Final answer:
If a planet is moved closer to Earth, its synodic period would change. The synodic period could be shorter for outer planets as Earth would overtake them more quickly. For inner planets, the period would change based on how their new orbits interact with Earth's orbit.
Step-by-step explanation:
When we discuss the synodic period of a planet, we are referring to the time it takes for that planet to return to the same position in the sky as seen from Earth, relative to the Sun. If a planet is moved closer to Earth, its synodic period would change depending on the planet's new position in its orbit and its orbital speed in relation to Earth's. Kepler's third law states that the closer an object is to the Sun, the faster it travels in its orbit.
Therefore, if the planet is moved closer to Earth in its orbit around the Sun, the planet would most likely orbit the Sun faster. For planets outside Earth's orbit, this would result in a shorter synodic period because Earth would overtake them more quickly.
Conversely, for planets inside Earth's orbit, they would overtake Earth more quickly, which could also alter their synodic period but whether it increases or decreases would depend on the specifics of their new orbits.