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# 16.8: The Hubble Constant

Hubble's relation expresses how the recession velocity increases with distance from the observer. The slope of the plot, the ratio of velocity to distance, is known as the Hubble constant. The Hubble constant, usually denoted by the symbol H0, measures the rate of the expansion, and it also indicates the size and age of the universe. It is one of the most important quantities in astronomy.

How do astronomers estimate the Hubble constant? It is a very difficult research undertaking. Measuring the red shift for a galaxy is the easy part. The hard part is the measurement of distance. Astronomers must avoid galaxies that are too close to the Milky Way because they are bound to each other by gravity and do not take part in the general expansion. They also must avoid galaxies elsewhere in space that are bound in a group or cluster, because they are all at roughly the same distance. Otherwise, it does not matter how the galaxies are selected. Since the expansion is isotropic, the same Hubble relation should be measured in every direction in the sky. Once the red shift and distance of a galaxy have been measured, each galaxy can be plotted as a point on a Hubble relation. The slope of the line that gives the best fit to the data is the Hubble constant.

All methods of distance determination need an accurate calibration at small distances. This calibration sets the luminosity of each new distance indicator — all of the uncertainty in the Hubble constant comes from this procedure. Once the calibration is known, relative distances are easy to calculate using the inverse square law. You can understand distance calibration by analogy from everyday experience. Suppose you were measuring distances around your house using a tape measure. The accuracy of all your measurements would depend on how accurately an inch had been defined, or calibrated, when the tape measure was made at the factory. Or suppose you were trying to measure the distance between two towns using the odometer in your car. The accuracy of your measurement would depend on how accurately the circumference of your wheel had been calibrated at the factory, since an odometer essentially counts wheel rotations. Likewise, in astronomy, the bedrock of all astronomical distance measurements is parallax. Trigonometry has been used to measure the distances to many stars in the neighborhood of the Sun; the calibration of all other distance indicators depends on this information.

There are several crucial benchmarks in the extragalactic distance scale. One is the Large Magellanic Cloud, which has an accurate geometric distance from measurements of the dust shell of Supernova 1987A. The LMC also contains a large enough "stellar zoo" for the calibration of rare distance indicators such as Cepheid and RR Lyrae variables. Another benchmark is the Virgo cluster, about 15 Mpc distant. The Virgo cluster is close enough that Cepheid variables can be detected using the Hubble Space Telescope. Since the Virgo cluster contains both spiral and elliptical galaxies, astronomers can calibrate the Tully-Fisher and Faber-Jackson relations to use for even greater distances. The light curve of a supernova can be used to measure distances out to hundreds of mega parsecs. Every galaxy that has both a supernova and measurable Cepheid variables can be used to calibrate and use supernovas as distance indicators.

Astronomers have not yet reached a consensus as to the value of the Hubble constant. The subject of the distance scale has a long history of controversy and unrecognized systematic errors. Hubble’s first measurements yielded a very steep slope and a high value for H0: 540 km/s/Mpc. This high value was alarming because it implied such a rapid expansion that the universe had to be younger than the well-measured age of the Earth! It turns out that Hubble had used an erroneous calibration of the Cepheid luminosity. During a vast observational effort that started in the mid 1970s, using hundreds of nights of time on large telescopes, published values of the Hubble constant ranged from 50 to 100 km/s/Mpc.

Why this uncertainty of a factor of two? One reason is dust. When we look out to the galaxies, we look through a veil of stars and dust in the disk of our own Milky Way. This relatively nearby dust dims and reddens the light from distant galaxies. Dust therefore makes a galaxy of a particular luminosity appear farther away than if it were viewed through no dust. The radial velocity is unaffected. The result is a lower value of the Hubble constant. Moreover, Cepheids and supernovae in distant galaxies are dimmed by dust in the galaxy that contains them as well as by patchy dust in our own galaxy. While all researchers agree that this effect is important, they disagree on the size of the correction for dust. Another cause of uncertainty is our imperfect understanding of many distance indicators. Our knowledge of the Hubble constant is only as good as our understanding of the physics that sets the luminosity of a distance indicator.

Despite a checkered history, recent measurements have converged on a relatively narrow range for the Hubble constant. Most observations are consistent with a H0 value of 71 km/s/Mpc. The measurement error on this number is about 5%. (Of course the possibility of unrecognized systematic error remains.) What does this mean in everyday units? It means that galaxies are being carried away from us by the expansion of space at a rate of 22 km/s or 22 x (3/5) x 3600 = 47,500 mph for every million light years increase in distance! This number has an uncertainty of about 5%, so we can quote it as 71 ± 4 km/s/Mpc. In other words, it is unlikely that the Hubble constant is much lower than 67 km/s/Mpc or much higher than 75 km/s/Mpc. The fact that many different measurement techniques give a similar expansion rate is a wonderful validation of our basic understanding of how the universe works.