CHAPTER VII.
MOVEMENTS OF PLANETS, SATELLITES, AND COMETS.
APPARENT MOVEMENTS OF PLANETS.—It has already been pointed out that like the sun and moon, the planets also have an apparent movement with respect to the more distant stars. Mercury and Venus are never seen very far from the sun, while other planets, among which are Mars, Jupiter, and Saturn, may be seen in the part of the heavens opposite to the sun.
One point, and that a very important one, which we notice from our observations is that the planets never depart very far from the ecliptic, so that the planes in which they perform their movements are nearly coincident with the plane in which our own annual journey round the sun is performed. The apparent movements of the planets are such that it is quite impossible to regard these bodies as circulating in regular orbits round the earth itself. If they revolve round any other body it is manifest that their apparent or geocentric motions will be compounded of the real movements of the planets and that of the earth. It is not necessary here to trace the steps by which it has been determined that the planets revolve in regular orbits around the sun. Suffice it to say that their observed movements are simply and sufficiently explained by supposing that, like the earth, which may now be regarded as a planet, they travel in elliptic orbits with the sun at one of the foci. Besides this revolution, the planets have a rotatory motion about their axes, but this question cannot be studied apart from the telescopic features, and will therefore be treated in Section III. of the present work.
The circumstance that the planets Mercury and Venus are never seen long after sunset or before sunrise, indicates that their orbits must lie between us and the sun. Hence, they are distinguished as the _interior planets_, while those outside the earth’s orbit are called the _exterior planets_.
MOVEMENTS OF INTERIOR PLANETS.—Let us consider briefly the conditions under which we observe the interior planets. If such a planet be represented by M in Fig. 23, while the earth is represented by E traversing a larger orbit, the planet is said to be in _inferior conjunction_ with the sun, when it lies directly between the sun and earth. The actual movements of the planets being direct—that is, anticlockwise—the planet at M has an apparent westerly motion as seen by an observer situated on the earth, and from this we gather that it moves more rapidly than the earth. For simplicity let us regard the earth as being at rest at the point E. Then, as the planet reaches the position M′, where it is as far as possible to the west of the sun, it is said to be at its _greatest western elongation_. Proceeding in its orbit, the planet’s apparent movement is direct, and it eventually comes in line with the sun on the further side as seen from the earth; it is then said to be in _superior conjunction_. From this point the planet moves to the east of the sun until it comes to the point M, after which the motion becomes retrograde, and the planet proceeds to inferior conjunction again. When at its greatest distance to the east of the sun, as at M‴, the planet is said to be at its _greatest eastern elongation_. Taking the term _elongation_ in general, it may be regarded as a measure of the angular distance of a planet from the sun as observed from the earth.
[Illustration:
FIG. 23.—_Movement of an Interior Planet._ ]
If the orbits of the planets were perfect circles, the greatest elongation distances of an interior planet would always be the same; sometimes, however, we are nearer to the sun than at the other times, and the apparent separation of the planet from the sun would seem greater than at other times, even if there were no other cause at work. The variations of the elongation distances are greater than can be accounted for by our own varying distance, and are naturally attributed to the elliptical form of the orbits of the interior planets themselves. Mercury, for example, sometimes only departs 18° from the sun, while at other times it reaches as far as 28° east or west.
When we take account of the fact that the earth has also a movement along its orbit, it will be seen that the same conditions hold good with regard to elongations and conjunctions, except that the intervals between them will be longer.
[Illustration:
FIG. 24.—_Morning and Evening Stars._ ]
MORNING AND EVENING STARS.—From superior to inferior conjunction an interior planet is to the east of the sun. It then rises after the sun, and sets after the sun, so that it is visible for a short time in the early evening; in other words, it is an _evening star_ during this part of its path. Between inferior and superior conjunctions, the planet is conversely a _morning star_. This is illustrated in Fig. 24, where the position of an observer towards whom the sun is rising is shown at A. An interior planet at P is above the horizon at sunrise, but will be below at sunset, the observer having been carried to A′ by the earths rotation; it will thus be a morning star. When the planet occupies the position P′ it is below the horizon at sunrise, but will remain in sight after the sun has set in the evening, the observer then having been transferred to A′ by the earth’s rotation.
