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An elliptical orbit is a path followed by celestial objects where the shape of the path is an ellipse. This is quite common in our solar system. Planets, moons, and satellites often follow elliptical paths around larger celestial bodies, like the Sun or Earth. An ellipse is a type of rounded shape but is more stretched out compared to a perfect circle.
The distinguishing feature of an elliptical orbit is that it has two foci (the plural of focus). For Sputnik's orbit, one focus is the Earth’s center. The reason elliptical orbits are significant is due to gravitational forces. These forces cause the object to move faster when closer to the focus and slower when farther away.
Understanding elliptical orbits helps us predict the position and speed of orbiting bodies. This makes them vital for calculations in astronomy and space navigation.

The semi-major axis is one of the defining characteristics of an ellipse. It is the longest radius of an ellipse, extending from its center to the perimeter. In the context of orbits, it represents the average distance between the orbiting body and the body it orbits.
In our exercise with Sputnik, the semi-major axis is calculated using the maximum and minimum distances from the Earth, also known as apogee and perigee. The formula for the semi-major axis is \[ a = \frac{\text{maximum distance} + \text{minimum distance}}{2} \]
Plugging in the distances, \[ a = \frac{583 + 132}{2} = 357.5 \text{ miles} \]
The semi-major axis provides insights into many orbital characteristics, including orbital period and speed. This is a key parameter in both the structure of elliptical orbits and calculations involving gravitation.

Linear eccentricity is a measure of how much an ellipse deviates from being circular. It is essentially the distance between the center of the ellipse and one of its foci. A perfect circle has an eccentricity of zero, indicating no deviation at all.
In the exercise above, the linear eccentricity, denoted as \( c \), is calculated using the formula:\[ c = a - \text{(radius of Earth - minimum distance)} \] Plugging in the known values, we find: \[ c = 4000 + 132 - 357.5 = 3774.5 \text{ miles} \]
This correction in calculating \( c \) is crucial because any negative value in an eccentricity context points to miscalculation, as it physically should represent a positive distance. The goal of this calculation is to further determine eccentricity \( e \), by using the formula:\[ e = \frac{c}{a} \]
Understanding and correctly calculating the linear eccentricity is critical as it denotes how elongated the orbit is, influencing orbital dynamics and motion prediction.