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Stellar Oscillations in Giant Stars K giants Mira RV Tau stars. Progenitors are higher mass stars than the sun. K giants occupy a „messy“ region of the H-R diagram. Giant stars are A-K starts that have evolved off the main sequence and on to the giant branch.

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Stellar Oscillations in Giant Stars K giants Mira RV Tau stars

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  • Stellar Oscillations in Giant Stars

    • K giants

    • Mira

    • RV Tau stars


Progenitors are higher mass stars than the sun

K giants occupy a „messy“ region of the H-R diagram

Giant stars are A-K starts that have evolved off the main sequence and on to the giant branch


A 2 Mסּ star on the main sequence

A 2 Mסּ star on the giant branch

Giant stars are of particular interest to planet hunters. Why? Because they have masses in the range of 1-3 Mסּ

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:

  • Pulsations (radial or non-radial)

  • Spots (rotational modulation)

  • Sub-stellar companions


r

R

M

Rסּ

rסּ

Mסּ

What about radial pulsations?

Pulsation Constant for radial pulsations:

0.5

–1.5

0.5

(

)

(

)

(

)

Q =

P

=

P

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 Mסּ , 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),

VoP

p/2

DR =2 Vo sin (2pt/P) =

p

0

b Gem: P = 590 days, Vo = 40 m/s, R = 9 Rסּ

Brightness ~ R2

DR ≈ 0.9 Rסּ

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:

  • 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

TLS

McDonald 2.7m

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

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

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

RV (m/s)

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

The Star

M = 2.5 Msun

[Fe/H] = –0.34


g Dra


The Planet around g Dra

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 → p-modes. Observed Dn = 2.94 mHz


M1/2

135

mHz

Dn0≈

R3/2

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 Mסּ , close to the first frequency. But…

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

Mean Dn = 4.05 mHz

P2 = 6.4 hrs A2=6.35 m/s

n2 = 43.4 mHz

n2 = 47.8 mHz

P3 = 5.9hrs d A3=4 m/s


M1/2

135

mHz

Dn0≈

R3/2

nn,l=

Dn0 (n + l/2 + d)

Recall our Scaling Relations

M/Mסּ

3.05 mHz

nmax

=

(R/Rסּ)2√Teff/5777K

Frequency spacing:

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:

  • These are nonradial modes and the observed spacing is one-half the large spacing

  • These are radial modes and the observed spacing is the large spacing

Case 1:

R = 3.6 Rסּ

M = 0.17 Mסּ

Case 2:

R = 14.5 Rסּ

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


DFT Velocities

Amplitude (m/s)

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


7.12 mHz

Dn0≈

M1/2

135

mHz

Dn0≈

R3/2

Frequency Spacing

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

2K/Dm = 65 km/s/mag

For n = 87 mHz


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

HD 13189 short period variations

For M = 3.5 Mסּ

R = 38 R

F = 4.8 d

2H = 2.7 d

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 Mסּ→overtone?


The Radial Velocity – to – Photometric Amplitude Ratio

MOST: DI/I = 0.019 = 0.02 mag

Radial Velocity 2K ~ 300 m/s

2K/Dm ≈ 15Nonradial?


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 → 1.24Mסּfor radial modes

Dn = 2.32 mHz → 5 Mסּfor 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

Consistent with fundamental and low overtone radial modes

M* = 1.2 Mסּ

R* = 36.3 Rסּ

P1 = 4.1 d

P2 = 3.1 d

P3 = 7.1 d


Oscillations in 11 UMa in 2009

P1 = 6.2 d

P2 = 14.2 d

F = 10.8 d

1H = 6.2 d

2H = 4.1 d

3H = 3.7 d

M* = 1.2 Mסּ

R* = 36.3 Rסּ

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 foreOph


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


1

G =

g

pt

L(x,g) =

(p2 + g2)

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:

t = lifetime of mode

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.


  • Ori has dust shells surrounding it.

    These shells may be related to these incidents of changing pulsation modes.


L/Lסּ

(23.4 ± 1.4) cm/sec

vosc

=

M/Mסּ

M/Mסּ

=

nmax

3.05 mHz

(R/Rסּ)2√Teff/5777K

How well do the Scaling Relationships do?

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


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