Hubble’s Law Lab

IntroductionIn this lab you will measure the age and expansion rate of the Universe. Using images and spectra of real galaxies you will determine the value of the Hubble constant. You will follow Hubble’s original procedure to measure the angular size of galaxies and derive the distance to each galaxy. You will also measure the redshift of each galaxy from its spectrum, and will learn how the recessional velocity and the galaxy’s redshift are related. Based on your measured value of the Hubble constant, you will be able to estimate the expansion rate and age of the Universe.

Background & TheoryIn 1929, Edwin Hubble (the astronomer, not the space telescope) discovered the relationship between a galaxy’s distance from us and how fast it is moving away from us (or its recessional velocity). This law, now known as Hubble’s law, states that the recessional velocity of a galaxy is proportional to its distance from us:

v=H0×d

Where v is the galaxy’s velocity in km/sec, d is the distance to the galaxy in megaparsecs (Mpc; 1 Mpc = 1 million parsec), and H0 is a proportionality constant, called the Hubble Constant. For example, Hubble’s Law states that a galaxy moving away from us twice as fast as another galaxy will be twice as far away from us as the other galaxy, while a galaxy moving three times faster will be three times farther away. Accurately measuring the value of H0 is an important goal in modern astronomy. In order to do this, astronomers measure the distances and velocities of many galaxies in the Universe. Obtaining a more accurate measure of the Hubble Constant was a key project of the Hubble Space Telescope.

Figure 1 shows a reproduction of Hubble’s first published Hubble diagram. In his original work, Hubble was only able to measure velocities and distances of a few nearby galaxies, and so his original estimate of the Hubble Constant is different from our modern day value. But Hubble was able to correctly show that the Universe is expanding based on the fact that more distant galaxies have higher recessional velocities.

Figure 1: Hubble’s original diagram showing that more distant galaxies are moving away faster than less distant galaxies.

Why is the Universe expanding? You may have heard of the Big Bang: the Big Bang theory states that the Universe expanded from an extremely hot, dense state and continues to expand today. All of the galaxies, stars, planets, and everything else we see in the Universe today were once contained in this single point at the beginning of the history of the Universe.

Immediately after the Big Bang, the Universe began to expand, dragging all of the matter in the Universe along with it. Today it looks to us like every galaxy we can see is moving rapidly away from us (i.e., the light from the galaxy is “redshifted”). However, the galaxies we see today are not themselves flying apart through space; rather the space itself is what is expanding, carrying the galaxies along with it. Furthermore, even though it appears that the Milky Way is at the center of the expansion, an observer in any other galaxy would see the same universal expansion and would think they were at the center as well. No one galaxy is at the center of the expansion; this is because as the Universe expands as a whole, the expansion is occurring everywhere for all galaxies, and all galaxies in the Universe are moving away from each other.

Imagine running the expansion of the Universe backwards, like running a movie in reverse: if we could do that, we would see galaxies getting closer and closer together until they all occupied the same point in space. Because the Hubble Constant relates the expansion rate of the Universe to the size of the Universe, we can use the Hubble constant to find the time at which all galaxies would be concentrated back together at the same location. The value of 1 divided by the Hubble constant is known as the Hubble Time, and this can be used to estimate the age of the Universe.

Figure 2: Artist’s concept illustrating the expansion of the Universe.

In this lab, you will use Hubble’s Law to determine the expansion rate of the Universe and also to estimate the age of the Universe. To do this you will measure the distance to and the velocity of a number of galaxies. You will use these values to derive the Hubble Constant and use the Hubble Constant to estimate the age of the Universe.

ProcedureThe figure below contains images (on the left side of the figure) and spectra (on the right) for five galaxies. For each galaxy you will determine its distance from us and its recessional velocity.

Figure 2: Galaxies

Part 1 – Determining the Distance1. Measure the diameter, d in millimeters (mm) of all 5 galaxies. Make your measurements

as accurately as possible, preferably within 0.2 mm. For galaxies that are not circular in shape, determine the diameter along the longest and shortest dimensions and average the two values. The fifth galaxy is a double galaxy, only measure the galaxy on the left, indicated by the arrow. Record the diameters of each galaxy in the data table on the next page.

2. Next, we need to determine the scale of the images. To do this, measure the horizontal linear distance of each image in millimeters (mm). Average the 5 horizontal image sizes and record your average value in your lab notebook in mm.

3. The scale of each image is indicated at the bottom of the bottom image. All of the images are the same scale. Calculate the scale of the images by measuring the length of this line in mm. Use the following formula: angular size/average linear size = ′′/mm. Record this value here, showing your work:

4. Convert the measured diameter (d) of your galaxy in mm to an angular diameter (A) in arcseconds using the scale you determined in step 2 and the following equation. Record these values in your data table.

A(' ' )=d (mm)×scale(' ' /mm)

5. All of the galaxies you have measured are giant elliptical galaxies. Hubble also used giant elliptical galaxies to do his work because these galaxies have a smooth, regular shape that is easy to measure. Astronomers know that, on average, all giant elliptical galaxies have about the same size. The galaxies you have measured in this lab look smaller or larger because they are at different distances: if all five galaxies are the same size, galaxies that are farther away look smaller than galaxies that are closer to us.

In this lab we will assume (just as Hubble did in his original work) that these galaxies all have a diameter of 32 kiloparsec (kpc) or 0.032 Mpc. Calculate the distance, D, to each galaxy using the following equation and record their distance (in Mpc) in your data table:

D (Mpc )=0.032 (Mpc )×206265( ' '

radian )A ( ' ' )

=206265×0.032A

Explain where the number “206265” in the equation above comes from!

