Planetary Latitudes

Up to now, we have neglected the fact that the orbits of the five visible planets about the sun are all slightly inclined to the plane of the ecliptic. Of course, these inclinations cause the ecliptic latitudes of the said planets to take small, but non-zero, values. In the following, we shall outline a model which is capable of predicting these values.

Figure 30 shows a top view of 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. Now, we can write (see Sect. 8)

$$F = \bar{F} + q,$$ (191)

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.,

$$\bar{F} = \bar{F}_0 + \breve{n}\,(t-t_0).$$ (192)

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

$$\beta = \frac{v}{r''},$$ (193)

where $r''$ is the length $GP$, and we have used the small angle approximation. However, it is apparent from Fig. 27 that

$$r'' = (r^2 + 2\,r\,r'\,\cos\mu+ r'^{\,2})^{1/2},$$ (194)

where $r'$ the length $G'P$, and $\mu$ the equation of the epicycle. But, according to the analysis in Sect. 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. (144). Thus, we obtain

$$\beta =h\,\beta_0,$$ (195)

where

$$\beta_0(F) = \sin i\,\sin F$$ (196)

is termed the deferential latitude, and

$$h(\mu,z) = \left[1 + 2\,(a\,z)^{-1}\,\cos\mu+ (a\,z)^{-2}\right]^{1/2}$$ (197)

the epicyclic latitude correction factor.

Figure 30: 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 Sect. 8), our procedure for determining the ecliptic latitude of a superior planet is summed up by the following formuale:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (198)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (199)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (200)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (201)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (202)

$$F = \bar{F} + q,$$ (203)

$$\beta_0 = \sin i\,\sin F,$$ (204)

$$\mu = \lambda_S - \bar{\lambda}-q,$$ (205)

$$\bar{h} = h(\mu,\bar{z})\equiv\left[1 + 2\,(a\,\bar{z})^{-1}\,\cos\mu+ (a\,\bar{z})^{-2}\right]^{1/2},$$ (206)

$$\delta h_- = h(\mu,\bar{z}) - h(\mu,z_{\rm max}),$$ (207)

$$\delta h_+ = h(\mu,z_{\rm min}) - h(\mu,\bar{z}),$$ (208)

$$z = \frac{1-\zeta}{1-\zeta_S},$$ (209)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (210)

$$h = \Theta_-(\xi)\,\delta h_-+ \bar{h} + \Theta_+(\xi)\,\delta\,h_+,$$ (211)

$$\beta = h\,\beta_0.$$ (212)

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 $0.08'$.

The ecliptic longitude of Mars can be determined with the aid of Tables 46, 67, and 68. Table 46 allows the mean argument of latitude, $\bar{F}$, of Mars to be calculated as a function of time. Next, Table 67 permits the deferential latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 68 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ to be calculated as functions of the epicyclic anomaly, $\mu$.

The procedure for using the tables is as follows:

1. Determine the fractional Julian day number, $t$, corresponding to the date and time at which the ecliptic latitude is to be calculated with the aid of Tables 1-3. Form $\Delta t = t-t_0$, where $t_0= 2\,451\,545.0$ is the epoch.
2. Calculate the planetary equation of center, $q$, ecliptic anomaly, $\mu$, and interpolation parameters $\Theta_+$ and $\Theta_-$ using the procedure set out in Sect. 8.
3. Enter Table 46 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta\bar{F}$. If $\Delta t$ is negative then the corresponding values are also negative. The value of the mean argument of latitude, $\bar{F}$, is the sum of all the $\Delta\bar{F}$ values plus the value of $\bar{F}$ at the epoch.
4. Form the true argument of latitude, $F=\bar{F} + q$. Add as many multiples of $360^\circ$ to $F$ as is required to make it fall in the range $0^\circ$ to $360^\circ$. Round $F$ to the nearest degree.
5. Enter Table 67 with the value of $F$ and take out the corresponding value of the deferential latitude, $\beta_0$. It is necessary to interpolate if $F$ is odd.
6. Enter Table 68 with the value of $\mu$ and take out the corresponding values of $\delta h_-$, $\bar{h}$, and $\delta h_+$. If $\mu > 180^\circ$ then it is necessary to make use of the identities $\delta h_\pm(360^\circ - \mu) = \delta h_\pm(\mu)$ and $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$.
7. Form the epicyclic latitude correction factor, $h = \Theta_-\,\delta h_-+ \bar{h} + \Theta_+\,\delta h_+$.
8. The ecliptic latitude, $\beta$, is the product of the deferential latitude, $\beta_0$, and the epicyclic latitude correction factor, $h$. The decimal fraction can be converted into arc minutes using Table 31. Round to the nearest arc minute.

