ANOMALY
, in Astronomy, is an irregularity in the motion of a planet, by which it deviates from the aphelion or apogee; or it is the angular distance of the planet from the aphelion or apogee; that is, the angle formed by the line of the apses, and another line drawn through the planet.
Kepler distinguishes three kinds of anomaly; mean, eccentric, and true.
Mean or Simple Anomaly, in the ancient astronomy, is the distance of a planet's mean place from the apogee. Which Ptolomy calls the angle of the mean motion.
But in the modern astronomy, in which a planet P is considered as describing an ellipse APB about the sun S, placed in one focus, it is the time in which the planet moves from its aphelion A, to the mean place or point of its orbit P.
Hence, as the elliptic area ASP is proportional to the time in which the planet describes the arc AP, that area may represent the mean anomaly.—Or, if PD be drawn perpendicular to the transverse axis AB, and meet the circle in D described on the same axis; then the mean anomaly may also be represented by the cir- cular trilineal ASD, which is always proportional to the elliptic one ASP, as is proved in my Mensuration, pr. 3, page 296, second edition.—Or, drawing SG perpendicular to the radius DC produced; then the mean anomaly is alse proportional to SG + the circular arc AD, as is demonstrated by Keil in his Lect. Astron.—Hence, taking DH = SG, the arc AH, or angle ACH will be the mean anomaly in practice, as expressed in degrees of a circle, the number of those degrees being to 360°, as the elliptic trilineal area ASP, is to the whole area of the ellipse; the degrees of mean anomaly, being those in the arc AH, or angle ACH.
Eccentric Anomaly, or of the centre, in the modern astronomy, is the arc AD of the circle ADB intercepted between the apsis A and the point D determined by the perpendicular DPE to the line of the apses, drawn through the place P of the planet. Or it is the angle ACD at the centre of the circle.—Hence the eccentric anomaly is to the mean anomaly, as AD to AD + SG, or as AD to AH, or as the angle ACD to the angle ACH.
True or Equated Anomaly, is the angle ASP at the sun, which the planet's distance AP from the aphelion, appears under; or the angle formed by the radius vector or line SP drawn from the sun to the planet, with the line of the apses.
The true anomaly being given, it is easy from thence to find the mean anomaly. For the angle ASP, which is the true anomaly, being given, the point P in the ellipse is given, and thence the proportion of the area ASP to the whole ellipse, or of the mean anomaly to 360 degrees. And for this purpose, the following easy rules for practice are deduced from the properties of the ellipse, by M. de la Caille in his Elements of Astronomy, and M. de la Lande, art. 1240 &c of his astronomy: 1st, As the square root of SB the perihelion distance, is to the square root of SA the aphelion distance, so is the tangent of half the true anomaly ASP, to the tangent of half the eccentric anomaly ACD. 2nd, The difference DH or SG between the eccentric and mean anomaly, is equal to the product of the eccentricity CS, by the sine of SCG the eccentric anomaly just found. And in this case, it is proper to express the eccentricity in seconds of a degree, which will be found by this proportion, as the mean distance 1: the eccentricity :: 206264.8 seconds, or 57° 17′ 44″.8, in the arch whose length is equal to the radius, to the seconds in the are which is equal to the eccentricity CS; which being multiplied by the sine of the eccentric anomaly, to radius 1, as above, gives the seconds in SG, or in the are DH, being the difference between the mean and eccentric anomalies. 3d, To find the radius vector SP, or distance of the planet from the sun, say either, as the sine of the true anomaly is to the sine of the eccentric anomaly, so is half the less axis of the orbit, to the radius vector SP; or as the sine of half the true anomaly is to the sine of half the eccentric anomaly, so is the square root of the perihelion distance SB, to the square root of the radius vector or planet's distance SP.
But the mean anomaly being given, it is not so easy to find the true anomaly, at least by a direct process<*> Kepler, who first proposed this problem, could not find | a direct way of resolving it, and therefore made use of an indirect one, by the rule of false position, as may be seen page 695 of Kepler's Epitom. Astron. Copernic. See also §628 Wolfius Elem. Astron. Now the easiest method of performing this operation, would be to work first for the eccentric anomaly, viz, assume it nearly, and from it so assumed compute what would be its mean anomaly by the rule above given, and find the difference between this result and the mean anomaly given; then assume another eccentric anomaly, and proceed in the same way with it, finding another computed mean anomaly, and its difference from the given one; and treating these differences as in the rule of position for a nearer value of the eccentric anomaly: repeating the operation till the result comes out exact. Then, from the eccentric anomaly, thus found, compute the true anomaly by the 1st rule above laid down.
Of this problem, Dr. Wallis first gave the geometrical solution by means of the protracted cycloid; and Sir Isaac Newton did the same at prop. 31 lib. 1 Princip. But these methods being unsit for the purpose of the practical astronomer, various series for approximation have been given, viz, several by Sir Isaac Newton in his Fragmenta Epistolarum, page 26, as also in the Schol. to the prop. above-mentioned, which is his best, being not only fit for the planets, but also for the comets, whose orbits are very eccentric. Dr. Gregory, in his Astron. lib. 3, has also given the solution by a series, as well as M. Reyneau, in his Analyse Demontrée, page 713, &c. And a better still for converging is given by Keil in his Prælect. Astron. page 375; he says, if the are AH be the mean anomaly, calling its sine e, consine f, the eccentricity g, also putting z = ge, and ; then the eccentric anomaly AD will be , supposing r = 57.29578 degrees; of which the first term rz/a is sufficient for all the planets, even for Mars itself, where the error will not exceed the 200th part of a degree; and in the orbit of the earth, the error is less than the 10000th part of a degree.
Dr. Seth Ward, in his Astronomia Geometrica, takes the angle AFP at the other focus, where the sun is not, for the mean anomaly, and thence gives an elegant solution. But this method is not sufficiently accurate when the orbit is very eccentric, as in that of the planet Mars, as is shewn by Bullialdus, in his defence of the Philolaic. Astron. against Dr. Ward. However, when Newton's correction is made, as in the Schol. abovementioned, and the problem resolved according to Ward's hypothesis, Sir Isaac affirms that, even in the orbit of Mars, there will scarce ever be an error of more than one second.
ANSÆ, Anses, in Astronomy, those seemingly prominent parts of the ring of the planet Saturn, discovered in its opening, and appearing like handles to the body of the planet; srom which appearance the name ansæ is taken.