Asteroseismological Asteroseismological determination of stellar determination of stellar

rotation axes:rotation axes:

Feasibility study (COROT AP+CP)Feasibility study (COROT AP+CP)

L. Gizon(1), G. Vauclair(2), S. Solanki(1), S. Dreizler(3)L. Gizon(1), G. Vauclair(2), S. Solanki(1), S. Dreizler(3)

(1) MPI for Solar System Research, Katlenburg-Lindau, D(1) MPI for Solar System Research, Katlenburg-Lindau, D

(2) Observatoire Midi-Pyrenees, Toulouse, F(2) Observatoire Midi-Pyrenees, Toulouse, F

(3) Goettingen Sternwarte, Goettingen, D(3) Goettingen Sternwarte, Goettingen, D

Science ObjectivesScience Objectives Measure angular velocity, Measure angular velocity, , and inclination of rotation axis to line of sight, , and inclination of rotation axis to line of sight, ii..

The angleThe angle i i can be determined seismically from the can be determined seismically from the visibilityvisibility of spheroidal modes of of spheroidal modes of pulsation.pulsation.

From From , , i,i, and the spectroscopically determined and the spectroscopically determined v v sin sin ii, it is possible to deduce the , it is possible to deduce the stellar radius, stellar radius, RR, without prior knowledge of stellar structure and evolution. (here , without prior knowledge of stellar structure and evolution. (here vv is is essentially the equatorial velocity.)essentially the equatorial velocity.)

A knowledge of A knowledge of ii for planet host stars can tell us about the planets themselves. for planet host stars can tell us about the planets themselves.

If If MpMp sin sin jj is known from periodic Doppler shifts, where is known from periodic Doppler shifts, where jj is the inclination of the is the inclination of the orbital plane of a planet with mass orbital plane of a planet with mass MpMp, then , then j=ij=i puts a constraint on puts a constraint on MpMp (Corot CP) (Corot CP)

If a planet is detected by COROT in If a planet is detected by COROT in transittransit, then , then jj is known with high accuracy and is known with high accuracy and it becomes possible to test the theoretical prediction that it becomes possible to test the theoretical prediction that jj and and ii are similar (Corot AP) are similar (Corot AP)

For stars that have very clean oscillation spectra, the latitudinal differential rotation For stars that have very clean oscillation spectra, the latitudinal differential rotation (mid-lat) (mid-lat) (eq)(eq) can also be estimated seismically. If not, latitudinal differential can also be estimated seismically. If not, latitudinal differential rotation may be constrained by comparing rotation may be constrained by comparing v v sin sin ii with with RR sin sin i i (this works only (this works only

when the stellar radius when the stellar radius RR is known from parallaxes). is known from parallaxes).

Mode visibility of solar-like oscillationsMode visibility of solar-like oscillations

Plots show expectation value of power in azimuthal (Plots show expectation value of power in azimuthal (mm) components of dipole () components of dipole (l=1l=1, , leftleft) and quadrupole () and quadrupole (l=2l=2, right, right) modes of global acoustic oscillations as a function of ) modes of global acoustic oscillations as a function of inclination angle, inclination angle, ii..

Assumption: energy equipartition between Assumption: energy equipartition between mm components at fixed values of components at fixed values of ll and and n; n; inin a statistical sense. A to first approximation, the visibility of the 2a statistical sense. A to first approximation, the visibility of the 2 ll+1 +1 mm-components is -components is determined by geometry only, at least when oscillations are measured in intensity.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 This is the case for solar oscillations, and it is likely to remain true for all stochastically excited solar-like oscillations.stochastically excited solar-like oscillations.

Plots show that it should be possible to estimate both Plots show that it should be possible to estimate both i i and and from sufficiently long from sufficiently long time-series, as long as time-series, as long as ii is more than about 15 deg (in order to be able to distinguish is more than about 15 deg (in order to be able to distinguish a star with a star with =0 from a star with =0 from a star with ii<15deg). Note that rotational splitting is not <15deg). Note that rotational splitting is not proportional to proportional to mm when the centrifugal distortion is taken into account. when the centrifugal distortion is taken into account.

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MonteCarlo simulationsMonteCarlo simulations

Pick fixed values of the observation duration (T), Pick fixed values of the observation duration (T), , , ii, and mode lifetime., and mode lifetime.

Simulate thousands of realizations of the oscillation Simulate thousands of realizations of the oscillation power spectrumpower spectrum

Frequency bins are independent when observations are Frequency bins are independent when observations are continuous.continuous.

At fixed frequency, the probability density function of the At fixed frequency, the probability density function of the power follows an exponential distribution.power follows an exponential distribution.

