillation spectrum of b Gem based on the 17 frequencies found via
Fourier analysis. The vertical dashed lines represent a grid of evenly-spaced
frequencies on an interval of 7.12 mHz
Frequency Spacing
M1/2
Dn0 ≈ 135 3/2 mHz
R
Dn0 ≈
7.12 mHz
Inteferometric Radius of b Gem = 8.8 R‫סּ‬
For radial modes → M = 1.89 ± 0.09 M‫סּ‬
For nonradial modes→ M = 7.5 M‫סּ‬
Evolutionary tracks give M = 1.94 M‫סּ‬
MOST Photometry for b Gem
For n = 87 mHz
2K/Dm = 65 km/s/mag
Observed Photometric Frequencies
in b Gem
The Radial Velocity – to – Photometric Amplitude Ratio
For modes for modes found in both photometry and
radial velocity the 2K/Dm ratio is consistent with values
found for Cepheids (2K/Dm ≈ 55) and thus radial
pulsators.
The first Tautenburg planet:
HD 13189
P = 471 d
Msini = 14 MJ
M* = 3.5 s.m.
P = 4.8 days
For M = 3.5 M‫סּ‬
R = 38 R
F = 4.8 d
2H = 2.7 d
HD 13189 short
period variations
P = 2.4 days
P = 5.8 days
Periodogram of RV residuals for a Tau after subtracting
the long period orbit
Aldebaran with MOST
5.8 days
Period consistent with fundamental radial mode for M = 2.5 M‫סּ‬
But isochrones give M = 1.2 M‫ →סּ‬overtone?
The Radial Velocity – to – Photometric Amplitude Ratio
MOST: DI/I = 0.019 = 0.02 mag
Radial Velocity 2K ~ 300 m/s
2K/Dm ≈ 15 Nonradial?
Radial Velocities of a Boo in
Estimates of the mass for Arcturus have been controversial and have
ranged from 0.1 to 3 M‫סּ‬. Can stellar oscillations resolve this?
Multi-period Fit
P1 = 3.57 d A1=34.7 m/s
P5 = 1.74 d A5 =6.93 m/s
P2 = 12.8 d A2=27.2 m/s
P6 = 5.77d A6 =6.23 m/s
P3 = 2.08 d A3=23.2 m/s
P7 = 1.38d A7 =6.27 m/s
P4 = 2.50 d A3=11.5 m/s
P8 = 1.19d A8 =5.4 m/s
The Oscillation Spectrum of Arcturus?
Mean spacing = 1.16 mHz
Mozurkowich et al. 2003:
Limb darkened diameter = 21.373 mas = 25.65 R‫סּ‬
Dn = 1.16 mHz → 1.24 M‫ סּ‬for radial modes
Dn = 2.32 mHz → 5 M‫ סּ‬for nonradial modes
The higher mass is inconsistent with the spectroscopic
analysis which indicate M ≈ 1 M‫סּ‬
a Boo in 2005
P = 3.36d
At any given time not all modes are visible → need
lots of observing time over a very long time
base → CoRoT and Kepler
A new planet hosting K giant star: 11 UMa
Döllinger et al. In preparation
P = 657 d
Msini = 3.6 MJupiter
e = 0.6
Oscillations in 11 UMa in 2007
M* = 1.2 M‫סּ‬
R* = 36.3 R‫סּ‬
P1 = 4.1 d
P2 = 3.1 d
P3 = 7.1 d
Consistent with
fundamental and low
overtone radial modes
Oscillations in 11 UMa in 2009
M* = 1.2 M‫סּ‬
R* = 36.3 R‫סּ‬
P1 = 6.2 d
P2 = 14.2 d
F = 10.8 d
1H = 6.2 d
2H = 4.1 d
3H = 3.7 d
We need a radius!
e Oph: G9.5 III (de Ridder et al. A&A 448, 689-695, 2006)
Amplitude Spectra of e Oph
Best Fitting Models for e Oph
Radial or Nonradial pulsations?
So far we have seen evidence for radial pulsations in K
giants, but are there nonradial modes?
Two tales of the same star, e Oph
MOST Photometry of e Oph
This power spectrum is typical for giants. You have a Gaussian envelope of
excess power due to the p-mode oscillations, and an exponential rise to low
frequencies believed to be due to convection motion.
Conclusion: Mean spacing of 5.3
mHz which are radial modes of
short lifetime (~3 days).
The autocorrelation function
shows peaks at possible
frequency spacings
Echelle diagram for e Oph
Conclusion using the same data set:
radial and nonradial modes but with
a long (10-20 d) lifetime.
The model reproduces e Oph
position in the HR diagram and the
interferometric radius
So why did two different groups get different answers using the
same data set?
The answer lies in how you interpret the wings of a
peak in the power spectrum.
The lifetime of a mode is not infinite and damping results in each
mode being split into a number of peaks under a Lorentzian profile
whose full width at half maximum (FWHM) is given by:
G=
1
pt
t = lifetime of mode
g
L(x,g) = 2 2
(p + g )
The shorter the mode
lifetime, the broader the
Lorentzian.
