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I'm trying to figure out the maximum possible density for an earth-sized planet before it would become an earth-sized star.

To try to figure this out, I found a reference star, VB 10, known for being among the smallest stars still fully capable of classifying as a star, and compared its features with those of the sun and earth.

Our sun has a radius of about 700,000 km and a mass of about 2E30 kg, giving it a density of around 1.4 g/cm^3. The earth has a radius of about 6,400 km and a mass of about 6E24 kg, giving it a density of around 5.5 g/cm^3. VB 10 has a radius of about 71,000 km and a mass of about 1.5E29 kg, giving it a density of around 100 g/cm^3.

Considering that VB 10 has a radius over 100 times that of the earth, and is very firmly a star at around 18 times the density, the earth could almost certainly be made at least 18 times more dense without becoming a star. But without more examples of high-density low-volume stars to compare, this is a very hard question to answer. Further, it is possible that VB 10 is very far beyond the threshold of becoming a star, meaning that it could be made, say, 100 times less dense and still be fully capable of carrying out fusion, so this method of prediction has some fairly significant flaws.

What does the math concerning conditions for fusion say?

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There are more variables than density that controls beginning of fusion. In particular elemental composition. Earth size planet with density of 5.5 g/cm^3 implies a planet composed of heavy elements which are very unlikely to fuse (and for elements like iron or heavier the fusion reaction are endothermic requiring additional energies far in excess of anything one could hope to achieve inside a planet-sized object). So even if one could compress Earth core to the densities of VB 10 there would be no fusion.

On the other hand if one would take a earth-sized planet worth of light elements (hydrogen mostly) potentially capable of fusion and keep adding material then the planet would become a gas giant long before it starts fusion at its core. Brown dwarves are capable of fusing deuterium if their mass exceeds 13 masses of Jupiter. The density then would be about 15 g/cm^3.

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  • $\begingroup$ It's not that elements such as iron are incapable of fusion; the issue is that fusion of elements heavier than iron is endothermic, and so wouldn't actually help you turn a planet into a star. $\endgroup$ – probably_someone Mar 4 '18 at 19:22
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    $\begingroup$ Agreed, and the energies required would in MeV range, which would not be possible to obtain. But I'll reword the phrasing. $\endgroup$ – A.V.S. Mar 4 '18 at 19:30
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Fusion can happen in an Earth like planet if the mass reaches 8 times that of the sun. That is when heavier elements are fused. However if a Jupiter like planet was to turn into a star the mass needed would be 0.08% of that of the sun. Hydrogen is easier to fuse as it is just one proton. Also only the silicon will be fused for the most part even with an Earth like planet 8 times mass of sun. This is because even this is not powerful enough. Since a planet can not get to 8 times solar mass an Earth like planet can not fuse elements. However there could be a temporary silicon fusion that produces elements like helium which could be fused so 4 to 5 masses is probably enough mass to let indirect silicon fusion to occur by fusion of different elements that might be present on a Earth like planet.

Earth like planet has a small amount of hydrogen(it is very small but still there so there might be enough hydrogen if an Earth like planet were to get bigger.)

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