Cusp or core? Revisiting the globular cluster timing problem in Fornax

Abstract

We use N-body simulations to revisit the globular cluster (GC) ‘timing problem’ in the Fornax dwarf spheroidal (dSph). In agreement with earlier work, we find that, due to dynamical friction, GCs sink to the centre of dark matter haloes with a cuspy inner density profile but ‘stall’ at roughly 1/3 of the core radius (rcore) in haloes with constant-density cores. The time-scales to sink or stall depend strongly on the mass of the GC and on the initial orbital radius, but are essentially the same for either cuspy (Navarro–Frenk–White) or cored haloes normalized to have the same total mass within rcore. Arguing against a cusp on the basis that GCs have not sunk to the centre is thus no different from arguing against a core, unless all clusters are today at ∼(1/3)rcore⁠. This would imply a core radius exceeding ∼3 kpc, much larger than seems plausible in any core-formation scenario. (The average projected distance of Fornax GCs is 〈RGC, Fnx〉 ∼ 1 kpc and its effective radius is ∼700 pc.) A simpler explanation is that Fornax GCs have only been modestly affected by dynamical friction, as expected if clusters started orbiting at initial radii of the order of ∼1–2 kpc, just outside Fornax’s present-day half-light radius but well within the tidal radius imprinted by Galactic tides. This is not entirely unexpected. Fornax GCs are significantly older and more metal-poor than most Fornax stars, and such populations in dSphs tend to be more spatially extended than their younger and more metal-rich counterparts. Contrary to some earlier claims, our simulations further suggest that GCs do not truly ‘stall’ at ∼0.3rcore⁠, but rather continue decaying towards the centre, albeit at reduced rates. We conclude that dismissing the presence of a cusp in Fornax based on the spatial distribution of its GC population is unwarranted.