Kepler's third law of planetary motion can be expressed in a simplified form for the solar system when considering periods in years and distances in astronomical units (AU). The law states that the square of the orbital period \( T \) of a planet (in years) is directly proportional to the cube of the semi-major axis \( r \) of its orbit (in astronomical units). This can be mathematically represented as:
\[ T^2 \propto r^3 \]
In the form of an equation, this can be expressed as:
\[ \frac{T^2}{r^3} = k \]
where \( k \) is a constant. For our solar system, this constant \( k \) is approximately equal to 1 for planets orbiting the Sun. Therefore, in practical terms, the relationship can be simplified to:
\[ T^2 = r^3 \]
This means that if you know the distance of a planet from the Sun in astronomical units, you can find its orbital period in years by taking the cube of the radius and then taking the square root. Likewise, if you know its period, you can calculate the radius using this relationship.