Question

# The square of the orbital period of a planet is proportional to the cube of its...

The square of the orbital period of a planet is proportional to the cube of its distance from the Sun. This is expressed in the formula T^2=a^3, where T is time, measured in years, and a is distance, measured in astronomical units (1 astronomical unit is the mean distance of Earth from the Sun). Use this information to answers questions 7-9.

7) Express a as a function of T. Express T as a function of a.

8) Pluto's orbital period is approximately 247.9 times that of Earth's. Estimate Pluto's mean distance from the Sun.

9) Venus's mean distance from the Sun is approximately 72.3% that of Earth's. Estimate Venus's orbital period.

7). If T2=a3 , then a = T2/3 and T = a3/2.

8).In case of Pluto, the orbital period say, Tp is approximately 247.9 times that of Earth's, say TE. Let Pluto’s mean distance from the Sun be ap. Then Tp2 = ap3 or, (247.9 TE)2 = ap3 or, 61454.41(TE)2 = ap3 or, 61454.41*13 = ap3 so that ap =(61454.41)1/3 = 39.46 astronomical units ( on rounding off to 2 decimal places).

9). Let Venus's orbital period be Tv and let its mean distance from the Sun be aV . Then TV2=av3 = (72.3/100)3*13 = 0.3777933067. Then Tv =(0.3777933067)1/2 = 0.61. ( on rounding off to 2 decimal places). Thus, Estimate Venus's orbital period is 0.61 times that of the Earth i.e. 224 days (approximately).