  A defining prediction of the cold dark matter (CDM) cosmological model is the
existence of a very large population of low mass haloes, down to planet-size
masses. However, their fate as they are accreted onto haloes many orders of
magnitude more massive remains fundamentally uncertain. A number of numerical
explorations have found subhaloes to be very resilient to tides, but resolution
limits make it difficult to explore tidal evolution at arbitrarily low masses.
What are the structural properties of heavily stripped subhaloes? Do tidal
effects destroy low-mass CDM subhaloes? Here we focus on cosmologically
motivated subhaloes orbiting CDM hosts, and show that subhaloes of any initial
mass can be stripped to arbitrarily small mass fractions. We show that previous
numerical results can be reproduced by a simple model that describes tidal
evolution as a progressive `peeling' in energy space and subsequent
re-virialization. Tidal heating can effectively be neglected at all masses,
because its importance i) does not increase for subhaloes of decreasing mass at
accretion, and ii) it decreases as stripping proceeds. This allows us to
predict analytically the structural properties of subhaloes with any initial
mass and arbitrary degrees of mass loss. Under the hypotheses that CDM haloes
have centrally divergent density profiles and approximately isotropic phase
space distributions, our results prove that subhaloes of very low masses are a
robust prediction of the CDM model: as for haloes, CDM subhalo populations
extend and are abundant down to very low masses.
