What Is The Minimum Size Of A Planet To Harbor Human Life?

The retention of gases such as oxygen, carbon dioxide, and water depends on the mass and radius of the planet, via its surface gravity. It also depends on the effective temperature, something we can calculate easily. I'd like to quote from an earlier answer of mine:

The planetary equilibrium temperature: $$T = \left(\frac{L_{\odot}(1-a)}{16 \sigma \pi D^2} \right)^{\frac{1}{4}}$$ We can approximation that $L_{\odot} \approx L_{\text{Sun}}=3.846 \times 10^{26}$. As another approximation, $a=0.3$. We also know that $\sigma=5.670 \times 10^{-8}$. Plugging this all in, $$T = \left(\frac{3.846 \times 10^{26}(1-0.3)}{16 \times 5.670 \times 10^{-8}\pi D^2} \right)^{\frac{1}{4}}=9.85856 \times 10^7 \times D^{-\frac{1}{2}}$$ At $D_V$, $D_E$, and $D_M$, this comes out to $$T_V=299.986 \text{ K}$$ $$T_E=254.547 \text{ K}$$ $$T_M=207.515 \text{ K}$$

Next, we can use a chart to figure out what gases will be in the atmosphere. First, however, we must calculate escape velocity.

Seager et al. (2007) proposed a mass-radius relationship for terrestrial planets. $$M\approx\frac{4}{3}\pi R^3\left[1+ \left(1-\frac{3}{5}n \right)\left(\frac{2}{3}\pi R^2\right)^n\right]$$ Here, $n$ is a parameter that depends on the composition of the planet. If we assume a planet composed mainly of silicates (like Earth), then $n\approx0.537$.

Escape velocity is given by $$v_{\text{esc}}=\sqrt{\frac{2GM}{R}}$$ Plugging in our expression, we have $$v_{\text{esc}}=\sqrt{2G\frac{4}{3}\pi R^2\left[1+ \left(1-\frac{3}{5}n \right)\left(\frac{8}{3}\pi R^2\right)^n\right]}$$

Let's go to the chart I mentioned earlier:

^{Image courtesy of Wikipedia user Cmglee under the Creative Commons Attribution-Share Alike 3.0 Unported license.}

At each of the three distances (now temperatures), we have the following requirements to keep all of the gases at the orbital radii of Venus, Earth, and Mars: $$\text{Venus:}\approx4\text{ km/s}$$ $$\text{Earth:}\approx3.5\text{ km/s}$$ $$\text{Mars:}\approx3.25\text{ km/s}$$ Using Wolfram Alpha, I get $$\text{Venus:}\approx177\text{ kilometers}$$ $$\text{Earth:}\approx162\text{ kilometers}$$ $$\text{Mars:}\approx154\text{ kilometers}$$ These numbers are substantially low. They result in part from complicating factors that drastically affect the temperature of the planet, including non-uniform albedo and potential greenhouse effects.

Non-quantitative summary

The main feature of a habitable planet is its composition: the composition of its atmosphere and the planet itself. Certain compounds must be present for human-like life to survive. These compounds can only exist at certain temperatures at certain surface gravities, or else they will escape the planet.

I used a chart and a mass-radius relation, as well as some assumed constants, to get various values for the radius necessary to keep some basic compounds (oxygen and carbon dioxide), assuming a silicate composition. The values differ based on how far away the planet is from its parent star. Note that the planet should remain in the circumstellar habitable zone. The three distances given here are near the inner edge, middle, and outer edge of the Sun's habitable zone.

posted almost 9 years ago

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