C Ho

where the time tH = H0-1 is called the Hubble time. Substitute the numerical value for H0 (30) into (37) to find the anticipated age,

the universe)

The WMAP observers report an age of 13.7 ± 0.2 Gyr.

Extragalactic "standard candles"

The quest for standard candles as independent indicators of distance continues to this day. Modern sensitive instrumentation and the ingenuity of contemporary astronomers are bringing about important advances. The objects now used as standard candles include supernovae at their peak brightness, galaxies of the spiral and elliptical types, globular clusters, novae, and planetary nebulae. Each of these can be very luminous and hence can be seen to very large distances.

These classes would not seem to be useful standard candles because they come with differing luminosities. Nevertheless, sometimes there are sufficiently large numbers of a given type of object in a given locale to use global properties of the entire sample as a distance indicator. The luminosity function discussed below is an example of such a global property.

Astronomers have also identified observable characteristics (e.g., spectral line broadening) of extragalactic objects that are related to the luminosity. Such objects can then be used as standard candles. This is completely analogous to cepheid or RR Lyrae variables which exhibit different luminosities, but the period of oscillation specifies the luminosity. It is also analogous to the spectroscopic classification of stars; the spectrum classifies the star and thus its luminosity is known. Such an approach is used with supernovae and galaxies as we now discuss.

Luminosity functions A hypothetical example of the use of global properties as a distance indicator would be to determine the average luminosity of all the globular clusters in a distant galaxy. Globular clusters are compact collections of 105 -106 stars of which several hundred surround the center of the Galaxy (Figs. 1.9 and 3.3). One can argue that the average luminosity of globular clusters should be the same from galaxy to galaxy, and that this average value could then be used as a standard candle.

In practice, one uses the entire shape of the luminosity function rather than simply the average luminosity. This function is the distribution of luminosities of a given type of object such as globular clusters in a given galaxy. Specifically, it is the number of objects per unit luminosity interval as a function of luminosity. As shown in Fig. 7, it is usually plotted on a log-log plot, in this case, log relative number per unit absolute magnitude vs. absolute magnitude; recall that magnitude is a logarithmic quantity.

It turns out that globular clusters in a typical galaxy have a distribution that peaks at some luminosity and has a characteristic width. The luminosity function can be characterized with these two parameters (peak luminosity and width). The two parameters serve as a standard candle, if they are calibrated to some known distance.

The luminosity function of planetary nebulae is also an important distance indicator. Planetary nebula are gas clouds ejected from stars of high temperature and luminosity late in their evolution. The ejected clouds of gas are illuminated by the ultraviolet radiation of the central star (Fig. 1.8). The cloud effectively absorbs and reradiates much of the luminosity, mostly as spectral lines in the optical part of the spectrum. The luminosity in these spectral lines can be seen from great distances, and a given galaxy might contain several hundred detectable objects.

Figure 9.7. Luminosity function of globular clusters from —2000 clusters in four elliptical galaxies in the Virgo cluster of galaxies. The ordinate is the log of the relative number of clusters in unit interval of absolute B -band magnitude, and the abscissa is the absolute magnitude in the blue band. The distribution is characterized by the magnitude, rnpeak and a width 2a which are used together as a standard candle for other galaxies. [From G. Jacoby, et al., PASP 104,599 (1992); courtesy W. Harris]

Figure 9.7. Luminosity function of globular clusters from —2000 clusters in four elliptical galaxies in the Virgo cluster of galaxies. The ordinate is the log of the relative number of clusters in unit interval of absolute B -band magnitude, and the abscissa is the absolute magnitude in the blue band. The distribution is characterized by the magnitude, rnpeak and a width 2a which are used together as a standard candle for other galaxies. [From G. Jacoby, et al., PASP 104,599 (1992); courtesy W. Harris]

The luminosity function or other characteristic of a distance-indicating object may vary with the type of galaxy with which it is associated. Galaxies come in a wide range of types from spiral to elliptical (Hubble types) with differing ages of stars and colors (reddish to bluish) and metal content. Distance indicators can, in principle, vary slowly with galaxy type. This effect can be calibrated with observations of the closer galaxies.

Supernovae

Supernovae are violent outbursts from stars that gravitationally collapse. The luminosity can approximate or exceed that of an entire galaxy in which it resides for a short time (days). One type of supernova (Type Ia) is believed to arise from a dense white-dwarf star that gradually accretes matter from a companion until it reaches its maximum allowed mass. At this point it undergoes runaway nuclear burning and incinerates itself. The nature of the triggering mechanism (slow accretion) suggests that every such event would release the same amount of energy; that is, the peak luminosity of the outburst should be a standard candle. This method is proving to be a valuable distance indicator for very large distances. In fact, as described below, it is now yielding rather surprising results.

