Problem 50 - 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 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

Step-by-Step Solution

Answer

Saturn takes approximately 29.46 Earth years to orbit the Sun.

Step 1: Understanding Kepler's Third Law

Kepler's Third Law of planetary motion states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit, i.e., \(T^2 \propto a^3\). Since Earth is one astronomical unit from the Sun, we have \(T^2 = k \cdot a^3\), where \(k\) is a constant.

Step 2: Establish Ratio Using Earth's Orbit

For Earth, \(a = 1\) A.U. and \(T = 1\) year. Therefore, \(T^2 = k \cdot a^3\) becomes \(1^2 = k \cdot (1)^3\). This simplifies to \(k = 1\). This means for Earth, \(1 = 1^2\).

Step 3: Setup Equation for Saturn

Using the same relationship, for Saturn, we have \(T^2 = 1 \cdot (9.54)^3\). Substitute \(a = 9.54\) to find \(T^2\).

Step 4: Calculate Cubic Value and Solve for T

Calculate \((9.54)^3\):\[9.54^3 = 868.2546.\]Now, substitute into the equation for \(T^2\):\[T^2 = 868.2546.\]

Step 5: Solving for T

Finally, take the square root of both sides to solve for \(T\):\[T = \sqrt{868.2546} \approx 29.46.\]Thus, the time for Saturn to complete an orbit around the Sun is approximately 29.46 Earth years.

Key Concepts

Orbital Period

The orbital period of a planet refers to the time it takes for the planet to complete one full orbit around the Sun. This concept is central to understanding planetary motion as described by Kepler's Law.
Kepler's Third Law states that the square of the orbital period ( T ) is directly proportional to the cube of the semi-major axis ( a ) of its orbit. This implies that as the distance from the Sun increases, the time taken for a revolution also increases. Specifically, T^2 is proportional to a^3 . This relationship helps us calculate the orbital period for any planet, provided we know its distance from the Sun. In our example, Earth's orbital period is one year, but Saturn's is much longer due to its greater distance.

Semi-Major Axis

The semi-major axis is essentially the average distance of a planet from the Sun during its orbit. It forms the major part of an elliptical orbit's length and is crucial in determining how long it takes for the planet to go around the Sun.
Kepler's Third Law shows a significant relationship between the semi-major axis and the orbital period: the larger the semi-major axis, the longer the orbital period. This is because the semi-major axis not only determines the size of the orbit but also affects the speed at which the planet travels through space.

• The semi-major axis for Earth is defined as 1 astronomical unit.
• For Saturn, it is 9.54 astronomical units.

Understanding the semi-major axis aids in predicting how these distances affect the planet's movement around the Sun.

Astronomical Unit

An astronomical unit (A.U.) is a standard unit of measurement in astronomy that represents the average distance from the Earth to the Sun, about 93 million miles or 150 million kilometers. This unit simplifies understanding and comparing distances within our solar system.
By defining Earth's distance from the Sun as 1 A.U., astronomers can easily express and compare planetary distances without dealing with excessively large numbers. In our exercise, Saturn's mean distance from the Sun is given as 9.54 A.U.
Using astronomical units allows for a straightforward application of Kepler's laws, as they provide a consistent scale to measure the vast distances involved in planetary orbits.

Planetary Motion

Planetary motion refers to the movement of planets around the Sun, which is governed by the laws of celestial mechanics, prominently Kepler's Laws. These laws help explain how planets travel in elliptical orbits with varying speeds depending on their distance from the Sun.
In particular, Kepler's Third Law provides a vital insight into this motion, establishing that a planet's orbital period is related to its average distance from the Sun. Our understanding of planetary motion has been crucial in developing models that predict the future positions of planets, enabling activities such as space missions and astronomical observations.

• Planets closer to the Sun, like Mercury, orbit faster and have shorter periods due to shorter semi-major axes.
• Planets further from the Sun, like Saturn, have longer orbital periods.

By comprehending planetary motion, we can appreciate the rhythm of our solar system and the intricate dance of celestial bodies.