Super Earths, Or How The Exponential Function Can Matter

We live in the times where exponentials have come to rule, as they never ruled before. Ignore at the risk of everything we claim to hold dear. As mathematically challenged Silicon Valley nerds put it, all too simplistically, the coming “singularity” looms. Simple minds do not much understanding create, though, so here a little elaboration…

An example of exponentials in action, is graciously offered by so-called “Super Earths“, giant versions of Earths, hundreds of which have been discovered in our neighborhood.

Before I get into this, a short lesson on the exponential.

The Ancient Greeks thought they knew mathematics, but they were prisoners of linear thinking (especially after the top intellectuals spurned non-Euclidean geometry and arithmetic). The exponential is the most obvious, most crucial to understand, most vital to handle example of nonlinear thinking.

An exponential is any function which grows proportionally to itself.

Our present “leaders” (Putin, Trump, Xi, Macron, etc.), and their underlings have no idea what an exponential is, and that it feeds on itself.

Civilizations get ambushed by exponentials. This is why they so often irresistibly decay: the effect is blatant, be it the Late Roman empire, Tang China, the Maya…  

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Socrates:“The unexamined life is not worth living“. That was HIS (wise) feeling. His own feeling. Others don’t have to share it. Actually vain, self-admiring, erroneous, hateful people detest nothing more than self-examination. They deeply dislike, hinder those, and what, promotes self-examination.

And tell me, Socrates, you who didn’t like knowledge you didn’t already have, and you thought everybody had, when did you learn about the exponential function? How can you know something that important you never even suspected existed? And, absent that tool of the spirit, you thought you could examine everything? How stupid was that? And you, out there, the ignorant admirers of Socrates and his ilk: you don’t even have the excuse to have been dead for 24 centuries! To extract you from the gutter, seize the exponential!

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After discovering a few thousands exoplanets, Super Earths are, so far, more frequent than simple Earths (it may be a bias from our present telescopes, but I don’t think so…). If the Super Earth is slightly bigger than Earth, depending upon the nature of its core, its surface gravity doesn’t have to be much higher than Earth (I computed). However, the present article considers Super Earths were the gravity is much higher than on Earth…

“Super-Earth” planets are gigantic versions of Earth. In some ways, they are more likely to be habitable than Earth-size worlds: their thicker atmospheres protect them better from radiations, either from their parent stars, supernovae, gamma ray bursts, galactic core explosions, etc.. However, it would be difficult for any inhabitants on these exoplanets to access to space. At least with known, or imaginable technologies.

To launch a vehicle as light as the Apollo moon mission capsule, a rocket on a super-Earth such as (potentially inhabitable) Kepler 20b would require more than double the escape velocity.

To leave Earth (“⊕”)’s gravitational influence, a rocket needs to achieve at minimum the escape velocity vesc = s 2GM⊕ R⊕ ∼ 11.2 km s−1 (2) for Earth, and vesc ∼ 27.1 km s−1 for a 10 M⊕, 1.7 R⊕ Super-Earth similar to Kepler-20 b. Computation shows one would need a mass of about 400,000 metric tons, mostly due to the exponential demand of fuel. That’s 5% of the mass of the Great Pyramid of Giza in Egypt (still by far the Earth’s most massive monument, excluding utilitarian walls and dams).  

That means a chemical rocket there should have one hundred times the mass of one here (Apollo’s Saturn V launcher was 3,000 tons). However, that’s not a show stopper: our largest ocean-going ships are more massive than that, and a massive rocket is imaginable. So Hippke is not correct when he says that:

“On more-massive planets, spaceflight would be exponentially more expensive,” said study author Michael Hippke, an independent researcher affiliated with the Sonneberg Observatory in Germany. “Such civilizations would not have satellite TV, a moon mission or a Hubble Space Telescope.“

This is of great practical interest. Research has revealed that Super Earths are abundant, and obvious targets for human colonization. They can reach up to 10 times the mass of our own Earth (after that, they retain light gases, and turn into mini Neptunes, unsuitable for direct colonization, although Pandora like scenarios are highly plausible). Many super-Earths apparently lie in the habitable zones of their stars, where temperatures can theoretically support liquid water on the planetary surface and thus, potentially, life as it is known on Earth. Although I have had reservations about this: I view the presence of a nuclear reactor inside the planet as necessary for life, since it provides with a magnetic shield, and the recycling of the atmosphere through plate tectonic, let alone continents… (Being in the water belt and the nuclear belt simultaneously is a miracle Earth’s biosphere profits from.)

