Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, use Kepler's Law, which states that the square of 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 Earth's distance of 150 million kilometers, find the equation relating \(T\) and \(a\).

Step-by-Step Solution

Answer

The equation relating \( T \) and \( a \) is \( T^2 = \left(\frac{1}{150^3}\right) a^3 \).

Step 1: Understand Kepler's Third Law

Kepler's Third Law states that the square of the orbital period (time taken for one complete orbit, denoted as \( T \)) of a planet is directly proportional to the cube of the semi-major axis (average distance from the Sun, denoted as \( a \)) of its orbit. Mathematically, this can be expressed as \( T^2 \propto a^3 \).

Step 2: Introduce the Proportionality Constant

To convert the proportionality into an equation, introduce a constant \( k \). The relationship then becomes \( T^2 = k \, a^3 \). \( k \) is known as the constant of proportionality.

Step 3: Use Earth's Known Values to Find \( k \)

We know that Earth's average distance from the Sun is 150 million kilometers and that its orbital period is 1 year. Substitute these values into the equation to find \( k \): \( 1^2 = k \, (150^3) \).

Step 4: Simplify to Find the Constant \( k \)

After substituting the known values, we get \( 1 = k \, (150^3) \). Simplifying this gives us, \( k = \frac{1}{150^3} \).

Step 5: Write the General Equation Relating \( T \) and \( a \)

Now that we have \( k \), the general equation for any planet's orbital period with respect to its mean distance from the Sun is \( T^2 = \left(\frac{1}{150^3}\right) a^3 \). This can be used to calculate the orbital period \( T \) for any other planet given its distance \( a \) from the Sun.

Key Concepts

Orbital period

The orbital period is a fascinating concept in astronomy that refers to the time a planet takes to complete one full orbit around the Sun. This duration can be quite different for each planet in our solar system, depending mainly on their distance from the Sun. Understanding the orbital period is key to comprehending how planets move through space. According to Kepler's Third Law, the orbital period squared (\( T^2 \) ) of a planet is directly proportional to the cube of the semi-major axis (\( a^3 \)), which denotes the average distance of the planet from the Sun. This relationship gives us a fundamental insight into how planets' positions and movements correlate with their distances from the Sun. Whether it's Mercury's quick 88-day orbit or Pluto's long 248-year journey, the orbital period helps us predict and understand these celestial paths. Keep in mind, when thinking about the orbital period, it sets the rhythm of a planet's year and influences other factors like climate cycles and exposure to solar energy.

Semi-major axis

The semi-major axis is essentially the planet's average distance from the Sun. To visualize it, imagine an elongated circle or ellipse, which is what a planet's orbit actually looks like. The semi-major axis is the longest radius of this ellipse, stretching from the center of the orbit to the edge. In Kepler's Third Law, the semi-major axis is denoted by \( a \). It plays a pivotal role in determining a planet's orbital characteristics. When considering the Earth's orbit, its semi-major axis is approximately 150 million kilometers. This average enables us to use the beautifully simple equation derived from Kepler's insights: \( T^2 = k \, a^3 \). This equation tells us how the semi-major axis (\( a \)) will influence the orbital period squared (\( T^2 \)), allowing us to predict one if we know the other. Understanding the semi-major axis helps us comprehend planetary motion, and, by extension, the time it takes for planets to revolve around the Sun, which is crucial for planning missions and understanding seasonal variations.

Constant of proportionality

The constant of proportionality, often represented as \( k \) in Kepler's Third Law, is a key factor in converting the direct proportion relationship into an equation. While the underlying principle suggests that the orbital period squared (\( T^2 \)) is just directly proportional to the cube of the semi-major axis (\( a^3 \)), using \( k \) transforms this proportionality into a usable formula: \( T^2 = k \, a^3 \). Finding \( k \) is an essential step. It allows us to create a specific equation for calculating the orbital periods of different planets in our solar system. For Earth, given its semi-major axis and known orbital period, \( k \) was calculated as \( \frac{1}{150^3} \). The role of \( k \) is vital because it adjusts the relationship to accurately reflect the distances in astronomical terms, thereby enabling precise calculations. Understanding the constant of proportionality essentially bridges the gap between theoretical proportions and practical, real-world data, allowing us to make accurate predictions about celestial movements.