The dramatic difference between what we can easily see and what really lies out in space is an important issue in astronomy. To use Hertzsprung's analogy, the whales are much easier to see than the fishes. We can see whales out to large distances, but in any volume of water, there are far fewer whales than fishes. The inverse square law and simple geometry can help us understand what the most common stars are in any region of space.
The apparent brightness of a star decreases with the square of the distance. This means that more luminous stars can be seen from farther away than feeble stars. A simple example will show the difference between counting stars to a certain apparent brightness, and counting all the stars in a volume. Imagine two stars, one like the Sun and one five times the luminosity of the Sun. We can calculate the level at which any star becomes invisible to the eye (or telescope). Let’s say that all stars like the Sun can be seen out to a distance of 1 parsec. However, by the inverse square law, we can see all stars of 5 times solar luminosity (L☉) out to √ (5/1) = 2.24 parsecs. In all directions from the Earth, this distance defines a sphere.
Let us call D the visibility distance, or the maximum distance we can see a star of a particular luminosity. The ratio of the spherical volume in which we can see the 1 solar luminosity (Sun-like) star to the spherical volume in which we can see the 5 solar luminosity stars is:
(4 π /3 (D5)3) / (4 π /3 (D1)3) = (2.23/1)3 ≈ 11
In this equation, D1 is our shorthand for the visibility distance of a star like the Sun and D5 is our shorthand for the maximum visible distance of a star with five times the luminosity of the Sun.
Therefore, if there were equal numbers of stars like the Sun and stars five times more luminous, we could see and count about 10 times more of the more luminous stars. Or, if the luminous stars were 10 times rarer in any chunk of space, we would count about equal numbers of each type. In other words, surveys to a certain brightness limit always lead to an overestimate of the true numbers of luminous stars. Intrinsically dim stars are counted over a much smaller volume than intrinsically luminous stars.
Notice that when we take a ratio, the numerical factors cancel out. We can also make this equation more useful by remembering that the inverse square law says that L ∝ D2, and so the limiting distance D ∝ √ L. If we substitute this in the equation above and keep everything in solar units, we find that the number of stars of luminosity L that we count compared to the Sun goes up proportionally to L3/2. If we call NV the true number of stars of luminosity L (compared to stars like the Sun) and NB the number we count to some apparent brightness limit, then
NV = NB L-3/2
Here is an even more extreme example. One star is like the Sun and the other is twenty times more luminous. We can see the more luminous star out to a distance of 20/1 = 4.46 parsecs. By the equation above, NV = NB (20)-3/2 = 0.01 NB. In this case, even if the stars were equally sprinkled through space, we would count nearly 100 times more of the more luminous kind! Or, if stars with 20 L? were 100 times less common than stars like the Sun, we would count very luminous stars and stars like the Sun with roughly equal frequency.
Another useful example involves a low luminosity, or dwarf, star. A star 1/10 the luminosity of the Sun can only be seen out to √ 0.1/1 = 0.32 parsecs. The smaller visibility distance corresponds to a smaller volume for seeing these dim stars. We would expect surveys to a certain brightness limit to underestimate the true numbers of low luminosity stars. Using the equation above, we find that NV = NB (0.1)-3/2 = 32 NB. If dwarf stars were equally sprinkled through space, we would undercount them by a factor of over 30 in a survey limited by apparent brightness. Or, if space were peppered with 30 times more 0.1 solar luminosity stars than Sun-like stars, we would count them with equal numbers.
Lists of nearby stars and prominent stars are very different. However, once we have a true census of the true numbers of high and low luminosity stars in a representative volume of space, we can correct any survey for the bias in favor of high luminosity stars. The result is a true census of whales and fishes.