PHASES OF INTERIOR PLANETS.—From the conditions which have been stated with regard to the movements of the interior planets, one is not surprised to find that telescopic examination reveals that these bodies put on phases similar to those of the moon. At superior conjunction the planets exhibit a fully illuminated disc, at greatest elongations they appear as a half moon, while at inferior conjunction their dark sides alone are presented to us. The apparent sizes of the planets, as measured with the aid of a telescope, are also found to vary according to their positions; when at inferior conjunction, the planet is much nearer to us than at other times, and it consequently appears larger. The apparent brightness of an interior planet also varies. At superior conjunction the whole of the disc is illuminated, but the planet is then so far removed from us that its light is very feeble. On the other hand, at inferior conjunction, when it is nearest to us, the dark side of the planet is turned towards us. The greatest brightness thus occurs at some intermediate point. In the case of Venus this is between the greatest elongations and inferior conjunction, when it is 40° from the sun. It is then bright enough to be seen with the naked eye in full sunshine, and has sometimes, on such occasions, been erroneously regarded by ignorant persons as the Star of Bethlehem.
TRANSIT OF VENUS.—If an inferior conjunction occurs when the planet is very near to a node—this term having the same significance as in the case of the moon (p. 94)—the planet, whether it be Mercury or Venus, will be seen projected as a dark spot upon the bright disc of the sun. Such an occurrence is called a _transit of Venus_ or of Mercury, as the case may be. Just as we do not get an eclipse of the sun every month, so we do not get a transit of Venus every time the earth and that planet have the same heliocentric longitude, and for the same reason, namely, that the plane of the orbit is inclined to the ecliptic. As we shall see in another chapter, a transit of Venus has a most important application in the determination of one of the fundamental constants of astronomy—the sun’s distance. The conditions as to the recurrence of transits are of great interest. In the case of Venus, the _synodic_ period is 584 days, this being the time which elapses between two successive inferior conjunctions. Five synodic periods are thus very nearly equal to eight years, and 152 synodic revolutions are even more nearly equal to 243 years. As seen from the earth, the sun crosses the nodes of the orbit of Venus on June 5 and December 7, and since there can be no transit when the planet is more than 4½° from the node, the transits will all occur about these dates. A transit will be followed by another after the lapse of 8 years, if the planet is not too far from the node; but there can be no other transit with the planet at the same node until 243 years have elapsed. There are, however, transits occurring at similar intervals when the planet is at the other node. The following dates on which transits have occurred, or will occur, will illustrate the foregoing statements:—
8 years│December 7, 1631,│243 years.│—————————— „ │December 4, 1639,│ „ │243 years. 8 years│December 9, 1874,│ „ │ „ „ │December 6, 1882,│——————————│ „ 8 years│June 5, 1761, │243 years.│—————————— „ │June 3, 1769, │ „ │243 years. 8 years│June 8, 2004, │ „ │ „ „ │June 6, 2012, │——————————│ „
[Illustration:
FIG. 25.—_Movement of an Exterior Planet._ ]
MOVEMENTS OF EXTERIOR PLANETS.—The exterior planets are at once recognised as such by their occasional appearance in the part of the sky opposite to that of the sun. They are then said to be in _opposition_. When in the same line as the sun, and on the remote side of it, as at P′ in Fig. 25, the planet is in _conjunction_. The apparent movements of such a planet are very complex. Neglecting for a moment the earth’s motion, it is evident that the apparent rate of movement of the planet with reference to the stars will vary very considerably according as the planet is near opposition or near conjunction, the movement appearing to be most rapid when the planet is nearest to us. Upon this unequal rate of motion is superposed a varying direction of motion produced by the changing position of the earth. When the planet is at P, and the earth at E, both are moving in the same direction, but as the earth has the greater angular velocity, the apparent motion of the planet will be retrograde, that is, the planet will appear to go backwards in its path. If the earth be near the point E′, its orbital movement will be directed away from the planet, and will scarcely affect its apparent position; accordingly, about this time the planet has a direct movement in the heavens. Between these two points the direction of the apparent movement of the planet has changed, so that at some intermediate position it would seem to have suspended its wanderings; here we have a _stationary point_. For a certain time, before and after conjunction, the linear directions of movements of the earth and planet will be opposed to each other, and on this account the _direct_ apparent motion of the planet will be accelerated. Presently, as the earth gains on the planet, another stationary point will be reached, and with the approach to opposition the planet will again retrograde.
