  Infrared observations of Sgr A$^*$ and M87$^*$ are incompatible with the
assumption that these sources have physical surfaces in thermal equilibrium
with their accreting environments. In this paper we discuss a general
parametrization of the energy balance in a horizonless object, which permits to
quantify how close a horizonless object is in its behavior to a black hole, and
analyze the timescale in which its surface can thermalize. We show that the
thermalization timescale is unbounded, growing large for objects that mimic
closely the behavior of a black hole (and being infinite for the latter). In
particular, the thermalization timescale is proportional to the time that
energy spends inside the horizonless object due to propagation and interactions
with the bulk. Hence, these observations can be used to quantitatively restrict
the dynamical behavior of horizonless objects, without being able to discard
the existence of a physical surface.
