Problem 39 - Precalculus Mathematics for Calculus

Question

Variable Stars The variable star Zeta Gemini has a period of 10 days. The average brightness of the star is 3.8 magnitudes, and the maximum variation from the average is 0.2 magnitude. Assuming that the variation in brightness is simple harmonic, find an equation that gives the brightness of the star as a function of time.

Step-by-Step Solution

Answer

The brightness function is: \( B(t) = 0.2 \cos\left(\frac{\pi}{5} t \right) + 3.8 \).

Step 1: Understand the Problem

The star's brightness varies in a simple harmonic manner around an average brightness of 3.8 magnitudes. The maximum deviation from this average is 0.2 magnitudes, with a period of 10 days. We need to find the equation that models this behavior.

Step 2: Model the Brightness Function

The brightness variation can be modeled using the cosine function for simple harmonic motion. The general form for this is: \[ B(t) = A \cos(\omega t + \phi) + C \]Where:- \(A\) is the amplitude of variation (0.2 magnitudes),- \(\omega\) is the angular frequency,- \(\phi\) is the phase shift (we can assume it to be zero for this problem),- \(C\) is the average brightness (3.8 magnitudes).

Step 3: Calculate the Angular Frequency

The angular frequency \(\omega\) is given by:\[ \omega = \frac{2\pi}{T} \]Where \(T\) is the period, which is 10 days. Therefore, \[ \omega = \frac{2\pi}{10} = \frac{\pi}{5} \]

Step 4: Write the Brightness Equation

Now substitute \(A = 0.2\), \(\omega = \frac{\pi}{5}\), \(\phi = 0\), and \(C = 3.8\) into the function:\[ B(t) = 0.2 \cos\left(\frac{\pi}{5} t \right) + 3.8 \]This equation represents the brightness of Zeta Gemini as a function of time.

Key Concepts

Angular Frequency

To understand simple harmonic motion, knowing angular frequency is crucial. Angular frequency, denoted as \(\omega\), determines how quickly the motion repeats itself, usually measured in radians per second. For periodic phenomena like the brightness variation of a variable star, it gives us a sense of the rate at which the cycle completes.

For our example with Zeta Gemini, the period \(T\) is given as 10 days. This means the star completes one cycle of its brightness change every 10 days. The formula for angular frequency is:
\[ \omega = \frac{2\pi}{T}, \]
which leads to:
\[ \omega = \frac{2\pi}{10} = \frac{\pi}{5}. \]
Each time unit this system moves by \(\frac{\pi}{5}\) radians on its brightness cycle. Understanding angular frequency helps in predicting the timing of brightness peaks and troughs.

Amplitude

Amplitude in simple harmonic motion describes the maximum extent of variation from the average position. In the context of Zeta Gemini, amplitude tells us how much the brightness of the star can vary.

Here, the amplitude \(A\) is 0.2 magnitudes. This is the maximum observed deviation above or below the average brightness of 3.8 magnitudes.

In physical terms, knowing the amplitude can give us insights into the range of fluctuation in the star's brightness.

• An amplitude of 0.2 means the star's brightness could be as high as \(3.8 + 0.2 = 4.0\) magnitudes and as low as \(3.8 - 0.2 = 3.6\) magnitudes.

Amplitude is a key parameter when modeling simple harmonic motion as it affects the overall range of the sine or cosine function used.

Brightness Variation

With variable stars, brightness variation is where the shift happens over time. This shift is modeled as simple harmonic motion in our case.

For Zeta Gemini, the brightness variation is characterized by a regular change around an average value, exhibiting periodic peaks and troughs.

This periodic change is a signature of many celestial bodies exhibiting simple harmonic motion.

• The average brightness of Zeta Gemini sits at 3.8 magnitudes.
• The brightness oscillates by a maximum of 0.2 magnitudes around this average.

In a graph representing brightness over time, this variation would appear as a smooth wave-like pattern, steadily repeating every 10 days.

Cosine Function

The cosine function is frequently used in modeling simple harmonic motion due to its periodic nature.

In our context, the brightness of Zeta Gemini can be described using a cosine function:
\[ B(t) = A \cos(\omega t + \phi) + C. \]
Here’s how each component plays a role:

• \(A = 0.2\): the amplitude representing the maximum brightness variation.
• \(\omega = \frac{\pi}{5}\): the angular frequency that defines the cycle's speed.
• \(\phi = 0\): the phase shift. In this case, it's assumed zero, meaning the cosine wave starts at its maximum.
• \(C = 3.8\): the average brightness of the star.

The function effectively models how the brightness of the star varies over time with a neat, predictable pattern.