What is the actual feasibility of interstellar travel?
For instance, a manned journey to Proxima Centauri, given the distance, what technology would need to be developed/perfected for such a journey?
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Given no time constraint, there are only two pieces missing for a manned journey to Proxima Centauri. Resources (money) and commitment. Getting there and back in a single human life time would be a totally different question/answer.
There are several, drives that can get us there, the solar sail probably being the simplest and easiest to maintain over the long time/distance. All of current propulsion options would take time measured in generations.
We have had humans constantly in space for decades now, so technologically speaking all the major hurdles of maintaining life in space are overcome.
There are some medical and psychological considerations, most of the medical having to do with adapting to space in a way that make re-adapting to earth difficult. The psychological considerations can likely not be fully addressed without going on the journey.
There are several logistical issues, to balance available resources to allow for a successful mission. You could capture an asteroid (preferably mostly metal) and hollow it out, put 500 people in it, and farms/gardens to support 5,000, and send it on its way. but the resources to accomplish this would be significantly greater then a simple trip to the moon, which we have not repeated for several decades.
Practically you could send a small envoy in a smaller man made craft, a crew of 3 to 10, with frozen sperm/egg/embryo DNA to maintain genetic diversity.
Edit: An alternative for building a huge ship to sustain a whole colony for thousands of years of travel would be to send cryogenized people, and unfreeze them once arrived; a middle term solution in case we do not manage to handle adult unfreezing shortly is to send frozen embryos and have robots hatch them once arrived on / near habitable land.
A baby girl recently was born from a 25 years frozen embryo. Plus babies as young as 24 gestational weeks are now able to live a normal life after months of careful mothering... nothing seems impossible when you've seen this.
I know this is horrible for the children who will come into life on this other stellar system.
The question assumes interstellar travel, so we're already indulging in far fetched science fiction. So I'll put aside my pessimism and indulge my inner space cadet.
We have a greater incentive to open a new frontier when our population growth becomes limited by resources available. Then the apparently exponential growth levels out and an S curve asymptotically approaches our ceiling. This is called logistic growth.
If we manage to break free of Cradle Earth, there is enough real estate and resources in the Main Belt to postpone bumping into the logistic growth ceiling for centuries or even millennia. This would be a great setting for gaining experience in maintaining biomes. Sunlight is still fairly abundant. The asteroid city states need not be entirely self sufficient as they could trade with other asteroid city states as well Earth, Moon and Mars.
Beyond the Main Belt are two healthy asteroid populations: the Trojans residing at the sun earth L4 and L5 points. The Hilda asteroids are natural cyclers between the Main Belt and the Trojans.
Building asteroid biomes would give some experience in burrowing. The moons of the Jupiter, Saturn, Uranus and Neptune could host subsurface biomes.
But as we get further from the sun, solar energy is harder to come by. At Jupiter and Trojan distance, insolation is about 1/25 what we enjoy on earth. At Saturn 1/100. At Neptune nearly a 1/1000.
I believe developing fusion energy sources would be prerequisite for developing real estate in the Kuiper Belt and the Oort Cloud. Perhaps non stellar fusion power is possible, who knows?
Oort city states would be more isolated than Main Belt or Trojan city states. There is more incentive for self sufficiency.
By the time human kinds expands to the outer Oort we would already have plenty of experience in maintaining biomes that are self sufficient for centuries or even millennia. These city states could easily drift off into interstellar space.
Given fusion power and lots of water ice (as many Oort bodies are thought to have), the city sates could send off hydrogen ions at a good fraction of light speed. The thrust would be pathetic but given centuries to do a burn, the interstellar city states could manage substantial delta V.
I'd say that we need the construction capability to build an O'Neill cylinder that is strong enough to survive thrust along it's axis, that contains a balanced biosphere, and an energy source that will last the journey.
This will give you a generation ship or, another way to look at it, an independent space station that can move from star to star.
We could do some of that today.
The cylinder itself is just an engineering and resource problem. Once we have the infrastructure in space to make it, we can make it.
The closed cycle biosphere is a bit tricky but there is no real new science involved, just experience balancing things and finding stable balances so things self correct instead of spinning out of control like we have seen in several experiments.
The kicker will be the energy source. Solar power only really helps out to the asteroid belt. After that, we need a different power source.
Point 3, while possible, is not something we should try to plan for. To use it in a story you have to invent handwavium (though less handwavium, possibly than FTL).