This being said, it is true that some ways to access space that we potentially have, won’t happen on Super Earths. Rockets work better in the vacuum of space than in an atmosphere: super-Earthlings might want to launch from a mountaintop. However, the strong gravitational pull of super-Earths would squash down super Alps (it’s a pure application of Quantum mechanics). Super towers won’t be be feasible, either…

Using space elevators traveling on giant cables rising out of the atmosphere depends upon the strength of the cable material. The strongest (per unit of mass) material known today, carbon nanotubes, is just barely strong enough for Earth’s gravity (it is not at this point possible to imagine stronger materials, putting in doubt the feasibility of space elevators on super-Earths). Here is Michael Hippke (Submitted on 12 Apr 2018):

Spaceflight from Super-Earths is difficult:

 

Many rocky exoplanets are heavier and larger than the Earth, and have higher surface gravity. This makes space-flight on these worlds very challenging, because the required fuel mass for a given payload is an exponential function of planetary surface gravity, ∼3.3exp(g0). We find that chemical rockets still allow for escape velocities on Super-Earths up to 10 times Earth mass. More massive rocky worlds, if they exist, would require other means to leave the planet, such as nuclear propulsion.

Comments:	Serious version of the April Fool’s idea (arXiv:1803.11384). Submitted on April 4th 2018
Subjects:	Popular Physics (physics.pop-ph); Earth and Planetary Astrophysics (astro-ph.EP)
Cite as:	arXiv:1804.04727 [physics.pop-ph]
	(or arXiv:1804.04727v1 [physics.pop-ph] for this version)

1. INTRODUCTION Do we inhabit the best of all possible worlds (Leibnitz 1710)? From a variety of habitable worlds that may exist, Earth might well turn out as one that is marginally habitable. Other, more habitable (“superhabitable”) worlds might exist (Heller & Armstrong 2014). Planets more massive than Earth can have a higher surface gravity, which can hold a thicker atmosphere, and thus better shielding for life on the surface against harmful cosmic rays. Increased surface erosion and flatter topography could result in an “archipelago planet” of shallow oceans ideally suited for biodiversity. There is apparently no limit for habitability as a function of surface gravity as such (Dorn et al. 2017). Size limits arise from the transition between Terran and Neptunian worlds around 2 ± 0.6 R⊕ (Chen & Kipping 2017). The largest rocky planets known so far are ∼ 1.87 R⊕, ∼ 9.7 M⊕ (Kepler-20 b, Buchhave et al. 2016). When such planets are in the habitable zone, they may be inhabited. Can “Super-Earthlings” still use chemical rockets to leave their planet? This question is relevant for SETI and space colonization (Lingam 2016; Forgan 2016, 2017).

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Pessimistically, Hippke considered another possibility, a staple of science-fiction which originated in the very serious “Orion” project of the 1950s, an apocalyptic period: nuclear pulse propulsion. It works by detonating thousands of atom bombs below a shield cum shock absorber attached to the vehicle, hurling it through space. This explosive propulsion has much more lifting power than chemical rockets, and might be the only way for a civilization to leave a planet more than 10 times Earth’s mass, Hippke (naively) said.

However, slaying the radioactive dragon he himself brought up, such a nuclear-powered spacecraft would pose not only technical challenges but political ones as well, he said: “A launch failure, which typically happens with a 1 percent risk, could cause dramatic effects on the environment. I could only imagine that a society takes these risks in a flagship project where no other options are available, but the desire is strong — for example, one single mission to leave their planet and visit a moon.”

Unwittingly, Hippke then demonstrates the danger of the single mind (in this case, his!) Indeed the most obvious way to use nuclear propulsion is simply to run a liquid, even water, through the core of a nuclear fission reactor. That was tested, and it works extremely well… and very safely! It’s much less prone to failure than a chemical rocket.  On a planet with ten times the Earth’s surface, there would be plenty of space to do such dirty launches by the thousands.

Besides, it may possible to engineer absolutely giant thermonuclear PROPULSION reactors (thermonuclear fusion is easier, the larger the reactor: the exponential at work again; if we just made a fusion reactor that was large enough, it would certainly work). The radioactivity generated would be neglectable.

So we don’t have to worry about colonizing Super Earths… We just have to worry about weight (that is, surface gravity)….

But, here, now, we have to worry about all those exponentials going crazy. Last I checked, the Arctic ice was running one million square miles below its old minimum: at some point the so-far linear decrease of Arctic ice is going to decrease exponentially, as warming there is highly self-feeding (that’s why it runs already at twice the rate of the rest of the planet…).

And as usual, let’s remember what the arrogant, stupid imperial Romans never learned, and the Maya never reached: inventing completely new, liberating, energizing technologies is how, and the only way how, to break the strangulation from the ecological, political, economical and moral exponentials which smother civilizations. A most recent example is diffuse, dim light solar cells, dye-sensitized solar cells (DSSCs), a tech already in full deployment, which has just made spectacular progress in the lab.

Even language acquisition is exponential… Let alone thought system acquisition. You want to examine life, in ultimate depth? Learn to think exponentially!

The coming “singularity” looms. How to manage it? First by understanding what makes it tick, exponentials.

Patrice Aymé

 

Tags: Exponential, Singularity, Socrates, Super Earths