  The evolution of the centre-of-mass wave-function for a mesoscopic particle
according to the Schr\"odinger-Newton equation can be approximated by a
harmonic potential, if the wave-function is narrow compared to the size of the
particle. It was noticed by Colin et al. [Phys. Rev. A, 93, 062102 (2016)]
that, in the regime where self-gravitational effects are weak, intermediate and
wider wave-functions may be approximated by a harmonic potential as well, but
with a width dependent coupling, leading to a time evolution that is determined
only by a differential equation for the width of a Gaussian wave-function as a
single parameter. Such an approximation results in considerably less
computational effort in order to predict the self-gravitational effects on the
wave-function dynamics. Here, we provide an alternative approach to this kind
of approximation, including a rigorous derivation of the equations of motion
for an initially Gaussian wave packet, under the assumption that its shape is
conserved. Our result deviates to some degree from the result by Colin et al.,
specifically in the limit of wide wave-functions.
