Asteroseismological determination of stellar rotation axes: Feasibility study (COROT AP+CP)

1. Asteroseismological determination of stellar rotation axes:Feasibility study (COROT AP+CP) L. Gizon(1), G. Vauclair(2), S. Solanki(1), S. Dreizler(3) (1) MPI for Solar System Research, Katlenburg-Lindau, D (2) Observatoire Midi-Pyrenees, Toulouse, F (3) Goettingen Sternwarte, Goettingen, D

2. Science Objectives • Measure angular velocity, W, and inclination of rotation axis to line of sight, i. • The angle i can be determined seismically from the visibility of spheroidal modes of pulsation. • From W, i, and the spectroscopically determined v sin i, it is possible to deduce the stellar radius, R, without prior knowledge of stellar structure and evolution. (here v is essentially the equatorial velocity.) • A knowledge of i for planet host stars can tell us about the planets themselves.  If Mp sin j is known from periodic Doppler shifts, where j is the inclination of the orbital plane of a planet with mass Mp, then j=i puts a constraint on Mp (Corot CP)  If a planet is detected by COROT in transit, then j is known with high accuracy and it becomes possible to test the theoretical prediction that j and i are similar (Corot AP) • For stars that have very clean oscillation spectra, the latitudinal differential rotation DW= W(mid-lat) -W(eq) can also be estimated seismically. If not, latitudinal differential rotation may be constrained by comparing v sin i with R* W sin i (this works only when the stellar radius R* is known from parallaxes).

3. Mode visibility of solar-like oscillations • Plots show expectation value of power in azimuthal (m) components of dipole (l=1, left) and quadrupole (l=2, right) modes of global acoustic oscillations as a function of inclination angle, i. • Assumption: energy equipartition between m components at fixed values of l and n; ina statistical sense. A to first approximation, the visibility of the 2l+1 m-components is determined by geometry only, at least when oscillations are measured in intensity. • This is the case for solar oscillations, and it is likely to remain true for all stochastically excited solar-like oscillations. • Plots show that it should be possible to estimate both i and W from sufficiently long time-series, as long as i is more than about 15 deg (in order to be able to distinguish a star with W=0 from a star with i<15deg). Note that rotational splitting is not proportional to mW when the centrifugal distortion is taken into account. Inclination angle, i

4. MonteCarlo simulations • Pick fixed values of the observation duration (T), W, i, and mode lifetime. • Simulate thousands of realizations of the oscillation power spectrum • Frequency bins are independent when observations are continuous. • At fixed frequency, the probability density function of the power follows an exponential distribution. • The expectation value of the power in a mode is assumed to be given by a Lorentzian function. • Measure oscillation parameters, including i, from fits using a maximum likelihood method. • Use distributions of measured parameters to estimate biases and standard deviations. • Conclude about feasibility of measuring a given parameter. • Plots show realisations (wiggly solid lines) of power spectra for l =0, 1, and 2. Thick gray lines are the expectation values, thick solid lines are the fits.

5. Example 1: one single l=1 triplet • Input parameters: T=6 months, S/N=100 (Corot ok), W=6WSun (where WSun=0.5mHz), and full width at half maximum (FWHM) of mode power G/2p=1mHz. • Plot shows distribution of values of i (left) and W (right) measured from synthetic spectra versus the true i value. The dashed line is the W guess. • Using one single l=1 triplet, i and W can be retrieved with a good precision if i>30deg. • In practice, the uncertainty on i will be reduced by a factor sqrt(N) where N is the number of observable dipole modes (N>10). The uncertainty scales like 1/sqrt(T). Fits to synthetic spectra True inclination angle, i

6. Example 2: three multiplets l=0,1,2 Simultaneous fit on one l=0 singlet, one l=1 triplet and one l=2 multiplet. Assumption: all peaks have the same linewidth, G. The l=0 mode helps constrain G, although it contains no information about rotation. Measured inclination angle True inclination angle • Input parameters: T=4 months, W=4WSun, G/2p=1mHz. • The plot shows the distribution of i values estimated from the fits as a function of the true value. • The fits are doing a much better job than for one l=1 alone: the uncertainty on i drops significantly. The value of i below which the results cannot be trusted is about 15 deg. • Once again, the error bar on i would be reduced by a factor of sqrt(N) where N is the number of radial orders that can be observed.

7. The modes must be resolved • Plot of measured inclination angle (symbols with error bars) for input values i=30deg and i=80deg (dashed lines) as a function of stellar angular velocity. • Only one l=1 triplet is fitted. • Other parameters, G/2p=1mHz and T=6 months are fixed. • The plot shows that the inclination angle can only be retrieved when W>2WSun=G, i.e. when modes are resolved. • This condition is independent of the observation duration, T. If individual modes are not resolved then a longer observation will not help. Measured inclination angle Input W/WSun at fixed G=2WSun

8. Conclusion & COROT targets • We have shown, using MonteCarlo simulations that the stellar angular velocity, W, and the direction of the rotation axis, i, can both be retrieve from solar-like pulsations. • The main condition is that individual modes of oscillation are resolved (W>G). • The observation duration must be at least 2 months, say (preferably more). • If these conditions are met, it should be not problem to measure i with a precision of a few degrees. That is unless the rotation axis points toward Corot (unlikely). Targets: • CP (seismo field): All solar-like pulsators selected for seismo long runs (e.g. F & G stars). • Of particular interest is HD 52265, which we first proposed for the Additional Program and was subsequently selected as a Prime Target. This G0V Sun-like star is known to have a planetary companion with Mp=1.13 MJupiter. • Red giants with solar-like pulsations are very interesting targets too. For example, the G6III star HD 50890 (V Mag=6) which will be observed by Corot. • AP (exoplanet field): all stars for which a planetary transit has been detected. Switch to 32s cadence. • This work is fully documented in the following papers: Gizon & Solanki, ApJ 589, 1009 (2003) Gizon & Solanki, Solar Phys. 220, 169 (2004)