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

Halley's comet's minimum distance from the sun is 0.57 AU.

Step 1: Identify the semi-major and semi-minor axes

The major diameter of the orbit is given as 36.18 AU. To find the semi-major axis \(a\), divide the major diameter by 2: \(a = \frac{36.18}{2} = 18.09\, \text{AU}.\)Similarly, the minor diameter is 9.12 AU, hence the semi-minor axis \(b\) is \(b = \frac{9.12}{2} = 4.56\, \text{AU}.\)

Step 2: Use the formula for distance from the center to a focus

In an elliptical orbit, the distance from the center to a focus \(c\) can be found using the equation \(c = \sqrt{a^2 - b^2}\).Substitute the values from Step 1 to find \(c\):\[ c = \sqrt{18.09^2 - 4.56^2} \approx \sqrt{327.6281 - 20.7936} = \sqrt{306.8345}\approx 17.52\,\text{AU}.\]

Step 3: Calculate the minimum distance from the sun (perihelion)

The minimum distance from the sun to a point on the ellipse (perihelion) is given by subtracting \(c\) from \(a\).Thus, the minimum distance is \(a - c\):\[ 18.09 \text{ AU} - 17.52 \text{ AU} = 0.57 \text{ AU}.\]

Key Concepts

Semi-Major Axis

The semi-major axis is a crucial element in understanding elliptical orbits. Imagine an ellipse as a stretched circle; the semi-major axis is the longest radius of this shape, spanning from the center to the farthest perimeter point. To find it, you take the major diameter and divide it by two. For Halley's comet, with a major diameter of 36.18 Astronomical Units (AU), the semi-major axis is calculated as follows:

• Formula: semi-major axis, \( a = \frac{\text{major diameter}}{2} \)
• Calculation: \( a = \frac{36.18}{2} = 18.09 \text{ AU} \)

Understanding this concept is essential as it plays a vital role in determining the properties of the orbit, including the perihelion and the overall shape of the orbit.

Semi-Minor Axis

The semi-minor axis is another fundamental component of an ellipse. It is the shortest radius, running perpendicular from the center of the ellipse to its edge. It is crucial for calculating how much the ellipse deviates from being circular.

• Formula: semi-minor axis, \( b = \frac{\text{minor diameter}}{2} \)
• For Halley's comet: \( b = \frac{9.12}{2} = 4.56 \text{ AU} \)

Together with the semi-major axis, the semi-minor axis helps to better describe the geometric shape of the orbit. Larger discrepancies between these axes suggest a more elongated, less circular orbit.

Astronomical Unit

An Astronomical Unit (AU) is a standard unit of measurement in astronomy, used to express distances within our solar system and defined as the average distance from the Earth to the Sun, roughly 149.6 million kilometers (or about 93 million miles). It provides a manageable way to describe and compare astronomical distances without resorting to massive numbers.

• This unit simplifies understanding distances in space, making it easier to discuss an object's position relative to the sun or other planets.
• Using AU standardizes discussion about celestial distances, promoting consistency in astronomical observations and calculations across the globe.

By using AU, scientists and hobbyists alike can appreciate the vastness of the solar system without getting bogged down in cumbersome figures.

Perihelion Distance

In celestial mechanics, the term perihelion refers to the point in an object's elliptical orbit that is closest to the Sun. Calculating perihelion distance is crucial, particularly for understanding how temperature and solar radiation affect an orbiting body at its closest approach.

• The minimum distance, or perihelion, is found by subtracting the distance from the center to a focus (denoted \( c \)) from the semi-major axis (\( a \)).
• For Halley's comet: given \( a = 18.09 \text{ AU} \) and \( c \approx 17.52 \text{ AU} \), the perihelion distance is \( a - c = 0.57 \text{ AU} \).

Understanding perihelion distance is pivotal in predicting climatic impacts and potential changes in an object's visibility and observational characteristics when it's close to or far from the Sun.