If both orbits were in the same plane, these apparent movements would all be backwards and forwards along a great circle of the celestial sphere coincident with the ecliptic, the eastward movement predominating. The planes in which the planets perform their revolutions are, however, inclined to the ecliptic, and the result is that they appear to us to travel in loops, some of which are illustrated in Fig. 26.
[Illustration:
FIG. 26.—_Apparent Paths of Ceres, Pallas, Juno and Vesta, in 1896._ ]
From the fact that we are constantly within the orbit of an outer planet, it is evident that we must always see more than half of the planetary hemisphere on which the sun is shining. Consequently, an exterior planet never puts on a crescent phase, or presents the appearance of a half moon. The nearer the planet the greater will be the dark area which it is possible for us to observe. In the case of Mars, for example, we sometimes see it gibbous like the moon about three days from full, but in the more distant planets this gibbosity is scarcely perceptible. The greatest phase of an exterior planet occurs when it is at _quadrature_, that is, when a line joining the earth and sun is perpendicular to one joining the earth with the planet.
FAVOURABLE AND UNFAVOURABLE OPPOSITIONS.—A little consideration of Fig. 25 will make it perfectly clear that an exterior planet is very much nearer to us at a time of opposition than at a conjunction. We are, in fact, then, nearer to the planet by the diameter of the earth’s orbit, a matter of some 186 millions of miles. Accordingly, the planets, more especially our neighbour Mars, are best studied in the telescope about a time of opposition. Now, if we had to deal with circular orbits, the distance of a planet at opposition would remain constant, and we should see the planet equally well at all oppositions. It is found, however, that this is not the case, and the ellipticity of the orbits of the earth and planets supplies a simple and sufficient explanation. Sir Robert Ball illustrates this in the case of Mars by a diagram similar to Fig. 27. It will be seen that, when the opposition occurs in August, the earth is much nearer to Mars than when it happens at other times. The least favourable oppositions are those which occur in February, the planet then being nearly twice as far removed from us as at the nearest approach during an August opposition.
[Illustration:
FIG. 27.—_Opposition of Mars._ ]
As regards the more distant planets, the diameter of the earth’s orbit and the variations of opposition distance are of less importance, since they form a much smaller proportion of the distances of those planets from the sun.
ELEMENTS OF A PLANETARY ORBIT.—A complete study of the apparent movements of the planets with which we are acquainted shows that their real movements are performed round the sun in ellipses, the sun being placed at a focus. Each orbit, like that of the earth, has its perihelion and aphelion points, and its apse line; not being coincident with the ecliptic, it will have a line of nodes, and an ascending and descending node. Each planet will further have a particular inclination to the ecliptic, and a period of revolution peculiar to itself. Consequently, to systematise our knowledge of any particular orbit, certain conventions are adopted, and the seven things we must know, in order that we may specify the size of the orbit, its position in space, and the situation of the planet in its orbit, are as follows:—
_a_ = Semi axis major of elliptic orbit.
_e_ = Eccentricity.
_i_ = Inclination to ecliptic.
Ω = Longitude of ascending node.
π = Longitude of perihelion.
P = Period of revolution. (_u_, the mean daily motion, sometimes replaces P.)
E = The epoch, giving the longitude of the planet at some particular time.[2]
The first two quantities indicate the size and shape of the orbit, the next three its position with regard to the ecliptic, and the last two are required to determine the situation of the planet in its orbit. Some of the elements are illustrated in Fig. 28.
