I mean at the end of TOS, they'd barely mapped a 1/4 of the alpha quadrant...
Where does this datapoint come from? We get the figures 11% and 19% at the start of TNG, which would be from 1/2 of Alpha (or a comparable volume) to 2/3 of it.
given the Enterprise's top speed back then, they would have mapped a tiny fraction of the alpha quadrant
We know the ship could do at least a couple of thousand lightyears per season, from the sum total of explicit distance references in episodes like "Squire of Gothos". A dozen ships like that would quickly cover much of Alpha, even if only in the sense of making UFP-explored space look like a really tiny hedgehog with really long spikes. The thing is, though, those other cultures would also be exploring, and it would be the sum total of these explorations that would bring together 800 cultures in an eyeblink.
The stellar disk of the Milky Way Galaxy is approximately 100,000 light-years in diameter, and is, on average, about 1,000 ly thick.
So, if we do that into quarters for the quadrants, we have 25,000 light years across and 25,000 light years in length, for the alpha quadrant, and still 1,000 light years thick... that gives a standard volume of 625,000,000,000 light years cubed just for the Alpha Quadrant... basic math there.
So given the Enterprise (Constitution class
), had a maximum speed of Warp 8, and a maximum safe speed of Warp 6 (As explained in TOS: That Which Survives
), we can avarage that the Enterprise did a comfortable speed of 216c
So again, back to maths...
Warp 6 = 216c...
So it would take about a week to reach Alpha Centauri (4.3 light years from earth)
Just to put that into context for you... it would take just over 115 years to fly from one side of the alpha quadrant to the other at Warp 6... Hardly fitting within the '5 year mission' perameters...