How Many Stars Are Brighter Than Each Planet?

Our word "planet" comes from a Greek word meaning "to wander," and the planets known since ancient times (Mercury through Saturn) stood out from the "fixed stars" as the planets shifted positions in their orbits. The classical planets are also pretty bright compared with most stars, so they stand out in that respect as well. However, the planets and other solar system bodies (notably asteroids) discovered in modern times are a lot dimmer, and I got to wondering how well hidden they are among the fainter stars.

The brightness of celestial bodies is measured in magnitudes. That term, as I understand, also dates back to antiquity; the brightest stars were of "first magnitude," a fainter tier was labeled "second magnitude," and so on, just as coarse groupings. In modern times, we can measure the amount of light coming from a celestial object, so we can impose some precision on that system. The definition used today says that a difference of five magnitudes denotes a factor of 100 difference in brightness (i.e., a difference of one magnitude is a factor of the fifth root of 100, or about 2.5, in brightness). Like the classical categories, modern magnitude numbers go down, not up, as objects get brighter; it also means that very bright objects like the sun or moon, or even the brighter stars or planets, can have negative magnitudes (e.g., a planet that's 100 times as bright as a star of magnitude 3.0 would have a magnitude of -2.0).

Planet vs. star magitudes

The figure above shows how the magnitudes of various solar system objects, which I obtained from this paper, fit among the distribution of magnitudes of the stars. The stepped line shows the number of stars in the sky, on the vertical axis, that are brighter than each value of magnitude, on the horizontal axis. The brighter stars are counted from the Yale Bright Star Catalog, and the fainter (higher magnitude) distribution is from this table summarizing the modern Tycho catalog. The brightest star, Sirius, brings the count to one starting at magnitude -1.46, Canopus is the second at magnitude -0.72, and so on with a rapidly increasing number of stars as magnitude increases (brightness decreases).

The sun and full moon are of course much brighter than any star or planet, but their light can be measured and thus large negative magnitudes can be assigned to them. Mercury is brighter than any star at its brightest; however, when it is nearly between Earth and the sun we are seeing its dark side, and its brightness plummets. It is lost in the glare of the sun under these circumstances, of course, and so whenever it is far enough away from the sun to the east or the west that it can be seen it is fairly bright. The same might be said of Venus, since it also passes between the Earth and the sun; however, because it has a thick atmosphere (unlike Mercury), light scatters around the edge (limb) of the planet and it remains quite bright, though measuring an exact value for the minimum (near the glare of the sun) is difficult.

The range is better defined for Mars, Jupiter, and Saturn. (The vertical positions of the lines for each planet are arbitrary; I just spaced them out to fit on the graph.) Jupiter is brighter than any star throughout its orbit; Mars is brighter than any star at its brightest, but at its dimmest there three dozen stars brighter than it is. Similarly, at its brightest Saturn is almost as bright as the second brightest star (Canopus), and seventeen stars are brighter than Saturn at its faintest. It is not difficult to pick out any of these planets, especially if one notices them at their greatest brightness and then follows them night after night as they dim.

The last few objects on the chart were discovered telescopically in the last few hundred years. The number of stars brighter than Uranus ranges from about 2500 to 5200, and there are tens of thousands of stars brighter than Neptune; a good map is needed to find these easily. The brightest asteroid, Vesta, is almost as bright as Uranus at its brightest, but nearly 100,000 stars are brighter than it at its faintest. (Pluto is off the right end of the plot at no brighter than magnitude 13.7, requiring an excellent map and a large telescope to spot.)

One very interesting possibility is suggested by the brightness of Uranus. From a dark location, the unaided eye can see down to magnitude 6 or so; this means that Uranus would have been just above the limit of naked-eye visibility in an ancient sky free of light pollution, and many people are likely to have spotted it. As noted, there are thousands of stars brighter than Uranus, but it is at least possible that some very careful observer might have noticed that one of those otherwise undistinguished stars was actually wandering slowly from month to month and year to year. (I have seen this myself from dark skies, but I of course had maps to point me to Uranus in the first place. Vesta is only bright enough to see without optical aid for a brief period every couple of years, so it would be harder to track it between disappearances.)

I am no scholar of the classics, but I always thought the Pythagoreans would have been good candidates for such a discovery. The story has it that they discovered the dodecahedron (a regular solid with twelve pentagonal faces) and irrational numbers (numbers like the square root of two that cannot be obtained by dividing one whole number by another, the way that one and a half is the result of dividing three by two, say), and kept these secret as religious mysteries. Wouldn't it be cool if somebody found an ancient record showing that they also knew about a secret planet, thousands of years before Herschel identified it with his telescope...?

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New 3 January 2019