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

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 center of dark matter halos with a cuspy inner density profile but ``stall'' at roughly 1/3 of the core radius ($r_{\rm core}$) in halos with constant-density cores. The timescales 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 (NFW) or cored halos normalized to have the same total mass within $r_{\rm core}$. Arguing against a cusp on the basis that GCs have not sunk to the center is thus no different from arguing against a core, unless all clusters are today at $\sim (1/3)\, r_{\rm core}$. This would imply a core radius exceeding $\sim 3$ kpc, much larger than seems plausible in any core-formation scenario. (The average projected distance of Fornax GCs is $\langle R_{\rm GC,Fnx}\rangle\sim 1$ kpc and its effective radius is $\sim 700$ 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 order $\sim 1$-$2$ 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 $\sim 0.3\, r_{\rm core}$, but rather continue decaying toward the center, 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.