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 of the orbit is 0.83.

Step 1: Understanding the Problem

We need to calculate the eccentricity of the asteroid Icarus's elliptical orbit. The perihelion is the closest distance from the Sun, and the aphelion is the farthest distance. The given values are 17 million miles for the perihelion and 183 million miles for the aphelion.

Step 2: Know the Definition of Eccentricity

Eccentricity (\(e\)) of an ellipse is calculated with the formula \(e = \frac{A - P}{A + P}\), where \(A\) is the aphelion and \(P\) is the perihelion. This formula measures how much the orbit deviates from being circular.

Step 3: Substitute Values into the Eccentricity Formula

Using the values given: \(A = 183\text{ million miles}\) and \(P = 17\text{ million miles}\), substitute them into the eccentricity formula: \(e = \frac{183 - 17}{183 + 17}\).

Step 4: Compute the Values

Calculate the numerator: \(183 - 17 = 166\). Calculate the denominator: \(183 + 17 = 200\). Substitute these back into the formula to get \(e = \frac{166}{200}\).

Step 5: Simplify the Eccentricity

Simplify \(\frac{166}{200}\) by dividing the numerator and the denominator by 2: \(\frac{166}{200} = \frac{83}{100} = 0.83\).

Step 6: Interpret the Result

The eccentricity \(e = 0.83\) is a measure of how elongated the orbit is. Since it is less than 1, the orbit is elliptical.

Key Concepts

Perihelion and Aphelion

When discussing orbital paths, particularly those involving the sun, two critical points need to be considered: perihelion and aphelion. These terms describe an object's closest and farthest points from the sun, respectively.
Examining these points can provide valuable insight into the dynamics of celestial bodies, such as asteroids, planets, or comets, as they move through space.

• Perihelion: This is the minimum distance between a celestial object and the Sun. For Icarus, this distance is 17 million miles, meaning this is when Icarus is closest to the sun.
• Aphelion: Conversely, this is the maximum distance from the Sun. For Icarus, the aphelion is 183 million miles, indicating when it's furthest from the sun.

Understanding these two terms helps us comprehend the shape of the orbit and the spatial dynamics involved over a complete orbit. It also lays the foundation for grasping how elliptical an orbit is.

Orbital Mechanics

Orbital mechanics is the field of study that deals with the motion of objects in space under the influence of gravitational forces. It's the backbone of many calculations concerning objects, such as planets and asteroids, as they travel through our solar system.
By analyzing orbital mechanics, scientists can determine future positions of celestial bodies, their velocity, and their trajectory relative to the sun or other bodies. Key principles include:

• Gravity: This is the fundamental force driving orbital motion, causing celestial objects like Icarus to follow a curved path around the sun.
• Elliptical Orbits: Most orbits are elliptical, not circular. This means that the distance between the planet or object and the sun changes as the object travels along its path.
• Kepler's Laws: These laws describe the motion of planets around the sun, dictating aspects like orbital speed and eccentricity.

The study of orbital mechanics provides insight into how and why celestial bodies move as they do, assisting in understanding both the past and potential future of such orbits.

Eccentricity Formula

The eccentricity formula is a crucial mathematical expression used to quantify the deviation of an orbit from a perfect circle. This value, denoted as \(e\), ranges from 0 to 1 for elliptical orbits.
It measures the amount of elongation in an orbit:

• Definition: Eccentricity is defined using the formula \(e = \frac{A - P}{A + P}\), where \(A\) is the aphelion and \(P\) is the perihelion. This equation helps visualize how stretched an orbit is.
• Application: In the case of Icarus, substituting the given values (\(A = 183\) million miles and \(P = 17\) million miles) into the formula gives \(e = \frac{183 - 17}{183 + 17} = \frac{166}{200} = 0.83\).
• Interpretation: Eccentricity of 0.83 implies a highly elliptical orbit, as values closer to 1 indicate more elongated paths, and values approaching 0 suggest circular orbits.

Understanding and calculating eccentricity is vital in orbital mechanics as it directly influences the trajectory shape and orbital characteristics.