Again, of all lines drawn from E to the circle, the least is E c, the greatest E C, and the lines are continually greater than each other as they recede from c and approach C; throughout the semicircle c Y, C, or the western semicircle, therefore the distances Ec, Ev,, EY, EV, continually increase, and the apparent diameters continually diminish; and throughout the semicircle CY, c, the eastern semicircle, the distances continually diminish, and the apparent diameters continually increase. This again corresponds with the results of observation. Lastly, it is evident that the motion of the planet being from west to east, while it moves through the arc Y, CY,, its apparent motion, which will be measured by the angle Y, E Y2, will be in that direction also; or it will be direct. At the points Y,, and Y2, the motion in the orbit for a short period is in the direction of the tangent; the position of the body, therefore, will appear for a time unchanged, or the body will be stationary. Throughout the arc Y, cY,, the apparent motion of the body will be measured by the angle Y, EY,, measured in the opposite direction from Y, E Y2, or from east to west; or the motion will then be retrograde. 2 These results have a considerable correspondence with those collected from observation. They give us direct and retrograde motions, and account for a stationary appearance. The motion, also, on the eastern side of the sun is at first direct from C to Y,, then the body appears stationary, and then the motion becomes retrograde from Y, till it disappears near c; and on its re-appearance on the western side of c, the body still has a retrograde motion, till it becomes stationary at Y,, and then has a direct motion between Y, and C. So far the results appear to correspond with those of observation; but they differ in two very material circumstances. On the supposition that we have made, the apparent direct and retrograde motions are measured by the same angle Y, EY,, measured in opposite directions; or they are equal, and the body appears stationary at Y,, Y2, the points of greatest elongation, and commences its retrograde motion as soon as it quits Y, and continues it till it arrives at Y1. In reality, as we have already seen, the amount of retrograde is less than that of direct motion; and the body continues a direct motion after it passes Y, and recommences it before it arrives at Yı, having its stationary points somewhere between those points, as at v and v1, and its retrograde motion confined to the arc between those points. Our supposition, therefore, though it gives us results corresponding to a considerable extent with those actually observed, does not at present represent them correctly. Indeed it is impossible that it should do so, for we have supposed the point S stationary, or that the motions of the planet are measured from S, the sun, considered as a fixed point; whereas S is itself in motion. Our results, therefore, drawn from fig. 22, cannot fully explain actual appearances until we take into account the motion of S. The motion of the sun itself, as seen from the earth, is a direct motion; the motion given to the planet itself, therefore the supposition of its partaking of the sun's motion would be a direct motion also. This will at once appear from the inspection of fig. 29. Let A B C D represent any different bodies, to all of which an equal motion in the same direction is given; and let the amount and direction of this motion for a short time be represented by the equal and parallel lines A a, B b, C c, D d. Suppose, also, that E, a point in the line d D produced, is the situation of the observer; and that the apparent motion Fig. 29. tion of C, a point on the opposite side of ED, is measured in the opposite direction, or it is a retrograde motion. The points D d are seen in the same line ED, and D therefore has no apparent motion at all. Now to apply these observations to the case of an inferior planet, we may suppose A to represent the sun, and A a its actual, or the angle A E a its apparent, motion. In this case, A a must be drawn perpendicular or very nearly so to A E, for the sun moves in an orbit very nearly circular, the earth being in the centre, and the direction of his motion is therefore very nearly perpendicular to his radius vector. For the sake of simplicity, we will suppose it accurately so. Now E D, the direction of the line in which there is no apparent motion, and beyond which the apparent motion becomes retrograde, is necessarily parallel to A a, the direction of the actual motion; in the case supposed, therefore, it is perpendicular to E A, for a A is so. But whenever the point B is on the same side of the line E D that A is, or whenever the angle A E B is less than AED, that is, in the present case, less than a right angle, the apparent motion of B is direct, though it varies in amount according to its distance and direction. Now if we suppose B to represent a planet, A E B is the elongation, and in the case of an inferior planet, the elongation is never so great as 90°. If, therefore, an inferior planet partake of the sun's