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: T^2 ∝ a^3, so (a_J / a_E) = (T_J / T_E)^{2/3}.

1) Relevant knowledge points and concepts
- Kepler's third law for planets around the same central mass states T^2 ∝ a^3, where T is the orbital period and a is the average orbital radius (semi-major axis). Using Earth's known values as a reference eliminates the gravitational constant and masses: (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.

2) Complete reasoning process (step by step)
- Start from (T_J/T_E)^2 = (a_J/a_E)^3.
- Rearranged: a_J/a_E = (T_J/T_E)^{2/3}.
- Substitute the periods: a_J/a_E = (12/1)^{2/3} = 12^{2/3}.

3) Calculations with intermediate steps
- Compute (T ratio)^2: 12^2 = 144.
- Take cube root: (a_J/a_E) = 144^{1/3}.
- Approximate cube root: 5^3 = 125, 5.2^3 ≈ 140.6, 5.24^3 ≈ 143.9, so 144^{1/3} ≈ 5.241.
- Multiply by Earth's distance: a_J = a_E × 5.241 = (1.5 × 10^11 m) × 5.241 = 7.8615 × 10^11 m.
- Round to two significant figures consistent with given choices: a_J ≈ 7.9 × 10^11 m.

4) Why the correct answer is correct
- The calculation directly follows Kepler’s third law and uses the given Earth-Sun distance as the reference scale. The numerical evaluation of 12^{2/3} yields ≈5.241, and multiplying by 1.5 × 10^11 m gives ≈7.86 × 10^11 m, which matches the provided choice 7.9 × 10^11 m.

5) Why the other options are incorrect (brief)
- 3.9 × 10^11 m: corresponds to a_J/a_E ≈ 2.6 → predicted period ≈ 4.2 years, far below 12 years.
- 5.2 × 10^11 m: corresponds to a_J/a_E ≈ 3.47 → predicted period ≈ 6.5 years, too small.
- 9.7 × 10^11 m: corresponds to a_J/a_E ≈ 6.47 → predicted period ≈ 16.5 years, too large.

6) Conclusion / confirmation
- The correct average distance from Jupiter to the Sun is 7.9 × 10^11 m, which matches 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.)Single choice

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