1. Dynamical Models for Galaxies Observed with SAURON Michele Cappellari
Leiden Observatory

2. Outline Extraction of kinematics: 2D-binning
Why Integral Field Spectroscopy is important?
SAURON example
Next steps

3. Adaptive 2D-binning: The SAURON Data of NGC 2273
Reconstructed image
S/N map
Barred Sa galaxy
The basic adaptive 2D-binning problem:
partition the data into bins having a minimum S/N

4. 2D-binning Requirements
Topological: partition the plane without holes or overlapping bins
Morphological: bins as compact or “round” as possible
Uniformity: minimal S/N scatter

5. 2D-binning by QuadTree Decomposition2x
Regular cells but:
large S/N scatter
border problems
Satisfies Topological and Morphological requirements ONLY

6. Voronoi Tessellation
Definition: each point in a bin is closer to its generator than to any other point
Satisfies Topological requirement ONLY

7. Centroidal Voronoi Tessellation
Cappellari & Copin 2002, astro-ph/
All Topological, Morphological and Uniformity requirements satisfied!

8. NGC 2273 Stellar Mean Velocity Field
Not binned
2D-binned velocity

9. NGC 2273 Stellar Velocity Dispersion Field
Not binned
2D-binned and interpolated

10. SAURON Dynamical Modeling
Requirements
no restriction on form of potential
arbitrary geometry
multiple components
no restriction on distribution function [but f(r,v)0]
full range of velocity anisotropy
no need to know analytic integrals of motion
This can be done with Schwarzschild’s orbit superposition method including all kinematic observables (V,,VP shapes)

11. Galaxy Orbits box short-axis tube inner long-axis tube
Axisymmetric potential
e.g. Cretton et al. (1999)
Triaxial Stäckel potential
e.g. Statler (1987)
inner long-axis tube
outer long-axis tube

12. Catastrophe Theory cusp fold   |x|-2/3   |x|-1/2
no inverse images
two inverse images
cusp
  |x|-2/3
fold
  |x|-1/2
When a smooth surface is projected onto a plane only these two singularities appear (Whitney 1955)

13. Orbital Tori Projection
Regular orbits in 3-dim can be reduced to translations on a 3-torus in 6-dim phase space
(x,v)(J,)
Orbital density is constant on the torus surface
folds
Projection
2D Torus
cusps

14. Catastrophe in the Orbital Surface Brightness
i=80o
i=60o
Symmetry axis
Symmetry axis
Regular orbits!
In an axisymmetric potential
fold
  |x|-1/2
cusp
  |x|-2/3

15. Orbital Surface Brightness
i=80o
i=60o Lz I3 E
Same orbit as before, but with a different colormap

16. Orbital Mean Velocity i=80o (x',y') V(x',y')
No observable contribution on the major/minor axis
(x',y')
V(x',y')

17. Orbital Velocity Dispersion
(x',y')
(x',y')

18. Consequences
The localized contribution of regular orbits to the observables implies:
Integral Field Spectroscopy is needed to constrain the internal dynamics
We can understand some known facts:
Easy to construct self-consistent models (fit the density with the orbital components)
Easy to reproduce any velocity field: higher order moments are needed to constrain the internal dynamics
The DF has to be smooth, otherwise cusps and folds would appear in the galaxy surface brightness

19. The E3 Galaxy M32 Small, inactive companion of Andromeda nebula
Many studies suggest presence of central BH
Best previous study ground-based long-slit data and HST/FOS aperture together with Schwarzschild axisymmetric dynamical models (van der Marel et al. 1998)
Results:
(M/L)I=2.0 ± 0.3
MBH=(3.4 ± 0.7)x106 Mo
55o < i < 90o

20. M32: Dynamical Modeling with SAURON Stellar Kinematics
STIS V  h3 h4 V  h3 h4
We use new data:
SAURON high resolution map of inner 9”x11” (de Zeeuw et al. 2002, MNRAS)
STIS major axis kinematics (Joseph et al. 2001, ApJ)

21. M32: Best-fitting Parameters
Stringent constraints for M/L, MBH, i
MBH in perfect agreement with van der Marel et al. (1998)
3 level
Verolme, Cappellari et al. 2002, MNRAS, in press (astro-ph/ )

22. M32: Importance of 2D Kinematics
Four slits + STIS
SAURON + STIS
3 level
Model parameters and internal structure strongly constrained

23. Next Steps
Get a deeper understanding on how our models compare to the real galaxies
Construct axisymmetric dynamical models: 26 of the 48 E/S0 in the SAURON survey, are consistent with axisymmetry
Construct triaxial dynamical models

24. Expected Results As a function of morphological type and luminosity
Orbital structure, including decoupled components
Intrinsic shapes, mass distribution
Rate of occurrence of gaseous and stellar disks
Star formation history and its connection with dynamical structure
Black hole demographics
Importance of BHs for shaping galaxies