So I'm wondering, how cheap would ftl travel have to get to become practical for transporting people and establishing bases for civil and military use?
- Why colonize? For science? To bring back raw materials? To escape a wrecked Earth? For the glory of your nation?
- Are there Earthlike worlds in your setting, where a colonist can breathe the air, drink water from a creek, build a log cabin, and plant crops?
- How many trips does the ship make per year, what is the payload, and how easy is the life support? Transporting 100 kg of human will be more expensive than transporting 100 kg of ironmongery.
Say a person with clothes and a few personal items is 100 kg. If water is readily available, dehydrated food should be less than 1 kg per day, 365 kg for the first year until crops come in. Add an axe, a saw, a hammer and a bunch of nails, a spade and agricultural tools, plus seed corn, and we're still talking about less than one ton per colonist.
If there are already farmers in place from the last colony ship, it might be possible to omit the food and to bring the tools of a trade instead. The colonist would buy food locally and start working as a carpenter, blacksmith, whatever.
How much would the colonist be prepared to spend on the ticket? Depends on conditions on Earth and on the colony world, of course, but I'm going to pull 50,000 out of thin air, or 100,000 for a family of two (no kids yet). It doesn't matter if that is EUR, USD, or GBP, this is just a ballpark. So $100 transport cost per kg would really get the colonization going.
If the ship has 10% payload, a round trip is one year, and it pays for itself after 20 years (with something to spare for operational costs and interest on the investment) the ship should be about $200 per kg of loaded weight.
Assuming by FTL the question implies interstellar travel, rather than around the solar system. Presumably our solar system will be well colonized before we advance very much outside of the solar system.
This would depend on many factors, but primarily the answer is "cheaper than it is currently to transport people into space at sub-light speeds."
Currently, most people, apart from multi-millionaires cannot afford to go into space at all. However many people can afford to fly across the country. Fewer, but still a significant portion of the population can afford to fly to another continent.
Presumably when the cost per person gets down in the range of going into space today, then we may see a lot of interest and an increase in activity surrounding colonization efforts.
Consider this from the perspective of circumnavigation in Columbus's day, or more accurately in the prime of the Viking civilization, a few hundred years before.
By the time the price comes down into the range of intercontinental travel today, exo-planetary colonies will be well established as were the North American colonies by the mid to late 16th century.
Other factors include capital interests which range from public and private logistics to government taxation and growth strategies - all surrounding the public interest in going to the new place. Parties involved in these factors invest in the necessary resources or subsidize those resources outright and that helps to bring down or pay the cost of transport respectively.
Cheaper than base yield: Even if FTL travel cost one millions dollars per flight, company would launch spaceship regularly if the operation is profitable. Asteroid tend to contain a lot of valuable mineral, so a mining colony can easily break even, especially if earth mine run out of minerals.
Cheaper than NASA budget: Space exploration get large amount of money from government and don't have to be profitable, just collect and exploit data. With FTL travel, NASA and other space agency would certainly launch probe and establish base on interesting point.
Well... "practical" isn't really a specific number so let's discuss that first...
For research; it's a 1 way trip and as we have seen with Nasa their programs are designed around a 10 or 20 year timeline.
For "realistic" colonization, back during the colonial days 1 to 2 year trips were the norm, however they were paid for by being able to carry trade goods back.
For Military practicality, to hold a territory you need to be able to get there within 1 month and back in medieval times any major conflicts would take 3 month travel times with an assumed return trip because of seasons, and travel time.
Commercial Flight travels 1 day or less travel time between nations and continents...
We also need to consider a crew.
Research: No crew needed.
Colonization: About 1000 people needed.
Military: About 150 or more people needed.
Commercial Planes: 1 to 100 people.
