example. First, you will need to determine how far away the galaxy is and how fast it is moving. For this exercise we will assume that:

1.

The absolute magnitude, M, for all of our selected galaxies is -22. (Don’t forget the minus sign!)

2.

The laboratory (“at rest”) value for the wavelength of the calcium K line is 3933.67 Å

3.

The laboratory (“at rest”) value for the wavelength of the calcium H line is 3968.847 Å

The distance to the galaxy is determined from comparing the apparent magnitude, m, with the absolute magnitude, M. The distance, D, to the galaxy is then found from

This is column 5 on your worksheet and will give you the distance in units of parsecs (pc), you will need to convert that to kilometers (km) for column 6 on your worksheet by multiplying by 3.09 X 10^{13 }km/pc.

Next you need to determine the speed of recession (radial velocity) of the galaxy. These results will go into columns 7 and 8 on your worksheet. For this we will use the Doppler shift of the spectral lines that we measured. The Doppler shift equation is

which means,

In this equation you should use the speed of light, c, as 300,000 km/s then the resulting velocity, v, will be in units of km/s. You will need to determine the Doppler shift velocity for the H and K lines independently and then average them.

You can now determine the age of the universe by dividing the distance, D, by the radial velocity, v. This will give you the age of the universe in seconds, column 10 on the worksheet. I.e.,

In particular, you should use the value of D that is in km and the averaged value of v that is in km/s. You will now need to convert the time from seconds to years. Do this by dividing the time (in seconds) by 3.15 X 10^{7 }seconds per year. This is column 11 on the worksheet.

Once you have done this for each galaxy you can average all the final values for the age of the universe as determined from each galaxy. In other words, average the values in column 11 on your worksheet. This will be your final result for the age of the universe. (Remember that 10^{10 }is the same thing as “times 10 billion.”)

Example: (I strongly suggest that you try typing this example in on your own calculator if you feel uncomfortable with the math. If your results don’t match mine then you should figure out why before you proceed with your data analysis. I have provided you with two sample results to check yourself.)

Under the first field of view (Ursa Major II) I have selected a galaxy that turns out to be UMa2-2 the apparent magnitude is 16.67 and the K and H lines are measured at 4484 Å and “off scale”, respectively. All this data is entered in columns 1-4 as shown on your worksheet.

# Next, use equation (1) to determine the distance from the brightness information,

This result is entered into column 5. Now we need to convert the distance from parsecs into meters by multiplying by 3.09 X 10^{13 }km/pc, the result of that (1.67×10^{22 }km) is placed in column 6.

5

(4)