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The semimajor axis of an orbit in astronomy is a crucial concept as it defines the size of the orbit. For elliptical orbits, which most bodies in our solar system follow, the semimajor axis is half of the longest diameter of the ellipse. This is a central parameter when it comes to calculating various orbital properties, including the orbital period and the distances at various points around the orbit.
Kepler's third law provides a simple relationship between the semimajor axis and the orbital period. It states that the square of the period of orbit (T^2) is proportional to the cube of the semimajor axis (a^3). For orbits with the period measured in Earth years and the semimajor axis in astronomical units (AU), Kepler's third law simplifies and becomes\(\left(\frac{a}{1 AU}\right)^3 = \left(\frac{T}{1 yr}\right)^2\).
In the context of Halley's comet, which has a period of 76.2 years, applying Kepler's law gives us a semimajor axis of approximately 17.9 AU.

Eccentricity measures how much an orbit deviates from being a perfect circle. It is a dimensionless parameter that ranges from 0 to 1, where 0 indicates a circular orbit and values closer to 1 indicate more elongated orbits. Knowing the eccentricity helps us understand the shape and dynamics of an orbit.
The formula to calculate eccentricity (e) is derived from the distances of closest approach (perihelion, q) and furthest point (aphelion, Q) from the Sun:\(e = \frac{Q-q}{Q+q}\)
For Halley's comet, where the perihelion distance is 0.56 AU and the aphelion distance is approximately35.24 AU, we find the eccentricity to be about 0.97. This high eccentricity suggests that Halley's comet travels a highly elongated path around the Sun, spending much of its time far away from the Sun, only coming close every 76.2 years.

Astronomical Units (AU) are a standard unit of measurement used in astronomy to describe distances within our solar system. One AU is defined as the average distance between the Earth and the Sun, approximately 149,597,870.7 kilometers. This unit allows astronomers to easily express and compare distances in the vast expanses of space.
Using AU simplifies many calculations and allows for easier comprehension of relative distances. When the semimajor axis of an orbit is expressed in terms of AU, it relates directly to Earth’s orbit size, offering an intuitive comparison.
In the study of Halley's comet, its semimajor axis being 17.9 AU means it is 17.9 times further from the Sun than the average Earth-Sun distance, indicating extensive travel through space both near and far from our planetary neighborhood.