Elliptic orbits
Let us determine
the radial and angular coordinates, and , respectively, of
a planet in an elliptical orbit about the Sun as a function of time.
Suppose that the planet
passes through its perihelion point, and , at .
The constant is termed the time of perihelion passage. It
follows from the previous analysis that

(4.51) 
and

(4.52) 
where ,
, and
are the orbital eccentricity, angular momentum per unit mass, and
energy per unit mass, respectively. The preceding equation can be rearranged to
give

(4.53) 
Taking the square root, and integrating, we obtain

(4.54) 
Consider an elliptical orbit characterized by . Let us write

(4.55) 
where is termed the eccentric anomaly. In fact, is an angle that
varies between and . Moreover, the perihelion point corresponds to
, and the aphelion point to . Now,

(4.56) 
whereas

(4.57) 
Thus, Equation (4.54) reduces to

(4.58) 
where
. This equation can immediately be integrated to give

(4.59) 
Here,

(4.60) 
is termed the mean anomaly,
is the mean orbital angular velocity, and
the orbital period. The mean anomaly is an angle that increases
uniformly in time at the rate of radians every orbital period.
Moreover, the perihelion point corresponds to
, and the aphelion point to
. Incidentally, the angle
, which determines the true angular location of the planet relative to its perihelion point, is called the
true anomaly.
Equation (4.59),
which is known as Kepler's equation, is a transcendental equation
that does not possess a convenient analytic solution. Fortunately, it is fairly straightforward to
solve numerically. For instance, when we use an iterative approach,
if is the th guess then

(4.61) 
this iteration scheme converges very rapidly when
(as is the case for planetary orbits).
Figure 4.5:
Eccentric anomaly.

Equations (4.51) and (4.55) can be combined to give

(4.62) 
This expression allows us to give a simple geometric interpretation of the eccentric anomaly, .
Consider Figure 4.5. Let represent the elliptical orbit of a planet, , about the Sun, .
Let be the major axis of the orbit, where is the perihelion point, the aphelion point, and
the geometric center. It follows that and (see Section A.9), where is the orbital major radius and the
eccentricity. Moreover, the distance and the angle correspond to the radial distance, , and the
true anomaly, , respectively. Let be a circle of radius centered on . It follows that is
a diameter of this circle. Let be a line, perpendicular to , that passes through and joins the
circle to the diameter. It follows that . Let us denote the angle as . Simple trigonometry
reveals that
and
. But, , or
, which can be
rearranged to give
, which is identical to Equation (4.62). We, thus, conclude that the
eccentric anomaly, , can be identified with the angle in Figure 4.5.
Equations (4.51) and (4.55) can be combined
to give

(4.63) 
Thus,

(4.64) 
and

(4.65) 
The previous two equations imply that

(4.66) 
The eccentric anomaly, , and the true anomaly, , always lie in the same quadrant (i.e., if
then
, etc.)
We conclude that, in the case of a planet in an elliptical orbit around the Sun, the radial distance, , and the
true anomaly, , are specified as functions of time via the solution of the following set of equations:
Here, ,
, and
. Incidentally, it is clear that if
then
,
, and
. In other words, the motion is periodic with period
.