Assignment 1
Hubble Law
Due: Wednesday, 2024 April 17

Introduction
The Hubble Law is one of the fundamental pieces of observational evidence upon which the modern cosmological theory, the Big Bang Theory, is based. Here, we see how Hubble and Slipher first found the Hubble Law.
1. The first step is the determination of the redshift of a set of galaxies.

Spectrum
The above is a sample stellar spectrum. Because the light from galaxies (in the visual part of the spectrum) is roughly the sum of the light from billions of stars, normal galactic spectra will resemble those of a set of stars. For stars like the Sun, the two lines produced by Calcium II (ionized calcium), known as the H and K lines, are very strong and are expected to appear prominently in galactic spectra. The wavelengths of these two lines are 0.3934 and 0.3968 microns (1 micron = 10^-6 meters) for K and H lines, respectively. In the spectrum above, the H and K lines can be seen at the left end (the blue end) of the spectrum.

Get the redshifts for the following galaxy clusters using the spectrum of the representative galaxy shown for each cluster,

For the spectra shown above, note the following:
The observed locations (wavelengths) of the K and H lines are marked by the head of the arrows (the right end of the arrow).
The location of the rest wavelength of the lines is marked by the left end of the arrow.
The length of the arrow then gives the difference in the observed wavelength of the K and H Ca II lines compared to the rest avelengths of the lines.

The redshift of a galaxy is defined as the relative change in the wavelength of a spectral line of a given element. That is, we compare the difference in the observed wavelength of a line and the rest wavelength of the line to rest wavelength of the line. The redshift z is then formally given by
where the Greek letter λ stands for wavelength.

To calculate the redshift of each galaxy, first measure the length of each arrow in centimeters.
Next, note that the positions marked by the letters a and e on the spectra are at wavelengths 0.3889 and 0.4472 microns so that points a and e are separated by 0.4472 microns - 0.3889 microns = 0.0583 microns.
Use this information and how far apart (in centimeters) a and e sit to get the scale for the spectrum, that is, find out how many microns correspond to each centimeter on the figure.
The lengths of the arrows and the scale for the spectra, determine the change in wavelength of the K and H Ca II lines in microns.
Fill in the Change in K and H wavelength, and Redshift columns in the following Table.
For the rest wavelength, use the average of the Calcium II K and H lines, 0.3951 microns.

Galaxy Cluster
Change in K and H wavelength
Redshift
Distance

Virgo

59,000,000 ly or 18,000,000 pc

Ursa Major

Corona Borealis

Bootes

Hydra

2. We supplied the Distance to Virgo galaxy cluster in the above Table. Under the assumption that all galaxies have the same diameter, use the relative sizes of the images given above to estimate the distances to the other galaxies by comparing how large they are in comparson to the size of M87 (the galaxy shown for the Virgo cluster of galaxies). Enter your estimates in the above Table.
How good is your assumption that galaxies have the same diameter? Scan the web and find typical sizes of typical galaxies. Based on your search, estimate how accurate are the distance estimates for the galaxies given in your table.
3. Make a plot of the redshift (actually plot cz, where c is the speed of light) versus distance for the 5 galaxy clusters using cz as the vertical axis and Distance, D, for the horizontal axis. This plot forms Hubble's Law.
4. The Hubble relation is cz = H_oD, where the Hubble constant H_o is the slope of the Hubble relation. Based on your plot, what is the value of the Hubble constant? Estimate the uncertainty in your measurement of the Hubble constant. What are your sources of error?
5. Check sites (reputable sites) on the net for the current best value for the Hubble constant, H_o. What is the source for the current best value for the Hubble constant, H_o? How recent is the measurement?
6. State a question that occured to you from the first two weeks of classes.