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We are locusts consuming everything, then move on to consume abother planet
but fysically.. realisticly we could build vessels that reach say half of light speed.
with solar sails.. and dyson rings (that do not have to be solid constructs but swarms of satelites).. and lasers to power said solarsails.. and perhaps electric engines... that catch intersteller matter (space is not empty just very very low density but the faster you go the more your matter scoop will scoop)and use ionisation to speed it up
with that speed the closest star would take 8 years...
surely the issue is that life excists on no other planet.. and we have no knowledge of terraforming.. but well we might in 200 years have biodomes.. closed of domed cities or spacestations with the means to produce food water and oxigen to sustain us..
from there another mission could eventually be launched...
if we presume any star ove 100 lightyears far is out of reach (since it needs to be reachable in a few humanlifes.. generational ships too have their limit)
and it takes upon arrival a reasonable 500 years before that base is populated enough and ready to start launching rockets out on their own..
well now.. in about 100.000 years we could than have bases on every star in the milkyway.. not that there be any sort of space empire.. the distaces still be very vast and thats longer than modern humans have been around...
but going true interstellar as in to another stellar system... no.. not realisticly.
there is a small change we might get zero point energy, and a warpengine.. which are theoretically possible.. but it remains to be see if they ever can be turned a reality.
andomeda, the closest other galaxy is 2.5 million lightyears away.. with the milky way being 0.1 milliuon lightyears acoss and no nice stars to hop on as stops in between.
it also makes the distance for the lasers and solarsails.,. the amount of supplies the durability of shielding.. the generations aboard.. truelly insane...
it would be a single trip of 5 million year...
unrealisticly we ever would be able to make it with conventual physics.
We're pretty much doomed to reside in this system.
"Any sufficiently advanced technology is indistinguishable from magic."
Arthur C. Clarke
well we move at half the speed of light in ALL directions... so that 100000 light years is the diameter while essentially we only have to travel the radius..
earth is 26000 ly from the center... so it's 24000 ly to one edge and about 76000ly to the other.
basicly you have earth launches in all directions .. than all those colonies launch in all directions.. it's a snowball effect.. not a straight line..
it might take a little longer to fully reach that furthest edge than 100000 ly if we are truelly stuck to half light speed... if locked at that speed it would take 150.000 years.. but we might be able to go a bit faster without relativistic effects to much a problem and invent ways to do that as we get mor experience with hopping one star to the next..
spreading at 0.76c would be enough to fill ALL the milky way in 100k years
but spreading at 0.5c would already cover most of it..
Anyway, I think interstellar travel is always going to be extremely difficult. Unless we discover FTL travel (impossible unless there are yet-undiscovered new physics), we're probably never going to be a truly interstellar civilization, much less an intergalactic one.
Every solar system will have to be fully self-sufficient, since communications are limited by the speed of light and transportation is even harder.