Hubble’s Law
Velocities of galaxies can be determined by measuring the amounts of shifts of positions of
certain lines in galactic spectra. When information about the velocity of a galaxy is coupled
with the distance to the galaxy, we can gain some insight into the motions of galaxies with
respect to each other.
Question:
If you surveyed a large number of galaxies and determined their velocities would you expect that the
velocities would be randomly distributed (with no apparent trends) or would you expect to find a
systematic trend? If the latter, what might that trend be and what would it indicate?
Concept:
Just as sound waves appear to change their pitch based on motions of the source of the sound
and the observer with respect to each other (the well-known Doppler Effect), light waves
emitted by a source may appear to change their wavelength depending on the motions of the
source of the light and the observer with respect to each other. If the source of light is
approaching the observer (based on the net motion of the source and the observer), the
emitted light is shifted to shorter wavelengths (the blue end of the visible spectrum) and we say
that the spectrum is blue-shifted. If the source of light is receding from the observer (based on
the net motion of the source and the observer), the emitted light is shifted to longer
wavelengths (the red end of the visible spectrum) and we say that the spectrum is red-shifted.
Useful relation:
Imagine a particular line in the spectrum of a galaxy. Suppose that – if the galaxy were at rest –
the wavelength of this particular line would be 𝜆0 . Now suppose that the galaxy is in motion
such that the galaxy is either approaching or receding from an observer: the line in the
spectrum of the galaxy would be shifted to a different wavelength λ (now we can define the
total change in the wavelength of the line 𝛥𝜆 = 𝜆 − 𝜆0 . We can determine the velocity of the
galaxy (in units of kilometers per second or km/s) using the following relation
(where c is the velocity of light: 3 × 108 m/s):
𝛥𝜆 𝑣
=
𝜆0 𝑐
Note that if a source were moving toward us (thus the line would be blue-shifted), the
calculated velocity would be negative. If a source were moving away from us (thus the line
would be red-shifted), the calculated velocity would be positive.
In the case of distant galaxies receding from us due to the expansion of the universe – we call
this red shift in wavelength Cosmological Redshift. This is technically different that a Doppler
Redshift.
Stars, Galaxies and Cosmology
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Hubble’s Law
Exercises:
1. An astronomer makes an observation of the spairal galaxy NGC 4414. In the spectrum
of this galaxy, the astronomer notices that a spectral line of hydrogen that normally has
a rest wavelength 𝜆0 = 656.3 nm has been shifted to the wavelength λ = 657.9 nm.
a. Is this line red-shifted or blue shifted? Is the galaxy moving toward the Earth or
moving away from the Earth?
b. Calculate the recessional velocity (in km/s) of the galaxy.
2. In 1929, Edwin Hubble (the namesake of the Hubble Space Telescope) conducted a
survey that measured the distances and velocities of a sample of approximately 20
galaxies. Data from his survey are provided below:
Galaxy
NGC 278
NGC 584
NGC 936
NGC 1023
NGC 2681
NGC 2683
NGC 2841
NGC 3115
NGC 3368
NGC 3379
NGC 3489
NGC 3521
NGC 3623
NGC 4111
NGC 4526
NGC 4565
NGC 4594
NGC 5005
NGC 5866
Stars, Galaxies and Cosmology
Distance (Mpc)
11.1
25.5
19.8
10.3
10.0
5.2
9.2
6.8
10.9
11.2
8.1
9.2
9.9
11.6
5.2
16.8
12.4
13.6
11.2
Recessional Velocity (km/s)
808
1837
1446
754
692
376
674
493
797
814
589
669
723
848
381
1225
904
995
819
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Hubble’s Law
Create: (10 points: 5 for the graph and 5 for the paragraph)
Use Microsoft Excel to create a scatter plot of distance versus velocity for the galaxies. Your
graph must have the following:
1. An appropriate title (related to the contents of the graph)
2. Labeled axes with units (Example: Recessional Velocity (km/s))
3. a trend line (this is a line of best fit that is calculated by Excel – select add trend line)
4. aesthetically pleasing (that means to make it look good and be user friendly)
Create a word document titled “Hubble’s Law” with your name in the footer. Copy and paste
the graph into the word document. Make it as large as will fit on the page.
Write a paragraph below the graph commenting on your results. Is the scatter plot truly a
“scatter” or is a general trend apparent? Are all of the measured recessional velocities
indicative of blue-shifts, red-shifts or a mixture of the two? As the distance to the galaxy
increases, is there an obvious trend with the observed velocity? Identify the equation of this
line as Hubble’s Law and discuss the meaning of the slope. You should call this slope 𝐻0 (in
units of km/s/Mpc) and describe the relation between 𝑯𝟎 , the distance to the galaxy d and the
observed recessional velocity 𝒗𝒓𝒆𝒄 of the galaxy (in units of km/s) as
𝑣𝑟𝑒𝑐 = 𝐻0 𝑑
Comment on how your value of Hubble’s constant compares to the current accepted value of
𝐻0 = 73 𝑘𝑚/𝑠/𝑀𝑝𝑐.
𝑷𝒓𝒊𝒏𝒕 𝒚𝒐𝒖𝒓 𝒈𝒓𝒂𝒑𝒉 𝒂𝒏𝒅 𝒑𝒂𝒓𝒂𝒈𝒓𝒂𝒑𝒉 𝒂𝒏𝒅 𝒔𝒖𝒃𝒎𝒊𝒕 𝒕𝒐 𝒎𝒆 𝒃𝒚 𝑭𝒓𝒊𝒅𝒂𝒚.
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Hubble’s Law
Calculate:
5. Using the velocity that you calculated for NGC 4144 in Question 1, calculate the
distance to the galaxy using Hubble’s Law and the value for 𝐻0 that you determined in
questions 3.
6. If you inspect the units of 𝐻0 carefully, you may notice that there are units of distance
in both the numerator and the denominator (km and Mpc). It is therefore possible to
cancel out both units of distance and leave behind just a unit of time (seconds) in the
denominator. Furthermore, if you take the reciprocal of 𝐻0 (1/𝐻0 ), you can calculate a
period of time that (roughly) corresponds to the age of the universe
Using the value of 𝐻0 that you calculated in question 3, calculate the age of the
universe in years. How does your calculated value compare to the current accepted age
of the universe of 13.7 billion years?
Useful conversions: 1 Mpc = 3.086 × 1019 𝑘𝑚 and 1 year = 3.156 × 107 𝑠
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