1. Spot mapping in cool stars Andrew Collier Cameron University of St Andrews

2. Science goals Dynamo geometry –Solar-like or something different? –Polar spots and active belts Spot structure –Resolved or not? Differential rotation and meridional flows Lifetimes of individual spots and active regions Stellar “butterfly diagrams” Different stellar types –Pre-main sequence stars –Young main-sequence stars with[out] radiative interiors –Subgiants and giants

3. Doppler Imaging I: Basic Principles Intensity AA v sin i-v sin iv(spot) Velocity v sin i-v sin iv(spot) Velocity

7. Data requirements Time-series of hi-res (R > 30000) spectra: Good supply of unblended intermediate- strength lines (!) Broad-band light-curves. TiO and other temperature diagnostics.

8. Rotational broadening gives shallow, blended lines … … but LTE models yield estimates of their positions and strengths

9. Combining line profiles Assume observed spectrum = mean profile convolved with depth-weighted line pattern: De-convolve mean profile z k via least squares: S:N improves from ~100 to ~2500 per 3 km s –1 pixel with ~2500 lines.  = Mean profile, z (UNKNOWN) Depth-weighted line pattern,  - KNOWN Rotationally broadened spectrum, r – KNOWN

10. LSD profiles of Gl 176.3 and AB Dor No sidelobes –major advantage over cross-correlation! Internal errors –computed from diagonal elements of inverse matrix or from SVD –Multiplex gain allows us to go 4 – 5 mag fainter than previously –see e.g. Barnes et al 1998: imaging of G dwarfs with V = 11.5 in a Per cluster. Gl 176.3: normal K0 dwarf AB Dor: v sin i = 90 km/sec

11. Time series: deconvolved Stokes I profiles AB Dor 1996 Dec 23-29 AAT + UCLES +Semel polarimeter Sum of L & R circularly polarized line profiles

12. Choice of mapping parameter What are we trying to map on the stellar surface? Temperature: f ~ T 4 –Bolometric surface brightness –Form of spectrum varies continuously with f –No restriction on mix of bright and dark features –Needs grid of (synthetic) spectra –May give problems in blurred images of unresolved spots Spot filling factor: f = A spots / (A spots + A phot ) –Takes values 0 < f < 1 –Requires fixed photospheric and spot temperatures –Doesn’t allow other temperature components –Can use “template” spectra of real stars with appropriate T eff to represent “spot” and “photosphere” –Copes well with unresolved spots

13. Local specific intensities Spectrum synthesis of individual lines in spectral region to be fitted, OR Slowly rotating star of similar spectral type observed with same instrument. Synthetic spectral fits from Strassmeier et al (1999)

14. Image-data transformation -v sin i +v sin i Geometric kernel: –Position M(t) of pixel M at time t –Doppler shift  = 0 v r (M,t)/c of different parts of the stellar surface at different times. –Foreshortening angle  (M,t)/ and eclipse criteria. Specific intensities: –Spectrum I(f,  ) emerging from stellar atmosphere at local foreshortening angle, as modified by image parameter f. Surface integration: –Yields total flux spectrum at each time t of observation

15. Regularised least-squares solutions

16. Regularised least-squares strategies Compute synthetic data D k, k=1,…,M for trial images f = f j, j=1, …, N. Badness of fit: Shannon-Jaynes image entropy: –Minimizes information and spurious correlations in image. Tikhonov (1963) regularization: –Maximises smoothness of solution.

17. RX J1508.6 -4423 Deconvolved profiles and fitted model with unspotted profile subtracted. Data Model fit Residuals (From Donati et al 2000)

18. Dealing with nuisance parameters Radial velocity V sin i Line EW Inclination Binary orbital parameters Binary surface geometry Strategy: minimise lowest attainable  2 with respect to nuisance parameters. Barnes et al (2000)

19. A more systematic approach Hendry & Mochnacki ApJ 531, 467 (2000) : –Surface imaging of contact binary VW Cep –Nuisance parameters adjusted simultaneously with image: »Phase correction  »Velocity amplitude K »System centre-of-mass velocity g »Inclination i »Mass ratio q »Fill-out factor F »Unspotted primary T eff –Artefacts introduced by bad nuisance- parameter values decrease final image entropy. VW Cep, 1991 Mar to 1993 May

