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Chapter 37: Problem 36

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be \(0.4 \%\) longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching or receding from Earth?

Short Answer

Expert verified
(a) The radial speed is \(1.2 \times 10^6\) m/s. (b) The galaxy is receding from Earth.

Step by step solution

01

Identify the formula for redshift

When the wavelength of light from an object is shifted by a certain percentage, it is typically due to the Doppler Effect. In astronomy, this shift is given by the formula for redshift: \( z = \frac{\Delta \lambda}{\lambda} \) where \( \Delta \lambda \) is the change in wavelength and \( \lambda \) is the original wavelength. Here, the redshift \( z \) is given as \( 0.004 \) (or \( 0.4\% \)).
02

Calculate the radial velocity using redshift

The radial velocity \( v \) of a galaxy can be determined from the redshift \( z \) using the formula: \( v = c \times z \), where \( c \) is the speed of light, approximately \( 3 \times 10^8 \) m/s. Plugging in the values gives us \( v = 3 \times 10^8 \times 0.004 = 1.2 \times 10^6 \) m/s.
03

Determine the direction of motion

Since the wavelengths observed from the galaxy are longer than those emitted by Earth sources, the galaxy is experiencing a redshift. A redshift indicates an object is moving away from the observer. Therefore, the galaxy is receding from Earth.

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Key Concepts

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

Doppler Effect
The Doppler Effect is a fascinating phenomenon observed in waves, like sound and light. When a source of waves moves relative to an observer, the frequency and wavelength of the waves change. This effect is named after the Austrian physicist Christian Doppler.
The most familiar example of the Doppler Effect involves sound waves. When a car with a siren drives past, the pitch of the siren seems to change. It sounds higher as it approaches and lower as it moves away.
  • Sound Waves: The change in frequency explains why a passing siren changes pitch from high to low.
  • Light Waves: Similar shifts happen with light. When an object moves towards us, its light waves compress, creating a blueshift. When it moves away, the waves stretch, resulting in a redshift.
This effect is crucial in astronomy because it helps determine how objects like galaxies and stars are moving in relation to Earth.
Radial Velocity
In astronomy, radial velocity refers to the speed at which an object moves towards or away from us along our line of sight. It is measured using the Doppler Effect and is essential for understanding the motion of celestial bodies.
  • Receding Objects: When an object moves away, its light shows a redshift, indicating a positive radial velocity.
  • Approaching Objects: On the contrary, objects moving towards us exhibit a blueshift, showing a negative radial velocity.
Radial velocity is measured in meters per second (m/s) and provides insights into the cosmic dance of galaxies, stars, and planets. Understanding this movement allows scientists to map the dynamic universe.
Astronomy
Astronomy is the study of celestial objects and phenomena beyond Earth's atmosphere. It encompasses everything from stars and planets to galaxies and cosmology. This science has been vital in expanding our comprehension of the universe.
One of the critical tools in modern astronomy is spectroscopy, which involves analyzing the light from celestial objects. This allows astronomers to determine properties such as:
  • Composition: Identifying elements and compounds present.
  • Temperature: Gauging how hot a star or planet is.
  • Motion: Using the Doppler Effect to measure velocity and direction.
Astronomy strives to answer fundamental questions about the universe, such as its origin, evolution, and our place within it.
Light Wavelengths
Light is a form of electromagnetic radiation that travels in waves. Each wave has a characteristic wavelength, which is the distance between two consecutive peaks. Wavelength determines the type of light we see and is measured in nanometers (nm).
  • Visible Light: This spectrum ranges roughly from 400 to 700 nm, with violet having the shortest and red the longest wavelengths.
  • Beyond Visible: Light can also be in the form of ultraviolet (shorter wavelengths than violet) or infrared (longer than red).
The observation of light wavelengths is critical in astronomy. It allows scientists to infer various aspects of celestial objects, such as their speed and distance, utilizing tools like spectroscopy to analyze the changes in these wavelengths.

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Most popular questions from this chapter

To four significant figures, find the following when the kinetic energy is \(10.00 \mathrm{MeV}\) : (a) \(\gamma\) and (b) \(\beta\) for an electron \(\left(E_{0}=0.510998\right.\) \(\mathrm{MeV}\) ), (c) \(\gamma\) and (d) \(\beta\) for a proton \(\left(E_{0}=938.272 \mathrm{MeV}\right)\), and (e) \(\gamma\) and (f) \(\beta\) for an \(\alpha\) particle \(\left(E_{0}=3727.40 \mathrm{MeV}\right)\).

An electron of \(\beta=0.999987\) moves along the axis of an evacuated tube that has a length of \(3.00 \mathrm{~m}\) as measured by a laboratory observer \(S\) at rest relative to the tube. An observer \(S^{\prime}\) who is at rest relative to the electron, however, would see this tube moving with speed \(v(=\beta c)\). What length would observer \(S^{\prime}\) measure for the tube?

A Foron cruiser moving directly toward a Reptulian scout ship fires a decoy toward the scout ship. Relative to the scout ship, the speed of the decoy is \(0.980 c\) and the speed of the Foron cruiser is \(0.900 c\). What is the speed of the decoy relative to the cruiser?

Spatial separation between two events. For the passing reference frames of Fig. \(37-25\), events \(A\) and \(B\) occur with the following spacetime coordinates: according to the unprimed frame, \(\left(x_{A}, t_{A}\right)\) and \(\left(x_{B}, t_{B}\right) ;\) according to the primed frame, \(\left(x_{A}^{\prime}, t_{A}^{\prime}\right)\) and \(\left(x_{B}^{\prime}, t_{B}^{\prime}\right) .\) In the unprimed frame, \(\Delta t=t_{B}-t_{A}=1.00 \mu \mathrm{s}\) and \(\Delta x=\) \(x_{B}-x_{A}=240 \mathrm{~m} .\) (a) Find an expression for \(\Delta x^{\prime}\) in terms of the speed parameter \(\beta\) and the given data. Graph \(\Delta x^{\prime}\) versus \(\beta\) for two ranges of \(\beta:\) (b) 0 to \(0.01\) and (c) \(0.1\) to \(1 .\) (d) At what value of \(\beta\) is \(\Delta x^{\prime}=0 ?\)

A spaceship at rest in a certain reference frame \(S\) is given a speed increment of \(0.50 c .\) Relative to its new rest frame, it is then given a further \(0.50 c\) increment. This process is continued until its speed with respect to its original frame \(S\) exceeds \(0.999 c .\) How many increments does this process require?
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