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Kepler's Third Law provides a relationship between the orbital period of a planet and its average distance from the sun. This law is crucial for understanding the motion of satellites.
In mathematical terms, Kepler's Third Law is expressed as:

• For circular orbits: \( T^2 = \frac{4\pi^2 r^3}{GM_E} \)

Here:

• \( T \) is the orbital period of the satellite.
• \( r \) is the radius of the orbit.
• \( G \) is the gravitational constant.
• \( M_E \) is the mass of the Earth.

The law suggests that the square of the orbital period \( T \) is directly proportional to the cube of the semi-major axis \( r \), in this case, the radius for circular orbits.
The problem we encounter involves finding the radius given the period, and this law is the key starting point to solve the puzzle. By rearranging the formula, you find the radius \( r \) of the satellite's orbit once the period is known.

Once we know the radius of the satellite's orbit, calculating the orbital speed becomes straightforward.
The speed \( v \) is determined by the formula:

• \( v = \frac{2\pi r}{T} \)

Where:

• \( v \) is the orbital speed.
• \( r \) is the radius of the satellite's orbit.
• \( T \) is the orbital period.

This formula is derived by considering the circular path a satellite takes around the Earth. The distance traveled is the circumference of the orbit \( 2\pi r \), which is over a time period \( T \).
Substituting the previously found radius and known period into this equation will give you the satellite's speed. The speed reflects how fast the satellite is moving to maintain its circular path against Earth's gravitational pull.

The gravitational constant, denoted by \( G \), is essential in the calculations of orbital motion for celestial bodies.
It is a key value in Newton's law of universal gravitation:

• \( G = 6.674 \times 10^{-11} \text{ Nm}^2/\text{kg}^2 \)

This constant helps determine the gravitational attractive force between two masses. Specifically, when involved in orbital mechanics, it enables us to quantify the force keeping a satellite in orbit around a planet, like Earth.
The gravitational constant appears in Kepler's Third Law when calculating the radius of the satellite's orbit. It is among the components that allow us to relate the period of the orbit to the mass of Earth and the actual distance from the Earth to the satellite. Without this constant, calculating accurate orbital characteristics would be impossible.