## 6 Opposite Terminal Acquisition and Tracking
 \tag{14b}\]  
Since the distance between the planets can be huge, i.e., up to 40 astronomical units (AUs), the finite speed of light, combined with the planetary and Earth angular speeds around the Sun (\(\omega_{P}\) and \(\omega_{E}\)) and the narrow laser beam divergences (which are 4 to 5 orders of magnitude smaller than conventional microwave systems), can complicate the initial acquisition and subsequent tracking of the opposite terminal. One attractive acquisition approach would be to have each terminal initially track the sunlit image of the opposite planet in order to narrow the search angle and then, based on our excellent knowledge of the orbital velocities and eccentricities (if any), point ahead of the viewed position to compensate for the two-way time delay. If \(t_{0}\) is the time a light signal was transmitted by the planetary terminal, \(t_{1}=RPE1/c\) is the time that signal was observed at the Earth station, and \(t_{2}=(RPE1+RPE2)/c\) is the time that the Earth response reaches the planetary terminal; then, the cosine of the point ahead angle for the Earth station is given by the dot product of the interplanetary range vectors at times \(t_{0}\) and \(t_{2}\) divided by their respective magnitudes, \(RPE1\) and \(RPE2\). This leads to the following set of equations for the planetary positions at time \(t=t_{0}\):  
\[P(t_{0})=R_{P}\binom{\cos\theta_{P0}}{\sin\theta_{P0}},\,E(t_{0})=R_{E}\binom{1 }{0}\]  
the two components of the roundtrip pulse transit times \(t_{1}\) and \(t_{2}\),  
\[t_{1}=\frac{RPE_{1}}{c},\,t_{2}=\frac{RPE_{2}}{c}\]  
and the position vectors for Earth and the planet at times \(t_{1}\) and \(t_{2}\)  
\[E(t_{1})=R_{E} \begin{bmatrix}\cos(\omega_{E}t_{1})\\ \sin(\omega_{E}t_{1})\end{bmatrix} ,\,P(t_{1})=R_{P} \begin{bmatrix}\cos(\theta_ {P0}+\omega_{P}t_{1})\\ \sin(\theta_{P0}+\omega_{P}t_{1})\end{bmatrix} \] \[E(t_{2})=R_{E} \begin{bmatrix}\cos(\omega_{E}(t_{1}+t_{2}))\\ \sin(\omega_{E}(t_{1}+t_{2}))\\ \cos(\theta_{P0}+\omega_{P}(t_{1}+t_{2}))\\ \sin(\theta_{P0}+\omega_{P}(t_{1}+t_{2}))\end{bmatrix} ,\]  
where \(\omega_{E}\) and \(\omega_{P}\) are the angular revolution rates of the Earth and planet about the Sun. The latter are obtained by dividing the orbit circumference, \(2\pi R_{P}\), by the orbital period in Table 1. The value for Earth is \(\omega_{E}=0.199\)\(\upmu\)rad/s. The vectors between the Earth and planetary terminals at the beginning and end of the roundtrip are in turn given by  
\[EP(t_{0})=P(t_{0})-E(t_{0}),\,\,EP(t_{2})=P(t_{2})-E(t_{2})\]  
The dot product of the two vectors, divided by their magnitudes, yields the cosine of the point ahead angle. The latter is plotted for all of the planets in Figure 6 as a function of their combined start angles \(\theta_{P0}\) where, due to the assumed circular symmetry, we can arbitrarily set \(\theta_{E}=0\) deg. For most planetary geometries, the point ahead angle is a fraction of a milliradiian but still orders of magnitude greater than the angular radii of the planet as seen from Earth, as listed in Column 8, Table 1. It should also be noted that orbital geometries where the point ahead angle drops to near zero in Figure 6 would be optimum locations to initiate opposite terminal acquisition. These opportunities would occur twice per Earth year for all of the planets but with different time intervals between them. The highest peak at \(\theta_{P0}=180\) degrees occurs for two reasons: (1) the relative planetary orbital velocities are additive because they point in opposite directions; and (2) the planets are at their maximum distance during the two-way light transit time between \(t_{0}\) and \(t_{2}\). The height of the central peak falls for planets farther from the Sun because they revolve more slowly and therefore add less to the Earth angular velocity contribution. In decreasing order, the peaks in Figure 6 correspond to Mercury (black), Venus (red), Mars (green, Jupiter (pink), Saturn (aqua), Uranus (brown), Neptune (black), and Pluto (red).