1. The differential Tully-Fisher relation for spiral galaxies Irina Yegorova SISSA, Trieste, Italy

2. Outline: Observations of spiral galaxies The standard Tully-Fisher relation Differential Tully-Fisher relation

6. Evidence of dark matter comes from: WMAP X-Ray gravitational lensing rotation curves

7. blue shift red shift

10. The Tully-Fisher (TF) relation is an empirically established correlation between the luminosity L of a spiral galaxy and its rotational velocity V (Tully-Fisher, 1977)

11. New method of determining Distances to galaxies R.B.Tully and J.R.Fisher, A&A, 54. 661-673, 1977

12. TF-relation has two important applications: 1. It is used to obtain cosmological distances M=m - 5logD - 25 2. It can be used for studying the dynamical properties and the evolution of galaxies

14. Physical basis of the TF-relation From the equation of centrifugal equilibrium we get: V 0 – representative velocity, M - total mass, R c – characteristic radius of luminous matter  - structural parameter depending on the shape of the mass distribution

15. The first equation can be written in this form: This equation can be written in the form of the Tully-Fisher relation:

16. Differential Tully-Fisher relation

17. Vmax

18. 1 st sample: 967 spiral galaxies Mathewson (1992) 2 nd sample: 304 spiral galaxies Courteau (1997) 86 galaxies selected for analysis 3 sample: 329 spiral galaxies Vogt (2004) 81 galaxies selected for analysis Samples:

19. 1. Mathewson sample: R R/R opt ; bin=0.2 2. Courteau sample: R R/R d ; bin=0.2 3. Vogt sample: R R/R d ; bin=0.2 Ropt=3.2Rd, where R_d is the disk exponential length-scale, for Freeman (exponential) disk this corresponds to the 25 B-mag/arcsec^2 photometric radius.

20. 1 st sample: TF-relation for 967 galaxies

22. 2 nd sample: TF-relation for 86 galaxies

23. Slope of the TF-relation M B = a i + b i log V(R i )

24. Gap in the slopes of 2 samples: The slope of TF-relation is related to k k=0.1 for I band k=0.3 for R band No dark matter function of the band

25. Physical meaning of the slope The slope of the TF-relation steadily rises with distance due to the fact that the fractional amount of the dark matter in galaxies changes with the radius.  increases with R  decreases with L this has influence on the slope

26. Scatter of the TF-relation

27. TF-relation using Vmax slope=-5.54; scatter=0.49; number of galaxies=83 slope=-7,579; scatter=0,328; number of galaxies=843

28. Main results: 1.We found the new method Differential TF relation. 2. The slope decreases monotonically with the distance, while the scatter increases with distance. This implies the presence of a non luminous mass component (DM) whose dynamical importance, with respect to the stellar disk (baryonic matter) increases with radius. 3. We found small scatter in the TF-relation. This implies that galaxies have similar physical characteristics. 4. Minimum of the scatter 0.15 (1 st sample), 0.24 (2 nd sample), 0.29 (3 sample) at 2.2R/R d at the position of the maximum disk contribution We are planning to do mass modeling for these galaxies to reproduce the observable slope and scatter in the TF-relation

29. Work in progress