Paul Colby said:
Well, I only need a factor of 11,780. Seriously, stars can gain and lose mass. One of the first things I looked at with the telescope I bought a while back, was the ring nebula. A star that burped out a small fraction of its mass not that long ago. There are stars that do this periodically. Can one rule out from observation such slowish mass ejections as a means of avoiding BH at end of life? Along this line, do we see as many neutron stars as we’d expect?
You are exactly correct. As you saw from information above, a black hole is created when the core is above somewhere between 2.2 and 3 solar masses (no one knows the maximum neutron star mass). So one might expect any star above 3 solar masses to make a black hole. But that's not true, they don't even supernova at all, and everything below maybe 8 solar masses (Betelgeuse) creates a white dwarf instead of going supernova. Why is that?
It is, as you say, that the star loses its outer envelope before it can be added to the white dwarf. A white dwarf can only go up to 1.4 solar masses, so you see a lot is sneaking away: an 8 solar mass star leaves behind only about a 1.4 solar mass core. So some 6.6 solar masses, the majority of the stellar mass, leaks out before the star can go supernova, just as you suggest.
However, this fact is already included in the estimates
@Vanadium 50 is using. He is not taking every star above 3 solar masses, but rather every one above 10 or 12 solar masses, to be the ones that turn into black holes. But I think the resolution to the paradox is a selection effect. To know a black hole is there, we need to see it, and generally that means it has to be in a close binary. That doesn't sound like a serious limitation, lots of massive stars start out with close companions, the problem is the close companion has to still be there after the supernova. It's not that the supernova destroys the companion, a star is a tough nut, it's that it unbinds the companion, which becomes a "runaway star." The result is you don't get to see the black hole.
Now, there is still a kind of special dance a star can do such that the black hole sticks around, and this is why we see binaries with black holes in them. What happens is, the star that will become the black hole has its envelope expand and some of it is lost and some transfers to the companion, leaving a stripped core that is still going to go supernova even though it is way below 8 solar masses (it is the mass of the core that counts, not the mass of the whole star). So a star like that could explode and not lose its companion (because by then the companion has more mass than it does, so it's really the companion that's holding it, not the other way around). These kinds of systems in their presupernova state were predicted to happen but have only recently started being seen, so a full accounting of how many there are has not been done.
The upshot of all this is, the resolution is probably (it seems to me) a combination of two factors. The first is, most of the time it is actually still pretty hard to see a black hole even in a binary, but it does seem like a much more important effect must be that it is very very hard to make a black hole and keep a close companion around, because the creation of black hole disrupts the binary you thought you were going to use to be able to detect it. But I don't say there isn't a need to count the systems that should lead to black hole binaries, and connect that to the black hole binaries, every time we start encountering things we haven't seen we always get surprised by something we didn't know was happening. (An example is supernova "kicks," it was always assumed a supernova would be spherically symmetric so would not kick the star that was blowing up, but nature abhors spherical symmetry, and so the black hole can get a pretty good kick and who knows how much that might help it get away from its companion.)
Finally, we can note that a good way to find planets is to look for radial velocity variations of the star, so the more that goes on, the more we might expect to come across invisible companions like black holes. The problem is, if the separation is wide enough that we haven't seen it yet, it means the period is very long, so we may have to watch for a while, many decades perhaps. Could take some persistence.