Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

The perihelion and aphelion for the orbit of the asteroid Icarus are 17 and 183 million miles, respectively. What is the eccentricity of its elliptical orbit?

Step-by-Step Solution

Answer

The eccentricity is 0.83.

Step 1: Understanding the Problem

The eccentricity of an ellipse measures how much it deviates from being circular. It ranges from 0 (a circle) to 1 (a parabola). The perihelion is the closest distance from the focus of the ellipse (the sun in this case), and the aphelion is the farthest distance.

Step 2: Define the Variables

Let the perihelion distance be denoted by \( p = 17 \) million miles and the aphelion distance be denoted by \( a = 183 \) million miles. The semi-major axis \( A \) can be calculated as the average of the perihelion and aphelion: \( A = \frac{a + p}{2} \).

Step 3: Calculate the Semi-Major Axis

Compute the semi-major axis using the formula: \[ A = \frac{17 + 183}{2} = 100 \text{ million miles} \]

Step 4: Calculate the Semi-Focal Distance

The distance between the center of the ellipse and one of its foci (semi-focal distance) is denoted as \( c \), calculated by the formula: \[ c = a - A = 183 - 100 = 83 \text{ million miles} \]

Step 5: Calculate Eccentricity

The eccentricity \( e \) of the elliptical orbit is given by: \[ e = \frac{c}{A} = \frac{83}{100} = 0.83 \]

Step 6: Conclusion

The eccentricity of the orbit of asteroid Icarus is 0.83, indicating a highly elongated orbit.

Key Concepts

Elliptical Orbit

An elliptical orbit is a closed curve that revolves around two focal points, with one of these foci often being the central body, such as the Sun in our solar system. Unlike a circular orbit where points on the path are equidistant from the center, ellipticals vary in distance.
The shape of an elliptical orbit is characterized by its eccentricity. Eccentricity is a crucial parameter of an ellipse, determining how elongated or flattened it is.
- An eccentricity of 0 means the orbit is a circle. - Values greater than 0 and less than 1 indicate an elliptical shape. - An eccentricity close to 1 implies a very stretched ellipse, almost parabolic.
Orbits of celestial bodies, such as asteroids or planets, are typically elliptical. Understanding these orbits helps astronomers predict the celestial body's path and period.

Perihelion and Aphelion

Perihelion and aphelion describe specific positions in an object's elliptical orbit relative to the Sun. These terms apply broadly to all celestial bodies orbiting a star.
- **Perihelion** is the point in the orbit where the object is nearest to the Sun. At this position, it moves faster due to the stronger gravitational pull from the Sun. - **Aphelion** is the point where the object is farthest from the Sun. The object slows down here because the gravitational pull weakens.
The distances of perihelion and aphelion dictate the shape and dynamics of the orbit. For instance, the asteroid Icarus has large variations between its perihelion and aphelion, indicating a highly eccentric orbit. This knowledge is pivotal for calculating orbital parameters like the semi-major axis.

Semi-Major Axis

The semi-major axis is a fundamental dimension in the geometry of an elliptical orbit. It represents half the longest diameter of the ellipse, spanning from the center to the edge.
Calculating the semi-major axis gives insight into the orbit's size and energy. It is also effectively the average distance from the orbiting body (such as an asteroid) to the star it orbits.

• Formula: The semi-major axis \( A \) is the average of the perihelion (\( p \)) and aphelion (\( a \)) distances: \( A = \frac{p + a}{2} \).
• For Icarus, with a perihelion of 17 million miles and an aphelion of 183 million miles, the semi-major axis calculates to 100 million miles.

This value is essential, as it influences the shape and period of the object's orbit and plays a central role in the calculation of eccentricity.

Orbit of Asteroid Icarus

The orbit of asteroid Icarus is an excellent example of an elliptical and eccentric orbit. Icarus is known for its pronounced eccentricity of 0.83, meaning its path around the Sun is significantly elongated rather than circular.
Understanding Icarus's orbit involves calculating its perihelion, aphelion, and semi-major axis. With perihelion and aphelion values at 17 million miles and 183 million miles, respectively, these distances are crucial in understanding its distinctive path.

• Its eccentricity implies that at perihelion, Icarus travels much faster due to the stronger gravitational influence of the Sun, while at aphelion, it slows down.
• The high eccentricity also suggests that its solar orbit receives varied solar radiation intensity, affecting its surface conditions significantly between these two points.

This high level of eccentricity makes asteroid Icarus an object of interest for astronomers studying solar system dynamics and orbital mechanics.