Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Halley's comet has an elliptical orbit with major and minor diameters of \(36.18 \mathrm{AU}\) and \(9.12 \mathrm{AU}\), respectively ( \(1 \mathrm{AU}\) is 1 astronomical unit, the earth's mean distance from the sun). What is its minimum distance from the sun (assuming the sun is at a focus)?

Step-by-Step Solution

Answer

The minimum distance from the Sun to Halley's comet is approximately 0.58 AU.

Step 1: Understanding the Problem

We are given the major and minor axes of Halley's orbit. To find the minimum distance from the Sun, we need to determine where the closest point in the orbit (perihelion) is, assuming the Sun is located at one of the foci of the ellipse.

Step 2: Ellipse Properties

An ellipse's major axis is the line through the longest part of the ellipse. Its minor axis is perpendicular to the major axis at the center. In an ellipse, the sum of distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis.

Step 3: Calculating the Semi-Major and Semi-Minor Axes

The semi-major axis is half the length of the major axis, and the semi-minor axis is half the length of the minor axis. Thus,\[a = \frac{36.18}{2} = 18.09 \text{ AU}\]and \[b = \frac{9.12}{2} = 4.56 \text{ AU}.\]

Step 4: Calculating the Distance to the Foci

The distance from the center to a focus, known as the linear eccentricity \(c\), is found using the equation \(c = \sqrt{a^2 - b^2}\). Substituting the values:\[c = \sqrt{18.09^2 - 4.56^2} = \sqrt{327.2481 - 20.7936} = \sqrt{306.4545} \approx 17.51 \text{ AU}.\]

Step 5: Calculating the Minimum Distance

The minimum distance from the Sun to Halley's comet, or the perihelion distance, is equal to the semi-major axis length minus the distance to the foci \(c\). Thus,\[d_{min} = a - c = 18.09 - 17.51 = 0.58 \text{ AU}.\]

Key Concepts

Ellipse Properties

When dealing with orbits in celestial mechanics, ellipses play a vital role. An ellipse is a closed curve, where the sum of the distances from any point on the ellipse to two fixed points, known as foci, remains constant. This unique property is central to understanding the shape and behavior of planetary orbits.

• The **major axis** is the longest diameter, passing through the center and both foci of the ellipse. It's essentially the widest part of the ellipse.
• The **minor axis** is perpendicular to the major axis at the center, representing the narrowest part of the ellipse.
• The **semi-major axis** is half the length of the major axis, while the **semi-minor** is half of the minor axis's length.

In the context of orbital mechanics, these axes help define the shape and size of an orbit, crucial for calculating distances in space travel.

Orbital Mechanics

Orbital mechanics is the study of how objects behave when they are in orbit, especially under the influence of a central gravitational force, like the Sun's gravity. This field combines principles of physics and calculus to predict orbital paths.

• Objects in space move in elliptical paths, with one focal point often occupied by a large mass like a planet or star. This strict arrangement obeys Kepler's laws of planetary motion.
• The physics involved simplify complex movements into predictable models, thanks to the mathematical properties of ellipses.
• Understanding these mechanics allows scientists to predict the motion of planets and comets, essential for planning space missions and studying solar system dynamics.

Orbital mechanics give us a framework for interpreting the celestial ballet of objects moving through space, such as Halley's comet on its return journey to the Sun.

Astronomical Units

The concept of an Astronomical Unit (AU) is essential in the field of astronomy. An AU is the average distance between the Earth and the Sun, approximately 149.6 million kilometers (92.96 million miles).

• Using AU as a unit simplifies astronomical calculations and discussions because it relates distances within the solar system to a familiar scale.
• It provides a convenient unit for expressing vast distances, eliminating the need for large, cumbersome numbers.
• Measuring distances between celestial bodies in AU is standard practice, and it helps communicate concepts clearly to those studying astronomy.

For Halley's comet, expressing its orbital dimensions in AU gives scientists and enthusiasts alike a more intuitive understanding of its trajectory and distance relative to Earth.

Perihelion Distance

Perihelion distance refers to the closest point of an object's orbit to the Sun, a crucial factor in understanding the heating and solar characteristics experienced by the body. This measurement stems from the properties of elliptical orbits.

• For comets like Halley's, determining the perihelion is critical as it affects solar radiation exposure, which can impact a comet's physical properties.
• The perihelion distance can be calculated by subtracting the linear eccentricity from the semi-major axis, as these values denote the object's closest approach.
• This distance plays a role in predicting future observational and orbital dynamics, such as when inquisitive astronomers might next view Halley's comet from Earth.

Knowing the perihelion distance helps us comprehend how close celestial bodies come to the Sun, allowing us to infer more about their orbits and interactions with solar energy.