Eccentric anomaly

ASP is called the true anomaly of the planet. It is found that, in nstronoinieal calculations, the true anomaly is not a very convenient angle with which to deal. Instead we use the mean anomaly, which is defined to be 271 times the ratio of the area of the elliptic sector ASP to the area of the ellipse. Another angle of significance is the eccentric anomaly, u, of the planet defined to be the angle ACQ where Q is the point in which the ordinate through P meets the auxiliary circle of the ellipse. 

The problem set by Bessel was that of expressing the difference between the mean and eccentric anomalies, k — , as a series of sines of multiples of the mean anomaly, i.c. that of determining the coefficients cT r — 1, 2, 3,.. . ) such that... 

We conclude this section by showing the relation between the fictitious time and the eccentric anomaly. To this end, setting uj2 = —877, let us write equation (7) as... 

The angles are obtained with the usual equations. In the planar problem, the true longitude (ip) is given by the angle formed by the radius vector with the first axis of the reference system (to be obtained through arctany/x where x,y are the components of r). In the spatial problem, some rotations are necessary beforehand. The anomalies may also be easily obtained, starting with the eccentric anomaly (u), which is given... 

The mean values r-1, r, r2.. . are more easily calculated by means of the eccentric anomaly. Using (14) and (15)... 

The similarity between these equations and (15), 22, shows clearly the analogy between tp, X, and the eccentric anomaly. 

The mean anomaly M is the angle from perigee that would be traversed by a satellite moving at its mean angular velocity n. Given an initial value Mq, usually taken as 0 for a particular epoch (time) at perigee, the mean anomaly at time tis M = M<, + n t — to), where n = The eccentric anomaly E may then... 

If ti and J denote the eccentric and mean anomalies of a planet show that... 

The equation of the path differs from that of an ellipse with the parameter l and eccentricity e by the factor y. While r goes through one libration, the true anomaly 0 increases by 2ir/y. The path approaches more nearly to an ellipse the smaller the coefficient cl of the additional term in the potential energy, and for c1=0 it becomes an ellipse. For small values of c1 we can regard the path as an ellipse, whose perihelion slowly rotates with the angular velocity... 

Historically, the departures of the moon s motion from circularity have been among the driving anomalies that celestial mechanics was designed to address and explain. The inequalities were expressed as variations in the longitude of the moon relative to that expected in the simplest theory in which its rate about the center of the deferent circle, or later around the eccentric. The development of theory by Newton, and workers in the past two centuries, has derived the precise forms of all of the classical inequalities. All motion is referred to both the position of the moon along the ecliptic. X and that of the sun X. The most important terms are... 

---