Introduction
Now that we have seen how to determine distances from within our solar system out into our Galaxy, let's push our techniques a step further. Below are some schematic pictures showing our closest neighbor galaxies. Since astronomers first identified galaxies as objects outside the Milky Way Galaxy, they have sought to find the distances to individual galaxies.
The distances to other galaxies are so large that it is not convenient to express them in light-years, or parsecs. Instead we will use the unit megaparsec (Mpc), which is 1 million parsecs. To find the distance to a galaxy, we must search among its stars for a familiar object whose luminosity is known. Such objects are known as distance indicators or standard candles as we have seen before. Because their period is related to their luminosity, Cepheid Variable stars are good distance indicators. If we know the star's period, we can use the period-luminosity relation to learn its average absolute magnitude. By comparing its apparent magnitude with its absolute magnitude we can find the distance modulus which is related to the distance.
If we can locate Cepheids in a galaxy, we can determine the distance to the galaxy, but ground based telescopes cannot detect Cepheids beyond about 6 Mpc. Only a dozen or so galaxies are that close. To see Cepheids at much greater distances, one has to go above the Earth's atmosphere with a telescope such as the Hubble Space Telescope (HST). The HST can detect Cepheids to about 40 Mpc, over 6 times farther than ground based telescopes.
Beyond some limit, Cepheids are not visible, so astronomers must calibrate other distance indicators. With a known distance to a galaxy with a Cepheid, astronomers can calibrate other objects such as Red Giant stars, Globular Clusters, Supernovae, or even the galaxies themselves! Of course, astronomers would like to use the same distance indicator to measure all galaxies with, but techniques that work well with nearby galaxies can't be used for more distant galaxies. Thus, a distance scale or ladder must be pieced together from a few different methods.
Now that we have established some distance measures from
our classroom, and the Moon, to the Galaxy and beyond, we need to investigate
that fact that the universe is not just standing still, but that the galaxies
that we find distances to are for the most part speeding away from us at
great velocity.
Hubble's Law and the Expansion of the Universe
Astronomers have known since the 1920s that the universe is expanding. In that decade, many astronomers noted that any galaxies they observed were moving away from us. Apart from a few nearby galaxies, all the galaxies' spectra were redshifted indicating that most galaxies are moving away from us.
In 1929, Edwin Hubble was able to combine his knowledge of galaxy redshifts with an estimate of the distance to these galaxies. He found that the more distant a galaxy was, the faster the galaxy was moving away from us. This statement has become known as the Hubble Law. Mathematically, the Hubble Law is written as
where Vexp is the expansion, or recessional, velocity (how fast the galaxy is moving away from us), D is the distance to the galaxy, and Ho is the Hubble constant.
This Hubble Constant is one of the most sought-after numbers in astronomy today. The value of Ho can tell us about the future and the past of our universe. Astronomers have been trying to measure the Hubble constant ever since Edwin Hubble's discovery. The results still vary from about 55 to 80 km/s/Mpc.
In this lab, you will use CLEA software to demonstrate one method of finding the Hubble constant. To find Ho, you need to know the distance and the recessional velocity of at least one galaxy -- and as with all experimental data, the more galaxies the better. Then, plotting V vs. D gives you a series of data points which should fall close to a straight line. The slope of this line is the Hubble constant: voila!
Directions:
The CLEA Hubble program simulates a telescope, where you make your 'observations' of these distant galaxies. You will need to download and install it now if you have not done so already.
Step One: Gathering Data
In the next few paragraphs, any command that you have to run on the computer will be in bold print. First, print out this Data Table worksheet, then go double start your CLEA Hubble's Law software, Login, and choose the Run option to begin.
You will be presented with a view of the inside of the telescope dome. Open the Dome, turn the Tracking On (to keep your telescope following the stars around the sky), then choose the Field option to go to the first field of view, the Ursa Major II.
In each field, center one of the galaxies using the Slew Rate and Direction (E,W,N,S) controls. This is like using the finder scope on any telescope. Click on Change View so that you can center the slit of the spectrograph on the brightest part of the galaxy. The spectrograph slit is the two red lines. Now you are all set to use your spectrograph to get a spectrum of the galaxy.
