How Wide Would the Habitable Zone Be in a Singular Red Giant System?

Question

Astrobiologists looking for signs of life outside Earth usually take excitement over planets orbiting red giant stars because a larger, brighter star usually means a farther-away but still wider habitable zone, a slice of space where liquid water would be possible.

To put this in perspective, a single red giant can have a luminosity ranging anywhere between 1,000 and 10,000 times brighter than our sun. Would that mean that a habitable zone in a solar system orbiting only one red giant would be 1,000 to 10,000 times as wide as our own?

Answer 1

Habitable zones have a width that scales as the square root of the luminosity. So a 10000 solar luminosity star will have a zone about 100 times wider.

However, the temporal width is another thing: the luminosity changes relatively fast, so planets will not be in the zone over long timescales.

(The reason for the square root dependence is that a piece of surface at semimajor axis $a$ gets $L/4\pi a^2$ Watt of incoming energy, and if a blackbody with emissivity $\epsilon$ radiates away $\sigma \epsilon T^4$. Energy balance gives $L/4\pi a^2 = \sigma \epsilon T^4$, or rearranged, $a = \sqrt{L/4\pi \sigma \epsilon T^4}$, proportional to $\sqrt{L}$. The life zone is given by the range where $T$ allows liquid water. In practice this is more complex and fuzzy since it all depends on complex atmospheric properties, but as a first approximation it feels OK.)