## 7 Effects of Planetary Rotation or Planetary Orbits  
In addition to the relative motions of the planets about the Sun, all of the planets rotate about their polar axes which, like Earth, are not necessarily oriented perpendicular to their orbital plane about the Sun. Using the information in Columns 5 and 8 of Table 1, we can compute the angular rate of rotation for each of the planets and multiply by the light roundtrip transit time using the one-way transit time in Column 7 to compute the maximum angular point-ahead angle for a lander located on the planet's equator. This maximum occurs when the lander is in the middle of the planet when viewed through the Earth telescope and diminishes as the lander approaches the edge of the planet prior to disappearing from view. The maximum point ahead angles due to axial rotation of the innermost planets are quite large compared to angular widths of the laser beams, i.e., 195 \(\upmu\)rad for Mercury and 47 \(\upmu\)rad for Venus. The point ahead angles for the outer planets are significantly greater: Mars (17 mrad), Jupiter (145 mrad), Saturn (256 mrad), Uranus (487 mrad), Neptune (528 mrad), and Pluto (69 mrad). These are all much larger than the beamwidths needed to transmit MHz or GHz bandwidths between the planets using modest laser output powers, as outlined in Figure 5. For example, the overall laser beam divergence between points that see roughly 10% of the peak illumination at the center of the beam (corresponding to only a few photons per bit) is roughly equal to the  
Figure 6: Terminal point ahead angles in microradians for the various planets in the solar system versus the sum of their angular positions in their orbits about the Sun. Maximum point ahead angle occurs when Earth and the planet are on opposite sides of the Sun (180\({}^{\circ}\)) due to the fact that: (1) the range between them, and therefore the light travel time, is maximized; and (2) their orbital velocities I are in opposite directions and therefore the relative velocity is additive. The point ahead angles decrease for the outer planets because of their slower orbital angular velocities. Smaller secondary peaks occur at 0\({}^{\circ}\) (or 360\({}^{\circ}\)) for all the planets because: (1) the orbital velocities are in the same direction and therefore the relative velocity is reduced; and (2) the light travel time is minimized. Intermediate angles where the point ahead angle drops to near zero would be advantageous for mutual acquisition of the two terminals and occur twice per Earth year. wavelength to telescope diameter ratio, \(\lambda/D\). Thus, at a wavelength of 532 nm, telescope diameters between 0.25 m and 2.5 m create a full beam divergence of roughly 20 and 2 \(\upmu\)rad respectively, orders of magnitude smaller than the angular deviations caused by planetary rotations at the equator. Only near the poles do the planetary rotations result in near zero point ahead angles. As demonstrated by Equation (8), use of longer wavelength lasers increases the beam divergence for a given pair of telescopes but also increases the laser power required to achieve the desired bit rate. For example, for a wavelength of 1064 nm, the required laser power increases by a factor of 4. In the opinion of the author, it appears unlikely that a lander mission on the planet could gather sufficient data to require high bandwidth communications in the MHz to GHz range due to a lander's highly limited access to information compared to an orbiting satellite. In contrast to a lander, a satellite in a properly oriented polar orbit would have access to information from the entire planet and could communicate continuously with an orbiting Earth terminal for an extended period of time before the orientation of the satellite orbital plane would have to be adjusted in order to maintain long-term Earth visibility. Furthermore, a polar orbit would negate possible transmission and scattering losses in the planetary atmosphere which might degrade the received signal strength and subsequently the achievable bit error rate.