Problem 91 - A Graphical Approach to College Algebra

Question

Solve each application. Orbit of Earth The orbit of Earth is an ellipse with the sun at one focus. The distance between Earth and the sun ranges from 91.4 to 94.6 million miles. (Source: The World Almanac and Book of Facts.) Estimate the eccentricity of Earth's orbit.

Step-by-Step Solution

Answer

The eccentricity of Earth's orbit is approximately 0.0172.

Step 1: Understanding Eccentricity

Eccentricity (\( e \)) measures how much an orbit deviates from being circular. For an ellipse, it is given by the formula \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus, and \( a \) is the semi-major axis.

Step 2: Identifying Parameters

The minimum distance to the sun (\( 91.4 \) million miles) is the perihelion, and the maximum distance (\( 94.6 \) million miles) is the aphelion. The semi-major axis \( a \) is the average of these distances.

Step 3: Calculating the Semi-Major Axis

Calculate the semi-major axis: \[ a = \frac{91.4 + 94.6}{2} = 93 \text{ million miles} \]

Step 4: Calculating the Eccentricity

The distance between the furthest point and the closest point is twice the distance \( c \), the focal distance. Thus, \( 2c = 94.6 - 91.4 = 3.2 \). Hence, \( c = \frac{3.2}{2} = 1.6 \text{ million miles} \) and \( e = \frac{c}{a} = \frac{1.6}{93} \approx 0.0172 \).

Step 5: Conclusion

The eccentricity of Earth's orbit is approximately \( 0.0172 \), indicating it is nearly circular with a slight elongation.

Key Concepts

Ellipse

An ellipse is a unique and important shape in mathematics and astronomy. It looks like an elongated circle, kind of like a squashed circle. One of the defining features of an ellipse is its two focal points, or foci. If you take any point on the edge of the ellipse, the sum of the distances to these two points is constant. This characteristic makes the ellipse different from a circle, where any point on the circumference is equidistant from the center. The Earth's orbit is an excellent example of an ellipse where the sun is one of the foci. In this way, the planetary paths are elliptical due to gravitational forces, forming orbits that range from nearly circular to quite elongated, depending on the eccentricity.

Semi-Major Axis

The semi-major axis is a crucial part of understanding ellipses. It's the longest radius that runs from the center of the ellipse to its edge, passing through both foci. In simpler terms, it's half of the longest diameter of the ellipse. In the context of planetary orbits, such as Earth's, the semi-major axis is a key parameter. It represents the average distance from the planet to the sun.

• Average distance between perihelion (closest point) and aphelion (farthest point).
• Symbolized often as 'a' in the ellipse formulas.

Using the semi-major axis, one can determine many aspects of an orbit, like its eccentricity—how stretched out the orbit is.

Perihelion and Aphelion

Perihelion and aphelion refer to the nearest and farthest points, respectively, in an orbit around the sun. These terms are essential when describing elliptical orbits like Earth’s.

• Perihelion: The point in the orbit where the Earth is closest to the sun. In Earth's orbit, this is about 91.4 million miles.
• Aphelion: The point where the Earth is farthest from the sun. For Earth, this is about 94.6 million miles.

The perihelion and aphelion distances help to calculate the semi-major axis—by averaging these two distances—as well as determining the eccentricity of the orbit. The difference between these two distances is vital in determining the focal distance, which influences the orbit’s shape and elongation.