Question

The paths of the planets and certain comets are ellipses, with the sun at one focal point. The path of Halley's comet is an ellipse with a major axis of 36.18 AU and a minor axis of 9.12 AU. An astronomical unit, \(\mathrm{AU}\), is the distance between the earth and the sun, about 92.6 million miles. What is the greatest distance that Halley's comet gets from the sun?

Short answer

The greatest distance that Halley's comet gets from the sun is 18.09 AU, which is approximately 1,676.34 million miles.

Step-by-step solution

Identifying the semi-major and semi-minor axes

The major axis of the ellipse is given as 36.18 AU, which means the semi-major axis (a) is half of this value. Similarly, the minor axis of the ellipse is given as 9.12 AU, so the semi-minor axis (b) is half of the minor axis.

Calculating the semi-major axis

To find the semi-major axis, divide the major axis by 2: \( a = \frac{36.18 \, \mathrm{AU}}{2} \)

Calculating the distance from the sun

Since the sun is at one of the focal points of the elliptical path of Halley's comet, the greatest distance from the sun to the comet occurs at the other end of the major axis, which is just the length of the semi-major axis.

Converting Astronomical Units to miles

To express the greatest distance from the sun in miles, multiply the semi-major axis in AU by the distance of one AU: \( 18.09 \, \mathrm{AU} \times 92.6 \, \text{million miles} \).

Key concepts

Semi-Major and Semi-Minor Axes

When we talk about the path of a planet or a comet through space, we often describe it using the shape of an ellipse. An ellipse resembles an elongated circle and has two main measurements: the semi-major and the semi-minor axes. These two axes are essentially the 'radii' of the ellipse, representing half the length of the longest and the shortest diameters, respectively.

The semi-major axis, denoted as a, is of special importance because it helps us determine the shape of the orbit and the orbital period of the object. The larger the semi-major axis, the larger the ellipse and the longer it takes for a celestial body to complete one orbit around the Sun. On the other hand, the semi-minor axis, denoted as b, helps us understand the ellipse's eccentricity, which is a measure of how much it deviates from being a circle. The relationship between the semi-major and semi-minor axes is also indicative of how elongated an orbit is; the more similar they are in length, the more circular the orbit.

Elliptical Orbits of Comets

Comets are known for their spectacular travels through the solar system, often visible from Earth as they come closer to the Sun. Their orbits are usually highly elliptical, meaning they have a great distance between the farthest and closest approaches to the Sun.

For example, Halley's comet has an elliptical orbit with the Sun at one of its focal points. The significance of this is that at certain times, Halley's comet is very distant from the Sun, while at other times, it's quite near. Understanding a comet's orbit, such as that of Halley's, requires knowing its semi-major and semi-minor axes, which indicate the size and shape of the orbit. The point at which the comet is furthest from the Sun is called the 'aphelion' and the closest point is referred to as 'perihelion'. This extreme elongation of cometary orbits results in their sporadic appearances and long periods spent in the colder regions of our solar system.

Astronomical Unit Conversion

In astronomy, an astronomical unit (AU) is a standard unit of measurement defined as the average distance from the Earth to the Sun, approximately 93 million miles (150 million kilometers). Converting astronomical units to more familiar units, such as miles or kilometers, is essential for understanding the vast distances in space.

To convert AUs to miles, one would multiply the distance in AUs by the equivalent number of miles per AU. In our exercise, the semi-major axis is 18.09 AU, so multiplying this by 92.6 million miles gives us the greatest distance of Halley's comet from the Sun in miles. This conversion allows us to convey the colossal scales involved in celestial mechanics in terms more relatable to the human experience.

Focal Points of an Ellipse

An ellipse has two fixed points known as the focal points. In the context of celestial orbits, one of these focal points is where the Sun is located. The other focal point does not normally have a celestial body present.

The focal points play a crucial role in understanding the dynamics of elliptical orbits. The total distance from any point on the ellipse to the two focal points is constant. This means that as a planet or comet moves along its orbit, it maintains a consistent sum distance to both foci, even as its individual distances to each focus change. This unique property of ellipses is called the focal property, and it's one of the fundamental principles behind Kepler's laws of planetary motion which govern the motions of bodies in space.