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)

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