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152. Sun's apparent motion in his orbit. The sun's apparent motion or angular velocity, deduced from his longitude as determined from day to day, is found to be different in different parts of the orbit. The motion in longitude is least when the apparent diameter is least, which is about the 1st of July; and is then 57′ 11" in a mean solar day. Its daily motion is greatest when the apparent diameter is greatest, which is about the 1st of January, and is then 61′ 10". From a comparison of the daily motions and corresponding apparent diameters, it is found that the daily motion is always proportional to the square of the apparent diameter. But the apparent diameter is inversely proportional to the distance or radius vector (97). Hence, the daily motion or angular velocity is inversely proportional to the square of the radius vector.

Consequently, if r and r' be two radii vectores, and v and v' the corresponding angular velocities, we have v: v' :: p/2 : r2, or vr2

2. This is also found to be true in the motion of a planet round the sun, and of the moon round the earth.

153. The Sun's radius vector describes equal areas in equal times; and consequently, in unequal times, it describes areas proportional to the times.

Let a and b, Fig. 26, be the sun's places in his orbit at some short interval prior and subsequent to his being at S, as, for instance, half an hour. Let ce and mn be arcs described about E, the place of the earth, as a centre; the former with a radius equal to a unit, and the latter with a radius equal to the radius vector from E to S. Then will the elliptical sector Eab be the area described by the radius vector during an hour when the sun is in the part of his orbit contiguous to S, and the angle aEb, or its measure, the arc ce, will be the sun's hourly motion or angular velocity. Put ce = v, and the distance from E to S 2. Then 17: Hence the area of the circular sector Emn

v: mn = vr.

En

X mn = 2. But the sector Emn does not sensibly differ from the elliptical sector Eab. Consequently, Eab = v2. In like manner for another position of the sun as S', taking v' and r' for the angular velocity and radius vector, we have Ea'b' = v' r'2. But by the last article vr2 = v'r'2. Hence, Eab = Ea'b'. Kepler, who discovered this fact, found also that the radius vector

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of each planet describes about the sun equal areas in equal times This is called Kepler's second law.

154. Kepler's third law. In comparing the periods in which the planets revolve round the sun and their mean distances from him, Kepler discovered that the squares of the periodical times of the planets are proportional to the cubes of their mean distances from the sun.

155. To find the position of the line of the apsides of the solar orbit. Let B and D, Fig. 24, on opposite sides of the transverse axis AP, be corresponding points of the orbit. Then it is evident that the sun's daily or hourly motion at D must be the same as at B; the time in which he moves from P to D must be the same as that in which he moved from B to P; and the longitude of the perigee P must be midway between the longitudes of B and D. Hence, when from a series of the sun's longitudes determined from observation, two times and the corresponding longitudes are found, at which the sun's hourly or daily motion in longitude is the same, the longitude of the perigee and the time that the sun is at that point become known.

Another method. As AP, the line of the apsides, divides the orbit into two equal parts, the sun must be as long in passing from A to P as from P to A. The time in either case is therefore half a year; and in this time the sun passes through 180° of longitude. No other straight line through the earth's centre divides the orbit into two equal parts. It is, therefore, only in passing from one apsis to the other, that the sun employs just half a year in changing his longitude 180°. Hence two longitudes of the sun being found which differ 180°, and are separated by an interval of half a year, will be the longitudes of the perigee and apogee; and the corresponding times will be the times the sun is at those points.

156. Motion of the apsides. From observations made at distant periods it is found that the apsides have a slow direct motion. According to Prof. Bessel, from an examination of many observations made at various times, the longitude of the perigee at the beginning of the year 1800 was 279° 30' 8"; and its yearly increase of longitude is 61.52.

If from 61.5, the annual motion of the perigee from the vernal equinox, we subtract 50.2, the annual retrograde motion of the equinox, we have 11".3 for the annual motion of the perigee.

Taking 180° from the longitude of the perigee in the year 1800, we have 99° 30′ 8′′, for the longitude of the apogee at that time. If this be reduced to seconds and divided by 61".52, the yearly motion of the apogee in longitude, the quotient is 5823. Hence it appears that about 5823 years anterior to the year 1800, the longitude of the apogee was nothing; and consequently the line of the apsides then coincided with the line of the equinoxes. It may be remarked, that this is about the period on which chronologists have fixed, as the time of the creation of the world.

