The Big Bang
The Big Bang

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The Big Bang

3 Distances of galaxies

3.1 First steps towards a distance scale

As you will see from Table 2, when it comes to astronomy and cosmology, one is called on to deal with a wide range of distances. (Note that a light-year (ly) is the distance light travels in one year, i.e. 9.46 × 1015 m. The distances are also quoted in a very commonly used astronomical unit of distance: the megaparsec, Mpc, where a parsec (pc) is 3.26 ly or 3.09 × 1016m.)

Table 2

Distance
Earth–Moon 1.28 light-seconds 1.25 × 10−14 Mpc
Earth–Sun 8.3 light-minutes 4.8 × 10−12 Mpc
Nearest star ≈ 4 ly 1.23 × 10−6 Mpc
Diameter of our Galaxy ≈ 105 ly 3 × 10−2 Mpc
Distance to nearest galaxy ≈ 2 × 106 ly ≈ 1 Mpc
Farthest galaxy seen ≈ 4 × 109 ly ≈ 1.2 × 103 Mpc

To measure the distance of a far-off galaxy clearly requires a series of steps. The first of these, the Earth–Sun distance, is based on our knowledge of the orbital period of the Earth about the Sun, and that of some other planet. One can then readily obtain an estimate of their relative distances from the Sun by using Kepler's third law. To convert this to an absolute measurement, one needs a determination of the actual distance between the Earth and the planet at some time. In practice, this fix is gained by measuring the distances to Mercury, Venus and Mars, using radar. These then allow one to compute the Earth–Sun distance. It is currently estimated, with obvious high precision, to be 149 597 870.66 km.

Knowing this, the diameter of the Earth's orbit can now be used as a baseline for measuring the distance to nearby stars, using the surveyor's triangulation method. One makes angular measurements on a star from opposite ends of an Earth-orbital diameter, i.e. at intervals of 6 months. At the time of writing (2004), data from the Hipparcos satellite surpass in precision and scope all previous measurements of nearby stellar distances.

But no matter how good this satellite-gathered data, there soon comes a stage where the angles become too small to measure accurately using the triangulation method. Some other method must be employed to extend the distance scale to the more distant stars.

If a source has a known total light output, i.e. known luminosity, then it can be used as a standard source. The distance of the source can be found from the received light power. The further away the star is, the fainter it appears. The relationship follows an inverse square law (allowing for various corrections, such as that due to the absorption of light by interstellar dust). The light power received per unit area of detector is called the flux density. It follows, summing over the area of a sphere centred on the star and of radius, r, equal to the distance to the star, that in the absence of absorption:

Unfortunately stars differ widely in their luminosities, so if we simply look up at the sky and pick out a faint star, it may be either an intrinsically dim star, or a particularly distant star – or partly dim and distant.

This ambiguity can be partly overcome by recognising that, while it is impossible to directly measure the luminosity of a particular source, one can estimate the temperature of the star. This is done from measurements on the overall shape of its emitted spectrum – stars emitting predominantly in the red part of the spectrum being cooler than those that emit more in the blue region. Thus the spectrum, and hence temperature, can be determined with a fair degree of confidence. Now, it turns out that when observations are made on a compact cluster of stars (the cluster being small enough for all its stars to be considered equidistant from us), a plot of flux density versus temperature generally gives the same distribution no matter which cluster is chosen – apart from an overall constant factor depending on the distance to the cluster. Thus, the most likely luminosity of a star is related to its temperature. But in order to use the temperatures of stars to obtain luminosities, and hence distances, from Equation 2, we need to progress beyond a plot of flux density versus temperature to one of luminosity versus temperature.

In order to achieve this, studies have been made of a group of about 100 stars called the ‘Hyades star cluster’. This group is close enough to us for its distance to be determined by the Hipparcos satellite. Knowing this distance and the flux density of the stars, the luminosities could be established, and the plot of luminosity versus temperature calibrated. By studying the flux densities and colours of stars more distant than the Hyades star cluster, we can use this calibrated plot to infer their distances from us.

