Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate p² and a³, 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.

0.1 Solution

We can use the equation for Kepler’s third law,P² ∝ a³. For Venus, P² = 0.62 × 0.62 = 0.38 and a³ = 0.72 × 0.72 × 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, P² = 1.00 × 1.00 = 1.00 and a³ = 1.00 × 1.00 × 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.

0.2 Check Your Learning

Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate P² and a³, 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.

0.3 Answer

For Saturn, P² = 29.46 × 29.46 = 867.9 and a³ = 9.54 × 9.54 × 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].

References

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