Determination of Ecliptic Latitude of Inferior Planet

Figure 35 shows the orbit of an inferior planet. As we have already mentioned, the epicycle and deferent of such a planet have the same elements as the orbit of the planet in question around the sun, and the apparent orbit of the sun around the earth, respectively. It follows that the epicycle and deferent of an inferior planet are, respectively, inclined and parallel to the ecliptic plane. Let the plane of the epicycle cut the ecliptic plane along the line $NG'N'$. Here, $N$ is the point at which the epicycle passes through the plane of the ecliptic from south to north, in the direction of the mean planetary motion. This point is called the ascending node. Note that the line $NG'N'$ must pass through the guide-point, $G'$, since the sun (which is coincident with the guide-point) is common to the plane of the planetary orbit and the ecliptic plane. Now, it follows from simple geometry that the elevation of the planet $P$ above the guide-point, $G'$, satisfies $v= r'\,\sin i\,\sin F$, where $r'$ is the length $G'P$, $i$ the fixed inclination of the planetary orbit (and, hence, of the epicycle) to the ecliptic plane, and $F$ the angle $NG'P$. The angle $F$ is termed the argument of latitude. We can write (see Cha. 9)

$$F = \bar{F} + q,$$ (227)

where $\bar{F}$ is the mean argument of latitude, and $q$ the equation of center of the epicycle. Note that $\bar{F}$ increases uniformly in time: i.e.,

$$\bar{F} = \bar{F}_0 + \breve{n}\,(t-t_0).$$ (228)

Now, since the deferent is parallel to the ecliptic plane, the elevation of the planet above the said plane is the same as that of the planet above the guide-point. Hence, from simple geometry, the ecliptic latitude of the planet satisfies

$$\beta = \frac{v}{r''},$$ (229)

where $r''$ is the length $GP$, and we have used the small angle approximation. However, it is apparent from Fig. 31 that

$$r'' = (r^2 + 2\,r\,r'\,\cos\mu+ r'^{\,2})^{1/2},$$ (230)

where $r$ the length $GG'$, and $\mu$ the equation of the epicycle. But, according to the analysis in Cha. 9, $r'/r = a/z$, where $a$ is the planetary major radius in units in which the major radius of the sun's apparent orbit about the earth is unity, and $z$ is defined in Eq. (196). Thus, we obtain

$$\beta =h\,\beta_0,$$ (231)

where

$$\beta_0(F) = a\,\sin i\,\sin F$$ (232)

is termed the epicyclic latitude, and

$$h(\mu,z) = \left[z^2 + 2\,a\,z\,\cos\mu+ a^2\right]^{-1/2}$$ (233)

the deferential latitude correction factor.

Figure 35: Orbit of an inferior planet. Here, $G$, $G'$, $P$, $N$, $N'$, and $F$ represent the earth, guide-point, planet, ascending node, descending node, and argument of latitude, respectively. View is from northern ecliptic pole.

In the following, $a$, $e$, $n$, $\tilde{n}$, $\breve{n}$, $\bar{\lambda}_0$, $M_0$, $\bar{F}_0$, and $i$ are elements of the orbit of the planet in question about the sun, and $e_S$, $\zeta_S$, and $\lambda_S$ are elements of the sun's apparent orbit about the earth. The requisite elements for all of the superior planets at the J2000 epoch ( $t_0= 2\,451\,545.0$ JD) are listed in Tables 30 and 66. Employing a quadratic interpolation scheme to represent $F(\mu,z)$ (see Cha. 8), our procedure for determining the ecliptic latitude of a superior planet is summed up by the following formuale:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (234)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (235)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (236)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (237)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (238)

$$F = \bar{F} + q,$$ (239)

$$\beta_0 = a\,\sin i\,\sin F,$$ (240)

$$\mu = \bar{\lambda}+q-\bar{\lambda}_S,$$ (241)

$$\bar{h} = h(\mu,\bar{z})\equiv\left[\bar{z}^2 + 2\,a\,\bar{z}\,\cos\mu+ a^2\right]^{-1/2},$$ (242)

$$\delta h_- = h(\mu,\bar{z}) - h(\mu,z_{\rm max}),$$ (243)

$$\delta h_+ = h(\mu,z_{\rm min}) - h(\mu,\bar{z}),$$ (244)

$$z = \frac{1-\zeta_S}{1-\zeta},$$ (245)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (246)

$$h = \Theta_-(\xi)\,\delta h_-+ \bar{h} + \Theta_+(\xi)\,\delta\,h_+,$$ (247)

$$\beta = h\,\beta_0.$$ (248)

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

For the case of Venus, the above formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of $0.7'$ and a maximum error of $1.8'$. For the case of Mercury, with the augmentations to the theory described in Cha. 9, the mean error is $1.6'$ and the maximum error $5'$.