"Why is the
sky dark at night?"
the following logical thought sequence (mathematical
it should be horrendously bright.
apparent intensity of a light source decreases with the square of
from the observer. (Assuming no interstellar dust absorption,
true. Lumens received from a star will vary inversely as the
the distance to that star.)
2) If the
distribution of stars is uniform in space, then the number of
stars at a
particular distance, r, from the observer will be proportional
surface area of a sphere whose radius is that distance. This area
proportional to the distance squared. A = (pi)r^2
Therefore, at each and every possible radial distance, r, the amount
toward us should be both directly proportional to the radius
number of stars) and inversely proportional to the radius
(they get dimmer with distance).
4) These two
effects cancel each other.
5) So every
spherical shell of radius r should add the same additional
6) Ergo: In
an infinite universe, if we sum (integrate) the light coming
from all the
infinite number of possible values of r, the sky should be
But the sky
is not infinitely bright. Why?
resolution of this paradox can be achieved by considering how
solved the problem of defining the ABSOLUTE luminosity
of a star. Because of (1) above, the more distant a star is,
it appears to be. In order to set up a standard, astronomers
agreed that if a star was placed at a distance of 10 parsecs
(approximately 32 1*2 light-years) from us and if it looked like a
1.0 star at that distance, they would agree to say that its
LUMINOSITY was 1.0.
There is a
well-known relationship between distance and apparent magnitude
of a star.
For example, if we put that same 1st magnitude star at a
517 LY (light-years), its APPARENT MAGNITUDE would be only
cannot see any star whose magnitude is higher (less luminous)
The 200 inch Hale telescope at Mt. Palomar can see down to
magnitude 23 or so.
approximately 8400 stars in our night sky that are brighter than
6.4. We do not see the others; they are too dim. Yes, yes, Carl
to talk about millions and millions of stars ? but we can only
8400 with our naked eyes. Carl was well known for his tendency
exaggerate. We get the impression of millions and millions when we
look up at
the Milky Way, but we can see only 8400 stars ? that's it ? and
some stars are VERY much brighter than absolute magnitude 1.0
would be visible farther out than 517 LY. But, many are much
so as a rough approximation let us consider the average star.
If it is
farther away than 517 LY, we cannot see it (AT ALL). So it might
as well not
be there AT ALL. The total light in our night sky (at least
the way we
can see it with our naked eyes) is not affected by much of
that is dimmer than magnitude 6.4 (typical stars farther away
517 LY). Even for the blue-white giant stars whose absolute
puts them at ?10 or ?12 (much brighter than absolute magnitude
exists some finite distance beyond which they too become
us ? their apparent magnitude slips down beyond 6.4.
There are a
very few vastly distant objects that we can see such as the
Andromeda Galaxy M 31. It is over 3 million LYs away. But it is
concentrated collection of stars and plasma that it looks to us
bright as a single magnitude 4 star.
The point is
this ? the infinite sum implied in step (5), above, is
The sum STOPS (is truncated) at a distance of about 500+ light
the typical star (and somewhere beyond that even for the
ones). There is an upper limit on the absolute brightness of a
there is no such thing as an infinitely brilliant star. So
there is a
finite upper limit to the integration process described in step
It doesn't go out to infinity.
It may also
help to remember that the human eye is different from
film or a CCD chip. It does not integrate over time. The
expose a photographic plate to starlight the brighter the image
(There is a limit even to this process in film due to what is
reciprocity failure.) But, humans can stare at the night sky all
and not see anything they didn't see after the first few
Things don't get brighter for us the longer we look at them. So
theoretically the longer we expose our CCD camera chip, the brighter
(deeper into space we can see). This is not true for the human eye.
We can see
the 8400 or so stars that we can see, and all the zillions of
as well not be there AT ALL as far as our humble naked human
Paradox is not a paradox at all if you look at it correctly. It
another example of theoretical mathematics applied incorrectly to a
phenomenon. Or a mathematician might say, "They got the upper
limit on the