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Chapter 7: Problem 8

We define the redshift, \(z\), as the shift in wavelength, \(\Delta \lambda\) divided by the rest wavelength \(\lambda_{0}\) On the assumption that only radial motions are involved, find an expression for \(z\) as a function of \(v / c\).

Short Answer

Expert verified
The redshift \(z\) is given by \(z = \frac{v}{c}\).

Step by step solution

01

- Understand the Redshift Definition

The redshift, denoted by \(z\), is defined as the shift in the wavelength \(\Delta \lambda\) divided by the rest wavelength \(\lambda_0\). Formally, \(z = \frac{\Delta \lambda}{\lambda_0}\).
02

- Express the Wavelength Shift

The change in wavelength \(\Delta \lambda\) is given by \(\Delta \lambda = \lambda - \lambda_0\), where \(\lambda\) is the observed wavelength.
03

- Relation Between Wavelength and Velocity

For an object moving radially with velocity \(v\), the observed wavelength \(\lambda\) changes due to the Doppler effect. The relationship can be expressed as \(\lambda = \lambda_0 (1 + \frac{v}{c})\), where \(c\) is the speed of light.
04

- Substitute the Observed Wavelength

Substitute the expression for the observed wavelength into the redshift definition. This gives \(z = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda - \lambda_0}{\lambda_0}\).
05

- Simplify the Expression

Knowing \(\lambda = \lambda_0 (1 + \frac{v}{c})\), substitute it into the redshift formula: \(z = \frac{\lambda_0 (1 + \frac{v}{c}) - \lambda_0}{\lambda_0}\).
06

- Solve for Redshift

Simplify the equation: \(z = \frac{\lambda_0 + \lambda_0 \frac{v}{c} - \lambda_0}{\lambda_0} = \frac{\lambda_0 \frac{v}{c}}{\lambda_0} = \frac{v}{c}\). Thus, the expression for the redshift as a function of velocity is \(z = \frac{v}{c}\).

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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 phenomenon that explains how the frequency or wavelength of a wave changes if the source or observer is moving. The most common example is the change in pitch of a car horn as it zooms past you. In astronomy, this effect helps us understand the movement of stars and galaxies.

When a star or galaxy moves towards us, the wavelengths of light they emit are compressed, leading to a 'blueshift'. Conversely, if they move away, the wavelengths stretch, causing a 'redshift'. This shift allows astronomers to calculate the speed and direction of these celestial objects.
Wavelength Shift
Wavelength shift refers to the change in the observed wavelength of light due to motion. It is crucial for understanding the redshift in distant astronomical objects.

  • The rest wavelength (\( \lambda_0 \)) is the wavelength of light when the source is at rest with respect to the observer.
  • The observed wavelength (\( \lambda \)) changes if the source moves relative to the observer.
This change in wavelength (\( \Delta \lambda \)) is calculated by the difference between the observed and rest wavelengths:
\( \Delta \lambda = \lambda - \lambda_0 \)
Radial Velocity
Radial velocity is the component of an object's velocity that is directed towards or away from the observer. It's a crucial concept for understanding stellar and galactic motions.

In our context, the radial velocity (\( v \)) affects the observed wavelength due to the Doppler Effect. If an object moves away from us, its emitted light is redshifted, meaning the wavelengths grow longer. Conversely, if it approaches us, the light is blueshifted.

Using the Doppler Effect, we can determine the redshift (\( z \)) via radial velocity
\( z = \frac{v}{c} \)
Speed of Light
The speed of light (\( c \)) is a fundamental constant in physics, approximately equal to 299,792 kilometers per second. It plays a critical role in astronomical calculations.

When we talk about redshift in the context of radial velocity, we use the speed of light. For instance, the formula for redshift (\( z \)) derived from an object's velocity (\( v \)) is
\( z = \frac{v}{c} \).
This relationship shows how the object's speed relative to the speed of light affects the observed wavelength.

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Most popular questions from this chapter

The angular displacement of an image (in radians) due to aberration is approximately \(v / c,\) as long as \(v \ll c .\) Use the fact that the Earth orbits the Sun once per year at a distance of \(1.50 \times 10^{8} \mathrm{km}\) to find the maximum displacement of a star's image due to the motion of the Earth. Express your answer in arc seconds.

Some radioactive particles are traveling at \(0.999 c .\) If their lifetime is \(10^{-20}\) s when they are at rest, what is their lifetime at this speed? How far do they travel in that time (as viewed in the frame at which they are moving at \(0.999 c ?\)

You and your friend carry identical meter sticks and identical clocks. Your friend goes by on a fast moving train, holding the meter stick parallel to the direction of motion of the train. If in the time it takes your clock to tick off 1.00 s you see 0.5 s tick off on your friend's clock, how long does your friend's stick appear to you? How long does your stick appear to your friend, assuming the sticks are parallel to each other.

Show that if tachyons exist, their rest mass must be an imaginary number if the energy is to be real for \(v > c\). (An imaginary number is the square root of a negative number.)

You and your friend carry identical clocks. Your friend passes by in a rapidly moving train. As your clock ticks off 1.00 s, you see your friend's clock tick off 0.50 s. How much time would your friend see your clock tick off in the time it takes your friends clock to tick off 1.00 s?
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