Kallinger at al. Intepreted
the wings as being
individual modes that were
quite narrow in width, ie.
that had long lifetimes
Barban et al. Smoothed the
power spectrum and intepreted
the broad wings around each
peak as due to a short lifetime
modes.
So who is correct? We will
have to wait for CoRoT and
Kepler!
Stellar Oscillations in HD 20884 (K2III)
Observerved with MOST
(Kallinger et al. 2008, CoAst.153, 84K)
The Observed Frequencies
The Echelle Diagram and Best Fit Model
Conclusion: Photometric space-based observations show evidence
for radial and non-radial modes in giant stars
Mira Variables
• Red Giant Stars
• Mass less than 2 solar masses
• Pulsating in periods longer than 100 days
• Light amplitudes greater than 1 magnitude
Short History from Dorrit Hoffleit
David Fabricius (1564_1617), an amateur astronomer and native of Friesland, The Netherlands, is recognized as the
first to have discovered a long period variable in 1596, later called o (omicron) Ceti by Johann Bayer in 1603. Fabricius
(Wolf 1877) observed the star from August 3, when he had used it as a comparison star for the determination of the
position of the planet he assumed to be Mercury (later identified by Argelander, 1869, as more probably Jupiter), until
August 21, when it had increased from magnitude 3 to magnitude 2. In September it faded, disappearing entirely by
October (Clerke 1902). At the time Fabricius assumed the star was a nova. However, he observed it to reappear on
February 15, 1609. Although Pingré saw it October 14, 1631, the star was practically forgotten until Johann Fokkens
Holwarda (1618_1651), also of Friesland, rediscovered it in 1638 and determined its period as eleven months. Johannes
Hevelius of Danzig (1611_1687) also observed the star on November 7, 1639, and in 1642 named it Mira, "The
Wonderful." Fabricius unfortunately did not live to enjoy this appreciation for his discovery. Fabricius, a minister, had
been murdered by a peasant whom he had cited from the pulpit as having stolen one of the minister's geese
(Poggendorff 1863)!
Light curve of Mira Variables
Velocity and Light Curves for Mira from 1926
Integrating the radial
velocity curve, the
change in radius of the
star is ~70 R‫סּ‬, or 0.33
AU!
Joy, 1926:
Radial velocity curves of some Mira variables.
2K (peak to peak amplitude): 4 km/s
DV ≈ 1 mag
2K/Dm ~ 2-3, significantly different
from Cepheids
A Period-Luminosity Relationship for Miras
From I.S. Glass: Miras in the LMC
Miras do not show an obvious Period – Luminosity Relationship
in the Optical, but a clear one in the Infrared
Mira is not a symmetric star!
Asymmetry is most likely related to non-symmetric mass loss coupled to the
pulsations
RV Tau Variables
• Spectral Type G-K giants (F-G at minimum, G-K at
maximum)
• Pulsating in periods 60-100 days
• Light amplitudes 0.2 magnitudes or greater
• Stars in transition between the AGB and white dwarf
stars
Red Giant Branch (RGB) star
leaves the main sequence
and ascends the giant branch
Asymptotic Giant Branch
(AGB): After core burning He
(horizontal branch stars), the
star moves back up the giant
branch
RV Tau Variables in the HR Diagram
Oscillations in the M supergiant
Betelgeuse (a Ori)
3D simulation of convection in a Ori
Convection cells on a supergiant are large, only a few cells at any given time, whereas the
sun has millions (size~700 km). These cells are also long-lived (years)
http://www.aip.de/groups/sternphysik/stp/box_simulation.html
RV Measurements from
McDonald
AVVSO Light Curve
Fourier transform of red
points:
Fourier transform of blue
points:
Period of a Ori abruptly changed from 317 days to 714
days. This coincided with an abrupt drop in the
brightness of a Ori.
a Ori has dust shells
surrounding it.
These shells may be
related to these
incidents of changing
pulsation modes.
How well do the Scaling Relationships do?
vosc
=
nmax
=
L/L‫סּ‬
(23.4 ± 1.4) cm/sec
M/M‫סּ‬
M/M‫סּ‬
3.05 mHz
(R/R‫)סּ‬2√Teff/5777K
Star
b Gem
Period
3.2 hrs
Vosc
4 m/s
M
1.9
R
8.8
L
33
Vpred
4 m/s
Ppred
3.7 hrs
g Dra
4 days
42 m/s
2.9
47.4
516
41 m/s
3d
a Ori
330700 d
330 d
2 km/s
19
836
105000 1.2 km/s 174 d
6 km/s
0.4
500
8500
Mira
5 km/s
2370 d
In spite of the large range in mass and radius the scaling relationships are
reasonably good predictors
a Ori
Today we looked at stellar
oscillations of stars up the
giant branch.
In general: Periods get longer,
and amplitudes get higher as
the star evolves. Most modes
are dominated by radial modes.
Next week: The stellar
graveyard