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Chapter 39: Problem 20

The red shift. A light source recedes from an observer with a speed \(v_{\text {source }}\) that is small compared with \(c .\) (a) Show that the fractional shift in the measured wavelength is given by the approximate expression $$\frac{\Delta \lambda}{\lambda}=\frac{v_{\text {samec }}}{c}$$ This phenomenon is known as the red shift, because the visible light is shifted toward the red. (b) Spectroscopic measurements of light at \(\lambda=397 \mathrm{nm}\) coming from a galaxy in Ursa Major reveal a red shift of \(20.0 \mathrm{nm} .\) What is the recessional speed of the galaxy?

Short Answer

Expert verified
The recessional speed of the galaxy is approximately \(1.51 \times 10^6 \, m/s\).

Step by step solution

01

Understanding and deriving the redshift formula

When a light source moves from an observer, the light it emits experiences a Doppler shift, i.e., its frequency changes for the observer. When the source is retreating, the frequency decreases and wavelength increases, causing a redshift. Let us denote \(\Delta \lambda = \lambda' - \lambda\), where \(\lambda'\) and \(\lambda\) are the observed and emitted wavelengths, respectively. When \(v_{\text{source}} << c\) (speed much less than speed of light), we can use the non-relativistic Doppler effect equation \(\frac{\Delta \lambda}{\lambda}=\frac{v_{\text{source}}}{c}\).
02

Calculation of the Recessional speed

We can rearrange the formula from step 1 for \(v_{\text{source}}\): \(v_{\text{source}} = c \cdot \frac{\Delta \lambda}{\lambda}\). We have \(\Delta \lambda = 20.0 \, nm\) and \(\lambda = 397 \, nm\) given and the speed of light \(c = 3.00 \times 10^8 \, m/s\). After converting the wavelengths from nanometers to meters (1 nm = \(10^{-9}\) m), we can substitute these values into the formula to find the recessional speed of the galaxy.
03

Compute the Recessional speed

Convert the wavelengths to meters: \(\Delta \lambda = 20.0 \times 10^{-9} \, m\) and \(\lambda = 397 \times 10^{-9} \, m\). Putting these values in our formula: \(v_{\text{source}} = 3.00 \times 10^8 m/s \cdot \frac{20.0 \times 10^{-9} \, m}{397 \times 10^{-9} \, m} \approx 1.51 \times 10^6 \, m/s\).

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Key Concepts

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

Doppler Shift
Imagine you are standing by the roadside and an ambulance speeds by with its siren blaring. As it moves towards you, the sound pitch seems higher, but as it passes and moves away, the pitch lowers. This change in frequency, or shift, due to the movement of a sound or light source relative to an observer, is known as the Doppler shift.

The same principle applies to light: if a light source is moving away from an observer, the light appears to stretch out, increasing its wavelength and decreasing its frequency. This results in a redshift, so named because red is at the end of the visible spectrum with the longest wavelengths. If the source were moving towards the observer, the light would be blue-shifted, as the wavelengths would appear shorter and the frequencies higher.
Spectroscopic Measurements
Spectroscopy is the study of how matter interacts with electromagnetic radiation. When light from a source, like a distant galaxy, passes through a spectrograph, it is spread out into its constituent colors鈥攎uch like a prism spreads out white light into a rainbow. This spread or spectrum reveals lines known as absorption or emission lines, corresponding to specific wavelengths where a material has either absorbed or emitted light.

Scientists use these spectroscopic measurements to determine properties of astronomical objects, including their chemical composition, temperature, density, and relative motion towards or away from us. These measurements are vital for calculating the redshift of light from distant galaxies, as these spectral lines will be shifted from their standard positions if the galaxy is moving relative to us.
Recessional Speed of Galaxies
In astronomy, one of the key discoveries was that most galaxies are moving away from us, and their recessional speed can be measured by analyzing the redshift of the light they emit. The farther away a galaxy is, the faster it appears to be moving away鈥攖his is Hubble's Law in action.

By examining the redshift, astronomers can calculate how fast a galaxy is receding. This provides insights into the dynamics of the universe, including its expansion. The redshift can be converted into a velocity by using the formula derived from the Doppler effect for non-relativistic speeds, which applies to cases where a galaxy's speed is significantly less than the speed of light.
Non-Relativistic Doppler Effect
The non-relativistic Doppler effect applies when the speed of the source is much less than the speed of light (denoted by 'c'). In this scenario, the change in wavelength (\(\frac{\text{Δ}\text{λ}}{\text{λ}}\text{ }\text{ }\)) observed is proportionally related to the source's velocity divided by the speed of light (\(\text{}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\)), as expressed in the provided solution.

This approximation is valid for everyday scenarios and is commonly used in astronomy for objects moving at speeds that are not close to the speed of light. Since galactic scales can be colossal, even speeds that seem high to us on Earth are non-relativistic in these terms. Thus, we can use this simplified version of the Doppler effect to estimate the recessional speed of galaxies.

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Most popular questions from this chapter

Show that the speed of an object having momentum of magnitude \(p\) and mass \(m\) is $$ u=\frac{c}{\sqrt{1+(m c / p)^{2}}} $$

A spacecraft is launched from the surface of the Earth with a velocity of \(0.600 c\) at an angle of \(50.0^{\circ}\) above the horizontal positive \(x\) axis. Another spacecraft is moving past, with a velocity of \(0.700 c\) in the negative \(x\) direction. Determine the magnitude and direction of the velocity of the first spacecraft as measured by the pilot of the second spacecraft.

A physicist drives through a stop light. When he is pulled over, he tells the police officer that the Doppler shift made the red light of wavelength 650 nm appear green to him, with a wavelength of \(520 \mathrm{nm}\). The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony?

According to observer \(\mathrm{A}\), two objects of equal mass and moving along the \(x\) axis collide head on and stick to each other. Before the collision, this observer measures that object 1 moves to the right with a speed of \(3 c / 4\) while object 2 moves to the left with the same speed. According to observer \(\mathrm{B}\), however, object 1 is initially at rest. (a) Determine the speed of object 2 as seen by observer B. (b) Compare the total initial energy of the system in the two frames of reference.

A ball is thrown at \(20.0 \mathrm{m} / \mathrm{s}\) inside a boxcar moving along the tracks at \(40.0 \mathrm{m} / \mathrm{s} .\) What is the speed of the ball relative to the ground if the ball is thrown (a) forward (b) backward (c) out the side door?
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