Hubble's Law
In 1912 Vesto Slipher began measuring spectra of the class of objects known as spiral nebulae. To his surprise he discovered that of 14 nebulae analyzed, 12 exhibited spectra shifted towards the red end of the spectrum. This information provided further evidence that the spiral nebulae were not objects associated with our Milky Way, but were in fact separate galaxies external to our own. The debate on the nature of the spiral nebula—a debate that was argued to a draw in the Curtis-Shapley debate—was settled once and for all by Edwin Hubble when he succeeded in identifying examples of the class of stars known as Cepheid variables in the spiral nebula M31 and calculated the giant spiral neblua's distance to be over 1 million light-years.
The 100-inch Hooker telescope
In the mid-1920s Hubble and his assistant Milton Humason replicated Slipher's work by photographing the spectra of about two dozen galaxies using the 100-inch Hooker reflecting telescope on the summit of Mount Wilson near Pasadena, California. They used a variety of techniques to determine the distances to the galaxies, but the most astounding discovery came in 1929 when Hubble combined his spectral data with the distance determinations and found that the majority of galaxies seemed to be receding from our Milky Way at velocities that were proportional to their distances. This relationship has come to be known as Hubble's law.
Procedure
In this exercise we will attempt to verify Hubble's law and determine the value of the Hubble parameter. We will use spectra and photographs of five representative elliptical galaxies from five clusters of galaxies. The recession velocity of each galaxy will be calculated by measuring the displacements of the K and H lines of neutral calcium (Ca I) found in the absorption spectra of solar-type stars in the galaxies. The rest wavelengths of the K and H lines are 3,933.7 Å and 3,968.5 Å respectively. The comparison spectrum we will use to determine the displacement of the K and H lines is the spectrum of helium.
To determine the distance to each of the galaxies, we will assume that all galaxies of this type (E0) have approximately the same physical diameter. The accepted value of the diameter is 0.03 Mpc. Therefore, when we observe these galaxies, the angular size of each galaxy is directly related to its distance by the small angle formula D = dθ where D is the physical diameter of the galaxy, d is its estimated distance, and θ is its angular diameter in radians.
Determining the Recession Velocities of Galaxies
To calculate the velocities we must first determine the plate scale of
the photograph showing the spectra. This plate scale is also commonly
called the dispersion of the photograph. If we look at the comparison
lines of the helium spectra superimposed above and below each galactic
spectrum (see below), we see that there are two helium lines labelled "a" and "e".
The wavelengths of those two spectral lines are 3,888.7 Å and 4,471.5 Å according to the table of wavelengths given on page 300 of the lab exercise handout. If we measure the distance between the centers of these two lines t in millimeters, we are able to calculate the dispersion in Å/mm. In other words using my measurement of t = 27.4 mm, the dispersion if found to be
Dispersion = (4,471.5 Å – 3,888.7 Å) ÷ t = 582.8 Å ÷ 27.4 mm = 21.3 Å/mm .
Next, to determine the displacements of the K and H lines (marked by a vertical arrow in the spectrum of the Virgo cluster galaxy), we must choose a convenient starting point. Because it is in the middle of the spectrum's photograph and is quite prominent, the "e" line of helium is chosen . To ensure that accurate displacements are measured, it may be helpful to draw a light pencil line from the center of the top "e" line to the center of the lower "e" line (note red line on figure). The center is chosen because the "e" line has an obvious horizontal spread.
The measurements (in mm) from the "e" line of the K and H line in the Virgo galaxy's spectrum are –24.9 mm and –23.2 mm respectively. Because the "e" line is our origin, displacements to the left are identified as negative while displacements to the right will be considered positive. For the best results, the distances to the K and H lines are measured from a reference pencil mark to the center of each line's spread (note the blue line for the K line and the orange line for the H line in the figure above). For greatest accuracy, measure the displacements to the nearest 0.1 mm.
Next, the wavelength differences between the reference "e" line and the K and H lines are calculated by multiplying the data from the first column by the dispersion of the photograph. In other words, for the Virgo galaxy's K line the wavelength difference is
Wavelength difference of K line = – 24.9 mm x 21.3 Å/mm = – 530 Å .
This measurement and calcuation must be repeated for the H line (see results in table of velocity determinations).
To find the actual redshifted values of the K and H lines, the wavelength differences are added to the wavelength of the "e" line (4,471.5 Å, as indicated at the top of column 3 on page 300). For example, the redshifted wavelength of the K line in the Virgo galaxy's spectrum is
Redshifted wavelength of K line = 4,471.5 Å + (–530 Å) = 3,942 Å .
