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Ch. 44 - Astrophysics and Cosmology
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 44 - Astrophysics and CosmologyProblem 27
Chapter 39, Problem 27

If a galaxy is traveling away from us at 2.2% of the speed of light, roughly how far away is it?

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Identify the relationship between the speed of a galaxy moving away from us and its distance using Hubble's Law: \( v = H_0 d \), where \( v \) is the recession velocity, \( H_0 \) is the Hubble constant, and \( d \) is the distance to the galaxy.
Express the given velocity \( v \) in terms of the speed of light \( c \). Since the galaxy is traveling at 2.2% of the speed of light, \( v = 0.022c \).
Substitute the value of \( v \) into Hubble's Law: \( 0.022c = H_0 d \). Rearrange the equation to solve for \( d \): \( d = \frac{0.022c}{H_0} \).
Determine the value of the Hubble constant \( H_0 \). A commonly used approximate value is \( H_0 = 70 \; \text{km/s/Mpc} \). Convert \( H_0 \) to consistent units if necessary (e.g., meters per second per megaparsec).
Substitute the values of \( c \) (speed of light, \( 3 \times 10^8 \; \text{m/s} \)) and \( H_0 \) into the equation for \( d \) to calculate the distance. Ensure all units are consistent before performing the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hubble's Law

Hubble's Law states that the recessional velocity of a galaxy is directly proportional to its distance from us. This relationship is expressed as v = H₀d, where v is the velocity, H₀ is the Hubble constant, and d is the distance. This law provides a framework for estimating distances to galaxies based on their observed velocities.
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Gauss' Law

Speed of Light

The speed of light in a vacuum is approximately 299,792 kilometers per second (or about 186,282 miles per second). In the context of astronomy, it serves as a fundamental constant that helps in understanding the vast distances in the universe. When a galaxy is moving away from us at a fraction of this speed, it allows us to calculate its distance using Hubble's Law.
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The Doppler Effect (Light)

Cosmological Redshift

Cosmological redshift refers to the phenomenon where light from distant galaxies is shifted to longer wavelengths due to the expansion of the universe. As a galaxy moves away, the light it emits stretches, resulting in a redshift that can be measured. This redshift is directly related to the velocity of the galaxy, which can then be used to infer its distance from Earth.
Related Practice
Textbook Question

(II) Suppose that three main-sequence stars could undergo the three changes represented by the three arrows, A, B, and C, in the H–R diagram of Fig. 44–35. For each case, describe the changes in temperature, intrinsic luminosity, and size.

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Textbook Question

We saw earlier (Chapter 19) that the rate energy reaches the Earth from the Sun (the “solar constant”) is about 1.3 x 10³ W/m². What is (a) the apparent brightness b of the Sun, and (b) the intrinsic luminosity L of the Sun?

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Textbook Question

Starting from Eq. 44–3, show that the Doppler shift in wavelength is ∆λ/λᵣₑₛₜ ≈ v/c (Eq. 44–6) for v ≪ c. [Hint: Use the binomial expansion.]

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Textbook Question

Calculate the peak wavelength of the CMB at 1.0 s after the birth of the universe. In what part of the EM spectrum is this radiation?

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Textbook Question

At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons (M ≈ 500 MeV/ c²), (b) Y (M ≈ 9500 MeV/c²), and (c) muons (M ≈ 100 MeV/c²)?

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