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Chapter 5: Problem 51

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.

Short Answer

Expert verified
The brightness function is \(B(t) = 0.2 \cos\left(\frac{2\pi}{10} t\right) + 3.8\).

Step by step solution

01

Identifying Variables

Let's identify the key variables needed for this problem. The period of the brightness variation, \(T\), is 10 days. The average brightness is given as 3.8 magnitudes. The maximum variation from this average is 0.2 magnitudes. We can set the amplitude of the brightness variation, \(A\), to be 0.2 magnitudes.
02

Understanding Simple Harmonic Motion

In simple harmonic motion, a function that varies with time, \(t\), can be modeled by a sine or cosine function. Given the average brightness and variation, we can model the brightness, \(B(t)\), using a cosine function centered around the average brightness since we're given maximum and minimum variations.
03

Constructing the Brightness Function

The general form of a cosine function describing simple harmonic motion is \(B(t) = A \cos(\omega t) + C\), where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(C\) is the vertical shift which, in this case, is the average brightness. Given \(A = 0.2\), \(C = 3.8\), we need to find \(\omega\).
04

Calculating Angular Frequency

The angular frequency \(\omega\) is given by \(\omega = \frac{2\pi}{T}\), where \(T\) is the period. Substituting the period of 10 days, we get \(\omega = \frac{2\pi}{10}\).
05

Writing the Complete Function

Now that we have all the components, we can write the equation for brightness: \(B(t) = 0.2 \cos\left(\frac{2\pi}{10} t\right) + 3.8\). This equation represents the brightness of Zeta Gemini as a function of time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
In the study of simple harmonic motion, the cosine function plays a crucial role. It is often used to describe oscillatory phenomena, including the variation in the brightness of stars like Zeta Gemini. The basic form of a cosine function is given by:

\[ B(t) = A \cos(\omega t) + C \]This equation describes how a quantity varies over time. Here, the cosine function helps model the regular variation in brightness. It begins at a maximum or minimum value, making it suitable for problems where initial conditions reflect these extremes.

When using a cosine function:
  • "A" represents the amplitude or the peak variation from the average value.
  • "蠅t" is the phase angle that changes over time.
  • "C" is the average value around which the variation occurs, known as the vertical shift.
The use of the cosine function helps us anticipate the predictable nature of the oscillations in simple harmonic motion.
Amplitude
Amplitude refers to the maximum extent of a wave's oscillation, away from the midpoint or equilibrium position. In the context of Zeta Gemini's brightness, amplitude is critical in defining how much the brightness deviates from its average magnitude.

For example, in our brightness model for Zeta Gemini:
  • The amplitude, \(A\), is given as 0.2 magnitudes.
  • This indicates the brightness varies by plus or minus 0.2 from its average value.
Thus, amplitude directly influences the "height" of the wave graph generated by the cosine function. And it's crucial for predicting the extremes in brightness, whether at its highest or lowest points.
Angular Frequency
Angular frequency is a measure of how quickly a wave oscillates. It is denoted by \(\omega\) and is closely linked to the concept of the period in oscillatory motion.

To calculate angular frequency, the formula used is:
  • \( \omega = \frac{2\pi}{T} \), where \(T\) is the period.
In our Zeta Gemini example, with a periodic cycle of 10 days:
  • \(\omega = \frac{2\pi}{10}\)
  • This translates to radians per day, illustrating how quickly the brightness cycles through its full range of values.
Understanding angular frequency is essential when working with simple harmonic motions, as it impacts the pacing of oscillations.
Period
The period of a function in simple harmonic motion represents the time taken to complete one full cycle of oscillation. In celestial observations, like those of Zeta Gemini, the period is key in understanding repetitive changes in brightness.

The period, denoted \(T\), plays a significant role in the formulation of the harmonic equation:
  • For Zeta Gemini, \(T = 10\) days.
  • This indicates that every 10 days, the brightness pattern repeats itself.
Since period is integral in calculating angular frequency (\(\omega = \frac{2\pi}{T}\)), understanding it provides insight into the timing and duration of these astronomical cycles. This cyclic predictability is fundamental in modeling and studying stars and other periodic phenomena.

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