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TRANSITS OF MERCURY AND VENUS.
302. Definition. When either Mercury or Venus, at the time of inferior conjunction, is near to either node of its orbit, or, which amounts to the same, when the longitudes of the sun and node are at that time nearly equal, the planet must pass between the sun and earth, appearing as a well-defined black spot traversing the disc. This phenomenon is called a Transit of the planet.
303. Transits of Mercury. The longitudes of Mercury's nodes are about 46° and 226°, and these longitudes vary but little more than a degree in a century. In the present age, therefore, transits of Mercury can only occur in the months of May and November, for it is only in these months that the sun can have the same longitude as the nodes.
When a transit has occurred at one node, there cannot be another at the same node, till the lapse of a period of time composed of whole synodic revolutions, and also, of whole years or nearly So. For they occur only at inferior conjunction, and those at the same node, nearly at the same time in the year. Hence, taking s to represent a synodic revolution of Mercury, p the periodic revolution of the earth or sidereal year, if m and n be two whole numbers such that ns =
mp nearly, or
nearly, then will m be
the number of the years in the period between two consecutive transits at the same node. Different values for m and n, less or more exact, may be obtained by the method of continued fractions.* It is thus found that transits at the same node occur at intervals of 6 or 7 years, 13 years, 33 years, &c.
A transit at one node is generally preceded or followed, at an interval of 31 years, by one at the other node.
The last two transits of Mercury, both of which were visible in this country, occurred in May, 1845, and November, 1848.†
* This method is frequently given in treatises on Algebra. A practical rule is given in Lewis's Arithmetic.
† Other transits that will occur during the present century, will happen November, 1861, November, 1868, May, 1878, November, 1881, May, 1891, and November, 1894. Of these the third and last will be visible in this country.
304. Transits of Venus. The longitudes of the nodes of Venus are about 75° and 255°, and the sun has these longitudes in June and December. Hence, it is only in June and December that transits of Venus occur.
A synodic revolution of Venus being about 584 days (292), a period of 5 synodic revolutions differs but little from 8 years. Hence, a transit at one node is generally preceded or followed, at an interval of 8 years, by another at the same node. A full investigation, with reference to both nodes, shows that, commencing with the last transit, which occurred in June, 1769, succeeding transits must occur at the terminations of the periods 1051 years, 8 years, 121 years, and 8 years, taken in order and repeated in the same order. Thus, the last two transits were in June, 1761 and 1769, and the next two will occur in December, 1874 and 1882.
Transits of Venus occur, therefore, much less frequently than those of Mercury.
305. Computation of a Transit. The computation of a transit of Mercury or Venus, for any given place, is nearly like that of an eclipse of the sun; the data for the planet taking the place of those for the moon.
306. Sun's Parallax. A transit of Venus is a phenomenon of great interest and importance as affording the best means of determining with accuracy the sun's parallax, and thence, his distance from the earth. For a full investigation of the method by which the sun's parallax is deduced from observations of this phenomenon, the student must be referred to larger works. But the following illustration will enable him to understand the general principles on which the deduction depends.
Let the circle cDd, of which S is the centre, Fig. 51, represent the sun's disc, and let V be Venus, pq a part of her relative orbit, along which she appears to move in the direction from to q, p the earth, and A and B the places of two observers, supposed to be situated at the opposite extremities of that diameter of the earth which is perpendicular to the ecliptic. Then, disregarding the earth's rotation, or, which is the same, supposing the positions A and B to remain fixed during the transit, the centre of the pla
net will, to the observer at A, appear to describe the chord cd, and to the observer at B, the parallel chord ef. Also, when, to the observer at A, the centre of the planet appears to be at a, it will, to the observer at B, appear to be at b. As AB is perpendicular to the plane of the ecliptic, and the plane of the sun's disc is for each observer very nearly so, the line ba may be regarded as being parallel to AB; and as the relative orbit, and, consequently, the chords cd and ef make but a small angle with the plane of the ecliptic, it may be regarded as perpendicular to these chords, and, therefore, as expressing the distance between them.
Now, the observers at A and B may determine the duration of the transit of the planet's centre as seen at these places; that is, the times of its appearing to describe the chords cd and ef. Then, as the relative hourly motion of Venus may be very accurately found from tables of the sun and planet, the values of the chords ed and ef, expressed in seconds, and, consequently, their halves hd and kf, may be obtained. Hence, hd and kf, and the sun's semidiameter SD, being known, hD and kD, and, consequently, their difference hk or ab, are easily found.
