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An elliptical orbit is the path followed by an object revolving around another object under the influence of gravity. Most celestial objects, like planets and satellites, travel in elliptical orbits. An ellipse is a stretched circle with two foci, and in this case, one of the foci is the center of the Earth.
In an elliptical orbit, the distance between the orbiting object and the central body varies. This means that, for a spacecraft or a planet, there are points where it is closer to or farther from the central body.

• Apogee: The farthest point from the central body.
• Perigee: The nearest point to the central body.

The path was calculated using Kepler's Third Law, focusing on these distances to understand the movement dynamics around a planet such as Earth.

The orbital period is the time it takes for a celestial body to complete one full orbit around another body. In the context of our problem, it refers to the time taken by the spacecraft to travel from its nearest point to its farthest point from Earth and back again to the nearest point.
Kepler's Third Law plays a crucial role in determining this period. This law states that the square of an object's orbital period is proportional to the cube of the semi-major axis of its orbit. The relationship can be expressed as:

• \[ T^2 = rac{4 \pi^2}{GM} a^3 \]

Where:

• \( T \) is the orbital period
• \( G \) is the gravitational constant
• \( M \) is the mass of the central object, in this case, the Earth
• \( a \) is the semi-major axis

Knowing the semi-major axis allows us to determine the orbital period using this equation.

In the context of an elliptical orbit, the semi-major axis is a vital concept as it represents half of the longest diameter of the ellipse. For a spacecraft traveling from the Earth to the Moon, the semi-major axis is essentially the average distance from the Earth’s center to the nearest and farthest points of the spacecraft's planned path.
The semi-major axis is calculated using the formula:

• \( a = \frac{r_1 + r_2}{2} \)

Where:

• \( r_1 \) is the closest distance to Earth (6670 km)
• \( r_2 \) is the farthest distance, which is the Moon’s distance from Earth (385000 km)

By calculating this average, we gain a better understanding of the orbit shape and size, which is essential for applying Kepler's Third Law to find the orbital period.

The gravitational constant (\( G \)) is a universal value that centralizes the calculations of gravitational forces in the universe. It is one of the key components in Kepler's Third Law, specifically when finding the proportionality constant \( k \) as

• \( k = \frac{4 \pi^2}{GM} \).

\( G \) has a value of \( 6.67430 \times 10^{-11} \, m^3 \, kg^{-1} \, s^{-2} \). This constant is used to calculate the gravitational force between two masses. In our specific problem, it acts alongside the mass of the Earth (\( M = 5.972 \times 10^{24} \, kg \)) in determining the dynamics of the spacecraft's orbit. Without \( G \), it would be impossible to accurately predict the forces and the resulting orbital characteristics of bodies under gravitational attraction. Gravitational constant ensures that we have a universally consistent way to predict movements in celestial mechanics and applies to all objects with mass.