motion, the motion which it thus derives will appear to be direct in every position; and this apparent motion is to be combined with the apparent motion resulting from the planet's motion in its own orbit, before we can ascertain what the real apparent motion would be. Now the apparent motion resulting from the planet's motion round the sun is sometimes direct, sometimes nothing, sometimes retrograde. In the first case, the real apparent motion will be the sum of two direct motions, and will of course be direct; in the second, it will be the direct motion occasioned by the sun's motion only; and in the last, it will be the difference between the direct motion occasioned by the sun's motion round the earth, and the retrograde motion occasioned by the planet's motion round the sun, and will therefore be direct or retrograde as the former or the latter of these two quantities is the greater. The direct motion might be so great as to exceed, in all cases, the utmost value of the retrograde, and the whole apparent motion would then always be direct. If, however, this be not so, it is yet evident that, for some time, the direct would exceed the retrograde motion; for the retrograde motion is evidently very little for some distance near the points Y,, Y,, in fig. 28, the points of greatest elongation, the curve at first departing very little from its tangent; and on the other hand, it is evident, also, that the retrograde motion is greatest at c, the point of inferior conjunction: for then the planet is nearest the earth, and its motion also being perpendicular to the line joining them produces the greatest effect. It may, therefore, well happen that for some time after the periods of greatest elongation, the direct may overcome the stationary motion, and the whole apparent motion, therefore, be direct; that they may then become equal, in which case, the planet would appear stationary; and that then the retrograde may exceed the direct motion, or the whole apparent motion become retrograde, and continue so until, after passing through inferior conjunction, the direct and retrograde motions again become equal, or the planet apparently stationary, before arriving at its greatest elongation; and then, the direct motion exceeding the retrograde, the whole apparent motion will again become direct till the planet arrives at its greatest elongation. At this period the retrograde motion first disappears, and then is converted into a direct one, and of course the whole apparent motion is direct until the same course of appearances recommences; and it is the course which we have already seen to obtain in nature. We see, therefore, that the apparent motions of the inferior planets are not inconsistent with the supposition of their moving in an orbit round the sun. Their phases point strongly to the conclusion that they do so: and nothing could have prevented us from at once adopting that conclusion except the uncertainty, till the question were examined, whether it could be reconciled with their apparent motions. It is, however, worth while to examine somewhat more minutely the law by which these apparent motions are regulated, for the purpose of deducing one or two results accurately corresponding with observation; and also for the sake of connecting the phenomena we are at present discussing, with some others to which we shall presently advert. For this purpose we must again refer to fig. 29. The motion of the planet occasioned by its partaking of the sun's motion, being equal in all cases to that of the sun itself, is always the same, and may be represented by the line B b, of which the length will be known and constant, and the apparent motion by the angle B E b. Let b B be produced to meet A E in F: BF will, as we have seen before, be perpendicular to A E. Now, Bb being very small, BEb must be very small also, and, consequently, the sine of BEb and BEb very nearly equal and also BE and E very nearly equal. But sin. BE b = sin. Bb Eb 6ВЕ Bb Eb = Bb Eb sin FBE = cos. FEB (as FEB 90°) = cos. E FB very nearly. = Therefore BEb varies as Bb EB cos. FE B, EB or directly as the cosine of the elongation, and inversely as the distance. Now, when the elongation corresponds on the opposite sides of EC, the distances are equal: thus in fig. 28, EY, EY2, the two tangents are equal, and so are E v1, Ev2, EV1, EV,, if the angles SEVI, SEV, are so. The elongations, however, although equal, are measured in opposite directions, and therefore if one of them be considered as positive, the other will be negative. But the cosine of an arc is the same, whether the arc be positive or negative: the apparent motion B E b, therefore, which varies directly as the cosine of the elongation, and inversely as the distance, will be the same at equal elongations on each side of E A: for both its elements are equal in those cases. The direct motion, therefore, given to the planet by its accompanying the sun is the same on each side of the sun. Again, the motion, whether direct or retrograde, produced by the planet's motion in its own orbit round the sun, is also the same on each side of the sun. In