An uncertainty principle for star formation. I. Why galactic star
  formation relations break down below a certain spatial scale
  Galactic scaling relations between the (surface densities of) the gas mass
and the star formation (SF) rate are known to develop substantial scatter or
even change form when considered below a certain spatial scale. We quantify how
this behaviour should be expected due to the incomplete statistical sampling of
independent star-forming regions. Other included limiting factors are the
incomplete sampling of SF tracers from the stellar initial mass function and
the spatial drift between gas and stars. We present a simple uncertainty
principle for SF, which can be used to predict and interpret the failure of
galactic SF relations on small spatial scales. This uncertainty principle
explains how the scatter of SF relations depends on the spatial scale and
predicts a scale-dependent bias of the gas depletion time-scale when centering
an aperture on gas or SF tracer peaks. We show how the scatter and bias are
sensitive to the physical size and time-scales involved in the SF process (such
as its duration or the molecular cloud lifetime), and illustrate how our
formalism provides a powerful tool to constrain these largely unknown
quantities. Thanks to its general form, the uncertainty principle can also be
applied to other astrophysical systems, e.g. addressing the time-evolution of
star-forming cores, protoplanetary discs, or galaxies and their nuclei.
