Sun-grazing comets come so close to the Sun that their perihelion distances are essentially zero. Find the orbital periods of Sun-grazing comets whose aphelion distances are (a) 100 \(\mathrm{AU}\), (b) \(1000 \mathrm{AU}\), (c) \(10,000 \mathrm{AU}\), and (d) \(100,000 \mathrm{AU}\). Assuming that these comets can survive only a hundred perihelion passages, calculate their lifetimes. (Hint: Remember that the semimajor axis of an orbit is one-half the length of the orbit's long axis.)

Kepler's third law, also known as the Law of Harmonies, is a foundational principle that helps us understand the motion of planets and other celestial bodies around the sun. It states that the square of the orbital period (\(T\text{ in years}\)) of a planet is directly proportional to the cube of the semimajor axis (\(a\text{ in AU}\)) of its orbit. Mathematically, it is represented as \[ T^2 = a^3 \]
For objects within our solar system, this law allows us to predict their orbital behaviors once we know their distance from the Sun. When we apply Kepler's third law to comets, particularly to Sun-grazing comets, we can calculate how long it takes them to complete one orbit around the Sun. Keep in mind, the semimajor axis (a) is half the length of the orbit's major axis, which is important to consider when dealing with comets that have very elongated orbits.

An astronomical unit (AU) is a unit of measurement equivalent to the average distance from the Earth to the Sun, approximately 149.6 million kilometers or 93 million miles. This unit provides a convenient way to express and relate distances within our solar system.
When scientists discuss the orbits of comets, like the Sun-grazing comets in our exercise, they often use the AU to describe perihelion (closest approach to the Sun) and aphelion (farthest distance from the Sun) points. In our calculations, we treat the Earth's orbit as the basic yardstick for measuring celestial distances, where 1 AU is the baseline for using Kepler's third law. This is because the Earth's orbital period is one year, and its semimajor axis is defined as 1 AU.

Perihelion and aphelion distances refer to the closest and farthest points, respectively, in the orbit of a celestial body around the Sun. The perihelion distance is the point where the body, such as a comet or planet, is nearest to the Sun, while the aphelion is the point where it's most distant.
The Sun-grazing comets in our exercise have perihelion distances close to zero, implying they pass extremely close to the Sun. In contrast, their aphelion distances can be astoundingly large, as indicated in the exercise with comets reaching as far as 100,000 AU from the Sun. These distances crucially affect the shape and duration of the orbit, as longer aphelion distances result in more elongated orbits and longer orbital periods according to Kepler's third law.

Comet lifetimes are determined by a variety of factors, most notably by the number of orbits they can complete before losing material to the point of disintegration or by being ejected from the solar system. The close approach to the Sun, or perihelion passage, is particularly harsh due to intense solar radiation and gravitational forces. Each pass can erode the comet's material through a process called outgassing.
According to the exercise, Sun-grazing comets can survive up to a hundred perihelion passages. Their lifetimes are calculated by multiplying the number of survivable orbits by the comet's orbital period. For example, if a comet's orbital period is 500 years and it can survive 100 orbits, its lifetime will be \( 500 \text{ years/orbit} \times 100 \text{ orbits} = 50,000 \text{ years} \)To determine the lifespan of various Sun-grazing comets, we consider their orbital periods in relation to their semimajor axes and use Kepler's laws to predict the sustainability of their cosmic journeys.