Problem 59 - Algebra and Trigonometry

Question

Orbit of a Satellite Scientists and engineers often use polar equations to model the motion of satellites in earth orbit. Let's consider a satellite whose orbit is modeled by the equation \(r=22500 /(4-\cos \theta),\) where \(r\) is the distance in miles between the satellite and the center of the earth and \(\theta\) is the angle shown in the following figure. (a) On the same viewing screen, graph the circle \(r=3960\) (to represent the earth, which we will assume to be a sphere of radius 3960 \(\mathrm{mi}\) and the polar equation of the satellite's orbit. Describe the motion of the satellite as \(\theta\) increases from 0 to \(2 \pi .\) (b) For what angle \(\theta\) is the satellite closest to the earth? Find the height of the satellite above the earth's surface for this value of \(\theta .\)

Step-by-Step Solution

Answer

The satellite is closest at \( \theta = 0 \), with height 3540 miles above Earth's surface.

Step 1: Understanding the Problem

We have to graph two equations: the orbit of the satellite given by \( r = \frac{22500}{4 - \cos\theta} \) and the Earth's surface represented by the circle \( r = 3960 \). We will then describe how the satellite moves as \( \theta \) increases from \( 0 \) to \( 2\pi \), and find when the satellite is closest to the Earth.

Step 2: Graphing the Earth's Surface

To represent the Earth, we graph the circle \( r = 3960 \). This is a simple circle centered at the origin with a radius of 3960 miles in the polar coordinate system.

Step 3: Graphing the Satellite's Orbit

Graph the polar equation \( r = \frac{22500}{4 - \cos\theta} \). As \( \theta \) changes from \( 0 \) to \( 2\pi \), the value of \( r \) changes. The plot will show how the satellite moves in its orbit.

Step 4: Describe the Satellite Motion

Analyze the graph of \( r = \frac{22500}{4 - \cos\theta} \) and \( r = 3960 \). As \( \theta \) increases from \( 0 \) to \( 2\pi \), the satellite follows an elliptical path, and its distance from the Earth varies.

Step 5: Finding Closest Approach to Earth

The satellite is closest to the Earth when \( r \) is minimized. Set \( \cos\theta = 1 \) to find this minimum, which occurs at \( \theta = 0 \). Substitute \( \theta = 0 \) into \( r = \frac{22500}{4 - \cos\theta} \) to find \( r = \frac{22500}{3} = 7500 \).

Step 6: Calculate Height Above Earth's Surface

The closest approach means the distance from the Earth's center is 7500 miles. Subtract Earth's radius: \( 7500 - 3960 = 3540 \). This gives the height of the satellite above Earth's surface when it is closest.

Key Concepts

Satellite Orbit

Satellites orbit the Earth following specific paths governed by the principles of motion and gravity. The path or trajectory of a satellite is often depicted using polar coordinates, which present the satellite's position in terms of a distance, called radius ( ext{ in this problem}), and an angle ( heta ext{ in this problem}). The given equation, \( r = \frac{22500}{4 - \cos \theta} \), models the orbital path of a satellite around Earth. This equation describes how the radius, or distance from the center of the Earth to the satellite, changes as the angle \( heta\) varies.

Polar coordinates are useful for analyzing orbits as they allow for easy identification of changes in distance as the satellite travels. As \(\theta\) increases from 0 to \(2\pi\), it represents one complete revolution around the Earth.

The satellite reaches its closest point to Earth when the denominator \(4 - \cos \theta\) is minimized, which happens when \(\cos \theta = 1\) (i.e., \(\theta = 0\)). Understanding this calculation is key to determining the satellite's path and ensuring it remains in a stable orbit.

Graphing Polar Equations

Graphing in polar coordinates allows you to visualize complex motions such as the elliptical path of a satellite. It involves plotting points using a radius \(r\) and angle \(\theta\). For the Earth's surface, graphing the simple polar equation \( r = 3960 \) gives you a circle with a radius equivalent to the Earth's radius in miles.

Graphing the satellite's orbit using the equation \( r = \frac{22500}{4 - \cos heta} \) requires calculating various \(r\) values at different \(\theta\) values to plot the path. As \(\theta\) changes from 0 to \(2\pi\), the plot reveals the elliptical nature of the orbit. This graph helps us distinguish between areas where the satellite is closer or further from Earth by observing changes in \(r\).

The visual comparison of the satellite's orbit with the Earth's circular representation allows us to easily understand the satellite's changing distance relative to Earth's surface.

Elliptical Orbits

Elliptical orbits are a common path taken by satellites due to gravitational forces. In the case of the satellite modeled by the equation \(r = \frac{22500}{4 - \cos \theta}\), it follows an elliptical orbit. An ellipse is characterized by its eccentricity, which measures how stretched out it is compared to a circle. In polar coordinates, this is determined by the equation's form, particularly the terms in the denominator such as \(\cos \theta\).

Elliptical orbits are favored for satellites because they allow for efficient travel around a celestial body while maintaining a predictable path. The point where the satellite is closest to the Earth is known as perigee. According to our equation, the closest approach occurs when \(\theta=0\), resulting in the satellite being 7500 miles from the center of Earth. Subtracting the Earth's radius gives the satellite's height above the surface at perigee, which is 3540 miles.

Understanding elliptical orbits is crucial for satellite placement and functioning, ensuring they can maintain a constant path without unexpected deviations.