Why are nearby stars like Proxima Centauri and Barnard's star not visible to the naked eye?
Proxima Centauri is the closest star to the earth other than the sun yet it isn't visible to the naked eye, but stars that are further away are easily found in the night sky. Why is that?
Of the estimated 100 billion stars in our galaxy (1.2 billion currently mapped), only about 2000 are visible to the naked eye. As @john notes, the only reasons we see these 2000 stars is because they are either 1) fairly close to us, or 2) extremely large stars. Alpha Centauri (which we can see) is about the size of our sun. Proxima is much smaller.
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@barrycarter: are you sure its only 2000? I remember last year I was lyieng on the beach of denmarks shore the dune behind me blocking all the light from the village, and no other source of light around but a very clear sky. All light I had was comming from the stars and the moon. looking at the sky felt like there is a white powder spread all over the sky that close all the stars I was able to see had been to each other. My memories might trick me but if someone had asked me how many stars I had seen I would have bet on ~5000
It's about 5000 in the whole of the night sky, but of course you can only see a bit less than half at any one time.
@Zaibis I was going by the 1628 star "Catalogue of the Brightest Stars" (http://cdsarc.u-strasbg.fr/viz-bin/Cat?V/53A), the first star catalog I ever used, but you are correct: you can see stars fainter than magnitude 5 when conditions are good. The real number probably lies between 1628 and the 9110 stars listed in the "Yale Bright Star Catalog" (http://tdc-www.harvard.edu/catalogs/bsc5.html) down to magnitude 7.
@RobJeffries Slightly complicating the exact count if you want to be pedantic (and apparently I do) is the fact that the Andromeda galaxy and both Magellanic clouds can be seen by the naked eye, which is only possible because the light from all those stars combine to make them visible at that distance. So depending on how important it is that you be able to *distinguish* the stars from each other, the count is somewhere between 5000 and 1,000,000,000,000 stars.
John's answer is correct. For a few more details:
Stars brightness works out to roughly the 4th power of their relative mass. This falls off a bit for very large stars, but for smaller stars it's in the ballpark. (Red-giant phases not included.)
Proxima Centauri is about 12% the mass of our sun, and about 14% the sun's diameter. At 12% its mass, its brightness, using the 4th power rule, is about 0.02% or 1/5,000th as bright. That's total light. Because it's smaller and colder, much of that light is in the infra-red range that we can't see. In terms of visible light or visual magnitude, it's about 1/20,000th as bright as our sun. That means, for it to be as bright as our sun from Earth in visible light, it would need to be about 1 million KM away. If it was as far away as Pluto, some 40 AU on average, it wouldn't look very different than Venus at Venus' peak brightness, with a distinct red color.
Larger stars can have a brightness over 100,000 times brighter than our Sun. So, because the mass of the star affects the brightness to about the 4th power, what it really comes down to is size matters.
Distance matters too, but only to the square of the distance, so mass of the star matters more. Here's an article with a chart of the visually brightest stars in the sky. Negative numbers and smaller numbers correspond to greater brightness.
It might go without saying, but virtually none of the stars that can be seen with the naked eye are red dwarf stars even though they're the most common type of star. They're too dim for us to see at multiple light-years distance.
In fact *almost* all the stars visible to the naked eye (I'm sure you can name the exceptions) are more luminous than the Sun.
Stars vary a lot in both size and brightness. The nearby stars you mentioned are less bright than many stars that are much further away. We see the bright stars but not the less bright ones. Its like comparing the effectiveness of the old 2-battery flashlight to the 5-battery one. Both are flashlights but one is much brighter than the other.