Transcript Global Helioseismology and SDO

Internal rotation: tools of
seismological analysis and
prospects for asteroseismology
Michael Thompson
University of Sheffield
[email protected]
Equation of motion for waves in rotating star
Consider perturbations around non-oscillatory state:
For simplicity neglect perturbation to gravity, and define
Then can define
wave equation as
to be this operator in the non-rotating star; and write
Frequency up to second order (uniform rotation)
Schematic showing how
successive perturbative
terms contribute to the
frequency.
Left: l=0 and l=1 mode
Only.
Right: l=0-3 modes.
The spectrum would be
very difficult to interpret
if rotation were not
properly taken into account.
Linear approximation (non-uniform rotation)
The form of the kernels is given crudely by the square of
spherical harmonics times the radial eigenfunction.
Radial eigenfunctions of selected p modes of roughly the
same frequency. The degree increases from (a) to (d).
7
Optimally localized averages (OLA)
Let data be
Choose coefficients ci such that
is localized about r = r0 . Then
so
Function
is a localized average of
.
is called an averaging kernel.
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Regularized least squares (RLS)
Parametrize solution
Choose coefficients
in terms of chosen basis functions
to minimize e.g.
where L is some linear operator (e.g. the second-derivative).
Then once again, the solution at each point is a linear combination
of the data, so averaging kernels exist:
:
Averaging kernels
1-D example inverting
834 p modes with degrees
from l=1 to l=200.
OLA
RLS
RLS
OLA
2-D Rotational Averaging Kernels
(1 s.d. uncertainties on inversion are
indicated in nHz, for a typical MDI
dataset)
Close-Up
RLS
OLA
Deconvolving the averaging kernel
E.g., suppose the rotation profile contains a jet. If both the jet and the
averaging kernel are approximately Gaussian, with widths w and w0
respectively, then the solution will contain a convolution of these
two, which is another Gaussian but of width (w2 + w02)1/2.
If w0 is known and (w2 + w02)1/2 is measured, w can be inferred.
Likewise for the tachocline width: if the profile is an error-function
step of width w, then its convolution with the averaging kernel
is a step of width (w2 + w02)1/2.
Oscillating stars
in the HR diagram
Houdek et al. (1999)
Subgiants and dwarfs
with observed
solar-like oscillations
The low-degree p-mode spectrum has a regular pattern, with
large separation (big delta) between modes of the same degree,
and small separation (little delta) between modes whose degrees
differ by two.
Asteroseismic HR diagram
of small separation against
large separation.
Stellar models of different
masses and ages are plotted.
As above but using the
ratio of small to large
separations in place of
the small separation.
Observed p-mode spectra
of several solar-like stars.
(Stellar masses increase from
the bottom upwards.)
Courtesy H. Kjeldsen
Some First Results from Solar-like Stars
Mostly results from ground-based observations
Sun: G5 dwarf. About 100 low-degree frequencies; large sep. 135μHz, small sep. 9μHz.
η Bootis: G0 subgiant. 21 frequencies, large sep. 40.4μ Hz, small sep. 3.06μHz
(Kjeldsen et al. 1995, 2003).
β Hydri: G2 subgiant. Spectrum includes modes of mixed p- and g-mode character
(Bedding et al. 2001, Carrier et al. 2001).
ξ Hydrae: G7 giant. Only ℓ=0 (i.e. radial) modes (Frandsen et al. 2002).
α Cen A: near-solar twin. 28 frequencies, large sep. 106μHz, small sep. 5.5μHz (Schou &
Buzasi 2001, Bouchy & Carrier 2001). Inferred mass 1.1 Msun, radius 1.2 Rsun,
age 6.5x109 years (Eggenberger et al. 2004).
α Cen B: near-solar twin. Large sep. 161μHz (Carrier & Bourban 2003). Yields mass
0.934 Msun and radius 0.870 Rsun (Eggenberger et al. 2004).
Procyon: F5 subgiant. Controversial! Oscillates (Martic et al. 1999, 2004; Eggenberger
et al. 2004) or not (Matthews et al. 2004).
Shorter mode lifetimes (1-2 days) in α Cen A,B and Procyon compared
with Sun (3-4 days) a puzzle. Also mode amplitudes for higher-mass stars lower than
predicted.
Good prospects for progress with space-based observations: MOST (now), COROT
(launch 2006), Kepler (launch 2008).
A broad range of degrees is
not necessary to form well-localized
averaging kernels.
Here, 111 l=1,2,3 p- and g-modes
are inverted, for a solar model.
OLA
RLS
A range of p- and
g-modes may be
excited in a delta
Scuti star.
Here model characteristics
and mode kernels are
illustrated.
Goupil et al. (1996)
Averaging kernels
and synthetic inversion
for a delta Scuti model.
Goupil et al. (1996)
Sharp features also affect the large (and small) separations
Fractional difference in
squared sound speed
Potential of inversions with only low-l data
(Only inner 40% of star is shown)
Prospects for detailed seismology of stellar interiors
Remarkable recent progress from ground-based observations,
but much more will be achievable from space.
Even without the higher-degree modes, could learn much about
a sun-like star from the low-degree modes:
inferences from asteroseismic HR diagram
core stratification
convective boundaries
ionization zones
a measure of internal rotation
Hare&Hounds experiments point to difficulties of confusion with
rotational splitting and mixed / g-mode spectrum, but rich
information there also.