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The planet Neptune travels around the Sun with a period of 165 years. Show that the radius of its orbit is approximately thirty times that of Earth's orbit, both being considered as circular.

Note by Rishabh Kumar
2 years ago

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Using Kepler third law:

$$\frac{a(neptune)^{3}}{T(neptune)²} = \frac{a(earth)^{3}}{T(earth)²}$$

Where $$a$$ and $$T$$ are respectively radius (generally, half of ellipse greatest axis) and period of trajectory.

Thus:

$$\frac{a(neptune)}{a(earth)} = \frac{T(neptune)}{T(earth)}^\frac{2}{3} = 165^\frac{2}{3}$$

$$\frac{a(neptune)}{a(earth)} = 30.082$$ · 2 years ago