Too close and the water boils; too far and it freezes. Around every star is a thin green ring where a world can hold liquid water — the Goldilocks band. Brighten the star and the ring races outward; dim it and the ring crowds in. Set a planet down and ask the only question that matters: is it in the ring?
The habitable zone scales as the square root of luminosity: inner and outer edges r ∝ √(L/L☉) — because a planet's equilibrium temperature depends on the flux L/r², so holding temperature fixed sends r ∝ √L. Using the classic conservative bounds (~0.95–1.37 AU for the Sun), the edges are computed live and a placed planet is judged in-or-out of the band. Brighten the star and the whole ring moves out as √L, exactly as drawn.
The √L scaling and the flux argument are real, but the exact edges of a habitable zone are debated (cloud feedback, atmosphere, the 'moist greenhouse' — estimates range widely) and 'habitable' here means only 'liquid water possible at the surface', not inhabited. The green ring is a first-order equilibrium estimate, not a guarantee of a living world.