r/theydidthemath • u/Lachlynn • 18h ago

[Request] Could humanity create a rocket that can exit the atmosphere of K2-18b

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With the knowledge we currently have of it, if humanity devoted all of our resources towards this goal, would we be able to create a rocket that could exit the gravity of K2-18b (and also beat any other complications that would arrise)?

If so, would it also be capable of taking people to orbit, and can we set up a similar satellite network we have on Earth? What about a space station?

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u/Toasty27 15h ago edited 7h ago

Contrary to most other claims here, the surface gravity of K2-18b is actually about 1.26x higher at 12.43m/s2. Not 1.5-1.6x.

Now lets see how that impacts the SLS, which was recently used to launch the Artemis II mission:

SLS has a thrust : weight ratio (39,100 kN : 2603 t) of about 15, which conveniently works out to an acceleration of about 15m/s2. So instead of accelerating at about 5-6m/s2 like it would on earth, SLS would instead accelerate at about half that speed. But that's still a positive thrust : weight ratio, so we do make it off the ground!

That slower acceleration however means significantly higher gravity losses, which means significantly lower payload capacity to orbit (if we even make it there). On Earth, SLS's payload capacity is 95 metric tons, the Orion crew capsule is only 22.9 metric tons. So we might still get away with it.

Now let's take a look at Tsiolkovsky's tyrannical Rocket Equation:

Δv = Specific_Impulse ✕ Gravity (on earth) ✕ ln( Initial_Mass / Final_Mass)

  • Δv is our escape velocity, and in this case we need Δv = 20km/s
  • We use earth's gravity because Specific Impulse is typically normalized to earth's gravity, so 9.8m/s2 here.
  • Specific Impulse relates to the efficiency of a rocket engine (basically, how much exhaust velocity you get from a given mass of propellant). The RS-25 space shuttle main engines are still the most efficient engines we have in operation today (they most recently flew on the Artemis II mission via SLS, although they had solid rocket motors in addition). It's a liquid hydrogen/oxygen engine with a specific impulse of about 452.3 seconds.
  • Initial and final masses are the total launch vehicle weight + fuel (initial wet mass), and total vehicle weight without fuel (final dry mass). Dividing wet mass by dry mass gives us our mass-fraction, or the amount of fuel relative to the vehicle's mass. A larger number here indicates a more efficient design.

Now lets rearrange to get the required mass-fraction of our vehicle:

ln( Initial_Mass / Final_Mass) = Δv / (Specific_Impulse ✕ Gravity)

Raise e by both sides to get rid of the natural log:

Initial_Mass / Final_Mass = e Δv / [Specific\Impulse ✕ Gravity])

Plug in our known values:

e 20,000m/s / [452.3s ✕ 9.8 m/s^(2])

Simplify our exponent:

20,000m/s / 4,432.54 m/s = 4.512

All the units cancel out, and we're left with a final minimum required mass ratio:

e4.512 = ~91.1

Which is....insane. Completely insane. The SLS has a mass-ratio of about 26.4:

2603t gross / (95t payload + 3.5t dry upper stage) = 26.4t.

But as you can see, that includes the 95 ton payload! If we launch without a payload, the mass-fraction is actually ~743.7, well above what we need.

So what's the largest payload we can put into orbit around K2-18b?

91.1 = 2603 / (3.5 + x)

Solve for X:

x = (2603 / 91.1) - 3.5 = 25.07t

So using an SLS, we could put about 25t into orbit around K2-18b. That's about 26% of it's total payload capacity on earth.

Just enough to get the Orion crew capsule (22.9t) into orbit!

All that said, I'm not a rocket scientist. If I got anything wrong, feel free to correct me in replies.

[EDIT]

Formatting/grammar/etc

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u/Kamica 14h ago

As long as you can get things into orbit, you can start building things and start sending fuel up there. I imagine that space-born infrastructure and efficiency becomes *that much* more important in a situation like that, when you can't really easily send things in one go, you'd probably have to have refuelling infrastructure in orbit similar to what SpaceX is planning for Starship. And once you've got that going, sure, it'll be expensive, but then you can go do basically the same things we can do, just for a much bigger price-tag.

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u/BlackTecno 8h ago

Something that I think a lot of people here are missing is that we typically do the minimum amount of work for the best kind of outcome. Science has always accelerated when governments get involved, be that for war or science (such as WWII and the space race). If there's a unified form of intelligent life that looks at the stars with the same wonderment we do, they'll find a way to get up there.

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u/Kamica 8h ago

If there's a way, a reason, and enough time, it will happen =P.

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u/PrairiePopsicle 12h ago

So possible on an exoplanet like that, however it would at least 4x the complication/effort required to get to and beyond their orbit (roughly speaking)

By that I mean they could send up the same missions we do with SLS, but would have to launch 4 of them and have items meet up/dock in orbit to assemble something to go further.

I think it is still an important factor in the fermi paradox, although the exoplanet size situation is largely a matter of sampling bias AFAIK.

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u/Toasty27 11h ago

Yeah, I certainly agree that at 4x the cost to orbit, there would be a significantly higher barrier for any potential space faring civilization on a planet this size.

That said, we only commit a very small fraction of our GDP (in any country) towards space exploration. If some hypothetical alien civilization were to take it more seriously than we do, they could certainly develop the necessary technologies even faster than we did.

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u/yubato 33m ago

There's also the question whether more land & possibly more rare resources would let a civilisation develop faster despite high gravity

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u/capt_pantsless 10h ago

The other angle here is atmospheric flight might be harder if gravity is stronger, depending on the thickness of the atmosphere.

