Determination of Ecliptic Longitude

Figure 29 compares and contrasts heliocentric and geocentric models of the motion of a superior planet (i.e., a planet which is further from the sun than the earth), $P$, as seen from the earth, $G$. The sun is at $S$. In the heliocentric model, we can write the earth-planet displacement vector, ${\bf P}$, as the sum of the earth-sun displacement vector, ${\bf S}$, and the sun-planet displacement vector, ${\bf P}'$. The geocentric model, which is entirely equivalent to the heliocentric model as far as the relative motion of the planet with respect to the earth is concerned, and is much more convenient, relies on the simple vector identity

$${\bf P} = {\bf S}+{\bf P}' \equiv{\bf P}'+{\bf S}.$$ (155)

In other words, we can get from the earth to the planet by one of two different routes. The first route corresponds to the heliocentric model, and the second to the geocentric model. In the latter model, ${\bf P}'$ gives the displacement of the so-called guide-point, $G'$, from the earth. Since ${\bf P}'$ is also the displacement of the planet, $P$, from the sun, $S$, it is clear that $G'$ executes a Keplerian orbit about the earth whose elements are the same as those of the orbit of the planet about the sun. The ellipse traced out by $G'$ is termed the deferent. The vector ${\bf S}$ gives the displacement of the planet from the guide-point. However, ${\bf S}$ is also the displacement of the sun from the earth. Hence, it is clear that the planet, $P$, executes a Keplerian orbit about the guide-point, $G'$, whose elements are the same as the sun's apparent orbit about the earth. The ellipse traced out by $P$ about $G'$ is termed the epicycle.

Figure 29: Heliocentric and geocentric models of the motion of a superior planet. Here, $S$ is the sun, $G$ the earth, and $P$ the planet. View is from the northern ecliptic pole.

Figure 30: Planetary longitude model. View is from northern ecliptic pole.

Figure 30 illustrates in more detail how the deferent-epicycle model is used to determine the ecliptic longitude of a superior planet. The planet $P$ orbits (counterclockwise) on a small Keplerian orbit $\Pi'PA'$ about guide-point $G'$, which, in turn, orbits the earth, $G$, (counterclockwise) on a large Keplerian orbit $\Pi G'A$. As has already been mentioned, the small orbit is termed the epicycle, and the large orbit the deferent. Both orbits are assumed to lie in the plane of the ecliptic. This approximation does not introduce a large error into our calculations because the orbital inclinations of the visible planets to the ecliptic plane are all fairly small. Let $C$, $A$, $\Pi$, $a$, $e$, $\varpi$, and $T$ denote the geometric center, apocenter (i.e., the point of furthest distance from the central object), pericenter (i.e., the point of closest approach to the central object), major radius, eccentricity, longitude of the pericenter, and true anomaly of the deferent, respectively. Let $C'$, $A'$, $\Pi'$, $a'$, $e'$, $\varpi'$, and $T'$ denote the corresponding quantities for the epicycle.

Figure 31: The triangle $GBP$.

Let the line $GG'$ be produced, and let the perpendicular $PB$ be dropped to it from $P$, as shown in Fig. 31. The angle $\mu\equiv PG'B$ is termed the epicyclic anomaly (see Fig. 32), and takes the form

$$\mu =T'+\varpi' - T - \varpi = \bar{\lambda}' + q' - \bar{\lambda}-q,$$ (156)

where $\bar{\lambda}$ and $q$ are the mean longitude and equation of center for the deferent, whereas $\bar{\lambda}'$ and $q'$ are the corresponding quantities for the epicycle--see Cha. 5. The epicyclic anomaly is generally written in the range $0^\circ$ to $360^\circ$. The angle $\theta\equiv PGG'$ is termed the equation of the epicycle, and is usually written in the range $-180^\circ$ to $+180^\circ$. It is clear from the figure that

$$\tan\theta = \frac{\sin\mu}{r/r'+ \cos\mu},$$ (157)

where $r\equiv GG'$ and $r'\equiv G'P$ are the radial polar coordinates for the deferent and epicycle, respectively. Moreover, according to Equation (81), $r/r' = (a/a')\,z$, where

$$z = \frac{1-\zeta}{1-\zeta'},$$ (158)

and

$$\zeta = e\,\cos\,M -e^{\,2}\,\sin^2\,M,$$ (159)

$$\zeta' = e'\cos\,M' -e'^{\,2}\sin^2\,M'$$ (160)

are termed radial anomalies. Finally, the ecliptic longitude of the planet is given by (see Fig. 32)

$$\lambda = \bar{\lambda} + q + \theta.$$ (161)

