Determination of Ecliptic Latitude of Superior Planet

Figure 34 shows the orbit of a superior planet. As we have already mentioned, the deferent and epicycle 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 deferent and epicycle of a superior planet are, respectively, inclined and parallel to the ecliptic plane. (Recall that the ecliptic plane corresponds to the plane of the sun's apparent orbit about the earth.) Let the plane of the deferent cut the ecliptic plane along the line $NGN'$. Here, $N$ is the point at which the deferent 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 $NGN'$ must pass through point $G$, since the earth is common to the plane of the deferent and the ecliptic plane. Now, it follows from simple geometry that the elevation of the guide-point $G'$ above of the ecliptic plane satisfies $v= r\,\sin i\,\sin F$, where $r$ is the length $GG'$, $i$ the fixed inclination of the planetary orbit (and, hence, of the deferent) to the ecliptic plane, and $F$ the angle $NGG'$. The angle $F$ is termed the argument of latitude. We can write (see Cha. 8)

$$F = \bar{F} + q,$$ (205)

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

$$\bar{F} = \bar{F}_0 + \breve{n}\,(t-t_0).$$ (206)

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

$$\beta = \frac{v}{r''},$$ (207)

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

$$r'' = (r^2 + 2\,r\,r'\,\cos\mu+ r'^{\,2})^{1/2},$$ (208)

where $r'$ the length $G'P$, and $\mu$ the equation of the epicycle. But, according to the analysis in Cha. 8, $r/r' = a\,z$, 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. (158). Thus, we obtain

$$\beta =h\,\beta_0,$$ (209)

where

$$\beta_0(F) = \sin i\,\sin F$$ (210)

is termed the deferential latitude, and

$$h(\mu,z) = \left[1 + 2\,(a\,z)^{-1}\,\cos\mu+ (a\,z)^{-2}\right]^{-1/2}$$ (211)

the epicyclic latitude correction factor.

Figure 34: Orbit of a superior 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:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (212)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (213)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (214)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (215)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (216)

$$F = \bar{F} + q,$$ (217)

$$\beta_0 = \sin i\,\sin F,$$ (218)

$$\mu = \lambda_S - \bar{\lambda}-q,$$ (219)

$$\bar{h} = h(\mu,\bar{z})\equiv\left[1 + 2\,(a\,\bar{z})^{-1}\,\cos\mu+ (a\,\bar{z})^{-2}\right]^{-1/2},$$ (220)

$$\delta h_- = h(\mu,\bar{z}) - h(\mu,z_{\rm max}),$$ (221)

$$\delta h_+ = h(\mu,z_{\rm min}) - h(\mu,\bar{z}),$$ (222)

$$z = \frac{1-\zeta}{1-\zeta_S},$$ (223)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (224)

$$h = \Theta_-(\xi)\,\delta h_-+ \bar{h} + \Theta_+(\xi)\,\delta\,h_+,$$ (225)

$$\beta = h\,\beta_0.$$ (226)

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 $0.3'$ and a maximum error of $1.5'$. For the case of Jupiter, the mean error is $0.2'$ and the maximum error $0.5'$. Finally, for the case of Saturn, the mean error is $0.05'$ and the maximum error $0.08'$.