One example of this procedure is given below.

Example: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 8, $t-t_0=1\,950.5$ JD, $q= -7.345^\circ$, $\mu= 114.286^\circ$, $\Theta_-=0.101$, and $\Theta_+ = 0.619$. Making use of Table 46, we find:

$t$(JD)	$\bar{F}(^\circ)$
+1000	$164.041$
+900	$111.637$
+50	$26.202$
+.5	$0.262$
Epoch	$305.796$
	$607.938$
Modulus	$247.938$

Thus,

$$F = \bar{F} + q = 247.938-7.345 = 240.593\simeq 241^\circ.$$

It follows from Table 67 that

$$\beta_0(241^\circ) = -1.615^\circ.$$

Since $\mu\simeq 114^\circ$, Table 68 yields

$$\delta h_-(114^\circ) = -0.017,\mbox{\hspace{0.5cm}}\bar{h}(... ...c)=1.056, \mbox{\hspace{0.5cm}}\delta h_+(114^\circ) = -0.027,$$

so

$$h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = -0.101\times 0.017+1.056-0.619\times 0.027 = 1.038.$$

Finally,

$$\beta = h\,\beta_0 = -1.038\times 1.615 = -1.676 \simeq -1^\circ 41'.$$

Thus, the ecliptic latitude of Mars at 00:00 GMT on May 5, 2005 CE was $-1^\circ 41'$.

The ecliptic longitude of Jupiter can be determined with the aid of Tables 50, 69, and 70. Table 50 allows the mean argument of latitude, $\bar{F}$, of Jupiter to be calculated as a function of time. Next, Table 69 permits the deferential latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 70 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ to be calculated as functions of the epicyclic anomaly, $\mu$. The procedure for using these tables is analogous to the previously described procedure for using the Mars tables. One example of this procedure is given below.

Example: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 8, $t-t_0=1\,950.5$ JD, $q= -0.091^\circ$, $\mu= 208.192^\circ$, $\Theta_-=-0.469$, and $\Theta_+ = -0.121$. Making use of Table 50, we find:

$t$(JD)	$\bar{F}(^\circ)$
+1000	$83.081$
+900	$74.773$
+50	$4.154$
+.5	$0.042$
Epoch	$293.660$
	$455.710$
Modulus	$95.710$

Thus,

$$F = \bar{F} + q =95.710-0.091 = 95.619\simeq 96^\circ.$$

It follows from Table 69 that

$$\beta_0(96^\circ) = 1.297^\circ.$$

Since $\mu\simeq 208^\circ$, Table 70 yields

$$\delta h_-(208^\circ) = 0.014,\mbox{\hspace{0.5cm}}\bar{h}(2... ...rc)=1.197, \mbox{\hspace{0.5cm}}\delta h_+(208^\circ) = 0.016,$$

so

$$h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = -0.469\times 0.014 +1.197-0.121\times 0.016 = 1.188.$$

Finally,

$$\beta = h\,\beta_0 = 1.188\times 1.297 = 1.541 \simeq 1^\circ 32'.$$

Thus, the ecliptic latitude of Jupiter at 00:00 GMT on May 5, 2005 CE was $1^\circ 32'$.

