Question

Halley's Orbit. Halley's comet orbits the Sun every 76.0 years. (a) Find its semimajor axis distance. (b) Halley's orbit is an extremely eccentric (stretched-out) ellipse, so at perihelion it is only about 90 million kilometers from the Sun, compared to more than 5 billion kilometers at aphelion. Does Halley's comet spend most of its time near its perihelion distance, near its aphelion distance, or halfway in between? Explain.

Short answer

(a) Semimajor axis is 17.94 AU. (b) Halley's comet spends most of its time near aphelion.

Step-by-step solution

Understanding Kepler's Third Law

Kepler's Third Law relates the period of an orbit (T) to its semimajor axis (a), given by the formula \(T^2 = a^3\), where T is in Earth years, and a is in Astronomical Units (AU).

Solving for the Semimajor Axis

Halley's comet has an orbital period of 76.0 years. Using Kepler's Third Law, substitute T = 76 to find \(a\): \[(76)^2 = a^3 \ 5776 = a^3 \]Take the cube root to solve for \(a\): \[a = 5776^{1/3}\approx 17.94 \, \text{AU} \]

Understanding Orbit Distance Distribution

Halley's comet follows an elliptical orbit with its perihelion at 0.6 AU (90 million kilometers) and aphelion at 34.4 AU (5 billion kilometers). Generally, objects in highly elliptical orbits spend more time near aphelion due to slower velocities there, as described by Kepler's Second Law.

Explanation Using Kepler's Second Law

According to Kepler's Second Law, a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Halley's comet moves faster when it is closer to the Sun and slower when it is further away. This means it spends most of its time near aphelion, because it travels more slowly at that point.

Key concepts

semimajor axis

The semimajor axis is a critical aspect in understanding orbits. It is essentially the longest radius of an elliptical orbit and, when we speak about orbits in astronomy, it is often used to describe the size or scale of the orbit. The semimajor axis is half of the major axis, the line running through the longest part of the ellipse.
This value is crucial because Kepler's Third Law uses it to relate the orbital period of an object around a sun or planet. For instance, Halley's comet has a semimajor axis determined by its 76-year orbital period, which calculates to approximately 17.94 Astronomical Units (AU).
To better understand how this translates, one AU is the average distance from the Earth to the Sun, about 149.6 million kilometers. So, Halley's semimajor axis is nearly 18 times the distance between the Earth and the Sun.

orbital period

The orbital period is the time an object takes to make one complete orbit around another object. For planets and comets, it is often measured in Earth years. Understanding the orbital period helps astronomers predict the path and positioning of celestial bodies over time.
Halley's comet, with an orbital period of 76 years, makes less frequent appearances in our night sky, which in itself connects to how eccentric and elongated its orbit is. The orbital period directly influences how much time the comet spends near various points in its orbit, affecting its visibility from Earth.
It's fascinating to see that Kepler's Laws allow us to calculate essential details about celestial bodies, like how far they travel from the sun, using just the time it takes them to orbit.

elliptical orbit

An elliptical orbit is not a perfect circle; instead, it is an elongated oval, consisting of two focal points, one of which is usually occupied by the central body being orbited, such as the Sun in our Solar System. This is a core concept of planetary motion. An object in an elliptical orbit like Halley's comet doesn't maintain a constant speed because its distance from the Sun varies, according to its position along the ellipse. The closer it is to the Sun (the perihelion), the faster it moves. Conversely, it slows down significantly when it reaches the farthest point from the Sun (the aphelion).
This characteristic results from Kepler's Second Law of planetary motion, which states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This dynamic causes objects in highly elliptical orbits to spend longer durations at greater distances from the sun.

perihelion and aphelion distances

In elliptical orbits, the terms perihelion and aphelion are used to describe the closest and farthest points from the Sun, respectively. These distances are crucial in understanding an orbit's shape and how the object moves during its journey. In the case of Halley's comet, at perihelion, it comes as close as 0.6 AU from the Sun, which is about 90 million kilometers. In contrast, at aphelion, it travels to a staggering distance of 34.4 AU, equivalent to about 5 billion kilometers. Due to these extreme distances, and pursuant to Kepler's Second Law, Halley's comet travels much slower at aphelion compared to perihelion. This slower pace means that the comet spends most of its orbital period far from the Sun. Understanding these points, scientists can predict when and where comets become visible as they approach their nearest points to us on Earth.