LAW OF PLANETARY ROTATION

—

TO THE EDITOR OF THE TIMES

SIR: — Permit me to announce to the world, through your valuable paper, an astronomical discovery, made by me on the eleventh day of November, 1854. Firmly believing, from my early youth, that the diurnal motions of the planets were the results of some hidden law, I endeavored, at different times, to discover the same, so as to determine the periods of rotation by calculation instead of observation. After many fruitless researches in regard to the original causes of planetary rotation, I was led by the indications of certain hypotheses to seek for the law of rotation connected with the masses and diameters of the planets, or, in other words, with their densities. These investigations resulted in the development of the following beautiful law:

THE CUBE ROOTS OF THE DENSITIES OF THE PLANETS ARE AS THE SQUARE ROOTS OF THEIR PERIODS OF ROTATION.

Or which amount to the same thing – THE SQUARES OF THE CUBE ROOTS OF THE DENSITIES OF THE PLANETS ARE AS THEIR PERIODS OF ROTATION.

But as the densities of globes are proportional to their masses or quantities of matter, divided by their volumes or by the cubes of their diameters, it follows, by this law, that the rotation of the planets, considered as spheres, is proportional to their masses and diameters. The law, therefore, may be expressed in terms of the masses and diameters, as follows:

THE SQUARES OF THE CUBE ROOTS OF THE MASSES OF THE PLANETS, DIVIDED BY THE SQUARES OF THEIR DIAMETERS, ARE AS THEIR PERIODS OF ROTATION.

To express this law in general algebraical formula, applicable to the periods of the rotation of all the primary planets, let M, D, P, represent respectively the mass, diameter, and rotative period of the earth; and let m, d, p, represent the mass, diameter, and rotative period of any planet, then we will have –

Or in terms of the densities and periods, thus –

Or, if the earth’s rotative period and density be each taken as unity or 1, then,(III.) (Planet’s density)^{2/3} = rotative period

As the rotative periods depend upon the masses and diameters of the planets, any errors entering into these elements by the imperfections of observation, will necessarily affect the calculated periods of rotation in a proportionate degreee.

The masses and diameters of the planets are only approximately obtained. Astronomers differ in regard to the amount of these elements; yet it is pleasing to know that the results of a great number of observations confine the errors within very small limits.

The masses and diameters in the following table are, perhaps, as near the truth, as the imperfections of past observation will allow us to approximate:

Name of Planet. |
Masses (the Earth’s being 1). |
Diameters in Miles. |
Calculated Periods of Rotation. |

MercuryVenus
Earth Mars Jupiter Saturn Uranus Neptune |
0.062770.90433
1.00000 0.14534 371.75470 289.02810 20.62549 26.87671 |
31407800
7925.5 4108.26 88592.7 79160 34500 37500 |
H.M.S.24 05 0023 21 21
23 56 04.09 24 37 23 9 54 12 10 29 17 9 30 00 9 35 32 |

The periods of rotation, in the 4th column of this table, were calculated by the aid of formula (I), using the elements in the 2nd and 3rd columns. These calculated periods are precisely the same as the observed periods, so far as the latter have been ascertained.

The diameter of Neptune is probably not yet known within several thousand miles; for an error of observation of three-tenths of a second of an arc, would, at that great distance, produce an error of over 4000 miles in the calculated diameter of the planet. The same statement is equally applicable to the determination of the mass. An error of observation on the dimensions of the orbit of Neptune’s satellite, though it should be only a small portion of a second of an arc, yet would produce a great difference in the calculated mass.

It will be seen by the formula which I have given, that if the rotation is known by observation, the ratio of the mass and diameter can be calculated; and that if any two of the elements are known, the third can be determined.

Whether the law of planetary rotation can be extended to the rotative periods of the satellites, attending the four exterior planets of the solar system, is not known. It is supposed by some, from observation, that the periods of the rotation of the satellites are equal to their periods of revolution around their primaries: but this needs confirmation by further observation of greater perfection and accuracy than what the present powers of the telescope seem capable of affording.

From the masses and diameters of the four satellites of Jupiter, as given in Herschel’s “Outlines of Astronomy,” I find by the application of the law of rotation, the following relative or proportional periods. (Assuming the period of the rotation of the 1^{st} satellite, nearest to the planet, to be equal to unity or 1.)

Jupiter’s Satellites. |
Proportional Periods of Rotation, as calculated. |

1st.^{ }2nd.3rd.
4th. |
1.0000000001.7884784931.635488852
1.375555069 |

If it be true, that the rotative periods of these satellites are equal to the periods of their revolutions, then the law does not apparently hold good for this secondary system, unless the diameters and masses are affected with considerable errors; if the apparent angular diameter of the 2nd satellite be reduced the one-twentieth of a second of an arc, it would reduce the real diameter 116 miles, which would give a calculated period of rotation such as should exist in order to correspond precisely with the ratio of the periods of revolution in their orbits. In the cases of the 3rd or 4th satellites, there would have to be a greater correction in order to make the rotative and orbitual periods of the same length. A mistake of a small fraction of a second of an arc, might easily be made in the apparent angular diameters. Likewise as the masses are deduced from observations of the minute perturbations which the satellites exercise upon each other, it is evident that a minute error in such observations, would give a much greater error in the calculated masses. Therefore, masses and diameters might be assumed within the limits of unavoidable errors, which would give calculated periods of rotation for the four satellites of the same length as their orbitual periods.

The astronomer cannot fail to perceive that this remarkable law is somewhat analogous to the one discovered by Kepler, connecting the orbitual periods of the heavenly bodies with their distances from their centre of motion. When Kepler’s law was discovered, the distances of the planets were very imperfectly known; but notwithstanding the imperfection of these data, he clearly perceived the relation between them and the periodic revolutions around the sun. So likewise, in this new discovery, no one can fail to see the relation existing between the density of planets and their diurnal rotations, notwithstanding the former element is so imperfectly ascertained.

With the most sincere desire for the developement and diffusion of useful knowledge, I subscribe myself your most humble servant,

O. Pratt

42 Islington, Liverpool, August 21^{st} 1857.

G.S.L. City, Jan. 2^{nd}, 1860.

TO THE EDITORS OF THE MOUNTAINEER:

GENTLEMENTo the Editors of the Mountaineer:

Gentlemen: — The above is a copy of a letter written by me to the “London Times:” if you consider it worthy of an insertion in your paper, it is at your disposal,

Respectfully,

Orson Pratt, Sen.

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[Transcribed by Nora Fowers and Becca Staker; Jan. 2011]

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