Part 19

Let a denote the synodical period of the planet, Let b denote the sidereal period of the earth, Let x denote the sidereal period of the planet. Then will 360°/b = daily motion of the earth, And 360°/x = daily motion of the planet; Also 360°/b - 360°/x = daily gain of the earth. But 360°/a = daily gain of the earth: Hence 360°/b - 360°/x = 360°/a

1/b - 1/x = 1/a

ax - ab = bx

(a-b)x = ab

x = ab/(a-b).

137. The Relative Distance of a Superior Planet.--Let S, e, and m, in Fig. 153, represent the relative positions of the sun, the earth, and Mars, when the latter planet is in opposition. Let E and M represent the relative positions of the earth and Mars the day after opposition. At the first observation Mars will be seen in the direction emA, and at the second observation in the direction EMA.

But the fixed stars are so distant, that if a line, eA, were drawn to a fixed star at the first observation, and a line, EB, drawn from the earth to the same fixed star at the second observation, these two lines would be sensibly parallel; that is, the fixed star would be seen in the direction of the line eA at the first observation, and in the direction of the line EB, parallel to eA, at the second observation. But if Mars were seen in the direction of the fixed star at the first observation, it would appear back, or west, of that star at the second observation by the angular distance BEA; that is, the planet would have retrograded that angular distance. Now, this retrogression of Mars during one day, at the time of opposition, can be measured directly by observation. This measurement gives us the value of the angle BEA; but we know the rate at which both the earth and Mars are moving in their orbits, and from this we can easily find the angular distance passed over by each in one day. This gives us the angles ESA and MSA. We can now find the relative length of the lines MS and ES (which represent the distances of Mars and of the earth from the sun), both by construction and by trigonometrical computation.

Since EB and eA are parallel, the angle EAS is equal to BEA.

SEA = 180° - (ESA + EAS) ESM = ESA - MSA EMS = 180° - (SEA + ESM).

We have then

MS : ES = sin SEA : sin EMS.

Substituting the values of the sines, and reducing the ratio to its lowest terms, we have

MS : ES = 1.524 : 1.

Thus we find that the relative distances of Mars and the earth from the sun are 1.524 and 1. By the simple observation of its greatest elongation, we are able to determine the relative distances of an inferior planet and the earth from the sun; and, by the equally simple observation of the daily retrogression of a superior planet, we can find the relative distances of such a planet and the earth from the sun.

IV. THE SUN.

I. MAGNITUDE AND DISTANCE OF THE SUN.

138. The Volume of the Sun.--The apparent diameter of the sun is about 32', being a little greater than that of the moon. The real diameter of the sun is 866,400 miles, or about a hundred and nine times that of the earth.

As the diameter of the moon's orbit is only about 480,000 miles, or some sixty times the diameter of the earth, it follows that the diameter of the sun is nearly double that of the moon's orbit: hence, were the centre of the sun placed at the centre of the earth, the sun would completely fill the moon's orbit, and reach nearly as far beyond it in every direction as it is from the earth to the moon. The circumference of the sun as compared with the moon's orbit is shown in Fig. 154.

The volume of the sun is 1,305,000 times that of the earth.

139. The Mass of the Sun.--The sun is much less dense than the earth. The mass of the sun is only 330,000 times that of the earth, and its density only about a fourth that of the earth.

To find the mass of the sun, we first ascertain the distance the earth would draw the moon towards itself in a given time, were the moon at the distance of the sun, and then form the proportion: as the distance the earth would draw the moon towards itself is to the distance that the sun draws the earth towards itself in the same time, so is the mass of the earth to the mass of the sun.

Although the mass of the sun is over three hundred thousand times that of the earth, the pull of gravity at the surface of the sun is only about twenty-eight times as great as at the surface of the earth. This is because the distance from the surface of the sun to its centre is much greater than from the surface to the centre of the earth.

140. Size of the Sun Compared with that of the Planets.--The size of the sun compared with that of the larger planets is shown in Fig. 155. The mass of the sun is more than seven hundred and fifty times that of all of the planets and moons in the solar system. In Fig. 156 is shown the apparent size of the sun as seen from the different planets. The apparent diameter of the sun decreases as the distance from it increases, and the disk of the sun decreases as the square of the distance from it increases.

141. The Distance of the Sun.--The mean distance of the sun from the earth is about 92,800,000 miles. Owing to the eccentricity of the earth's orbit, the distance of the sun varies somewhat; being about 3,000,000 miles less in January, when the earth is at perihelion, than in June, when the earth is at aphelion.

"But, though the distance of the sun can easily be stated in figures, it is not possible to give any real idea of a space so enormous: it is quite beyond our power of conception. If one were to try to walk such a distance, supposing that he could walk four miles an hour, and keep it up for ten hours every day, it would take sixty-eight years and a half to make a single million of miles, and more than sixty-three hundred years to traverse the whole.

"If some celestial railway could be imagined, the journey to the sun, even if our trains ran sixty miles an hour day and night and without a stop, would require over a hundred and seventy-five years. Sensation, even, would not travel so far in a human lifetime. To borrow the curious illustration of Professor Mendenhall, if we could imagine an infant with an arm long enough to enable him to touch the sun and burn himself, he would die of old age before the pain could reach him; since, according to the experiments of Helmholtz and others, a nervous shock is communicated only at the rate of about a hundred feet per second, or 1,637 miles a day, and would need more than a hundred and fifty years to make the journey. Sound would do it in about fourteen years, if it could be transmitted through celestial space; and a cannon-ball in about nine, if it were to move uniformly with the same speed as when it left the muzzle of the gun. If the earth could be suddenly stopped in her orbit, and allowed to fall unobstructed toward the sun, under the accelerating influence of his attraction, she would reach the centre in about four months. I have said if she could be stopped; but such is the compass of her orbit, that, to make its circuit in a year, she has to move nearly nineteen miles a second, or more than fifty times faster than the swiftest rifle-ball; and, in moving twenty miles, her path deviates from perfect straightness by less than an eighth of an inch. And yet, over all the circumference of this tremendous orbit, the sun exercises his dominion, and every pulsation of his surface receives its response from the subject earth." (Professor C. A. Young: The Sun.)

142. Method of Finding the Sun's Distance.--There are several methods of finding the sun's distance. The simplest method is that of finding the actual distance of one of the nearer planets by observing its displacement in the sky as seen from widely separated points on the earth. As the relative distances of the planets from each other and from the sun are well known, we can easily deduce the actual distance of the sun if we can find that of any of the planets. The two planets usually chosen for this method are Mars and Venus.

(1) The displacement of Mars in the sky, as seen from two observatories which differ considerably in latitude, is, of course, greatest when Mars is nearest the earth. Now, it is evident than Mars will be nearer the earth when in opposition than when in any other part of its orbit; and the planet will be least distant from the earth when it is at its perihelion point, and the earth is at its aphelion point, at the time of opposition. This method, then, can be used to the best advantage, when, at the time of opposition, Mars is near its perihelion, and the earth near its aphelion. These favorable oppositions occur about once in fifteen years, and the last one was in 1877.