Launch Plane Change

A launch loop oriented along the 8 degree south latitude line launches into a plane defined by that vector and the center of the Earth. The launch plane rotates, and orbits launched into one plane will need some north/south delta V to transition to a different plane.

Over 24 sidereal hours (86164 seconds), the normal vector rotates around the pole with a circle with a radius of sin(8°) on the unit sphere. The angular spacing on that circle from the "prime" launch orbit is the plane change required from a launch at an earlier or later time.

For example, for a launch 900 seconds later, the Earth has turned 360*900/86164 = 3.76°. That is an angle of sin(8°)×sin(3.76°) or 9.13 milliradians around that circle. The plane change velocity near apogee is 2 x sin( 0.5 x 9.13 mrad ) x Vapogee. Vapogee is approximately 900 m/s for a one day construction launch orbit, so the plane change velocity near apogee would be approximately 8.2 m/s northward. Similarly, for a launch only 300 seconds after the prime launch orbit, the angle is 3.04 mrad and the plane change velocity is 2.7 m/s northward.

The velocities near perigee are much higher, and the appropriate time for the plane change is before apogee capture. Plane changes near perigee are much more expensive, because the perigee velocity is much higher and more costly to rotate.

Off topic speculation

The launch window can be reduced; this will likely be the case for a narrow window of launch velocities near the beginning of station construction, when assembly rates in orbit will be limited by the tools available, not launch rates. A 900 second window will eventually allow the delivery of 100 tonnes per day, 3000 tonnes per month, or 36,000 tonnes per year to each of 96 stations from each minimum launch loop.

ISS is 420 tonnes for a maximum crew of 6, 70 tonnes per crewmember. For a more spacious 200 tonnes per crew member, 36,000 tonnes per year is a capacity growth of 180 crew slots per year. If the crew rotates every three months, 5 tonnes per crewmember transit (including consumables), that is 20 tonnes per crew-year and a support capacity of 1800 crew per station, a total population for 96 stations of 170,000. One minimum launch loop (powered by 6 GWe) might support 100,000 crewmembers in orbit along with many large experimental tools and millions of robots assisted with telepresence from Earth; tens of millions could have "virtual day jobs" in orbit. 10 TWe of space power to an array of very large launch loops could support over 100 million visitors per year.

However, at that scale it will be better to build more permanent outposts, with centrifugal gravity and some approximation of closed-loop life support. The stations will still depend on the imported manufacturing output of a complex global economy; 25 million North Koreans still depend on the rest of the world to supply what their meager "independent" economy cannot.