Department of Physics University of Durham   Level One
School of Physics University of New South Wales   General Education

Measuring the Hubble Constant


Aims of the Experiment

  1. To illustrate the measurement of galaxy sizes.
  2. To introduce the measurement of galaxy redshifts from spectra.
  3. To plot the diameter-redshift relation for galaxies in the Hubble Deep Field.
  4. To estimate the Hubble Constant from the diameter-redshift relation.
  5. To discuss possible biases inherent in this method.


Introduction

 Background Information - During the 1920's, Edwin Hubble working with observations from the Mount Wilson Observatory discovered that the Universe is expanding. He compared recession velocities of galaxies measured from their spectra to their apparent brightness estimated from photographic plates. The original paper detailing this conclusion from 1929 can be seen here. This expansion causes objects at greater and greater distances to be receding from us at higher and higher speeds, at a velocity given by an expression now known as Hubble's Law: V = H x R. Here V represents the galaxy's recessional velocity, R is its distance away from Earth, and the constant of proportionality, H, is called the Hubble Constant and has units of km/sec per megaparsec (km/s/Mpc). The value of the Hubble Constant is important for both observations of the objects in the Universe, as it allows us to convert their recession velocities into true distances, and for estimating the age of the Universe. Below is shown Hubble's original figure from the 1929 paper - where he plotted the radial velocities in km/sec of the galaxies against his estimates of their distances (in parsecs) derived from the apparent magnitudes of what he thought were the bright supergiant stars. These `stars' turned out to be giant HII regions - rather than individual stars - placing the galaxies at much larger distances than Hubble originally estimated.

The recession of the galaxies from us causes a Doppler shift in their observed spectra, with spectral features (e.g. absorption and emission lines) being shifted redwards from the restframe wavelengths (measured in the laboratory). The strength of this shift, and hence the recession velocity, is measured in terms of the galaxy's redshift, usually denoted z. A small number of galaxies in the space near to our Galaxy actually have blueshifts, they are approaching us rather than receding. This results from a second component, the peculiar velocity, of any galaxy's motion due its gravitational attraction to surrounding galaxies and mass concentrations. Thus in the case of the nearby blueshifted galaxies their peculiar velocity towards us, caused by the gravitational pull of our Galaxy, exceeds their expansion velocity due to the Hubble flow and so they are actually approaching us.

The exact value of the Hubble Constant is still not accurately known, but is generally believed to be between 50-100 km/sec/Mpc. This means that a galaxy 1 Mpc away will be moving away from us at a speed of between 50-100 km/sec, while another galaxy 10 Mpc away will be receding at 10 times this speed. The value of the Hubble constant therefore determines the rate at which the Universe is expanding, and equivalently allows us to roughly estimate the length of time it has been expanding, since the Big Bang, and thus the age of the Universe.

On very large scales general relativity predicts departures from a strictly linear Hubble law (in addition to the small-scale noise caused by the random velocities of individual galaxies). The degree of departure, and the sense, depends on the value of the total mass of the Universe. Hence a plot of recession velocity (or redshift) versus distance, called a Hubble diagram, which is a straight line for objects at small distances, can for observations of sources at large redshifts tell us about the amount of matter in the Universe and provide crucial information about the future evolution of the Universe: whether the Universe will end in a Big Crunch or keep on expanding forever.


Standard Rods

Two of the key scientific justifications for building the Hubble Space Telescope were to use it to measure the sizes of distant galaxies and to detect the variations in the luminosity of variable stars in distant galaxies. Both of these observations can be used to place limits on the Hubble Constant, a number of approaches have also been used to estimate the Hubble Constant from a variety of observational data. The key to all these techniques is to find the true distance of an object and then compare this to its recession velocity to estimate the Hubble Constant.

