One of the limitations of astronomy is the fact that we are trapped on the Earth. Even our best space probes have only covered a tiny fraction of the distance to the nearest star. Much of astronomy deals with overcoming this limitation through remote sensing and clever inference based on limited data from remote sources of light. We cannot roam through space and discover the size and shape of the Milky Way galaxy directly. However, we can understand the size and shape of the Milky Way, and our place in it, by counting stars.
Isotropic means the same in every direction. Anisotropic means different in different directions. You can use anisotropy to learn about your surroundings. If you are somewhere in a forest, you might notice fewer trees (or the trees thinning out) in one direction. This direction is probably the shortest distance to the edge of the forest. If you see equal numbers of trees in every direction, you have no clue where the edge is. Or imagine that you are lost in a fog bank. As you wander, you might find a place where sunlight penetrates. This anisotropy — the fact that the level of light was not the same in every direction — is your clue to the location of the edge of the fog.
We are "lost" in the Milky Way galaxy. If there were equal numbers of stars in every direction, we would have no clue where the edge of the Milky Way is. There would be a limit to the number of stars seen in any direction, but that could just be due to limited light gathering power and the stars being too faint to detect. We might be tempted to believe that we were at the center of the galaxy. In fact, the distribution of stars in the sky is anisotropic, and this anisotropy helps us measure our galactic environment. For simplicity we can ignore the obscuring effects of dust.
Imagine how we might view the sky if the stars around us were arranged in a disk. At the center of the distribution, we would see few stars looking directly out of the disk on either side, but larger and equal numbers of stars and looking in any direction through the disk. In this simple "toy" model, the band of the Milky Way would look the same in every direction. We could tell that we live in a slab-like arrangement of stars, but not whether it is a disk or an enormous sheet, and we could not locate the edge. Now imagine that we are offset from the center of the disk. We would see more stars looking through the disk toward its center and less looking through the disk toward its edge.
Suppose we lived in a spherical distribution of stars. At the center of the distribution, we would see roughly equal numbers of stars in every direction. We would conclude either that we lived in the center of the universe or that the universe extended so far that we could not see its edge. But if we are offset from the center of the sphere, we would see more stars in all directions toward the center of the sphere and less stars in all directions away from the center. Each of these examples shows how we can use anisotropy in the counts of stars to deduce the size and shape of our galactic environment. In the real sky, hot young stars or open clusters trace the shape of the galactic disk and globular clusters trace the shape of the spherical galactic halo.
We can quantify the change in the number of stars with the change in the viewing angle as we look through a disk, a slab, or a sheet. Imagine we are situated in a slab-like arrangement of stars that extends very far in every direction. The shortest distance to the edge of the slab is X. Using simple geometry, the distance through the stars at any angle Θ from the vertical is:
Y = X / cos Θ
This relationship defines the way that the number of stars will increase looking in directions closer and closer to the plane of the Milky Way. In terms of our galactic coordinates, b = 90 - Θ. Consider X the minimum number of stars we see at the galactic pole, where Θ = 0° and Y = X. At a galactic latitude of b = 45° (so Θ = 45°), Y = X / cos 45 = 1.4 X, so we see 40% more stars in that direction. At a galactic latitude of b = 10° (so Θ = 80°), Y = X / cos 80 = 5.8 X, so we see 5.8 times more stars at such a low galactic latitude. The number of stars increases sharply as we approach the galactic plane.
It is an oversimplification to treat our galaxy as a uniform slab of stars. First, the Milky Way is a not a perfectly uniform disk of stars. Stars are grouped and clustered, which will cause variations in the counts of stars in adjacent directions. A more important complication is the effect of dust. Dust is found in the regions between stars. So when we look through areas with more stars, we will also be looking through more dust. Since dust dims starlight, this leads to an underestimate of the numbers of stars at low galactic latitude. This explains why it took so long to correctly map out the Milky Way.
The same slab-like geometry in can be used to explain why stars dim when they set in the night sky. Suppose that the slab is the Earth’s atmosphere, and the dots are dust particles in the atmosphere (we ignore Earth’s curvature and treat the atmosphere as a flat slab). The dust will dim and redden the light from all stars; recall that the opacity and amount of extinction just depend on the number of particles encountered or the path length through the particles. The minimum amount of extinction occurs looking directly overhead. This is the distance X, and an elevation angle of 90°. Stars overhead typically have their red light dimmed by 10% and their blue light dimmed by 30% — the extra loss of blue light shows the effect of reddening. A star at an elevation angle of 45° is dimmed by an extra 40% since Y = X / cos 45 = 1.4 X. A star at an elevation angle of 70° (only 20° above the horizon) is seen through an even larger amount of atmosphere, Y = X / cos 70 = 2.9 X. It is therefore dimmed by 65% compared to a star that is overhead.
Astronomers prefer to observe targets that are high in the sky and now we can understand why. As a star sets toward the horizon, its light is rapidly dimmed and reddened by dust particles in Earth’s atmosphere. Chasing a star as it sets in the western sky leads to diminishing returns, as its light is steadily extinguished by an increasing path through the atmosphere. Astronomers try to plan their observations to catch stars as they transit the meridian. In the case of a circumpolar star, it can be observed all night.