Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Earth's orbit around the sun is an ellipse of eccentricity \(0.0167\) and major diameter \(185.8\) million miles. Find its perihelion.

Step-by-Step Solution

Answer

The perihelion of Earth's orbit is approximately 91.36 million miles.

Step 1: Understanding key terms

We need to find the perihelion of Earth's orbit. The perihelion is the point in the orbit where the Earth is closest to the Sun.

Step 2: Identify given data

We know the major diameter of the Earth's orbit is 185.8 million miles and the eccentricity is 0.0167. The major diameter is the longest diameter across the ellipse.

Step 3: Calculate the semi-major axis

The major diameter is twice the length of the semi-major axis. Thus, the semi-major axis \(a\) can be calculated as: \( a = \frac{185.8}{2} = 92.9 \text{ million miles} \).

Step 4: Calculate perihelion distance formula

The formula for the perihelion distance is \( a(1 - e) \), where \(a\) is the semi-major axis and \(e\) is the eccentricity.

Step 5: Solve for the perihelion distance

Substitute \( a = 92.9 \text{ million miles} \) and \( e = 0.0167 \) into the formula \( a(1 - e) \): \[ 92.9 \times (1 - 0.0167) \].

Step 6: Calculate the result

Evaluate the expression: \[ 92.9 \times (1 - 0.0167) = 92.9 \times 0.9833 = 91.36057 \text{ million miles} \]. Thus, the perihelion distance is approximately 91.36 million miles.

Key Concepts

Eccentricity of an Ellipse

Eccentricity is a measure that describes how much an ellipse deviates from being a perfect circle. For an ellipse, this value is denoted by the letter "e", and it ranges between 0 and 1.
Eccentricity can be visualized as the "flatness" of the ellipse. If the eccentricity is 0, the ellipse is actually a perfect circle. When the eccentricity is closer to 1, the ellipse appears more stretched or elongated.
To calculate eccentricity, one can use the formula:

• For a specific focus and vertex: \( e = \frac{c}{a} \), where \( c \) is the distance from the center to the focus, and \( a \) is the semi-major axis.

In the context of planetary orbits, like Earth around the Sun, the eccentricity is often very small, making the orbit nearly circular. Earth's eccentricity of 0.0167 implies its orbit is very close to a circle, with just a slight outward bulging.

Semi-Major Axis

The semi-major axis of an ellipse is one of its most important characteristics, defining the overall size of the ellipse. It is half of the longest diameter of the ellipse, known as the major axis.
You can think of the semi-major axis as the average distance from the center of the ellipse to its outer edge along the longest direction.
In astronomy, the semi-major axis of an orbit is crucial because it determines key attributes of the orbit:

• It helps in calculating the orbital period of a body, such as a planet or moon.
• It's used in determining the distance at perihelion (closest point to the Sun) and aphelion (farthest point from the Sun).

For Earth's orbit, the semi-major axis is approximately 92.9 million miles. This measurement plays a key role in calculating distances like the perihelion using the formula \( a(1 - e) \). Here, \( a \) represents the semi-major axis, and \( e \) is the eccentricity.

Ellipse Geometry

Ellipse geometry is fascinating, as it balances between the familiar circle and the elongated, or squashed, curves. An ellipse is defined as the set of points where the sum of distances from two fixed points (the foci) is constant.
The main components of an ellipse include:

• Major Axis: This is the longest line that can be drawn through the center, touching both sides of the ellipse.
• Minor Axis: The shortest line that crosses through the center at right angles to the major axis.
• Foci: Two fixed points inside the ellipse that assist in its definition. The sum of distances from any point on the ellipse to these foci is always the same.

To visualize, think of a stretched circle with two pins (the foci) tugging it from inside.
Understanding ellipse geometry is vital for fields like astronomy, where celestial bodies often move in elliptical orbits, as well as engineering and computer graphics, where ellipses frequently occur in designs and simulations.