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We recently proposed that the star-forming potential of dense molecularclouds in the Central Molecular Zone (CMZ, i.e. the central few 100 pc) of theMilky Way is linked to their orbital dynamics, potentially giving rise to anabsolute-time sequence of star-forming clouds. In this paper, we present anorbital model for the gas stream(s) observed in the CMZ. The model is obtainedby integrating orbits in the observed gravitational potential and represents agood fit to the distribution of dense gas, reproducing all of its keyproperties. The orbit is also consistent with observational constraints notincluded in the fitting process, such as the velocities of Sgr B2 and theArches and Quintuplet clusters. It differs from previous models: (1) the orbitis 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) SgrA$^*$ coincides with the focus of the (eccentric) orbit rather than beingoffset. Our orbital solution supports the scenario in which the dust ridgebetween G0.253+0.016 ('the Brick') and Sgr B2 represents an absolute-timesequence of star-forming clouds, triggered by the tidal compression duringtheir recent pericentre passage. We position the clouds on a common timelineand find that their pericentre passages occurred 0.30-0.74 Myr ago. Given theirshort free-fall times (0.3-0.4 Myr), the quiescent cloud G0.253+0.016 and thevigorously star-forming complex Sgr B2 are separated by a single free-fall timeof evolution, implying that star formation proceeds rapidly once collapse hasbeen initiated. We provide several quantitative predictions of our model andconclude with a discussion of the model in the Galactic context, highlightingits 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)

The dynamical evolution of molecular clouds near the Galactic Centre - I. Orbital structure and evolutionary timeline