Socratica

Let’s compare Jupiter and Saturn, two gas giants in our solar system, using Kepler’s Third Law.

Jupiter:
Distance from the Sun: 5.2 AU
Orbital period: ~ 11.86 Earth years

Saturn
Distance from the Sun: 9.58 AU
Orbital period: about 29.46 Earth years

Using Kepler’s Third Law, we calculate the constant for both planets:

For Jupiter: \[ k_{Jupiter} = \frac{T_{Jupiter}^2}{a_{Jupiter}^3} = \frac{11.86^2}{5.2^3} \approx 1.00037 \]

For Saturn:
\[ k_{Saturn} = \frac{T_{Saturn}^2}{a_{Saturn}^3} = \frac{29.46^2}{9.58^3} \approx 0.98712 \]
Since Saturn is almost twice as far from the Sun as Jupiter, it takes significantly longer to complete one orbit. While Jupiter takes about 12 Earth years to orbit the Sun, Saturn, being farther away, takes nearly 30 Earth years.
The increase in orbital period is much greater than the increase in distance, which is why Kepler’s Third Law shows a relationship that scales with the cube of the distance and the square of the period.

These results also confirm that the constant \(k\) is approximately the same for both planets, verifying Kepler’s Third Law. The small difference arises due to rounding and slight variations in measurements.
The value of \(k\) for our solar system is approximately 1 when using years for the orbital period and astronomical units (AU) for the semi-major axis \(a\).