Astronomy

Such would be the conditions as to the phases of the moon, if the earth were at rest.

The Month.—If the earth were fixed in space with regard to the sun, the moon’s phases would be repeated in the time corresponding to its period of revolution round the earth. This is 27 days 7 hours 43 minutes, and measures the length of a sidereal month.

It is much more useful, however, to refer the month to the phases actually observed. If in Fig. 19 we have the sun, earth, and moon represented at a full moon by S, E, and M respectively, the next full moon will not occur until the three bodies occupy the positions S, E′, and M′, the earth having travelled about 30° along its orbit. Between two full moons, then, the moon must make a complete revolution round the earth, and through an additional angle, A E′ M′, which will be equal to the earth’s angular motion in the interval. This movement of the moon occupies 29 days 12 hours 44 minutes, and is the duration of a lunar month. It also determines the synodic period of our satellite, a term which, taken generally, signifies the period in which a planet or satellite recovers the same position with respect to the sun when observed from the earth.

A calendar month, of which there are twelve in a year, must of necessity consist of a whole number of days, and the average duration of such a month is longer than that of a lunar month.

A remarkable relation exists between the synodic month and the length of the year. In 19 Julian years of 365¼ days there are almost exactly 235 synodic months, so that after the completion of this period full moons again occur on the same days of the month. The discovery of this cycle is usually ascribed to Meton, a Greek astronomer, 433 B.C. It is accordingly known as the Metonic Cycle, and is still used in the calculation of the moveable festival of Easter.^[1]

Rotation and Librations.—Even observations made without instrumental assistance show that the surface of our satellite always presents the same face to us, and without further inquiry one might suppose that it had no axial movement corresponding to that of its primary. If there were no rotation, however, we should in turn see all parts of the moon, and the observed circumstances indicate that it must rotate on an axis, in the same direction as that of its orbital movement, and in the same time. In Fig. 20 let E represent the earth, and a b c the part of the moon which is turned towards us when it is at M. When the moon arrives at M′, observations show us that the same part is presented to our view, so that the part corresponding to that we saw in position M is represented by a′ b′ c′. Now, if the moon had not rotated in the interval, the line joining a and c would have retained the same direction, and would have been in the position d e; the part c′ e would thus have been carried out of sight, while another part which was not seen when the moon was at M would have come into view. In order that we may see the same part of the moon in two different positions, M and M′, the dividing-line a c between the visible and invisible portions must turn through an angle equal to that between the lines d e and a′ c′; and since this angle is equal to that described by the moon in the same time, the period of the moon’s rotation on its axis must be equal to that of its revolution round the earth.

On account of the elliptical form of its orbit, the angular movement of the moon is not quite uniform; like the earth, it is subject to the law of areas. Hence, as the rotation is equable, the foregoing explanation does not strictly hold. In fact, this varying velocity results in a libration in longitude, which means that we sometimes see a little more of the western edge and sometimes of the eastern edge. There is also a libration in latitude on account of the fact that the moon’s axis is inclined to the plane of its orbit, so that at different times we see more of the North or South Pole, as the case may be; in this respect the moon behaves to the earth somewhat as the earth does to the sun in regard to the seasons, but the inclination is not so great.

The moon is so near to us that the portion of it which we see depends to a slight extent upon our terrestrial location. When the moon is rising we see a little more of its western edge than will be seen by an observer to the east of us, where the moon is in the south, and more than we ourselves shall see when it has come to our own meridian. Just before the time of setting we get to see a little beyond the eastern edge. This is called the diurnal libration, and never amounts to more than a degree.

Thanks to these librations, we are enabled to make telescopic observations of 9 per cent. of the moon’s surface which would not otherwise be open to our investigations.

Changes of the Moon’s Orbit.—The moon’s orbit is by no means to be regarded as a hard and fast geometrical figure. Indeed, it is subject to such great distortions in consequence of “perturbations” that the computation of the moon’s position at any future time is one of great complexity. One of the most easily recognised changes in the orbit is the revolution of its nodes, that is, of the points where it crosses the plane of the ecliptic.

The latter being a plane of indefinite extent, to which the moon’s orbit is inclined at 5° 9′, the moon will be alternately above and below the ecliptic for about half its period of revolution. The point where it passes from south to north of the ecliptic, A in Fig. 21, is the ascending node, and the corresponding point on its southward path is the descending node of the orbit. Connecting these two points is the line of nodes (A B), and by observations of the points where the moon’s path intersects the ecliptic at different times it is found that the line of nodes regredes or moves backwards. The rate of this revolution of the moon’s nodes is very irregular, but a whole revolution is made in 18·6 years.

This retrogression of the moon’s nodes may be well illustrated by the following heliocentric longitudes of the ascending node as given in recent “Nautical Almanacs”:

The line of apsides of the moon’s orbit joins the perigee and apogee; the direction of this line in space changes in a very variable manner, but in the long run it makes a complete revolution in 8·9 years.

