Problem 39 - Algebra and Trigonometry

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 equation is \( B(t) = 0.2 \cdot \sin\left(\frac{\pi}{5}t\right) + 3.8 \).

Step 1: Understanding Simple Harmonic Motion

In simple harmonic motion, the brightness variation can be modeled by a sinusoidal function. A basic form of the equation is: \[ B(t) = A \cdot \, \sin(\omega t + \phi) + C \]Where: - \( A \) is the amplitude of the brightness variation,- \( \omega \) is the angular frequency,- \( \phi \) is the phase shift, and- \( C \) is the average brightness of the star.

Step 2: Define Known Parameters

From the given problem:- The average brightness \( C \) is 3.8 magnitudes.- The maximum variation from the average, or the amplitude \( A \), is 0.2 magnitude.- The period \( T \) of the brightness cycle is 10 days.

Step 3: Calculate Angular Frequency

The angular frequency \( \omega \) is related to the period \( T \) by the formula: \( \omega = \frac{2\pi}{T} \).Given \( T = 10 \) days:\[ \omega = \frac{2\pi}{10} = \frac{\pi}{5} \]

Step 4: Construct the Equation

Assuming no initial phase shift (\( \phi = 0 \)), the function becomes:\[ B(t) = 0.2 \cdot \sin\left(\frac{\pi}{5}t\right) + 3.8 \]This equation describes the variation of brightness of Zeta Gemini over time with respect to its average brightness and period.

Key Concepts

Sinusoidal Function

In mathematics and physics, a sinusoidal function is a type of mathematical function that describes waves or periodic motions. The most common form is the sine wave, often expressed as:

• \( y = A \cdot \sin(\omega t + \phi) + C \)

Here, the sinusoidal function is used to model the brightness variations of variable stars like Zeta Gemini. It is ideal for this type of motion because the brightness changes in a smooth and predictable manner over time.
Sinusoidal functions are fundamental in representing oscillations, such as sound waves, light waves, and the alternating current electricity. In the context of the exercise, it describes the repetitive brightness variation in a predictable cycle, where the peak and troughs are consistently spaced due to the periodic nature of the function.

Angular Frequency

Angular frequency, often denoted by \( \omega \), is an important concept that describes how fast something oscillates in simple harmonic motion. It is given by the formula:

• \( \omega = \frac{2\pi}{T} \)

Where \( T \) is the period, or the time it takes to complete one full cycle.
In our exercise regarding Zeta Gemini, the period of the star's brightness cycle is 10 days, giving us an angular frequency of \( \frac{\pi}{5} \) radians per day. Angular frequency is expressed in radians per unit time, which provides a direct understanding of how quickly the oscillations are occurring in a given timeframe.

Amplitude

Amplitude (\( A \)) signifies the maximum extent of a periodic oscillation, measured from the position of equilibrium.

• In our example: The amplitude is the maximum variation in brightness from the average, which is 0.2 magnitude for Zeta Gemini.

A higher amplitude would mean more intense variations in brightness, while a lower amplitude suggests more subtle changes.
The concept of amplitude is crucial as it determines the "loudness" of the oscillation. In simple terms, it's the distance from the center line to the peak (or trough) of the wave. For a simple harmonic oscillator like Zeta Gemini's brightness, the amplitude is what defines the extent to which the star's brightness fluctuates from its mean value.

Variable Stars

Variable stars are a type of star whose brightness as seen from Earth fluctuates over time. This variation can be due to changes in the star itself or by external factors such as eclipsing binaries.

• For Zeta Gemini, it exhibits a simple harmonic motion in its brightness due to intrinsic properties, making it an excellent candidate for using sinusoidal functions.

These stars are of great interest to astronomers because changes in brightness can reveal information about their size, mass, luminosity, and sometimes even the existence of planets orbiting around them.
By understanding the dynamical models of variable stars, scientists can gain deeper insights into stellar behaviors and the life cycle of stars. Knowing the cycle of brightness can help astronomers predict future variations and analyze the underlying causes of such changes.