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Chapter 21: Problem 38

You observe a distant quasar in which a spectral line of hydrogen with rest wavelength \(\lambda_{\text {rest }}=121.6 \mathrm{nm}\) is found at a wavelength of \(547.2 \mathrm{nm}\). What is its redshift? When the light from this quasar was emitted, how large was the universe compared to its current size?

Short Answer

Expert verified
The redshift is 3.5, and the universe was 4.5 times smaller when the light was emitted.

Step by step solution

01

Identify Given Data

Note the values provided in the problem: - Rest wavelength, \(\lambda_{\text{rest}} = 121.6 \text{ nm}\) - Observed wavelength, \(\lambda_{\text{observed}} = 547.2 \text{ nm}\).
02

Calculate the Redshift

Use the formula for redshift \(z\): \[ z = \frac{\lambda_{\text{observed}} - \lambda_{\text{rest}}}{\lambda_{\text{rest}}} \] Substitute the values: \[ z = \frac{547.2 \text{ nm} - 121.6 \text{ nm}}{121.6 \text{ nm}} \] Solve the equation: \[ z = \frac{425.6}{121.6} \] \[ z \approx 3.5 \]
03

Interpret Redshift to Universe Expansion

The redshift can also be used to understand how much the universe has expanded since the light was emitted. The formula for this is: \[ 1 + z = \frac{a_{\text{now}}}{a_{\text{then}}} \] where \(a_{\text{now}}\) is the current size of the universe, and \(a_{\text{then}}\) is the size of the universe when the light was emitted. With \(z = 3.5\): \[ 1 + 3.5 = \frac{a_{\text{now}}}{a_{\text{then}}} \] \[ 4.5 = \frac{a_{\text{now}}}{a_{\text{then}}} \] Solve for \(a_{\text{then}}\): \[ a_{\text{then}} = \frac{a_{\text{now}}}{4.5} \]

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Key Concepts

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

cosmology
Cosmology is the scientific study of the large scale properties of the universe as a whole. It involves understanding the origins, evolution, and large-scale structures of the universe. Cosmologists examine various phenomena such as the cosmic microwave background, galaxy formation, and the rate of universe expansion.
One of the key observations in cosmology is the redshift of light from distant galaxies and quasars. Redshift provides evidence that the universe is expanding. When studying the cosmos, astronomers rely heavily on this concept to draw conclusions about the age, size, and overall behavior of the universe.
The study of cosmology helps us understand where the universe is heading and its ultimate fate. Key tools in this field include telescopes, satellites, and theoretical models.
universe expansion
The universe expansion refers to the idea that the universe has been growing since the Big Bang. This concept was first discovered by observing that distant galaxies are moving away from us, which is evidenced by the redshift of their spectral lines. As light from these galaxies travels to us, the space through which it travels is expanding, stretching the light to longer wavelengths.
Key points to understand about universe expansion:
  • Hubble's Law: This states that the speed at which a galaxy moves away is proportional to its distance from us.
  • The Scale Factor: This represents the size of the universe at any given time compared to its current size.
  • Redshift and Scale Factor Relationship: Redshift can be used to determine how much the universe has expanded since the light left the galaxy or quasar.
By measuring redshift, we can derive the rate at which the universe is expanding and gain insights into the constants and parameters that govern cosmic evolution.
hydrogen spectral line
Hydrogen, the most abundant element in the universe, emits light at specific wavelengths when its electrons change energy levels. These specific wavelengths are known as spectral lines. The most familiar hydrogen line is the 121.6 nm wavelength, also known as the Lyman-alpha line.
When studying distant cosmic objects like quasars, astronomers observe these hydrogen spectral lines to determine their redshift. By comparing the observed wavelength of the spectral line with its rest wavelength, they can calculate how much the light has been stretched.
Key points about hydrogen spectral lines:
  • Rest Wavelength: The original wavelength of the light emitted by hydrogen.
  • Observed Wavelength: The wavelength detected by telescopes, shifted due to the expansion of the universe.
  • Redshift Calculation: Using the difference between the observed and rest wavelengths, astronomers calculate the redshift to understand the object's distance and the rate of universe expansion.
Understanding hydrogen spectral lines is crucial for astronomers because it helps to map the large-scale structure of the universe and trace its evolutionary history.

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Most popular questions from this chapter

Assume that the most distant galaxies have a redshift \(z=10\) The average density of normal matter in the universe today is \(4 \times 10^{-28} \mathrm{kg} / \mathrm{m}^{3} .\) What was its density when light was leaving those distant galaxies? (Hint: Keep in mind that volume is proportional to the cube of the scale factor.)

Some galaxies have redshifts \(z\) that if equated to \(v_{\mathrm{r}} / \mathrm{c}\) correspond to velocities greater than the speed of light. Special relativity is not violated in this case a. because of relativistic beaming. b. because of superluminal motion. c. because redshifts carry no information. d. because these velocities do not measure motion through space

The Hubble time \(\left(1 / H_{0}\right)\) represents the age of a universe that has been expanding at a constant rate since the Big Bang. Calculate the age of the universe in years if \(H_{0}=80 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\) (Note: 1 year \(=3.16 \times 10^{7}\) seconds, and \(1 \mathrm{Mpc}=3.09 \times 10^{19} \mathrm{km} .\)

Why is the Milky Way Galaxy not expanding together with the rest of the universe? a. It is not expanding because it is at the center of the expansion. b. It is expanding, but the expansion is too small to measure. c. The Milky Way is a special location in the universe. d. Local gravity dominates over the expansion of the universe.

The general relationship between recession velocity \(\left(v_{r}\right)\) and redshift \((z)\) is \(v_{r}=c z .\) This simple relationship fails, however, for very distant galaxies with large redshifts. Explain why.
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