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, , and inclination of rotation axis to line of sight, i. Measure angular velocity, , 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. The angle i can be determined seismically from the visibility of spheroidal modes of pulsation. From , 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.) From , 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. 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 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  (mid-lat)   (eq) can also be estimated seismically. If not, latitudinal differential rotation may be constrained by comparing v sin i with R   sin i (this works only when the stellar radius R  is known from parallaxes). For stars that have very clean oscillation spectra, the latitudinal differential rotation  (mid-lat)   (eq) can also be estimated seismically. If not, latitudinal differential rotation may be constrained by comparing v sin i with R   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. 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; in a 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. Assumption: energy equipartition between m components at fixed values of l and n; in a 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. 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  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  =0 from a star with i<15deg). Note that rotational splitting is not proportional to m  when the centrifugal distortion is taken into account. Plots show that it should be possible to estimate both i and  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  =0 from a star with i<15deg). Note that rotational splitting is not proportional to m  when the centrifugal distortion is taken into account. Inclination angle, i

4. MonteCarlo simulations Pick fixed values of the observation duration (T), , i, and mode lifetime. Pick fixed values of the observation duration (T), , i, and mode lifetime. Simulate thousands of realizations of the oscillation power spectrum Simulate thousands of realizations of the oscillation power spectrum Frequency bins are independent when observations are continuous. Frequency bins are independent when observations are continuous. At fixed frequency, the probability density function of the power follows an exponential distribution. 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. 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. Measure oscillation parameters, including i, from fits using a maximum likelihood method. Use distributions of measured parameters to estimate biases and standard deviations. Use distributions of measured parameters to estimate biases and standard deviations. Conclude about feasibility of measuring a given parameter. 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. 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),  =6  Sun (where  Sun =0.5  Hz), and full width at half maximum (FWHM) of mode power  =1  Hz. Input parameters: T=6 months, S/N=100 (Corot ok),  =6  Sun (where  Sun =0.5  Hz), and full width at half maximum (FWHM) of mode power  =1  Hz. Plot shows distribution of values of i (left) and  (right) measured from synthetic spectra versus the true i value. The dashed line is the  guess. Plot shows distribution of values of i (left) and  (right) measured from synthetic spectra versus the true i value. The dashed line is the  guess. Using one single l=1 triplet, i and  can be retrieved with a good precision if i>30deg. Using one single l=1 triplet, i and  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). 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). True inclination angle, i Fits to synthetic spectra

6. Example 2: three multiplets l=0,1,2 Measured inclination angle True inclination angle Simultaneous fit on one l=0 singlet, one l=1 triplet and one l=2 multiplet. Assumption: all peaks have the same linewidth, . The l=0 mode helps constrain  although it contains no information about rotation. Input parameters: T=4 months,  =4  Sun,  =1  Hz.Input parameters: T=4 months,  =4  Sun,  =1  Hz. The plot shows the distribution of i values estimated from the fits as a function of the true value.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.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.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. 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. Only one l=1 triplet is fitted. Other parameters,  =1  Hz and T=6 months are fixed. Other parameters,  =1  Hz and T=6 months are fixed. The plot shows that the inclination angle can only be retrieved when  Sun = , i.e. when modes are resolved. The plot shows that the inclination angle can only be retrieved when  Sun = , 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. This condition is independent of the observation duration, T. If individual modes are not resolved then a longer observation will not help. Input  Sun at fixed  Sun Measured inclination angle

8. Conclusion & COROT targets We have shown, using MonteCarlo simulations that the stellar angular velocity, , and the direction of the rotation axis, i, can both be retrieve from solar-like pulsations. We have shown, using MonteCarlo simulations that the stellar angular velocity, , 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 (  ). The main condition is that individual modes of oscillation are resolved (  ). The observation duration must be at least 2 months, say (preferably more). 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). 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: Targets: CP (seismo field): All solar-like pulsators selected for seismo long runs (e.g. F & G stars). 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 M Jupiter. 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 M Jupiter. 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. 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. 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: This work is fully documented in the following papers: Gizon & Solanki, ApJ 589, 1009 (2003) Gizon & Solanki, ApJ 589, 1009 (2003) Gizon & Solanki, Solar Phys. 220, 169 (2004) Gizon & Solanki, Solar Phys. 220, 169 (2004)