  Using analytic calculations and N-body simulations we show that in constant
density (harmonic) cores, sinking satellites undergo an initial phase of very
rapid (super-Chandrasekhar) dynamical friction, after which they experience no
dynamical friction at all. For density profiles with a central power law
profile of log-slope, $-\alpha$, the infalling satellite heats the background
and causes $\alpha$ to decrease. For $\alpha < 0.5$ initially, the satellite
generates a small central constant density core and stalls as in the $\alpha =
0$ case.
  We discuss some astrophysical applications of our results to decaying
satellite orbits, galactic bars and mergers of supermassive black hole
binaries. In a companion paper we show that a central constant density core can
provide a natural solution to the timing problem for Fornax's globular
clusters.
