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For the purposes of this question, I wish to consider active, hydrogen-burning stars, not deuterium-burning brown dwarfs, or stellar remnants like black holes or neutron stars. (Though including those would be interesting questions in and of themselves)

What is the upper limit of distance that we can say, "To a high degree of certainty, every star that exists within this sphere centered on the Solar System, has been discovered?"

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    $\begingroup$ The answer depends what you mean by "discovered". If you just mean - is present in some catalogue as a point source then likely around 100 pc. If you mean identified as a nearby star out to some limiting distance, then much closer. $\endgroup$
    – ProfRob
    Aug 8 at 21:16
  • $\begingroup$ @ProfRob By "discovered" I meant the latter: found, distance measured, and known to be inside the limiting sphere. Apologies if I was unclear. $\endgroup$
    – notovny
    Aug 10 at 13:22
  • $\begingroup$ The title of your question is confusing and misleading. You should say 'distance' instead of 'radius' there. $\endgroup$
    – Thomas
    Aug 11 at 7:14
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Your best bet is probably a distance-limited catalog designed to include everything within a specific distance. The most recent such compilation I'm aware of is Reylé et al. (2021), which has a limit of 10 pc and includes slightly more than 300 (hydrogen-burning) stars, along with about 20 white dwarfs and several dozen brown dwarfs.

They note that they are probably incomplete for very late/cool/faint brown dwarfs and also faint white dwarfs, but they seem pretty confident they have all the (hydrogen-burning) stars.

So the safe upper limit would be 10 pc.

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  • $\begingroup$ This is nice, but as explained in my revised answer I think it actually answers a different question than the one asked by the OP. (In my notation, it gives an estimate of $r_{bd}$ rather than $r_s$, which is a different quantity.) In any case, I've enjoyed the intellectual back and forth of discussing this stuff with you in comments. $\endgroup$
    – user15381
    Aug 8 at 17:37
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    $\begingroup$ @BenCrowell The question asked about stars (not brown dwarfs) that "have been discovered", not "could potentially be discovered". So the 10 pc limit is appropriate, since that's a published volume that has been thoroughly searched. $\endgroup$ Aug 8 at 18:17
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Thanks for asking this question. I had fun looking up information and learning more by trying to figure this out.

As we get to the lowest masses for hydrogen-burning stars, the luminosity as a function of mass is either discontinuous or nearly so. The derivative becomes either infinite or very large. (From now on I'll just say words like "discontinuous" without the qualifiers.) The following graph, from Kroupa 2002, shows this feature:

enter image description here

The graph has mass on the y axis and M_V on the x axis, so the bad behavior shows up as a horizontal graph (derivative of the inverse function blows up). For more on stars right at the hydrogen-burning boundary, see this article by Lodieu. Figure 1.2 is helpful, showing how the luminosity tracks bifurcate for the two classes of stars. The critical mass for hydrogen burning depends strongly on metallicity, as do temperature and luminosity. For a star with solar metallicity, a star at the critical mass has $M_V\approx 19.5$ and $M_K\approx 11.5$, but these numbers are really sort of ill-defined because one is essentially zooming in on a discontinuity (or near-discontinuity) in a graph and trying to pick off a $y$ value. The way this mathematical ambiguity manifests itself in reality is that $y$ depends very strongly on other factors, such as metallicity.

As we approach this critical mass, the luminosity drops discontinuously, and therefore so does the maximum radius of detection. So if the question is to be well posed, we have to distinguish between two questions. Let $m$ be the mass of the star, $m_0$ the critical mass for hydrogen burning, and $r$ the greatest distance at which it can be detected. Then we have two different limits. There is one for stars,

$r_s = \lim_{m\rightarrow m_0^+} r,$

and a different one for brown dawrfs,

$r_{bd} = \lim_{m\rightarrow m_0^-} r.$

Looking through the list of nearest stars on WP, we get stuff like M8.5 stars with an absolute J magnitude of about 11.5, per a correction by Peter Erwin. The J band is centered at about 1.2 μm. It looks to me like an M8 is very close to the minimum mass for hydrogen fusion. The WISE survey is a fairly recent full-sky infrared survey, and it used a set of filters called W1 through W4. W1 is centered at about 3.4 μm. Looking at this thesis by Silverstein, usually the W1 magnitudes of cool objects like this are about 1 magnitude brighter than their J magnitudes, so maybe a star like this would have a W1 magnitude of about 10.5. WISE's sensitivity varies a lot in different parts of the sky, but they seem to be confident that they can detect anything with a W1 magnitude lower than 16.6. This gives a distance of roughly 160 pc. This is probably a reasonable estimate of $r_s$.

From Peter Erwin's answer, we know that $r_{bd}\gtrsim 10$ pc, but the question is really asking about $r_s$, not $r_{bd}$.

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    $\begingroup$ It's not correct to say "the luminosity drops to zero" for the low-mass end of the hydrogen-burning main sequence. See, e.g., 2MASS J05233822-1403022, which has an absolute visual magnitude of 20.6 (and a distance of 12 pc). $\endgroup$ Aug 7 at 21:33
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    $\begingroup$ The M8.5 stars in the Wikipedia list have absolute $V$ magnitudes of around 19; their absolute $J$ magnitudes are about 11.5. $\endgroup$ Aug 7 at 21:43
  • $\begingroup$ @PeterErwin: Thanks for the correction about the J magnitude. That makes a huge difference! Re the luminosity dropping to zero, as I said, that was an idealization. $\endgroup$
    – user15381
    Aug 7 at 22:32
  • $\begingroup$ It's not ideal, it's wrong and simplifies away physics crucial to getting a correct answer for this type of question. It's simplification at best, if you want. Yet there is a minimum mass, and thus a minimum magnitude as quoted by Peter $\endgroup$ Aug 7 at 23:44
  • $\begingroup$ @planetmaker how is it wrong? A star that's minimally above the mass limit for burning hydrogen could have an arbitrarily low burn rate. I can imagine stars burning something else (D was mentioned) and fall out of the classification; how about a gen-1 star that is just H and He, barely above the H fusion threshold - could such a star still exist today, with very little other elements picked up so it still burns "essentially just H" but at a really low rate? $\endgroup$
    – user132372
    Aug 8 at 8:30

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