# Habitable zones around other stars

I have a question about measuring the boundaries of habitable zones on other planets.

Is it okay to assume that, if Sun's habitable zone starts at a distance $R_0$ and its luminosity is $L_0$, we can calculate any other star's with luminosity $L$ habitable zone's inner boundary as $$R= R_0 \sqrt{\frac{L}{L_0}}$$?

Formula was derived from $$F=\frac{L}{4\pi R^2}$$ where $F$ is the Flux, $R$ is the distance to the star. I assume that the Flux at the boundary should remain the same as it was in the solar system, thus $F_1=F_2$;

If thpse were incorrect assumptions I would like to know what am I missing.

Thank you!

• I see no problem with it, and Wikipedia implies you're correct. Nov 22, 2014 at 0:21

This looks fine, BUT, indeed have a good read of the wikipedia pages on the topic of habitable zones to see all the complications there are in deducing where around a star the conditions may be "habitable". In addition I recommend, if you are serious about this calculation, reading the short article by Kane & Gelino (2012) and visiting the accompanying Habitable Zone Gallery.

The formula you have gives a reasonable baseline and estimate, but should probably not be extrapolated too far away from solar-like stars.

Complications include:

Do you want your habitable zone to be habitable for a particular length of time? Massive stars change their luminosity fast.

You are assuming a circular orbit, but planet orbits may be elliptical.

Your calculation assumes that your planet has a similar atmosphere (and therefore gravity etc.) as the Earth. Different atmospheres lead to different surface temperatures. Low metallicity stars will have planets without any carbon dioxide! There are also external factors to do with stellar magnetic activity that can drastically influence atmospheres (and have done so in our solar system); that means stars of different type and age could have altered habitable zones.

Very small stars would have very close-in habitable zones, but then tidal effects could be very important.

Both very massive stars and low mass stars (especially when younger) can have strong ultraviolet radiation fields that may preclude life (as we know it).

The calculation assumes that all the heat required comes from the star. But it could also come from the radioactive decay of rocks or by tidal heating in the case of a moon orbiting a larger planet (think Io, Europa).

• Yeah, I know I'm making heavy assumptions that life on the other star would be similar or exact to ours, but otherwise I think calculations become increasingly sophisticated. As for massive stars I think that O, B, A spectral type ones should be ruled out at all because of their short life spans. As for the eccentricity of the orbits, I kind of need to look into that sep1arately. Thank you for the articles! Nov 22, 2014 at 10:22

Let me see if I understand the derivation.

$$F=\frac{L}{4 \pi R^2}$$ becomes $$F_\odot=\frac{L_\odot}{4 \pi R_\odot^2}$$ and $$F_{\text{ other star}}=\frac{L_{\text{ other star}}}{4 \pi R_{\text{ other star}}^2}$$ and so setting them equal means $$\frac{L_\odot}{4 \pi R_\odot^2}=\frac{L_{\text{ other star}}}{4 \pi R_{\text{ other star}}^2}$$ and $$\frac{L_\odot}{R_\odot^2}=\frac{L_{\text{ other star}}}{R_{\text{ other star}}^2}$$ We re-arrange to get $$\frac{R_{\text{ other star}}}{R_\odot^2}=\frac{L_{\text{ other star}}}{L_\odot}$$ $$\frac{R_{\text{ other star}}}{R_\odot}=\sqrt{\frac{L_{\text{ other star}}}{L_\odot}}$$ which is your equation $$R_{\text{ other star}}=R_\odot \sqrt{\frac{L_{\text{ other star}}}{L_\odot}}$$ So the derivation seems to check out mathematically.

Logically, you also seem to be fine. It makes total sense that the luminous flux should be equal in both cases, and Wikipedia agrees with you twice, here:

Whether a body is in the circumstellar habitable zone of its host star is dependent on the radius of the planet's orbit (for natural satellites, the host planet's orbit), the mass of the body itself, and the radiative flux of the host star.

and here

Astronomers use stellar flux and the inverse-square law to extrapolate circumstellar-habitable-zone models created for the Solar System to other stars.

I think you're fine.