# A Rock the Size of Jupiter?

For some reason, the topic of really big rocks came up at dinner the other night, and SteelyKid declared that she wanted to find “A rock as big as the solar system.” We pointed out that that was pretty much impossible, more or less by definition, rocks being sub-parts of the solar system.

“OK, how about a rock as big as Jupiter?” That’s a much harder question to answer, and required a trip to the Internet. Not during dinner, of course– it’s hard enough to get her to eat when we’re all sitting at the table– but a day or so later, which led to this blog post.

So, there are a couple of different ways to interpret the request for a rock the size of Jupiter. I’m going to take the most literal first, and ask about constructing an object with the physical size of Jupiter but made out of rock. Jupiter, Wikipedia helpfully tells us has a radius of (in round numbers) about 70,000 km, or 7×107m. So, a rock the size of Jupiter would be a sphere of the same radius, having a volume of about 1.4×1024m3. (If I were a real astronomer, I’d convert this to grams just to make the numbers a million times bigger, because astronomers are very insecure that way…)

What would the mass of such a thing be? Well, looking at this big list of densities, things that I would identify as rocks tend to have a density of around 2,000 kg/m3. A Jupiter-sized sphere of rock would thus have a mass of around 2.8×1027kg. That’s only about half again the mass of the actual existing Jupiter, which is 1.9×1027kg. Which makes a Jupiter-sized rock seem surprisingly plausible.

So what’s going on here? Well, Jupiter has a very low density, as it’s basically a giant ball of gas. Mostly hydrogen, helium, methane, and other fairly light things. The overall density of Jupiter is around 1,400 kg/m3, on the light side for rocks. It’s down in the range of plastic materials, so if you wanted to make a Jupiter-sized ball of plexiglass, it would be roughly the mass of the real Jupiter.

But then, if you look up the density of gases, you’ll find numbers way lower than that– methane is probably the heaviest of the gases I mentioned above, and its density is less than half that of Jupiter, according to the Internet. So what gives? Well, the desity Google gives for methane is at atmospheric pressure here on Earth. But if you had a giant ball of it in space, gravity would make that contract, compressing the central regions to a much higher density. The density reported for Jupiter is an average density, obtained by dividing the total mass by the total volume. The outer atmosphere is light and fluffy, but in the core, the gas is compressed to extremely high density.

(This, by the way, is it’s silly to claim that Saturn would float in water…)

We can also see this happen with rock– we think of Earth and the other inner-solar-system objects as “rocky planets” but in fact the average density of Earth is considerably higher than that average rock density I used above– more like 5,500 kg/m3. Again, the gravity of such a large mass would compress the whole thing to a much smaller radius, increasing the density of the central region. So, if you’re trying to make something the size of Jupiter whose outer layers would be recognizable as rock, you probably need a much higher average density. If you used the average density of Earth (which is the largest and densest of the “rocky” planets), you’d end up with an object having about four times the mass of Jupiter. But, of course, that would compress further under the gravitational attraction of all that mass, giving a yet higher average density, and so on.

Since I don’t want to calculate the actual mass distribution of a thing the size of Jupiter with a rocky surface, let’s take a different slant on this and ask at what point this whole thing would fall apart? Well, as you get bigger than Jupiter, you start pushing toward becoming a “brown dwarf,” not quite big enough to be a star, but big enough to start nuclear reactions happening in the core. A quick scan of Wikipedia suggests that this happens around 13 Jupiter masses, which would mean a Jupiter-sized object with 13 times Jupiter’s average density, or a bit more than three times the average density of Earth.

Given that Earth is more than twice as dense (on average) as generic rock, and we’re making an object with several hundred times the mass of Earth, I suspect that we’d pass that limit well before making a Jupiter-sized body with a rocky surface. It’s not clear to me exactly what would happen, given that the composition of generic rock is much different than the composition of a typical brown dwarf, but I doubt very much that the resulting object would be recognizable as a “rock.” So, disappointingly for SteelyKid, a rock the size of Jupiter isn’t something we’re likely to find any time soon.

(An alternate interpretation would be to take “size” to mean “mass,” and then ask whether you could have a rocky planet with the mass of Jupiter, and what the radius of such a ting would be. That’s a pretty easy calculation, which you can do for extra credit. Send your results to Rhett for grading.)

