Hubble’s Law

Measuring the Age of the Universe Goals:

Using images and spectra of a number of galaxies you will determine their distance from us and their velocity away from us. From these distances and velocities, we will calculate an approximate age of the Universe.

Background: In the 1920s Edwin P. Hubble discovered a relationship that is now known as Hubble’s Law. This law states that the velocity of a galaxy is proportional to its distance from us:

V = H0 d where v is the velocity of the galaxy (in km/s), d is its distance (in megaparsecs, Mpc; 1Mpc = 3.086x1019 km) and H0 is Hubble’s constant.

So, for example, a galaxy that is moving away from us at twice the speed of another galaxy is twice as far away. The exact value of Hubble’s constant is still uncertain in astrophysics today, and during this practical we are going to begin to understand why.

To calculate the value of Hubble’s constant we need to

1. Measure the distance of a number of galaxies

2. Measure the velocity of those same galaxies. 1. Measuring Distance:

• In the handout, you will find a series of black and white images of galaxies. By

measuring the size of the galaxies in these images with a ruler we are going to estimate their distance. A brief explanation of how this is done can be found in the Appendix.

• We have to assume that all galaxies of the same shape are approximately the same size – this allows us to produce a graph of their size in millimeters versus their distance from us. This graph has been plotted for you and is Graph 1 in your handout.

• Using a ruler, measure in millimeters the size of each of the galaxies in the black and white images along their 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. Enter your results in the table provided.

• Convert each of your measurements in millimeters to a distance in megaparsecs (Mpc) using Graph 1 and enter the results into the table.

• This completes part one of your investigation. Now on to measuring the galaxy velocities.

2: Measuring Velocity:

• The velocity of a galaxy is measured using the Doppler Effect. The radiation coming from a moving object is shifted in wavelength because of its motion:

λ − λ0 = v

λ0 c

• λ0 is the wavelength in the lab (rest wavelength) and λ is the wavelength of the moving galaxy.

• Wavelengths are usually measured in Angstroms (Å). The speed of light has a constant value of 300,000 km/s.

• Astronomers refer to this change in wavelength as a redshift, z:

z = λ − λ0 = v λ0 c

• So by measuring the shift in lines in a galaxy’s spectrum we are able to measure the speed of the galaxy.

• Underneath the image of each galaxy is the spectrum of that galaxy. On the spectrum, there are two reference lines (representing the rest wavelengths) labelled close to the axes and two troughs in the plotted line. The troughs are the absorption lines corresponding to the CaK and CaH lines in the galaxy’s spectrum. Measure the wavelength of each trough and enter it in the data table for the respectiveline.

• The table will help you to then calculate the value of the redshift of the CaK line and the CaH line for each of the galaxies.

The rest wavelength, l0, for CaK is 3933.7 angstroms and for CaH is 3968.5 angstroms.

• Using both the CaK and CaH line will give you two values for z so that you may take an average to get y our best estimate for the velocity of the galaxy.

• In the final column of the table, use the average value of Z to calculate, v, by multiplying Z by the speed of light, c.

The speed of light is 300,000 km/s.

Galaxy Size(mm)

Distance(Mpc)

CaK CaH AverageZ

Velocity(km/s)

l Z1 l Z2 (Z1+Z2)/2 V=cZNGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC

Results & Conclusions:

• Using the graph paper titled “Hubble’s Law”, plot the distances you calculated in Part 1 against the velocities you calculated in Part 2 for each galaxy.

• Once you have plotted all your values draw a best fit line through the points. Think about whether the line should go through (0,0).

• When you are satisfied with your best fit line, calculate the slope of the line. Since v = H0 d, the slope you calculate is equal to Hubble’s constant.

Slope = H0 = ____________________ km/s/Mpc

• Hubble’s constant is equal to 1 over time, where time is the age of the Universe.

H0=1/t

• Using your value for the Hubble constant, calculate the age of the Universe as shown below. The 3.086 x 1019 converts between km and Mpc, so they cancel and leave your result in seconds. The 1/(60*60*24*365.25) converts seconds into years.

Age of the Universe = 3.086 x 1019 / H0 = ____________________ seconds Age of the Universe = __________________ seconds / (60*60*24*365.25) = __________________ years

• The currently accepted age of the Universe is 13.8 billion years. Calculate the percent difference between your result and the known age of the Universe.

• Discuss how you could improve your results.

Hubble’s Law

Distance(Mpc)

Velocity(k

m/s)

Appendix

Finding the distance knowing a galaxy's actual size and angular size

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 roughly the same physical size, no matter where in the Universe they are. This is known as "the standard ruler" assumption. To use this assumption, however, we have to know the actual size of the "ruler" and to do that, we need the distances to the galaxies that form our standard ruler. So, since we are working with spiral galaxies, we choose nearby galaxies such as Andromeda, Triangulum, Messier 81, and others to which we have found an accurate distance measure using variable stars or other reliable distance indicator. We find that, on average, the average size of these standard ruler galaxies is 22 kiloparsecs (22 kpc or 22,000 parsecs or ~72,000 light years).

Theoretically, then, to determine the distance to a galaxy one would need to measure only its angular size and use the small angle formula: a = s / d, where a is the measured angular size (in radians), s is the galaxy's true size (diameter, 22 kpc in our work here), and d is the distance to the galaxy.

a = s/d or d = s/a [Eqn. (3)]

There is another caveat, however. Each telescope and detector scales the images of celestial objects differently. While the actual angular size of a galaxy does not change, that galaxy might take up 30,000 pixels squared on one detector while on another one (having larger sized pixels) it takes up 5400 pixels squared. In the data used for this investigation, all images were taken by the same telescope and detector. So no worries there.

Recall that if we see a galaxy that is 1/2 or 1/3 the angular (apparent) size of another galaxy, we would like to be able to state that that galaxy is 2 times or 3 times farther away. To do this, we must assume that if galaxies resemble each other, then they are approximately the same actual size. We must first calibrate the actual size by using a galaxy to which we know the true distance. We are looking for galaxies in the sample that are Sb galaxies, as we would use the nearby Sb galaxy, M31 the Andromeda galaxy, to calibrate the distances. We know the distance to the Andromeda galaxy through observations of the Cepheid variables in it. Then, to determine the distance to more distant, similar galaxies, one would only need to measure their apparent (angular) sizes, and use the small angle formula.

We have created a “calibration curve” for you, which allows you to directly convert between the millimeters you will measure for the galaxy size and the distance away that galaxy would need to be in order to appear that size. This is shown in Graph 1.

NGC2903

NGC2775

NGC3147

NGC1357

NGC3368

NGC5548

NGC3623

NGC6217