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I'm writing a story in which a civilization previously capable of (firmly technobabble-based) FTL travel has become stranded, orbiting an isolated star with no rocky planets. (There was a planet, but it was destroyed by their crash-landing in the star system.) The civilization is comfortably capable of forging an existence in artificial habitats orbiting their new star, but for the plot to function, it must be prohibitively difficult for them to leave the star system (to return home, or colonize elsewhere).

Their best chance would be to get their hands on an already-constructed FTL travel device, but I'm having trouble constructing a convincing (within-universe) reason why they couldn't simply build a new one from stuff they have at hand.

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    $\begingroup$ Hint: read why we can't land on our Moon anymore. $\endgroup$ – Mołot Jan 3 '19 at 11:35
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    $\begingroup$ @Mołot You piqued my interest. Do you have a link to a good resource on the subject? $\endgroup$ – Jasper Jan 3 '19 at 15:15
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    $\begingroup$ If you really want to strive for realism but also have a completely destroyed planet, please read about the ludicrous amount of energy needed to actually destroy a planet. The Earth is a is a 4,550,000,000-year-old, 5,973,600,000,000,000,000,000-tonne ball of iron $\endgroup$ – vsz Jan 3 '19 at 18:31
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    $\begingroup$ How can they crash-land into a planet with enough energy to destroy the planet and still have survivors on the ship? $\endgroup$ – NomadMaker Jan 3 '19 at 23:56
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    $\begingroup$ @Molot That article doesn't say we couldn't figure out how to land on the moon if we really wanted to; it says we're figuring out how to do more than just land on it next time, and also don't have the same kind of budget we did last time. $\endgroup$ – Ray Jan 6 '19 at 23:14

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There are basically two possible answers :

  1. They do not actually know the technology.

  2. They lack necessary resources.

Possible reasons for not knowing the technology are :

Security

As something that can destroy habitable planets by accident the technology was highly classified and even its basic theory was only available on need to know basis to people with high security clearance in secure facilities. Basically in the facilities used develop and manufacture the drives. Only time it would be on spaceships would be in encrypted data vaults rigged to self-destruct if anything unusual happens.

This sounds as a bit of a cop out but remember that they did in fact destroy a planet. Any mumbo-jumbo FTL system is wide open to highly destructive and dangerous abuse almost by definition. There is a reason why so many authors source their FTL from advanced aliens who refuse to share the details of how it works with a species as irrational and violent as humans.

It would be entirely reasonable that nobody surviving the accident had access to FTL technology or the encryption keys needed to read the information stored in the surviving data storage. The data might as mentioned even have been only in a separate highly secure data bank that was destroyed or simply inaccessible due to nobody having anywhere near the necessary clearance. Or the authority to grant such clearance.

It does not work

FTL really is quite tricky. In theory an FTL drive accident might see you travelling backwards or forwards in time, thrown so far that nothing you recognize is within your observable universe, or maroon you in another universe with subtly different physics.

The last is one option for losing FTL technology. If you succeed in going FTL but fail to return normal in controlled fashion you should be stuck in FTL until something yanks you out. Alternate universe where your type of FTL is not sustainable would qualify as such a something.

So they would still have the technology. They could build the drives. They could turn them on. And then they'd get yanked back to normal space roughly five meters away from where they started. Probably as a rapidly expanding ball of plasma.

Obviously this works for more localized "no fly zones" as well. But alternate universe is harder to detect and you cannot circumvent it with STL travel. And it would not prevent other people from visiting with functional FTL drives compatible with the current universe and puzzling your people greatly as it has been well established that those drives could not possibly work. Every study done before the accident (in another universe) confirmed that!

So you just need to prevent the people from doing expensive theoretical physics research to confirm things that are already well proven and accidentally noticing that rules have changed. Basically that amounts to supplying an alternative theory to why drives keep exploding. Terrorist? Manufacturing quality issues? Maybe the drives required insanely precise fine tuning to work properly and that information really was highly classified (a trade secret of the manufacturer) and lost.

Possible resources they might lack :

Unobtainium

Already covered in the top answer but included for completeness. This is another common solution authors use to limit FTL abuse. Note that precursor artefacts, rare psionic talents, arcane magical runes and divine blessings amount to the same. They are all excuses why FTL cannot be used freely and they all work as excuses to prevent it entirely.

Infrastructure

You need the machines to build the machines. Realistically FTL engines might be very high in that hierarchy and making them would require an industrial base well beyond what your survivors can manage in the next thousand years or so.

Astrogation data

In many universes a successful FTL trip requires having a very good idea where you are now. This is a serious issue if you have an FTL accident that throws you to some random unknown location in time and space. Especially if you are absolutely certain that you are somewhere else. Simplest reason is an alternate universe that looks exactly like something familiar but actually has different hyperspace coordinates.

For extra entertainment have people realize they are in another universe and need to remap their coordinates but not realize that their form of FTL simply does not work in the current universe. The difference between the two might be hard to notice and it would be very simple for somebody to get it entirely wrong early on and nobody to ever question it again. Especially if the one doing that initial check is the same person responsible for the initial "accident".

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To directly answer the question, building a starship will take more resources in terms of manufacturing support than can be carried on a starship. This means it will take the development of an advanced economy at the destination, which could take decades if not centuries. So, they have the knowledge to build a new ship, just not the ability to build a new ship.

The notion, though, that the crash of a starship into a planet can destroy the planet is very weak. The gravitational binding energy of a planetary mass is proportional to the square of the planet's mass divided by the radius of the planet. For the Earth, this energy is roughly equivalent to 12 days of the total output of the sun compressed into a ten second delivery. That is over 4 trillion tons of mass converted to energy in 10 seconds or less, all of the resulting energy delivered to the Earth.

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