Question

A idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta$$ The distance from the moon to the planet is taken to be 1 , the distance from the planet to the Sun is \(a\), and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4\) c. \(a=4, n=5\)

Short answer

Answer: For a fixed value of a, in this case, 4, it appears that every positive integer value of n produces loops in the path of the moon. The number of loops present in the path increases with increasing values of n.

Step-by-step solution

Define the parametric equations

Given the parametric equations, $$x(\theta)=a \cos \theta+\cos n \theta$$ $$y(\theta)=a \sin \theta+\sin n \theta$$ We will plot the graph of the path of the moon for each given set of constants \(a\) and \(n\).

Plot the graph for a = 4, n = 3

To plot the graph, we substitute the values of \(a=4\) and \(n=3\) into the parametric equations and use a graphing software or calculator to visualize the graph. $$x(\theta)=4 \cos \theta+\cos 3 \theta$$ $$y(\theta)=4 \sin \theta+\sin 3 \theta$$ Plot the graph using these equations for values of \(\theta\) ranging from \(0\) to \(2\pi\). Observe the presence of loops in the graph.

Plot the graph for a = 4, n = 4

Similarly, substitute the values of \(a=4\) and \(n=4\) into the parametric equations to plot the graph: $$x(\theta)=4 \cos \theta+\cos 4 \theta$$ $$y(\theta)=4 \sin \theta+\sin 4 \theta$$ Plot the graph using these equations for values of \(\theta\) ranging from \(0\) to \(2\pi\). Observe the presence of loops in the graph.

Plot the graph for a = 4, n = 5

Lastly, substitute the values of \(a=4\) and \(n=5\) into the parametric equations to plot the graph: $$x(\theta)=4 \cos \theta+\cos 5 \theta$$ $$y(\theta)=4 \sin \theta+\sin 5 \theta$$ Plot the graph using these equations for values of \(\theta\) ranging from \(0\) to \(2\pi\). Observe the presence of loops in the graph.

Conjecture about loops

After observing the graphs for all three situations, make a conjecture about which values of \(n\) produce loops for a fixed value of \(a\). The conjecture could be based on the relationship between \(n\) and the number of loops present in the path, or any pattern observed in the loops.

Key concepts

Circular Orbit

In the context of astronomical bodies, a circular orbit refers to the motion of a celestial object around another due to gravitational attraction, following a path that is a perfect circle. Usually, this idealization assumes constant speed and no external forces acting other than gravity. This concept is crucial in the example of a moon orbiting a planet, while the planet itself follows a circular orbit around the Sun. In reality, most orbits are elliptical, but circular orbits provide a simpler model to understand basic orbital dynamics and make computations easier.

To visualize a circular orbit, consider a planet moving around the Sun at a fixed distance, always maintaining the same speed. For a moon orbiting a planet, imagine that at every instance of its revolution, its distance from the planet doesn't change. This ideal model helps us predict positions and movements of moons or satellites over time. The uniform motion in a circular path simplifies our understanding of gravitational forces and interactions between celestial bodies.

Parametric Equations Graph

Parametric equations play a vital role in visualizing complex trajectories, such as the path of a moon orbiting a planet. Using parametric equations, we describe a curve by expressing the coordinates of the points on the curve as functions of a parameter, typically \(\theta\).

In this exercise, parametric equations are given as \(x(\theta)=a \cos \theta + \cos n \theta\) and \(y(\theta)=a \sin \theta + \sin n \theta\). Here, \(a\) represents the average distance from the planet to the Sun, and \(n\) indicates the moon's orbital frequency around the planet compared to the planet's orbit around the Sun. By plotting these equations, the combination of the planet's and moon's relative motions are graphically represented.

• When \(a\) is changed, it affects the size and rotation range of the primary orbit.
• The value of \(n\) impacts the secondary motion of the moon, creating various loops.

These graphs help us observe how altering parameters influences the pattern of loops, essential in deducing behavior in planetary movements.

Orbital Mechanics

Orbital mechanics, also known as celestial mechanics, is the branch of physics that deals with the motions of celestial objects under the influence of gravitational forces. It forms the basis for understanding how objects like moons, planets, and satellites move in space.

This fundamental concept focuses on applying the principles of physics, particularly Newton's laws of motion, to predict and analyze trajectories within a gravitational field. Essential topics include understanding Kepler's laws of planetary motion and the influence of gravitational interactions. The exercise involving a moon and planetary paths integrates these principles, illustrating how parametric definitions capture complex orbital dynamics.

• Kepler's laws explain the shape and timing of orbits, indicating elliptical paths rather than strictly circular.
• Gravitational influences dictate the speed and path adjustments of celestial bodies.

Comprehending orbital mechanics is crucial for topics such as satellite placement, space exploration, and predicting astronomical events. Here, analyzing the parametric equations helps break down these complex movements and underscores the intricate dance of celestial bodies in our universe.