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16:05
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A: What would be a good reason for space colonists to lose their technology?

Richard SmithI feel that the premises of your question are somewhat unrealistic. Firstly, radiation fades very quickly. https://emilms.fema.gov/IS3/FEMA_IS/is03/REM0504050.htm In that source, it says "for every 7-fold increase in time after detonation, there is a 10-fold decrease in the exposure rate." In a...

Most of these points have slightly flawed premises.
On #1, #2: A nuclear winter will still stop any reasonable way to produce energy. --- A 10-fold radiation decrease won't help you if if's one million above the "you can survive this" threshold, or if you cannot survive until the radiation is back down. --- Radiation that is already in your environment is far more difficult to control than radiation from the outside. (Of course stellar radiation is still nasty. Intermediate on Earth are still preferrable to interplanetary radiation.)
On #3: While you have plenty of energy if you can navigate space, there might be other restrictions. You might not have the technology to accelerate asteroids, or under time pressure (need to get out of space before the pursuing Vogons detect you, or before the last supplies run out). So while you're right in general, the story situation might be different, particularly if it is a really small spacefaring fleet that lives off supplies (something you allude to yourself in other points).
On #4 and #5: I completely agree on this one, unless the escape fleet has some handwavium - a teleporter built by precursor civilisations, for example. Going to another galaxy doesn't give you anything that you don't have in the current one though. Even a different star cluster might be hard to justify.
On #6: That's actually less of an issue. Distribute a few handfulls of organic dirt in various places, and exponential growth and evolution will make it occupy all the ecological niches that the native flora/fauna cannot use as efficiently.
On #6 (continued): Food: Carbohydrates and fat are chemically pretty simple, so if they worked for our ecology, you can expect to find them in native fruits and animals. I.e. energy intake is not a problem. You will have problems with essential amino acids unless the local ecology happens to use the same aminos, but these are too varied so you likely will have issues. This is where you need to bring your own ecology; fortunately, we're not talking about large quantities (mere grams per day).
On #6 (continued again): Viruses and bacteriae will be mostly incompatible, but the few that get through can be devastating (which would be a pretty rare thing to happen though, so there's a chance that this won't hit the first colony).
On #7: At interstellar speeds, you don't need an asteroid to kill a ship, a fist-sized stone, a pebble, or a grain could be enough. Even a single proton could smash a ship if you are going fast enough, and the proton happens to hit an atomic nucleus and the fragments happen to hit neighbouring nuclei. But even asteroids can be very hard to detect, some are really black and you detect them only when they obscure a star, and you might not have enough time for evasive maneuvers. Details depend a lot on the speed and where you are navigating though.
1, 2. Read this: quora.com/Who-would-win-in-a-war-between-Russia-and-the-US/… It appears very well sourced. TLDR: Nuclear winter is a myth, there are not nearly enough nukes to kill even a billion people, and radiation quickly becomes nonhazardous. The danger of nuclear weapons is constantly overstated and exaggerated.
However, even if there was a nuclear winter, living on earth would still be easier than living in space. Even gamma radiation will be stopped by a few meters of dirt and concrete, so bunkers are definitely viable. There would still be at least aquifer water, desalination (energy expensive, but cheaper than space travel), oxygen, etc easily available. Also, how would nuclear winter stop energy generation? Solar might become nonviable, but you could still use fossil fuels, geothermal, fission, or fusion (which you probably have if you have interstellar spacecraft).
3. Why would you not have the technology to move asteroids? If you can move a ship the size of New York, you could attach the ship to an asteroid and move it towards Mars. Also, there probably isn't an invading alien fleet, or else invading Mars would be pointless, as they would just come and kill you anyways. Also, if an ancient civilization is trying to kill you, you are certainly doomed, and running away will not change that. There is no stealth in space (look it up), so no matter if you flee or not, they will know where you went.
Continued: If you flee into space, they will simply chase you down with a larger fleet or with a massive volley of intelligent RKKVs and kill you. If you stay where you are, they will melt your planet with a Nicoll-Dyson beam or slam a few billion RKKVs each with the energy of the Chicxulub asteroid into your planet. If you try to expand into your solar system, they will melt every planet in your solar system or hit every one with an RKKV, then follow it up with a massive invasion force of millions to billions of self replicating war machines.
Interstellar civilizations are incomprehensibly vast. Even a Kardashev 1.something civilization with a partial dyson swarm would outnumber us billions to one and would likely have an economy hundreds of billions or trillions of times larger than ours, as well as better technology.
They could exterminate us with the same effort we use to go to the supermarket. In fact, if there was a homicidal civilization near us, we'd already be dead. It is possible to detect atmospheric compositions of exoplanets with telescopes, and they would presumably go about melting every planet with an atmosphere capable of supporting life that they detect with a Nicoll-Dyson beam before anything more advanced than a bacterium emerges.
See Isaac Arthur's Interstellar Warfare and the first few minutes of his Space Warfare videos for a sense of the scale of these civilizations.
6. It does not matter if they also use carbohydrates and fats and proteins. Their versions will probably be completely different. Remember, even if their basic sugar monomer is C6H12O6 like ours, it will probably have slight structural differences that make it indigestible to us. We need specific enzymes to catalyze the breakdown of every type of molecule, and we will not have the ones needed to break down their molecules.
Even if they use the exact same monomers of glucose, fructose, and galactose, their polysaccharides will probably have those linked together in different ways, and we will similarly be incapable of digesting them. The same goes for proteins and fats.
7. A proton moving at 0.2c has something like 10^-9 J of energy. Our current nuclear reactors release neutrons at 0.1c, and those don't cause huge explosions. No proton would destroy a ship. Also, a 1kg rock would only have about 100 kilotons of explosive force if moving at 0.1c. That is somewhat impressive, but would not be fatal to a ship the size of New York State.
Remember, modern spacecraft have no armor, as they do not travel at relativistic speeds, as lifting things into orbit is expensive, and and as modern engines are inefficient. With fusion, antimatter, or black hole engines, moving much more mass will be possible. Also, larger spacecraft will be able to have much thicker armor, due to the square cube law (doubling dimensions gives you eight times the volume and only four times more area, so thicker armor will be proportionally lighter).
Because of this, I would expect such a large spacecraft to have very thick armor, with Whipple shields to fragment incoming projectiles on top. It would probably also have multiple layers of internal armor, so even penetrating objects would not go through all the way. The Castle Bravo nuclear test of 15 megatons, which released more energy than this 1kg rock, only made a 250 ft deep hole, and that was in soil and rock, not in ultratough futuristic spacecraft hullmetal.
Any large asteroid will be detected easily by the ship's sensors at massive ranges, while any small debris, though harder to detect, will be easier to vaporize at close ranges with either lasers or nuclear missiles, and won't cause fatal damage to the ship.
 
