## 6 Opposite Terminal Acquisition and Tracking  
The orbits of the planets in our solar system are all nearly circular and coplanar because during the solar system's formation the planets formed out of a disk of dust which surrounded the Sun. Pluto, however, deviates from coplanar by about 17\({}^{\circ}\). Nevertheless, all of the planets in our solar system have been meticulously observed by astronomers for more than a century and their orbital characteristics (including slight deviations from the circular coplanar orbits assumed here) are therefore very well known, but the precise position of the planetary probe and the synchronization of the onboard clock with Earth  
Figure 5: The product of the transmitted laser power emitted by the planetary probe and the two telescope diameters squared (\(y\)-axis) is plotted as a function of the planetary distance (\(x\)-axis) for a wavelength of 532 nm and four candidate bit rates: 1 MHz (black), 10 MHz (red), 100 MHz (brown), and 1 GHz (pink). The dots along the lines indicate the average distance of the planet from Earth, as listed in Table 1. The chart can be used for other wavelengths by multiplying the \(y\)-axis number by (\(\lambda\)/532 nm)\({}^{2}\). clocks may not be. This is an area where the instrument's secondary functions of precise ranging and time transfer via two-way transponder can be helpful in precisely guiding the spacecraft to its desired destination and during the initial acquisition and tracking of the opposite terminal. The latter is made much more difficult due to the huge interplanetary distances, the finite velocity of light, and 4 to 5 orders of magnitude reduction in beam divergence relative to conventional microwave-based systems. Sunlight reflected off the opposite planet can help guide the initial acquisition provided one also compensates for the planetary orbital motions and finite velocity of light. In the following analysis, we will ignore the small ellipticities in some of the planetary orbits. Under our planetary circular orbit approximation, the Earth and planet positions can be written as  
\[E=R_{E} \begin{bmatrix}\cos(\theta_{E})\\ \sin(\theta_{E})\end{bmatrix} ,\ P=R_{P} \begin{bmatrix}\cos(\theta_{P})\\ \sin(\theta_{P})\end{bmatrix}  \tag{9}\]  
where \(R_{E}\) and \(R_{P}\) are the orbital radii, and therefore the Earth-planet distance at time \(t\) is equal to  
\[ \begin{split} R_{PE}(t)=\sqrt{\left[R_{P}\cos(\theta_{P})-R_{E} \cos(\theta_{E})\right]^{2}+\left[R_{P}\sin(\theta_{P})-R_{E}\sin(\theta_{E}) \right]^{2}}\\ =\sqrt{R_{E}^{2}+R_{P}^{2}-2R_{E}R_{P}\cos[\theta_{P}(t)+\theta_{E}(t) ]}\end{split}  \tag{10}\]  
where the time dependence of the combined angular change during the one-way laser pulse time of flight (TOF) is given by  
\[\theta_{P}(t)+\theta_{E}(t)=\theta_{P}(t_{0})+\theta_{E}(t_{0})+(\omega_{P}+ \omega_{E})(t-t_{0}) \tag{11}\]  
and \(\omega_{P}\) and \(\omega_{E}\) are the angular rates of the two planetary orbits. The cosine in Equation (10) can now be approximated by  
\[ \begin{split}\cos[\theta_{P}(t)+\theta_{E}(t)]=\\ \cos[\theta_{P}(t_{0})+\theta_{E}(t_{0})]\cos\left[(\omega_{P}+ \omega_{E})\frac{R_{PE}}{c}\right]-\sin[\theta_{P}(t_{0})+\theta_{E}(t_{0})] \sin\left[(\omega_{P}+\omega_{E})\frac{R_{PE}}{c}\right]\end{split}  \tag{12}\]  
Substitution of Equation (12) into Equation (10) results in a quadratic equation for the interplanetary distance traveled by the laser pulse arriving at time \(t\), i.e.,  
\[R_{PE}^{2}(t)+\left\{2R_{E}R_{P}\cos[\theta_{P}(t_{0})+\theta_{E}(t_{0})] \left(\frac{\omega_{P}+\omega_{E}}{c}\right)\right\}R_{PE}(t)-\left(R_{E}^{2} +R_{P}^{2}\right)=0 \tag{13}\]  
with the solution  
\[R_{PE}(t)=\frac{-B\pm\sqrt{B^{2}-4 AC}}{2A}\] (14a) where \[ \begin{split} A&=1\\ B&=2R_{E}R_{P}\cos[\theta_{P}(t_{0})+\theta_{E}(t_{0})]\frac{ \omega_{P}+\omega_{E}}{c}\\ C&=-\left(R_{E}^{2}+R_{P}^{2}\right)\end{split}