Take a Reading, Start Count and run the exposure until the signal to noise is over 10 before you Stop Count. You can stop and restart the exposure at any time, so experiment with what happens if you stop the exposure at a low value of signal to noise. What does the spectra look like? What does it look like if you go to a higher value of signal to noise?
Two absorption lines should be visible in your galaxy
spectra. These are the Ca II H and K lines. If you click the
mouse anywhere on your spectra, the program will tell you what that wavelength
is. Use the mouse to measure the wavelengths of the deepest part
of the H and K lines. Note: K has a shorter wavelength than H.
Before you Return to the telescope environment:
Record in Table 1:
and in Table 2, Record:
1. Wavelength, 
,
you measure for both the H and K lines.
and in Table 3, Record:
1. Apparent Magnitude (m) of the galaxy.
Because this is a simulation, and not real life, you only
have to get spectra from one galaxy in each field. The software will
give you the same data from every galaxy in the field, so you only have
to look at one. However, you do have to look at each of the five
fields (Coma Berenices, Ursa Major 1 and 2, Bootes, and Corona Borealis).
Use the Field option to switch between fields. You have to
get back to the telescope view by clicking off the Change View button
before you can change fields.
Step Two: Determining Velocity
The two pieces of data you need to find Ho are recessional velocity and distance. The velocity can be calculated from the redshift of the spectral lines, Ca II H and K. Measured in the laboratory (i.e., at rest), the wavelengths of these lines would be
Use these rest wavelengths, 
,
to find the redshift of the lines, 
,
and then the velocity, V. is:
Since we want the velocity to have units of km/s (to match up with the Hubble constant), use the speed of light = 3 x 105 km/s. Calculate the velocity using the H line and then the K line, and then find an average velocity using both of those lines. Record this information in Table 2.
Step Three: Calculating Distance
This is usually the stumbling block when trying to find a value for the Hubble constant. While the recessional velocity can be measured fairly easily if the galaxy's motion is mostly unperturbed by other nearby galaxies, distance is a hard thing to measure. You have already used some methods to find distances to astronomical objects: parallax, main sequence fitting and standard candle usage such as Cepheid Variable stars. However, none of these methods work very well for really distant galaxies because they are so far away that we often can't resolve individual stars. We can measure apparent magnitude of an entire galaxy fairly easily, so if we knew its absolute magnitude we could simply use the distance modulus. The problem is then finding the absolute magnitude of the galaxy. This is where 'standard candles' come into the picture.
Simply put, a standard candle is a type of object which always has the same intrinsic brightness, or varies in brightness in a way we can predict (like a Cepheid). In the case of our galaxies here, we are going to assume that the big bright elliptical galaxies all have the same absolute magnitude, and that the absolute magnitude (Mv) is -22. These big elliptical galaxies will be our standard candles.
With the absolute magnitude and the apparent magnitude, the distance to each galaxy, in parsecs, is
Change the units of the result from parsecs to megaparsecs, since Mpc is what we will use to create our Hubble diagram. Record this information in Table 3.
Step Four: Creating our Hubble Diagram
We will now go into Excel to plot our data. We want to plot recessional velocity, V (i.e., your average V), versus distance, D, for the five galaxies you gathered data on. Once Excel is open, put the recessional velocties in the first column, and the distance in the second column. Then highlight the data and click on the chart icon in the toolbar, or alternately by choosing Insert, and then Chart, Excel will plot the data for you. Choose XY Scatter for the type of chart, and click Next which will take you to Step 2 of the Chart Wizard. Click Finish and the plot will appear.
Your points should fall close to a straight line.
Remembering that we here on Earth are at zero distance and zero velocity,
so you can add one more data point (0,0) to your plot to represent us.
Now make a best-fit line to your data, by choosing the Chart option and
Add Trendline. Choose the linear The Trend/Regression line, and then click
Options and check off the Display Equation on Chart. The equation given will be in y=mx+b format, where y are the
y values, x are the x values, b is the y intercept, and b is the slope
of the line. The slope of this best-fit line is Ho, the
Hubble constant! Record this to your Data Sheet.
Discussion Questions: