Question

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.

Answer 1

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

8).In case of Pluto, the orbital period say, T_p is approximately 247.9 times that of Earth's, say T_E. Let Pluto’s mean distance from the Sun be a_p. Then T_p² = a_p³ or, (247.9 T_E)² = a_p³ or, 61454.41(T_E)² = a_p³ or, 61454.41*1³ = a_p³ so that a_p =(61454.41)^1/3 = 39.46 astronomical units ( on rounding off to 2 decimal places).

9). Let Venus's orbital period be T_v and let its mean distance from the Sun be a_V . Then T_V²=a_v³ = (72.3/100)³*1³ = 0.3777933067. Then T_v =(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).