Problem 78 - Fundamentals of Physics

Question

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of \(434 \mathrm{~nm}\) when observed in the laboratory, has a wavelength of \(462 \mathrm{~nm}\). (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

Step-by-Step Solution

Answer

The radial speed is approximately \(1.935 \times 10^7\) m/s, and the galaxy is receding from Earth.

Step 1: Understand the Concept of Redshift

The redshift is a phenomenon where the observed wavelength of light from a distant object, such as a galaxy, is longer than the actual wavelength measured in the laboratory. This increase in wavelength indicates the object is moving away from us. The formula for redshift is given by: \[ z = \frac{\lambda_{observed} - \lambda_{original}}{\lambda_{original}} \] where \( \lambda_{observed} \) is the observed wavelength, and \( \lambda_{original} \) is the actual wavelength.

Step 2: Calculate the Redshift (z)

Substitute the given values into the redshift formula: \[ z = \frac{\lambda_{observed} - \lambda_{original}}{\lambda_{original}} = \frac{462 \mathrm{~nm} - 434 \mathrm{~nm}}{434 \mathrm{~nm}} = \frac{28 \mathrm{~nm}}{434 \mathrm{~nm}} \] Calculating this gives: \( z \approx 0.0645 \).

Step 3: Use Redshift to Find Radial Speed

The radial speed \( v \) of the galaxy relative to Earth can be calculated using the formula: \[ v = zc \] where \( c \) is the speed of light, approximately \( 3.00 \times 10^8 \) m/s. Thus, \( v = 0.0645 \times 3.00 \times 10^8 \) m/s, which results in \( v \approx 1.935 \times 10^7 \) m/s.

Step 4: Determine the Galaxy's Movement Relative to Earth

Since a longer observed wavelength than the laboratory wavelength indicates a redshift, and redshift corresponds to the galaxy moving away, the galaxy is receding from Earth.

Key Concepts

Wavelength

The concept of "wavelength" is fundamental when discussing light and electromagnetic radiation. Wavelength is the distance between two consecutive peaks or troughs of a wave. It's usually measured in meters, but for light, it is often more convenient to use nanometers (nm), where 1 nm is one-billionth of a meter. The visible spectrum of light ranges from about 400 nm (violet) to 700 nm (red). Each color of visible light has a distinct wavelength; for example, light with a 434 nm wavelength appears blue. The concept is crucial in understanding how we can tell whether a distant object in the universe, such as a galaxy, is moving toward or away from us by the apparent change in the wavelength of the light (redshift or blueshift). In the context of astronomical observations, if the wavelength of the light we observe is longer than the one measured in a lab, it suggests a redshift, indicating the object is moving away from us.

Doppler Effect

The Doppler Effect is an essential principle that helps us understand how waves behave when the source and the observer are moving relative to each other. You might have noticed the siren of an approaching ambulance sounding higher-pitched than when it moves away. This same effect occurs with light. For light waves, an approaching source compresses the wavelengths (shifting them towards the blue end of the spectrum), while a receding source stretches the wavelengths (shifting them towards the red end of the spectrum). In astronomy, the Doppler Effect allows scientists to assess the speed and direction of stars and galaxies. By observing shifts in the wavelength of the light from these celestial objects, we can determine if they are approaching or moving away from us—an invaluable tool for understanding the dynamics of the universe.

Radial Speed

Radial speed refers to the speed at which an object moves directly toward or away from an observer. In the context of galaxies and other astronomical objects, it measures how fast these objects are moving along the line of sight from Earth.This can be calculated using the redshift or blueshift phenomena. The formula for finding radial speed, in this case, is derived from the redshift formula: \[ v = zc \]Here, \( v \) is the radial velocity, \( z \) is the redshift, and \( c \) is the speed of light (about \( 3.00 \times 10^8 \) meters per second). A positive radial speed indicates the object is moving away (receding), while a negative value would suggest it's approaching. This calculation is significant because it helps us understand the movement patterns and expansion of the universe.

Cosmology

Cosmology is the scientific study of the large-scale properties of the universe as a whole. It includes the study of the universe's origin, evolution, structure, dynamics, and ultimate fate. The fascinating thing about cosmology is that it possibly holds the answers to some of the most fundamental questions about existence. The observation of redshifts and blueshifts plays a crucial role in cosmology. These shifts in the electromagnetic spectrum of distant galaxies provide evidence for the expanding universe, as described by the Big Bang Theory. Discovering that galaxies are generally moving away from us suggests that the universe is expanding. This expansion was a groundbreaking revelation that changed our understanding of the universe's nature and future. Thus, by merging observations with theory, cosmology attempts to answer questions about the beginning of time, the size of the universe, and the fundamental physics that govern all cosmic entities.