1902 Encyclopedia > Tides > [Tidal Friction] Amount of Tidal Retardation of Earth’s Rotation

## Tides (Part 48)

IX. TIDAL FRICTION (cont.)

48. Amount of Tidal Retardation of Earth’s Rotation

With respect to the actual amount of retardation of the earth’s rotation, we quote the following from Thomson and Tait’s Nat. Phil. (1883), § 830. [Footnote 377-1]

"In observational astronomy the earth’s rotation serves as a time-keeper, and thus a retardation of terrestrial rotation will appear astronomically as an acceleration of the motion of the heavenly bodies. It is only in the case of the moon’s motion that such an apparent acceleration can be possibly detected. Now, as Laplace first pointed out, there must be a slow variation in the moon’s mean motion arising from the secular changes in the eccentricity of the earth’s orbit around the sun. At the present time, and for several thousand years in the future, the variation in the moon’s motion is and will be an acceleration. Laplace’s theoretical calculation of the amount of that acceleration appeared to agree well with the results which were in his day accepted as representing the facts of observation. But in 1853 Adams showed that Laplace’s reasoning was at fault, and that the numerical results of Damoiseau’s and Plana’s theories with reference to it consequently require to be sensibly altered. Hansen’s theory of the secular acceleration is vitiated by an error of principle similar to that which affects the theories of Damoiseau and Plana; but, the mathematical process which he followed being different from theirs, he arrived at somewhat different results. From the erroneous theory Hansen found the value of 12″·18 for the coefficient of the term in the moon’s mean longitude depending on the square of the time, the unit of time being a century; in a later computation given in his Darlegung he found the coefficient to be 12″·56. [Footnote 377-2]

"In 1859 Adams communicated to Delaunay his final result, namely, that the coefficient of this term appears from a correctly conducted investigation to be 5″·7, so that at the end of a century the moon is 5″·7 before the position it would have had at the same time if its mean angular velocity had remained the same as at the beginning of the century. Delaunay verified this result, and added some further small terms which increased the coefficient from 5″·7 to 6″·l.

"Now, according to Airy, Hansen’s value of the ‘advance’ represents very well the circumstances of the eclipses of Agathocles, of Larissa, and Thales, but is if anything too small. Newcomb, on the other hand, is inclined from an elaborate discussion of the ancient eclipses to believe Hansen’s value to be too large, and gives two competing values, viz., 8″·4 and 10″·9. [Footnote 377-3]

"In any case it follows that the value of the advance as theoretically deduced from all the causes, known up to the present time to be operative, is smaller than that which agrees with observation. In what follows 12″ is taken as the observational value of the advance, and 6″ as the explained part of this phenomenon. About the beginning of 1866 Delaunay suggested that the true explanation of the discrepancy might be a retardation of the earth’s rotation by tidal friction. Using this hypothesis, and allowing for the consequent retardation of the moon’s mean motion by tidal reaction, Adams, in an estimate which he has communicated to us, founded on the rough assumption that the parts of the earth’s retardation due to solar and lunar tides are as the squares of the respective tide-generating forces, finds 22 sec. as the error by which the earth, regarded as a time-keeper, would in a century get behind a perfect clock rated at the beginning of the century. Thus at the end of a century a meridian of the earth is 330″ behind the position in which it would have been if the earth had continued to rotate with the same angular velocity which it had at the beginning of the century. . .

Footnotes

377-2 "It appear not unusual for physical astronomer to use an abbreviated phraseology, for specifying accelerations, which needs explanation. Thus, when they speak of the secular acceleration being, e.g. '12″·26 in a century,' they mean by 'acceleration' what is more properly 'the effect of acceleration on the moon's mean longitude.' The correct unabbreviated statement is 'the acceleration is 25″·12 per century per century.' Thus Hansen's result is that in each century the mean motion of the moon is augmented by an angular velocity of 25″·12 per century, so that at the end of a century the mean longitude is greater by ½ of 25″·12 than it would have been had the moon's mean motion remained the same as it was at the beginning of the century. Considering how absurd it would be to speak of a falling body as experiencing an acceleration of 16 feet in a second, or of 64 feet in two seconds, and how false and inconvenient it is to speak of a watch being 20 seconds fast when it is 20 seconds in advance of where it ought to be, we venture to suggest that, to attain clearness and correctness without sacrifice of brevity, 'advance' to be substituted by 'acceleration' in the ordinary astronomical phraseology."

377-3 Researches on the Motion of the Moon, Washington, 1878.