157. True and Mean Anomalies and Equation of the Centre. The angular distance of a body from the perihelion or perigee of its orbit, reckoned to the eastward through the whole circumference of the circle, is called the true anomaly.* The angular distance from the perihelion or perigee, at which the body would at any time be, if it moved with its mean or average angular velocity, is called its mean anomaly. The difference between the true and mean anomalies at any time is called the equation of the centre. Thus if S, Fig. 24, be the sun's place at any time, and s the place at which he would have been at that time, if he had moved from P to s with his mean angular velocity, then will the angle PES be his true anomaly, PEs' his mean anomaly, and SEs the equation of the

centre.

The equation of the centre expresses the difference between the mean and true longitudes. For, let EQ be the direction of the vernal equinox. Then the angle QES is the sun's true longitude when he is at S, and QEs his mean longitude; and these differ by the angle SES.

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158. Anomalistic year. The interval between two consecutive returns of the sun to the perigee is called an anomalistic year. Hence as 360°-50.2: 360° + 11.3: length of tropical year : length of anomalistic year; which is thus found to be 365d. 6h. 13m. 46sec.

* Formerly the anomaly was reckoned from the aphelion or apogee.

159. Sun's mean daily motions in longitude and anomaly. As the length of the tropical year (145) : 1 day :: 360° : sun's mean daily motion in longitude; which is thus found to be 59' 8".33. In a similar manner, taking the length of the anomalistic year (158), we find the sun's mean daily motion in anomaly to be 59′ 8".16.

When the longitude of the perigee of the solar orbit, and the time he is at that point, have been obtained (156), his mean longitude and mean anomaly may be found for any given time, by means of the known mean motions in longitude and anomaly.

160. Greatest equation of the sun's centre. At the perigee and apogee, the equation of the centre is evidently nothing. When the sun moves from the perigee, his true motion is greater than his mean motion, and therefore his true anomaly will be greater than his mean anomaly. The difference between these or the equation of the centre must necessarily increase, till the true motion diminishes so much as to become equal to the mean motion. It will then have its greatest value. For, from that time, the mean motion being greater than the true, the mean anomaly will gain on the true anomaly, and the equation of the centre must decrease till it becomes nothing at the apogee. The same in a reverse order takes place from the apogee to the perigee. As the orbit is symmetrical on each side of the line of the apsides, the greatest equation on one side must be equal to the greatest on the other.

By examining a daily series of the sun's true longitudes when on the meridian, obtained from observations, it will be found that about the first of April, the sun's true daily motion in longitude, which is then decreasing, is for several days nearly equal to the mean daily motion. The same will be found to have place about the first of October, except that then the true daily motion is increasing. At each period, let that noon at which the true and mean motions appear to be most nearly equal, be selected. Let L and L' be the true longitudes of the sun at these noons respectively, M and M', the mean longitudes, and E, the greatest equation of the centre. Then, since the difference between the true longitude and mean longitude is the same as that between the true anomaly and mean anomaly (157), we have

or,

EL-M, and also E M'L'.

Hence, 2E (M'-M) - (L'-L),

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In this expression for the greatest equation, (M' — M) is the sun's mean motion in longitude during the interval between the two noons selected, and becomes known from the mean daily motion in longitude; and (L'-L) is known from the given true longi

tudes.

The greatest equation of the sun's centre is thus found to be 1° 55'.3, nearly.

161. Eccentricity of the solar orbit. Let r = EP (Fig. 24), the radius vector for the perigee, r' = EA = radius vector for the apogee, v = sun's true daily motion at the perigee, v'

the same

at the apogee, and e = the eccentricity. Then v and v', being the greatest and least daily motions, are known (152). By the same article, rr: √/v: √v'.

r'

r'

√v'

Hence, r+r: r' — r : : √ v + √ v : √v — √v',

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The eccentricity and greatest equation in any elliptical orbit evidently depend on each other. If either is given, the other may be obtained by mathematical investigation. From such investigation, it has been ascertained, that when the orbit does not differ greatly from a circle, the eccentricity is nearly equal to the quotient of half the greatest equation, expressed in seconds, divided

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