Problem 90 - A Graphical Approach to College Algebra

Question

Solve each application. The famous Halley's comet last passed Earth in February 1986 and will next return in \(2062 .\) Halley's comet has an elliptical orbit of eccentricity 0.9673 with the sun at one of the foci. The greatest distance of the comet from the sun is 3281 million miles. (Source: The World Almanac and Book of Facts. Find the least distance between Halley's comet and the sun.

Step-by-Step Solution

Answer

The least distance is approximately 54.73 million miles.

Step 1: Understanding the Problem

To find the least distance between Halley's comet and the sun, we need to understand that this involves finding the perihelion distance of the comet's elliptical orbit. In an ellipse, the distance from the center to each focus is given by the product of the semi-major axis and the eccentricity.

Step 2: Identify Known Quantities

We know that the greatest distance (aphelion) from the comet to the sun is 3281 million miles, and the eccentricity of the ellipse is 0.9673.

Step 3: Calculate Semi-major Axis

The aphelion distance in an ellipse is given by the equation: \[ a(1 + e) = d_{aphelion} \]where \(a\) is the semi-major axis and \(e\) is the eccentricity. Plugging in the known values:\[ a = \frac{3281}{1 + 0.9673} \approx 1669.61 \text{ million miles} \]

Step 4: Compute Perihelion Distance

The perihelion distance is given by the formula:\[ d_{perihelion} = a(1 - e) \]Substituting the known values:\[ d_{perihelion} = 1669.61(1 - 0.9673) \approx 54.73 \text{ million miles} \]

Step 5: Conclusion

The least distance between Halley's comet and the sun is approximately 54.73 million miles.

Key Concepts

Eccentricity

An elliptic orbit, such as the one that Halley's comet follows around the sun, is not perfectly circular, and this shape is characterized by its eccentricity. The eccentricity is a number between 0 and 1 and it describes how much the shape of the path deviates from being a perfect circle.
When the eccentricity is close to 0, the orbit is almost circular. If it's closer to 1, as it is for Halley’s comet with an eccentricity of 0.9673, the orbit is stretched and elongated.
In mathematical terms, eccentricity \[ e \] is calculated using the formula:

• \( e = \frac{c}{a} \)
• where \( c \) is the distance from the center to a focus of the ellipse
• and \( a \) is the semi-major axis.

Thus, knowing the eccentricity helps us understand how stretched or squished the orbit is.

Perihelion Distance

The perihelion distance in an ellipse refers to the closest point that an orbiting body reaches with the sun. For Halley's comet, determining this distance helps us understand how near it gets to the sun during its orbital period.
To find the perihelion distance \( d_{\text{perihelion}} \), we use the semi-major axis \( a \) and the eccentricity \( e \) in the formula:

• \( d_{\text{perihelion}} = a(1 - e) \)

This formula shows that the perihelion distance can be thought of as the shrinking of the full semi-major axis due to the orbit's eccentric shape.
In the case of Halley's comet, using its known semi-major axis and eccentricity, the least distance from the sun comes out to be approximately 54.73 million miles.

Semi-major Axis

The semi-major axis is a very important parameter when it comes to understanding elliptical orbits. It represents half the longest diameter of the ellipse, stretching from the center of the ellipse to its farthest edge.
For any orbiting body, including Halley's comet, the semi-major axis \( a \) can be calculated if we know either its closest or farthest distance from the focal point, along with the eccentricity.
In simple terms,

• \( a(1 + e) = d_{\text{aphelion}} \) gives the greatest or aphelion distance.
• \( a(1 - e) = d_{\text{perihelion}} \) gives the closest or perihelion distance.

By rearranging the formula for the aphelion distance, we derived the semi-major axis of Halley's comet to be approximately 1669.61 million miles. This value isn't just a measurement; it's key to further calculations in analyzing the comet's path around the sun.