Summary

According to the preceding analysis, the Cartesian components of the Earth-Sun displacement vector, ${\bf S}$, in the ($x$, $y$, $z$) coordinate system are

$$S_x = -a\,\left[\cos(\omega+\psi) - e\,\cos\omega\right],$$ (28)

$$S_y = -a\,\left[\sin(\omega+\psi)-e\,\sin\omega\right],$$ (29)

$$S_z = 0,$$ (30)

where

$$\psi = \phi + e\,\sin\phi,$$ (31)

with

$$\phi = n\,t +\phi_0,$$ (32)

and $n=2\pi/T$. Note that $\phi$ is the Earth's mean anomaly, and $\psi$ is (approximately) its eccentric anomaly. Here, $a$, $e$, $T$, and $\omega$ are the semi-major radius, eccentricity, mean orbital period, and argument of the perihelion, respectively, of the Earth's orbit about the Sun. The initial position of the Earth is determined by its initial mean anomaly, $\phi _0$.

The Cartesian components of an Earth-Planet displacement vector, ${\bf P}$, in the ($x$, $y$, $z$) coordinate system are

$$P_x = S_x + a\left(\left[\cos(\omega+\psi)-e\,\cos\omega\right]\cos\theta -\left[\sin(\omega+\psi)-e\,\sin\omega\right]\sin\theta\,\cos i\right),$$ (33)

$$P_y = S_y+ a\left(\left[\cos(\omega+\psi)-e\,\cos\omega\right]\sin\theta +\left[\sin(\omega+\psi)-e\,\sin\omega\right]\cos\theta\,\cos i\right),$$ (34)

$$P_z = a\,\left[\sin(\omega+\psi)-e\,\sin\omega\right]\sin i,$$ (35)

where

$$\psi = \phi + e\,\sin\phi,$$ (36)

with

$$\phi = n\,t +\phi_0,$$ (37)

and $n=2\pi/T$. Here, $a$, $e$, $T$, $\omega$, $\theta$, and $i$ are the semi-major radius, eccentricity, mean orbital period, argument of the perihelion, longitude of the ascending node, and inclination to the ecliptic, respectively, of the orbit of the planet in question about the Sun. The initial position of the planet is determined by its initial mean anomaly, $\phi _0$. Note that the parameters $a$, $e$, $T$, $\omega$, $\theta$, $i$, and $\phi _0$ uniquely specify a planetary orbit, and are known collectively as orbital elements.

Ecliptic latitude, $\beta$, and ecliptic longitude, $\lambda$, are the angular components of a spherical polar coordinate system based on the Cartesian system ($x$, $y$, $z$), and are a convenient way of specifying the position of the Sun and planets in the Earth's sky relative to the stars. In fact, $x=r\,\cos\lambda\,\cos\beta$, $y=r\,\sin\lambda\,\cos\beta$, and $z=r\,\sin\beta$. Thus, once the Cartesian components of some heavenly body's vector displacement from the Earth, ${\bf R}$, say, have been calculated, the object's ecliptic latitude and longitude are obtained from

$$\beta = \tan^{-1} \left(\frac{R_z}{[R_x^{\,2}+R_y^{\,2}]^{1/2}}\right),$$ (38)

$$\lambda = \tan^{-1}\left(\frac{R_y}{R_x}\right),$$ (39)

respectively. Finally, ecliptic latitude and longitude can be related to the more conventional system of declination, $\delta$, and right ascension, $\alpha$, via

$$\sin\delta = \sin\epsilon\,\sin\lambda\,\cos\beta + \cos\epsilon\,\sin\beta,$$ (40)

$$\cos\alpha\,\cos\delta = \cos\lambda\,\cos\beta,$$ (41)

$$\sin\alpha\,\cos\delta = \cos\epsilon\,\sin\lambda\,\cos\beta - \sin\epsilon\,\sin\beta,$$ (42)

and

$$\sin\beta = \cos\epsilon\,\sin\delta - \sin\epsilon\,\sin\alpha\,\cos\delta,$$ (43)

$$\cos\lambda\,\cos\beta = \cos\alpha\,\cos\delta,$$ (44)

$$\sin\lambda\,\cos\beta = \sin\epsilon\,\sin\delta+\cos\epsilon\,\sin\alpha\,\cos\delta,$$ (45)

where $\epsilon=23.4^\circ$ is the inclination of the Earth's equatorial plane to the ecliptic.