Line-broadening in galaxies

Galaxies themselves have been used as standard candles. It turns out that large spiral galaxies are more luminous if they are more massive. This is not surprising because a greater mass indicates more stars. Also, the more massive galaxies cause the gas in their outer regions to rotate around their centers faster than for lower-mass galaxies. This follows directly from F = ma and Newton's gravitational law.

This speed of rotation shows up as a Doppler broadening of the spectral lines emitted by the rotating hydrogen gas at the radio wavelength of 21 cm (1420 MHz) if the spectrum includes radiation from the entire galaxy. In this case the receding and approaching parts of the galaxy will yield, respectively, red and blue shifted lines. Taken together in one spectrum the 21-cm line thus appears broadened. Because both the luminosity and the broadening are correlated with the mass, the line broadening specifies the luminosity of the galaxy in question. In short, a spiral galaxy with a known Doppler broadening is a standard candle. This method for the determination of distances is known as the Tully-Fisher method.

A similar relation is used for galaxies which are lacking in interstellar gas and for which there is no organized rotation; the angular momentum is low. In these elliptical galaxies, the stars move around in the gravitational well with more or less random velocities. Again, if the total mass is greater, the star velocities and the overall galaxy luminosity are greater. The absorption lines in the optical spectra of the individual stars contribute to the overall galaxy spectrum. This overall spectrum exhibits broadened spectral lines because of the random velocities of the stars contributing to it.

Thus, the line widths are again a measure of the luminosity. Elliptical galaxies with a specific line width are thus standard candles of a known luminosity. This is known as the Faber-Jackson method. It has been extended to include an angular-diameter parameter, and is now called the Dn-a method, where the two symbols refer to the angular diameter (containing a specified average surface brightness) and the line width.

Surface brightness fluctuations A very different method invokes pixel to pixel fluctuations of intensity in the images of elliptical galaxies. A distant galaxy will generally appear as a diffuse blob due to the thousands of unresolved stars in each image element, that is, in each pixel of a CCD. (Charge-coupled devices are described in Section 6.3.) However, a relatively small number of extremely luminous stars (giants) in the galaxy will give the galaxy a roughened or mottled appearance because each pixel contains only a few such objects.

Since there are typically several such stars in a pixel, one can not use a single such object as a standard candle. Happily though, it turns out that the flux from a single "average" star is directly obtainable from the pixel-to-pixel fluctuations of light in the CCD image. Thus these stars can be used as standard candles if their average luminosity is known from studies of closer examples.

Consider the situation of Fig. 8a wherein a galaxy is imaged onto a CCD at the focus of a telescope. Each cell in the figure represents a single pixel, and the dots are the giant stars in the galaxy. The galaxy is presumed to be larger than the pixel arrays shown. If the galaxy is removed to twice the distance, the galaxy will be reduced in angular size and likewise the angular spacing between giants will be reduced. In the image plane, the giants will thus be closer together, and more of them will be imaged onto each pixel; see Fig. 8b.

The data from such observations consist of a single number from each pixel representing the total number of photons recorded in that pixel during the exposure. In Figs. 8c,d, we represent this number with a shading. The shading takes into account the numbers of giant stars in each pixel as well as the fact that the stars at the farther distance each yield only 1/4 the flux of those at the closer distance. It is important to note that the calibration of the shading with photon number is the same for Figs. 8c and 8d.

The number of photons recorded per pixel in Fig. 8, averaged over many pixels, is independent of distance to the galaxy because the larger (average) number of giant stars in each pixel at the farther distance exactly cancels the decrease in flux from each such star. This is just the distance independence of specific intensity discussed in Section 8.4. We thus argue that the "average" shadings in the two frames (Figs. 8c,d) are the same. The average of the pixel values provides no distance information.

However, there is an obvious and important difference in the two cases of Figs 8c,d. In the closer case, the lesser number of giant stars in each pixel yield greater fluctuations about the mean flux. The fluctuations are thus a strong distance indicator. We now show quantitatively how Poisson fluctuations provide the desired distance information.

If the giant stars all have similar spectra, the photon number recorded in some bandwidth in each pixel is a measure of the energy recorded in each pixel. For a single point source, this is a measure of a flux density (W m-2 Hz-1 in SI units) because we know the effective area and bandwidth of the telescope system. We therefore designate the energy per pixel from a single star to be the "flux" f.

