The Sun-Planet displacement vector

The determination of the elements of the Sun-Planet displacement vector, ${\bf P}'$, is complicated by the fact that the orbits of all the planets (except for the Earth) are slightly inclined to the ecliptic plane. Fig. 12 shows a diagram of a planetary orbit. The orbit intersects the ecliptic plane on the so-called line of nodes, $NN'$. This line must pass through the Sun, $S$, since the Sun is common to both the ecliptic and orbital planes. Here, $N$ is termed the ascending node, since the planet passes through the ecliptic plane from below at this point. Likewise, $N'$ is termed the descending node. The angle $\omega$ between the direction of the perihelion, $M$, and the ascending node, $N$, as seen from the Sun, is termed the argument of the planet's perihelion. The angle $\psi$ is (approximately) the planet's eccentric anomaly.

Figure 12: The elements of the Sun-Planet displacement vector. Here, $S$ is the Sun, $E$ the Earth, $P$ the planet, $C$ the geometric center of the orbit, $MM'$ the line of apsides, and $NN'$ the line of nodes.

The Sun-Planet displacement vector, ${\bf P}'$, is again the sum of two vectors: i.e., ${\bf P}' ={\bf c}' +{\bf R}'$. The fixed vector ${\bf c}'$ is of magnitude $e\,a$, where $e$ and $a$ are the eccentricity and semi-major radius of the planet's orbit around the Sun, respectively, and points in the direction of the planet's aphelion, $M'$, as seen from the Sun. The rotating vector ${\bf R}'$ is of magnitude $a$.

Let us adopt a set of right-handed Cartesian coordinates, ($x'$, $y'$, $z'$), such that the $x'$-$y'$ plane lies in the plane of the orbit, the $x'$-axis points in the direction of the ascending node, as seen from the Sun, and the $z'$-axis would point towards the northern ecliptic pole if the planetary orbit were not slightly inclined to the ecliptic. These coordinates are illustrated in Fig. 12. It is clear, from this figure, that the components of ${\bf c}'$ and ${\bf R}'$ are

$${\bf c}' = e\,a\,\left(-\cos\omega,\, -\sin\omega,\,0\right),$$ (19)

$${\bf R}' = a\,\left[ \cos(\omega+\psi),\,\sin(\omega+\psi),\,0\right],$$ (20)

respectively.

Figure 13: Ecliptic coordinates. Here, $S$ is the Sun, $NN'$ the line of nodes, and $\vernal$ the vernal equinox.

We now need to relate the ($x'$, $y'$, $z'$) coordinate system shown in Fig. 12 to the ($x$, $y$, $z$) system shown in Fig. 11. Fig. 13 illustrates how this is achieved. If the plane of the paper represents the ecliptic plane, and the circle lies in the plane of the planetary orbit, then the circle intersects the page on the line of nodes, $NN'$. Thus half the circle is above the page, and half below, as indicated in the figure. The direction to the ascending node, $N$, makes an angle $\theta$ with respect to the direction to the vernal equinox, $\vernal$, as seen from the Sun, $S$. Here, $\theta$ is called the longitude of the planet's ascending node. In order to transform between the ($x$, $y$, $z$) and ($x'$, $y'$, $z'$) coordinate systems, we first rotate anti-clockwise about the $z$-axis (looking down the axis) by an angle $\theta$, and then rotate anti-clockwise about the new $x$-axis (looking down the axis) by an angle $i$. Here, $i$ is the angle of inclination of the planetary orbit to the ecliptic. Hence, using standard matrix transformation theory,

$$\left(\begin{array}{c}x\\ y\\ z\end{array}\right) = \left( \... ...}\right) \left(\begin{array}{c}x'\\ y'\\ z'\end{array}\right).$$ (21)

It, thus, follows that the components of ${\bf c}'$ and ${\bf R}'$ in the ($x$, $y$, $z$) coordinate system are

$$c_x' = e\,a\left( -\cos\omega\,\cos\theta + \sin\omega\,\sin\theta\,\cos i\right),$$ (22)

$$c_y' = e\,a\left( -\cos\omega\,\sin\theta -\sin\omega\,\cos\theta\,\cos i\right),$$ (23)

$$c_z' = e\,a\left(-\sin\omega\,\sin i\right),$$ (24)

and

$$R_x' = a\left[ \cos(\omega+\psi)\,\cos\theta -\sin(\omega+\psi)\,\sin\theta\,\cos i\right],$$ (25)

$$R_y' = a\left[ \cos(\omega+\psi)\,\sin\theta +\sin(\omega+\psi)\,\cos\theta\,\cos i\right],$$ (26)

$$R_z' = a\left[\sin(\omega+\psi)\,\sin i\right],$$ (27)

respectively.

It should be noted that the inclination of planetary orbits to the ecliptic plane (ideally) manifests itself in the Ptolomaic model as an inclination of the deferents of superior planets, and the epicycles of inferior planets. All other circles in the model remain parallel to the ecliptic plane.