During the latter half of the 1 9 th century, a few astronomers thought there might be a planet circling the Sun inside Mercury's orbit. They even gave it a name: Vulcan. We now know that Vulcan does not exist. If a planet with an orbit one-fourth the size of Mercury's actually existed, what would be its orbital period relative to that of Mercury?

Orbital period refers to the time it takes for a planet or any celestial object to complete one full orbit around its star. Often measured in Earth years, it tells us how long an object takes to circle its star once.
For example, Mercury has an orbital period of about 88 Earth days. This means that Mercury completes one trip around the Sun in 88 days.

Understanding the orbital period is crucial for studying the movement of planets and other objects in space. It helps astronomers make predictions about where these objects will be at any given time.
Using Kepler's Laws, especially the Third Law, we can relate the orbital period to other factors like the semi-major axis of the orbit. This relationship allows us to find the period of any planet if we know its orbit's size.

The semi-major axis is a key element in understanding an object's orbit. It is the longest radius of an elliptical orbit and runs from the center of the orbit to its farthest edge.
To put it simply, if you imagine the planet’s orbit as an oval, the semi-major axis is half the length of the longest line you can draw through the center of this oval.
In mathematical terms, if a planet’s orbit is perfectly circular, then the semi-major axis is the same as the radius of the circle.
For instance, let’s consider Mercury. If we say that Mercury's semi-major axis is 'a_M', and another hypothetical planet, Vulcan, has a semi-major axis one-fourth of Mercury's, we can use this ratio to simplify calculations about Vulcan’s orbit.

Kepler's Third Law is all about proportional relationships. It tells us that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit.
This can be written as: \ T^2 \propto a^3 \
Where \( T \) is the orbital period and \( a \) is the semi-major axis.
Basically, if you know one value, you can find the other. For example, if a planet's orbit is four times smaller (1/4 the size), according to Kepler's Third Law, its orbital period will be: \ (T_V / T_M)^2 = (a_V / a_M)^3 = (1/4)^3 = 1/64 \
Taking the square root: \ (T_V / T_M) = 1/8 \
This shows that if the hypothetical planet Vulcan had an orbit one-fourth the size of Mercury's, its orbital period would be around 1/8th of Mercury’s. Understanding these relationships helps us predict and compare orbital behaviors of different planets easily.