Astronomy is a mathematical science. That, however, doesn't need to scare us, for the beautiful heavens can be appreciated by any one, whether one can add 2 and 2 or balance a check book or not. However mathematics can reveal relationships, tell us the time or navigate to the planets. For this article I'll stick to the first.
Ever wonder why the planets are where they are in distance from the sun? In 1766 Johann Titius, a German mathematician was able to find a mathematical series that closely predicted the distances of the known planets of the time, with one little problem, as we will see. This formula was published by Johann Bode 6 years later, who made it famous, and unfortunately for Titius, it became forever known as Bode's Law.
Titius started with the series 0, 3, 6, 12, 24, 48, 96 etc. doubling the each number. To each he added 4, giving the resultant series: 4, 7, 10, 16, 28, 52, 100, and so on. The earth, being the third planet, would be 10. Now, astronomers put the earth's mean distance as 1 astronomical unit (au.)., So to get the series in astronomical units, simply divide each by 10 to get the series 0.4, 0.7, 1, 1.6, 2.8, 5.2 and 10.0. If you check a handy astronomy book, you'll find Mercury at .39 au., Venus at .72 au., the earth at 1.0 au., Mars at 1.5 au., Jupiter at 5.2 au., and Saturn at 9.6 au.
The one little problem alluded to above is that there is no planet at 2.8 au. At least there wasn't at the time when Titius put forth this mathematical series. On New Years night, 1801, a Sicilian monk Guiseppi Piazzi discovered a star that didn't belong. It moved over the next days like a planet. Its distance from the sun, it turned out, was 2.8 au. The object was named Ceres, goddess of agriculture, the first of a new class of planets called asteroids, meaning "star shaped". Most asteroids orbit the sun between Mars and Jupiter.
Uranus, when it was discovered, fit the series, but Neptune and Pluto diverge from it as can be seen in the table below.
Bode-Titius Law and Reality |
|||
Planet |
Bode (a.u.) |
Actual (a.u.) |
% |
Mercury | 0.40 |
0.39 |
3.36 |
Venus | 0.70 |
0.72 |
-3.18 |
Earth | 1.00 |
1.00 |
0.00 |
Mars | 1.60 |
1.52 |
5.26 |
Ceres | 2.80 |
2.77 |
1.08 |
Jupiter | 5.20 |
5.20 |
0.00 |
Saturn | 10.00 |
9.58 |
4.38 |
Uranus | 19.60 |
19.20 |
2.08 |
Neptune | 38.80 |
30.10 |
28.90 |
Pluto | 77.20 |
39.30 |
96.44 |
I have played in the past with applying a Bode-like rule for the Galilean satellites of Jupiter. I thought at one time, that I came up with a better fit than this, but here goes. This is a variation on Bode's Law. I start with the series 2, 4, 8 and 16, and add 3, then multiply each by 100 to get units of thousands of kilometers (megameters?). The result can be seen on the next page. It's not too bad, with the exception of Io.
Bob's Law and Reality | |||
Moon | Bob (km x 10^{3}) |
Actual (km x 10^{3}) |
% Error |
Io | 500 |
422 |
18.48 |
Europa |
700 |
671 |
4.32 |
Ganymede |
1,100 |
1,070 |
2.80 |
Callisto |
1,900 |
1,833 |
0.90 |
Does Bode's Law work because of some underlying principal? Or is it just a coincidence? Try for yourself. can you get a better series for Jupiter's large moons? How about Saturn's?
Perhaps the most amazing and wonderful coincidence of all is the apparent size of the sun and moon as seen by observers on the earth. The moon's mean distance from the earth is 240,000 miles, while that of the sun is 93,000,000 miles. That makes the sun 387.5 times farther away than the moon. The moon is 2060 miles in diameter while the sun is 865,000 miles in diameter. The sun is then 400 times larger than the moon. Thus the two bodies appear to us about the same size.
Since the moon's orbit of the earth and the earth's orbit of the sun are both ellipses, with the moon's slightly more eccentric, the moon sometimes appears slightly smaller that the sun sometimes, and other times slightly larger than the sun. To illustrate this fact is a photograph, above, I took of the annular eclipse of May 30, 1984. An annular eclipse is one in which the moon isn't quite big enough to cover the face of the sun.
This annular eclipse was special. The sun and moon were almost exactly the same apparent size. Never did the annulus or ring completely form, as crater rims and mountain peaks broke the ring at many points. The width of the path of annularity on the earth's surface was less than a mile wide at Piedmont Park in Atlanta, Georgia where we viewed it from.
Sometimes you have to work a bit to have your coincidences turn out.
Questions? Comments? Send Email to me at bob@bjmoler.org
Uploaded: 08/07/2001