Lab #4 Hubble’s Law
Purpose: To familiarize the student with Hubble’s law.
Apparatus and Materials: Pencil, calculator, and Microsoft® Excel
Introduction: In astronomy, since typical distances are much larger than those usually dealt with on Earth, different units are used. The average distance between the Earth and the Sun is 93,000,000 miles (1.50x10^{8} km). This distance is given the special name of the astronomical unit (AU). An even larger distance is the light year which represents the distance that electromagnetic radiation, traveling at 3.00x10^{8} m/s, would travel in one year. The parsec (pc) is even larger and represents 3.26 ly. These values are summarized in Table 1.
Table 1
Unit  Value  Astronomical Unit (AU)  1.50x10^{8} km  Light Year (ly)  9.46x10^{12} km  Parsec (pc)  3.09x10^{13} km 
Edwin Hubble was an accomplished American astronomer. When he was investigating the spectral shifts of galaxies, he noticed that galaxies located far away from Earth displayed redshifted spectra. In addition, the further the galaxy was from Earth, the greater the observed redshift. Thus, he concluded that the universe was indeed expanding.
Hubble noticed that the radial velocities of some galaxies were approximately proportional to their distance from Earth. Mathematically, this can be written as
(Equation 1)
where v is the recessional velocity of the galaxy, H is a constant of proportionality called Hubble’s constant, and d is the distance between the observed galaxy and Earth. This equation is known as Hubble’s law. Equation 1 is of the form
. (Equation 2)
Thus, by plotting recessional velocity v versus distance d, one can obtain Hubble’s constant by simply finding the slope of the line. If v has units of km/s and d is in Mpc, then H will have units of km/(s·Mpc). Typical values obtained for H range between 50 and 80 km/(s·Mpc), depending on what galaxies are being observed and what experimental techniques are used.
Table 2 lists the distance and recessional velocity for five galaxies along with the calculated Hubble’s constant H for each galaxy. Note how the recessional velocity v increases with distance d. To calculate H one simply rearranges Equation 1 to get
. (Equation 3)
As an example using the data from Virgo, Equation 3 gives
.
Table 2

Distance Mpc 
Recessional Velocity km/s  Hubble’s Constant
 Virgo  17  1,210  71.2  Ursa Major  210  15,000  71.4  Corona Borealis  310  21,600  69.7  Bootes  560  39,300  70.2  Hydra  870  61,200  70.3 
Finally, Hubble’s constant can be used to calculate the age of the universe. To do this, one must first convert the units of Hubble’s constant from km/(s·Mpc) to 1/yr. This is done by using conversion factors:
,
or
.
We also know that velocity is distance divided by time, or in symbols:
. (Equation 4)
Substituting this into Equation 1 gives
, (Equation 5)
which can be simplified to get
(Equation 6)
or
. (Equation 7)
Thus, Hubble’s constant is inversely proportional to the calculated age of the universe. The age of the universe can easily by calculated from Equation 7 to get
.
Based on this data, one can deduce that the universe is approximately 13.9 billion years old.
Exercise:
1. Using the data in Table 2, calculate the average Hubble’s constant value.
2. Using the data in Table 2 and Microsoft® Excel, plot recessional velocity versus distance. Fit the data with a trendline and display the equation of the line along with the R^{2} value on the plot.
Questions and Discussion:
1. Compare the average value of Hubble’s constant from Step 1 above with the value obtained from the trendline in Step 2. Are they the same or different? Why?
2. Using Hubble’s law, what happens to the calculated age of the universe if Hubble’s constant increases?
3. Nothing has been said regarding the force of gravity between galaxies. What effect does the force of gravity have on the recessional velocities of galaxies? How would the value of Hubble’s constant be changed? What impact would this have on the calculated age of the universe? Give an explanation of your answers.
References:
1. An Introduction to Physical Science Laboratory Guide, 10^{th} Edition, James T. Shipman and Clyde D. Baker, “Hubble’s Law”, Experiment 44, Houghton Mifflin Company (2003).
2. An Introduction to Physical Science, 11^{th} Edition, James T. Shipman, Jerry D. Wilson, and Aaron W. Todd, Houghton Mifflin Company (2006).
3. Astronomy: A Beginner’s Guide to the Universe, 5^{th} Edition, Eric Chaisson and Steve McMillan, Pearson Education Inc., (2007).
