Low eccentricity planetary orbits

The first six planets in the solar system all possess relatively small eccentricities, ranging from $0.21$ (Mercury) to $0.0068$ (Venus). The squares of the eccentricities are even smaller, the largest being $0.044$. It is, therefore, an excellent approximation to neglect terms involving $e^2$ in our equations of planetary motion. It immediately follows, from Eq. (3), that $s\simeq a$. In other words, the planetary orbits are approximately circular. Recall, however, from Fig. 4, that the Sun is displaced from the geometric center of each orbit, along the line of apsides, by a non-negligible distance $e\,a$. A typical low eccentricity planetary orbit is illustrated in Fig. 5. The planet is at $P$, the Sun at $S$, the equant at $Q$, the perihelion at $M$, and $MM'$ is the line of apsides.

Figure 5: A low eccentricity planetary orbit illustrating various polar angles. Here, $S$ is the Sun, $P$ the planet, $C$ the geometric center of the orbit, $Q$ the equant, and $MM'$ the line of apsides.

Figure 5 also illustrates various different choices of polar angle: the angle $\theta$, measured from the Sun; the angle $\psi$, measured from the geometric center; and the angle $\phi$, measured from the equant. Note, from Eqs. (2), and (3), that, up to ${\cal O}(e)$, the distance $SP$ is $a/(1+e\,\cos\theta)$, whereas the distance $CP$ is $a$. It is easily demonstrated that the distance $QP$ is $a/(1-e\,\cos\phi)$. Thus, standard trigonometry yields

$$\frac{\sin\theta}{\sin\psi} \simeq 1+ e\,\cos\theta,$$ (7)

and

$$\frac{\sin\phi}{\sin\psi} \simeq 1 -e\,\cos\phi.$$ (8)

Neglecting ${\cal O}(e^2)$, these equations can be solved to give

$$\phi\simeq \theta -2\,e\,\sin\theta\simeq \psi - e\,\sin\psi.$$ (9)

Expanding Eq. (6) in orders of $e$, and again neglecting ${\cal O}(e^2)$, we obtain

$$\int_0^\theta (1-2\,e\,\cos\theta)\, d\theta = \theta -2\,e\,\sin\theta \simeq h'\,(t-t_0).$$ (10)

However, $\theta = 2\,\pi$ when $t=t_0+T$, where $T$ is the planet's mean orbital period. Hence, $2\pi=h'\,T$, giving

$$\theta - 2\,e\,\sin\theta \simeq n\,(t-t_0),$$ (11)

where

$$n = \frac{2\pi}{T}$$ (12)

is the planet's mean orbital angular velocity. A comparison of Eqs. (9) and (11) yields

$$\phi \simeq n\,t+\phi_0,$$ (13)

where $\phi_0=-n\,t_0$. We, thus, conclude that the radius vector connecting the planet to the equant rotates uniformly at the planet's mean orbital angular velocity.

Our final picture of a low eccentricity planetary orbit is shown in Fig. 6. The orbit is circular, with radius $a$. The Sun, $S$, is displaced from the orbit's geometric center, $C$, a distance $e\,a$ along the line of apsides, $MM'$. The equant, $Q$, is displaced the same distance in the opposite direction. Finally, the radius vector, $QP$, connecting the equant to the planet, rotates at a uniform rate. This simple picture, which is surprisingly accurate, lies at the heart of Ptolemy's model of the solar system

Figure 6: A low eccentricity planetary orbit. See previous caption.

It is most convenient, for our purposes, to parameterise planetary rotation in terms of the polar angle $\psi$, rather than $\phi$ (see Fig. 5). According to Eqs. (9) and (13), the angle $\psi$ increases at a non-uniform rate specified by

$$\psi\simeq \phi + e\,\sin \phi.$$ (14)

The angle $\phi$ is called the mean anomaly, whereas the angle $\psi$ is (approximately) equal to the so-called eccentric anomaly. It is also easily seen that

$$\theta \simeq \phi + 2\,e\,\sin \phi.$$ (15)