Now with regards to the energy requirements, what you're asking is really more about mass and volume, not so much energy itself. For example, I can't carry Jupiter areound with me so it's impractical, it's not because of the "energy", but because of the mass/volume associated with that energy. If you could compress Jupiter down to a cubic meter, it would be practical with Warp and Gravity (maybe), but not hyperdrive, or wormhole drives... and it might be a hard sell to make a warp ship with no sub-light capabilities.
So what we're really looking for is a volume of matter whose mass is low enough that can be moved with sublight thrusters and can produce high energy output.
So the first thing, lets say we have a thruster that can direct itself in any way (it would be a horribad design but we could do it) the best thruster we have is a Vulcain which has thrust capability to 1140000 Newtons.
This means the maximum wight of any ships we produce must be no more than 116247.64794 kg per thruster added.
The engine itself weight 1686 kg and every second of thrust you want you have to include 235kg.
Let's say I want 2 minutes of thrust available. That's 28,200 kg, which leaves me with 88,047.64794 kg. Let's just make this easy and say sensors and the other components are 47.64794kg. This is unlikely but, we're trying to get a minimum. so we are working with less than 88,000 kg of fuel for FTL per thruster.
Assuming you can fill that with all Uranium, which you can't in a practical set up the amount of energy that would be contained is 7,098,349,140,000 Mj.
Now the next thing is going back to the Practical for application thing...
If we assume the average density of 1 star per 4ly that means for:
research - 0.25 to 0.5c speed is fine for research speeds, but that's not FTL so moving on.
Colonization - 4c would be "ok", but with our set up you couldn't do a return trip which means this isn't practical.
Military - There isn't much to use military out that way so, again not practical.
Commercial Planes - Adjust the weight to average person is 80kg, round up to 100kg and multiply by 100 to get 10,000kg reduced from our previous answser, to 78,000kg of uranium and you are left with 6,288,360,000,000 Mj. (we can further refine this by getting the weight of the fusalage of a plane, but meh)
Now remember we said this type of travel takes 24 hours, and we want a return trip as well so the minimum for this is 1,460.97c
Which means we need fuel efficiency of 2,152,118,113.30828148421938848847 Mj per factor of c per day or reduced further 89,671,588.054511728509141187019583 Mj per hour. This would need to be modified to include the weight of the ship and the fact that Uranium isn't a practical fuel source, especially not in this quantity and not in this proximity to humans...
If you increase the number of thrusters and look towards a colony ship you can change these numbers though because you only need to be able to travel 8ly at 4c rather than 8ly at 1461c.
And the amount of Uranium this would take would cost \$17,196,056.45.
Commercial plane fuel costs \$140,000.
So with current technology you need to get 100 times cheaper at minimum...
It really does depend on what society is willing to pay.
The closest comparison you're going to get is the Moon Shot project from the 1960's.
The US President made a promise to put a man on the moon by the end of the decade, and he gave NASA the budget to do so. Adjusted for inflation, the amount spent would be over $200 billion in today's money. That's something in the order of 2% of the entire US government budget.
History would show us that it is practical to pour that kind of money into a space program.
I'm guessing that an FTL project would an order of magnitude bigger than that (though it does very much depend on the nature of the FTL that you're devising). At that price, it would clearly need to be a global endeavour with most of the world's nations joining together to pay for it. One nation alone would struggle, even a nation like the US.
So it's a global project. The question is: Would the nations be willing to pay for it?
To me, the answer to that depends on what they're getting out of it. If we know that there will be inhabitable planets to colonise, or alien races to meet, then the answer is an absolute Yes. The opportunities inherent in that are enormous. But more importantly, the risk of being the country left out of such a project would make it unthinkable for any of the major developed nations not to be involved.
So I would say that even if your project costs multiple trillions of dollars, it would still be "practical" for human society of achieve it, as long as the perceived rewards were high enough. In fact, I'd go so far as to say that there is no practical upper limit to the cost. If human society is motivated enough to do it, then it could be done. At extremely high costs, your limiting factor isn't money but resources and labour. But hitting those limits would mean that the project would take longer, not that it couldn't practically be done at all.