Dynamical friction vs. subhalo heating in Cold Dark Matter haloes
The orbital evolution of massive stellar systems embedded in dark matter haloes is governed by competing processes. Dynamical friction removes orbital energy, whereas fluctuations in the gravitational field generated by dark matter subhaloes inject energy through stochastic heating. We investigate the balance between these two mechanisms using analytical arguments and numerical experiments. We show that, for a given host halo and subhalo population, there exists a critical mass $M_{\rm crit}$ at which heating and friction balance. Clusters with masses $M_{\rm cl}>M_{\rm crit}$ lose orbital energy and sink towards the centre of the halo, whereas those with $M_{\rm cl}<M_{\rm crit}$ gain energy and expand outwards. At $M_{\rm cl}\simeq M_{\rm crit}$, the two processes balance, leading to a cluster population that, on average, neither sinks nor expands within the host halo. In CDM haloes, the critical mass is primarily controlled by the upper end of the subhalo mass function. Since massive subhaloes are intrinsically rare, the balance between heating and friction shows substantial halo-to-halo scatter. For dark matter haloes in the mass range associated with dwarf spheroidal galaxies (dSphs), we find $M_{\rm crit}\gtrsim 10^5M_\odot$, comparable to the masses of globular clusters in the Fornax dSph. Stochastic heating by dark substructure may therefore significantly delay the orbital decay of globular clusters and help alleviate the Fornax timing problem.