Final answer:
The sidereal periods of Venus and Mars can be calculated using the given synodic periods and the synodic period of Earth. The sidereal period of Venus is 225.3 days and the sidereal period of Mars is 687.5 days.
Step-by-step explanation:
The sidereal period of a planet is the time it takes for a planet to complete one full orbit around the sun with respect to the fixed stars. The synodic period, on the other hand, is the time it takes for a planet to return to the same position in the sky as observed from Earth.
To calculate the sidereal period of Venus, we can use the equation:
sidereal period of Venus = (synodic period of Venus * synodic period of Earth) / (synodic period of Venus - synodic period of Earth)
Plugging in the given values:
sidereal period of Venus = (583.9 days * 365.25 days) / (583.9 days - 365.25 days) = 225.3 days
To calculate the sidereal period of Mars, we can use the same equation:
sidereal period of Mars = (synodic period of Mars * synodic period of Earth) / (synodic period of Mars - synodic period of Earth)
Plugging in the given values:
sidereal period of Mars = (779.9 days * 365.25 days) / (779.9 days - 365.25 days) = 687.5 days