Planetizing the World of Jack and the Beanstalk: How Small is Too Small?

Question

We all know the tale of Jack and the Beanstalk--a farmboy sells his cow for some "magic beans", which in turn grow into a mountainously tall stalk that led him to the land of a giant. Now, scientifically speaking, the only way for Jack to meet the giant is if the beanstalk led him to a habitable world with lower gravity.

Which was the sort of thing that "Jack", a member of the mad scientist illegal terraforming community, wanted to take advantage of when the real scientific community discovered that one of the alternate Earths has an iron-rich core the exact same size as ours is, but hidden beneath a smaller crust, which compresses the mantle. This combination of smaller crust and same-size core means that the planet would be warm enough to support liquid water, therefore life. For him, it is the perfect place to turn the story of "Jack and the Beanstalk" into an actual ecosystem, complete with smuggled stores of fabacean seeds, or "beans", just to see if the lower gravity would be enough to turn these small herbs into giant plants that anyone named "Jack" could actually climb up.

But here is the thing--if the crustal diameter is too small, therefore compressing the mantle too tightly, then the crust would liquefy, turning the surface into a volcanic hellscape. So in an alternate Earth where the core is the same size, how much smaller would the crustal diameter be for this world to be livable instead of hell?

Answer 1

I'm going to be honest, I think this sounds a little too far outside of science for this to work.

Base requirement is that the Earth and moon would have to be completely different because they would need to be closer and tidally locked so that the moon doesn't immediately rip the beanstock out of the ground. Pluto and Charon are a great candidate for that, but they are frozen because of the distance from the sun, and are too small to hold an atmosphere. Of course, if you increase the size and bring them closer to the sun, they end moving too far away to be feasible.

The tidal locking would also cause the Earth to be dramatically colder on the moon side because it would experience much colder nights, while the other side would be warmer and drier because the moon produces the tides, so oceans may be likely to accumulate more on the moon side. When I say warmer and colder I don't mean like greater difference than the Sahara and Antarctica.

I guess it might be feasible with those parameters, but that's a hell of a lot of handwavium. There's a reason fantasy and scifi don't usually overlap well unless you just kind of drop real science.

I know you were primarily asking about the crustal diameter, but I think you have a few more bigger issues before that really comes up. Is the mantle size different in anyway? That would definitely have a greater affect, because the crust is so thin around the edges of the Earth, that changing that alone would essentially have next to no affect except make the surface molten as suggested. A much smaller mantle, which would likely be necessary for this scenario would however have drastic affects on the Earths internal heating. I guess this is just more of a few things to consider.

Answer 2

I think a good reference would be the inner earth temperature by depth which clearly shows that you can't shave off more than a few kilometers (and that is negligible in the big picture) of crust before you cross the uncomfortable 100°C region where life would be severely unlikely. If you want to take away the largest amount of mass with minimal effect to temperature and all the other crap I would suggest shaving off large amounts off the mantle and maybe the liquid outer core. The temperature does not increase that much after a few hundred or thousand kilometers and what really brings the temperature down is the solid crust anyways.

There is a good chance that a smaller earth would turn completely solid though, since the volume/surface ratio would change drastically in the favor of cooling off. (Yes, homogenization is also bad for keeping temperature, but you still need to consider that a 10 fold decrease in diameter for example would decrease gravity a thousand fold and increase the cooling rate by 10 times)

Last thing you have to consider is that gravity is not the only limiting factor for animal growth on a planet! A very important measure is the volume to surface ratio of an animal as defined by rubners surface law or square cube law in general. It basically means that a larger animal will have to slow down its metabolism in turn in order to avoid overheating which is also why we see large animals with artificially raised (unnecessary) surface (look at elephant ears for example). So your giants would, depending on their size, be either very dead after a few seconds or have an incredibly slow metabolism as in, they pretty much don't move at all.

Answer 3

Intro

There are a lot of physical issues with this planet being connected by the end of a bean-stock that is also connected to Earth, but if its magical anything is possible.

I'm going to first examine the composition of our new planet and how we might approach the minimization of its radius.

Earth-Like Gravity

The most obvious issue for minimizing the radius of this planet is mass, since I'm assuming you want the gravity to be at least somewhat similar to Earth's. For the smallest dimensions of a Earth like planet that are physically possible the density will be the most important parameter, which of course constrains composition.

Now there are a few issues that really constrain the possible solutions to the problem. First if the core has to have the same diameter as that of Earth and has to have the same iron based composition, then we'll start there. The core is composed of an inner and outer component, so I'll assume the core in it's entirety.

