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The mining ship Red Dwarf left the solar system after the accidental radiation leak that killed the crew.

We know it travelled for 3 million years - and by the time season 1 was underway they hit light speed.

Do we know how far away from Earth the ship got?

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According to the script for Back in the Red, Part I, the Red Dwarf is 3 million light years from Earth. I think we can assume this number has been rounded up or down slightly.

CMO: What's this rumour that we're three million light years into Deep Space, and Red Dwarf's changed shape?

HOLLISTER: That is classified information, Karen! Who the hell told you that?

Presumably the ship was traveling at a very appreciable fraction of lightspeed for the majority of its outward journey, then continued at an unspecified speed above lightspeed for several hours before finally coming to a halt. Their return journey was at a much lower speed, allowing them to stop the ship for fun and frolics.

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    That being said, the ship has traveled through time (on numerous occasions), been stolen, traveled into an alternate universe and navigated a system of wormholes. The fact that they even have the faintest clue where they are is nothing short of amazing.
    – Valorum
    Apr 1 '16 at 18:43
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    This also neglects to mention the fact that the Milky Way galaxy is only 100,000 light years wide.
    – Valorum
    Apr 1 '16 at 18:55
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Currently, not a clue. I don't think anyone knew.

Red Dwarf has, presumably, been travelling in a straight line for three million years when Lister awakes from stasis, presumably to stay out of the space lanes to stop radiation contamination. Then, of course, it hit light speed and Holly tried to turn around. I don't think he managed it (Lister did say it would take two weeks to turn, but he may have been exaggerating).

After that, we know that they found remnants from Earth - simulants, gelfs, derelicts etc, and of course the Nova 5 and Kryten. In "Infinity Welcomes Careful Drivers" it alludes to how far the Nova 5 came out from Earth in order to ignite stars for the Cola company, but I forget how far that was (and it's a different continuation).

After that, of course, Red Dwarf is stolen by the nanites so the crew move to Starbug and enter stasis for 200 years, after which they turn around and enter stasis for presumably another 200 years. We also have the period where the crew go into stasis so they can get to the Nova 6.

Put bluntly, I don't think anybody has a clue, but right now (as of The Promised Land) they're probably roughly back where the Nova 5 was, assuming the Nova 6's mission and area it was in was similar.

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