Problem 204 - Calculus Volume 1

Question

As implied earlier, according to Kepler's laws, Earth's orbit is an ellipse with the Sun at one focus. The perihelion for Earth's orbit around the Sun is \(147,098,290\) \(\mathrm{km}\) and the aphelion is \(152,098,232 \mathrm{km} .\) a. By placing the major axis along the \(x\) -axis, find the average distance from Earth to the Sun. b. The classic definition of an astronomical unit (AU) is the distance from Earth to the Sun, and its value was computed as the average of the perihelion and aphelion distances. Is this definition justified?

Step-by-Step Solution

Answer

The average distance is 149,598,261 km; the AU definition is justified.

Step 1: Understand Kepler's Laws and Definitions

Kepler's laws of planetary motion describe the elliptical orbits of planets with the Sun at one focus. An ellipse has two foci, and Earth's orbit around the Sun is an ellipse with the Sun at one focus. The perihelion is the point in the orbit where Earth is closest to the Sun, and the aphelion is where it is farthest from the Sun.

Step 2: Calculate the Semi-Major Axis

The average distance from Earth to the Sun in an elliptical orbit is given by the semi-major axis of the ellipse. The semi-major axis, denoted as 'a', is the average of the perihelion and aphelion distances. Calculate it using the formula:\[ a = \frac{\text{perihelion} + \text{aphelion}}{2} \]Substitute the given values:\[ a = \frac{147,098,290 + 152,098,232}{2} \]

Step 3: Compute the Average

Perform the addition and division to find the semi-major axis:1. Add the perihelion and aphelion distances: \(147,098,290 + 152,098,232 = 299,196,522\)2. Divide by 2 to find the average:\[ a = \frac{299,196,522}{2} = 149,598,261\text{ km} \]

Step 4: Verify the Astronomical Unit Definition

The classic definition of an astronomical unit (AU) is the average distance from Earth to the Sun, originally defined as the mean of the perihelion and aphelion distances. From Kepler's understanding, this average distance corresponds to the semi-major axis of the orbit, justifying the definition of the AU.

Key Concepts

Understanding an Ellipse in Orbital Mechanics

In the realm of astronomy, an ellipse is a geometric shape that plays a crucial role in understanding the orbits of planets and celestial bodies. Specifically, an ellipse is an elongated circle, defined by its two foci. When discussing orbits, one of these foci is occupied by a massive body, like the Sun in the case of Earth's orbit.

This implies that Earth doesn't orbit in a perfect circle, but follows an elliptical path. This fundamental property is outlined in Kepler's first law of planetary motion. As a result, this elliptical nature affects how far or how close Earth gets to the Sun during different times of the year.

Perihelion and Aphelion: Closest and Farthest Points

Two specific points in an elliptical orbit are known as perihelion and aphelion. The perihelion is the point where a planet is closest to the Sun. Conversely, the aphelion represents the point where the planet is farthest away from the Sun.

These points are critical in understanding the varying gravitational forces exerted on the Earth by the Sun. This does not only affect the Earth's temperature but also the length of the seasons. For instance, the Earth experiences perihelion around January, making that part of the year when Earth is slightly closer to the Sun than at other times.

Understanding the Semi-Major Axis

The semi-major axis of an elliptical orbit is perhaps one of the most significant measurements in astronomy. It represents half of the longest diameter of the ellipse, or simply, the average of the closest and farthest distances from the Sun, which are the perihelion and aphelion distances, respectively.

Mathematically, it is calculated as follows:

• Add the perihelion and aphelion distances.
• Divide the sum by two.

This calculation provides the average distance from Earth to the Sun for the given elliptical orbit, which aids in understanding Earth’s orbital dynamics and how it affects its climate and duration of the year.

Defining the Astronomical Unit (AU)

The astronomical unit (AU) is a standard unit of measurement used by astronomers to describe distances in space. It is defined as the average distance from Earth to the Sun. Historically, this was calculated using the average of the Earth's perihelion and aphelion distances — essentially the semi-major axis of the Earth's orbit.

This unit provides a simple way to compare distances within our solar system. By utilizing the AU, scientists can easily convey information about distances in a more comprehensible manner. It simplifies calculations when describing orbits that would otherwise be buried in details of large numerical values.