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[parent] applying Kepler's third law (Application)

Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate $p^2 $ and $a^3$, and verify that they obey Kepler's third law. Venus' orbital period is $0.62$ years, and its semimajor axis is $0.72$ AU. Earth's orbital period is $1.00$ year, and its semimajor axis is $1.00$ AU.

Solution

We can use the equation for Kepler's third law, $P^2 \propto a^3$. For Venus, $P^2 = 0.62 \times 0.62 = 0.38$ and $a^3 = 0.72 \times 0.72 \times 0.72 = 0.037$ (rounding numbers sometimes causes minor discrepancies like this). The square of the orbital period $(0.38)$ approximates the cube of the semimajor axis $(0.37)$. Therefore, Venus obeys Kepler's third law. For Earth, $P^2 = 1.00 \times 1.00 = 1.00$ and $a^3 = 1.00 \times 1.00 \times 1.00 =1.00$. The square of the orbital period $(1.00)$ approximates (in this case, equals) the cube of the semimajor axis $(1.00)$. Therefore, Earth obeys Kepler's third law.

Check Your Learning

Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate $P^2$ and $a^3$, and verify that they obey Kepler's third law. Saturn's orbital period is $29.46$ years, and its semimajor axis is $9.54$ AU. Jupiter's orbital period is $11.86$ years, and its semimajor axis is $5.20$ AU.

Answer

For Saturn, $P^2 = 29.46 \times 29.46 = 867.9$ and $a^3 = 9.54 \times 9.54 \times 9.54 = 868.3$. The square of the orbital period $(867.9)$ approximates the cube of the semimajor axis $(868.3)$. Therefore, Saturn obeys Kepler's third law.

This article is a derivative work of the creative commons share alike with attribution in [1].

Bibliography

[1] Fraknoi, Andrew, David Morrison, and Sidney Wolff. The Sky Above. In Astronomy 2e. Houston, Texas : OpenStax, 2022. The Sky Above



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Physics Classification: 45.50.Pk (Celestial mechanics )
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