The expectation value of the power in a mode is The expectation value of the power in a mode is assumed to be given by a Lorentzian function.assumed to be given by a Lorentzian function.

Measure oscillation parameters, including Measure oscillation parameters, including ii, from , from fits using a maximum likelihood method. fits using a maximum likelihood method.

Use distributions of measured parameters to Use distributions of measured parameters to estimate biases and standard deviations.estimate biases and standard deviations.

Conclude about feasibility of measuring a given Conclude about feasibility of measuring a given parameter. parameter.

Plots show realisations (wiggly solid lines) of power Plots show realisations (wiggly solid lines) of power spectra for spectra for l l =0, 1, and 2. Thick gray lines are the =0, 1, and 2. Thick gray lines are the expectation values, thick solid lines are the fits.expectation values, thick solid lines are the fits.

Example 1: one single Example 1: one single l=l=1 triplet1 triplet

Input parameters: T=6 months, S/N=100 (Corot ok), Input parameters: T=6 months, S/N=100 (Corot ok), =6=6Sun Sun (where (where SunSun=0.5=0.5Hz), Hz), and full width at half maximum (FWHM) of mode power and full width at half maximum (FWHM) of mode power =1=1Hz.Hz.

Plot shows distribution of values of Plot shows distribution of values of ii (left) and (left) and (right) measured from synthetic (right) measured from synthetic spectra versus the true spectra versus the true ii value value. . The dashed line is the The dashed line is the guess. guess.

Using one single Using one single l=l=1 triplet, 1 triplet, ii and and can be retrieved with a good precision if i>30deg. can be retrieved with a good precision if i>30deg. In practice, the uncertainty on In practice, the uncertainty on ii will be reduced by a factor sqrt(N) where N is the 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). number of observable dipole modes (N>10). The uncertainty scales like 1/sqrt(T).

True inclination angle, i

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Example 2: three multiplets l=0,1,2Example 2: three multiplets l=0,1,2M

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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, Input parameters: T=4 months, =4=4SunSun, , =1=1Hz.Hz.•The plot shows the distribution of The plot shows the distribution of ii values estimated from the fits as a function of the values estimated from the fits as a function of the true value.true value.•The fits are doing a much better job than for one l=1 alone: the uncertainty on The fits are doing a much better job than for one l=1 alone: the uncertainty on ii drops drops significantly. The value of significantly. The value of ii below which the results cannot be trusted is about 15 deg. below which the results cannot be trusted is about 15 deg.•Once again, the error bar on Once again, the error bar on ii would be reduced by a factor of sqrt(N) where N is the would be reduced by a factor of sqrt(N) where N is the number of radial orders that can be observed.number of radial orders that can be observed.

The modes must be resolvedThe modes must be resolved Plot of measured inclination Plot of measured inclination

angle (symbols with error angle (symbols with error bars) for input values bars) for input values i=i=30deg 30deg andand i= i=80deg (dashed lines) as 80deg (dashed lines) as a function of stellar angular a function of stellar angular velocity. velocity.

Only one Only one l=l=1 triplet is fitted. 1 triplet is fitted. Other parameters, Other parameters, =1=1Hz Hz

and T=6 months are fixed.and T=6 months are fixed.

The plot shows that the The plot shows that the inclination angle can only be inclination angle can only be retrieved when retrieved when SunSun==, , i.e. when modes are resolved.i.e. when modes are resolved.

This condition is independent This condition is independent of the observation duration, T. of the observation duration, T. If individual modes are not If individual modes are not resolved then a longer resolved then a longer observation will not help. observation will not help.

Input Sun at fixed Sun

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Conclusion & COROT targetsConclusion & COROT targets We have shown, using MonteCarlo simulations that the stellar angular velocity,We have shown, using MonteCarlo simulations that the stellar angular velocity,, and the , and the

direction of the rotation axis, direction of the rotation axis, ii, can both be retrieve from solar-like pulsations., 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 measureIf these conditions are met, it should be not problem to measure i i with a precision of a few with a precision of a few

degrees. That is unless the rotation axis points toward Corot (unlikely).degrees. That is unless the rotation axis points toward Corot (unlikely).

Targets:Targets: CP (seismo field):CP (seismo field): All solar-like pulsators selected for seismo long runs (e.g. F & G stars). 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 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 subsequently selected as a Prime Target. This G0V Sun-like star is known to have a planetary companion with Mp=1.13 Mcompanion with Mp=1.13 MJupiterJupiter..

Red giants with solar-like pulsations are very interesting targets too. For example, the G6III star 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.HD 50890 (V Mag=6) which will be observed by Corot.

AP (exoplanet field):AP (exoplanet field): all stars for which a planetary transit has been detected. Switch to 32s all stars for which a planetary transit has been detected. Switch to 32s cadence.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)