91Ó°ÊÓ

Motion toward or away from us distorts the pitch of sound, and it also distorts the wavelength of light. This phenomenon is known as the Doppler effect. In the case of light, the distortion is measurable only for objects moving at extremely high velocities. Motion of objects toward us produces a blue shift in the spectrum, whereas motion of objects away from us produces a red shift. Quantitatively, the red shift \(S\) is the change in wavelength divided by the unshifted wavelength, and thus red shift is a pure number that has no units associated with it. Cosmologists believe that the universe is expanding and that many stellar objects are moving away from Earth at radial velocities sufficient to produce a measurable red shift. Particularly notable among these are quasars, which have a number of important properties (some of which remain poorly understood). \({ }^{11}\) Quasars are moving rapidly away from us and thus produce a large red shift. The radial velocity \(V\) can be calculated from the red shift \(S\) using $$ V=c \times\left(\frac{(S+1)^{2}-1}{(S+1)^{2}+1}\right) $$ where \(c\) is the speed of light. a. Most known quasars have a red shift greater than 1. What would be the radial velocity of a quasar showing a red shift of 2 ? Report your answer as a multiple of the speed of light. b. Make a graph of the radial velocity (as a multiple of the speed of light) versus the red shift. Include values of the red shift from 0 to 5 . c. The quasar \(3 C 48\) shows a red shift of \(0.37\). How fast is \(3 C 48\) moving away from us? d. Find approximately the red shift that would indicate a radial velocity of half the speed of light. e. What is the maximum theoretical radial velocity that a quasar could achieve?

Step by step solution

01

Calculate Radial Velocity for Red Shift of 2

Given the red shift \( S = 2 \), substitute into the formula:\[V = c \times \frac{(S+1)^{2}-1}{(S+1)^{2}+1}\]Substitute \( S = 2 \):\[V = c \times \frac{(2+1)^{2}-1}{(2+1)^{2}+1} = c \times \frac{8}{10} = 0.8c\]Thus, the radial velocity of the quasar is \( 0.8c \).

02

Graph Radial Velocity vs. Red Shift

Construct a graph with the red shift \( S \) on the x-axis, ranging from 0 to 5, and the radial velocity as a multiple of the speed of light \( V/c \) on the y-axis. For each \( S \), calculate \( V \) as in Step 1, and plot the points. The function is given by:\[V = c \times \frac{(S+1)^{2}-1}{(S+1)^{2}+1}\]The graph will show \( V/c \) increasing with \( S \).

03

Calculate Radial Velocity for 3C 48 (Red Shift of 0.37)

For \( S = 0.37 \), use the formula:\[V = c \times \frac{(S+1)^{2}-1}{(S+1)^{2}+1}\]Substitute \( S = 0.37 \):\[V = c \times \frac{(0.37+1)^{2}-1}{(0.37+1)^{2}+1} = c \times \frac{(1.37)^{2}-1}{(1.37)^{2}+1} \approx c \times 0.300\]Thus, the radial velocity is approximately \( 0.300c \).

04

Find Red Shift for Radial Velocity of 0.5c

Set \( V = 0.5c \) and solve for \( S \):\[0.5 = \frac{(S+1)^{2}-1}{(S+1)^{2}+1}\]Multiply both sides by \((S+1)^{2}+1\):\[0.5((S+1)^{2}+1) = (S+1)^{2}-1\]Solve the quadratic equation to find \( S \). Upon solving, \( S \approx 0.75 \).

05

Maximum Theoretical Radial Velocity

As the red shift \( S \) approaches infinity, the term \( \frac{8}{10} \) in the expression for \( V \) approaches 1. Thus, the maximum theoretical radial velocity \( V \) is potentially approaching \( c \), but cannot exceed \( c \).

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Key Concepts

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

Red Shift

Red shift is a crucial concept in understanding how we can detect the movement of celestial objects. When an object in space moves away from us, the light it emits stretches out, leading to a shift towards the red part of the light spectrum.
This change in wavelength is not just a theoretical concept but a measurable one, calculated as the ratio of the change in wavelength to the original wavelength. Red shift values are unitless numbers that can tell us a lot about the movement of stars, galaxies, and other cosmic entities.

The higher the red shift value, the faster the object is receding from us. This phenomenon is pivotal in cosmology, playing a significant role in our understanding of the universe's expansion. The observations of red shifts in distant galaxies provide strong evidence supporting the Big Bang theory.

Radial Velocity

Radial velocity refers to the speed at which an object moves toward or away from us along our line of sight.
It's an essential metric for astronomers because it helps them determine the motion of stars and galaxies relative to Earth.

Radial velocity is intrinsically linked to red shift, as the red shift values can be used to calculate how fast an object is moving away. For example, the formula given in the original problem,\[ V = c \times \left(\frac{(S+1)^{2}-1}{(S+1)^{2}+1}\right) \]allows the conversion of a measured red shift to a radial velocity, illustrating the speed of the object as a fraction of the speed of light.

Recognizing radial velocity helps astronomers trace the movements within the universe, further confirming the theory of its expansion.

Speed of Light

The speed of light, often denoted by the letter \'c\', is approximately \(299,792,458\, \text{meters per second}\).
It’s not just a physical limit but a cornerstone in physics and astronomy.

When we talk about objects in space, the speeds involved are so vast that they're often compared to the speed of light. In calculations like determining radial velocity, the speed of light provides a standard benchmark.
Since nothing can travel faster than light, expressions like "a radial velocity of \(0.8c\)" tell us that a quasar is moving away at 80% of the speed of light. This comparison makes it easier to grasp how significant the velocity is on a cosmic scale.

Understanding the speed of light in these terms gives us a universal yardstick to measure astronomical phenomena.

Cosmology

Cosmology is the scientific study of the large-scale properties of the universe as a whole.
It has evolved dramatically with the aid of observations like red shift and radial velocity, giving us insights into the universe's origin, current structure, and potential fate.

One of the fundamental theories in cosmology is the Big Bang theory, which postulates that the universe began from an extremely dense and hot state and is expanding over time.
Observations, especially those involving red shift, show that galaxies are moving away from us, corroborating this expansion theory.

Moreover, measuring red shifts and radial velocities helps cosmologists map the universe's structure and learn about the dynamics of celestial bodies.

• This involves understanding phenomena like black holes, cosmic microwave background radiation, and dark energy.
• All of which contribute to a more complete picture of our universe.

Leap beyond our galaxy and delve into cosmology to unlock the mysteries of the cosmos.