Spot mapping in cool stars

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Spot mapping in cool stars

• Andrew Collier Cameron
• University of St Andrews

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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 without radiative
  interiors
• Subgiants and giants

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Doppler Imaging I Basic Principles
A
A
Intensity
Intensity
v sin i
-v sin i
v(spot)
v sin i
-v sin i
v(spot)
Velocity
Velocity
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Data requirements

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

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

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Combining line profiles

• Assume observed spectrum mean profile convolved
  with depth-weighted line pattern
• De-convolve mean profile zk via least squares
• SN improves from 100 to 2500 per 3 km s1
  pixel with 2500 lines.

Depth-weighted line pattern, ? - KNOWN
?
Mean profile, z (UNKNOWN)
Rotationally broadened spectrum, r KNOWN

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LSD profiles of Gl 176.3 and AB Dor
AB Dor v sin i 90 km/sec
Gl 176.3 normal K0 dwarf

• 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.

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Time seriesdeconvolved Stokes Iprofiles

• AB Dor
• 1996 Dec 23-29
• AAT UCLES Semel polarimeter
• Sum of L R circularly polarized line profiles

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Choice of mapping parameter

• What are we trying to map on the stellar surface?
• Temperature f T4
• 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 Aspots /
  (Aspots Aphot)
• Takes values 0 lt f lt 1
• Requires fixed photospheric and spot temperatures
• Doesnt allow other temperature components
• Can use template spectra of real stars with
  appropriate Teff to represent spot and
  photosphere
• Copes well with unresolved spots

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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)
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Image-data transformation

• Geometric kernel
• Position M(t) of pixel M at time t
• Doppler shift Dl l0vr(M,t)/c of different parts
  of the stellar surface at different times.
• Foreshortening angle q(M,t)/ and eclipse
  criteria.
• Specific intensities
• Spectrum I(f, l,q) 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

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Regularised least-squares solutions
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Regularised least-squares strategies

• Compute synthetic data Dk, k1,,M for trial
  images f fj, j1, , N.
• Badness of fit
• Shannon-Jaynes image entropy
• Minimizes information and spurious correlations
  in image.
• Tikhonov (1963) regularization
• Maximises smoothness of solution.

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RX J1508.6 -4423

• Deconvolved profiles and fitted model with
  unspotted profile subtracted.

(From Donati et al 2000)
Data Model fit
Residuals
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Dealing with nuisance parameters

• Radial velocity
• V sin i
• Line EW
• Inclination
• Binary orbital parameters
• Binary surface geometry
• Strategy minimise lowest attainable c2 with
  respect to nuisance parameters.

Barnes et al (2000)
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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 Df
• Velocity amplitude K
• System centre-of-mass velocity g
• Inclination i
• Mass ratio q
• Fill-out factor F
• Unspotted primary Teff
• Artefacts introduced by bad nuisance-
  parameter values decrease final image
  entropy.

VW Cep, 1991 Mar to 1993 May
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Heterogeneous datasets

• Spectral data s1, s2, ... from different
  observatories
• Broad-band photometry p
• Need to maximize Q S(f) - Lp(c2p) - Ls1(c2s1) -
  Ls1(c2s2) - ...

(Unruh, CameronCutispoto 1995 Barnes et al
2000 Hendry Mochnacki 2000)
SAAO data
AAT data
E.g. PZ Tel Barnes et al 2000
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Surface resolution and noise

• Error bars on images
• adjacent pixels are correlated (blurring)
• regularised least squares methods dont yield
  error estimates directly.
• Consistency tests e.g. HR 1099 images in
  different lines by Strassmeier Bartus (1999)

Ca I 6439
Fe I 6430
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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

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The Occamian approach

• Applied to spot imaging problem by Berdyugina
  (1998)
• Astron. Astrophys. 338, 97105 (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

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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 (c2 rules OK!)
• Perform these tests in data space where errors
  are understood!
• Always remember the First Law of Doppler Imaging
• If you cant see it in the trailed spectrum, it
  probably isnt there.

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Starspots as flow tracers

• Latitude-dependent rotation in 3 images of AB Dor
  (AAT 1996 Dec 2329, Donati et al 1999)

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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!
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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
• c2 of fit to May 10 observations as a function
  of W0, DW
• Cross-correlation image with best-fit shear
  pattern shown

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Spot lifetimes dwarfs vs (sub)giants

• Barnes et al (1998) No correlation of fine-scale
  spot structure between 2 images of a 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.

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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 als II Peg maps

Latitude
Polar view
Longitude
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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 Donatis 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.