Velocities of Galaxies Can Be Determined by Measuring the Amounts of Shifts of Positions

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 . 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 . 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: m/s):

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 Page 1 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 = 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 Distance (Mpc) Recessional Velocity (km/s) NGC 278 11.1 808 NGC 584 25.5 1837 NGC 936 19.8 1446 NGC 1023 10.3 754 NGC 2681 10.0 692 NGC 2683 5.2 376 NGC 2841 9.2 674 NGC 3115 6.8 493 NGC 3368 10.9 797 NGC 3379 11.2 814 NGC 3489 8.1 589 NGC 3521 9.2 669 NGC 3623 9.9 723 NGC 4111 11.6 848 NGC 4526 5.2 381 NGC 4565 16.8 1225 NGC 4594 12.4 904 NGC 5005 13.6 995 NGC 5866 11.2 819

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 (in units of km/s/Mpc) and describe the relation between , the distance to the galaxy d and the observed recessional velocity of the galaxy (in units of km/s) as

Comment on how your value of Hubble’s constant compares to the current accepted value of .

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 that you determined in questions 3.

6. If you inspect the units of 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 (1/), you can calculate a period of time that (roughly) corresponds to the age of the universe

Using the value of 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 = and 1 year =

Stars, Galaxies and Cosmology Page 4