Question

Use Kepler's third law to answer the following questions: a. Determine the period of a planet with a solar distance of \(10 \mathrm{AU}\). b. Determine the distance between the Sun and a planet with a rotational period of 5 years. c. Imagine two bodies, one twice as large as the other, orbiting the Sun at the same distance. Which of the bodies, if either, would move faster than the other?

Short answer

a. 31.62 years. b. 2.924 AU. c. Neither body moves faster.

Step-by-step solution

Understanding Kepler's Third Law

Kepler's third law states that the square of the period of orbit (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (a), where the axis is measured in Astronomical Units (AU) and the period in Earth years. Mathematically, it is expressed as: \[ T^2 = a^3 \]

Solving Part (a)

Given the solar distance is \( 10 \text{ AU} \). Substitute \( a = 10 \) into the equation:\[ T^2 = 10^3 \]\[ T^2 = 1000 \]Taking the square root of both sides, we find:\[ T = \sqrt{1000} \approx 31.62 \text{ years} \]

Solving Part (b)

Given the period \( T = 5 \) years. We need to find \( a \).Using the equation \( T^2 = a^3 \), substituting \( T = 5 \):\[ 5^2 = a^3 \]\[ 25 = a^3 \]Taking the cube root of both sides:\[ a = \sqrt[3]{25} \approx 2.924 \text{ AU} \]

Solving Part (c)

According to Kepler's laws, the speed of the orbiting body is determined by its distance from the Sun and not its mass. Given both bodies are at the same distance from the Sun, they will have the same orbital speed regardless of their sizes. Thus, neither body moves faster than the other.

Key concepts

Orbital Period

Kepler's Third Law is crucial in understanding planetary motion, specifically when discussing the orbital period. The orbital period of a planet is the time it takes to complete one full orbit around the Sun. Like the hands of a clock tracing a circular path, a planet moves along its orbital trajectory. According to Kepler, if you know the semi-major axis (more on this soon), you can determine the orbital period.

The relationship is described by the formula: \[ T^2 = a^3 \]Where:

• \( T \) represents the orbital period measured in Earth years,
• \( a \) denotes the semi-major axis measured in Astronomical Units (AU).

Hence, each planet's orbital dance is predictably choreographed by its distance from the Sun.

Semi-Major Axis

The semi-major axis is a vital component of an elliptical orbit. Imagine an ellipse with the Sun at one focus. The longest diameter of this ellipse is called the major axis. Half of this diameter—stretching from the center to the edge of the ellipse—is the semi-major axis. This axis essentially functions as the orbit's average distance from the Sun.

In Kepler's equation, the semi-major axis provides a critical piece of the puzzle for predicting a planet's orbital period. Larger values of the semi-major axis correspond to longer periods, akin to stringing a ball on a longer tether and watching it revolve more slowly.

The semi-major axis is a crucial tool for determining various orbital dynamics within a solar system.

Solar Distance

Solar distance refers to the mean distance between a celestial body and the Sun. In the context of Kepler's laws, we often express this distance in Astronomical Units (AU). An AU is the average distance between Earth and the Sun, approximately 93 million miles or 150 million kilometers.

In practical terms, if a planet has a solar distance of 10 AU, like in our example, it is 10 times farther from the Sun than Earth is. Because of Kepler's Second Law, planets with greater solar distances take longer to orbit the Sun, as demonstrated by the increased orbital periods.

Thus, solar distance is not merely a measurement but a dynamic factor in orbital mechanics.

Astronomical Units

Astronomical Units (AU) simplify the measurement of vast cosmic distances. Earth is the cosmic yardstick with 1 AU representing the distance from our planet to the Sun. This standard unit is crucial for communicating large distances during scientific discussions.

One might wonder why we use AU instead of, say, kilometers or miles. In the realm of astronomy, the numbers become extraordinarily large, and AU offers a more comprehensible scale. Especially when discussing how far away planets or other celestial objects are from the Sun.
Thus, understanding and using AU helps astronomers to easily speak about distances within our solar system without losing sight of the comparisons between planetary orbits.