1. #1 Eric Lund
June 30, 2014

The other wrinkle is that if you should happen to assemble a Jupiter-sized pile of rock, it will pull in a bunch of gas from the surrounding cloud. That’s the main physical difference between an Earth and a Neptune (let alone something the size of Jupiter): hydrogen can escape from Earth’s gravity, but not Neptune’s. Bottom line: If you did manage to assemble a Jupiter-sized rock, it’s not a question of whether you end up with a planet or a brown dwarf, it’s whether you end up with a brown dwarf or an M class star (the outer parts of such a star are cool enough for dust grains to not be obliterated).

June 30, 2014

Yeah, I was imagining this not as an object that would form via normal planetary formation mechanisms in the middle of a cloud of gas and dust, but something aliens with godlike powers might build on a bet in space that had been largely cleared already. If you somehow got a Jupiter-sized rock in the process of forming a planetary system, I’m pretty sure you’d just end up with a binary start system after it sucked up a vast amount of gas to go with all the rock.

3. #3 Stareyes
June 30, 2014

Actually, one of the interesting things is that, until we started measuring the sizes of exoplanets, astronomers assumed that ‘a bit bigger than Jupiter’ was the largest radius a planet or brown dwarf could be: around a couple Jupiter masses, piling on more stuff caused enough compression that the planet got smaller.

(I actually had a chart that showed the theoretical radius versus mass relationship of planets of different materials. It’s in my grad school notes somewhere.)

Of course, some of the current ‘hot Jupiters’ are far bigger than our old models say any planet should be, so there’s some theorists playing with ways to add enough heat to make them puffy.

4. #4 My one cent
June 30, 2014

For anyone who cares, S. Seager and colleagues actually did calculate the maximum radius of rocky planets in their 2007 paper “Mass-Radius Relationships for Solid Exoplanets” available on the arxiv (http://arxiv.org/abs/0707.2895)

5. #5 Uncle Al
June 30, 2014

If you wish to assemble a big rock, you must dissipate gravitational binding energy versus thermal conductivity and radius. One can cast a transparent spherical glass ball three meters in diameter, but cooling is a pisser, plus differential thermal contraction.

Assembling optically flat cold horizontal sections (e.g., a sliced egg) via vacuum welding is plausible. Let us instead make a fused silica moon a national priority, and go on from there.

6. #6 Michael Fisher
http://en.gravatar.com/michaeljfisher
June 30, 2014

How about a Jupiter-sized, very high spin, oblate spheroid rock?

7. #7 Pete A
June 30, 2014

Okay, but the most important question is: What was SteelyKid’s take on what you said?

Convincing yourself and your adult audience is totally missing SteelyKid’s point :-)

8. #8 agm
July 1, 2014

5.5 g/cc (equal to 5500 lg/m^3).

Wow, I did not realize just how much of the earth is essentially pure refined metal. A density value that high in the oil field means your density log is reading casing and not the rock formation.

July 1, 2014

SteelyKid’s response to the explanation was a drastic change of scale: “Can there be a rock as big as you?”

I gave her a look, and said “Am I the size of Jupiter?”

“Well, no. But you are a really big Daddy!”

10. #10 Eric Lund
July 1, 2014

“Can there be a rock as big as you?”

You live within day trip distance of the Catskills and Adirondacks. Maybe you should ask a colleague in Earth Sciences (or whatever they call that subject at Union) to recommend a suitable trip for inquisitive kids like SteelyKid and the Pip. Because there you could point to a mountain and truthfully tell SteelyKid that it’s a big rock.

11. #11 Pete A
July 1, 2014

“Well, no. But you are a really big Daddy!”

This is a sobering lesson for us all to remember. Even for adults trained in science it is very hard to comprehend quantities that are vastly different from our own point of reference, such as height and mass. I can just about relate to metres and kilograms over the range from 1E-6 to 1E+6, but I cannot begin to comprehend the size of a water molecule or the number of stars in the Milky Way, let alone the entire universe. Such numbers are mind-boggling.

To non-scientists, a grain of sand is minute and a forest pine tree is enormous. The terms really small, small, big, and really big usually span well under two orders of magnitude; often less than one order of magnitude!

The power output of the Sun is circa 3.8E26 watts yet it generates only 193 microwatts per kilogram. This makes it very difficult to fathom just how big it really is :-)

12. #12 cheem
July 6, 2014

Would a cooled white dwarf count as a ~Jupiter-sized rock? It would effectively be a giant ball of carbon, and diamond is a kind of rock…