When humanity developed gravitic technology in the later half of the 21st century they suddenly had easy access to the Solar System. Being human that wasn't enough for them. They used breakthroughs in suspended animation to create the Low Berth and sent sleeper ships to Barnard's Star and the Centauri system.
A ship could have as little as 20% fuel to make a one way trip. The fuel would allow a 32 week boost and deceleration at 1 gee using 16% of the fuel. The trip to the Centauri system was made at .6 c and took 7 years, 2 months. For most of this time (372 weeks) the ship's systems would be powered down and 1% fuel mass would last about 100 times normal or 400 weeks. Note artificial gravity, heating, and such were shut down. Most ships carried double this amount, at least initially to let them return if necessary.
Alpha Centauri A had a habitable garden world with a dense atmosphere and shallow seas. Colonization began when no evidence of intelligent life was found. Colonization was far from easy. The native life was extremely aggressive and almost impossible for humans to digest. Fortunately the opposite was true. In fact the toxins and chemical weapons the local flora and fauna evolved were rarely dangerous to humans and were even useful in the manufacture of drugs.
The colonists had several decades of little contact with Earth and they liked it that way developing a libertarian sort of society with a decentralized government. Then the jump drive brought Earth practically to their doorstep. The colonists were suddenly faced with a mother world that was hungry for their bio resources. After several riots broke out over the waves of new immigrants pouring in Earth ousted the colonial government and installed a military governor. Many colonists continue the oppose this government from the wilds. The resistance is young and lacks real organization and is waiting for a leader to step forward.
Many of the STL ship crews never settled down on Prometheus, instead building outposts on the uninhabitable worlds or building stations all the way out into the Oort Cloud. These outposts are remote and scattered enough for it to be a pain for Earth to police constantly. Many colonials resisting the Earth government base themselves here. Catching the flare of a ship jumping is not a sure thing at fractions of a light year. finding their bases means jumping among hundreds of dwarf planets. Sometimes a sensor will produce evidence of a base but the patrol ship will arrive to find the pirates or rebels vacated it weeks or months before.
Barnard's World is an icy moon of Barnard's Giant. Barnard's Farther Giant has several moons but no permanent colonies as yet. The moon has large subsurface ice deposits making colonization easier. Its inhabitants are mostly employed by StarGen, a company researching stellar phenomena such as flares and variable luminosity. Most people simply refer to it as the Company.
Stargen operates several research stations at other moons. BGG has relatively safe and low power radiation belts for a gas giant. This can change fast when the star has a flare. Monitor stations keep a close watch. A flare can mean several hours of people retreating to 'storm cellars' protected bunkers till the period of high radiation has passed. Most local ships have similar shelters.
The company had very strict rules in place for their workers from the start. As a result when Company FTL ships arrived very little changed. Immigration is tightly controlled and non-company visitors are restricted to the starport. A number of people immigrated from Prometheus with the Earth take over (some via STL ships) and as a result the port has a very libertine atmosphere. The Company turns a blind eye to most of the goings on there. Company security patrol access to the starport but don't enter it. Corporate employees who enter it are warned they're on their own.
Besides research water mining and tunnel engineering are major occupations. The planet also has a small but growing tourist trade due to the rough pleasures of its starport and incredible views of its gas giant.
Recent discoveries point to the planet having several shells of alternating ice and water. Initial research shows some of these layers may have biochemical or even life. This has raised concerns regarding quarantine procedures and possible infection.