The ecliptic longitude of Saturn can be determined with the aid of Tables 54, 71, and 72. Table 54 allows the mean argument of latitude, $\bar{F}$, of Saturn to be calculated as a function of time. Next, Table 71 permits the deferential latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 72 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ to be calculated as functions of the epicyclic anomaly, $\mu$. The procedure for using these tables is analogous to the previously described procedure for using the Mars tables. One example of this procedure is given below.

Example: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 8, $t-t_0=1\,950.5$ JD, $q= 2.561^\circ$, $\mu= 286.625^\circ$, $\Theta_-=0.071$, and $\Theta_+ = 0.759$. Making use of Table 54, we find:

$t$(JD)	$\bar{F}(^\circ)$
+1000	$33.478$
+900	$30.130$
+50	$1.674$
+.5	$0.017$
Epoch	$296.482$
	$361.781$
Modulus	$1.781$

Thus,

$$F = \bar{F} + q =1.781+2.561 = 4.342\simeq 4^\circ.$$

It follows from Table 71 that

$$\beta_0(4^\circ) = 0.173^\circ.$$

Since $\mu\simeq 287^\circ$, Table 72 yields

$$\delta h_-(287^\circ) = -0.002,\mbox{\hspace{0.5cm}}\bar{h}(... ...c)=0.966, \mbox{\hspace{0.5cm}}\delta h_+(287^\circ) = -0.003,$$

so

$$h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = -0.071\times 0.002 +0.966-0.759\times 0.003 = 0.964.$$

Finally,

$$\beta = h\,\beta_0 = 0.964\times 0.173 = 0.167 \simeq 0^\circ 10'.$$

Thus, the ecliptic latitude of Saturn at 00:00 GMT on May 5, 2005 CE was $0^\circ 10'$.

Figure 31: 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.

Figure 31 shows a top view of 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. Now, we can write (see Sect. 9)

$$F = \bar{F} + q,$$ (213)

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.,

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

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

$$\beta = \frac{v}{r''},$$ (215)

where $r''$ is the length $GP$, and we have used the small angle approximation. However, it is apparent from Fig. 27 that

$$r'' = (r^2 + 2\,r\,r'\,\cos\mu+ r'^{\,2})^{1/2},$$ (216)

where $r$ the length $GG'$, and $\mu$ the equation of the epicycle. But, according to the analysis in Sect. 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. (182). Thus, we obtain

$$\beta =h\,\beta_0,$$ (217)

where

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

is termed the epicyclic latitude, and

$$h(\mu,z) = \left[z^2 + 2\,a\,z\,\cos\mu+ a^2\right]^{1/2}$$ (219)

the deferential latitude correction factor.

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 Sect. 8), our procedure for determining the ecliptic latitude of a superior planet is summed up by the following formuale:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (220)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (221)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (222)

$$q = 2\,e\,\sin \,M + (5/4)\,e^2\,\sin\,2M,$$ (223)

$$\zeta = e\,\cos M - e^2\,\sin^2 M,$$ (224)

$$F = \bar{F} + q,$$ (225)

$$\beta_0 = a\,\sin i\,\sin F,$$ (226)

$$\mu = \bar{\lambda}+q-\bar{\lambda}_S,$$ (227)

$$\bar{h} = h(\mu,\bar{z})\equiv\left[\bar{z}^2 + 2\,a\,\bar{z}\,\cos\mu+ a^2\right]^{1/2},$$ (228)

$$\delta h_- = h(\mu,\bar{z}) - h(\mu,z_{\rm max}),$$ (229)

$$\delta h_+ = h(\mu,z_{\rm min}) - h(\mu,\bar{z}),$$ (230)

$$z = \frac{1-\zeta_S}{1-\zeta},$$ (231)

$$\xi = \frac{\bar{z}-z}{\delta z},$$ (232)

$$h = \Theta_-(\xi)\,\delta h_-+ \bar{h} + \Theta_+(\xi)\,\delta\,h_+,$$ (233)

$$\beta = h\,\beta_0.$$ (234)

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 Sect. 9, the mean error is $1.6'$ and the maximum error $5'$.

