Final answer:
Using Kepler's third law, which relates the square of a planet's orbital period to the cube of its semimajor axis, we find Mars' period to be 1.88 Earth years. This is calculated from its semimajor axis of 1.52 AU, obtained by dividing its average mean distance from the Sun by the mean distance of Earth from the Sun.
Step-by-step explanation:
To calculate the period of Mars using Kepler's third law, we start with the known values for Earth and then find Mars' period relative to Earth's. The average mean distance of Earth from the Sun is 1 AU (astronomical unit), and its period is 1.0 year. Kepler's third law states that the square of the period (P²) is proportional to the cube of the semimajor axis (a³) of the planet's orbit. Since we are given Mars' average mean distance as 227.9 × 106 km, we can convert this to AU by noting that 1 AU = 149.6 × 106 km. Doing so yields 227.9 × 106 km / 149.6 × 106 km = 1.52 AU for Mars' semimajor axis.
To find the period of Mars (PMars), we use Earth's period as a reference. Since Mars is 1.52 AU away from the Sun, we can calculate its period by cubing the semimajor axis, which gives us 1.52³ = 3.53. According to Kepler's third law, P² = a³ for any planet in the solar system, so taking the square root of 3.53 gives us the period of Mars. Thus, PMars is approximately 1.88 Earth years, or Mars takes about 1.88 years to complete one orbit around the Sun.