INWIT™ — TIME AND TIDE

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Time and Tide

by Vincent Mallette
Copyright © 1999 Inwit Publishing, Inc.

TIME AND TIDE

For billions of years the earth has lost rotational energy to the moon and sun by tidal friction. (Tides have existed for at least 2.8 billion years).¹ Eons ago, before the tides robbed us of so much energy, the earth spun faster — that is, the day was shorter. There is good evidence that 900 million years ago the day was about 18 hours long.² At present the earth’s rotation is slowing down between one and two thousandths of a second per century.³ Because the earth weighs 6 1/2 sextillion tons, even this slight slowing represents a tremendous transfer of energy. How much energy? Calculations show that the loss is about a million kilowatt-hours per second.⁴ In more familiar units, that’s about 4 billion horsepower that drains from our planet day and night. And keep in mind...that 4 billion horsepower is the result of the earth slowing down by no more than half a picosecond⁵ per second! But don’t complain about the tides — there’s good reason to think that the moon is essential to life on earth: "Without the moon...the earth’s obliquity⁶ would oscillate even more than that of Mars, leading to far greater climatic instability than we presently experience and endangering the course of biological evolution".⁷ This author goes on to say that the moon may be responsible for the earth’s relatively powerful magnetic field, which screens us from harmful particles ejected by the sun.

Historical Note

Galileo never believed that the moon caused the tides. Two hundred years later Napoleon was told by his admirals that he had nothing to fear from the British navy because it was run by superstitious people who thought the moon caused the tides. Well, the moon does cause the tides (the sun contributes⁸ about 32%; Jupiter contributes about 0.0001%). Long before Napoleon, Newton had proved the moon’s role in raising the tides, and had correctly explained the math and physics of the situation. We may excuse Galileo, though. Living before Newton’s systematic treatment of the subject, he thought he saw some serious objections to the moon theory. For example, there are two high tides per day, not one, as a simple attractive moon theory would predict. Even more fatal, high tide does not occur when the moon is directly overhead, but usually about 6 hours later. Both these objections can be answered by modern physics,⁹ but Galileo just didn’t have the tools. It is all the more credit, though, to the Greek explorer Pytheas,¹⁰ who realized the tides were caused by the moon — 300 years B.C.!

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¹ Astronomy: The Cosmic Journey, Third Edition — by William K. Hartmann (Belmont, CA: Wadsworth Publishing Company, 1985), p. 86

² And the year was 481 of these 18-hour days: C. P. Sonett et al, "Late Proterozoic and Paleozoic Tides, Retreat of the Moon, and Rotation of the Earth" Science, 273, 103 (in particular Table 2) (1996). This is the most recent and best available information. Disregard older figures given in astronomy books, popular magazines, etc.

³ A calculation prior to 1955 gave 1.6 milliseconds, but modern sources just say "about a millisecond." By the way, forget about "leap" seconds. That’s a completely different matter and will be dealt with in another article.

⁴ For comparison, all the electric utilities in the United States generate about 100,000 kilowatt-hours per second — source: The Universal Almanac 1996 — ed. by John W. Wright (Kansas City: Andrews and McMeel, 1995), p. 273. "A million kilowatt-hours per second" — source: Handbook of Chemistry and Physics, 64th Edition — ed. by Robert C. Weast (Boca Raton, FL: CRC Press, Inc., 1983), p. F-148. The value was given as 1.1 • 10²⁹ ergs/century. A value about one-fifth smaller (0.865 • 10²⁹ ) is given in Physics of the Earth — by Frank D. Stacey (New York: John Wiley & Sons, Inc., 1969), p. 36, equation (2-55). Not all of this energy goes to the moon — some of it goes into churning the earth’s liquid core. But in any case enough goes to the moon to drive the moon nearly 4 centimeters farther from the earth every year [3.82 cm from Apollo lunar laser ranging: C. P. Sonett et al, "Late Proterozoic and Paleozoic Tides, Retreat of the Moon, and Rotation of the Earth" Science, 273, 103 (1996)]

For reference: The value was given as 1.1 • 10²⁹ ergs/century.

That’s 1.1 • 10²² joules/century, or 1.1 • 10²⁰ joules/year, or 3.49 • 10¹² joules/sec, or 3.49 • 10¹² watts.

⁵ A picosecond is a trillionth of a second.

⁶ The famous 23 1/2° which is responsible for the seasons

⁷ Allen L. Hammond, Science, 24 January 1975, p. 245. The specifics are: "The oscillations [of the obliquity] are due to an interaction between two dynamic phenomena — the precession of the equinox as the tilted axis describes a conical motion and the precession of the planet’s orbit plane as the entire orbit wobbles in and out of alignment with the rest of the solar system. The earth’s obliquity changes very little, at present, because the presence of the moon shortens the equinoctial precession period, precluding a resonant interaction with the orbit plane precession." (p. 245)

⁸ The most careful calculation of this I know is in Newtonian Mechanics — by A. P. French (New York: W. W. Norton, 1971), p. 537. Many books give a (slightly incorrect) figure which amounts to 29% for the sun. French states, "...the tide-raising ability of the moon exceeds that of the sun by a factor of about 2.15. ...When they are on the same line through the earth (whether on the same side or on opposite sides) there should be a maximum tide equal to 1.465 times that due to the moon alone. ...[W]hen the angular positions of sun and moon are separated by 90°, the tidal amplitude should fall to a minimum value equal to 0.535 times that of the moon [alone]. The ratio of maximum to minimum values is thus about 2.7 " — p. 537

⁹ The moon pulls the seawater closest to it, but it pulls the earth away from the seawater on the other side, making two high tides at the same time on opposite sides of the earth. Meanwhile the earth is rotating, so about 12 hours after high tide you’ll see the another high tide. Actually, the physics is not as simple as this, but you’d be sorry if I gave you the full explanation. As for the 6 hour delay, it’s due to the inertia of the seawater, the friction of the seabed, the local geography of the coast, etc.

¹⁰ The Mediterranean Sea, where most of the Greeks were content to stay, has no tides. But Pytheas ventured out into the Atlantic, where the tides are strong. The Timetables of Science — by Alexander Hellemans and Bryan Bunch (New York: Simon and Schuster, 1988), p. 37

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