CO emission and variable CH and CH

Context. The runaway star HD 34078, initially selected to investigate small scale structure in a foreground diffuse cloud, has been shown to be surrounded by highly excited H2, the origin of which is unclear. Aims. We first search for an association between the foreground cloud and HD 34078. Second, we extend previous investigations of temporal absorption line variations (CH, CH+, H2) in order to better characterize them and understand their relation to small-scale structure in the molecular gas. Methods. We have mapped the $^{12}\mathrm{CO}(2$–1) emission at 12´´ resolution around HD 34078's position, using the 30 m IRAM antenna. The follow-up of CH and CH+ absorption lines has been extended over 5 more years: 26 visible spectra have been acquired since 2003 at high or intermediate resolution. In parallel, CH absorption towards the reddened star $\zeta~{\rm Per}$ has been monitored to check the instrumental stability and homogeneity of our measurements. Three more FUSE spectra have been obtained to search for N(H2) variations. Results. CO observations show a pronounced maximum near HD 34078's position, clearly indicating that the star and diffuse cloud are associated. The optical spectra confirm the reality of strong, rapid and correlated CH and CH+ fluctuations (up to 26% for N(CH+) between 2007 and 2008). On the other hand, N(H2, $J=0$) has varied by less than 5% over 4 years, indicating the absence of marked density structure at scales below 100 AU. We also discard N(CH) variations towards $\zeta~{\rm Per}$ at scales less than 20 AU. Conclusions. Observational constraints from this work and from 24 μm dust emission appear to be consistent with H2 excitation but inconsistent with steady-state bow shock models and rather suggest that the shell of compressed gas surrounding HD 34078 or lying at the boundary of a small foreground clump is seen at an early stage of the interaction. The CH and CH+ time variations as well as their high abundances are likely due to chemical structure in the shocked gas layer located at the stellar wind/ambient cloud interface. Finally, the lack of variation in both N(H2, $J=0$) towards HD 34078 and N(CH) towards $\zeta~{\rm Per}$ suggests that quiescent molecular gas is not subject to pronounced small-scale structure