  Context: Mass loss plays a dominant role in the evolution of low mass stars
while they are on the Asymptotic Giant Branch (AGB). The gas and dust ejected
during this phase are a major source in the mass budget of the interstellar
medium. Recent studies have pointed towards the importance of variations in the
mass-loss history of such objects. Aims: By modelling the full line profile of
low excitation CO lines emitted in the circumstellar envelope, we can study the
mass-loss history of AGB stars. Methods: We have developed a non-LTE radiative
transfer code, which calculates the velocity structure and gas kinetic
temperature of the envelope in a self-consistent way. The resulting structure
of the envelope provides the input for the molecular line radiative
calculations which are evaluated in the comoving frame. The code allows for the
implementation of modulations in the mass-loss rate. This code has been
benchmarked against other radiative transfer codes and is shown to perform well
and efficiently. Results: We illustrate the effects of varying mass-loss rates
in case of a superwind phase. The model is applied to the well-studied case of
VY CMa. We show that both the observed integrated line strengths as the
spectral structure present in the observed line profiles, unambiguously
demonstrate that this source underwent a phase of high mass loss (~ 3.2E-4
Msun/yr) some 1000 yr ago. This phase took place for some 100 yr, and was
preceded by a low mass-loss phase (~ 1E-6 Msun/yr) taking some 800 yr. The
current mass-loss rate is estimated to be in the order of 8E-5 Msun/yr.
Conclusions: In this paper, we demonstrate that both the relative strength of
the CO rotational line profiles and the (non)-occurrence of spectral structure
in the profile offer strong diagnostics to pinpoint the mass-loss history.

Astrophysics