1. A new technique of detection and inversion
of magnetic fields in stars:
PCA-ZDI
Julio Ramirez Velez
Meir Semel

2. It is possible to decrease the noise level by:
- Adding spectral lines (e. g., Zeeman Doppler Imaging)
- Using PCA-ZDI
The polarization that we can detect in solar-type stars is the one integrated over the whole surface.
It is not surprising that in the majority of the cases the circular or linear polarization signals per spectral line are hidden below the noise level.
Multiline analysis
How do we analise them??
What are these signals??

3. Scalar product

4. Scalar product

5. Scalar product

6. Scalar product

7. Scalar product

8. Scalar product

9. Scalar product

10. Scalar product
The addition of lines gives reliable results if the profiles are SIMILAR
The technique is restricted to the NO BLENDED lines that would be in the
WEAK FIELD REGIME.
Restricted to Stokes V profiles.
In order to overcome all these difficulties, we propose the PCA-ZDI technique.
 We use the eigenvectors coming from a data base of synthetic Stokes profiles instead
of delta functions.

11. Why correlations??
Scalar product
The addition of lines gives reliable results if the profiles are SIMILAR
The technique is restricted to the NO BLENDED lines that would be in the
WEAK FIELD REGIME.
Restricted to Stokes V profiles.
In order to overcome all these difficulties, we propose the PCA-ZDI technique.
 We use the eigenvectors coming from a data base of synthetic Stokes profiles instead
of delta functions.

12. Why correlations??
Scalar product
The addition of lines gives reliable results if the profiles are SIMILAR
The technique is restricted to the NO BLENDED lines that would be in the
WEAK FIELD REGIME.
Restricted to Stokes V profiles.
Why principal components??
In order to overcome all these difficulties, we propose the PCA-ZDI technique.
 We use the eigenvectors coming from a data base of synthetic Stokes profiles instead
of delta functions.

13. Why correlations??
To add coherently the spectral lines that have the same (or very similar) Doppler shifts.
 We can differenciate the distincts plasma movements coming from different parts
of the star of from the different components of a star system.
Why principal components??
Illustrative examples:
ZDI will give a null signal.
The principal components contain the physical information and the correlations will be different from zero  we will detect the signal!

14. Why correlations??
To add coherently the spectral lines that have the same (or very similar) Doppler shifts.
 We can differenciate the distincts plasma movements coming from different parts
of the star of from the different components of a star system.
Why principal components??
We have no restrictions !
Illustrative examples:
ZDI will give a null signal.
The principal components contain the physical information and the correlations will be different from zero  we will detect the signal!

15. B Modelling of the stellar atmosphere to construct the data base
The PCA-ZDI technique
Modelling of the stellar atmosphere to construct the data base
We use the COSSAM (Stift 2000) code to calculate the synthetic spectra
of the star covering around 3000 A
Following the original ideas expressed in ZDI (Semel 1989),
we consider a unique isolated magnetic agent.
LOS B
Position of the magnetic element
in the stellar surface (µ)‏
Strength and orientation
of the magnetic field.
(B, 

16. A wavelength range going from 450 to 750 nm.
The PCA-ZDI technique
Let Y be a vector containing the variables of our stellar atmosphere Y =[ µ, B,  ].
The data base consists in 255 Stokes vectors with 255 combinations of the parameters at
A wavelength range going from 450 to 750 nm.
S (Y) = (I,Q,U,V)‏
Very high resolution
10 mA.
 points in wavelength !!
Spectral Range
=[450,750] nm

17. Pn will be the DETECTORS
The PCA-ZDI technique
Let M the matrix containing the 255 Stokes vectors
M =[300000,255,4]
We diagonalise the matrix M in order to obtain the Principal Components in which the Stokes vectors of the data base can be decomposed: Y
N = 0 , 1 ,.., 254
The {Pn} is the base of the eigenvectors (principal components).
Pn will be the DETECTORS

18. The spectral dispersion is different for each wavelength
The PCA-ZDI technique
The spectral dispersion is different for each wavelength
 We change the wavelength axis to the velocity one:
Where X is the velocity.
We assume that our data base is complete enough (contains enough physical ingredients)
and we perform the cross-correlation beween the observed (VERY NOISY)
Stokes vector and one of the detectors:

19. The PCA-ZDI technique
We assume that our data base is complete enough (contains enough physical ingredients)
and we perform the cross-correlation beween the observed (VERY NOISY)
Stokes vector and one of the detectors:
Pseudo profiles
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IIpeg

20. The PCA-ZDI technique
We assume that our data base is complete enough (contains enough physical ingredients)
and we perform the cross-correlation beween the observed (VERY NOISY)
Stokes vector and one of the detectors:
Pseudo profiles
hd155555
IIpeg
WE ARE ABLE TO DETECT SIGNALS
USING THE PCA-ZDI TECHNIQUE

