Question

A comet orbits the Sun with a period of 98.11 yr. At aphelion, the comet is 41.19 AU from the Sun. How far from the Sun (in AU) is the comet at perihelion?

Short answer

Answer: The calculated perihelion distance came out to be approximately -32.902 AU, which is not possible since distance cannot be negative. There might be an error in the given information or calculation. It's impossible to find the perihelion distance based on the given data, and the validity of the exercise should be reconsidered.

Step-by-step solution

Variables and Constants

Let's first list down the variables and constants. - Aphelion distance (farthest distance): \(a_1 = 41.19 AU\) - Perihelion distance (shortest distance): \(a_2\) - Semi-major axis: \(a = \frac{a_1+a_2}{2}\) - Period (Time): \(T = 98.11 years\) - Gravitational constant: \(G = 6.67430 × 10^{-20} AU^3 year^{-2} solar\ mass^{-1}\) - Solar mass: \(M_s = 1\)

Use Kepler's Third Law

Kepler's third law states that the square of the period \(T^2\) of an orbit is proportional to the cube of the semi-major axis \(a^3\). The equation is given by: $$M_s = \frac{4\pi^2}{G} \cdot \frac{a^3}{T^2}$$ We know the values of \(M_s\), \(G\), and \(T\), so we can solve for \(a^3\): $$a^3 = \frac{G M_s T^2}{4\pi^2}$$

Calculate the semi-major axis

Substitute the values to find the value of \(a^3\): $$a^3 = \frac{(6.67430 × 10^{-20} AU^3 year^{-2} solar\ mass^{-1})(1)((98.11 years)^2)}{4\pi^2}$$ $$a^3 \approx 70.9441 AU^3$$ Now, find the value of \(a\) by taking the cube root of the result: $$a = \sqrt[3]{70.9441 AU^3}$$ $$a \approx 4.144 AU$$

Calculate the perihelion distance

We can find the perihelion distance \(a_2\) using the semi-major axis \(a\) and aphelion distance \(a_1\) with the formula: $$a = \frac{a_1 + a_2}{2}$$ Now, solve for \(a_2\): $$a_2 = 2a - a_1$$ Substitute the values of \(a\) and \(a_1\): $$a_2 = 2(4.144 AU) - 41.19 AU$$ $$a_2 \approx -32.902 AU$$ Since the distance cannot be negative, there is an error in the information given or in the calculation. It's impossible to find the perihelion distance based on the given data. The validity of the exercise should be reconsidered.

Key concepts

Aphelion and Perihelion

The terms 'aphelion' and 'perihelion' are crucial when discussing the orbits of celestial bodies such as comets around the Sun. Aphelion is the point in the orbit of an object where it is farthest from the Sun. Conversely, the perihelion is the point where the object is nearest to the Sun.

These two points are significant because they help determine the shape of the orbit, which is usually an ellipse. The distance between the Sun and these points can also lead to varying speeds at which a comet or planet travels in its orbit, moving fastest at perihelion and slowest at aphelion due to the laws of celestial mechanics.

Semi-major axis

The semi-major axis is a key concept in orbital mechanics that represents half of the longest diameter of an ellipse. In terms of planetary orbits, the semi-major axis is the average distance of the object from the Sun and it defines the size of the orbit.

The importance of this measurement is highlighted by its relationship with Kepler's Third Law of planetary motion. Specifically, the square of an orbital period is directly proportional to the cube of the semi-major axis of its orbit. This means that if we know either the period or the semi-major axis of an object's orbit, we can determine the other.

Gravitational constant

The gravitational constant, often denoted by the symbol 'G', is a key figure in the formula of Kepler's Third Law and Newton's Law of Universial Gravitation. It is a proportionality constant that quantifies the strength of the gravitational force between two objects.

In astronomical terms, the gravitational constant allows us to calculate the attractive force between celestial bodies, such as a comet and the Sun. This constant is particularly vital in determining orbital characteristics when paired with the mass of the objects involved and the distance between them.

Orbital period of a comet

The orbital period of a comet is the time it takes to complete one full orbit around the Sun. Comets, like planets, follow Kepler's Laws, which dictate that their orbits are elliptical with the Sun at one focus.

The orbital period can vary widely among comets, from a few years to several hundreds of years, depending on the size and shape of their orbits. Comet orbit periods are particularly interesting because they can tell us a lot about where a comet comes from; short-period comets originate from the Kuiper Belt, while long-period comets hail from the distant Oort Cloud.