Internal rotation: tools of seismological analysis and

prospects for asteroseismology

Michael Thompson

University of Sheffield

michael.thompson@sheffield.ac.uk

Equation of motion for waves in rotating star

For simplicity neglect perturbation to gravity, and define

Consider perturbations around non-oscillatory state:

Then can define to be this operator in the non-rotating star; and writewave equation as

Frequency up to second order (uniform rotation)

Schematic showing howsuccessive perturbative terms contribute to the frequency.Left: l=0 and l=1 modeOnly.Right: l=0-3 modes.

The spectrum would be very difficult to interpretif rotation were notproperly 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).

Optimally localized averages (OLA)

7

Choose coefficients ci such that

is localized about r = r0 . Then

so is a localized average of .

Function is called an averaging kernel.

Let data be

Regularized least squares (RLS)

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Choose coefficients to minimize e.g.

Parametrize solution in terms of chosen basis functions :

where L is some linear operator (e.g. the second-derivative). Then once again, the solution at each point is a linear combinationof the data, so averaging kernels exist:

OLA

RLS

Averaging kernels

1-D example inverting834 p modes with degrees from l=1 to l=200.

RLS OLA

OLA

RLS

Close-Up

2-D Rotational Averaging Kernels

(1 s.d. uncertainties on inversion are indicated in nHz, for a typical MDIdataset)

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 thesetwo, which is another Gaussian but of width (w2 + w0

2)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-functionstep of width w, then its convolution with the averaging kernelis a step of width (w2 + w0

2)1/2.

Oscillating starsin the HR diagram

Houdek et al. (1999)

Subgiants and dwarfswith observedsolar-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 degreesdiffer by two.

Asteroseismic HR diagram of small separation againstlarge separation.Stellar models of different masses and ages are plotted.

As above but using the ratio of small to largeseparations in place ofthe small separation.

Observed p-mode spectraof several solar-like stars.

(Stellar masses increase fromthe bottom upwards.)

Courtesy H. Kjeldsen

Some First Results from Solar-like Stars

Mostly results from ground-based observationsSun: 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 mass0.934 Msun and radius 0.870 Rsun (Eggenberger et al. 2004).Procyon: F5 subgiant. Controversial! Oscillates (Martic et al. 1999, 2004; Eggenbergeret al. 2004) or not (Matthews et al. 2004).Shorter mode lifetimes (1-2 days) in α Cen A,B and Procyon comparedwith Sun (3-4 days) a puzzle. Also mode amplitudes for higher-mass stars lower thanpredicted.

Good prospects for progress with space-based observations: MOST (now), COROT (launch 2006), Kepler (launch 2008).

A broad range of degrees isnot necessary to form well-localizedaveraging kernels.Here, 111 l=1,2,3 p- and g-modesare inverted, for a solar model.

OLA

RLS

A range of p- andg-modes may be excited in a deltaScuti star.

Here model characteristicsand mode kernels are illustrated.

Goupil et al. (1996)

Goupil et al. (1996)

Averaging kernels and synthetic inversionfor a delta Scuti model.

Sharp features also affect the large (and small) separations

Potential of inversions with only low-l data

(Only inner 40% of star is shown)

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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 abouta sun-like star from the low-degree modes:

inferences from asteroseismic HR diagramcore stratificationconvective boundariesionization zonesa measure of internal rotation

Hare&Hounds experiments point to difficulties of confusion withrotational splitting and mixed / g-mode spectrum, but rich information there also.