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Ch 44: Particle Physics and Cosmology
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
Chapter 43, Problem 30b

The spectrum of the sodium atom is detected in the light from a distant galaxy. Use the Hubble law to calculate the distance of the galaxy from the earth.

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1
Identify the redshift (z) of the sodium spectrum by comparing the observed wavelength (λ_observed) of the sodium spectral line to its rest wavelength (λ_rest). Use the formula: z = (λobserved - λrest) / λrest.
Relate the redshift (z) to the recessional velocity (v) of the galaxy using the formula: v = z × c, where c is the speed of light (approximately 3.00 × 10⁸ m/s).
Apply Hubble's law to find the distance (d) of the galaxy from Earth. Hubble's law is given by: v = H0 × d, where H0 is the Hubble constant (approximately 70 km/s/Mpc). Rearrange the formula to solve for d: d = v / H0.
Convert the units of the Hubble constant to match the units of velocity (v) and ensure consistency. For example, if H0 is given in km/s/Mpc, convert it to m/s/Mpc if necessary.
Substitute the values of the recessional velocity (v) and the Hubble constant (H0) into the formula for distance (d) to calculate the galaxy's distance from Earth.

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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 velocity at which a galaxy is receding from an observer is directly proportional to its distance from that observer. This relationship is expressed mathematically as v = H₀d, where v is the recessional velocity, H₀ is the Hubble constant, and d is the distance. This law is fundamental in cosmology for determining distances to faraway galaxies based on their redshift.
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Redshift

Redshift refers to the phenomenon where light from an object is shifted to longer wavelengths as it moves away from an observer. In the context of galaxies, this shift can be measured in the spectrum of light emitted by elements like sodium. The amount of redshift provides information about the velocity of the galaxy's recession, which is crucial for applying Hubble's Law.

Spectroscopy

Spectroscopy is the study of the interaction between light and matter, allowing scientists to analyze the composition and properties of celestial objects. By examining the spectrum of light emitted or absorbed by elements, such as sodium, astronomers can identify specific wavelengths and determine redshift. This technique is essential for understanding the physical conditions of distant galaxies and their movement relative to Earth.