# How large could a planetary diamond be?

Back in the days when men were real men, women were real women, and small, furry creatures from Alpha Centauri were real small furry creatures from Alpha Centaury, an obscenely rich woman decided she wanted to have the biggest diamond of all.

Realizing that this would mean a diamond of planetary size, she went to Magrathea and ordered it there. She had only two conditions: It should be a perfect single-crystalline diamond, and it should be impossible to make a larger one, because trying to do so would make it stop being a perfect diamond. Of course, Magrathea delivered what was ordered.

Now as long as the diamond planet was in the hyperspace construction halls of Magrathea, all natural laws could be effectively put out of force as needed, but as soon as the planet was put into normal space, it was under the full force of the laws of physics.

Now my question: How large was the diamond which was ultimately delivered by Magrathea?

• I believe diamond is the tightest possible packing so the only limit I see is when electron degeneracy makes it cease to be a compound at all--you're talking a decent portion of a stellar mass. Commented Mar 12, 2015 at 20:36
• Probably the point where it would collapse under its own gravity. No idea how to calculate that though. Commented Mar 12, 2015 at 20:38
• @DanSmolinske - not quite...diamonds compress into something else around 10 million atmospheres of pressure. Commented Mar 12, 2015 at 22:04
• I'll leave this for someone else to use, but carbon planets, while hypothetical, seem to very much suit what you're looking for. Oh, and +1 for the Magrathea premise. Tell Slarty I said hi. Commented Mar 12, 2015 at 22:12
• This seems like a question better suited at Physics. Commented Mar 13, 2015 at 1:14

Between 253,000km and 573,000km in radius.

It certainly would not be either of those values, but between them, closer to the upper end. I'll say just under 500,000km in radius. About 0.72 times the size of the Sun.

It would be about 1.3 solar masses of diamond. Large enough that it doesn't begin the CNO cycle. This is also assuming that the constructed nature of the planet means it does not have extreme temperatures at its core.

The mass is the easier part. The compression of diamond is the hard part.

Carbon stays as diamond under very high pressure.

It's the least compressible material known. However, in a recent study it was pressurized up to five terapascals. Turns out, it does compress.

Just over one solar mass of carbon would have an internal pressure of about 70.96 terapascals. We don't have data for what carbon does at those pressures. However, at five terapascals is compresses to about three times its standard density. With some very dirty street-fighting-math we can extrapolate the polynomial and guesstimate that, at 71 terapascals, the density will be up to $38 {{g}\over{cm^3}}$.

Since I really don't want to set up an integral where the density is a function of pressure as a function of radius, I'll just look at the extremes. A planet of density $38 {{g}\over{cm^3}}$ will be about $6.805×10^{25}$ cubic meters in volume. Or about 0.048 times the volume of the Sun and 48 times the volume of Jupiter and a radius of 253,000km. A planet of density $3.513 {{g}\over{cm^3}}$ will be about $7.361×10^{26}$ cubic meters in volume. Or about 0.52 times the volume of the Sun and 510 times the volume of Jupiter and a radius of 560,000km.

• Could you add the calculations/formulas used for this? Commented Mar 12, 2015 at 20:52
• What about the point at which the carbon begins to undergo fusion due to its mass? I have a suspicion that this is a little larger than that... and I rather think that it would not be a pure diamond at that point. Commented Mar 12, 2015 at 21:42
• @MontyWild Yep, good point. I've recalculated. Commented Mar 12, 2015 at 22:06
• @samuel - we've got conflicting numbers...I found a paper that suggest around 10 million atmosphere is the upper limit before the carbon rearranges into something other than diamond. A diamond with the mass of jupiter would see around 100 million atmospheres of pressure at it's center, and that would no longer be diamond rather some other exotic carbon compound Commented Mar 12, 2015 at 22:16
• @Samuel - it's somewhat symantics, but the question states " It should be a perfect single-crystalline diamond, and it should be impossible to make a larger one, because trying to do so would make it stop being a perfect diamond." When does the compression in the core start to prevent it from being the perfect diamond? Commented Mar 12, 2015 at 23:05

I think the answer is significantly smaller than you may think...ultimately a diamond is not the most compressed form of carbon.

In 2014 an experiment to recreate pressures at Jupiter core was preformed...

http://www.astrobio.net/news-brief/bright-like-diamond-lasers-compressed-carbon-recreate-jupiters-core/

quoting the article:

“At progressively higher pressures, the carbon atoms will change their configuration,” said Smith. “At over 10 million atmospheres of pressure, the carbon atoms are predicted to rearrange themselves so they would no longer be in a diamond structure, but rather they will assume a different [higher density] structural arrangement. It will still be carbon, but, so the theory predicts, the crystal structure and the mechanical and chemical properties will be different.”