Now,

$$\theta(\mu,z) \equiv \tan^{-1}\left[ \frac{\sin\mu}{(a/a')\,z+ \cos\mu}\right]$$ (162)

is a function of two variables, $\mu$ and $z$. It is impractical to tabulate such a function directly. Fortunately, whilst $\theta(\mu,z)$ has a strong dependence on $\mu$, it only has a fairly weak dependence on $z$. In fact, it is easily seen that $z$ varies between $z_{\rm min} = \bar{z} - \delta z$ and $z_{\rm max} = \bar{z}+\delta z$, where

$$\bar{z} = \frac{1+e\,e'}{1-e'^{\,2}},$$ (163)

$$\delta z = \frac{e+e'}{1-e'^{\,2}}.$$ (164)

Let us define

$$\xi = \frac{\bar{z}-z}{\delta z}.$$ (165)

This variable takes the value $-1$ when $z=z_{\rm max}$, the value $0$ when $z=\bar{z}$, and the value $+1$ when $z=z_{\rm min}$. Thus, using quadratic interpolation, we can write

$$\theta(\mu,z)\simeq \Theta_-(\xi)\,\delta\theta_-(\mu) + \bar{\theta}(\mu) + \Theta_+(\xi)\,\delta\theta_+(\mu),$$ (166)

where

$$\bar{\theta}(\mu) = \theta(\mu,\bar{z}),$$ (167)

$$\delta\theta_-(\mu) = \theta(\mu,\bar{z}) - \theta(\mu,z_{\rm max}),$$ (168)

$$\delta\theta_+(\mu) = \theta(\mu,z_{\rm min}) - \theta(\mu,\bar{z}),$$ (169)

and

$$\Theta_-(\xi) = - (1/2)\,\xi\,(\xi-1),$$ (170)

$$\Theta_+(\xi) = + (1/2)\,\xi\,(\xi+1).$$ (171)

This scheme allows us to avoid having to tabulate a two-dimensional function, whilst ensuring that the exact value of $\theta(\mu,z)$ is obtained when $z=\bar{z}$, $z_{\rm min}$, or $z_{\rm max}$. The above interpolation scheme is very similar to that adopted by Ptolemy in the Almagest.

Our procedure for determining the ecliptic longitude of a superior planet is described below. It is assumed that the ecliptic longitude, $\lambda_S$, and the radial anomaly, $\zeta_S$, of the sun have already been calculated. The latter quantity is tabulated as a function of the solar mean anomaly in Table 33. In the following, $a$, $e$, $n$, $\tilde{n}$, $\bar{\lambda}_0$, and $M_0$ represent elements of the orbit of the planet in question about the sun, and $e_S$ represents the eccentricity of the sun's apparent orbit about the earth. (In general, the subscript $S$ denotes the sun.) In particular, $a$ is the major radius of the planetary orbit in units in which the major radius of the sun's apparent orbit about the earth is unity. The requisite elements for all of the superior planets at the J2000 epoch ( $t_0= 2\,451\,545.0$ JD) are listed in Table 30. The ecliptic longitude of a superior planet is specified by the following formulae:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (172)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (173)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (174)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (175)

$$\mu = \lambda_S - \bar{\lambda}-q,$$ (176)

$$\bar{\theta} = \theta(\mu,\bar{z})\equiv \tan^{-1} \left(\frac{\sin\mu}{a\,\bar{z}+\cos\mu}\right),$$ (177)

$$\delta\theta_- = \theta(\mu,\bar{z}) - \theta(\mu,z_{\rm max}),$$ (178)

$$\delta\theta_+ = \theta(\mu,z_{\rm min}) - \theta(\mu,\bar{z}),$$ (179)

$$z = \frac{1-\zeta}{1-\zeta_S},$$ (180)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (181)

$$\theta = \Theta_-(\xi)\,\delta\theta_-+ \bar{\theta} + \Theta_+(\xi)\,\delta\theta_+,$$ (182)

$$\lambda = \bar{\lambda} + q+ \theta.$$ (183)

Here, $\bar{z} = (1+e\,e_S)/(1-e_S^{\,2})$, $\delta z = (e+e_S)/(1-e_S^{\,2})$, $z_{\rm min} = \bar{z} - \delta z$, and $z_{\rm max} = \bar{z}+\delta z$. The constants $\bar{z}$, $\delta z$, $z_{\rm min}$, and $z_{\rm max}$ for each of the superior planets are listed in Table 44. Finally, the functions $\Theta_\pm$ are tabulated in Table 45.

For the case of Mars, the above formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of $3'$ and a maximum error of $14'$. For the case of Jupiter, the mean error is $1.6'$ and the maximum error $4'$. Finally, for the case of Saturn, the mean error is $0.5'$ and the maximum error $1'$.