Table 1: Data Table

Galaxy Name

Measured Diameter d

(in mm)

Angular Diameter

A (in arcsec)

True Diameter (in Mpc)

Distance (in Mpc)

Spectral Shift

(in nm)

z

(redshift)Velocity

(km/sec)

Virgo 0.032

Ursa Major

0.032

Corona Borealis

0.032

Bootes 0.032

Hydra 0.032

Part 2 – Determining the VelocityThe right side of the galaxy figure is the observed spectrum of each galaxy. Think back to Lab 3 in which we learned about spectra produced by astronomical objects. The spectrum from each galaxy is an absorption spectrum. The wavelength scale of each spectrum is shown at the bottom of each spectrum. The blue/violet side of each spectrum is on the left and has the shortest wavelengths, ~300 nanometers (1 nanometer, nm, = 10−9 m) and the red end of the spectrum is on the right side which has the longest wavelengths, ~500 nm.

You should notice that in each galaxy the absorption lines in the galaxy’s spectrum is redshifted (to the right side of the spectrum) by some amount. Remember that the redshift of an object or a spectral feature can be used to measure how fast an object is moving away from us. In this case, the redshift is due to the galaxy’s recessional velocity away from us as it is pulled away from us due to the overall expansion of the Universe.

7. Measure the shift of the spectral features for each galaxy. We will use two strong calcium lines. These lines appear as the two dips near 395 nm in the “restframe” spectrum.

8. Measure the wavelength shift of each spectrum (using the Calcium lines) in nanometers (nm). Record these values (in nm) in your data table.

9. Now you can calculate the redshift (z) of each galaxy. The average rest (un-redshifted) wavelength of the calcium lines is 395 nm. Use the equation below to calculate the redshift (z) of each galaxy and record these values in your data table:

z=λmeasured−λrest

λrest=Wavelength Shift (¿nm)

395

10.Calculate the recessional velocity of each galaxy in km/sec and record these values in your data table. The velocity is just the speed of light (c) in km/s multiplied by the redshift, where c = 300,000 km/s, and is given by:

vrec=c × z=3.0×105 km /s× z

Part 3 – Determining the Hubble Constant11.Using the graph paper provided, make a plot of the recessional velocity (km/s) of the

galaxies versus their distance (Mpc). The velocity should be on the vertical axis and the distance on the horizontal axis. Be sure to label your axes.

12.Next, draw a single straight line from the origin of your plot that best fits all five of your data points. The line should be straight, and does not need to go through each individual data point, but should represent the average trend of your 5 points.

13.Determine the slope of your best-fit line. The slope of the line can be found by picking a point on your graph that the line goes straight through (it does not have to be one of your data points). From this point, read the y-value on the vertical axis to find the velocity associated with this point. Then read the x-value on the horizontal axis to find the distance associated with this point. Divide the velocity by the distance to get the slope of your line. The slope of your line will be in units of (km/s)/Mpc. This is your estimate of the Hubble Constant. Show your calculations for the slope of your line and record your value of the Hubble constant in your lab notebook.

Part 4 – The Age of the UniverseImagine that you have been driving in your car on a trip for some length of time, but you don’t remember what time you left. Your car’s trip odometer tells you that you have driven 180 miles, and you know that your cruise control has been keeping you at a steady speed of 60 miles per hour since you left. Then you can certainly determine how long you’ve been gone:

Travel time = distance traveled/velocity = 18 miles / 60 miles per hour = 3 hours

This same type of information is contained in the Hubble constant, and we can use it to determine the age of the Universe, or the amount of time that has passed since all galaxies and all the matter in the Universe was together at the same place. This is assuming that the Universe has been moving apart (expanding) at the same speed the whole time.

The point on your line used to determine the Hubble Constant represents the distance and recessional velocity of a hypothetical galaxy that is exactly following Hubble’s law. The distance associated with it is how far that galaxy has separated from us during the age of the Universe, and the velocity is the rate at which it is separating from us. We can therefore use the same formula for travel time as above to estimate the age of the Universe, assuming that the rate has always been constant. However, we must first convert the distance units (km and Mpc) to be the same, and it will also be more useful to have time in years not seconds.

14.Use the point on your line corresponding to this hypothetical galaxy to determine the age of the Universe. Follow the calculations below for your data point. Show all your work in your lab notebook and record your answer in your notebook.

Change the units of velocity and distance as follows:

Velocity(¿km / year)=velocity (¿km /sec)×3.1×107 se c / year

Distance(¿ km)=distance(¿Mpc)×3.1×1019km /Mpc

Now you can calculate the travel time (i.e., the age of the Universe) by dividing distance by the velocity:

Ageof theUniverse=Distanc e(¿km)/Velocity(¿ km / year)

Additional Questions1. Compare your value of the Hubble Constant to the one astronomers use today. The

accepted value of the Hubble Constant is 73 km/sec/Mpc (measured by the Hubble Space Telescope). What is the percent error in your determination of H0? Show your work:

Percent error=(73− your value)73

×100%

2. Name two possible uncertainties that could be a source of error in your calculation of the Hubble Constant (Hint: think about the scales of the images, as well as any assumptions we had to make).

3. How does the value you derived for the age of the Universe compare to the age of the Sun (approximately 5 billion years)?

4. The age of most globular clusters in the Milky Way (some of the oldest objects in the Milky Way) is inferred to be about 12 billion years, because stars with main-sequence lifetimes less than ~12 billion years are no longer on the main sequence. Is your age of the Universe consistent with the age of the globular clusters, and therefore with the age of the Milky Way? Explain your reasoning.

5. Originally Hubble measured H0 = 500 km/s/Mpc. Based on this value, were galaxies believed to be much closer or much farther apart than they are now? Would the age of the Universe have been believed to be much larger or much smaller? Explain your reasoning.