  When a fast moving star or a protostellar jet hits an interstellar cloud, the
surrounding gas gets heated and illuminated: a bow shock is born which
delineates the wake of the impact. In such a process, the new molecules that
are formed and excited in the gas phase become accessible to observations. In
this article, we revisit models of H2 emission in these bow shocks. We
approximate the bow shock by a statistical distribution of planar shocks
computed with a magnetized shock model. We improve on previous works by
considering arbitrary bow shapes, a finite irradiation field, and by including
the age effect of non-stationary C-type shocks on the excitation diagram and
line profiles of H2. We also examine the dependence of the line profiles on the
shock velocity and on the viewing angle: we suggest that spectrally resolved
observations may greatly help to probe the dynamics inside the bow shock. For
reasonable bow shapes, our analysis shows that low velocity shocks largely
contribute to H2 excitation diagram. This can result in an observational bias
towards low velocities when planar shocks are used to interpret H2 emission
from an unresolved bow. We also report a large magnetization bias when the
velocity of the planar model is set independently. Our 3D models reproduce
excitation diagrams in BHR71 and Orion bow shocks better than previous 1D
models. Our 3D model is also able to reproduce the shape and width of the broad
H2 1-0S(1) line profile in an Orion bow shock.
