Is there a theoretical maximum size for rocky planets?

Estimates vary, but I'll be cautious and say that a radius of roughly two Earth radii is most likely the upper limit for rocky planets.

There are many studies, both theoretical and empirical, that have tried to attack the problem. I'll attempt to summarize the results of a few of them:

• Lammer et al. 2014: This group focused on planets losing their "hydrogen envelopes" - gaseous layers of hydrogen that they may accrete during the early parts of their lives. Their calculations indicate that planets of less than one Earth mass ($M_{\oplus}$) would accumulate envelopes of masses between $2.5 \times 10^{16}$ and $1.5 \times 10^{23}$ kilograms. The latter is about one-tenth of Earth's mass. Planets with masses between $2M_{\oplus}$ and $5M_{\oplus}$ could accumulate envelopes with masses between $7.5 \times 10^{20}$ and $1.5 \times 10^{28}$ kilograms - substantially more massive than Earth! This is the peak envelope mass, though; the group calculated that planets with masses of less than $1M_{\oplus}$ would lose their envelopes within about 100 million years. They found that planets with masses greater than $2M_{\oplus}$ will keep their envelopes, and so become "gas dwarfs" or "mini-Neptunes."
• Lopez & Fortney 2013: Lopez and Fortney worked off of data from Kepler and modeled the radii of planets. They determined that planets with radii of less than $1.5R_{\oplus}$ will become super-Earths, and planets with radii of greater than $2_{\oplus}$ will become mini-Neptunes. That suggests a radius limit of $2R_{\oplus}$, though most terrestrial planets will probably be under $1.5R_{\oplus}$.
• Seager et al. 2008: This group tied mass and radius together based on theoretical calculations. They eventually came to the equation $$M_s \approx \frac{4}{3} \pi R_s^3 \left[1+ \left(1-\frac{3}{5}n \right)\left(\frac{2}{3} \pi R_s^2 \right)^n \right]$$ where $n$ is a certain given parameter and $M_s$ and $R_s$ are the mass and radius scaled by composition-dependent values. It is therefore possible to compare the papers by Lammer et. al. and Lopez and Fortney if $n$ is known. The resulting values are dependent on the material the planet is made of (see Table 3 for examples), but it seems that a pure silicate planet would have an upper limit of $3R_{\oplus}$, while an ocean world could reach $4\text{-}5R_{\oplus}$.

I'd go with about $2R_{\oplus}$ as the upper limit for terrestrial planets, though there may be exceptions in certain extenuating conditions.

That's for planets that form as terrestrial planets from the start. Curiously enough, gas planets can become terrestrial planets by having their outer layers blown away by their parent star, leaving behind an object called a chthonian planet. These "planets" aren't much more than the core of the gas planet. No chthonian planets have been confirmed to exist, but they're possible.

I should add that Samuel also proposed the $2M_{\oplus}$-limit in his answer below.

posted about 9 years ago

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