Problem 47 - Precalculus with Calculus Previews

Question

Use a graphing utility to obtain the graphs of the given three polar equations on the same rectangular coordinate system. Use different colors for each graph. $$ \begin{aligned} &\text { Inner orbit } r=9 \text { , Hohmann transfer }\\\ &r=\frac{15.3}{1+0.7 \cos \theta}, \text { outer orbit } r=51 \end{aligned} $$

Step-by-Step Solution

Answer

Plot circles \( r = 9 \) and \( r = 51 \) with an ellipse \( r = \frac{15.3}{1+0.7 \cos \theta} \).

Step 1: Understand the Polar Equations

We are given three polar equations to plot. The first is a constant polar equation, \( r = 9 \), representing a circle with a radius of 9. The second is a polar equation of the form \( r = \frac{c}{1 + e \cos \theta} \), which is the equation of a conic section. Here, \( c = 15.3 \) and \( e = 0.7 \), representing an ellipse for \( e < 1 \). The third is also a constant, \( r = 51 \), representing another circle with a radius of 51.

Step 2: Set Up Graphing Utility

Choose a graphing utility such as Desmos or GeoGebra. Set up the graphing tool to work with polar coordinates rather than Cartesian coordinates. Ensure the graphing window covers enough \( r \)-values to accommodate the largest radius, \( r = 51 \).

Step 3: Plot Inner Orbit \( r = 9 \)

Input the equation \( r = 9 \). This represents a circle centered at the origin with a radius of 9. Use a specific color, say red, to distinguish this plot from others.

Step 4: Plot Hohmann Transfer \( r = \frac{15.3}{1 + 0.7 \cos \theta} \)

Input \( r = \frac{15.3}{1 + 0.7 \cos \theta} \) into your graphing utility. This equation represents an ellipse with a focus at the origin. Plot this ellipse in a different color, like blue, to differentiate it from the circles.

Step 5: Plot Outer Orbit \( r = 51 \)

Input the equation \( r = 51 \). This forms another circle centered at the origin with a radius of 51. Use a different color again, such as green, for this plot.

Step 6: Analyze the Combined Graph

Look at all three graphs together in the same coordinate system. You should see two circles and an ellipse, each distinct in color. This set of graphs visually represents the inner orbit, transfer orbit, and outer orbit on a single plot.

Key Concepts

Hohmann Transfer Orbit

The Hohmann Transfer Orbit is a concept from astrodynamics that describes the most energy-efficient way to move a spacecraft from one circular orbit to another. It involves an elliptical orbit called a transfer orbit, which is tangent to both the initial and final orbits.

• The transfer orbit uses two impulse maneuvers: one to transfer the spacecraft onto the Hohmann ellipse, and another to move it from the ellipse to the target orbit.
• This method minimizes fuel consumption, making it a preferred technique for satellite and mission planning.

Understanding this concept in polar coordinates involves plotting equations like the one given: \( r = \frac{15.3}{1+0.7\cos(\theta)} \).
The equation describes a conic section that, depending on certain parameters, forms an ellipse critical for visualizing orbits in polar plots.

Graphing Utility

A Graphing Utility, like Desmos or GeoGebra, is a tool that simplifies plotting complex mathematical equations. These utilities allow you to work in various coordinate systems, including polar, which is crucial for our exercise.

• Start by entering the equations in polar format and adjusting the settings to accommodate these radial systems effectively.
• Ensure the window size is sufficient to display all necessary ranges, such as the largest radius in our exercise, which is \( r = 51 \).

Using different colors for each plot is essential for clarity, helping you distinguish between different equations such as the inner orbit, outer orbit, and the Hohmann Transfer ellipse.
Overall, these graphing utilities not only aid visualization but also deepen the understanding of spatial relationships in polar coordinate systems.

Conic Sections

Conic Sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas, each having distinct properties and equations.

• The equation \( r = \frac{c}{1 + e \cos \theta} \) represents conic sections in polar coordinates: if \( e < 1 \), it is an ellipse; if \( e = 1 \), a parabola; and if \( e > 1 \), a hyperbola.
• In our exercise, \( e = 0.7 \) which, since it is less than 1, describes an ellipse.

Understanding these sections is essential as they frequently represent planetary orbits in celestial mechanics.
Applying these concepts to graphing polar equations allows one to visualize the shape and orientation of orbits, an essential skill in both mathematics and physics.