Mars revolves around the sun in 687 Earth days, which represents a Martian year. How far is Mars from the sun? Express your answer in astronomical units.
(Show a complete solution)
An astronomical unit (AU) is the average distance between the Earth and the Sun, which is about 93 million miles or 150 million kilometers. Using Kepler's third law of planetary motion, we can relate the time period of a planet's revolution around the Sun to its average distance from the Sun.
Kepler's third law states that the ratio of the squares of the periods of any two planets revolving around the Sun is equal to the ratio of the cubes of the semi-major axes of their orbits. Mathematically, this can be expressed as:
(T1/T2)^2 = (R1/R2)^3
where T1 and T2 represent the periods of the two planets, and R1 and R2 represent the semi-major axes (average distances) of their orbits. Let T1 be the period of Earth (365 days) and T2 be the period of Mars (687 days).
(365/687)^2 = (1/R2)^3
Now, we rearrange the equation to solve for R2, the distance of Mars from the Sun in astronomical units.
R2 = 1/((365/687)^2)^(1/3)
Plugging the numbers into a calculator, we find that:
R2 ≈ 1.52 AU
So Mars is approximately 1.52 astronomical units away from the Sun.