The ecliptic longitude of Venus can be determined with the aid of Tables 58, 73, and 74. Table 58 allows the mean argument of latitude, $\bar{F}$, of Venus to be calculated as a function of time. Next, Table 73 permits the epicyclic latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 74 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ to be calculated as functions of the epicyclic anomaly, $\mu$.

The procedure for using the tables is as follows:

1. Determine the fractional Julian day number, $t$, corresponding to the date and time at which the ecliptic latitude is to be calculated with the aid of Tables 1-3. Form $\Delta t = t-t_0$, where $t_0= 2\,451\,545.0$ is the epoch.
2. Calculate the planetary equation of center, $q$, ecliptic anomaly, $\mu$, and interpolation parameters $\Theta_+$ and $\Theta_-$ using the procedure set out in Sect. 9.
3. Enter Table 58 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta\bar{F}$. If $\Delta t$ is negative then the corresponding values are also negative. The value of the mean argument of latitude, $\bar{F}$, is the sum of all the $\Delta\bar{F}$ values plus the value of $\bar{F}$ at the epoch.
4. Form the true argument of latitude, $F=\bar{F} + q$. Add as many multiples of $360^\circ$ to $F$ as is required to make it fall in the range $0^\circ$ to $360^\circ$. Round $F$ to the nearest degree.
5. Enter Table 73 with the value of $F$ and take out the corresponding value of the epicyclic latitude, $\beta_0$. It is necessary to interpolate if $F$ is odd.
6. Enter Table 74 with the value of $\mu$ and take out the corresponding values of $\delta h_-$, $\bar{h}$, and $\delta h_+$. If $\mu > 180^\circ$ then it is necessary to make use of the identities $\delta h_\pm(360^\circ - \mu) = \delta h_\pm(\mu)$ and $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$.
7. Form the deferential latitude correction factor, $h = \Theta_-\,\delta h_-+ \bar{h} + \Theta_+\,\delta h_+$.
8. The ecliptic latitude, $\beta$, is the product of the epicyclic latitude, $\beta_0$, and the deferential latitude correction factor, $h$. The decimal fraction can be converted into arc minutes using Table 31. Round to the nearest arc minute.

One example of this procedure is given below.

Example: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 9, $t-t_0=1\,950.5$ JD, $q= -0.712^\circ$, $\mu= 21.689^\circ$, $\Theta_-=-0.355$, and $\Theta_+ = -0.125$. Making use of Table 58, we find:

$t$(JD)	$\bar{F}(^\circ)$
+1000	$162.138$
+900	$1.924$
+50	$80.107$
+.5	$0.801$
Epoch	$105.253$
	$350.223$
Modulus	$350.223$

Thus,

$$F = \bar{F} + q = 350.223-0.712 = 349.511\simeq 350^\circ.$$

It follows from Table 73 that

$$\beta_0(350^\circ) = -0.423^\circ.$$

Since $\mu\simeq 22^\circ$, Table 74 yields

$$\delta h_-(22^\circ) = 0.008,\mbox{\hspace{0.5cm}}\bar{h}(22^\circ)=0.591, \mbox{\hspace{0.5cm}}\delta h_+(22^\circ) = 0.008,$$

so

$$h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = -0.355\times 0.008+0.591-0.125\times 0.008 = 0.587.$$

Finally,

$$\beta = h\,\beta_0 =- 0.587\times 0.423 = -0.248\simeq -0^\circ 15'.$$

Thus, the ecliptic latitude of Venus at 00:00 GMT on May 5, 2005 CE was $-0^\circ 15'$.

The ecliptic latitude of Mercury can be determined with the aid of Tables 62, 75, and 76. Table 62 allows the mean argument of latitude, $\bar{F}$, of Mercury to be calculated as a function of time. Next, Table 75 permits the epicyclic latitude, $\beta_0$, to be determined as a function of the true argument of latitude, $F$. Finally, Table 76 allows the quantities $\delta h_-$, $\bar{h}$, and $\delta h_+$ to be calculated as functions of the epicyclic anomaly, $\mu$. The procedure for using the tables is analogous to the previously described procedure for using the Venus tables. One example of this procedure is given below.

