1. Doppler Imaging of Stars
The Doppler imaging technique
Doppler imaging of Spotted stars
Doppler imaging of Magnetic Ap stars
Doppler imaging of pulsating stars

2. For stars whose spectral line profiles are dominated by rotational broadening there is a one to one mapping between location on the star and location in the line profile:
V = –Vrot
V = +Vrot
V = 0

3. What kind of star can be Doppler Imaged?
Stars whose spectral line profile is dominated by rotational broadening (V sin i > 10 – 15 km/s)
Number of resolution elements
N = V sin i / (cDl/l)
V sin i = projected rotational velocity of star
cDl/l = intrinsic width of spectral line (~ 3-5 km/s) in velocity units

4. What can be Doppler Imaged?
Temperature structure
RS CVn stars
FK Comae stars
Young stars (T Tauri stars)
Abundance features
Ap stars
Velocity distribution
Nonradially pulsating stars

5. Getting spatial information from line shapes
Latitude information
Shape information f1 f2 f3
Rotation phase

6. Doppler Imaging as a Matrix Equation
D = I ·R
D = data vector
I = image vector
R = transfer matrix
Solution:
I = D •R–1
But R is not invertible because this is a projection problem (i.e. there are many solutions)

7. I1R11 + I2 R21 + ··· + In Rn1 = D1
I1R12 + I2 R21 + ··· + In Rn2 = D2
• • •
I1R1m + I2R2m + ··· + InRnm = Dm
m equations and n variables

9. Constructing the image vector•
In is the local quantity you are mapping (z.B. temperature, equivalent width)

10. Constructing the Data vector
Rotation phase f f1 f2 f3 f4 f5 m
points
The observed spectral line profiles placed end-to-end

11. Constructing the Transfer Matrix Rjk
Ndata f1 f2 f3 f4 f5
Nimage
For pixel in this is the local line profile marginal response
Ndata = m x Nphases m
Marginal response, i.e. when when in is multiplied by the matrix local profile you recover the full spectral line:
Rjk = Il/Temperature or Il/Equivalent width
Il = spectral line profile (produced by model atmospheres)

12. I • R = D = x Ndata Nimage Nimage Data vector Image vector I1 I2 I3 I4IN • = x
Nimage m
Rotation Phase m

13. Solution of the Matrix Equation with Maximum Entropy Method (MEM)
Since there are many solutions available we need to chose one
Maximum entropy choses the one with the simplest image, or one with the least information content that is still consistent with the data
Entropy: S = – Σ
pj log (pj)
Constraint:
c2 =
(gk – dk)2 /sk2
g = model, d = data, s = noise level

14. We want to maximize a function (S) subject to a constraint (c2).
This is done with Lagrange multipliers.
We maximize the functional:
Q = S – l c2
Where l is the Lagrange multiplier
In practice we use a commercial MEMSYS package developed by Gull and Skilling that iteratively solves the matrix equation using the transpose as a guess of the inverse matrix

15. Formation of the Pseudo-emission bumps

16. 7 Spot input distribution

17. 7 Spot MEM reconstruction

18. ‚VOGT-star‘ input distribution

21. Solution is insensitive to wrong assumptions about the inclination

24. Doppler Images of Spotted stars
Magnetic activity caused by interaction between rotation and convection
Rapid rotation, deep convection => large spots
Types of rapidly rotating active stars:
RS CVn stars: evolved stars in binary systems that are tidally spun up
Young stars that have not yet lost their angular momentum

29. Migration of Spots on HR 1099

30. Doppler image of V 410 Tau

31. Are the Polar Spots Real?
Some have argued that the polar spots are not real. That they are due to:
Artefact of the technique (highly unlikely)
That polar spots cannot be real because there is no Doppler information for polar features (false)
That it is due to uncertainties in the atmospheric profiles
That it is due to chromospheric emission filling in the cores

32. Spectral line profiles behave as if the spots are at high latitudes:

33. There appears to be an inclination effect in the mean shapes of line profiles:

37. Doppler Imaging of Ap stars Origins of the Abundance Anomalies
Diffusion Mechanism of Michaud
Atmospheres of A stars have no convection
Gravity provides a downward force
This can produce under-abundances
Radiation pressure provides an upward force
This can create over-overabundances
Magnetic fields inhibit motion across field lines

38. Radiation force dominates
Gravity dominates
Overabundant
Line forming region
depleted
Radiation force dominates

39. Horizontal field lines suppress motion, but can also provide additonal support
Where the field lines are vertical ions can move freely

40. The Oblique Rotator model of Stibbs (1950)
Magnetic axis
Rotation axis
Expect distributions in rings and caps

45. Simple treatment of abundance spots is to simply scale a single line profile
But stronger lines have a different shape
An improved Doppler imaging should take into account the shape of the profiles as a function of line strength

49. How do we know that the spectral variations are due to abundance and not temperature differences?
Photometric variations show opposite phase in the ultraviolet compared to the infrared => photometry due to variations in opacity
Line profiles show equivalent width variations, but variations due to spots show little variations in the equivalent width

50. Cr on e UMa

52. Cr on q Aur

53. Si on q Aur

54. Si on 11 Ori

57. Decentering dipole will cause rings to shift towards one pole
Star center
dipole center

61. Doppler Imaging of the Velocity Distribution
Local variations in the velocity field can also cause distortions in the line profiles
Nonradial pulsations create asymmetric velocity distributions
In rapidly rotating stars these can be Doppler imaged

62. Nonradial Pulsations: Spherical Harmonics
Spherical Harmonics with „Quantum numbers“ n,l,m (radial, angular, azimuthal)
l = number of nodes
m = number of nodes intersecting pole
l-m = number of nodes parallel to equator

69. HD
NRP: l=m=2

70. Limitations of the Doppler Imaging Technique
Two temperature (abundance) distributions will be smoothed out:
Minimizes differences between photosphere and continuum
Two temperature distributions cannot be recovered
Increases area slightly
Input T
Reconstruction x
2. Smearing along longitude lines (projection effect)

71. Limitations of the Doppler Imaging Technique
Mirroring
For stars viewed equator-on there is an equal probability that a spot is in the northern or southern hemisphere
For inclined stars Doppler imaging favors features in the northern hemisphere (+l) since these are visible more, but weak features are still placed at –l
Low latitude features are shifted to slightly higher latitudes
Not effective for slow rotators (v sin i < 10 – 15 km/s)

72. Many Doppler Imaging techniques have been developed and used with success:
Trial and error (Vogt and Penrod 1982)
Tikhonov Regularization (Goncharsky et al. 1977)
Minimizes the function: dt dl ( ) 2 dt db ( ) 2
f(M) = +
t = pseudo-optical depth (i.e. line strength)
l = longitude
b = latitude

73. Maximum Entropy (Vogt, Penrod, & Hatzes 1987)
CLEAN-like algorithm (Kürster et al. 1994)
Get an approximate distribution, change the coolest pixels to a single temperature, and compare to the line profiles
Occamiam approach (Berdyugina 1983)

74. Others that most likely will work:
Neural Networks
Genetic algorithms
The various technques produce the same answer (more or less), differences are neglible
Conclusion: Problem is better constrained than we thought and although many solutions exist they probably are in the same region of the solution space

76. Image vector I1 I2 I3 I4 I5 •
Data vector
Image vector I1 I2 I3 I4 I5 •
Rotation Phase