When the sun is passing through the moon’s line of apsides it temporarily increases the eccentricity of the orbit; when at right angles to this line, the orbit becomes more nearly circular. This disturbance of the moon has accordingly a period equal to that required for two successive passages of the sun over the apse line of the moon’s orbit.

Such are a few of the movements which come within the province of the lunar theory, a fuller treatment of which is beyond our scope.

The Harvest Moon.—The full moon which occurs nearest to the autumnal equinox is called the harvest moon, for the reason that it rises very nearly at the same hour for several nights together, and so gives us a greater share of moonlight, by which harvest operations may be extended. At that time the sun will be at the autumnal equinoctial point, and when it is setting in the west, the vernal equinoctial point, and the moon with it, must be rising due east. The part of the ecliptic then above the horizon will extend from the east to the west point, but will lie wholly below the celestial equator (Fig. 22). As the moon’s path is very slightly inclined to the ecliptic, its movement will thus make only a small angle with the horizon, and for several nights together it will rise at nearly the same time.

In March, when the sun is near the vernal equinox, the full moon will be near the autumnal equinoctial point; when the sun is setting, the moon will be rising as before, but in this case the part of the ecliptic which is above the horizon lies wholly above the celestial equator. The ecliptic is thus inclined at an angle to the horizon greater by 47° than when the vernal equinox is rising in autumn; the moons path being near the ecliptic, its movement during a day will at this time carry it a long way below the Equator, and it will rise much later the following day.

In the Southern Hemisphere, the conditions are reversed, the harvest moon occurring at our vernal equinox, which, however, is the commencement of the southern autumn quarter.

The phenomena of the harvest moon recur, but are not so marked, in the month of October, and it is then called the hunter’s moon.

It is important to bear in mind that this rising of the moon at nearly the same hour for several days occurs every month, but as the risings then occur either in daylight or after midnight, and the moon is not full, no special attention is drawn to them.

Again, since the phenomenon of the harvest moon depends upon the small inclination of the path of the full moon to the horizon when it is at the equinoctial point, the circumstances will be modified by the latitude of the place of observation. At the Equator, for example, there will be no harvest moon, as there the ecliptic is always greatly inclined to the horizon; in fact, it will be inclined at the same angle in spring as in autumn.

The moon’s path being inclined to the ecliptic, the conditions as to the harvest moon will depend to a small extent upon the position of the moon’s nodes, which, as we have seen, revolve in a period of a little less than 19 years. At times, then, the moon’s path will be inclined 5° more, and 9 years afterwards 5° less, than is the plane of ecliptic, and under the latter conditions the harvest moon will be most pronounced.

High and Low Moons.—At the time of full moon, the moon is in the opposite part of the heavens to that occupied by the sun, sometimes being 5° above and other times 5° below. Manifestly, then, if the sun be high in the heavens at mid-day, it will be only a little below the northern horizon at midnight, and the moon, consequently, will be only a small distance above the southern horizon. In summer, then, quite apart from the fact that the nights are shorter, there is less moonlight. In winter, on the other hand, the sun descends far below the northern horizon at midnight, and the full moon has a high elevation in the southern part of the sky. By this happy arrangement, the full moon is longest above the horizon when its light is of greatest benefit to mankind.

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.

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.

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:—

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.

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.

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.

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

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.

Eclipses of the Moon.—As the various members of the solar system shine only by virtue of the light which they receive from the sun, they will cease to be visible if by any means they are deprived of the sun’s rays. Each planet or satellite must evidently cast a shadow which is turned directly away from the sun, and any other body passing wholly or partially within such a shadow will be proportionately debarred from receiving the direct light of the sun.

Were the sun a mere point of light these shadows would be parts of cones, the apex always being at the sun, and they would be prolonged indefinitely into space. As a matter of fact, every individual point upon the sun’s disc is competent to cast a conical shadow, and the net result is that only a relatively small space behind a planet or satellite is really in total darkness. This will be readily understood from Fig. 29, in which S is the sun, and E the earth. The total shadow now becomes a cone, with the apex turned directly away from the sun, but round this there is a region of partial shadow which is only illuminated by portions of the sun. If we imagine a section of the shadow across the line a b, we should find a central disc of total darkness called the umbra, and surrounding this a ring of half shadow called the penumbra.

From the known dimensions of the sun and earth, and the distance between them, it is easy to calculate the size of the earth’s shadow-cone, and its length is found to be greater than the distance of the moon. The axis of this shadow will, of course, always be in the plane of the ecliptic. If, then, at the time of opposition, the moon is sufficiently near the plane of the ecliptic, it will pass through the shadow, and we shall have the phenomena of a lunar eclipse. When the moon is wholly immersed in the umbra, the eclipse is total, and if it further passes quite symmetrically through the shadow, the eclipse is said to be central. This would always be the state of affairs if the moon performed its monthly journey in the plane of the ecliptic, and a total eclipse would occur every month. The moon’s orbit, however, is inclined to the ecliptic, so that for a central eclipse, the moon must be simultaneously at opposition and at a node. If the moon be near the node when at opposition, a total eclipse may occur, but it cannot be central, and the duration of the total obscuration will be reduced. Still further from the node, the moon will be above or below the ecliptic, and will be only partially involved in the shadow-cone; such an eclipse is called a partial one. Beyond a certain distance from the node, the inclination of the moon’s orbit will take the moon entirely out of the umbral shadow, and no eclipse will be possible.