21. Not only circular but also LINEAR polarization
Has been detected using this technique!!
SPW4  The first detections of magnetics fields in stars using the
PCA-ZDI technique

22. + THEY ARE NOT THE REAL PROFILES!
The information of the atmospheric variables is encoded in these profiles (that we will call the pseudo profiles) in a more complicated way than in the typical Zeeman profiles.
Gives rise to polarization due to a
PHYSICAL MECHANISM that we can model. +
PCA-ZDI method
ZEEMAN
STOKES PROFILES
PSEUDO-PROFILES

23. How do we analise the PCA-ZDI pseudo-profiles?
Second Part of the talk
How do we analise the PCA-ZDI
pseudo-profiles?
Is there the information of the
magnetic field?

24. The DISCRET multi line case
(ZDI)‏
The center gravity method
The pseudoline PCA inversions
The CONTINUM case
(PCA-ZDI)
PCA inversions of the pseudo-profiles

25. In order to study if it is possible to
The DISCRET case
In order to study if it is possible to
retrieve the magnetic field vector from
the observations by means of the “addition
of lines” (ZDI) technique, we perform some
synthetic tests considering the following
spectral lines:
Adding all these Fe lines, we construct the pseudoline:

26. The DISCRET case

27. The DISCRET multi line case
(ZDI)‏
The center gravity method
The pseudoline PCA inversions
The CONTINUM case
(PCA-ZDI)
PCA inversions of the pseudo-profiles

28. The center gravity method
The DISCRET case
B (G)‏
Using the pseudo line
The center gravity method

29. The DISCRET multi line case
(ZDI)‏
The center gravity method
The pseudoline PCA inversions
The CONTINUM case
(PCA-ZDI)
PCA inversions of the pseudo-profiles

30. Using the Milne-Eddington approximation, we compute a data base
The DISCRET case
Using the Milne-Eddington approximation, we compute a data base
for each one of the spectral lines.
 Adding each profile of all data bases we have the data base for the pseudo-line.
We consider as free parameters:
B=[0,5000] Gauss, Angle Azimuthal=[0,180] et Angle LOS=[0,90]
We synthesize 500 profiles of the pseudo-line and we invert them developing
a PCA inversion code.

31. Using the Milne-Eddington approximation, we compute a data base
The DISCRET case
Using the Milne-Eddington approximation, we compute a data base
for each one of the spectral lines.
 Adding each profile of all data bases we have the data base for the pseudo-line.
We consider as free parameters:
B=[0,5000] Gauss, Angle Azimuthal=[0,180] et Angle LOS=[0,90]
We synthesize 500 profiles of the pseudo-line and we invert them developing
a PCA inversion code. 
When no noise
is added, the errors..

32. The DISCRET case
Inversions of noisy profiles

33. The DISCRET case
Inversions of noisy profiles

34.  In the most noisy case, the best retrieval of the
The DISCRET case 
In the most noisy case,
the best retrieval of the
magnetic field is the one
of the pseudo-line.

35.  In the most noisy case, the best retrieval of the
The DISCRET case 
In the most noisy case,
the best retrieval of the
magnetic field is the one
of the pseudo-line.
THE INFORMATION OF THE MAGNETIC FIELD VECTOR IS WELL CONTAINED
IN THE PROFILES OF THE PSEUDO-LINE

36. The DISCRET multi line case
(ZDI)‏
The center gravity method
The pseudoline PCA inversions
The CONTINUM case
(PCA-ZDI)
PCA inversions of the pseudo-profiles

37. Inversions are performed in the space of
The CONTINUM case
DATA BASE of S(Y)
DATA BASE of fS
Inversions are performed in the space of
the pseudo-profiles of the principal components.

38. Example of a pseudo-profile of the data basis.
The CONTINUM case
Example of a pseudo-profile of the data basis.

39. * position of the magnetic agent in the stellar surface, (Mu)‏
The CONTINUM case
Using (I,V) is possible to obtain
* position of the magnetic agent in the stellar surface, (Mu)‏
* the magnetic field strength
* the Angle in the LOS

40. Detections are possible for all the Stokes parameters
CONCLUSIONS
- We do not employ the weak field approximation (we are not restricted by the psysical
conditions that MIGHT be included in the data base)
- We are able to detect LINEAR polarization signals.
- The most important fact is that the analysis of both profiles coming from the addition
of lines or from the PCA-ZDI techniques are RELIABLE.
The information of the magnetic field vector is contained in the pseudo-profiles.
Detections are possible
for all the Stokes parameters
Inversions are
also possible.