Mars

The ecliptic latitude 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 27-29. 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 Cha. 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 UT:
 
From Cha. 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 UT on May 5, 2005 CE was $-1^\circ 41'$.