Directions to follow if doing all calculations by hand and using a calculator

The “by-hand” spread sheet (example shown below) has all of the appropriate columns.

Find the distance in megaparsecs to each galaxy by following these steps:

  1. Measure the X and Y values for determining the angular size of each galaxy.
  2. Use the Pythagorean theorem to find the long axis of each galaxy.
  3. Calcuate the angular size of each galaxy by multiplying its length in pixels by 0.0093 micro-radians per pixel.
  4. Find the distance to each galaxy in megaparsecs by dividing the actual size of the galaxy (assumed to be 22 kiloparsecs) by the angular size in micro-radians.

Find the radial (recessional) velocity to each galaxy by following these steps:

  1. Find the X pixel value for the rest wavelength of each line.
  2. Find the X pixel value for the red-shifted wavelength of each line.
  3. Subtract the rest wavelength pixel value from the red-shifted pixel value and multiply the difference in pixels by 0.76 Ångstroms per pixel.
  4. Calculate the redshift, z for each of the lines (divide the result from step 3 just above by the rest wavelength of the line), and enter these data in the z boxes on the worksheet.
  5. Find the average of the redshifts for each galaxy, and enter it in the table.
  6. Use this average redshift to find the velocity and enter this in the table in the radial velocity column:
    velocity(v) = speed of light(c) times redshift(z) .
  7. As stated in the background and theory section, we assume that all of these galaxies are about the same size. From other methods we know that galaxies of the type used in this lab are about 22 kpc (1 kpc = 1000 pc) across. Find the distance to each galaxy using the small angle formula, adapted for the units we are using here and record this distance in your data table under Distance:

    d (Mpc) = s (kpc) / a (micro-rad) .

Graph your results and find the Hubble constant by following these steps:

  1. Make a graph of your data (please use graph paper), with distance on the x-axis, and velocity on the y-axis. Draw a straight line that best fits the points on the graph; remember that this line must pass through the origin (the 0,0 point).
  2. Measure the slope of this line (rise/run); this is the value of the Hubble constant, in the funny units of km/sec/Mpc.
  3. Your graph probably does not make a perfect line, and you will notice that you had to make a guess as to where to draw your line. One simple way to estimate the uncertainty in the value of Ho is to draw the steepest reasonable line and the shallowest reasonable line on the graph, and measure their slopes. Half of the difference between these two slopes is your uncertainty in Ho.