Problem 50 - College Algebra

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 orbits the Sun approximately every 29.46 years.

Step 1: Understanding Kepler's Law

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (mean distance) of its orbit. Mathematically, it can be expressed as \( T^2 \propto a^3 \), where \( T \) is the orbital period and \( a \) is the mean distance from the Sun.

Step 2: Establish the Proportional Relationship for Earth

For Earth, we know that the mean distance \( a = 1 \) A.U. and the orbital period \( T = 1 \) year. Plug these into Kepler's Law: \( T^2 = a^3 \). Therefore, \( 1^2 = 1^3 \). The constant of proportionality is effectively 1 in this system of units.

Step 3: Apply Kepler's Law to Saturn

For Saturn, the mean distance \( a = 9.54 \) A.U. Write Kepler's Law for Saturn: \( T_{ ext{Saturn}}^2 = (9.54)^3 \). Calculate \( (9.54)^3 \) to get the right side of the equation: \( (9.54)^3 = 868.254064 \). So, \( T_{ ext{Saturn}}^2 = 868.254064 \).

Step 4: Solve for Saturn's Orbital Period

To get the orbital period \( T \) for Saturn, solve for \( T \) by taking the square root of 868.254064: \( T_{ ext{Saturn}} = \sqrt{868.254064} \approx 29.46 \) years.

Step 5: Conclusion

Based on the calculations, Saturn takes approximately 29.46 years to orbit the Sun once.

Key Concepts

Orbital Period

The orbital period is a crucial component of Kepler's Third Law. It refers to the amount of time a planet takes to make one complete orbit around the Sun. This time is usually measured in Earth years.
The length of the orbital period tells us how far a planet travels and how fast it moves through space. For example, Earth has an orbital period of 1 year because it travels around the Sun in 365 days.
But for planets further from the Sun, like Saturn, the orbital period is longer. In fact, using the information provided by Kepler's Law, we see that Saturn's orbital period is about 29.46 Earth-years. This calculation takes into account the fact that Saturn's average distance from the Sun (its semi-major axis) is much greater than Earth's.
Kepler's Third Law helps astronomers predict these periods based on the planet's distance from the Sun.

Astronomical Unit

An astronomical unit (A.U.) is a standard unit of measurement used in astronomy. It represents the average distance from Earth to the Sun.
One A.U. is approximately 93 million miles or 150 million kilometers.
This unit provides a simple way to express large distances within our solar system without using very large numbers.
For example, Earth's mean distance from the Sun is 1 A.U., which is the basis for measuring the distances of other planets in the solar system.

• Mars is about 1.52 A.U. from the Sun, making it farther away than Earth.
• Saturn, much further out, is 9.54 A.U. from the Sun.

Using the A.U. simplifies calculations like determining the orbital period using Kepler's Third Law.

Semi-major Axis

The semi-major axis is a critical part of understanding a planet's orbit. In an elliptical orbit, which is the shape of most planetary orbits, the semi-major axis is the longest diameter of the ellipse.
It represents half of the longest dimension across the ellipse and serves as an average distance from the planet to the Sun.
For example, the semi-major axis of Earth's orbit is 1 A.U., which means Earth's orbit is averagely 1 astronomical unit from the Sun.

• Jupiter's semi-major axis is about 5.2 A.U., indicating it's more than five times farther from the Sun compared to Earth.
• Saturn's semi-major axis is about 9.54 A.U., which is used to determine its orbital period using Kepler's Law.

The larger the semi-major axis, the longer the orbital path, which usually results in a longer orbital period. This is why planets like Saturn take many years to complete a full orbit around the Sun.