Barnard's classical habitable-zone estimation relies, I think, on this 2013 paper. But a couple years ago, we were looking into Alfvén systems. This is before we even had planets around Barnard's star, Gliese's #699. It turns out this star - as "GJ 699" - had convection calculations in 2018. For its colour V − Kₛ ≈ 4.99 (thank you copilot), logτ=2.07. Convective turnover should be 124 days based just on their own calc of 130.4/1.05.
We do have to correct the Rossby, given rotation period has since 2018 been updated from 130 to 140 days. That pushes Rossby to 1.13.
Plugging into Equation 5, 17.46 * (1.13^0.41) = 18.36 Sol radius maximum; 17.46 * (1.13^0.03) = 17.5 Sol radius minimum. In terms of semimajor, 0.08-0.085 AU. Barnard e - outer, 0.0382 AU - rolls well inside the minimum.
Like Barnard's inner worlds didn't have enough problems. A ten-day "HZ" planet would still be fried. 22.6-to-42-day on the other hand standeth a chance out to 0.13 AU.
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