∙Consider a spiral galaxy that is moving directly away from...

$\bullet$ Consider a spiral galaxy that is moving directly away from Earth with a speed $V=3.240 \times 10^{5} \mathrm{m} / \mathrm{s}$ at its center, as shown in FIGURE $25-40 .$ The galaxy is also rotating about its center, such that points in its spiral arms are moving with a speed $v=5.750 \times 10^{5} \mathrm{m} / \mathrm{s}$ relative to the center. If light with a frequency of $7.308 \times 10^{14} \mathrm{Hz}$ is emitted in both arms of the galaxy, what frequency is detected by astronomers observing the arm that is moving (a) toward and (b) away from Earth? (Measurements of this type are used to map out the speed of various regions in distant, rotating galaxies.)

$\bullet$ Consider a spiral galaxy that is moving directly away from Earth
with a speed $V=3.240 \times 10^{5} \mathrm{m} / \mathrm{s}$ at its center, as shown in FIGURE
$25-40 .$ The galaxy is also rotating about its center, such that points
in its spiral arms are moving with a speed $v=5.750 \times 10^{5} \mathrm{m} / \mathrm{s}$ relative to the center. If light with a frequency of $7.308 \times 10^{14} \mathrm{Hz}$ is
emitted in both arms of the galaxy, what frequency is detected by
astronomers observing the arm that is moving (a) toward and
(b) away from Earth? (Measurements of this type are used to map
out the speed of various regions in distant, rotating galaxies.)

Key Concepts

Doppler Effect

The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the source of the wave. In astronomy, it is used to determine whether objects such as stars or galaxies are moving toward (resulting in a blueshift) or away from us (resulting in a redshift), based on the analysis of their emitted light.

Radial Velocity

Radial velocity is the component of an object's velocity that is directed along the line of sight of the observer. It is crucial in interpreting the Doppler shift because only the motion along the direction towards or away from the observer contributes directly to the change in the observed frequency of the light.

Rotational Dynamics in Galaxies

Rotational dynamics in galaxies refer to the orbital motion of stars and gas around the galactic center. In spiral galaxies, different parts of the galaxy can have different velocity components relative to Earth due to this rotation; one side may be moving toward the observer while the opposite side moves away. This differential motion leads to variations in the observed spectral lines, which astronomers use to map the rotation and mass distribution within the galaxy.

Spectroscopic Analysis

Spectroscopic analysis is the method of studying the properties of light, including its wavelength and frequency components, to extract information about the physical properties and motion of celestial objects. By analyzing the shifts in spectral lines caused by the Doppler effect and rotational motion, astronomers can determine the speeds of various regions within galaxies and gain insights into their kinematic and dynamic properties.

Transcript

00:01 Alright, so here we have the earth, and we're interested in looking at a spiral galaxy that's emitting a particular frequency of light.

00:13 And since it's a spiral galaxy, the galaxy is over here, some of it is spinning towards us, and some of it is spinning away from us.

00:25 Now, the whole galaxy altogether is moving at some particular speed v, and when we're looking at some particular speed, v, and when we're looking, at this arm of the galaxy, the net speed is going to be v plus the spinning of the galaxy.

00:44 But when we're looking at this arm of the galaxy, the net speed is going to be v minus whatever s is.

00:54 So this is the net speed away from us, this is the net speed away from us.

00:59 Okay, so the first thing we're going to do is look at v and s.

01:03 V is equal to 3 .2 .4 times 10 to the fifth meters per second.

01:15 S, the spinning of this galaxy, is also just known as little v, but i'm going to call it s because it's easier for me to write down.

01:26 5 .75 times 10 to the fifth meters per year.

01:37 Second.

01:39 So actually we're going to have a net speed coming towards us.

01:43 So i'm just going to call this u towards us.

01:49 This is the net speed, is 3 .24 minus 5 .75 times 10 to the minus 5 .75, which is going to be minus 2 .54, 2 .51 times 10 to the 5th.

02:11 So it's minus in the away direction, which means that the net u coming towards us is actually a positive 2 .51 times 10 to the 5th.

02:24 So if we want to solve for a doppler shift, we know that f prime from our doppler shift equation is going to be equal to f times 1 plus because the spiral arm has a net direction towards.

02:44 Us 2 .51 times 10 to the 5 over 3 times 10 to the 8th...

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