ASTRONOMY.
Plain and Easy Rules
For finding the Periodic Time of a Planet
or Comet from its Mean Distance; also to
find its Distance from its Period.
BY ORSON PRATT, SEN.
TO THE EDITOR OF THE DESERET NEWS:–
SIR:–I saw an article in your paper of the 19th inst.. headed, “Samuel Elliot Coues and his theory of Astronomy,” copied from the United States Magazine. It is with no small degree of pleasure that I learn, for the first time, of the laudable exertions of Mr. Coues, in endeavoring to trace out and simplify the laws that govern the workings of the Universe. I am happy also to learn that he has cast away the vague and foolish idea of attracting and projectile forces, and adopted in some measure, my theory of the Self-Moving Forces of the University, which I published in January 1851.
Altho’ he has endeavored to supercede the “Third Law of Kepler,” in regard to the relations existing between the distances and periods of the bodies of the solar system, yet in the introduction of his new rules, he has been obliged, most unquestionably, to resort to that very law, in obtaining his unit of distance, namely 870.167 miles,–which he assumes to be the diameter of the sun. At that precise distance, the periodic time of a body passing around the sun, is equal to 3182 of a solar day. Now .3182 is the ratio of the diameter of a circle to its circumference (the circumference being equal to 1). A periodic time, identical with this ratio, is ascertained by the application of Kepler’s law, to require a distance the same as assumed by Mr. Coues as his unit.
The diameter of the sun may, nor may not, agree with this assumed unit; it may vary 10,000 miles from it. Indeed, there is no cause, that we are aware of, which would make the periodic time, represented by this ratio, to agree with time as measured by the earth’s diurnal rotation in preference to the diurnal periods of the other planets. The days, as measured on other planets, would each alter the assumed unit; but the diameter could not agree with but one. For the want of generality therefore, in this assumed law, I see not the least grounds for believing in the coincidence said to exist, between the diameter and the assumed unit.
Dividing the distance of the earth from the sun by the unit distance of Mr. Coues, the earth’s mean distance, expressed in these units, is obtained, namely, 109.61. The unit distance of all the other planets is obtained in the same manner.
From these distances, and with the aid of the ratio of the circumference of a circle to its diameter, Mr. Coues deduces the periodic time of each planet from its own distance, without employing the lengthy and tedious process connected with Kepler’s law, which requires three terms given to find a fourth.
The ratio of the circumference of a circle to its diameter, is altogether unnecessary in the calculation, and renders the process much more complex than if not used. Any distance may be assumed as a unit, and calculations founded upon it, will produce the same results, but the assumption of some units will make the calculation more tedious than others.
I will now give the following Rules, which I invented several years ago for my own use:–
RULE I
To find the mean sidereal period of a planetary body around the sun, in mean solar days, when only the mean distance in miles is given:
Divide the distance by 1859195, and the quotient multiplied into its own square root, will be the required period:
Or in other words, divide the distance by 1859195, and the cube of the square root of the quotient will be the required period:
Or this may be stated algebraically thus:
( 4/1859195)3/2 =P
RULE II.
To find the distance in miles, when only the periodic time is given:
Multiply the square of the cube root of the period by 1859195, and the product will be the required distance:
Algebraically thus, 1859195XP3/2=d
1859195 miles from the center of the sun, a body will revolve around it in 1 mean solar day.—I have therefore, chosen that distance as the unity of distance for the solar system.
If the distance of each planet, expressed in miles, be divided by 1859195, the proportional distance of each will be obtained, expressed in terms [column break] of the assumed unit, as in the second column in the following table:–
| Let 1859195 be assumed=1 or the unity of distance | ||
| Name | Mean distance from the Sun in Units | Mean Sidereal Period in mean solar days |
| 1859195MercuryVenus…Earth….
Mars…. &c. |
1.0000019.7797036.9603551.09738
77.85664 &c. |
1.00087.969224.701305.256
686.980 &c. |
If any of the mean distances in the second column be multiplied into its own square root, the product will express the solar days in its annual period, as in the third column. For instance, the square root of the earth’s unit distance 51.09738 is 7.14824; these two numbers multiplied into each other, give 365.2563 days.
If any one of the periods in the third column are given, the unit distance in the second column may be found, by squaring the cube root of its period.
This process is far less complex than that by Mr. Coues, or of Kepler. It is founded on the following beautiful law that governs the planets in their orbits, namely:
The mean velocity of the planets in different orbits are inversely as the square roots of their mean distances. That is, one planet, four times more distant from the centre of the motion than another, has one half the velocity; and 9 times more distant 1-3 the velocity; and 16 times more distant 1-4 the velocity, &c.
Now 1-2, 1-3, 1-4, are the inverse square roots of their respective distances. And bodies having 4, 9, or 16 times greater orbits, and moving with only 1-2, 1-3 or 1-4 the velocity, will require 8, 27, or 64 times as long a period to complete their revolutions; but 8, 27, 64, are the cubes of the square roots of the distances 4, 9, 16.
From this law are derived the two very plain rules which I invented for my own convenience in astronomical investigations. I present them here, hoping that they may be of some utility in facilitating the researches of others.
The periodic time of a body only 1 mile distant from the centre of motion in the sun (assuming all his gravitating forces collected in that point) would be equal to a very small fraction of a day expressed by the following decimal, .000000030[unreadable number]39446855, which multiplied into the cube of the square root of any planetary distance, expressed in miles, will give its mean sidereal period in mean solar days. Thus,
(95000000)3/2 X.00000000039446855 = 365.2563
Mercury’s distance is 36774320 miles, therefore,–
(36774320)3/2 X.00000000039446855 = 87.969
If the periodic time of any planet or comet be divided by .00000000039446855 the square of the cube root of the quotient, will be its mean distance in miles. Thus,
365.2563_____ 2/3 = 95000000 miles
.00000000039447855
87.969______ 2/3 = 36774320 miles
.00000000039446855
If the mean distance of the earth from the sun be assumed as unity or 1, then the cube of the square root of the proportional distance of each planet, multiplied into 365.2563612 will give its mean sidereal period in mean solar days. For example, Mars’ proportional mean distance from the sun is 1.5236923, and
Mars, (1.5236923)3/2 x 365.2563612 = 686.9796d
Mercury, (.3870981)3/2 X 365.2563612 = 87.9593d
Venus, (.7233316)3/2 X 365.2563612 = 224.701d
The proportional distances are found as follows:–
Mercury, 87.9692580 2/3 = .3570981 , the proportional distance.
. 365.2563612
Venus, 224.7007869 2/3 = .7233316, its proportional distance.
. 365.2563612
Mars, 686.9796458 2/3 = 1.5236923, its proportional distance.
. 3652563612
Any of the Units which we have assumed in the foregoing process, has rendered the calculation much more easy and simple, than to employ as in Kepler’s law, the squares and cubes of three known terms to find the square or cube of the fourth. It is also much less complicated than the simplifications of Kepler’s law, introduced by Mr. Coues.
These simple rules are applicable in principle, to the periodic times and distances of all the heavenly bodies, whether planets, asteroids, comets, satellites, or stars.
Yours most respectfully,
ORSON PRATT, Sen.
Great Salt Lake City, Oct. 24, 1854
[Orson Pratt, Astronomy, Deseret News, Oct. 26, 1854]
[Transcribed by DeeAnn T. Pratt and Mauri Pratt; June 2012]