For the following exercises, use Kepler's Law, which states...

For the following exercises, use Kepler's Law, which states that the time, $T$ , required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a$ , that the planet is from the Sun. Using the Earth's time of 1 year and mean distance of 93 million miles, find the equation relating $T$ and $a$ .

For the following exercises, use Kepler's Law, which states that the time, $T$ , required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a$ , that the planet is from the Sun.
Using the Earth's time of 1 year and mean distance of 93 million miles, find the equation relating $T$ and $a$ .

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Key Concepts

Kepler's Law

Kepler's Law, particularly the third law in this context, establishes a relationship between the orbital period of a planet and its distance from the Sun. It posits that the time required for a planet to orbit the Sun is proportional to a specific power of its mean distance. This law provides the basis for constructing an equation that can determine one variable using the other when calibrated with known planetary data.

Direct Variation

Direct variation is a mathematical concept where one quantity is directly proportional to another, meaning that when one quantity increases or decreases, the other does so by a constant multiplicative factor. In the case of the problem, time (T) is directly proportional to the cube of the mean distance (a), implying that any change in distance will cause a corresponding cubic change in the orbital period.

Proportionality Constant

The proportionality constant is the fixed number that relates two quantities in a direct variation equation. By using known values (like Earth's orbit time and distance), one can calculate this constant, which then allows for the formulation of a general equation. This constant is crucial for transforming the theoretical proportional relationship into a practical and usable formula that relates orbital period to distance.

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