Problem 67 - University Physics with Modern Physics

Question

One of the wavelengths of light emitted by hydrogen atoms under normal laboratory conditions is \(\lambda=656.3 \mathrm{nm},\) 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 \(\lambda=953.4 \mathrm{nm}\) , 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?

Step-by-Step Solution

Answer

The atoms are moving away from Earth at about 1.071 × 10⁸ m/s.

Step 1: Identify the Problem

We are given the wavelengths of light \( \lambda_0 = 656.3 \, \text{nm} \) (emitted wavelength) and \( \lambda = 953.4 \, \text{nm} \) (observed wavelength). We need to find the velocity of the source using the Doppler effect formula for light.

Step 2: Write Doppler Effect Formula for Light

The relativistic Doppler shift formula is given by:\[ \frac{\lambda}{\lambda_0} = \sqrt{\frac{1 + v/c}{1 - v/c}} \]where \( \lambda \) is the observed wavelength, \( \lambda_0 \) is the original wavelength, \( v \) is the velocity of the source relative to the observer, and \( c \) is the speed of light \( c \approx 3 \times 10^8 \, \text{m/s} \).

Step 3: Rearrange the Formula to Solve for Velocity

First, we express the equation:\[ \lambda = \lambda_0 \sqrt{\frac{1 + v/c}{1 - v/c}} \]Then, rearrange to solve for \( v \):\[ \left(\frac{\lambda}{\lambda_0}\right)^2 = \frac{1 + v/c}{1 - v/c} \]\[ \frac{\lambda^2}{\lambda_0^2}(1 - v/c) = 1 + v/c \]

Step 4: Solve for \( v \)

Continuing from the rearranged equation, we multiply both sides by the denominator:\[ \frac{\lambda^2}{\lambda_0^2} - \frac{\lambda^2}{\lambda_0^2} \cdot \frac{v}{c} = 1 + \frac{v}{c} \]Rearranging gives:\[ \frac{\lambda^2}{\lambda_0^2} - 1 = \left( \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}} \right) \]Let's isolate \( v/c \). Define \( x = \frac{\lambda}{\lambda_0} \), then:\[ x^2 - 1 = \frac{v}{c} \left( x^2 + 1 \right) \]Solving for \( v/c \):\[ \frac{v}{c} = \frac{x^2 - 1}{x^2 + 1} \]Substitute \( x = \frac{953.4}{656.3} \approx 1.453 \):\[ \frac{v}{c} = \frac{1.453^2 - 1}{1.453^2 + 1} \approx \frac{2.111209 - 1}{2.111209 + 1} \approx \frac{1.111209}{3.111209} \approx 0.357 \]

Step 5: Calculate the Velocity

Now, calculate the velocity \( v = 0.357 \times 3 \times 10^8 \, \text{m/s} \approx 1.071 \times 10^8 \, \text{m/s} \). The positive sign of \( v \) indicates that the galaxy is moving away from the Earth.

Step 6: Conclusion

The emitting atoms are moving at approximately \( 1.071 \times 10^8 \, \text{m/s} \) away from the Earth, as observed from the redshift in the spectral line.

Key Concepts

Wavelength Shift

The concept of a wavelength shift is a critical component of understanding the Doppler effect, especially in astrophysics and cosmology. It refers to the change in wavelength of light or other electromagnetic waves as observed from a moving source relative to an observer.

• If the source is moving towards the observer, the wavelengths get shorter – this is known as a blueshift.
• Conversely, if the source is moving away, the wavelengths become longer, resulting in what is known as a redshift.

Significant shifts in wavelengths can tell us a lot about the velocity and direction of movement of distant stars and galaxies. In the context of the original problem with hydrogen spectral lines shifting from the visible to the infrared spectrum, it indicates that the galaxy is moving away from Earth, accompanied by lengthening (redshifting) of the wavelength.

Relativistic Physics

Relativistic physics, particularly as described by Einstein's theory of relativity, is essential when examining bodies moving at speeds close to the speed of light. In the case of the relativistic Doppler effect, the classical Doppler effect is modified to account for the high velocity of astronomical objects.

• The formula involves the speed of light, \(c\), as a constant, and incorporates the velocity \(v\) of the moving source.
• The equation used in the solution, \( \frac{\lambda}{\lambda_0} = \sqrt{\frac{1 + v/c}{1 - v/c}} \), shows how wavelengths are altered at such high speeds.

This equation highlights how motion affects time and space, especially at velocities approaching the speed of light. Calculating the velocity using the relativistic Doppler shift is crucial for accurate measurements when observing celestial objects that might be receding or approaching relative to the Earth.

Spectral Lines

Spectral lines are unique to different elements and occur when atoms emit or absorb specific wavelengths of light. These lines are used in astrophysics to identify elements present in stars and galaxies. Each element has a 'fingerprint' of spectral lines under specific conditions.

• For example, hydrogen emits a distinct line at \(656.3 \text{nm}\) known as the H-alpha line observed under laboratory conditions.
• When viewed through the prism of a telescope, these lines can shift due to motion, thus providing information on the velocity and direction of the source.

The shift of these lines to longer or shorter wavelengths helps scientists measure the Doppler effect precisely. In the provided problem, we analyzed a shift from the red to the infrared portion of the spectrum, indicating the galaxy's motion relative to Earth.

Redshift

Redshift is a term commonly associated with the expansion of the universe and the movement of objects within it. When a spectral line moves from a shorter wavelength to a longer wavelength, it undergoes a redshift.

• This implies the source of the light is moving away from the observer, a fundamental observation supporting the theory of an expanding universe.
• The degree of redshift can provide insights into how fast an object, like a galaxy, is receding.

In the exercise, the observed redshift indicates the galaxy's significant velocity away from Earth. Redshifts are not just useful for determining speed but also for calculating the distance to far-off galaxies, utilizing the correlation with the cosmological principle that more distant galaxies move away faster due to the universe's expansion. This helps astronomers map the universe's scale and shape.