Question

Which one of the following statements represents the law of periods? group of answer choices

O the rotational period of the sun equals the sum of the rotational periods of the planets the square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.
O the orbital period of a satellite in orbit of a planet is inversely proportional to its mass.
O every planet sweeps out the same area in a one earth year period, making one complete orbit about the sun.
O the period of a planet in its orbit about the sun is directly proportional to the radius of its orbit.

Answer 1

Kepler's Third Law, the law of periods, states that the square of a planet's orbital period is directly proportional to the cube of its semimajor axis, expressing a fundamental relationship in planetary motion.

The statement that represents the law of periods is:

"The period of a planet in its orbit about the sun is directly proportional to the radius of its orbit."

This statement encapsulates Kepler's Third Law of Planetary Motion, also known as the law of periods. The law of periods is a fundamental principle that describes the relationship between the orbital period (time taken to complete one orbit) and the semimajor axis (average distance from the sun) of a planet's orbit.

Mathematically, the law of periods can be expressed as:

`\[ T^2 \propto r^3 \]`

where:

• T is the orbital period of the planet,

• r is the radius (semimajor axis) of the planet's orbit,

• The symbol
  `\(\propto\)` denotes proportionality.

The square of the orbital period
`(\( T^2 \))` is directly proportional to the cube of the semimajor axis
`(\( r^3 \))`. In other words, the ratio of
`\( T^2/r^3 \)` is a constant for all planets in our solar system.

This law implies that planets farther from the sun have longer orbital periods, and the relationship holds true for all celestial bodies orbiting a central mass, not just planets. Johannes Kepler derived this law based on observational data provided by Tycho Brahe, and it laid the groundwork for understanding the geometry of planetary motion.