Hubble Law Lab, the Short Version

Summary
The student will determine a value for Hubble's constant, based on their observations of the images and spectra of 12 spiral galaxies.

[For a longer, more in-depth lab, see The Hubble Law: An Introductry Astronomy Lab]

Background and Theory
In the 1920's, Edwin P. Hubble discovered a relationship that is now known as Hubble's Law. It states that the recessional velocity of a galaxy is proportional to its distance from us:

v = H_o * d,

where v is the galaxy's velocity (in km/sec), d is the distance to the galaxy (in megaparsecs; 1 Mpc = 1 million parsecs), and H_o proportionality constant, called "The Hubble constant". Hubble's Law states that a galaxy moving away from us twice as fast as another galaxy is twice as far away. The Hubble constant is a hotly contested quantity in astrophysics. In order to precisely determine the value of H_o, we must determine the velocities and distances to many galaxies.

The velocity of a galaxy is measured using the Doppler effect. The radiation coming from a moving object is shifted in wavelength:

where

is the rest wavelength of the radiation, and

is the amount the radiation has been shifted (the observed wavelength minus the rest wavelength).

Wavelengths are usually measured in Angstroms (Å). The speed of light has a constant value of 300,000 km/sec. The quantity on the left side of the equation above is usually called the redshift, and is denoted by the letter z.

We can determine the velocity of a galaxy from its spectrum: we measure the wavelength shift of a known absorption line and solve for v. Example: An absorption line that is found at 5000Å in the lab is found at 5050Å when analyzing the spectrum of a particular galaxy. Therefore this galaxy is moving with a velocity v = (50/5000) * c = 3000 km/sec away from us.

A trickier task is to determine a galaxy's distance, since we must rely on more indirect methods. One may assume, for instance, that all galaxies of the same type are the same physical size, no matter where they are. This is known as "the standard ruler" assumption. To determine the distance to a galaxy one would only need to measure its apparent (angular) size, and use the small angle equation: a = s / d, where a is the measured angular size (in radians!), s is the galaxy's true size (diameter), and d is the distance to the galaxy.

Procedure
Print out the worksheet and plot sheet.

From the Galaxy List, choose a galaxy from the list on your worksheet.
Find the angular size of the galaxy using its image. The images used in this lab are negatives, so that bright objects -- such as stars and galaxies --appear dark. There may be more than one galaxy in the image; the galaxy of interest is always the one closest to the center.
To measure the size, click on opposite ends of the galaxy, at either end of the longest diameter. Be sure to measure all the way to the faint outer edges. Otherwise, you will dramatically underestimate the size of the galaxy, and introduce a systematic error. The angular size of the galaxy (in milliradians; 1 mrad = 0.057 degrees = 206 arcseconds) will be displayed; record this number on your worksheet.
Repeat step 2 for all 10 of the galaxies on the worksheet.
The full optical spectrum of the galaxy is shown at the top of the spectrum page. Below it are enlarged portions of the same spectrum, in the vicinity of some common spectral features. The small dark bar near the lower left corner of the sub-spectrum indicates the rest wavelength of the line. Measure the wavelength by clicking at the middle of the spectral line in the galaxy's spectrum. Find the red-shifted wavelength for Ca K, Ca H and H- lines for each galaxy on the worksheet using its spectrum.
Repeat step 4 for all 10 of the galaxies on the worksheet.
Calculate the redshift, z for each of the lines, and enter these data in the z boxes on the worksheet.
Find the average of the redshifts for each galaxy, and enter it in the table.
Use this average redshift to find the velocity:
v=cz, and enter this in the table under Velocity.
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:
d (Mpc) = s (kpc) / a (mrad),
and record this distance in your data table under Distance.
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). (Why?)
Measure the slope of this line (rise/run...) -- this is the value of the Hubble constant, in the funny units of km/sec/Mpc. 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 H_o.
And now for the age of the universe!: If the universe has been expanding at a constant speed since its beginning, the universe's age would simply be 1/H_o.
Convert H_o to inverse-seconds (1/sec) by cancelling out the distance units: 1 Mpc = 3.09X10¹⁹ km.
The "expansion age" of the universe is t = 1/H_o. This is a very simple model for the expansion of the universe. A better model would account for the deceleration caused by gravity. Models like this predict the age of the universe to be: t = 2/(3H_o).
Re-calculate the age using this relation.
How does this value compare to the age of the Sun?