fig. 30 let S represent the sun, E the earth, and V W part of the orbit of an inferior planet, considered as circular. Let S E be joined, and let V W be two points in the orbit, having equal elongations on each side of S E. As before, therefore, V E, WE, the distances of the planet from the earth at the two positions are equal, and so also are the angles SVE, SW E, for the the portions of its orbit described by the planet in a very short time; these arcs will, if the motion is uniform, be equal. Besides this, for a very short space, the arc will coincide with the tangent, and may be considered as a straight line; if at W a tangent W Z be drawn in the opposite direction to the arc W w, W w may be considered as a continuation of ZW. Treating then the triangles V E v, WEw, as rectilinear, V v sin. V Ev = sin. EV v = sin E Vv Ev sin. WE w = sin. EW w sin. EWZ W w = E w W w very nearly, (sin. EWZ EW =sin. EW w) and sin. VEv=sin WE w: for Vv=Ww, EV=EW: and EV v EVS-SV v=EWS-SWZ= E W Z. The angles VEv, WEw, or the apparent motions, depending upon the motion of the planet round the sun, are therefore themselves equal in corresponding positions on different sides of the sun. It is evident, therefore, that the whole apparent motions of the planet must correspond on each side of the sun for these are always the sum, or the difference of the apparent motions derived from the planet's motion round the sun, and from the sun's motion round the earth: and as each of these is the same at equal elongations each way, their sum or difference, or the whole apparent motion, must be equal. We should therefore find the retrograde motion, before and after inferior conjunction, equal: the stationary points at the same elongation, the rate of motion at equal distances always equal-and so we very nearly do. In these deductions, however, we have supposed the orbits circular, and the motion uniform: if either of these suppositions be inaccurate, our results will not accurately apply. With respect to the sun, we know them to be inaccurate, for his orbit is an ellipse, and his motion unequal, depending on the variation in his distance. Still, as the ellipse in which he moves does not differ much from a circle, and as the inequality of his motion is small, our conclusions, deduced on the erroneous suppositions we have adopted, will differ but little from the truth; and will sufficiently explain the manner in which his motion will affect the apparent motion of a planet revolving round him, though they will no longer be accurately correct as a representation of the amount to which it does so. In the same way, if we were to suppose the orbit of the planet some curve, differing not very considerably from a circle, the general effect of this motion in such an orbit would be sufficiently represented by the conclusions we have already deduced; though many particulars would cease to be accurately correct. For instance, the point of greatest elongation might not accurately correspond to that when exactly half the disk of the planet was visible; for the line drawn from the sun to that point might not be exactly perpendicular to the tangent then: the apparent motions at equal opposite elongations might not be exactly equal; for neither the distances, the amount of the actual motions, nor their direction, might accurately correspond. Still they would not differ much, and the particular conclusions we have come to would not be far from the truth: and the general principles by which direct and retrograde motion, and the exist ence and situation of the stationary points are accounted for, would evidently remain unaffected. SECTION III. On the Phases and apparent Motions of the Superior Planets. THE phases of the superior planets present but little that is observable. Mars, indeed, is occasionally very perceptibly gibbous; but even he never assumes the appearance of a semicircle, or of a crescent: and the other superior planets are hardly seen to present less than their full face to an observer at the earth. This constitutes a marked difference between their appearances and se of the inferior planets. Another marked difference consists in this: that, whereas the inferior planets are never seen beyond a certain distance from the sun, and occasionally pass between the sun and the earth, these planets are seen at all distances from the sun, and never pass between the sun and earth. They are, indeed, seen on the line joining the sun and the earth; but it is on the side opposite to the sun, or at the distance of 180° from him. At this time they are said to be in opposition. Their motions may also be ascertained, so as to enable us to compute that they are, at given times, in the line joining the earth and sun, on the same side as the sun, or in conjunction: but their distances from the earth are then greater than the sun's distance, or they are beyond him, just as the inferior planets are when in superior conjunction. The points of opposition, in the case of a superior planet, and of inferior conjunction in an inferior one, are in the same line with these, but on the opposite side of the sun: