To solve this problem, we need to consider the meaning and implications of Kepler's third law, [tex]\( P^2 = k A^3 \)[/tex], where [tex]\( P \)[/tex] is the orbital period (the time it takes for a planet to complete one orbit around the Sun), [tex]\( A \)[/tex] is the radius of the semi-major axis of the orbit, and [tex]\( k \)[/tex] is a constant that depends on the units of [tex]\( P \)[/tex] and [tex]\( A \)[/tex].
Let’s analyze each statement:
A. The orbital period is measured in units of time, and the semi-major axis is measured in units of mass.
- This statement is incorrect.
- The orbital period [tex]\( P \)[/tex] is indeed measured in units of time, such as years or days.
- However, the semi-major axis [tex]\( A \)[/tex] is measured in units of distance, such as astronomical units (AU) or meters, not units of mass.
B. The value [tex]\( k \)[/tex] is constant for each of the eight planets in our solar system.
- This statement is correct.
- For objects orbiting the Sun, [tex]\( k \)[/tex] is the same when [tex]\( P \)[/tex] is measured in years and [tex]\( A \)[/tex] is measured in astronomical units. For all planets orbiting the Sun, the value [tex]\( k \)[/tex] is a constant value representing the gravitational parameter in those units.
C. For a body orbiting the Sun, increasing the orbital period increases the length of the semi-major axis.
- This statement is also correct.
- According to Kepler's third law, [tex]\( P^2 = k A^3 \)[/tex]. If the orbital period [tex]\( P \)[/tex] increases, the semi-major axis [tex]\( A \)[/tex] must also increase to maintain the equality, assuming [tex]\( k \)[/tex] is constant.
D. To calculate the value of [tex]\( k \)[/tex] for a planet in our solar system, find [tex]\( A^3 + P^2 \)[/tex] for the planet.
- This statement is incorrect.
- The correct relationship given by Kepler's third law is [tex]\( P^2 = k A^3 \)[/tex] and not [tex]\( A^3 + P^2 \)[/tex]. To calculate [tex]\( k \)[/tex], one would use the ratio [tex]\(\frac{P^2}{A^3}\)[/tex], not the sum [tex]\( A^3 + P^2 \)[/tex].
Given this analysis, the correct statement about Kepler's law is:
C. For a body orbiting the Sun, increasing the orbital period increases the length of the semi-major axis.
