Final answer:
To launch a probe into orbit around the Sun with a 4.00-year period, its orbital radius should be 2.52 AU, which means its orbital speed would be less than Earth's.
Step-by-step explanation:
If you want to launch a probe with an orbital period of 4.00 years around the Sun, you will need to calculate the orbital radius using Kepler's third law, which states that the square of the period (P) of an orbit is proportional to the cube of the semi-major axis (a) of that orbit. When P is measured in years and a is in astronomical units (AU), the relationship is given by P² = a³. With the Sun-Earth mean distance defined as 1 AU, you can determine the probe's orbital radius in AU.
To find the orbital radius a for a period P = 4.00 years, we calculate:
(4.00)² = a³
16 = a³
a = ∛(16)
a = 2.52 AU (to three significant figures)
Since the orbital speed of a body decreases with distance from the Sun, the orbital speed of this probe would be less than the orbital speed of Earth.