Extended envelopes around Galactic Cepheids. V. Multi-wavelength and time-dependent analysis of IR excess.

We aim to investigate the IR excess of 45 MW Cepheids combining different observables to constrain the presence of CSE. We used the SPIPS algorithm, a robust implementation of the parallax-of-pulsation method that combines photometry, angular diameter, stellar effective temperature, and RV measurements in a global modelling of the pulsation. We obtained new photometric measurements at mid-IR with the VLT/VISIR complemented with literature data. We then compared the mean magnitude from 0.5$\mu$m to 70$\mu$m with stellar atmosphere models to infer the IR excess, which we attribute to the presence of a CSE. We report that at least 29% of our sample have a detected IR excess. We estimated a mean excess of 0.08mag in K and 0.13mag in N. Other Cepheids possibly have IR excess, but they were rejected due to their low detection level compared to a single-star model. We do not see any correlation between the IR excess and the pulsation period as previously suspected for MW Cepheids, but a rather constant trend. We also do not find any correlation between the CO absorption and the presence of a CSE, but rather with the stellar effective temperature, which confirms that the CO features previously reported are mostly photospheric. No bias caused by the presence of the CSE is detected on the average distance estimates from a SPIPS analysis with a fitted colour excess. We also do not find correlation between the presence of IR excess and the evolution stage. We report a fraction of 29% of Cepheids with an IR excess likely produced by the CSE. Longer period Cepheids do not exhibit greater excess than short periods as previously suspected from observations and theoretical dusty-wind models. Other mechanisms such as free-free emission, among others, may be at the origin of their formation. We also show that not fitting the colour excess leads to a bias on the distance estimates in our Galaxy.