the points, therefore, of superior conjunction in the one case, and of conjunction in the other; and those of inferior conjunction in the one case, and of opposition in the other, seem to have a certain degree of correspondence. They correspond also in another remarkable respect. The superior planets, as well as the inferior, appear sometimes to have a direct, sometimes a retrograde motion; and they also, like the others, are stationary for an interval between the two. In the case of the superior planets, the conjunction takes place during the period of direct motion, as we have already seen that the superior conjunction of an inferior planet does; and the opposition takes place during the period of retrograde motion, as we have already seen that the inferior conjunction of an inferior planet does. Besides this, we find the apparent diaposition, and continually diminishing as meters of these planets greatest in opthey approach conjunction, when they are least another circumstance of the same kind of correspondence. Taking the whole course of a superior planet from one opposition to another, we find its apparent motion at first retrograde, then at a certain elongation the planet becomes stationary, then its motion is direct, and continues so while it passes through conjunction, and till it arrives at very nearly the same elongation on the other side of the sun as it before had at its stationary point: then it again becomes stationary, and then its motion again becomes retrograde, till it is again in opposition. These appearances, as far as they go, exactly correspond with those of an inferior planet, substituting only the words conjunction for superior conjunction, and opposition for inferior conjunction. They are not, however, all the appearances of an inferior planet, for the remarkable circumstances of the points of greatest elongation, and of the recurrence of all the phases exhibited by the moon, are wanting. The want of these makes it less easy, than in the case of an inferior planet, to divine the law of their motions; and, as these planets are seen at all angular distances from the sun, there is not the same obvious reason as before to conjecture that their motions depend upon his. Still there is a general similarity in the two sets of appearances, which makes it natural that we should inquire whether they cannot be explained in the same manner. For this purpose, let us again take the case of a planet supposed to move in a circular orbit round the sun; and as before, let us omit for the present, for the sake of simplicity, all consideration of the sun's motion. The earth, being between the sun and the planet when the latter is in opposition, must then be at a point within the orbit of the planet. Let, then, in fig. 31, the circle represent the orbit of the planet; S, its centre, the sun, and E, the earth; greater as they recede from EO and approach EC, and then again continually less as they recede from EC on the other side, and approach E O. The distance, therefore, of the planet from the earth is least in opposition, and continually increases thence till the planet is in conjunction, when it is greatest; and the apparent diameter is of course greatest in opposition, and continually less thence till the planet is in conjunc tion, when it is least. Again, if Mm be points on opposite sides of E C, at equal distances from C, EM, Em, are equal; and of course the apparent diameter at M and m are so also. The same course of phenomena, therefore, succeed each other in a reverse order, at corresponding distances on the other side of C, and the apparent diameter continually decreases from opposition to conjunction. The phenomena, therefore, as in the case of inferior planets, correspond on this supposition with observation. Supposing the planet itself to be spherical, the portion of the disk visible would vary (p. 76) as the versed sine of the exterior angle of elongation. At the points O and C, this angle is evidently 180°, or its versed sine is the whole diameter, and the full enlightened face of the planet would be turned towards the earth. At any intermediate point M, this will not be so: the sun and earth not being in the same line from the planet would have different parts of its surface turned towards them; and phases would be occasioned. The amount of these variations depends on the extent by which the angle SMY differs from 180°, or it depends on the magnitude of the angle E MS (the angle subtended at the planet by the distance between the sun and earth, or the earth's elongation at the planet), for EMS and SMY together make up 180°. Now if we draw the lines C M, MO, the angle C MO, whenever the point M is taken between C and O, is the angle in a semicircle, and therefore a right angle. And it will obviously, in all cases, be composed of three parts, CMS, SME, EMO; and, consewill be less than a right angle. SMY, quently, EMS, being one of these parts, therefore, will be in all cases greater than 90°, and the planet, though it may become gibbous, can never have half its face hidden, or become a crescent. Again, sin. SME sin. SEM; ES SM' |