Jupiters core see's 100 million atmospheres of pressure...which actually should compress the diamond into some other form of carbon that is not diamond. This puts the upper mass of about 1/10th the mass of jupiter before the diamonds structure beings to collapse into something else at the center of the massive gemstone.

With a density of about 3 times that of Jupiter, you are looking at a diamond about 1/30th the mass of jupiter, which is still around 10 times that of the earth.

• Also from the article you cited: "The result was a cool diamond under more than 50 million atmospheres of pressure, just like the carbon in Jupiter’s core." Seems like it's still a diamond... Commented Mar 12, 2015 at 22:22
• @Samuel But, she's asking for a perfect, single crystalline diamond. No flaws or deformities whatsoever. I would think that having the core at a significantly different compression than the crust would distort the crystalline structure. Also, wouldn't the optical qualities change? Commented Mar 12, 2015 at 23:01
• @EmmettR. The change in density is a seamless change, any diamond that can be called a planet will have different densities between its surface and core. The optical properties of the core will not be visible from the surface. I doubt you could see very far into the planet. Commented Mar 12, 2015 at 23:22
• Following links in your linked article and links therein, I got to this article. Apparently nothing is strange enough that you wouldn't find it out there ;-) BTW, I wouldn't consider pure compression a flaw, however a different crystalline structure definitely would be. Commented Mar 13, 2015 at 1:02
• A crystal assumes a regular uniform lattice throughout. I think this would be impossible if the lattice scale changes as you move towards the the core. You would need to introduce flaws to fit in the extra atoms. Perhaps there is an alternative arrangement. Commented Mar 13, 2015 at 22:38

I'm not a physicist but I'd assume that it would be at the point that the gravitational force of the diamond was equal to the compressive limit of diamond. Given that the diamond's compressive strength is about 110 GPa: http://www.chm.bris.ac.uk/motm/diamond/diamprop.htm and the formula for the pressure at the center of a uniform-density sphere (to make the calculation simpler): $$M= \left(\frac{4 \pi}{3}\right) \times \left({{\rho}R^3}\right)$$ $$P= \left(\frac{3}{8 \pi}\right) \times \left(\frac{GM^2}{R^4}\right)$$ $$= \left(\frac{2 \pi}{3}\right) \times\left({G{\rho}^2R^2}\right)$$ $$R= \left(\frac{1}{\rho}\right)\times \sqrt{\frac{3 P}{2 \pi G}}$$ https://www.physicsforums.com/threads/pressure-at-center-of-planet.66257/

I get a sphere with a radius of about 7,970 kilometers, or about 9,905 miles in diameter. This is about 25 percent bigger than Earth, in both diameter and mass.

I think a point has been missed. As Samuel points out, diamond is compressible, with an increase in density of roughly 10 at the center for 1 solar mass. As this increase in density implies a reduction in volume, the whole diamond will experience significant gravitational heating as it settles into its new dimensions. And although the thermal conductivity is very high, I suspect that the core temperature will undergo significant increase. This would obviously decrease the upper limit on mass before the CNO cycle kicks in. Furthermore, since the compression varies with depth, I'm not entirely certain that the monocrystaline character of the mass would be preserved.

• Any heating issue could be resolved by building the planet sufficiently slowly. The monocrystaline nature is, of course, a possible problem, see also Twelfth's answer. Commented Mar 13, 2015 at 1:05
• I'm assuming the third paragraph in the OP produces a monocrystalline mass with uniform density. It's only when the mundane laws of physics take hold that gravitational effects come into play. Commented Mar 13, 2015 at 1:11
• The Magratheans are professional planet builders. They know how to build planets so that they have the desired properties (if physically possible) after being put in normal space. If gravitation only sets in after being put in normal space, the obvious solution would be to cool the inner core before releasing into normal space, so the gravitational heating just gives the right temperature. Commented Mar 13, 2015 at 1:16
• Unless they can cool to less than absolute zero, that doesn't work. Commented Mar 13, 2015 at 3:07
• They would add gravity gradually before bringing it to normal space. Or while constructing it. Otherwise the stresses would break the crystalline structure. This would give the heat from compression time to dissipate. Commented Mar 13, 2015 at 11:43