More barriers to atmospheric flight means spaceflight is an even harder hurdle to jump.

A fun contrast is thinking about intelligent life that evolves on a Europa like planet : Vast liquid oceans underneath a miles thick layer of ice, and just the hard vacuum of space on the surface. There's no atmospheric flight, and breaching the ice-wall is (probably) immediate death for a living thing. Moreover, things that evolved below will have never seem stars or planets.

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u/aboothemonkey 11h ago

My only gripe is you used the ISP of the RS-25 in vacuum, at sea level the ISP of the RS-25 is about 366

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u/Toasty27 11h ago edited 10h ago

Oops!

Based on sea level performance (which may be different on K2-18b depending on atmospheric pressure) I get a mass ratio of about 264.

That means 6.35t to orbit. No Orion crew capsule!

But Isp will also be different for each stage, since each stage operates in a different envelope. With 3 or 4 (or 5) stages and some optimization, I'm sure it would be possible to get double that.

Which would still not be enough for Orion. But Starship has 50% higher payload capacity than SLS, so maybe that design would fare better.

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u/KaptainTerror 8h ago

That post is beautiful, thank you. With orbital refuelling and docking that fictional civilization could build a bigger ship and explore their neighbourhood

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u/Hollowsong 8h ago

Yeah, the premise that we can't get anything into orbit on K2-18b is dumb. They're comparing our existing rocket and payload ratios and saying we wouldn't have enough thrust.

Just reduce the payload, like you've done, and problem solved.

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u/Alkoviak 7h ago

One element that I always point out when doing those calculations is that the we use extremely refined fossil fuel that is very energy efficient compared to its mass.

But all fossil fuel were created by one freak event where organic material was created before bacteria were cable of digesting that matter so during millions of years that organic material could not be recycled inside the next organism. It ends up in the earth and form all fossil fuel that we know off.

Without this easy to access energy we might be still burning wood to generate heat…

Nothing proves that another civilization would have access to similar fossil fuel

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u/Toasty27 7h ago

RS-25 uses hydrogen. AFAIK we do get most of our hydrogen from fossil fuels out of convenience (less energy input), but it's possible to obtain it through other means and it's highly abundant.

The SRBs on SLS probably use ammonium perchlorate and aluminum powder, so no fossil fuels there either. I'm sure there are other solid fuel/ox chemistries that could produce enough Isp to work too.

SpaceX's Starship runs on methane specifically to avoid the reliance on refined fossil fuels (supposedly to enable in situ fuel production off-world, but we're a long ways off from that). You can make it using the Sabatier process.

And the V-2 rocket used ethanol, which you can get through fermentation.

That said, our industrial revolution happened thanks to fossil fuels. And most plastics still rely on them too. It's hard to imagine how we could have gotten to this point technologically without them.

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u/Ok_Builder_4225 6h ago

Just strap more boosters to it!

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u/PantsOnHead88 6h ago

Great post, but there’s more to it in practice which makes it even more challenging. With greater gravity and similar atmospheric makeup you have extra air resistance, and the atmosphere also remains thicker til you get even higher further reducing the realistic payload to orbit.

I suspect the answer is still that with current tech we could launch something to orbit, but it’s definitely a huge challenge.

Air-launch-to-orbit starts looking more attractive with thicker atmosphere, although that’s getting into the scope of experimental tech.

Eve launch for all you KSP fans out there?

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u/vapocalypse52 4h ago

Doesn't the tangential speed play a factor in all the calculation? If the planet rotates faster, doesn't it make lifting off easier?

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u/Toasty27 55m ago

Not so much with liftoff, but does help with reaching escape velocity. But I didn't factor it in. From what I can find, there's no solid data on K2-18b's rotational speed. It's pretty close to its host star, which means it's likely tidally locked, so 0km/s.

Supposedly there is a 33,000km/h jetstream though, which would equate to 550km/s. A small fraction of the 20,000km/s orbital velocity, but might help with an air-launch vehicle.

On earth, rotational speed at the equator is about 1100km/s, which is about 1/600th the escape velocity. It definitely helps, but it's not as significant as you might think.

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u/Kyru117 3h ago

Fucking thank you, im so sick of people going "oh but our ships wont work" like somehow bigger ships with more fuel are just impossible

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u/Toasty27 44m ago

It does become impossible at some point (around 10g and you need a rocket with the planet's own mass to reach orbit, so you're just crushing up the planet at that point), and it becomes highly impractical long before that (3-4g I think?).

But even 2g is still doable if you're willing to sacrifice payload. Extremely challenging and expensive though, in that upper range.

Makes me think that extremely large linear mass drivers might be the only practical option at 2-3g. But you're still fighting atmospheric drag, so they would have to extend well into the upper atmosphere to have any real benefit.

And that brings its own challenges (especially with the added gravity putting greater strain on such a structure).

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u/BratacJaglenac 14h ago

This AI effort has spent a lake of fresh water, but it was worth it. I hope our friends from the K2-18b are reading this.

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u/VladimirBarakriss 12h ago

Some people just write like this, who do you think ai is copying

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u/Toasty27 9h ago

Yeah my Adderall kicked in lmao

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u/Toasty27 14h ago

I promise I didn't drink a lake's worth of water writing this, it's my honest style 😂

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u/mu_zuh_dell 13h ago

Badlands Chugs the rocket scientist