For Earth, \(P^{2} / A^{3}=1.0\) (in appropriate units). Suppose a new dwarf planet is discovered that is 14 times as far from the Sun as Earth is. For this planet, a. \(P^{2} / A^{3}=1.0\) b. \(P^{2} / A^{3} > 1.0\) c. \(P^{2} / A^{3} < 1.0\) d. one can't know the value of \(P^{2} / A^{3}\) without more information.

The orbital period is the time it takes for a planet or any celestial body to complete one full orbit around a star, such as the Sun. In calculations, it's usually represented by the letter \(P\). For Earth, one complete orbit around the Sun takes about 365.25 days, which is our year.

Orbital periods can vary greatly depending on the distance from the star. The farther away the body is, the longer the orbital period. This relationship is crucial in understanding planetary motion.

Kepler's third law, which we will discuss later, provides a straightforward way to calculate the orbital period if we know the distance. For example, if a dwarf planet is found farther from the Sun than Earth, it will have a longer orbital period.

• For a planet closer to the Sun, the orbital period is shorter.
• For a planet farther from the Sun, the orbital period is longer.
• Orbital periods help in understanding the timing of planetary events like transits and eclipses.

The semimajor axis, often denoted as \(A\), is a measure of the size of an orbit. Specifically, it's the average distance from the orbiting body to the star it revolves around. For Earth, this distance is about 93 million miles or 150 million kilometers, which defines one Astronomical Unit (AU).

When Kepler's third law mentions \(A^3\), it is referring to the cube of this distance. For instance, if a newly discovered dwarf planet is said to be 14 times farther from the Sun than Earth, its semimajor axis would be \(14A_{Earth}\).

The distance affects various factors like gravitational forces and orbital periods. Here are some highlights:

• The larger the semimajor axis, the longer the orbital path.
• For Earth, this value is 1 AU, setting a standard unit for measuring other planets' distances.
• Knowing the semimajor axis is essential for calculating other orbital properties.

Planetary motion describes how planets and other celestial bodies move in their orbits around a star. Kepler's laws of planetary motion provide a foundational understanding of these movements.

Kepler's third law states that the square of a planet's orbital period \((P^2)\) is directly proportional to the cube of the semimajor axis of its orbit \((A^3)\). This relationship is expressed as:
\[ \frac{P^2}{A^3}=1.0 \]

For Earth, \((P^2 / A^3 = 1.0)\). This same ratio applies to any planet orbiting the same star, like a new dwarf planet discovered 14 times farther from the Sun than Earth. Regardless of the distance, the ratio \((P^2 / A^3)\) remains constant.

With the discovery of a new body in our solar system:

• Its motion can be predicted using Kepler’s laws.
• The semimajor axis of 14 times the distance of Earth leads to an equivalent orbital period ratio.
• Understanding planetary motion helps us predict planetary positions at various times.

Kepler's third law aids not just in understanding distant planets but also moons and artificial satellites.