4 hours later…
20:21
I challenge your frame challenge: That is is impossible to have a nuclear winter. That the current-day nuclear stockpiles are below winter level is pretty irrelevant.
With a nuclear winter, civilisation collapses first because of all industrial centers being demolished, then because there is no way to operate all the energy sources without industrial centers, and finally because you cannot grow enough food without artificial light. A nuclear winter is survivable only for those who have stockpiled enough; for worldbuilding purposes, just assume that this winter is long enough. Or assume that
3. Sure you can move asteroids, just not quickly.
The Martians may be content to just fend you off. Even humans don't do genocide usually just because some native shot an arrow at your battleship, unless they are really bored or have been waiting for a genocide excuse. (I find the premise that Mars has an advanced civilization a quite labored and often-ridiculed trope, so maybe that particular worldbuilding isn't very enlightened - or, on the other hand, the author may have thought of something really great here.)
6. No, carbohydrates and fatty acids are pretty standard. There are only so many ways you can build a carbohydrates, and all of them exist in our ecosphere; fatty acids have more variation, but they are build from a very small set of small blocks so the human physiology should be able to digest them as well. Maybe they need to apply chemistry, such as cooking, but it should work.
20:38
6 (cont'd) Proteins are a very different matter. They are basically a chain of N-CR-CO-CR peptides, where R is a variation; this is one of the simplest chain that can be assembled and disassembled with little energy, so it' s pretty ideal as a basic building block and likely to be present. The rests (R) do vary, so here we have indeed a potential for food compatibility.
7. If the stellar fleet is moving at 0.9 or 0.99 c, a single proton hitting it can do significant damage even if the proton is motionless (relative to the galaxy). Any slower fleet would be unable to navigate to a different galaxy in an interesting timeframe. For interstellar travel, you'd go for 0.5 c, where individual particles don't carry much impact but dust grains do.
BTW I read the Quora link, and looked the nuclear winter material up. It seems that Allan E. Hall is doing a very elaborate misleading representation; in particular, his dismissal of Nuclear Winter being a thing seems pretty fishy (he completely ignores the effects of dust on solar input, a hilarious omission).

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