Keplerian Orbit Theory

According to Kepler's first law of planetary motion, all planetary orbits are ellipses lying in a fixed plane which are confocal with the sun. Moreover, according to Kepler's second law, the radius vector which connects the sun to a given planet sweeps out equal areas in equal time intervals.

Figure 2: A Keplerian orbit.

Consider Figure 2. $\Pi PBA$ is the arc of an ellipse which corresponds to a planetary orbit. Furthermore, $C$ is the geometric center of the orbit, $S$ the focus at which the sun is located, $P$ the instantaneous position of the planet, $\Pi$ the perihelion point (i.e., the point of closest approach to the sun), and $A$ the aphelion point (i.e., the point of furthest distance from the sun). The ellipse is symmetric about $\Pi A$, which is termed the major axis, and about $CB$, which is termed the minor axis. The length $CA\equiv a$ is called the orbital major radius. The length $CS$ represents the displacement of the sun from the geometric center of the orbit, and is generally written $e\,a$, where $e$ is termed the orbital eccentricity. The length $CB\equiv b = a\,(1-e^2)^{1/2}$ is called the orbital minor radius. The length $SP\equiv r$ represents the radial distance of the planet from the sun. Finally, the angle $RSP\equiv T$ is the angular bearing of the planet from the sun, relative to the major axis of the orbit, and is termed the true anomaly.

$\Pi QDA$ is the arc of a circle whose geometric center is $C$, and whose radius is $a$. Hence, the circle passes through the perihelion and aphelion points. $R$ is the point at which the perpendicular from $P$ meets the major axis $\Pi A$. The point where $RP$ produced meets circle $\Pi QDA$ is denoted $Q$. Finally, the angle $SCQ\equiv E$ is called the elliptic anomaly.

Now, the equation of the ellipse $\Pi PBA$ is

$$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1,$$ (1)

where $x$ and $y$ are the perpendicular distances from the major and minor axes, respectively. Likewise, the equation of the circle $\Pi QDA$ is

$$\frac{x'^{\,2}}{a^2} + \frac{y'^{\,2}}{a^2} = 1.$$ (2)

Hence, if $x=x'$ then

$$\frac{y}{y'} = \frac{b}{a},$$ (3)

and it follows that

$$\frac{RP}{RQ} = \frac{b}{a}.$$ (4)

It is easily demonstrated that $SR=r\,\cos T$, $RP=r\,\sin T$, $CR= a\,\cos E$, $RQ= a\,\sin E$, and $CS= e\,a$. Consequently, Eq. (4) yields

$$r\,\sin T = b\,\sin E.$$ (5)

Also, since $SR = CR-CS$, we have

$$r\,\cos T = a\,(\cos E - e).$$ (6)

Taking the square root of the sum of the squares of the previous two equations, we obtain

$$r = a\,(1-e\,\cos E),$$ (7)

which can be combined with Eq. (6) to give

$$\cos T = \frac{\cos E - e}{1-e\,\cos E}.$$ (8)

Now, according to Kepler's second law,

$$\frac{{\rm Area}\,\Pi PS}{\pi\,a\,b} = \frac{t-t_\ast}{\tau},$$ (9)

where $t$ is the time at which the planet passes point $P$, $t_\ast$ the time at which it passes the perihelion point, and $\tau$ the orbital period. However,

$${\rm Area} \,\Pi PS = {\rm Area}\, SRP + {\rm Area} \,\Pi RP$$

$$= \frac{1}{2}\,r^2\,\cos T\,\sin T + {\rm Area}\,\Pi RP.$$ (10)

But,

$${\rm Area}\,\Pi RP = \frac{b}{a}\,{\rm Area}\,\Pi RQ,$$ (11)

since $RP/RQ = b/a$ for all values of $T$. In addition,

$${\rm Area}\,\Pi RQ = {\rm Area}\,\Pi QC - {\rm Area}\,RQC$$

$$= \frac{1}{2}\,E\,a^2 - \frac{1}{2}\,a^2\,\cos E\,\sin E.$$ (12)

Hence, we can write

$$\left(\frac{t-t_\ast}{\tau}\right) \pi\,a\,b= \frac{1}{2}\,r... ... T\,\sin T + \frac{b}{a}\,\frac{a^2}{2}\,(E - \cos E\,\sin E).$$ (13)

It is easily demonstrated that $r\,\sin T = b\,\sin E$, and $r\,\cos T = a\,(\cos E - e)$, so the above equation reduces to

$$M = E - e\,\sin E,$$ (14)

where

$$M = \left(\frac{2\pi}{\tau}\right)\,(t-t_\ast)$$ (15)

is an angle which is zero at the perihelion point, increases uniformly in time, and has a repetition period which matches the period of the planetary orbit. This angle is termed the mean anomaly.

In summary, the radial and angular polar coordinates, $r$ and $T$, respectively, of a planet in a Keplerian orbit about the sun are specified as implicit functions of the mean anomaly, which is a linear function of time, by the following three equations:

$$M = E - e\,\sin E,$$ (16)

$$r = a\,(1- e\,\cos E),$$ (17)

$$\cos T = \frac{\cos E - e}{1-e\,\cos E}.$$ (18)

It turns out that the earth and the five visible planets all possess low eccentricity orbits characterized by $e\ll 1$. Hence, it is a good approximation to expand the above three equations using $e$ as a small parameter. To second-order, we get

$$E = M + e\,\sin M + (1/2)\,e^2\,\sin 2M,$$ (19)

$$r = a\,(1-e\,\cos T - e^2\,\sin^2 T),$$ (20)

$$T = E + e\,\sin E + (1/4)\,e^2\,\sin 2 E.$$ (21)

These equations can be combined to give $r$ and $T$ as explicit functions of the mean anomaly:

$$r = a\,(1 -e\,\cos M + e^2\,\sin^2 M),$$ (22)

$$T = M + 2\,e\,\sin M + (5/4)\,e^2\,\sin 2M.$$ (23)

It is interesting to note that the above formulae are almost identical to those obtained by expanding Ptolemy's equant theory to second-order in the eccentricity: i.e.,

$$r = a\,[1 -e\,\cos M + (3/2)\,e^2\,\sin^2 M],$$ (24)

$$T = M + 2\,e\,\sin M + e^2\,\sin 2M.\$$ (25)