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Eccentricity is a measure that describes how much an orbit deviates from being circular. In celestial mechanics, eccentricity ( \(e\) ) helps determine the shape of the orbit.

• If \(e = 0\) , the orbit is a perfect circle.
• If \(0 < e < 1\) , the orbit is elliptical.
• If \(e = 1\) , the orbit is parabolic.
• If \(e > 1\) , the orbit is hyperbolic.

In this particular problem, the given eccentricity is \(e = 1.057\) , which is greater than 1. This hyperbolic eccentricity indicates that the comet C/1980 E1 is on an escape trajectory. It implies that it is not gravitationally bound to the Sun and, therefore, is not expected to return once it passes by. The hyperbolic trajectory means the comet was likely a visitor from outside the solar system.

The perihelion is the point in the orbit of a celestial body (such as a comet or planet) where it is closest to the Sun. It plays a crucial role in determining various parameters of the orbit, including the semi-major axis.
For the comet C/1980 E1, the perihelion is given as \(3.364 \, \mathrm{AU}\) (Astronomical Units). This is the closest distance between the comet and the Sun during its orbit.
Knowing the perihelion distance provides insights into how close the object will get to the Sun. In this case, because the comet has a high eccentricity and is following a hyperbolic trajectory, the perihelion represents just one point along its path as it swings close to the Sun before continuing out into space, not to return.

The semi-major axis is one half of the longest diameter of an ellipse. It plays a significant role in defining the size of the orbit.
The relationship between the semi-major axis \(a\) , perihelion \(q\) , and eccentricity \(e\) for hyperbolic trajectories is given by: \[ q = a(1 - e) \]Solving for \(a\) with \(q = 3.364 \, \mathrm{AU}\) and \(e = 1.057\) , we get: \[ a = \frac{3.364}{1 - 1.057} = \frac{3.364}{-0.057} \approx -59.03 \, \mathrm{AU} \]The negative value of \(a\) reflects the nature of the hyperbolic orbit, indicating the path extends infinitely away from the Sun. Unlike elliptical orbits, semi-major axis in hyperbolic orbits doesn't signify a physical length but rather a way to quantify the trajectory's escape characteristics.

The semi-minor axis is the shortest radius along an ellipse or hyperbola. Although not present in non-elliptical paths, it helps in the mathematical representation of these trajectories.
In hyperbolic trajectories, the semi-minor axis \(b\) is related to the semi-major axis \(a\) and eccentricity \(e\) by the equation: \[ b = a \sqrt{e^2 - 1} \]Using the previously calculated \(a = -59.03 \, \mathrm{AU}\) and \(e = 1.057\), we find: \[ b \approx -59.03 \times \sqrt{0.116449} \approx -59.03 \times 0.3413 \approx -20.14 \, \mathrm{AU} \]Again, the negative sign simply underscores the characteristics of a hyperbolic orbit, pointing to the fact that unlike circles and ellipses, hyperbolas do not enclose an area but extend infinitely.