In Fig. 51, AVBU represents the equinoctial, EVFU the ecliptic, P and their respective poles, V the vernal equinox, U the autumnal equinox, M a star, PMR the hour circle through the star, QMT the circle of latitude through the star, and TVR = e. The earth's diurnal motion causes all the heavenly bodies to appear to rotate from east to west at the uniform rate of 15° per hour. If in Fig. 50 we conceive the observer placed at the centre O, and his zenith, horizon, and celestial meridian fixed in position, and all the heavenly bodies rotating around PP' as an axis from east to west at the rate of 15° per hour, we form a correct idea of the apparent diurnal motions of these bodies. When the sun or a star in its diurnal motion crosses the meridian, it is said to make a transit across the meridian; when it passes across the part NWS of the horizon, it is said to set; and when it passes across the part NES, it is said to rise (the effect of refraction being here neglected). Each star, as M, describes daily a small circle of the sphere parallel to the equinoctial, and called the Diurnal Circle of the star. The nearer the star is to the pole the smaller is the diurnal circle; and if there were stars at the poles P and P', they would have no diurnal motion. To an observer north of : the equator, the north pole P is elevated above the horizon (as shown in Fig. 50); to an observer south of the equator, the south pole P' is the elevated pole. § 66. SPHERICAL CO-ORDINATES. Several systems of fixing the position of a star on the surface of the celestial sphere at any instant are in use. In each system a great circle and its pole are taken as standards of reference, and the position of the star is determined by means of two quantities called its spherical co-ordinates. I. If the horizon and the zenith are chosen, the co-ordinates of the star are called its altitude and its azimuth. The Altitude of a star is its angular distance, measured on a vertical circle, above the horizon. The complement of the altitude is called the Zenith Distance. The Azimuth of a star is the angle at the zenith formed by the meridian of the observer and the vertical circle passing through the star, and is measured therefore by an arc of the horizon. It is usually reckoned from the north point of the horizon in north latitudes, and from the south point in south. latitudes; and east or west according as the star is east or west of the meridian. II. If the equinoctial and its pole are chosen, then the position of the star may be fixed by means of its declination and its hour angle. The Declination of a star is its angular distance from the equinoctial, measured on an hour circle. The angular distance of the star, measured on the hour circle, from the elevated pole, is called its Polar Distance. The declination of a star, like the latitude of a place on the earth's surface, may be either north or south; but, in practical problems, while latitude is always to be considered positive, declination, if of a different name from the latitude, must be regarded as negative. If the declination is negative, the polar distance is equal numerically to 90° + the declination. The Hour Angle of a star is the angle at the pole formed by the meridian of the observer and the hour circle passing through the star. On account of the diurnal rotation, it is constantly changing at the rate of 15° per hour. Hour angles are reckoned from the celestial meridian, positive towards the west, and negative towards the east. III. The equinoctial and its pole being still retained, we may employ as the co-ordinates of the star its declination and its right ascension. The Right Ascension of a star is the arc of the equinoctial included between the vernal equinox and the point where the hour circle of the star cuts the equinoctial. Right ascension is reckoned from the vernal equinox eastward from 0° to 360°. IV. The ecliptic and its pole may be taken as the standards of reference. The co-ordinates of the star are then called its latitude and its longitude. The Latitude of a star is its angular distance from the ecliptic measured on a circle of latitude. The Longitude of a star is the arc of the ecliptic included between the vernal equinox and the point where the circle of latitude through the star cuts the ecliptic. let For the star M (Fig. 50), 1= latitude of the observer, h=DM = the altitude of the star, 2 ZM the zenith distance of the star, a=∠PZM= the azimuth of the star, p=PM = the declination of the star, r= N P In many problems, a simple way of representing the magnitudes involved, is to project the sphere on the plane of the horizon, as shown in Fig. 52. NESW is the horizon, Z the zenith, NZS the meridian, WZE the prime vertical, WAE the equinoctial projected on the plane of the horizon, P the elevated pole, M a star, DM its altitude, ZM its zenith distance, ∠PZM its azimuth, MR its declination, PM its polar distance, ZPM its hour angle. Z W M D R A S FIG. 52. E § 67. THE ASTRONOMICAL TRIANGLE. The triangle ZPM (Figs. 50 and 52) is often called the astronomical triangle, on account of its importance in problems in Nautical Astronomy. The side PZ is equal to the complement of the latitude of the observer. For (Fig. 50) the angle ZOB between the zenith of the observer and the celestial equator is obviously equal to his latitude, and the angle POZ is the complement of ZOB. The arc NP being the complement of PZ, it follows that the altitude of the elevated pole is equal to the latitude of the place of observation. The triangle ZPM then (however much it may vary in shape for different positions of the star M) always contains the following five magnitudes : PZ= co-latitude of observer = 90°-l, hour angle of the Since the hourly A very simple relation exists between the sun and the local (apparent) time of day. rate at which the sun appears to move from east to west is 15°, and it is apparent noon when the sun is on the meridian of a place, it is evident that if hour angle = 0°, 15°, -15°, etc., time of day is noon, 1 o'clock P.M., 11 o'clock A.M., etc. In general, if t denote the absolute value of the hour angle, according as the sun is west or east of the meridian. § 68. PROBLEM. Given the latitude of the observer and the altitude and azimuth of a star, to find its declination and its hour angle. In the triangle ZPM (Fig. 52), in which the - sign is to be used if a > 90°. The hour angle may then be found by means of [44], whence we have sin t sin a cos h sec d. |