Thomas Hackman: Stellar differential rotation1 Detecting stellar differential rotation NORDITA – Solar and stellar dynamo cycles Thomas Hackman,

Thomas Hackman: Stellar differential rotation 2 Detecting stellar differential rotation from photometry and spectroscopy Contents:  Effects of differential rotation on photometry and spectroscopy  Time series analysis of photometry  Doppler imaging  Combining spectroscopy and photometry

Thomas Hackman: Stellar differential rotation 3 1. Observing the solar differential rotation Quantification of the surface differential rotation: Surface differential rotation was first observed by tracing sunspots: The angular velocity vs. the radial distance from the solar core was retrieved by helioseismology

Thomas Hackman: Stellar differential rotation 4 1. Sun spot latitudes vs. angular velocity Observations of sunspot groups (Howard 1994)

Thomas Hackman: Stellar differential rotation 5 2. Estimating stellar differential rotation Some available methods:  Study changes in the rotation periods derived from UBV-photometry or Ca II H&K:n fluxes  Modelling of star spots using photometry  The profile of photospheric absorption lines  Doppler-imaging  Combining Doppler-imaging and photometric time series analysis

Thomas Hackman: Stellar differential rotation 6 3. Differential rotation and photometry Spots determine the photometric rotation period P phot of active late-type stars Surface differential rotation => P phot will depend on the spot latitude  Changes in the latitudinal location of the main spot(group) will be seen as changes in the period P phot

Thomas Hackman: Stellar differential rotation 7 TSPA = Three stage period analysis (Jetsu & Pelt, 1999, A&AS 139, 629) Model: Results: Period ( P ), amplitude ( A ), mean magnitude ( M ), photometric minimum  min 3.1 Time series analysis of photometry

Thomas Hackman: Stellar differential rotation Differential rotation from photometry From the variations in P phot we can derive a lower limit for the differential rotation Z : where P W is the weighted mean of the periods and  P W their standard deviation Assuming a solar differential rotation curve we can estimate  from:

Thomas Hackman: Stellar differential rotation Estimate of the differential rotation of a FK Comae type star HD (Jetsu et al., 1999, A&A 351, 212): => larger  than expected for rapid rotator

Thomas Hackman: Stellar differential rotation Estimate of the differential rotation of a young solar analogue HD (Lehtinen, 2009): Again: Larger  than expected

Thomas Hackman: Stellar differential rotation Is a varying period really a proof of differential rotation? Problem: Any period analysis will give a scatter in P phot, even if the real period is constant … but this scatter can be estimated and is in general much smaller than the measured  P W Signs of differential rotation:  A (cor)relation between P phot and the amplitude A of the photometric light curve  Drifts in the photometric minimum (obtained with constant P phot )

Thomas Hackman: Stellar differential rotation Drifting minimum phases Latitudinal migration of spot activity + differential rotation => drifts in the photometric minimum phases Wobbling of the minima in LQ Hya (Berdyugina et al. 2002)

Thomas Hackman: Stellar differential rotation Drifting minimum phases Minimum phases of HD (Jetsu et al. 1999)

Thomas Hackman: Stellar differential rotation Spot modelling of stellar light curves Using star spot models one can retrieve spot latitudes and rotation rates from photometry Ex. MOST satellite observations (Croll et al. 2006, Walker et al. 2007) Problem: Non- uniqueness of the solution Croll et al. 2006

Thomas Hackman: Stellar differential rotation Differential rotation and line profiles Surface differential rotation will alter the rotationally broadened profile of photospheric absorption lines (Bruning 1981):   > 0 => a ”sharper” line profile   a more ”flat-bottomed” line Fourier transform can be used to detect differential rotation (Gray 1977, Reiners & Schmitt 2003) … but spots will complicate the analysis

Thomas Hackman: Stellar differential rotation Differential rotation from line profiles HD (Reiners & Schmitt 2003)

Thomas Hackman: Stellar differential rotation Differential rotation from Doppler images Locate spots from time separated Doppler images and study if their longitudinal migration dependence on their latitudes Cross-correlate consecutive Doppler images and solve the differential rotation curve Include differential rotation as a parameter in the Doppler imaging solution

Thomas Hackman: Stellar differential rotation The problem of artefacts Doppler images always include artefacts  ”Shadows” of appearing at some longitudes but lower latitudes of the real features  Longitudinal ”stripes”  High contrast alternating cool and hot features, often ”arches” or ”ovals”  High contrast small features  Axisymmetric artefacts: Latitudinal bands  Latitudinal shifts of real features Including artefacts in the analysis => usually nearly rigid body rotation

Thomas Hackman: Stellar differential rotation Doppler images of HD DI:s of HD (Dunstone et al. 2008): Spots and radial magnetic field

Thomas Hackman: Stellar differential rotation Doppler images of HD Low latitude “shadows” of high latitude spots (Hackman 2004)

Thomas Hackman: Stellar differential rotation Combining photometry and spectrometry The spot latitudes from Doppler imaging The corresponding spot rotation period from time series analysis of photometry Differential rotation can be included as a stellar parameter in the Doppler imaging inversion  To correct the rotation profile …  … but not necessarily to correct the surface shear, because we only need to track the main spot

Thomas Hackman: Stellar differential rotation Observations Spectrometry:  Nordic Optical Telescope + SOFIN Échelle spectrograph Photometry:  Different APT:s

Thomas Hackman: Stellar differential rotation Estimate of differential rotation of HD (Hackman 2004) Spot latitude  (= b ) vs. P phot yields … but the range in  is very limited and is this negative  physically possible? Best fit with the “solar” differenential rotation curve to line profiles yields

Thomas Hackman: Stellar differential rotation Estimate of differential rotation of FK Comae Average spot latitudes vs. W yield nearly rigid body rotation (Korhonen et al. 2007): …but again not very convincing

Thomas Hackman: Stellar differential rotation Conclusions Time series analysis of photometry often gives a higher  than expected from theoretical models Doppler imaging combined with photometric time series analysis should be the best method to estimate differential rotation … but more data is needed (same old story)

Thomas Hackman: Stellar differential rotation 26 Acknowledgements Thanks to prof. Ilkka Tuominen, without whom I would probably not be here today!

Thomas Hackman: Stellar differential rotation Solar differential rotation in depth Angular velocity as a function of r/R sol (R. Howe / GONG/ NOAO)