Problem 53 - University Physics with Modern Physics

Question

How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 10.0\(\%\) higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

Step-by-Step Solution

Answer

The star is moving toward us at ~9.5% the speed of light.

Step 1: Understanding the Problem

We need to determine the speed of a star as a percentage of the speed of light such that the frequency of its light observed on Earth is 10% higher compared to the frequency at which it is emitted. This involves using the Doppler effect for light.

Step 2: Doppler Effect Formula for Light

The relativistic Doppler effect formula for frequency shift is given by \( f' = f \sqrt{\frac{1 + \beta}{1 - \beta}} \), where \( f' \) is the observed frequency, \( f \) is the emitted frequency, and \( \beta = \frac{v}{c} \) represents the speed of the star as a fraction of the speed of light.

Step 3: Setting Up the Equation

Since the observed frequency is 10% higher, we set \( f' = 1.1f \). Plugging this into the Doppler effect formula gives us \( 1.1 = \sqrt{\frac{1 + \beta}{1 - \beta}} \).

Step 4: Solving the Equation

Square both sides of the equation: \( 1.21 = \frac{1 + \beta}{1 - \beta} \). Cross-multiply to find \( 1.21(1 - \beta) = 1 + \beta \). Simplifying, we have \( 1.21 - 1.21\beta = 1 + \beta \). Rearrange to get \( 0.21 = 2.21\beta \).

Step 5: Calculate the Star's Speed

Solve for \( \beta \): \( \beta = \frac{0.21}{2.21} \approx 0.095 \). This means the star is moving at approximately 9.5% the speed of light.

Step 6: Direction of Movement

Since the observed frequency is higher, the star must be moving toward us (resulting in a blueshift).

Key Concepts

Relativistic Doppler Effect

The relativistic Doppler effect occurs when an object emitting waves, like light from a star, moves relative to an observer. Unlike the classical Doppler effect, the relativistic version must consider the effects of Einstein's theory of relativity.
In this context, the object is moving at a significant fraction of the speed of light, so the effects due to the time dilation and Lorentz contraction must be factored in.
For light, the effect can be described with the formula: \[ f' = f \sqrt{\frac{1 + \beta}{1 - \beta}} \]where:

• \( f' \) is the observed frequency.
• \( f \) is the emitted frequency.
• \( \beta = \frac{v}{c} \) is the ratio of the object's speed \( v \) to the speed of light \( c \).

Depending on whether the source is moving towards or away from the observer, the frequency observed will either increase (blueshift) or decrease (redshift). This effect explains how astronomers determine whether celestial bodies are moving towards or away from us.

Frequency Shift

Frequency shift refers to the change in frequency of a wave as measured by an observer, compared to the frequency at which the wave was emitted by the source.
In our case, involving light from a star, this is linked to the Doppler effect.
If the star moves towards the observer, the waves compress, leading to an increase in frequency or blueshift. Conversely, if the star is moving away, the waves stretch, causing a decrease in frequency or redshift.
In the exercise, the frequency observed is 10% higher than emitted, indicating a blueshift, as calculated using:\[ 1.1 = \sqrt{\frac{1 + \beta}{1 - \beta}} \]This translates a star's motion toward us at roughly 9.5% the speed of light.
This measurable shift is crucial for studying galaxy movements and understanding the universe's expansion.

Speed of Light

The speed of light is a fundamental constant in physics, denoted by \( c \) and is approximately 299,792,458 meters per second (or roughly 300,000 km/s).
This speed is crucial in the theory of relativity and serves as the ultimate speed limit in the universe, beyond which nothing can travel faster.
In the context of the Doppler effect for light, a star's velocity as a percentage of the speed of light (\( \beta \)) is calculated to understand the frequency shift of the light:
- Stars moving at significant fractions of this speed manifest noticeable relativistic effects such as time dilation and length contraction.- Equation transformations into speeds express the substantial influence on observed frequencies, allowing astrophysicists to deduce relative motions of interstellar objects and their velocities precisely.