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Achilles and the Tortoise

With the exception of Pluto, the planets travel in nearly circular concentric orbits centered on the Sun. All sorts of numerical relationships exist between these orbits. For example: p2=a3 says that the square of a planet's period (in years) is the cube of it's axis measured in AUs. [One AU is equal to the distance of the Sun to the Earth.] However, the relationship I want to cover in this article is how long it takes for two planets to pass each other. It isn't quite as obvious as it might seem.

Mercury takes 88 days to circle the Sun; Earth takes 365¼ days. By the time Mercury circles the Sun, Earth has moved 87 degrees along its orbit. So Mercury has some more catching up to do. By the time Mercury moves these extra 87 degrees, Earth has moved an additional 21 degrees. As Mercury races to catch up, Earth keeps moving but steadily loses ground. If this sounds a little like the fable of Achilles and the Tortoise, it should. Both are forms of Zeno's Paradox which befuddled mathematicians for nearly two millennia.

The formula isn't as frightful as all this seems to suggest. It is (S•F)÷(S-F) where F and S are the periods of the Faster and Slower bodies respectively. Well, the upshot of all of this is that Mercury will pass Earth after 115.9 days, since 115.9=(88•365¼)÷(365¼-88).

This may be all very well, you may ask, but so what? Well, it explains a very interesting event which will dominate our evening skies for the next half year. The event I refer to is Venus staying put in the southwestern sky for the next half year as the other planets drift past Venus into the evening twilight. Venus has a period (year) which is 2242/3 days long. Using the formula above, it will take almost 584 days for Venus to catch up with the Earth. Only Mars at 1095 days takes longer. However, even at its most brilliant, Mars is never as eye catching as Venus.

Let me give you a couple of examples of this formula before I relent and talk about something else. This formula can be used when two systems are rotationally connected.

Example 1: At exactly 6:00, the two hands of a dial clock are in a straight line. How long before the hands are next in a straight line? As you can guess, this will occur slightly after 7:05. The hour hand makes a revolution in twelve hours (720 minutes) while the minute hand takes 60 minutes. Using our formula , we see the time span is 655/11 = (60•720) ÷ (720-60) minutes, so that between 7:05:27 and 7:05:28 the hands are straight.

Example 2: Achilles runs a hundred times faster than the Tortoise. Since Achilles is such a magnificent athlete (and a bit of a show off), he cedes the Tortoise a full lap head start. How far does Achilles have to go before he draws along side the Tortoise? Well not very far. Shortly after the end of the first lap, Achilles passes the tortoise: 1.0101...=(1•100)÷(100-1).

While Venus hovers in the south western sky, the planets that decorated the evening skies during the Fall, slip past Venus into the glare of the Sun. On January 5, Neptune passes Venus, followed by Uranus on January 13th, a few days later they disappear behind the Sun. Mercury jumps up in early February to join Venus for a couple of weeks only to disappear by mid March. Jupiter passes Venus on February 23rd, just a tiny 1/3 degree apart and then it too disappears into the sunlight. Finally, Saturn passes Venus on March 19th only to be lost into the Sun a few weeks later. Pluto has already passed Venus into the Sun's glare, leaving only Mars to keep Venus company.

At the start of January, Mars will rise just before 1 AM. By the end of February , it will rise between 10 PM and 11PM. By the end of April, Mars will rise before the Sun sets. For the remainder of the year, Mars will be visible in the evening sky, first in the south east, then in the south and finally in the south west as the sky gets dark. Enjoy Mars and Venus this spring and early summer. We will have to wait until late summer for the reappearance of the outer planets.

Leslie Coleman
Author:
Leslie Coleman
Entry Date:
Jan 1, 1999
Published Under:
Leslie Coleman's Columns
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