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2.2 Basic principle

We first outline the method briefly, before discussing each stage in more detail. Nearby stars have a reliable distance measurement in the form of the parallax effect. This effect arises because the earth’s motion around the sun produces an apparent shift in the position of nearby stars compared to background stars at much greater distances. The shift has a period of a year, and an angular amplitude on the sky of the Earth-Sun distance divided by the distance to the star. The definition of the parsec is the distance which gives a parallax of one arcsecond, and is equivalent to 3.26 light-years, or 3.09 × 1016 m. The field of parallax measurement was revolutionised by the Hipparcos satellite, which measured thousands of stellar parallax distances, including observations of 223 Galactic Cepheids; of the Cepheids, 26 yielded determinations of reasonable significance [41].

Some relatively nearby stars exist in clusters of a few hundred stars known as “open clusters”. These stars can be plotted on a Hertzsprung–Russell diagram of temperature, deduced from their colour together with Wien’s law, against apparent luminosity. Such plots reveal a characteristic sequence, known as the “main sequence” which ranges from red, faint stars to blue, bright stars. This sequence corresponds to the main phase of stellar evolution which stars occupy for most of their lives when they are stably burning hydrogen. In some nearby clusters, notably the Hyades, we have stars all at the same distance and for which parallax effects can give the absolute distance. In such cases, the main sequence can be calibrated so that we can predict the absolute luminosity of a main-sequence star of a given colour. Applying this to other clusters, a process known as “main sequence fitting”, can also give the absolute distance to these other clusters.

The next stage of the bootstrap process is to determine the distance to the nearest objects outside our own Galaxy, the Large and Small Magellanic Clouds. For this we can apply the open-cluster method directly, by observing open clusters in the LMC. Alternatively, we can use calibrators whose true luminosity we know, or can predict from their other properties. Such calibrators must be present in the LMC and also in open clusters (or must be close enough for their parallaxes to be directly measurable).

These calibrators include Mira variables, RR Lyrae stars and Cepheid variable stars, of which Cepheids are intrinsically the most luminous. All of these have variability periods which are correlated with their absolute luminosity, and in principle the measurement of the distance of a nearby object of any of these types can then be used to determine distances to more distant similar objects simply by observing and comparing the variability periods.

The LMC lies at about 50 kpc, about three orders of magnitude less than that of the distant galaxies of interest for the Hubble constant. However, one class of variable stars, Cepheid variables, can be seen in both the LMC and in galaxies at much greater distances. The coming of the Hubble Space Telescope has been vital for this process, as only with the HST can Cepheids be reliably identified and measured in such galaxies. It is impossible to overstate the importance of Cepheids; without them, the connection between the LMC and external galaxies is very hard to make.

Even the HST galaxies containing Cepheids are not sufficient to allow the measurement of the universal expansion, because they are not distant enough for the dominant velocity to be the Hubble flow. The final stage is to use galaxies with distances measured with Cepheid variables to calibrate other indicators which can be measured to cosmologically interesting distances. The most promising indicator consists of type Ia supernovae (SNe), which are produced by binary systems in which a giant star is dumping mass on to a white dwarf which has already gone through its evolutionary process and collapsed to an electron-degenerate remnant; at a critical point, the rate and amount of mass dumping is sufficient to trigger a supernova explosion. The physics of the explosion, and hence the observed light-curve of the rise and slow fall, has the same characteristic regardless of distance. Although the absolute luminosity of the explosion is not constant, type Ia supernovae have similar light-curves [1195148] and in particular there is a very good correlation between the peak brightness and the degree of fading of the supernova 15 days4 after peak brightness (a quantity known as Δm15 [11853]). If SNe Ia can be detected in galaxies with known Cepheid distances, this correlation can be calibrated and used to determine distances to any other galaxy in which a SN Ia is detected. Because of the brightness of supernovae, they can be observed at large distances and hence, finally, a comparison between redshift and distance will give a value of the Hubble constant.

There are alternative indicators which can be used instead of SNe Ia for determination of H0; all of them rely on the correlation of some easily observable property of galaxies with their luminosity. For example, the edge-on rotation velocity v of spiral galaxies scales with luminosity as L ∝ v4, a scaling known as the Tully–Fisher relation [162]. There is an equivalent for elliptical galaxies, known as the Faber–Jackson relation [36]. In practice, more complex combinations of observed properties are often used such as the Dn parameter of [34Jump To The Next Citation Point] and [91], to generate measurable properties of elliptical galaxies which correlate well with luminosity, or the “fundamental plane” [3432] between three properties, the luminosity within an effective radius5, the effective radius, and the central stellar velocity dispersion. Here again, the last two parameters are measurable. Finally, the degree to which stars within galaxies are resolved depends on distance, in the sense that closer galaxies have more statistical “bumpiness” in the surface-brightness distribution [157]. This method of surface brightness fluctuation can again be calibrated by Cepheid variables in the nearer galaxies.


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