 Solution

The habitable zone is the range of orbital distances where an appropriate planetary atmosphere could sustain liquid water at the surface; it is not a guarantee of inhabited or even habitable conditions for every planet. At a fixed habitable incident flux, $a_{\mathrm{HZ}}\propto L_{\star }^{1/2}$. With the supplied $L_{\star }\propto M_{\star }^3$, Kepler's third law givesTaking Earth as a rough central reference, a $0.1M_{\odot }$ star has $a_{\mathrm{HZ}}\sim 0.0316\mathrm{AU}$ and $P_{\mathrm{HZ}}\sim 6.5$ days. Its six-day, Earth-sized companion receives about $1.11$ times terrestrial irradiance and could therefore lie in the habitable zone in this simplified model. Atmosphere, stellar activity, tidal locking and volatile retention remain unknown. The six-day companion around the Sun-like star receives about $239$ times terrestrial irradiance and is approximately Jupiter-sized; it is not a plausible surface-water Earth analogue. The stellar spectral dependence of actual habitable-zone edges is omitted by the simple scaling.