As ba is parallel to AB, the triangles ABV and abV are similar, and we have, aV : AV :: ab: AB. But, from the tables, we know the ratio of aV to aA, and, consequently, of aV to AV. This ratio is, at a mean, 72 to 28 very nearly, or 5 to 2 nearly. Hence, we have, approximately, 5: 2 :: ab: AB = ab. But, AB, which is the measure of the angle AaB, is double the sun's horizontal parallax. Consequently, the sun's horizontal parallax =ab, nearly. It follows that, whatever small error may be made in determining ab, the error in the parallax obtained will be only about one-fifth as great.
It is not necessary that the observers should be situated as supposed above; but it is important that they should be at places far distant from each other, in rather a north and south direction. The places being known, the complete investigation of the subject furnishes a method of deducing the parallax, taking into view the earth's rotation and every other circumstance that can influence the accuracy of the result.
307. Determination of the sun's parallax. Astronomers having made known the importance of having accurate observations of
the transits of Venus at different and distant places, expeditions on the most efficient scale were fitted out for the purpose previous to the last transit, in 1769, by the British, French, Russian, and other governments. From the observations then made, combined with some of those made in 1761, Professor Encke has found the sun's mean horizontal parallax to be 8'.5776.
SUPERIOR PLANETS-SATELLITES OF JUPITER, SATURN AND URANUS.
308. Superior Planets. The superior planets, revolving in orbits without that of the earth, cannot exhibit to us phases similar to those of Mercury and Venus. The disc of Mars, however, about the period of his quadratures, appears decidedly gibbous. The other planets revolve so far without the earth's orbit that their enlightened surfaces are always turned almost entirely towards the earth, and the gibbous form is not perceptible.
309. General remarks. Mars is easily distinguished from the other planets by the ruddy colour of his light. He is a small planet, next larger than Mercury. His apparent diameter varies from about 3" to 18". In consequence of this great variation in apparent diameter, he appears at different times, except with regard to colour, as quite a different body.*
310. Period, distance, &c. Mars revolves round the sun in a little less than 23 months, at a distance of 144 millions of miles. His diameter is about 4000 miles, and his volume that of the earth. He revolves in 24 h. 39 m., about an axis that is inclined to the axis of the ecliptic, in an angle of 30° 18'.
* The change in the apparent diameter of Venus is still greater (297); but, in consequence of her phases, the change in the light received from her, while sufficiently remote from the sun to be visible, is much less.
311. Spheroidal form. According to the observations of some astronomers, Mars has perceptibly a spheroidal form. Arago makes his polar diameter to be less than the equatorial by of the latter.
JUPITER AND HIS SATELLITES.
312. General remarks. Jupiter is the largest of the planets, his volume exceeding the sum of all the others; and, with the exception of Venus, he is the most brilliant. His apparent diameter varies from 30" to 45′′.
313. Period, distance, &c. Jupiter revolves round the sun in rather less than 12 years, at a distance of 494 millions of miles. His diameter is 90,000 miles, which is more than 11 times the earth's diameter. Consequently, his volume is more than 1300 times that of the earth. He revolves in 9h. 56 m. about an axis nearly perpendicular to the plane of the ecliptic.
314. Spheroidal form of Jupiter. The form of Jupiter is decidedly spheroidal. According to accurate micrometrical measurements, the polar diameter is less than the equatorial by nearly 15 of the latter.
Investigations, founded on the principles of mechanics, prove, that if a globular body composed of yielding matter, such as may some time have been the state of the earth and planets, be made to rotate about an axis, it must assume a spheroidal form; differing less or more from a sphere, according to its magnitude and the rapidity of its rotation. These investigations, applied to the earth and Jupiter, assign to each, very nearly, the degree of oblateness it is found to have.
315. Belts of Jupiter. When Jupiter is examined with a telescope of considerable power, his disc is observed to be crossed in a direction parallel to his equator, by several dark bands which are called his belts. These belts do not always present exactly the same appearance. They vary slightly in breadth and position; but always remain parallel to one another and to the equator.
316. Jupiter's Satellites. It has already been mentioned (9), that Jupiter is attended by four moons or satellites. These revolve