Problem 42 - Algebra and Trigonometry

Question

Kepler's Third Law Kepler's Third Law of planetary motion states that the square of the period \(T\) of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance \(d\) from the sun. (a) Express Kepler's Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about \(2.79 \times 10^{9}\) mi from the sun. Find the period of Neptune.

Step-by-Step Solution

Answer

The period of Neptune is approximately 518 years.

Step 1: Express Kepler's Third Law as an Equation

Kepler's Third Law states that the square of the period \(T\) of a planet is directly proportional to the cube of its average distance \(d\) from the sun. This relationship can be written in the form of an equation: \( T^2 = k \cdot d^3 \), where \(k\) is the constant of proportionality.

Step 2: Calculate the Constant of Proportionality

We know the period \(T\) of Earth is about 365 days, and the average distance \(d\) is 93 million miles, or \(9.3 \times 10^7\) miles. Substitute these values into the equation to find \(k\): \[(365)^2 = k \cdot (9.3 \times 10^7)^3\]Calculate \(k\) using the above values:\[133225 = k \cdot (9.3 \times 10^7)^3\]Solving for \(k\), we have:\[k = \frac{133225}{(9.3 \times 10^7)^3}\]

Step 3: Solve for the Constant

Calculate the value of \((9.3 \times 10^7)^3\):\[(9.3 \times 10^7)^3 = 9.3^3 \times (10^7)^3 = 804.357 \times 10^{21}\]Substituting back:\[k = \frac{133225}{804.357 \times 10^{21}}\]\[k \approx 1.656 \times 10^{-16}\]

Step 4: Find Neptune's Period using the Equation

With \(k\) found, use Neptune's average distance \(d = 2.79 \times 10^9\) mi to find its period. Plug \(d\) and \(k\) into the equation:\[T^2 = 1.656 \times 10^{-16} \times (2.79 \times 10^9)^3\]Calculate \((2.79 \times 10^9)^3\) first:\[(2.79 \times 10^9)^3 = 21.688539 \times 10^{27}\]Substitute back:\[T^2 = 1.656 \times 10^{-16} \times 21.688539 \times 10^{27} \]Simplify:\[T^2 = 3.592 \times 10^{12}\]

Step 5: Solve for Neptune's Period

To find \(T\), take the square root:\[T = \sqrt{3.592 \times 10^{12}}\]Calculate:\[T \approx 1.895 \times 10^6 \text{ days}\]Convert days into years (since 1 year ≈ 365.25 days):\[T \approx \frac{1.895 \times 10^6}{365.25} \approx 518 \, \text{years}\]

Key Concepts

Planetary Motion

Planetary motion refers to the consistent and predictable movements of planets as they orbit around a star, such as the sun in our solar system. These orbits are not random; rather, they follow specific laws, most notably formulated by Johannes Kepler in the early 17th century. Kepler's three laws describe how planets move in elliptical paths, how they sweep out equal areas in equal times, and how the time of orbit is related to their distance from the sun.
Understanding these principles is essential as they form the basis for predicting planetary positions or analyzing satellite motions. In essence, planetary motion provides insights into the gravitational attraction between celestial bodies. This gravitational force is key to maintaining the harmonious dance of planets within a solar system, ensuring a delicate balance that allows for dynamic yet stable orbits.

Proportionality Constant

A proportionality constant in the context of Kepler's Third Law is a specific value that links the period of a planet's orbit to its distance from the sun. This constant, denoted as \(k\), is essential for converting abstract proportional relationships into tangible calculations.
Kepler's Third Law is mathematically expressed as \(T^2 = k \cdot d^3\), where \(T\) represents the orbital period and \(d\) the average distance from the sun. The constant \(k\) ensures that regardless of the units used for \(T\) and \(d\), the equation balances perfectly.
For Earth, with a known year and distance, this constant can be derived, allowing us to make predictions about other planets. By integrating known data into the formula, \(k\) becomes a tool for exploring new orbits and even other planetary systems.

Period Calculation

Calculating the orbital period of a planet involves determining the time it takes for a planet to make a complete orbit around the sun. Using Kepler's Third Law, the calculation begins with the equation \(T^2 = k \cdot d^3\). Knowing the distance (\(d\), in this instance, Neptune's is \(2.79 \times 10^9\) mi), and the previously calculated proportionality constant \(k\), we can substitute the values into the formula to find \(T^2\):

• First, determine \(d^3\) for your particular planet.
• Multiply this result by the constant \(k\).
• Finally, take the square root of the outcome to find \(T\), the orbital period.

For Neptune, the final period converts to roughly 518 Earth years, informing us of the immense time Neptune takes to orbit the sun.

Distance from Sun

A planet's distance from the sun is a crucial factor in determining its orbital period as per Kepler's Third Law. The farther a planet is, the longer it will take to complete an orbit, since its path is larger and the gravitational force diminishes with distance.
For instance, Earth is about 93 million miles from the sun, which determines its one-year orbital period. Neptune, at a staggering \(2.79 \times 10^9\) miles away, has a period that spans about 518 Earth years. This dramatic difference emphasizes how distance influences the time taken for a full revolution.
Understanding these distances is vital for contextualizing the dynamic scale of our solar system. It allows astronomers to compare planets not only within our own system but also across different stellar systems, using basic measurements to estimate orbits far beyond our immediate reach.