Question

Explain your reasoning with one or more complete sentences. According to Kepler's third law, (a) Mercury travels fastest in the part of its orbit in which it is closest to the Sun. (b) Jupiter orbits the Sun at a faster speed than Saturn. (c) all the planets have nearly circular orbits.

Short answer

Statements (a) and (b) are true; statement (c) is approximately true.

Step-by-step solution

Understanding Kepler's Third 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 of its orbit. Mathematically, it can be expressed as \( T^2 \propto a^3 \), where \( T \) is the orbital period and \( a \) is the semi-major axis of the orbit.

Analyzing Mercury's Fastest Travel

According to Kepler's laws, a planet orbits faster when it is closer to the Sun due to the gravitational forces being stronger. This is Kepler's Second Law which states that planets sweep equal areas in equal times. Therefore, statement (a) is true: Mercury travels fastest when it is closest to the Sun.

Comparing Orbital Speeds of Jupiter and Saturn

According to Kepler's Third Law, planets closer to the Sun have shorter orbital periods and therefore travel at faster average speeds. Jupiter, being closer to the Sun than Saturn, has a shorter orbital period and travels faster. Thus, statement (b) is true: Jupiter orbits the Sun faster than Saturn.

Evaluating Planetary Orbits

Kepler's First Law states that planets travel in elliptical orbits with the Sun at one focus, not necessarily along a perfect circle. However, for practical purposes, many planetary orbits, including those of the planets in our solar system, are nearly circular. Statement (c) is approximately true: most planetary orbits are nearly circular.

Key concepts

Orbital Period

The orbital period refers to the time it takes for a planet to complete one full orbit around the Sun. According to Kepler's Third Law, the orbital period (\( T \)) of a planet is directly related to the size of its orbit. Specifically, it states that the square of a planet's orbital period is proportional to the cube of the semi-major axis (\( a \)) of its orbit. This means that if we know the length of the semi-major axis, we can determine the orbital period and vice versa.
This relationship helps explain why planets further from the Sun, which have larger semi-major axes, also have longer orbital periods and move more slowly through space compared to those closer to the Sun.

Semi-Major Axis

The semi-major axis of a planet's orbit is a critical component in understanding Kepler's Laws. It is the longest radius of an elliptical orbit, stretched out halfway between the closest and furthest points from the Sun. This axis effectively determines the size of the orbit.
Kepler's Third Law mathematically uses the semi-major axis to relate it to the orbital period of a planet. The larger the semi-major axis, the longer it takes for the planet to complete its orbit, following the equation \( T^2 \propto a^3 \).
Understanding the semi-major axis is vital in comparing the orbits of different planets, as it provides insight into both the duration of their orbits and their average distance from the Sun.

Planetary Orbits

Planetary orbits are shaped as ellipses rather than perfect circles. Kepler's First Law describes that planets move in these elliptical orbits with the Sun located at one focus. While the orbits aren't exactly circular, many, including Earth's, have a low eccentricity, making them appear nearly circular.
Elliptical orbits mean that the distance between a planet and the Sun changes throughout the orbit. This is why planets may sometimes be closer (periapsis) or further (apoapsis) from the Sun during their journey.
Remembering that planetary orbits are nearly circular simplifies understanding, but recognizing their true elliptical nature provides a deeper grasp of celestial mechanics.

Gravitational Forces

Gravitational forces play a central role in maintaining the structure and dynamics of planetary orbits. The gravitational attraction between the Sun and each planet keeps the planet in orbit, pulling it towards the Sun.
Kepler's Second Law, or the law of equal areas, relates to gravitational forces by explaining how a planet moves faster when it is closer to the Sun than when it is further away. This is due to stronger gravitational attraction resulting in a greater orbital speed at the perihelion.

• The gravitational force depends on the masses of the Sun and the planet.
• The force becomes stronger as the planet gets closer to the Sun.
• This varying force causes changes in the speed of the planet within its orbit.

Understanding gravitational forces is crucial as it explains how orbits are maintained and why they vary in speed throughout the journey.