79205 Astronomy

Studio Week 12: Hubble's Law and the Hubble Constant

Hubble's Law says that the "recession velocity" of distant galaxies is proportional to their distance from us. This is tricky to test, since it is very difficult to accurately determine the distance to the galaxies in the first place, although it is relatively straightforward to determine their recession velocity.

In this exercise you will use bright elliptical galaxies near the center of galaxy clusters as "standard candles" to attempt to verify Hubble's Law. You will also extract a value of the Hubble Constant, and a range of uncertainty, so you can get a feel for how difficult it is to accurately measure the expansion rate of the universe. Record all work on the worksheet.

As discussed in class, and as seen in Kutner Fig.18.7, if you choose a set of objects with the same absolute magnitude M then Hubble's Law implies that a plot of m versus logz should be a straight line with a slope of "5". Furthermore, the y-intercept of this line allows you to determine the Hubble Constant, assuming you know absolute magnitude M. In particular

This is the principle you will use to test Hubble's Law and to extract the Hubble Constant.

The data you will use lists several so-called "Brightest Cluster Galaxies" (BCG's) along with their redshift and apparent magnitude. In some cases, more than one magnitude value (obtained from different observations of the galaxy) are listed.

The last two entries in the table, galaxies M87 and NGC4874, are special. We believe we know the distances to these galaxies independently, and by different means. In the case of M87, the globular cluster luminosity function tells us it is 16.6Mpc away. (Note that the distance to M100, another Virgo cluster galaxy, nearby M87, has been determined to be 17.2Mpc, in good agreement.) For NGC4874, a Type I Supernova has been observed, allowing us to determine its distance to be 109.6Mpc.

Verify Hubble's Law. Make a plot of m versus logz using this list of BCG's. You can expect a lot of scatter, because you are assuming that all BCG's have the same absolute magnitude, and that's not a great assumption for this data set. Still, a correlation between apparent magnitude and redshift should be clear.

Draw the best straight line you can through these points. What is the slope? Draw the best straight line again, but this time force it to have slope=5. Is this reasonable?

Be aware that galaxy M87 is rather close to us, at the center of the Virgo cluster, and peculiar motions make it rather difficult to use it for determining the Hubble constant. You may want to exclude it from your "fits".

Determine the Hubble constant. From your plot of m versus logz, and your best straight line with slope=5, find the y-intercept, either graphically or algebraically. Determine the value of M using the distance and apparent magnitude of either M87 or NGC4874. (Do these two galaxies give the same answer?)

Get some idea of the uncertainty in the Hubble constant, as you've determined it, by asking yourself how much you can let the "straight line with slope=5" move up or down on your plot. That is, what is a reasonable range for the intercept? Turn this into a range of values for the Hubble constant. Comment on your conclusions.

If you don't want to download the data to analyze it, it is also listed here for your convenience.