Stellar winds can affect gas dynamics in debris disks and create observable belt winds

Context: Gas is now detected in many extrasolar systems around mature stars aged between 10 Myr to $\sim$ 1 Gyr with planetesimal belts. Gas in these mature disks is thought to be released from planetesimals and has been modelled using a viscous disk approach. At low densities, this may not be a good assumption as the gas could be blown out by the stellar wind instead. Methods: We developed an analytical model for A to M stars that can follow the evolution of gas outflows and target when the transition occurs between a disk or a wind. The crucial criterion is the gas density for which gas particles stop being protected from stellar wind protons impacting at high velocities on radial trajectories. Results: We find that: 1) Belts of radial width $\Delta R$ with gas densities $< 7 \, (\Delta R/50 {\rm \, au})^{-1}$ cm$^{-3}$ would create a wind rather than a disk, which would explain the recent outflowing gas detection in NO Lup. 2) The properties of this belt wind can be used to measure stellar wind properties such as their densities and velocities. 3) Debris disks with low fractional luminosities $f$ are more likely to create gas winds, which could be observed with current facilities. Conclusions: The systems containing low gas masses such as Fomalhaut or TWA 7 or more generally, debris disks with fractional luminosities $f \lesssim 10^{-5} (L_\star/L_\odot)^{-0.37} $ or stellar luminosity $\gtrsim 20 \, L_\odot$ (A0V or earlier) would rather create gas outflows (or belt winds) than gas disks. Gas observed to be outflowing at high velocity in the young system NO Lup could be an example of such belt winds. The detection of these gas winds is possible with ALMA (CO and CO$^+$ could be good wind tracers) and would allow us to constrain the stellar wind properties of main-sequence stars, which are otherwise difficult to measure.