Let the average orbital radius of a planet be [math: r]. Let the orbital period be [math: T]. What quantity is constant for all planets orbiting the Sun?

Question

BlackTom AI · Accepted Answer

Kepler’s third law gives the key relationship: T^2 ∝ r^3, so the ratio T^2 / r^3 is constant for all planets orbiting the same central mass.

1) Relevant knowledge points and concepts
- Newton’s law of gravitation: F = G M m / r^2 (M is Sun’s mass). 
- Centripetal force for circular motion: F = m v^2 / r. 
- Orbital speed in terms of period: v = 2π r / T. 
Combining these yields Kepler’s third law in precise form.

2) Complete reasoning process step by step
Start by equating gravitational force to centripetal force:
G M m / r^2 = m v^2 / r.
Cancel m and rearrange:
G M / r^2 = v^2 / r  =>  v^2 = G M / r.
Substitute v = 2π r / T:
(2π r / T)^2 = G M / r.

3) Calculations with intermediate steps
(2π r / T)^2 = 4π^2 r^2 / T^2.
Set equal to G M / r:
4π^2 r^2 / T^2 = G M / r.
Multiply both sides by T^2 r:
4π^2 r^3 = G M T^2.
Solve for T^2 / r^3:
T^2 = (4π^2 / (G M)) r^3  =>  T^2 / r^3 = 4π^2 / (G M).

4) Why the correct answer is correct
The right-hand side 4π^2/(G M) contains only universal constants G and the Sun’s mass M, so it is the same for every planet orbiting that Sun. Therefore T^2 / r^3 is constant.

5) Why the other options are incorrect
- TR or TR^2: no theoretical relation predicts T·r or T·r^2 is constant; dimensions and derived law disagree.
- T^3 / R^2: exponent relationship is wrong (Kepler gives T^2 ∝ r^3, not T^3 ∝ r^2).

6) Conclusion
The constant quantity is T^2 / r^3, which matches the provided correct answer [T2R3].

Let the average orbital radius of a planet be [math: r]. Let the orbital period be [math: T]. What quantity is constant for all planets orbiting the Sun?单项选择题

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