1. David Cole, University of Leicester Walter Dehnen; Mark Wilkinson – University of Leicester; Justin Read – ETH Zurich 29 June 2012

2. The Local group dwarfs Intensively studied Identify substructure in cosmological simulations with satellite galaxies Dark matter dominated Deduce the mass structure

3. Measuring the DM density Good kinematic data Should be able to infer the density profile Jeans modelling Problems Walker et al MNRAS 2009

4. Distinct stellar populations Some dSphs have more than one identifiable stellar population. Sculptor data (Amorisco and Evans MNRAS 2011) Use methods which do not require an assumed dark matter profile Surface brightness for metal poor pop. (blue), metal rich pop. (red) CuspedCored

5. Fornax One of the more massive dSphs with 5 Globular Clusters (GCs) Unique in having GCs Sagitarius and Canis Major have some but tidally disrupted (d~24 & 7 kpc) The GCs are old and metal poor Age ~old MW GCs 1 2 3 5 4

6. The Timing Problem DM cusp GCs should fall to the centre of Fornax due to dynamical friction Form a Nuclear Star Cluster – Tremaine et al 1975 No central star cluster seen From Goerdt et al MNRAS 2006 Circular orbits & cusped density profile

7. Is there a failure of dynamical friction? N-body simulations show that dynamical friction ceases at the edge of a density core Harmonic core effect Could explain why we see GCs at a finite distance from centre of Fornax Goerdt et al 2006 Can we improve on this study? From Read et al MNRAS 2006

8. Two issues Long Term timing problem Immediate timing problem

9. Evidence for dynamical friction? Distribution of globular clusters in mass and projected distance from the centre of Fornax Dashed vertical line indicates the stellar half- light radius of the dSph Similar distribution to the stars Trend with mass?

10. Examine using best observations Distance and velocity data cannot place the GCs with sufficient accuracy Distance to Fornax ~138+/- 8 kpc These all overlap => line of sight separation uncertain Alternatives?

11. Statistical method Plausible models GC models: Have projected distances Make kinematics same as stars Have line of sight velocities Uniform distribution of line of sight distances Can create a range of plausible mass models consistent with observations Run thousands of simulations

12. Create mass models Models based on MCMC modelling (Mark Wilkinson to be published) Best fit Cusp (SC) Best fit Core (WC) Best fit Intermediate (IC) Density profile : Also model with large core based on Walker and Penarrubia MNRAS 2011 – Large core (LC)

13. Match to kinematic data Feed back models into the kinematic data as a consistency check BUT matching our models to the kinematics is not the aim of this project Data points from Walker et al MNRAS 2009

14. Results Apo-centric radii after 2 and 10 Gyr. Shaded region indicates the current tidal radius of Fornax. The thin horizontal lines indicate the observed projected radius

15. Density Reduction SC and IC models, the central density profiles are significantly reduced Only model SC is reduction stronger when clusters have reached the core of Fornax

16. Results Orbits with large initial r apo are not significantly affected by dynamical friction Cluster GC3 most affected by dynamical friction, followed by GC4 and GC2, while GC1 and GC5 least affected after 2Gyr Cluster GC3 always reaches the core of Fornax within 10Gyr (except for model LC) Dynamical friction effect at 2Gyr is increasing with the central mass density from model WC to SC, as expected The effect of dynamical fricion after 10Gyr is more similar for the three halo models with weak to steep cusps than after 2Gyr

17. Probability of Clusters Sinking Need quantity for each simulated cluster which would follow a known distribution with orbital phase and projection angle drawn randomly. Use P(RR p | orbit) Our initial distribution of P(RR p | orbit) is non-uniform Weight simulated cluster orbits consistent with uniform sampling.

18. Results Weak Cusp Steep Cusp Colours show different GCs Red – GC1; Blue – GC2; Green – GC3; Magenta – GC4; Cyan – GC5

19. Correlation of p(R R p |orbit) and r apo Correlation between p(R R p |orbit) and r apo at later times Applies over a wide range of eccentricities e<0.4 open symbols; e0.4 crosses; [e=(r apo r peri )/(r apo +r peri )] For models IC and SC, some differentiation between these two groups of initial orbits At t = 2Gyr eccentric orbits smaller r apo because they have smaller initial r peri and hence suffer more dynamical friction) Exception: if the observed R was initially untypically small (when they spend most of their time at large radii).

20. Quantitative estimates Probability (r apo < 2.8kpc) falls in Depends on the mass model and the eccentricity of the initial orbit. Doesnt depend on distribution function

21. Two Solutions Fornax has a large core Fornax has a small core or shallow cusp

22. Where did the GCs originate? If we have an evolving solution GCs at or near tidal radius a Hubble time ago Fits with weak evidence of mass segregation The GCs have not formed within Fornax, but are most likely accreted

23. Caveats Our models all assume a spherical mass distribution for Fornax The tidal field of the Milky Way The inner dynamics of the GCs and tidal interaction with Fornax

24. Large core behaviour In the large core if the GC starts inside the core the orbit moves out (!) to the edge of the core Under investigation Paper by Tremaine and Weinberg 1984 may offer partial explanation time Gyr r kpc Orbit for GC3

25. The Case of GC1 Why should the one cluster vulnerable to tides be on an orbit where it would hardly ever suffer disruption? Steady-state solution: Fornax once had a richer globular- cluster system and we only see the survivors. Evolving solution: low-mass clusters, such as GC1, would not be dragged down much, and there is no need to postulate a large early population of clusters. It is a collisional system and so it has expanded by internal 2-body relaxation => could have had a higher density in the past Gieles et al 2010.

26. Conclusions The more cusped density profiles are much more likely to cause GCs to fall to the centre of a dwarf galaxy For cusped mass models clusters GC3 or GC4 will sink into the centre of Fornax within 1-2Gyr with 90% probability Fornax has a large core and dynamical friction is slow or has stalled a long time ago. Fornax has a small core or shallow cusp and dynamical friction is still ongoing, albeit slowly and the clusters must have been further away from Fornax in the past than today.

27. The cusp/core problem Navarro et al 2010 Oh et al 2008 IC 2574 Observations Theory

28. Large Core model Walker and Penarrubia 2011, ApJ 742, 20 Model as two chemodynamically distinct stellar subcomponents constrain model parameters using MCMC Estimates of mass enclosed at the half-light radius

29. Results after 2 Gyr Initial distribution is uniform in line of sight distance between 0 and 2 kpc (~tidal radius) Bin the GC instantaneous apocentre Colours show different mass models Cyan – Steep Cusp (SC) Red – Intermediate Cusp (IC) Black – Weak Cusp (WC) Green – Large Core (LC)

30. Results after 10 Gyr Uniform line of sight distance distribution Cyan – Steep Cusp (SC) Red – Intermediate Cusp (IC) Black – Weak Cusp (WC) Green – Large Core (LC)