  Spherically symmetric, time-periodic oscillatons -- solutions of the
Einstein-Klein-Gordon system (a massive scalar field coupled to gravity) with a
spatially localized core -- are investigated by very precise numerical
techniques based on spectral methods. In particular the amplitude of their
standing-wave tail is determined. It is found that the amplitude of the
oscillating tail is very small, but non-vanishing for the range of frequencies
considered. It follows that exactly time-periodic oscillatons are not truly
localized, and they can be pictured loosely as consisting of a well
(exponentially) localized nonsingular core and an oscillating tail making the
total mass infinite. Finite mass physical oscillatons with a well localized
core -- solutions of the Cauchy-problem with suitable initial conditions -- are
only approximately time-periodic. They are continuously losing their mass
because the scalar field radiates to infinity. Their core and radiative tail is
well approximated by that of time-periodic oscillatons. Moreover the mass loss
rate of physical oscillatons is estimated from the numerical data and a
semi-empirical formula is deduced. The numerical results are in agreement with
those obtained analytically in the limit of small amplitude time-periodic
oscillatons.