20. Heterogeneous datasets Spectral data s1, s2,... from different observatories Broad-band photometry p Need to maximize Q = S(f) -  p (  2 p ) -  s1 (  2 s1 ) -  s1 (  2 s2 ) -... SAAO data AAT data E.g. PZ Tel: Barnes et al 2000 (Unruh, Cameron&Cutispoto 1995; Barnes et al 2000; Hendry & Mochnacki 2000)

21. Surface resolution and noise Error bars on images: –adjacent pixels are correlated (blurring) –regularised least squares methods don’t yield error estimates directly. Consistency tests: e.g. HR 1099 images in different lines by Strassmeier & Bartus (1999): Ca I 6439: Fe I 6430:

22. Surface resolution and noise – 2 Images derived from simultaneous, independent datasets (Barnes et al 1998): –Full dataset –Odd-numbered spectra only –Even-numbered spectra only

23. The Occamian approach Applied to spot imaging problem by Berdyugina (1998) –Astron. Astrophys. 338, 97–105 (1998) Matrix P defines “PSF” of forward problem: Approximation to Hessian matrix: Eigenvectors of H define principal axes of error ellipsoid in image space. Principal components with small eigenvalues are poorly-constrained by data A subset of those principle components exhausts the information content of image f Use SVD to solve for f; error estimates are:

24. Future prospects: The perfect spot code The perfect code would have: –Simultaneous fitting of nuisance parameters –Error bars on images and nuisance parameters –Full utilisation of temperature-dependent profile information in thousands of lines –Correct treatment of heterogeneous data types (spectra, photometry, TiO,...) The perfect user of such a code would: –Use well-understood statistical methods to test hypotheses (    rules OK!) –Perform these tests in data space where errors are understood! –Always remember the First Law of Doppler Imaging: – If you can’t see it in the trailed spectrum, it probably isn’t there. -v sin i +v sin i

25. Starspots as flow tracers Latitude-dependent rotation in 3 images of AB Dor (AAT 1996 Dec 23–29, Donati et al 1999)

26. Surface shear: 1996 December 23 - 29 CCF for surface- brightness images CCF for magnetic images: Equator pulls one rotation ahead of polar regions every ~ 120 d or so -- very similar to solar shear!

27. Data-space fits to differential rotation Donati et al (2000) fitted 2-parameter differential rotation law to PTT star RX J1508.6 -4423 Differential rotation law used to shear image derived from May 06 data and generate synthetic May 10 data    of fit to May 10 observations  as a function of    Cross-correlation image with best-fit shear pattern shown:

28. Spot lifetimes: dwarfs vs (sub)giants Barnes et al (1998): No correlation of fine-scale spot structure between 2 images of  Per G dwarf He 699 taken 1 month apart. –But overall active-region positions unchanged? Berdyugina et al (1999): Major spot complexes on II Peg persist for 2-3 months, but fine structure unresolved. 1998: JulyOctoberNovember 1997: JuneAugustDecember

29. Do spots drift poleward on HR 1099? Strassmeier & Bartus 1999 Spectra on 57 consecutive nights, 1996 Nov-Dec Movie constructed from “running mean” sequences of 12 consecutive spectra. Main spot and transient neighbours form and dissolve on timescales consistent with Berdyugina et al’s II Peg maps: Polar view Longitude Latitude

30. So how far have we got? Differential rotation: –Young solar-type stars have solar-like latitudinal shear even at rotation rates 50 times solar. –Study of meridional flows needs better-sampled data over timescales of weeks to months. Lifetimes of individual spots and active regions –Weeks to months (respectively?) Stellar cycles and “butterfly diagrams” –Wait for Jean-François Donati’s talk! Different stellar types –Pre-main sequence stars -- more needed –Young main-sequence stars -- well studied, but what do fully-convective M dwarfs look like? –Subgiants and giants -- longer-lived spots? –Binaries -- the next big challenge.