The Earth-Sun displacement vector

The elements of the Earth-Sun displacement vector, ${\bf S}$, are shown in Fig. 11. The vector is the sum of two vectors: i.e., ${\bf S} = {\bf c} + {\bf R}$. The fixed vector ${\bf c}$ is of magnitude $e\,a$, where $e$ and $a$ are the eccentricity and semi-major radius of the Earth's orbit about the Sun, respectively, and points in the direction of the Earth's perihelion, $M$, relative to its aphelion, $M'$. The rotating vector ${\bf R}$ is of magnitude $a$.

In the figure, $\omega$ is angle subtended between the direction of the Earth's perihelion, relative to its aphelion, and the direction of the vernal equinox (i.e., the direction of the point in the Earth's sky at which the Sun appears to cross the extension of the Earth's equatorial plane from below). The latter direction is denoted $\vernal$. Note that $\omega$ is known as the argument of the Earth's perihelion. The angle $\psi$ is (approximately) the Earth's eccentric anomaly.

Figure 11: The elements of the Earth-Sun displacement vector. Here, $S$ is the Sun, $E$ the Earth, $C$ the geometric center of the orbit, $MM'$ the line of apsides, and $\vernal$ the vernal equinox.

Let us adopt a set of right-handed Cartesian coordinates, ($x$, $y$, $z$), such that the $x$-$y$ plane corresponds to the ecliptic plane (i.e., the plane of the Earth's orbit), the $x$-axis points towards the vernal equinox, and the $z$-axis points towards the northern ecliptic pole. These coordinates are illustrated in Fig. 11. It is clear, from this figure, that the components of ${\bf c}$ and ${\bf R}$ are

$${\bf c} = e\,a\,\left(\cos\omega,\, \sin\omega,\,0\right),$$ (17)

$${\bf R} = a\,\left[ -\cos(\omega+\psi),\,-\sin(\omega+\psi),\,0\right],$$ (18)

respectively.