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Chapter 37: Problem 59

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda\) = 656.3 nm, in the red portion of the electromagnetic spectrum. In the light emitted from a distant galaxy this same spectral line is observed to be Doppler-shifted to \(\lambda\) = 953.4 nm, in the infrared portion of the spectrum. How fast are the emitting atoms moving relative to the earth? Are they approaching the earth or receding from it?

Short Answer

Expert verified
The emitting atoms are moving away with a speed of approximately \( 1.36 \times 10^8 \) m/s.

Step by step solution

01

Identify Given and Required Quantities

The given quantities are the emitted wavelength \( \lambda_{0} = 656.3 \text{ nm} \) and the observed wavelength \( \lambda = 953.4 \text{ nm} \). We are required to find the speed \( v \) of the source and determine if the source is moving towards or away from the Earth.
02

Apply Doppler Effect Formula for Light

The Doppler Effect formula for light when velocities are much less than the speed of light \( c \) is given by \[ \lambda = \lambda_{0} \left( 1 + \frac{v}{c} \right) \] where \( v \) is the speed of the source. Rearrange to solve for \( v \): \[ v = c \left( \frac{\lambda}{\lambda_{0}} - 1 \right) \].
03

Convert Wavelengths to Meters

Convert the given wavelengths from nanometers to meters: \( \lambda_{0} = 656.3 \times 10^{-9} \text{ m} \) and \( \lambda = 953.4 \times 10^{-9} \text{ m} \).
04

Substitute Values into Doppler Formula

Substitute the values into the formula: \[ v = 3.00 \times 10^{8} \text{ m/s} \left( \frac{953.4 \times 10^{-9}}{656.3 \times 10^{-9}} - 1 \right) \].
05

Calculate the Velocity

Perform the calculation: \[ \frac{953.4}{656.3} \approx 1.453 \], then calculate \( v = 3.00 \times 10^{8} \times (1.453 - 1) \approx 1.36 \times 10^{8} \text{ m/s} \).
06

Determine Direction of Motion

Since the observed wavelength is longer than the emitted wavelength, the spectral line is redshifted, indicating the source is moving away from the Earth.

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Key Concepts

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

Redshift
The term "redshift" refers to the phenomenon where the wavelength of light or other electromagnetic radiation from an object is increased, or shifted to the red part of the spectrum. This occurs when the object emitting the light is moving away from the observer. In simpler terms, the light's wavelength becomes longer, which we perceive as redder.

In the context of astronomy, redshift is a crucial concept because it helps scientists determine the velocity at which galaxies and other celestial objects are receding from Earth. The further away a galaxy is, the more significantly its light is stretched and redshifted.
  • An observed redshift in a galaxy's spectral lines, like the hydrogen line shifting from 656.3 nm to 953.4 nm, implies it is moving away from us.
  • This characteristic shift offers insights into the expansion of the universe, as it suggests a tendency for galaxies to move apart.
  • Moreover, the measurement of redshift can help estimate the speed at which a galaxy is receding, utilizing the Doppler Effect.
Speed of Light
The speed of light in a vacuum is approximately \( 3.00 \times 10^8 \text{ m/s} \). This fundamental constant of nature is denoted by the symbol \( c \). The constancy of the speed of light is one of the key postulates of Albert Einstein's theory of relativity, and it plays an essential role in the calculations involving the Doppler Effect for light.
  • In the Doppler Effect formula, the speed of light serves as a reference scale that relates changes in wavelength to the velocity of an astronomical object.
  • Because light speed is a constant, it allows us to precisely calculate the velocity of distant galaxies and cosmic objects when we observe how much the light is redshifted or blueshifted.
  • This also means any calculation involving light must consider that nothing can travel faster than light, adding constraints on our understanding of velocity within the universe.

The speed of light is pivotal not just in the context of the Doppler Effect but in numerous areas of physics, establishing a baseline velocity that underpins much of modern physics.
Wavelength
Wavelength is the distance between successive crests of a wave, particularly electromagnetic waves such as light. It is typically expressed in meters, but for light in the visible spectrum, nanometers (nm) are often used due to their smaller scale.
  • The wavelength of light is inversely proportional to its frequency: \[ \lambda = \frac{c}{f} \] where \( \lambda \) represents the wavelength, \( c \) is the speed of light, and \( f \) is the frequency.
  • Light with longer wavelengths, like red light, is found at one end of the visible spectrum, while shorter wavelengths, like blue or violet light, are at the other end.
  • Changes in wavelength, due to factors such as the Doppler Effect, indicate shifts in frequency and thus the velocity and direction of the source objects.

Understanding wavelength shifts is essential for inferring how fast and in which direction an object in space is traveling relative to Earth. These shifts enable astrophysicists to better comprehend the dynamics of the universe, such as galaxy expansion and star movement.

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Most popular questions from this chapter

Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of 0.9380c as measured in the laboratory. What is the magnitude of the velocity of one particle relative to the other?

An imperial spaceship, moving at high speed relative to the planet Arrakis, fires a rocket toward the planet with a speed of 0.920c relative to the spaceship. An observer on Arrakis measures that the rocket is approaching with a speed of 0.360c. What is the speed of the spaceship relative to Arrakis? Is the spaceship moving toward or away from Arrakis?

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth at a speed of 0.520c. If the radiation has a frequency of 8.64 \(\times\) 10\(^{14}\) Hz in the rest frame of the star, what is the frequency measured by an observer on earth?

The negative pion (\(\pi^-\)) is an unstable particle with an average lifetime of 2.60 \(\times\) 10\(^{-8}\)s (measured in the rest frame of the pion). (a) If the pion is made to travel at very high speed relative to a laboratory, its average lifetime is measured in the laboratory to be 4.20 \(\times\) 10\(^{-7}\) s. Calculate the speed of the pion expressed as a fraction of c. (b) What distance, measured in the laboratory, does the pion travel during its average lifetime?

Two particles in a high-energy accelerator experiment approach each other head-on with a relative speed of 0.890c. Both particles travel at the same speed as measured in the laboratory. What is the speed of each particle, as measured in the laboratory?
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