Problem 65 - Precalculus: Mathematics for Calculus

Question

The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earth's distance from the sun is \(147,000,000 \mathrm{km}\) at perihelion and \(153,000,000 \mathrm{km}\) at aphelion. Find an equation for the earth's orbit. (Place the origin at the center of the orbit with the sun on the \(x\) -axis.) (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

The equation of Earth's orbit is \(\frac{x^2}{(150,000,000)^2} + \frac{y^2}{(149,939,917.2)^2} = 1\).

Step 1: Understand the Properties of an Ellipse

In an elliptical orbit, the sum of the distances from any point on the ellipse to the two foci is constant. The center of the ellipse is at the origin, the earth's orbit is symmetrical about the x-axis with the sun at one focus. The perihelion and aphelion distances correspond to the vertices, and are located on the x-axis.

Step 2: Determine the Semi-Major Axis

The semi-major axis, denoted as \(a\), is the average of the perihelion and aphelion distances. Calculate \(a = \frac{147,000,000 + 153,000,000}{2} = 150,000,000\) km.

Step 3: Calculate the Distance Between the Center and the Focus (c)

The distance \(c\) from the center of the ellipse to a focus (the position of the sun) is given by \(c = \frac{153,000,000 - 147,000,000}{2} = 3,000,000\) km.

Step 4: Compute the Semi-Minor Axis (b)

Use the relationship \(b^2 = a^2 - c^2\) to find the semi-minor axis. \(b^2 = (150,000,000)^2 - (3,000,000)^2\). Compute \(b = \sqrt{(150,000,000)^2 - (3,000,000)^2}\), resulting in \(b = 149,939,917.2\) km.

Step 5: Formulate the Equation of the Earth's Elliptical Orbit

The equation of an ellipse centered at the origin with the major axis along the x-axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substitute \(a=150,000,000\) km and \(b=149,939,917.2\) km into the equation: \(\frac{x^2}{(150,000,000)^2} + \frac{y^2}{(149,939,917.2)^2} = 1\).

Key Concepts

Perihelion

In the context of elliptical orbits, the term "perihelion" refers to the point where a planet or other celestial body is closest to the Sun during its orbit. This is a significant concept as it affects the planet's speed and energy due to gravitational interactions.

• The word perihelion is derived from Greek, where "peri" means "near" and "Helios" means "Sun."
• This point is a vertex of the elliptical orbit, situated on the major axis.
• At perihelion, a planet moves faster along its orbit due to the conservation of angular momentum.
• For Earth, the perihelion distance is approximately 147,000,000 km.

Understanding perihelion is important in calculating orbital parameters and forming the elliptical path of celestial bodies, explaining why seasons occur, and how the distance to the Sun varies over the year.

Aphelion

The term "aphelion" signifies the opposite of perihelion. It is the point in an elliptical orbit where a celestial object is farthest from the Sun. This concept is crucial for understanding how the orbit's shape influences the climate and conditions on planets.

• "Aphelion" also finds its roots in Greek, with "apo" meaning "away, off, apart" and "Helios" meaning "Sun."
• Similarly to perihelion, aphelion lies along the orbit's major axis, representing a second vertex of the ellipse.
• At aphelion, the orbital speed of a celestial body decreases, again due to the conservation of angular momentum.
• For Earth, the sun is roughly 153,000,000 km away at aphelion.

Grasping the concept of aphelion helps students understand not only orbital mechanics but also how the duration of seasons and temperatures might vary as Earth swings farther from or closer to the Sun.

Equation of an Ellipse

The equation of an ellipse is foundational to describing the shape and orientation of orbits in space. When celestial bodies travel around the Sun, their paths can often be described as ellipses. Here’s how we deduce this crucial equation:

• For an ellipse centered at the origin, with its major axis aligned along the x-axis, the general equation is given by: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
• Here, \(a\) represents the semi-major axis, the average of the distances to perihelion and aphelion.
• The semi-minor axis, \(b\), is calculated using the relationship \(b^2 = a^2 - c^2\), where \(c\) is the distance from the center to the focus (where the Sun is located).
• Substituting Earth's orbital parameters, we have:\[\frac{x^2}{(150,000,000)^2} + \frac{y^2}{(149,939,917.2)^2} = 1\]

Describing celestial orbits using the equation of an ellipse helps students appreciate the mathematical symmetry that governs planetary movement and aids in predicting positions in astronomy.