Most techniques for estimating the Hubble Constant therefore rely on using either standard candles or standard rods. Standard Candles - these objects or events are thought to have fixed luminosities. When observed across a range of distances their apparent luminosities can therefore be used to estimate their relative distances. This class includes: variable stars such as Cepheids (see the time-lapse sequence of a Cepheid variable in M100 above), Type 1a Supernova and Luminous Radio Galaxies. Standard Rods - these are objects with a fixed physical length, using simple geometry the apparent angular size of this feature can be used to estimate the true distance of the source. An example of a standard rod is used in the following experiment - which assumes that all galaxies have the same physical size. While not a good approximation across all classes of galaxy, a variation on this technique has been used to estimate the Hubble Constant in the past. In the following lab you will be using observations taken by Hubble Space Telescope of a crude `standard rod' - the apparent angular diameter of luminous galaxies - to compare the sizes of galaxies at different distances (and hence redshifts) from us. As you will see, this is not a particularly robust way to measure the Hubble Constant, but it is a good illustration of the pitfalls which may bias some of the more complex techniques used.


Redshifts and Spectra

The figure below is an example spectrum, on this are marked the various absorption and emission lines commonly seen in the spectra of normal galaxies. The most prominent absorption lines are produced by metal ions (Calcium and Magnesium) in the atmospheres of stars, these same lines can be identified in spectra of the Sun, thus these features should be identifiable in the spectra of all galaxies (which are necessarily composed of stars). In contrast, the emission lines are formed by the recombination of ionized atoms in more tenuous gas in star-forming regions, these will thus be only seen in galaxies which are actively forming stars. The relative prominence of the various emission and absorption lines will vary depending upon the current and past star formation history of the galaxy. In particular galaxies which are not currently forming stars (passive, early-type galaxies such as Ellipticals) will not show the emission lines. The galaxy spectrum is shown in the restframe (a redshift of z=0).

Listed below are the restframe wavelengths for the various spectral features marked in the figure above. These are the main emission and absorption lines you should be trying to identify in the spectra analysed in the following pages. Remember that not all the features will be visible in a given spectrum and that because of the redshifting of the spectrum the lines will fall at redder wavelengths than the restframe values listed here. Nevertheless, the pattern of emission and/or absorption can be used to identify individual features. In all of the cases shown in the next pages you should be able to identify a minimum of at least 2 spectral features in each spectrum. You can then estimate the galaxy redshift from the each line and confirm that you identified the correct features.

Emission Lines
[OII]
3727Å
H-beta
4861Å
[OIII]
5007Å
H-alpha
6563Å
Absorption Lines
Ca K
3933Å
Ca H
3969Å
Ca G
4303Å
Mg b
5174Å

The redshift, z can be estimated from the observed wavelength of a spectral feature (measured from a galaxy spectrum) for which you know the restframe wavelength of the line (given in the table above) as follows:


 

The redshift is related to the recession velocity, V, through the Doppler formula. When the recession velocity is only a small fraction of the speed of light, c, this further simplifies to:

Use the above simplification in the following analysis, but you should comment on whether you feel the approximation is valid.


Estimating the Hubble Constant

Having measured the angular size of a galaxy and its redshift, and knowing the average diameter of a galaxy, we can combine these measurements to estimate the Hubble Constant from this observation (this is dealt with here). Obviously the observations of a single galaxy will not provide a good estimate of the Hubble Constant, both because of random errors in the measurements and because the galaxies have a distribution of physical sizes. To reduce the random measurement errors you should repeat your measurements and combine these to get a better determination of the angular size and redshift of the galaxy, as well as errors on these values. To better estimate the Hubble Constant you should also measure a number of galaxies, to better sample the distribution of galaxy sizes. Keep these points in mind as you go through the exercise in the following pages.

When you come to the end of this exercise you should have produced a table listing the measurements of the angular sizes and redshifts for the 15 galaxies in the sample, as well as estimated errors for the individual measurements of both quantities. The final step will be to convert these observations into an estimate of the Hubble Constant.


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