Problem 63 - Thomas Calculus

Question

a. The space engineer's formula for eceentricity The space engineer's formula for the eccentricity of an elliptical orbit is $$ e=\frac{r_{\max }-r_{\min }}{r_{\max }+r_{\min }} $$ Drawing ellipses with string You have a string with a knot in each end that can be pinned to a drawing board. The string is 10 in. Iong from the center of one knot to the center of the other. How far apart should the pins be to use the method illustrated in Figure 10.5 (Section 10.1\()\) to draw an ellipse of eccentricity 0.2? The resulting ellipse would resemble the orbit of Mercury.

Step-by-Step Solution

Answer

The pins should be 2 inches apart.

Step 1: Understand the Relationship Between String Length and Semi-Major Axis

The total length of the string used in drawing the ellipse corresponds to the sum of the long axis, also known as the major axis, of the ellipse. Given that the string is 10 inches, this length corresponds to \(2a\), where \(a\) is the semi-major axis of the ellipse.

Step 2: Express a in Terms of Total String Length

Given that \(2a = 10\), we can solve for the semi-major axis \(a\) by dividing both sides by 2. Therefore, \(a = \frac{10}{2} = 5\) inches.

Step 3: Use Eccentricity Formula to Relate the Axes

The eccentricity formula \(e = \frac{r_{\max} - r_{\min}}{r_{\max} + r_{\min}}\) simplifies to \(e = \frac{a - b}{a + b}\), where \(a\) and \(b\) are the semi-major and semi-minor axes. For an ellipse, the distance between the pins (the foci) is \(2c\), where \(c\) is related to \(a\) and \(b\) via \(c = \sqrt{a^2 - b^2}\).

Step 4: Express c in Terms of Eccentricity and Semi-Major Axis

Given the eccentricity \(e = 0.2\), we can express \(c\) as \(c = ae = 5 \times 0.2 = 1\). Thus, \(c = 1\) inch.

Step 5: Determine Distance Between the Pins

Since each pin represents a focus of the ellipse and they are separated by \(2c\), the distance between the pins is \(2 \times 1 = 2\) inches.

Key Concepts

Semi-Major Axis

The semi-major axis is a crucial part of understanding elliptical shapes, especially when applied to space engineering. The semi-major axis (usually represented by \(a\)) is the longest diameter of an ellipse. It stretches from one end of the ellipse to the other, passing through the center. This concept is vital in determing the overall size of the ellipse and is often associated with the path a celestial body takes as it orbits.When we talk about ellipses in space, such as planetary orbits, the length of the semi-major axis tells us about the scope of the orbit. This axis reflects the average distance from the focal point, commonly a star or planet, to the orbiting body. In practical scenarios, like drawing an ellipse using string as shown in the original exercise, the string's total length represents the major axis—twice the semi-major axis (i.e., \(2a\)). So if our string is 10 inches, then the semi-major axis measures 5 inches. It provides a foundational understanding to solve for eccentricity and other parameters.

Semi-Minor Axis

In any ellipse, the semi-minor axis (noted as \(b\)) complements the semi-major axis. While the semi-major axis defines the maximum stretch of the ellipse, the semi-minor axis describes the shortest diameter that also passes through the center. Together, these axes define the ellipse's shape and dimensionality.The ellipse's eccentricity, which measures how stretched or elongated it is, can be expressed in terms of both the semi-major and semi-minor axes. The smaller the value of the eccentricity, the closer the shape is to a circle, meaning both axes would be closer in length.Conceptually, if an elliptical orbit's eccentricity formula \(e = \frac{a-b}{a+b}\) is applied, the semi-minor axis comes into play to describe the orbit's distortion from a perfect circle. When plotting an ellipse with string and pins as per the exercise, knowing both semi-major and semi-minor axes helps in determining the positions of the foci (pin positions), and subsequently, the eccentricity and shape of the resulting ellipse.

Space Engineering

Space engineering uses concepts of elliptical orbits extensively, as most celestial orbits are not perfect circles but ellipses. In this context, both the semi-major and semi-minor axes play vital roles in designing and understanding the paths objects take in space. Space engineers calculate eccentricity to plan satellite launches, predict orbital periods, and determine the influence of gravitational forces. The semi-major axis is crucial in these calculations. For instance, spacecraft orbits around planets must be planned considering the semi-major axis and eccentricity to ensure the craft maintains a stable path. By plotting an ellipse's axes, engineering calculations for orbit paths become more straightforward. For example, this helps in optimally positioning satellites to provide the best coverage and efficiency. Thus, through understanding the geometric principles of ellipses, space engineers can develop robust and efficient spatial trajectories.