Example: May 5, 2005 CE, 00:00 GMT:
 
From Sect. 9, $t-t_0=1\,950.5$ JD, $q= -16.974^\circ$, $\mu= 252.692^\circ$, $\Theta_-=0.107$, and $\Theta_+ = 0.583$. Making use of Table 62, we find:

$t$(JD)	$\bar{F}(^\circ)$
+1000	$132.342$
+900	$83.108$
+50	$204.617$
+.5	$2.046$
Epoch	$204.436$
	$626.549$
Modulus	$266.549$

Thus,

$$F = \bar{F} + q = 266.549-16.974= 249.575\simeq 250^\circ.$$

It follows from Table 75 that

$$\beta_0(250^\circ) = -2.511^\circ.$$

Since $\mu\simeq 253^\circ$, Table 76 yields

$$\delta h_-(253^\circ) = 0.184,\mbox{\hspace{0.5cm}}\bar{h}(2... ...rc)=1.037, \mbox{\hspace{0.5cm}}\delta h_+(253^\circ) = 0.272,$$

so

$$h = \Theta_-\,\delta h_- + \bar{h}+\Theta_+\,\delta h_+ = 0.107\times 0.184+1.037+0.583\times 0.272= 1.215.$$

Finally,

$$\beta = h\,\beta_0 =- 1.215\times 2.511 = -3.051\simeq -3^\circ 03'.$$

Thus, the ecliptic latitude of Mercury at 00:00 GMT on May 5, 2005 CE was $-3^\circ 03'$.

Table 66: Additional Keplerian orbital elements for the five visible planets at the J2000 epoch (i.e., 12:00 GMT, January 1, 2000 CE, which corresponds to $t_0= 2\,451\,545.0$ JD). The elements are optimized for use in the time period 1800 CE to 2050 CE. Source: Jet Propulsion Laboratory (NASA), http://ssd.jpl.nasa.gov/.

Object	$i(^\circ)$	$\breve{n}\,(^\circ/{\rm day})$	$\bar{F}_0\,(^\circ)$
Mercury	$6.9190$	$4.09234221$	$204.436$
Venus	$3.3692$	$1.60213807$	$105.253$
Mars	$1.8467$	$0.52404094$	$305.796$
Jupiter	$1.3044$	$0.08308122$	$293.660$
Saturn	$2.4860$	$0.03347795$	$296.482$

Table 67: Deferential ecliptic latitude of Mars. The latitude is minus the value shown in the table if the argument is in parenthesies.

$F (^\circ)$	$\beta_0(^\circ)$	$F (^\circ)$
000/180	0.000	(180)/(360)
002/178	0.064	(182)/(358)
004/176	0.129	(184)/(356)
006/174	0.193	(186)/(354)
008/172	0.257	(188)/(352)
010/170	0.321	(190)/(350)
012/168	0.384	(192)/(348)
014/166	0.447	(194)/(346)
016/164	0.509	(196)/(344)
018/162	0.571	(198)/(342)
020/160	0.631	(200)/(340)
022/158	0.692	(202)/(338)
024/156	0.751	(204)/(336)
026/154	0.809	(206)/(334)
028/152	0.867	(208)/(332)
030/150	0.923	(210)/(330)
032/148	0.978	(212)/(328)
034/146	1.032	(214)/(326)
036/144	1.085	(216)/(324)
038/142	1.137	(218)/(322)
040/140	1.187	(220)/(320)
042/138	1.235	(222)/(318)
044/136	1.283	(224)/(316)
046/134	1.328	(226)/(314)
048/132	1.372	(228)/(312)
050/130	1.414	(230)/(310)
052/128	1.455	(232)/(308)
054/126	1.494	(234)/(306)
056/124	1.531	(236)/(304)
058/122	1.566	(238)/(302)
060/120	1.599	(240)/(300)
062/118	1.630	(242)/(298)
064/116	1.660	(244)/(296)
066/114	1.687	(246)/(294)
068/112	1.712	(248)/(292)
070/110	1.735	(250)/(290)
072/108	1.756	(252)/(288)
074/106	1.775	(254)/(286)
076/104	1.792	(256)/(284)
078/102	1.806	(258)/(282)
080/100	1.818	(260)/(280)
082/098	1.828	(262)/(278)
084/096	1.836	(264)/(276)
086/094	1.842	(266)/(274)
088/092	1.845	(268)/(272)
090/090	1.846	(270)/(270)

