The voyage duration of c. 50,000 years for the Centauri Princess is largely governed by the fact that the Alpha Centauri system is passing by our Solar System in its travels about the center of the Milky Way galaxy, and any high-mass/slow-cruising craft launched from Earth needs to get there within a reasonable time window.

It is a matter of scientific fact that between now and the next 30,000 years, the Alpha Centauri system (and any planets that may exist therein) will gradually draw closer toward our Solar System by about 1 light-year from the present day 4.3 light-years distance separation. This is made clear in the following kinematics diagram (sourced from the German Wikipedia):


Relative distances between the Sun and nearby stars over thousands of years. Click the image to see a full resolution of this graphic.[Source: German Wikipedia]

Thereafter, the system will start to recede away from us again, widening the gap and making any journeys between Sol and Alpha Centauri increasingly dis-optimised (see original discussions on this by Ahad).

A ball pak of circa 50,000 years then tends toward "the longest" timescale, travelling at the "slowest speed" that a spaceship can ever take to reach the system, given the relative independent proper motions between Sol and Alpha Centauri.
Escape sequence three

The relative motions between the Sun (Sol) and Alpha Centauri in the next 50,000 years, showing an optimised journey by an interstellar ark at slow speed. Click the image to see a full resolution of this graphic. [Source: Abdul Ahad]

 For example, if the journey were to take 100,000 years instead of 50,000 years then Alpha Centauri will have drifted away to a far range of as much as 8 light years - twice its present day distance.

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