Now, the human race has been expanding exponentially at the historic average of 2% per year. That means that, on average, the number of people doubles every 35 years. It's crowded here, and we've got a starship and an empty planet only 4 light years away. So we load half the population and take them to Alpha Centauri. It took (according to some estimates) 20,000 years for homo sapiens to get where we are today. Do you know how long it will take us to populate Alpha Centauri to today's levels? Only 35 years.
Okay, it's 39 years later (Four years transit time plus 35 years of growth), 2050, and now you have two crowded planets. No problem, Barnard's Star is only 6 years away from Earth, and Wolf 359 is 8 years from Alpha Centari. So we pack up half the population of Earth and send them to Barnard's Star, and we take half the population of Alpha Centauri and send them to Wolf 359. Again, it will only take 35 years to fill each of the planets. By 2093 we will need to find 8 more planets. We now have a colony on each of the stars within ten light years. 35 years after that, and we will need 16 planets, 70 years and we'll need 32, then 64. By 2200 we will have colonized all the stars within 20 light years.
By 2360ish we'll hit a snag. We will have populated all of the stars within 35 light years of Earth. Colony ships leaving Earth at this point will not arrive at their destination before it is time to send out another colony ship. Of course, all the other colonies will be sending out their colony ships as well. We'll need another 512 planets. At the end of another 35 year cycle, we'll need 1024, another cycle and we'll have used up all the stars within 50 light years.
Scientists estimate that there is about one star per 280 cubic light years. In 800 years or so, our empire will need 34 million new planets. However there are only some 19 million stars within 800 light years. In other words, we will have outgrown our ability to travel.
Today we have 7 billion people on the planet. By 2150, your target date, we will have 36 billion people. Your 50/50 by 2150 plan would result in each person having only half an acre of land on which to live and support themself. This [netdna-cdn.com] suggests 2 acres per person are needed. 50/50 by 2150 would result in 3/4 of the population starving to death.
It's basic mathematics. A fixed resource cannot supply an ever increasing population. Any plan that does not include zero population growth and 100% recycling will eventually fail.
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