Table 68: Epicyclic latitude correction factor for Mars. $\mu$ is in degrees. Note that $\bar{h}(360^\circ - \mu) = \bar{h}(\mu)$, and $\delta h_{\pm}(360^\circ-\mu) = \delta h_{\pm}(\mu)$.

$\mu$	$\delta h_-$	$\bar{h}~~~~$	$\delta h_+$	$\mu$	$\delta h_-$	$\bar{h}~~~~$	$\delta h_+$	$\mu$	$\delta h_-$	$\bar{h}~~~~$	$\delta h_+$	$\mu$	$\delta h_-$	$\bar{h}~~~~$	$\delta h_+$
0	-0.025	0.604	-0.028	45	-0.025	0.652	-0.029	90	-0.025	0.836	-0.031	135	0.015	1.410	0.003
1	-0.025	0.604	-0.028	46	-0.025	0.654	-0.029	91	-0.025	0.843	-0.031	136	0.018	1.433	0.006
2	-0.025	0.604	-0.028	47	-0.025	0.656	-0.029	92	-0.024	0.850	-0.031	137	0.021	1.457	0.009
3	-0.025	0.604	-0.028	48	-0.025	0.659	-0.029	93	-0.024	0.857	-0.031	138	0.025	1.482	0.013
4	-0.025	0.605	-0.028	49	-0.025	0.661	-0.029	94	-0.024	0.865	-0.031	139	0.028	1.507	0.017
5	-0.025	0.605	-0.028	50	-0.025	0.664	-0.030	95	-0.024	0.872	-0.031	140	0.032	1.533	0.021
6	-0.025	0.605	-0.028	51	-0.025	0.666	-0.030	96	-0.024	0.880	-0.031	141	0.037	1.560	0.026
7	-0.025	0.605	-0.028	52	-0.025	0.669	-0.030	97	-0.024	0.888	-0.031	142	0.041	1.588	0.031
8	-0.025	0.606	-0.028	53	-0.025	0.672	-0.030	98	-0.023	0.896	-0.031	143	0.046	1.616	0.036
9	-0.025	0.606	-0.028	54	-0.025	0.674	-0.030	99	-0.023	0.904	-0.031	144	0.051	1.646	0.043
10	-0.025	0.606	-0.028	55	-0.025	0.677	-0.030	100	-0.023	0.912	-0.030	145	0.057	1.676	0.049
11	-0.025	0.607	-0.028	56	-0.025	0.680	-0.030	101	-0.023	0.921	-0.030	146	0.063	1.708	0.056
12	-0.025	0.607	-0.028	57	-0.026	0.683	-0.030	102	-0.022	0.930	-0.030	147	0.069	1.740	0.064
13	-0.025	0.608	-0.028	58	-0.026	0.686	-0.030	103	-0.022	0.939	-0.030	148	0.076	1.773	0.073
14	-0.025	0.609	-0.028	59	-0.026	0.689	-0.030	104	-0.022	0.948	-0.030	149	0.084	1.807	0.083
15	-0.025	0.609	-0.028	60	-0.026	0.693	-0.030	105	-0.021	0.958	-0.030	150	0.092	1.843	0.093
16	-0.025	0.610	-0.028	61	-0.026	0.696	-0.030	106	-0.021	0.968	-0.029	151	0.100	1.879	0.104
17	-0.025	0.611	-0.028	62	-0.026	0.699	-0.030	107	-0.021	0.978	-0.029	152	0.109	1.916 <