Let us further designate the expected (or mean) number of bright giants per pixel to be m. The expected total energy recorded in each pixel is thus the product fm. Here, for our purposes, we assume all the stars have the same flux. Since each pixel corresponds to a known solid-angle element on the sky, the product fm is thus a measure of the specific intensity (W m-2 Hz-1 sr-1 in SI units). Recalling that

(a) Actual view, distance r (b) Actual view, distance 2r

(c) CCD view, distance r

(d) CCD view, distance 2r

(c) CCD view, distance r

(d) CCD view, distance 2r

Figure 9.8. Giant stars (black circles) in a galaxy as viewed by a charge-coupled device (CCD). Each cell is a pixel. (a,b) Stars for two distances of the galaxy, one twice the other. Note that the star patterns of (a) are in the central 4 boxes of (b). (c,d) Shadings indicating the number of detected photons recorded in each pixel for the two cases (a,b). The calibrations of shading to photon number are the same for (c) and (d). The recorded photon number averaged over all pixels is almost the same (within statistics) for the two cases, but the fluctuations about the mean are much greater for the closer case (c). Table 3 applies directly to this example. [Adapted from J. Tonry, personal communication; also see G. Jacoby et al, PASP 104, 599(1992)]

I = B (8.42), we call the product a surface brightness b, b = fm (J/pixel; mean surface brightness) (9.39)

This is the expected (or mean) surface brightness of the galaxy measured in units of recorded joules per pixel.

The fluctuations of this brightness from pixel to pixel reflect the fluctuations in the number of giants per pixel. The fluctuation in this number will obey the Poisson distribution which has an rms deviation of m1/2 (6.8). This latter number must be multiplied by the flux from a single star to obtain the rms deviation in the surface ab = fm1/2 (W; rms dev. in brightness) (9.40)

Eliminate m from (39) and (40) and solve for f, the flux from a single giant star in the galaxy, ab

+ f = — (J/pixel; flux from single giant star) (9.41)

Thus the two measured quantities ab and b yield the flux density f (J/pixel) from a single star. As noted this can be converted to SI units. As we know, the flux density from a single star yields its distance, given independent knowledge of its luminosity. Thus (41) can be used to find the distance to the galaxy containing the giant stars. In practice the giants will have a spread of luminosities which must be taken into account in the analysis.

The quantity b in (41) is the expected (or mean) surface brightness. The mean value is obtained by averaging the values recorded in the pixels of the image. Similarly ab is obtained by taking the rms deviation of these same recorded values. As noted, the average surface brightness b is independent of distance to the galaxy (more stars per pixel but less flux from each star). In contrast, the rms brightness deviation ab decreases with distance since the flux f in (40) decreases faster with distance than the rms star number deviation m1/2 increases. The flux f from a single star (41) thus decreases with distance, as it must.

The detected giant stars in elliptical galaxies turn out, from independent determinations, to have an average luminosity Lav (per giant star) that is quite constant from galaxy to galaxy. Thus indeed they can serve as standard candles. The flux recorded by a pixel from a point source of luminosity L av at distance r from earth is f = (Lav/4^r2)A, where A is the effective area of the telescope mirror and where we adopt a non-relativistic Euclidean universe. We use the area A of the mirror because all the light that impinges on this area from a point source is deposited on the pixel where the point source is imaged. This expression can be equated to (41) to yield the distance r to the average giant star in terms of known quantities, ibL av AN1/2 1

This discussion illustrates the principle of this fluctuation method. It has in fact proved to be a practical method of finding distances independent of the redshift. In turn these yield values of the Hubble constant, from H0 = v/r where v is obtained from the redshift. The method requires that the galaxy under study be sufficiently large in angular extent to extend across a substantial number of telescope resolution elements. The average luminosity L av of a single giant star may be obtained from cepheid distance indicators in nearby galaxies, e.g., Andromeda (M31).

Table 9.3. Surface brightness fluctuations, examplea

Parameter Distance r Distance 2r

Mean no. stars per pixel m 4m

Flux from one star per pixel f f /4

a Refer to Fig. 8; courtesy of J. Tonry (personal communication). b The flux in line 5 agrees with that in line 2.

The illustration of Fig. 8 can be put in the context of the above derivation with Table 3, which shows how the several quantities change when the distance is increased a factor of two. The last row shows that indeed the expression (41) yields the flux from a single star, in agreement with line 2.

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