The Hubble Constant and the Age of the Universe
In 1924, Edwin Hubble showed that spiral nebulae were distant galaxies containing billions of stars. He also found that these galaxies were moving away from us with speeds that were proportional to their distance from us. What this means is that the more distant galaxies are traveling away from us with greater speeds. Nearer galaxies are traveling away from us with smaller speeds. This relationship is called Hubble's law.
v = H0d
where v is the speed at which the galaxies are receding (i.e. moving away) from us, d is the distance from us to the galaxy, and H0 is the Hubble constant. If you graph the speed of the galaxies (v) on the vertical axis and the distance to the galaxies (d) on the horizontal axis, then the slope of the graph (H0) is the Hubble constant. Hubble plotted speed in units of km/s and distance in Mpc (megaparsecs). The presently accepted value of the Hubble constant is about 72 km/s/Mpc (with an uncertainty of about 7 km/s/Mpc which means that the value may be as low as 65 km/s/Mpc and as high as 79 km/s/Mpc).
Knowing Hubble's constant, we can determine an approximate value for the age of the universe. The activity below will help you understand how this is possible.
In this lab, you will measure the Hubble constant for an expanding rubber band. Follow these directions below in order to analyze the expansion of the rubber band.
1. Take a rubber band, flatten it out without stretching it, and place it next to a ruler.
2. Using a marker or pen, make at least 7 marks on the rubber band, 1 cm apart as shown below.
3. Record the distance of each of the Marks (2 – 7) from Mark 1 in millimeters (Note that 1 cm = 10 mm) in the data table on the last page of this activity. For example, Mark 2 is 1 cm away from Mark 1, so the distance is 10 mm. Note that this entry has already been put in the table for you.
4. Keeping the Mark 1 aligned with 0 on the ruler, stretch the rubber band until Mark 2 is at the 2 cm (20 mm) point on the ruler as shown below.
5. Record the new positions of marks 2 – 7 in the data table. For example, in the picture above, Mark 2 is at 20 mm (2cm), Mark 3 is at 60 mm (6 cm), Mark 4 is at 120 mm (12 cm), and so on. Note that the new position of Mark 2 is already recorded for you in the table.
6. Now, calculate the distance that each of the marks (2 – 7) moved during the stretching process. For example, Mark 2 started 10 mm away from Mark 1 and ended up 20 mm away from Mark 1 after stretching. This means that Mark 2 moved a distance of 20 mm – 10 mm = 10 mm. Note that this first entry has been recorded in the table for you already.
7. We can calculate the speed at which each of the Marks (2 – 7) was moving away from Mark 1 during the stretching process. Speed is calculated by taking the distance something travels and dividing by the time it takes. For example, if a car travels 100 miles in 2 hours, then its speed is (100 miles)/(2 hours) = 50 miles/hour. Let’s assume that it took 2 seconds for you to stretch the rubber band. We can then calculate the speed of each Mark by dividing the distance traveled by 2 seconds. For example, Mark 2 moved a distance of 10 mm, so its speed would be (10 mm)/(2 s) = 5 mm/s. Note that the speed of Mark 2 has already been entered into the table for you.
Plotting Your Data on a Graph
8. Print out the graph paper from the link on the course website and use it to plot a graph of Speed vs. Original Distance From Mark 1 (**the first column in your data table). Note that Speed is on the vertical axis and Distance is on the horizontal axis.
9. Once you have all of your data points on the graph, use a ruler to draw a “best fit” line through the middle of your data points. DO NOT CONNECT THE DOTS. This should be a single straight line, drawn such that it is as close as possible to each data point. The figure at right is an example of a best-fit line.
Finding the Hubble Constant
The Hubble constant for the universe is the slope of the speed vs. distance graph for distant galaxies (and other such objects). Let's calculate the Hubble constant for our expanding rubber band. To do this, you must calculate the slope using the following procedure.
10. Select two points on your best-fit line that are spaced fairly far apart. DO NOT USE THE DATA POINTS you plotted if they are not actually on the best-fit line. You will want to be able to read the coordinates of these points off the graph, so it is usually best to pick points where the graph paper lines intersect each other.
11. Use the two points you just found to draw a right triangle as shown in the figure below.
12. Measure the vertical “rise” of this triangle (this is the difference in speed in the two points) and record it below. Note that each block on the graph is worth 1 mm/s.
Rise = ______mm/s
13. Measure the horizontal “run” of this triangle (this is the difference in distance between the two points) and record it below. **CAREFUL: Do not just count the number of blocks on the graph. Each block is worth 2 mm. So, if there were 4 blocks, this means that the run would actually be 8 mm.
Run = ______mm
14. Calculate the slope using “rise divided by run” and record it below. Note that since the units of the rise (speed) are mm/s and the units of the run (distance) are mm, the units of the slope (rise/run) are mm/s/mm.
Slope = ______mm/s/mm
Answer the Following Questions
1. Do any of the marks on the rubber band get closer together as it stretches?
2. Which Mark (2 – 7) moved the fastest relative to Mark 1?
3. Suppose that, instead of Mark 1, we chose Mark 4 as our reference for measuring distances and speeds. Would the results be the same or different? Why?
4. What is the Hubble constant, H0, for the expanding rubber band? Make sure you include the units.
- Note that the units of the rubber band’s Hubble constant (slope of graph) are mm/s/mm. Since the second set of mm units is divided into the first set of mm units, they actually cancel out. This leaves the units as really being (1/s), which is read as “one over seconds”, or “inverse seconds”. If we were to take the inverse of the Hubble constant (i.e. flip it upside down), what would the units be then?
- Now, use a calculator to flip your Hubble constant upside down. That is, take 1/H0 and see what you get. Remember, the units are in seconds. (You should obtain 2 seconds, which is the amount of time the rubber band was stretching).
The Hubble Constant and the Age of the Universe
As mentioned before, the Hubble constant for the Universe is measured to be about 72 km/s/Mpc. Unlike in our rubber band activity, the two distance units (km and Mpc) are not the same. However, we can convert Mpc to km so that they are the same. If we do this, and at the same time convert the seconds to years, the Hubble constant for the Universe comes out to be 7.33×10−11 km/yr/km. And, as we already know, the second set of km units cancels out the first set of km units, leaving us with a value of 7.33×10−11 1/yr. So the units are in inverse years. Use a calculator to invert this number. What do you get? What are the units? What does this value represent?
Recall that there is some uncertainty in the measured Hubble constant. It can be as low as 6.62×10−11 1/yr or as high as 8.04×10−11 1/yr. If we use these values, what are the largest and smallest possible values for the age of the universe?
To determine the age of the Universe, we have used the Hubble constant which tells us something about the expansion rate of the Universe. However, in doing so, we have made an assumption. We have assumed that since the Big Bang happened, the Universe has been expanding at a constant rate. Imagine for a minute that future scientists find evidence that the rate of expansion of the Universe was not constant, but has in fact been changing. Explain what each of the following findings would mean for the age of the Universe.
Case 1: What if future scientists find that the Universe was actually expanding more quickly in the distant past and the rate of expansion has been slowing down ever since. Would this mean that the Universe is actually younger or older than we currently think it is? Explain your reasoning.
Case 2: What if future scientists find that the Universe was actually expanding more slowly in the distant past and the rate of expansion has been speeding up ever since. Would this mean that the Universe is actually younger or older than we currently think it is? Explain your reasoning.
Data Table for Rubber Band Measurements
(after stretching) / Distance Moved in mm / Speed (mm/s)
2 / 10 / 20 / 10 / 5
3
4
5
6
7