  We recently proposed that the star-forming potential of dense molecular
clouds in the Central Molecular Zone (CMZ, i.e. the central few 100 pc) of the
Milky Way is linked to their orbital dynamics, potentially giving rise to an
absolute-time sequence of star-forming clouds. In this paper, we present an
orbital model for the gas stream(s) observed in the CMZ. The model is obtained
by integrating orbits in the observed gravitational potential and represents a
good fit to the distribution of dense gas, reproducing all of its key
properties. The orbit is also consistent with observational constraints not
included in the fitting process, such as the velocities of Sgr B2 and the
Arches and Quintuplet clusters. It differs from previous models: (1) the orbit
is open rather than closed due to the extended mass distribution in the CMZ,
(2) its orbital velocity is twice as high as in previous models, and (3) Sgr
A$^*$ coincides with the focus of the (eccentric) orbit rather than being
offset. Our orbital solution supports the scenario in which the dust ridge
between G0.253+0.016 ('the Brick') and Sgr B2 represents an absolute-time
sequence of star-forming clouds, triggered by the tidal compression during
their recent pericentre passage. We position the clouds on a common timeline
and find that their pericentre passages occurred 0.30-0.74 Myr ago. Given their
short free-fall times (0.3-0.4 Myr), the quiescent cloud G0.253+0.016 and the
vigorously star-forming complex Sgr B2 are separated by a single free-fall time
of evolution, implying that star formation proceeds rapidly once collapse has
been initiated. We provide several quantitative predictions of our model and
conclude with a discussion of the model in the Galactic context, highlighting
its relation to large-scale gas accretion, the dynamics of the bar, the $x_2$
orbital family, and the origin of the Arches and Quintuplet clusters.
(Abridged)
