Transcript Differential rotation in young low

Spot mapping in cool stars
Andrew Collier Cameron
University of St Andrews
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
Doppler Imaging I: Basic Principles
A
A
Intensity
Intensity
-v sin i
v(spot)
Velocity
v sin i
-v sin i
v(spot)
Velocity
v sin i
Data requirements
• Time-series of hi-res (R > 30000) spectra:
• Good supply of unblended intermediatestrength lines (!)
• Broad-band light-curves.
• TiO and other temperature diagnostics.
• Rotational broadening gives shallow, blended lines …
• … but LTE models yield estimates of their positions and strengths
Combining line profiles
• Assume observed spectrum = mean profile convolved
with depth-weighted line pattern:
Depth-weighted line pattern,  - KNOWN

Mean
profile, z
(UNKNOWN)
Rotationally broadened spectrum, r – KNOWN
=
• De-convolve mean profile zk via least squares:
rj    jk zk 


2
Solve :
 0, where   
2
zk

j
j
2
2
• S:N improves from ~100 to ~2500 per 3 km s–1 pixel with
~2500 lines.
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.
Time series:
deconvolved
Stokes I
profiles
• AB Dor
• 1996 Dec 23-29
• AAT + UCLES
+Semel
polarimeter
• Sum of L & R
circularly
polarized line
profiles
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 < f < 1
Requires fixed photospheric and spot temperatures
Doesn’t allow other temperature components
Can use “template” spectra of real stars with appropriate
Teff to represent “spot” and “photosphere”
– Copes well with unresolved spots
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)
Image-data transformation
• Geometric kernel:
– Position M(t) of pixel M at time t
– Doppler shift  = 0vr(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:
Dk   I kj ( f j ; j ;  k   j )s j cos  j
j
– Yields total flux spectrum at each time t of
observation
-v sin i
+v sin i
Regularised least-squares solutions
IMAGE- DATA TRANSFORMATION
Image values : f , j  1,...N
j
Synthetic data: D   V ( f )
k
kj j
kj
REGULARIZING FUNCTION (IMAGE ENTROPY)

S  f     w  f log
i i
i

and gradients
S
f
1 f 

i
1 f log

i
m
1m 
f
i

etc.
j
D
'
k
Linearized marginal responses : V 
kj  f
j
GOODNESS - OF - FIT
 F D  2
k
k
C( f )  


k
k 
C
NEXT ITERATION TOWARD SOLUTION OF

 f j
S f  C f   0
f
j

2
k k
F  D V'

k kj
2 k
OBSERVATIONS
New image values : f , j  1,...N
j
F , k 1,...M
k
Errors :  , k1,...M
k
Data:
Regularised least-squares strategies
• Compute synthetic data Dk, k=1,…,M for trial
images f = fj, j=1, …, N.
2
• Badness of fit:
M
 (f)  
2
k 1
Fk  Dk ( f )
 k2
• Shannon-Jaynes image entropy:S( f )  
fj
N
w j f j ln

m
j1
– Minimizes information and spurious correlations in image.
• Tikhonov (1963) regularization:
– Maximises smoothness of solution.
T( f )    f ds
2
RX J1508.6 -4423
• Deconvolved profiles and
fitted model with unspotted
profile subtracted.
Data
(From Donati
et al 2000)
Model fit
Residuals
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)
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 f
» 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 nuisanceparameter values decrease final image
entropy.
QuickTime™ and a
Video decompressor
are needed to see this picture.
VW Cep, 1991 Mar to 1993 May
Heterogeneous datasets
• Spectral data s1, s2, ... from different observatories
• Broad-band photometry p
• Need to maximize Q = S(f) - Lp(2p) - Ls1(2s1) - Ls1(2s2) - ...
(Unruh, Cameron&Cutispoto 1995; Barnes et al 2000; Hendry & Mochnacki 2000)
SAAO
data
AAT
data
E.g. PZ Tel: Barnes et al 2000
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:
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
The Occamian approach
• Applied to spot imaging problem by Berdyugina (1998)
– Astron. Astrophys. 338, 97–105 (1998)
• Matrix P defines “PSF” of forward problem: D  P.f
• Approximation to Hessian matrix:
H(f)  P T Q 1 P, where Q  Diag[ 12 ,  22 ,... 2M ]
Pkj  Dk / f j even if D = P( f ) is nonlinear
• 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:
 2 ( fˆ )  Diag[H –1 ]
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 (2 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
Starspots as flow tracers
• Latitude-dependent rotation in 3 images of AB
Dor (AAT 1996 Dec 23–29, Donati et al 1999)
QuickTime™ and a
Video decompressor
are needed to see this picture.
Surface shear: 1996 December 23 - 29
• CCF for
surfacebrightness
images
Equator pulls one rotation
ahead of polar regions
every ~ 120 d or so -- very
similar to solar shear!
• CCF for
magnetic
images:
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
• 2 of fit to May 10
observations as a
function of W0, W
• Cross-correlation
image with best-fit
shear pattern shown:
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.
1997:
June
August
December
July
October
November
1998:
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:
QuickTime™ and a
Video decompressor
are needed to see this picture.
Polar view
Longitude
Latitude
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.