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 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. Both move at the same speed.

Step-by-step solution

Understanding Kepler's Third Law

Kepler's third law states that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit: \( T^2 = a^3 \), where \( T \) is the orbital period in Earth years and \( a \) is the semi-major axis given in astronomical units (AU).

Calculate Orbital Period for 10 AU

To find the period of a planet situated 10 AU from the Sun, substitute \( a = 10 \) into Kepler's third law equation: \( T^2 = 10^3 = 1000 \). Solving for \( T \) gives \( T = \sqrt{1000} \approx 31.62 \) years. Thus, the orbital period is approximately 31.62 years.

Calculate Distance for 5-Year Period

For a planet with a period of 5 years, solve \( T^2 = a^3 \) with \( T = 5 \). This yields \( 5^2 = a^3 \), so \( 25 = a^3 \). Solving for \( a \) gives \( a = \sqrt[3]{25} \approx 2.924 \). Thus, the solar distance is approximately 2.924 AU.

Compare Velocities of Two Bodies with Different Masses

Kepler's third law implies that the orbital period is independent of the object's mass. Thus, two bodies orbiting the Sun at the same distance have the same orbital period and velocity, regardless of mass. Therefore, neither body would move faster than the other.

Key concepts

Orbital Period

The orbital period is the time it takes for a planet or celestial body to complete one full orbit around the Sun. It is a key concept in understanding the motion of planets within our solar system. According to Kepler's third law, the orbital period squared (\(T^2\)) is directly proportional to the semi-major axis cubed (\(a^3\)) of its orbit, expressed mathematically as \(T^2 = a^3\). This equation provides a way to calculate how long it takes for a planet to travel around the Sun based on its distance from the Sun.

• Orbital period is measured in Earth years.
• It is linked to the size of the planet's orbit.
• Larger orbits result in longer orbital periods.

To illustrate, if a planet's semi-major axis is 10 AU, solving the equation \(T^2 = 10^3\) results in an orbital period of about 31.62 years.

Semi-Major Axis

The semi-major axis is half of the longest diameter of an elliptical orbit, representing the average distance of the planet from the Sun. In calculations involving Kepler's third law, the semi-major axis is a crucial component, as it helps determine the orbital period. The relationship is clearly seen in the formula \(T^2 = a^3\). Using this, if you know a planet's orbital period, you can find its semi-major axis.

• Measured in astronomical units (AU), where 1 AU is the average distance from the Earth to the Sun.
• Larger semi-major axes indicate greater distances from the Sun.
• Key factor in calculating orbits in the solar system.

For example, if a planet has an orbital period of 5 years, solving \(a^3 = 5^2\) yields a semi-major axis of approximately 2.924 AU.

Astronomical Unit

An astronomical unit (AU) is the standard unit of measurement used to express distances within our solar system. It is defined as the average distance from the Earth to the Sun, roughly 93 million miles or 150 million kilometers. This measurement simplifies understanding and comparing planetary distances without dealing with large numbers.

• 1 AU is the baseline for measuring all other distances in the solar system.
• Helps compare distances between various planets and celestial objects.
• Used in equations like Kepler's third law for determining orbital periods and semi-major axes.

So, when we say a planet is 10 AU from the Sun, it means it is ten times the distance from the Earth to the Sun.

Planetary Orbits

Planetary orbits describe the path that planets follow around the Sun. These orbits are primarily elliptical, with the Sun at one focus of the elliptical path. According to Kepler's laws, every planet travels in such an orbit, making the understanding of orbital dynamics essential for astronomy and astrophysics. Planetary orbits involve several key principles:

• They help predict planetary positions at any given time.
• Orbital speed varies, being faster when closer to the Sun and slower when farther away due to gravitational forces.
• These principles apply to all celestial bodies orbiting a central mass.

Interestingly, the mass of two orbiting bodies does not affect their speed. This means, even if one planet is twice as massive as another while maintaining the same distance from the Sun, both planets will orbit at the same velocity. This is due to the fact that the orbital period and speed depend solely on the distance to the Sun, not the masses of the orbiting bodies.