It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)

Question

BlackTom AI · Accepted Answer

Kepler's third law relates orbital period T and average orbital radius a: T^2 ∝ a^3. For two planets orbiting the same star we can write (T_J/T_E)^2 = (a_J/a_E)^3.

Given: T_J = 12 years, T_E = 1 year, a_E = 1.5 × 10^11 m. Solve for a_J:
1) Rearrange: a_J = a_E × (T_J/T_E)^(2/3).
2) Substitute numbers: a_J = 1.5 × 10^11 m × (12/1)^(2/3).
3) Evaluate exponent: 12^(1/3) ≈ 2.28943, so 12^(2/3) = (12^(1/3))^2 ≈ (2.28943)^2 ≈ 5.2415.
4) Multiply: a_J ≈ 1.5 × 10^11 m × 5.2415 ≈ 7.8623 × 10^11 m.
5) Round to two significant figures consistent with the choices: a_J ≈ 7.9 × 10^11 m.

Why this is correct: using Kepler’s law ensures the period–distance relationship is satisfied. If a_J = 7.9 × 10^11 m, then (a_J/a_E) = 7.9/1.5 ≈ 5.2667 and (a_J/a_E)^(3/2) ≈ 5.2667^(1.5) ≈ 12, which matches Jupiter’s 12-year period.

Why the other choices are wrong:
- 3.9 × 10^11 m is far too small: (3.9/1.5)^(3/2) ≈ 4.0 years, not 12 years.
- 5.2 × 10^11 m yields (5.2/1.5)^(3/2) ≈ 6.5 years, too short.
- 9.7 × 10^11 m gives (9.7/1.5)^(3/2) ≈ 16.7 years, too long.

Therefore the correct average distance is 7.9 × 10^11 m, matching the provided answer.

It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)单项选择题

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