Problem 17 - Thomas Calculus

Question

In Exercises 16 and \(17,\) two planets, planet \(A\) and planet \(B\) , are orbiting their sun in circular orbits with \(A\) being the inner planet and \(B\) being farther away from the sun. Suppose the positions of \(A\) and \(B\) at time \(t\) are $$ \mathbf{r}_{A}(t)=2 \cos (2 \pi t) \mathbf{i}+2 \sin (2 \pi t) \mathbf{j} $$ and $$ \mathbf{r}_{B}(t)=3 \cos (\pi t) \mathbf{i}+3 \sin (\pi t) \mathbf{j} $$ respectively, where the sun is assumed to be located at the origin and distance is measured in astronomical units. (Notice that planet \(A\) moves faster than planet \(B . )\) The people on planet \(A\) regard their planet, not the sun, as the center of their planetary system (their solar system). Using planet \(A\) as the origin, graph the path of planet \(B .\) This exercise illustrates the difficulty that people before Kepler's time, with an earth- centered (planet \(A\) ) view of our solar system, had in understanding the motions of the planets (i.e., planet \(B=\) Mars). See D. G. Saari's article in the American Mathematical Monthly, Vol. 97 (Feb. \(1990 ),\) pp. \(105-119\) .

Step-by-Step Solution

Answer

Planet B traces a complex path (not a circle) around planet A.

Step 1: Find the Position of Planet B Relative to Planet A

To find the position of planet B relative to planet A, we need to calculate the vector from A to B. The position vector of planet A is \( \mathbf{r}_A(t) = 2 \cos(2 \pi t) \mathbf{i} + 2 \sin(2 \pi t) \mathbf{j} \) and the position vector of planet B is \( \mathbf{r}_B(t) = 3 \cos(\pi t) \mathbf{i} + 3 \sin(\pi t) \mathbf{j} \). To find the relative position of B, we subtract the position of A from the position of B: \( \mathbf{r}_{B/A}(t) = \mathbf{r}_B(t) - \mathbf{r}_A(t) \).

Step 2: Compute Vector r_{B/A}(t)

Substitute the expressions for \( \mathbf{r}_A(t) \) and \( \mathbf{r}_B(t) \) to compute the vector \( \mathbf{r}_{B/A}(t) = (3 \cos(\pi t) - 2 \cos(2\pi t)) \mathbf{i} + (3 \sin(\pi t) - 2 \sin(2\pi t)) \mathbf{j} \).

Step 3: Express Each Component of r_{B/A}(t)

We have \( x_{B/A}(t) = 3 \cos(\pi t) - 2 \cos(2 \pi t) \) for the \( \mathbf{i} \) component and \( y_{B/A}(t) = 3 \sin(\pi t) - 2 \sin(2\pi t) \) for the \( \mathbf{j} \) component. These expressions represent the coordinates of planet B as observed from planet A.

Step 4: Graph the Path of Planet B Around A

Using the expressions \( x_{B/A}(t) \) and \( y_{B/A}(t) \), plot the points for various values of \( t \) between 0 and 1 to determine the path traced by planet B in the frame of reference of planet A. This path will not be a simple circle due to the differences in angular velocities and radii of the orbits.

Key Concepts

Circular Orbits

In our solar system, many celestial bodies, like planets, tend to orbit in paths that are nearly circular. Imagine a planet moving around the Sun along a fixed circular path. This movement defines a circular orbit. A circular orbit is special because the distance from the planet to the central body (like the Sun) remains constant at all points along the orbit.

There are some key characteristics of circular orbits:

• Constant Speed: The planet moves at a uniform speed, meaning there's no speeding up or slowing down along its path.
• Uniform Direction: Always facing toward the center, creating a perfect centripetal motion that keeps the planet on its path.
• Same Radius: The distance from the planet to the central body never changes.

In the original exercise, planets A and B demonstrate circular orbits around their sun (the origin in our coordinates), meaning they each maintain a constant radius throughout their circular paths.

Planetary Motion

Planetary motion refers to how planets move in their orbits around stars, like how Earth orbits the Sun. These motions can be regular and predictable, described often by factors like speed, direction, and type of orbit.

In this exercise, the motion of two planets, A and B, is described by trigonometric equations. These equations tell us how each planet's position changes over time:

• Planet A: Orbits with position \( \mathbf{r}_{A}(t)=2 \cos (2 \pi t) \mathbf{i}+2 \sin (2 \pi t) \mathbf{j} \), indicating a faster orbit with a smaller radius.
• Planet B: Orbits with position \( \mathbf{r}_{B}(t)=3 \cos (\pi t) \mathbf{i}+3 \sin (\pi t) \mathbf{j} \), having a wider orbit but moving slower.

Each planet moves in a distinct path influenced by its velocity and distance from the sun. Observing how these paths vary helps in understanding planetary motion, revealing why one planet might seem to move differently from another depending on your point of observation.

Kepler's Laws

Kepler's Laws are fundamental principles that describe planetary motion using three key laws formulated by Johannes Kepler. Much of our understanding of planetary orbits is guided by these principles.

Let's look at these laws in simple terms:

• First Law (Law of Ellipses): Planets orbit stars in elliptical paths, with the star at one of the two foci. In circular orbits, this is a special case where the ellipse becomes a circle.
• Second Law (Law of Equal Areas): A line between a planet and its star sweeps out equal areas during equal intervals of time. This means a planet moves faster when closer to the star and slower when farther away, which governs how speeds vary in elliptical orbits.
• Third Law (Law of Harmonies): The square of a planet's orbital period (time taken for one complete orbit) is proportional to the cube of the semi-major axis of its orbit. This relationship helps to understand how the orbit's size influences its duration.

In the original exercise, customary circular motions might appear chaotic when analyzed from different frames of reference, illustrating the historical challenges faced without Kepler's insights. Observations appeared more erratic using geocentric views before the heliocentric (Sun-centered) understanding made matters clearer.