Suppose a galaxy is moving away from Earth at a speed 0.78c. It emits radio waves with a wavelength of 0.489 m. What wavelength would we detect on Earth?
Suppose a galaxy is moving away from Earth at a speed 0.78c. It emits radio waves with a wavelength of 0.489 m. What wavelength would we detect on Earth?
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Question
![**Understanding Doppler Effect in Astronomy**
**Scenario:**
Suppose a galaxy is moving away from Earth at a speed of 0.78c, where "c" is the speed of light. The galaxy emits radio waves with a wavelength of 0.489 meters. What wavelength would we detect on Earth?
**Explanation:**
This problem deals with the Doppler Effect, which describes how the wavelength of waves (such as light or sound) changes due to the motion of the source relative to the observer. When an astronomical object like a galaxy moves away from Earth, the light it emits is stretched to longer wavelengths, a phenomenon known as redshift.
To calculate the observed wavelength on Earth, the following relativistic Doppler shift formula can be used:
\[
\lambda' = \lambda \times \sqrt{\frac{1 + \beta}{1 - \beta}}
\]
where:
- \(\lambda'\) is the observed wavelength,
- \(\lambda\) is the emitted wavelength (0.489 m),
- \(\beta = \frac{v}{c}\), where \(v\) is the speed of the galaxy (0.78c).
By substituting the known values into the formula, we can determine the wavelength detected on Earth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F440b7269-170b-4024-8d0e-4b8a2f594bf7%2F32ac4b76-0b75-43c8-8f68-0d42c4002c64%2Fmj1gavi_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Doppler Effect in Astronomy**
**Scenario:**
Suppose a galaxy is moving away from Earth at a speed of 0.78c, where "c" is the speed of light. The galaxy emits radio waves with a wavelength of 0.489 meters. What wavelength would we detect on Earth?
**Explanation:**
This problem deals with the Doppler Effect, which describes how the wavelength of waves (such as light or sound) changes due to the motion of the source relative to the observer. When an astronomical object like a galaxy moves away from Earth, the light it emits is stretched to longer wavelengths, a phenomenon known as redshift.
To calculate the observed wavelength on Earth, the following relativistic Doppler shift formula can be used:
\[
\lambda' = \lambda \times \sqrt{\frac{1 + \beta}{1 - \beta}}
\]
where:
- \(\lambda'\) is the observed wavelength,
- \(\lambda\) is the emitted wavelength (0.489 m),
- \(\beta = \frac{v}{c}\), where \(v\) is the speed of the galaxy (0.78c).
By substituting the known values into the formula, we can determine the wavelength detected on Earth.
Expert Solution
Concept and Principle:
- Doppler effect is the change in frequency and wavelength of a wave due to the motion of the source and observer. In the case of light since light requires no medium to travel the Doppler effect only depends on the relative velocity of the observer and source.
- When the source is moving away from the observer the observed wavelength will be longer than the emitted wavelength. This is known as redshift. When the source is moving towards the observer the wavelength will be shorter and is called blueshift.
- The observed wavelength is given by,
Here λ is the observed wavelength, λ0 is the emitted wavelength, v is the relative velocity, and c is the speed of light.
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