Problem 70 - College Algebra

Question

Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two ellipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles? Earth's orbit: \(\quad\) Length of major axis: 186 Length of minor axis: 185.8 million miles Mars's orbit: Length of major axis: 283.5 Length of minor axis: 278.5 million miles

Step-by-Step Solution

Answer

The equations for the elliptical orbits of Earth and Mars are \(\frac{x^2}{93^2} + \frac{y^2}{92.9^2} = 1\) and \(\frac{x^2}{141.75^2} + \frac{y^2}{139.25^2} = 1\) respectively.

Step 1: Determine the semi-major axis and semi-minor axis

The semi-major axis \(a\) and semi-minor axis \(b\) are half of the length of the major and minor axis, respectively. For Earth, \(a = 186/2 = 93\) million miles and \(b = 185.8/2 = 92.9\) million miles. For Mars, \(a = 283.5/2 = 141.75\) million miles and \(b = 278.5/2 = 139.25\) million miles.

Step 2: Write the equations for the ellipses

Following the standard equation of an ellipse, the equation of Earth's orbit is \(\frac{x^2}{93^2} + \frac{y^2}{92.9^2} = 1\) and the equation for Mars's orbit is \(\frac{x^2}{141.75^2} + \frac{y^2}{139.25^2} = 1\).

Step 3: Graph the two ellipses using a graphing utility

Typically it would be necessary to insert these equations into a graphing utility and observe both orbits graphically in the same viewing rectangle. The scales along x and y axes should be kept in millions of miles.

Step 4: Explain the difficulty for early astronomers

After plotting the graphs, it quickly becomes evident that these orbits, despite being elliptical, appear almost circular due to the minor difference between the semi-major and semi-minor axes. This would make it difficult for early astronomers to accurately detect their elliptical nature.

Key Concepts

Major and Minor Axis

The major and minor axes are the longest and shortest diameters of an ellipse, respectively. They intersect at the center of the ellipse, and their lengths are key in defining its shape. The major axis runs along the widest part of the ellipse, while the minor axis is perpendicular to it and spans the narrowest part. In the context of planetary orbits, such as those of Earth and Mars, the length of these axes can be vast, with millions of miles in between. When determining an ellipse from these measurements, what's actually needed are the semi-major and semi-minor axes, which are half the lengths of the major and minor axes, respectively. This reflects the nature of an ellipse being symmetrical about its center. For example, the Earth’s orbit has a semi-major axis of 93 million miles and a semi-minor axis of 92.9 million miles.

Understanding the relationship between the major and minor axes not only helps in writing the equation of an ellipse but also is essential in scientific fields such as astronomy, where it is used to describe planetary orbits.

Standard Equation of an Ellipse

The standard equation of an ellipse is a mathematical representation that enables us to plot its shape on a coordinate plane. It is written in the form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] where \(a\) represents the length of the semi-major axis and \(b\) represents the length of the semi-minor axis. If the ellipse is oriented such that the longer axis is horizontal, \(a\) and \(b\) are used in the denominators under \(x^2\) and \(y^2\), respectively. Alternatively, if the longer axis is vertical, the denominators are reversed. The right side of the equation being equal to 1 signifies that all the points \((x, y)\) that satisfy the equation will lie on the ellipse.

Using the given values for the major and minor axes of Earth's and Mars's orbits, we came up with their respective standard equations. By properly understanding and applying this equation, one can accurately describe and graph any ellipse as long as the lengths of the semi-major and semi-minor axes are known.

Graphing Utility

A graphing utility is a tool, often software-based, that allows users to plot graphs, visualize mathematical functions and data sets, and analyze statistical information. When it comes to ellipses, graphing utilities are incredibly useful for visualizing their shapes and positions, particularly when the differences between the major and minor axes are subtle. With the equations of the ellipses derived from the major and minor axes of Earth and Mars, a graphing utility can render the orbits visually. By inputting the standard equations into the utility and setting the appropriate scale, students can observe how the orbits differ and resemble each other.

For example, by graphing Earth's and Mars's orbits using the standard equations derived in the solutions, it becomes clear why early astronomers faced challenges in discerning the elliptical nature of the orbits. The visually similar, near-circular appearance made it difficult to establish the true shape without meticulous observation and measurement. Thus, graphing utilities not only aid in educational purposes but also in historical analysis of scientific observations.