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Galactic scaling relations between the (surface densities of) the gas massand the star formation (SF) rate are known to develop substantial scatter oreven change form when considered below a certain spatial scale. We quantify howthis behaviour should be expected due to the incomplete statistical sampling ofindependent star-forming regions. Other included limiting factors are theincomplete sampling of SF tracers from the stellar initial mass function andthe spatial drift between gas and stars. We present a simple uncertaintyprinciple for SF, which can be used to predict and interpret the failure ofgalactic SF relations on small spatial scales. This uncertainty principleexplains how the scatter of SF relations depends on the spatial scale andpredicts a scale-dependent bias of the gas depletion time-scale when centeringan aperture on gas or SF tracer peaks. We show how the scatter and bias aresensitive to the physical size and time-scales involved in the SF process (suchas its duration or the molecular cloud lifetime), and illustrate how ourformalism provides a powerful tool to constrain these largely unknownquantities. Thanks to its general form, the uncertainty principle can also beapplied to other astrophysical systems, e.g. addressing the time-evolution ofstar-forming cores, protoplanetary discs, or galaxies and their nuclei.

An uncertainty principle for star formation. I. Why galactic star formation relations break down below a certain spatial scale