Internal rotation: tools of seismological analysis and prospects for asteroseismology Michael Thompson University of Sheffield

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 write wave equation as

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).

Optimally localized averages (OLA) 7 Choose coefficients c i such that is localized about r = r 0. Then so is a localized average of. Function is called an averaging kernel. Let data be

Regularized least squares (RLS) 8 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 combination of the data, so averaging kernels exist:

OLA RLS Averaging kernels 1-D example inverting 834 p modes with degrees from l=1 to l=200.

RLS OLA RLS Close-Up 2-D Rotational Averaging Kernels (1 s.d. uncertainties on inversion are indicated in nHz, for a typical MDI dataset)

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 w 0 respectively, then the solution will contain a convolution of these two, which is another Gaussian but of width (w 2 + w 0 2 ) 1/2. If w 0 is known and (w 2 + w 0 2 ) 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 (w 2 + w 0 2 ) 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 6.5x10 9 years (Eggenberger et al. 2004). α Cen B: near-solar twin. Large sep. 161μHz (Carrier & Bourban 2003). Yields mass Msun and radius 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.

Sharp features also affect the large (and small) separations

Potential of inversions with only low-l data (Only inner 40% of star is shown) Fractional difference in squared sound speed

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.