[Illustration:
FIG. 28.—_Elements of an Elliptic Orbit._ ]
DETERMINATION OF A PLANET’S PERIOD.—Observations enable us to determine the synodic period of a planet, and knowing that the earth’s period is a year, it is a simple matter to determine that of the planet. In the case of an exterior planet, the interval from opposition to opposition furnishes the best means of determining the synodic period. The exact moment of an opposition cannot usually be directly observed, and what one actually does is to measure the R.A. and declination of the sun on several days about the time of opposition, as also those of the planet; then, by reducing these co-ordinates to celestial longitude and latitude, it is not difficult to determine at what moment the longitudes differed by 180°, that is, the moment at which opposition took place. The problem of finding the planet’s sidereal period, then, amounts to this: at what rate must the planet be moving in order that the earth may make a complete revolution, and move, in addition, through the same angle as the planet? In other words, what must be the period of the planet in order that the earth may gain a whole revolution in the interval corresponding to the synodic period? The daily movement of the planet will be 360°/P, and that of the earth 360°/365¼, if P denote the number of days in the planet’s sidereal period. The earth’s gain per day will thus be the difference between these two quantities, and since a whole revolution is gained in the synodic period, the gain per day can be expressed as 360°/S, where S represents the synodic period; thus we get
360°/365¼ − 360°/P = 360°/S or 1/365¼ − 1/P = 1/S
The synodic period of Mars is 780 days, and the application of the foregoing formula leads us to 687 days as the time of its revolution round the sun.
A single determination of a synodic period does not give precise results, for the reason that the orbits of the planets are elliptical, and the intervals consequently dependent upon whether the planet is near perihelion, or far removed from it when an opposition is observed. It is, therefore, necessary to determine the time of opposition at long intervals, and so reduce the errors in measuring the length of a single period.
MOVEMENTS OF SATELLITES.—Telescopic observations show that some of the planets are accompanied by _satellites_, which revolve round their primaries as the moon revolves round the earth. The apparent movements of these bodies, with regard to the planets, are very similar to those of the interior planets with regard to the sun, having similar points of greatest eastern and western elongations. The facts which have been collected show that each satellite, like our own moon, moves in an elliptical orbit, with the planet in one of its foci. With one exception, the satellites attending the planets of our system have a direct movement; those of Uranus, however, have apparently a movement in the same direction as the hands of a watch, but this can be regarded as direct, if we consider the plane of the orbit to be inclined more than 90° to the plane of the ecliptic.
THE ORBITS OF COMETS.—Another class of bodies which circulate round the sun now claims our attention. These are the _comets_, some of which are never seen without the aid of telescopes, while others have been brilliant enough to excite a widespread wonder and interest. They usually have a very rapid movement relatively to the stars; and to learn something as to their real motions, we commence by measuring their right ascensions and declinations as frequently as possible. When such observations are plotted, they give us the geocentric movement of a comet, which generally seems very irregular, and gives one the idea that it is subject to no law. Unlike the planets, comets do not usually keep near the ecliptic, but move in planes inclined at all angles to it. Their rates of apparent movement also change very rapidly.
When the effect of the earth’s movement upon that of a comet is eliminated, it is found that the movement of the comet is performed either in an ellipse, a parabola, or an hyperbola, the sun in each case occupying one of the foci.
From our definition of the eccentricity of an ellipse, it will be seen that, when the eccentricity is zero, we have a circle. When the eccentricity becomes unity, the ellipse becomes a parabola, so that the latter curve may be regarded as part of an ellipse, of which the foci are at an infinite distance apart. In the case of the hyperbola, the eccentricity is greater than unity.
Comets which move round the sun in ellipses are called _periodic comets_, for the reason that they return regularly into the sun’s neighbourhood. Those which traverse parabolic or hyperbolic paths will pass once round the sun and continue to journey into the depths of interstellar space until their movements are changed by the proximity of other bodies into the neighbourhood of which their wanderings may take them.
When a new comet is observed, one of the things which astronomers endeavour to do is to determine its orbit, so that its path may be predicted with sufficient accuracy to enable it to be picked up readily with a telescope when it becomes so feeble that it is no longer visible to the naked eye. In the first instance, the motion is assumed to be parabolic, and any deviation from such an orbit forms the subject of a rigorous calculation by means of which the precise form is determined.