1. Planetary Nebulae and Wavelet Transform Francois Cuisinier GEMAC/OV – UFRJ ( Brazil) Co-workers: A.P. Moises (ON/MCT, IAG/USP) M.L. Ferreira (GEMAC/OV – UFRJ) C.R. Rabaca (GEMAC/OV – UFRJ) D.R. Goncalves (IAG/USP)

2. Wavelet Transform: What for?  Wavelet Transform is a mathematical transform that analyzes the spatial frequencies content of images  Wavelet Transform is invariant to scaling (e.g. particularly fit to put into evidence fractal structures)  The decomposition functions have finite support  no artificial high frequencies (like in Fourier Transform)  the frequency information is kept locally, and can easily be visualized  Noise can be removed according to the spatial scales

3. NGC7662 HST F658N ([NII]) raw image

4. NGC7662 HST F658N ([NII]) wavelet processed image

5. Our Sample  14 PN selected from HST archive  Selected on the basis of their size: Angular diameter between 5 and 30 arcsec, in order (1) to have a sufficient size to present detectable substructures (2) to fit into the planetary camera of the WFPC2

6. Hubble 4

7. Hb4 - Halo F656N (H  ) Mass Halo = 1/3 Mass total

8. Hen 2-402 F658N [NII] raw imagewavelet processed image

9. Hen 2-402 F658N [NII]

10. HST Sample – Morphological Structures NP knotsfil.halo shell Hen 2-402 X Hubble 4 XXX Hen 2-389 X? M1 - 46 XX M2 - 43 XXX PC19 XX NGC6790 XXX Hen 2-447 XX NP knotsfil.halo shell BD+30°3639 XX M3 - 35 XXX PN G211.2-03.5 XXX Hen 2-1 XX Hen 2-104 X Hen 2-194 XXX M1 - 30 XX

11. NGC 6891 NOT images – Guerrero et al. 2000

12. NGC 6891 NOT images – Guerrero et al. 2000

13. Application of Wavelet Transform to Temperature Fluctuations  Temperature Fluctuations have historically been put into evidence by the comparison of temperatures derived by different methods, sensitive to different regions (e.g. [OIII] and Bac).  Some direct measurements (e.g. through the mapping of one temperature indicator ([OIII] or Bac) exist (HST imaging or ground based /HST spectroscopy), but generally indicate fairly low Temperature Fluctuations  Temperature fluctuations are however usually described by a unique weighted mean indicator, t 2  t 2 does not yield any indication on the location of the Temperature Fluctations

14. How can Wavelets help ?  Can put into evidence faint structures, as TF from [OIII] maps ratios  T e = f ( TW (I 4363 ) / TW (I 5007 ) )  The amplitude of the variations is however small, possibily at the limit of the detection of Wavelet Transform  Need to find some objective argument to segregate artifacts from true structures

15. NGC6210 Green:significative TF White: no significative TF

16. Region without significative TF Region with significative TF NGC6210 Observadas Simuladas Observadas Simuladas [OIII] 4363/5007 Å Null information: Simulation of intrinsic [OIII] 4363/5007 Å through (5007 + noise)/5007 Å

17. NGC6818 Green:significative TF White: no significative TF

18.  Wavelets allow to put into evidence faint substructures, even in dense areas in planetary nebulae which have not been much explored before.  Though our HST sample is far from complete, much more planetary nebulae seem to present substructures than quoted in previous studies.  This needs however to be verified, since many of these substructures have scales of 1-2 pixels.  Wavelet Transform has a potential to assess temperature fluctuations from [OIII] images ratios. Conclusions