The circumstances of an eclipse of the moon thus vary very considerably, and there is still another reason why we may expect them to be different. We have seen that the earth’s distance from the sun changes throughout the year, and, in consequence, its shadow will be of varying length, and the diameter of the shadow at any specified distance will not be constant. The moon, again, is not always at the same distance from the earth, and it will, therefore, pass through varying depths of shadow in different eclipses, and with different velocities.

The breadth of the earth’s umbral shadow at the point where the moon passes through it is, on the average, about three times the moon’s diameter, and the time taken for the moon to traverse this distance is about two hours. The duration of totality in a central eclipse may, therefore, amount to two hours, while an additional two hours may be occupied by the partial phases.

The Lunar Ecliptic Limit.—The greatest distance of the moon from a node at which a partial eclipse is possible, is called the lunar ecliptic limit, and is very easily calculated. In Fig. 30, let E N represent a part of the ecliptic, N being the node of the moon’s orbit, and E the centre of the earth’s shadow. As the orbit of the moon is inclined about 5° 9′ to the ecliptic, it may be represented by the line N M, inclined at an angle to N E. If E A be the radius of the earth’s shadow, which, on the average, is about three-quarters of a degree, and M A the moon’s apparent semi-diameter (about a quarter a degree), it is clear that the point beyond which no eclipse is possible is that in which the line M E, perpendicular to N M, is equal to the sum of the semi-diameters. All the quantities for solving the triangle N E M are thus known, and it can be readily calculated that N M, the greatest distance of the moon from the node at which an eclipse would be possible, under average conditions is about 11°.

Taking into account the varying distances between the sun, earth, and moon, it is found that an eclipse must always occur if the moon is within 9° of the node, and may occur if it be 12° from the node. These figures refer to the passage of the moon through the umbra, as the effect of its entrance into the penumbra is too slight to be observed.

The entrance of the moon into the earth’s shadow is a definite phenomenon, which is independent of the observer’s position on the earth, and the phases of the eclipse are seen at exactly the same moment from all places where the moon is above the horizon. The computation of the circumstances at a given place is accordingly a simple one.

When a lunar eclipse is not total at any of its phases, it is usual to specify its magnitude by the ratio of the greatest measurement of the obscured part to the moon’s diameter. Thus the magnitude of the partial eclipse of February 28th, 1896, is given in the “Nautical Almanac” as 0·870, the moon’s diameter being taken as unity.

The conditions of lunar eclipses which have been stated have reference to the moon’s passage through the earth’s geometrical shadow, but the actual conditions are greatly modified by the fact that the earth is surrounded by an atmosphere which refracts the suns light so much that the moon is seldom quite obscured during totality. The commencement of the total phase is also rendered difficult of observation by the somewhat indefinite boundary between the umbra and penumbra.

Eclipses of the Sun.—If the moon performed its revolution in the plane of the ecliptic, it is evident that it must always come between us and the sun once in each month. This it does not do, but occasionally it happens to be in the ecliptic when in conjunction, and the moon is then seen to be projected upon the sun. In other words, there is an eclipse of the sun. Let us consider the circumstances, in the first instance, to an observer placed at the centre of the earth. If the centres of the moon and sun appear in the same straight line, the eclipse will be total or annular, according as the moon or sun has the greater apparent diameter. Both these forms of eclipses are possible, on account of the varying apparent diameters of the sun and moon consequent upon their variable distances from the earth. If the moon appear the larger it will evidently cover up the whole of the sun, but when it is the smaller, a ring of sunlight will be visible round the dark holy of the moon, and the eclipse will be an annular one. These conditions are illustrated in Fig. 31, a and b representing a total and an annular eclipse respectively. If the moon and sun be not quite in the same straight line, the moon may still be seen partially projected on the sun’s disc, in which case there will be a partial eclipse of the sun, as in Fig. 31, c.

In a total eclipse there are four so-called contacts: the first when the moon is seen to encroach upon the sun’s disc, the second when the advancing edge of the moon reaches the opposite limb, the third when the following edge of the moon first touches the sun’s boundary, and the fourth when the projected moon finally passes off the sun. The interval between the second and third contacts marks the duration of totality. As referred to our supposed observer at the centre of the earth, the duration evidently depends upon the apparent rate of the moon’s eastward movement as compared with that of the sun, as well as upon the differences of the apparent diameters of the two bodies.

The production of eclipses of the sun may also be considered as arising from the immersion of an observer in the shadow of the moon. This shadow has its axis turned from the sun, but is so short that it does not always reach the earth. If an observer comes near the axis of the conical shadow, and within the apex, the eclipse will be total; if he is in the axis, but outside the apex, the eclipse will be annular.