Not parsecs from Earth. Seconds of arc from Rigel on the celestial sphere. In a cone with a base radius of 1 parsec from Rigel (Beta Orionis, some 800 light years from Earth), there is a volume of around 8000 cubic light years. Expand the radius to 2 seconds of arc, and you get around 36000 cubic light years.
The terms "parsec" and "second of arc" are not synonymous. An arcsecond is a unit of angular separation; a parsec is a unit of real, physical distance equal to 3.2616 light years. It's the distance at which something 1 AU in diameter would subtend an angle of one arcsecond. You appear to be using "parsec" interchangeably with "arcsecond," and that's gibberish.
And you still don't seem to understand what I was trying to do. I'm critiquing Mandel's idea of Beta Rigel as a star that happens to be in the direction of Rigel, but much closer. The Beta Rigel in
Star Charts is more or less exactly aligned with Rigel as seen from Earth (at least in the plane of the map, but implicitly in three dimensions); if such a star existed, it would form a visual binary with Rigel in the sky and we'd know it was there. My point is that there is no such star that close to Rigel in the sky; that the nearest non-red-dwarf star to Rigel (in terms of angular separation) that's relatively close to Earth (in terms of physical distance) is 29 Orionis, which is about one and a half degrees away from Rigel in the sky (in other words, over 5500 arcseconds away), and is much farther from Earth than Mandel's Beta Rigel.
Within 12.5 light years of Earth, there are 33 stars of various sizes. That's 8200 cubic light years.
In the sphere of space within 20 light years of Earth there are 109 stars of various sizes. This is a sphere of 33500 cubic light years.
So in the a cone extending to Rigel with a one parsec radius, we can expect to find at least 30 stars, some of which should statistically be amenable to life.
Assuming you mean to say "arcsecond" instead of "parsec"... you're making the false assumption that the stars are evenly distributed. What you're saying is true on the average, but in actual fact, any such cone could just as easily contain 50 or 10 or 0 stars.
Besides, what I'm talking about is not a theoretical estimate. I'm telling you, I'm looking at a graphical representation of what actually
is there in the sky in the direction of Rigel. There are no horizons in space -- you can see everything that's there if your telescopes are sensitive enough. We already know every bright star that's out there in that direction, because we can
see them. And the
documented fact is that 29 Orionis is the (angularly) closest bright star to Rigel. You can see for yourself by going outside and
looking through a telescope. This isn't speculation, it's something we know for a fact because people have actually gone out and
looked.
And yes, it's true that there are probably dimmer stars within that cone that we haven't discovered yet. I've already acknowledged that. And I've already explained why they couldn't possibly be ST's "Rigel." The Rigel system in ST has at least 12 planets, at least 7 of which are at least marginally habitable. That's hard enough to justify in the case of a large, hot star with a wide habitable zone. It's impossible to justify for a tiny, dim red dwarf whose habitable zone would be barely wide enough to fit one planet.
I don't know how fine your catalog's resolution is, but if there aren't a few golden stars between here and Rigel, then it's incomplete.
If there are any G-type stars not yet discovered in that direction, then they'd have to be pretty far away. Red dwarfs are dim, so there are ones quite nearby that we still haven't discovered. There are still new ones being found within 30 light-years. But Sunlike stars are considerably brighter, so obviously they'd have to be farther away to fall below our current detection threshold. So any such unknown G-type star as you propose would have to be much, much closer to Rigel than to Earth, which makes it useless as a candidate for Mandel's Beta Rigel. You keep trying to broaden the parameters beyond the range that's relevant to the specific point I'm making. Mandel specifically proposed an A5V dwarf that's within forty light-years of Earth and is roughly in the direction of Rigel. My point is simply that there is no real star that corresponds even roughly to that. Whether there's, say, an unknown G star 500 light-years away in the southern part of Orion is completely irrelevant to that point. Besides, if it were that far away, what would be the point? Why not just use Beta Orionis itself? The whole idea of a second Rigel is that it's substantially closer to Earth.
And as I said (twice!), any previously or currently unknown stars within a reasonably close range would have to be too small and dim to support the large number of inhabited planets Rigel has in Trek.
Who said we're only talking about one star? Do they ever describe all twelve planets as in orbit around a single star?
Again, I'm responding specifically to Mandel's premise from
Star Charts, which does portray Beta Rigel as a single star with twelve planets. More generally, to get back to the core topic, "The Doomsday Machine" specifically refers to the Rigel Colonies, plural, suggesting that they occupy a Rigel system with more than one inhabited planet.