This method applies to stars confined to our own Galaxy. As for stars in other galaxies, most of these cannot be distinguished separately. So, the next step is to try to find a particularly bright type of star – one that can be recognised not only in our Galaxy but also in neighbouring galaxies. If this were possible, it would give us a method of extending the distance scale out to other galaxies. Fortunately there does exist a type of star that can be recognised at least in neighbouring galaxies – a star known as aCepheid variable. The important characteristic of a Cepheid is that its light output varies in a regular fashion, with a period which is directly related to its mean luminosity (see Figure 6a). The variation is due to cyclical changes in diameter. The relationship between mean luminosity and period can be calibrated by studying Cepheids that are sufficiently close for their distances to have been measured by methods mentioned earlier. So whenever such stars can be distinguished from the other types of variable star in the more distant galaxies, their luminosities can be deduced (see Figure 6b). Comparing luminosity with the measured flux density then enables the distance of the star to be calculated. Cepheids have been used extensively to measure the distances to nearby galaxies -those belonging to a cluster of galaxies known as the Local Group (to be described later). Following the advent of the Hubble Space Telescope with its superior resolution of individual stars, it became possible in 1996 to extend Cepheid-based measurements as far as a galaxy known as M100, in the Virgo cluster, yielding a distance of 5.6 × 107 ly.

Figure 6
based on H. Arp (1960) in Astronomical J., 65, 426
Figure 6 (a) An 18-day section of the light curve of the typical Cepheid variable, Delta Cephei, which has a 5.37-day oscillation. (b) The observed relationship between the period and luminosity (in watts) of Cepheid variables (1 W = 1 J s−1). Notice that both scales are logarithmic, so the straight line implies that period ∝ (luminosity)m, where 1/m is the slope of the line.

Another type of star that can be recognised in other galaxies is a Type Ia supernova. What happens is this: when a medium-sized star, such as the Sun, approaches the end of its active life, it shrinks down to a small star called a ‘white dwarf’. If the white dwarf happens to belong to a binary system of two stars, it can, from time to time, capture material from the atmosphere of its companion, so increasing its own mass. But this is a process that cannot continue indefinitely. The maximum mass for a white dwarf is 1.4 solar masses; anything above that limit and its inner forces cannot resist the inward pull of gravity, and the star has to collapse down to the next stable form (called a ‘neutron star’). Thus, the white dwarf in the binary system can capture material only up to this limiting mass. Once it exceeds this limit, the collapse occurs with the excess energy emitted as an explosion – the Type Ia supernova. Because white dwarfs are always of the same limiting mass when this happens, the explosions are similar, yielding essentially the same spectrum and variation of light intensity over time. (It is these characteristics that allow the Type Ia supernovae to be distinguished from other types.) Why they are important in the present context is that, to within ±30%, they have the same luminosity. Taking into account that the variation in luminosity with time does show a difference in characteristic decay times, and these are correlated to somewhat different values of peak luminosity, allowance can be made for this, and the uncertainty spread in peak luminosity reduced to ±20%. This in turn leads to relative distances measurable to ±10%.

Type Ia supernovae provide us with standard lamps (or standard explosions!), that, at their peak, are 100 000 times brighter than a Cepheid, and are visible hundreds of times further away. Having established their luminosity by measuring the few that occur in galaxies for which the distance is already known from measurements on Cepheids, they can be used to extend the distance scale to very great distances.

This however still does not go far enough. Since normal stars cannot be resolved in the farther galaxies, the additional methods of estimating distance will have to be based on the properties of galaxies as a whole. We therefore interrupt the distance story to describe some properties of whole galaxies, showing first why they are important to cosmology, and then how they have led to new ways of measuring distance.

Question 4

Suppose an astronomer using a telescope of 2 m diameter has a detector whose limit of sensitivity is 3.2 ×10−17 W. Use Figure 6 to deduce, for this instrument, the period of the faintest Cepheid variable that can be observed at a distance of 2 × 106 ly [light years].

(Remember: The value of a light-year in SI units is given on the data sheet)

Answer

At 2 × 106 ly, the fraction of light intercepted is:

The energy received from the faintest star is 3.2 × 10−17 W.

Therefore the output of the star must be

From Figure 6(b), the corresponding period of the Cepheid variable is about 3 days.

S357_1

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