Why is the synodic period observed from Earth longer than the actual sidereal period of the Sun's rotation?
Answer
Because the Earth is simultaneously orbiting the Sun.
The sidereal period is the actual time required for a feature to rotate 360 degrees relative to distant background stars. The synodic period, however, is the time it takes for that feature to return to the exact same line of sight as seen from Earth. Since the Earth is constantly moving in its orbit around the Sun during this observation period, the Sun must rotate slightly further than a full 360 degrees (e.g., $\approx 360^\circ + \text{Earth's orbital travel}$) for the observer on our planet to see the feature return to the same apparent location in the sky. This necessary extra rotation makes the observed synodic period consistently longer than the true sidereal period.
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