# What is the theoretical maximum variability a pulsating red giant can have such that a habitable planet can stay habitable for long periods of time?

Red giants have luminosities of $$\sim 3000 L_\odot$$. According to the inverse square law, the habitable zone must be $$1^{+1.5}_{-0.2} \cdot \sqrt{3000} \approx 54.772^{+82.158}_{-10.954} \text{AU}$$ away from the star. If the star is exhibiting variability, its luminosity may change by about/over $$50\%$$. This will make the habitable zone vary by a factor of $$2\sqrt{0.5}=\sqrt2 \approx 1.414$$. So do any stable orbits (in terms of habitability) exist for planets orbiting such a red giant star, assuming the variability timescale is within the orbital period of such a planet?