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# Extragalactic Distance Measures

The trick to determining the distance to a galaxy is to find in that galaxy a standard candle, an object that has a known luminosity. If such a class of objects can be found, and if it can be calibrated, preferably by measuring the parallax of one such object within our own galaxy, we can calculate the distance to the galaxy by measuring the brightness of the object and applying the inverse square law.

The primary standard candle in astronomy is the Cepheid variable, a star with a luminosity that is set by its pulsation period. A second important standard candle is the type 1a supernova, which has a peak luminosity that can be used as a standard candle. Because type 1a supernovae are rare in any given galaxy, their use is limited to testing theories of cosmology and calibrating a third important distance measure?the cosmological redshift. Very distant galaxies are moving away from us with a velocity that is proportional to distance. The redshift of the light from these galaxies is therefore a measure of their distance. This distance measure, however, can only be calibrated against standard candle distance indicators.

### Cepheid Variables

The best standard candle for determining the distance to the nearby galaxies is the Cepheid variable star. These are bright and reasonably common, with strong identifying signatures, so their observation in other galaxies is not too difficult. Many of those observed in our own Galaxy have measured parallaxes, so this standard candle is calibrated to physical units; 273 Cepheid type variables having been observed by the Hipparcos satellite.

The Cepheid variable has a luminosity that is a function of period alone. If you observe one and determine its period of variability, then you can assign it the luminosity of nearby Cepheids with similar periods. By measuring the brightness of the Cepheid in the distant galaxy, one can derive the distance using the inverse square law; in terms of absolute magnitude M and apparent magnitude m,1 the distance is given by R = 101 + 0.2( m - M ) parsecs.

### Type 1a Supernovae

The standard candle of choice in cosmological studies is the type 1a supernova. It is as bright as any event in the universe, so it can be seen in the most distant galaxies. A supernova is brighter than its host galaxy, and on many occasions, the host galaxy of an observed type 1a supernova is too dim to observe.

The characteristic of type 1a supernovae that make them standard candles is that low redshift supernovae with similar durations and spectra have similar peak luminosities. Those observed at low redshift can be calibrated with Cepheid variables.

The basic theory behind this type of supernova is that we are seeing the aftermath of the explosion of a carbon-oxygen thermonuclear bomb. The progenitor, a degenerate dwarf (white dwarf), is pushed over the Chandrasekhar mass limit; as the star starts to collapse, the oxygen and carbon in the star undergo nuclear fusion, releasing the energy in the supernova.

Theory, however, cannot provide the observed behavior from first principles, so it is unable to show whether the standard candle behavior, which is seen in nearby supernovae, persists at large redshift, where the universe is younger, and therefore somewhat different from the current universe in its galactic structure and chemical composition. If changing conditions within the universe make the luminosity of a type 1a supernova change with redshift, then the distance that is derived will be systematically too large or too small. This has a direct bearing on the application of this standard candle to cosmology.

Beyond the question of whether conditions in the early universe affect the luminosity of a type 1a supernova, a second problem besets this standard candle that severely limits its use: supernovae are rare in any given galaxy. They therefore cannot be used to determine the distance to any galaxy that we may be interested in. Their only uses are in calibrating other distance measures, such as the cosmological redshift, in testing cosmological theories, and in studying the surrounding of these supernovae. Current studies with these standard candles are examining the expansion of the universe at redshifts between z = 0.01 and 1.

### Cosmological Redshift

The light from distant galaxies is shifted to lower frequencies. This observed behavior is well established, and is the motivation behind the theory that the universe is expanding: the redshift is interpreted as a consequence of the galaxies moving away. Independent of theory, the magnitude of the shift to lower energies, which is called a redshift, is a measure of the distance to a galaxy that can be calibrated through comparison to standard candle measures of distance.

The standard way of expressing the redshift of a galaxy in astronomy is through the variable z, which is defined by the equation

 νobs = νemit /( 1 + z ),

where νobs is the observed frequency of an emission line, and νemit is the emitted frequency of the emission line.

For z much less than unity, the distance is found to be proportional to z; This relationship is given by the Hubble constant H0, which is the ratio of the implied velocity to the distance. The distance is then related to the redshift by d = c z/H0, where c is the speed of light. The value of the Hubble constant that is determined using type 1a supernovae as standard candles is H0 = 65 km s-1 Mpc-1 (Mpc stands for megaparsec), a value that is believed correct to 10%. From this we see that objects with a redshift of 0.1 are about 4.6 gigaparsecs way.

For redshifts approaching unity, the dependence of distance on redshift is set by the precise nature of our cosmology. The distance versus the redshift at large redshift is an active field of research that impacts our theories for the evolution of the universe.

1The magnitude scale is a logarithmic scale of brightness. As the magnitude of a star increases, its brightness decreases. The apparent magnitude m is the brightness of a star measured at Earth, and it depends both on the luminosity and the distance of the star. The absolute magnitude M is the brightness of a star that is placed 10 parsecs from Earth. Normally one must state the frequency range over which the measurement is made. The Sun has an absolute visual magnitude of 4.83, and the brightest star in the sky, Sirius, has an absolute visual magnitude of 1.4 and an apparent magnitude of -4.6. 