Again, repeat the calcuation for the redshift of the H line.
Finally, the velocity of the Virgo galaxy can be calculated using the well-known Doppler equation
v = c Δλ/Δλo
where
v = c x (redshifted wavelength – "rest" wavelength) ÷ "rest" wavelength .
If we use a value of 300,000 km/sec for the speed of light c and 3,933.7 Å for the rest wavelength of the K line of Ca I, we find that the velocity of recession of the Virgo galaxy is
v = (300,000 km/sec) x (3,942 Å – 3,933.7 Å) ÷ 3,933.7 Å = 633 km/sec .
The calculation must be repeated for the H line using a value of the rest wavelength 3,968.5 Å. Note that in the calculation of the recession velocities of the K and H lines, the factor of c ÷ rest wavelength does not change. You can use 300,000 km/sec ÷ 3,933.7 Å for the K line and 300,000 km/sec ÷ 3,968.5 Å throughout the exercise. The example calculations are shown in the table below:
Determining the Distance to the Galaxies
Just as we had to determine a disperion for the photographs of the spectra,
we must also determine a value of the disperson for the photos of the
galaxies. To facilitate this, the handout has a convenient scale marking
labelled "150
arcseconds" and all we need to do to
find the dispersion in arcsec/mm is measure this line segment to the nearest
0.1 mm (see figure below).
So, using my measurement of s = 17.7 mm, the dispersion if found to be
150" ÷ s = 150" ÷ 17.7 mm = 8.47"/mm .
To determine the linear size of the galaxies in Virgo and Ursa Major, it is necessary to measure the diameter along two dimensions (such as the major and minor axes of these elliptical figures). The remaining three galaxies are nearly circular and only one measurement of diameter is necessary. For the Virgo galaxy, averaging the two measurements of diameter—going out into the fuzzy "halo" of the galaxy as far as possible—we find that the Virgo galaxy has a diameter of 19.5 mm (note on the figure at right that only the major axis w of the Virgo galaxy is shown as an example).
After recording the averaged diameter of the Virgo galaxy, the angular diameter of the galaxy in arcseconds must be calculated using the formula
Angular diameter = 19.5 mm x 8.47"/mm = 167" .
Next, the galaxy's diameter d in radians—a dimensionless unit that allows us to perform calculations using angular measurements)—can be calculated using the following formula described on page 301 of the handout. The number of arcseconds in a radian can be found by calculating
180° x 60'/1° x 60"/1' x π = 206.265 .
Therefore, the diameter of the Virgo galaxy in radians is
Diameter in radians = Angular diameter ÷ 206,265 .
Using our data, we have
d = 167 ÷ 206,265 = 0.00081 .
Finally, the distance D to the galaxy in the Virgo cluster is found using the formula on page 301, which includes the accepted value of the diameter (in megaparsecs) of the typical E0 galaxy. Thus,
D = 0.03 ÷ d .
Therefore
D = 0.03 Mpc ÷ 0.00081 = 37 Mpc
for the distance to the Virgo cluster galaxy. A summary of the calculations is shown in the following figure:
Results
The table below summarizes the results of our determination
of the recession velocity and distance to all of the galaxies studied.
Since we actually calculated the recession velocity of both the K line
and H line of Ca I, we must average the two numbers (476 and 584 km/sec)
to get a single recession velocity for each galaxy.
After measuring the spectral lines and diameters of the representative galaxies in the other four clusters, a plot of the data with a least-squares fit line allows us to calculate a slope of the data (Excel's Chart Wizard was used to construct the graph, determine the best fit line, and calculate the slope). That slope is the Hubble parameter Ho. In order to find the correct slope of the line, the y-intercept must be the origin (0,0).
If a hand-drawn method is used, an approximation of the least-squares fit can be made by drawing a line from the origin of the graph that extends through what appears to be the "average" position of the data points. If the line is correct, it would theoretically be possible to balance the graph on a knife edge if you position the least-squares line on the edge of the blade.
Calculate the value of the slope.
Finally, using a conversion factor of 9.78x1011 to change megaparsecs to kilometers and seconds to years, we can make a rough estimate of the expansion age of the Universe. Our result is
Expansion Age = 9.78x1011 ÷ Ho = 1.32x1010 yrs = 13.2 billion years.