From my handwavey calculations I guestimated the mass to be about $8x10^{23}$ kg, from researching it a bit I found the core is estimated to be about $1.9 x 10^{24}$ kg which is about a third of the Earth's total mass. [O.G. Sorokhtin et. al, Dev. in E&ES, 2011] I'll use this later figure.

The total radius so far is about 3500 km. We need to account for a remaining $4x10^{24}$ kg of mass. We want a dense material, but it must not be denser than the core. If we add another 1850-2000 km for the mantel we can get an average density between 7.7-8.7 $g/cm^{3}$. This is not unreasonable for a metallic composition. However, this is another issue altogether; metal is way too thermally conductive to work if the core is to be the same as Earth's. A particularly cold core wouldn't actually affect the surface temperature (after all this is determined atmospherically and not from the internal temperature). However, a cold core would result in a weak or lack of internal dynamo effect and no magnetic field, which of course make life impossible. To resolve this, you could force it to work, of course the solution is completely contrived and result in some rather bizarre things happening; a mix of heavy metalloids and oxygen might do the trick.

So now we have a reasonably dense planet with the same gravity as Earth. Coming out with a planet about 5500 km including the crust, our planet will still have near Earth gravity while being a bit smallar. However, our planet is still much larger than Mars. The circumference of this planet would just be a few thousand km smaller than Earth. While this is quite a bit, it probably wouldn't feel too much different when on the surface. The curvature of this planet would be noticeable at lower altitudes, but a few thousand kilometers isn't as much as it sounds when we consider that the the Earth's circumference is 40,000 km! If someone is in the US for example, any place within the continental US is no more than 5000 km away.

Other options

Now, you mentioned a lower gravity. But the question is how much lower do you want to go. The lower the gravity, the more bizarre the inhabitants of this planet. I'm not much of a biologist, but I'm aware of the basic effects of lower gravity might have upon organisms. If the sort of animals and plants on this planet were similar to the ones on Earth, they would probably tend to be taller. They probably would be more massive as well since the limit to being crushed under ones' own weight would increase. But the animals wouldn't be the only problem here.

Lowering the gravity allows for less extra planetary mass as well and possibly less gravitational acceleration. But we'll still need a sizable mantle for a variety of reasons.

If we just isolate the Core, we actually get more gravitational acceleration than on the surface of the Earth: about 11 $m/s^{2}$ compared to our 9.81 $m/s^{2}$. A fairly thick mantle will still be necessary or else the mantle will be crushed under the gravitational attraction as well as being melted due to the heat transfer. There's also the problem of the core losing all the heat and another problem of no magnetic field as described above. Furthermore the outer core is fluid (the inner core is solid) and the Earth's dynamo is primarily driven by the outer core, with convection cycles running up into the lower mantel.

It is often tempting to cobble together basic calculations using hand-wavy equations for heat transfer, perhaps using the density of a candidate composition to estimate how thick our mantel might be, maybe we might go from there to solve the gravitational acceleration. This unfortunately would result in patent nonsense no better than just making something up which sounded nice. To properly get a sense of the interactions which go into these calculations we would need to create an equation of state for this planet using the interplay among pressure, heat and temperature. From this we can get a sense of how the pressure and density interact and how heat would transfer for various materials which are allowed under the density/pressure constrains. To properly do these calculations would require an entire thesis unto itself.

If you are interested in seeing an example of the sort of calculations required, see

[http://articles.adsabs.harvard.edu//full/1980LPSC...11.1999A/0001999.000.html]

which contains the article "Equations of State in Planet interiors" by Orson Anderson and John Baumgardner.

All in all, insisting that the core of the bean-stock linked planet must be the same as Earth's core, in both composition and heat, seriously restricts the possibiities for how realistic the size can be. For the resulting planet to be realistically suitable to an Earth surface, it will end up having a size not to far from Earth's current size. As we've explored, keeping the Earth like gravity will result in a very Earth sized planet. Allowing the gravity to be less than Earth's will decrease circumference a bit, but I would not suspect the results to be too extreme.

The interesting thing about systems such as these, is that the interplay found in the EoS really restricts the possibilities. I'd estimate that at the smallest, you'd end up with a planet not too differently sized that the one described above with Earth like gravity.

Small Earth-like planet with low gravity

A possibility for envisioning an Earth-like planet with low gravity would be to have a much smaller core. The pressure of Earth's core is still too low for iron/nickel to be created from pressure alone. As regarding the dynamo effect, many materials can produce dynamo effects in theory, Iron is not special at core temperatures as it loses its ferromagnetic properties above the Curie temperature which is well below the temperature in Earth's core. So there are lots of choices for materials. Mars is a great example, as it is about half of Earths radius, but about a tenth of Earth's mass. If the radius were even smaller given it's mass, the gravity, which is a little over a third of Earth's, would increase.