Global Propertiessizes Luminosities And Masses

Most astronomers have had, at times, to contend with physicists who like to remark that astronomy is the "science" where the error bars go on the exponents. Against this background, it is slightly galling that we cannot claim to have very precise measurements of the masses or brightness of any galaxies, and that estimates of their distances are only slightly better. Galaxies do present formidable obstacles to what would normally be good laboratory measurement practice, so astronomers can perhaps be forgiven this lapse. The radiation from galaxies comes from extended areas on the sky, without sharp edges, and is measured against various kinds of foreground and background radiation. Lacking sharp edges, they will not allow us to draw a circle around some region and say that it includes all the light.

Despite this fuzziness, we need some ways to describe the sizes of galaxies. There are several useful approaches, tied to various extents either strictly to observable quantities or theoretical concepts. It is straightforward to measure the brightness within some particular aperture size, so that there are measures of galaxy magnitude which integrate the flux within either a particular isophote or within an elliptical approximation to this isophote. Holmberg introduced a system integrating the light within an isophote corresponding to photographic (essentially blue) magnitude 26 per square arcsecond. This level proved too faint for routine photographic photometry, so de Vaucouleurs adopted a system based on B25—the magnitude integrated above a surface brightness of B = 25 per square arcsecond. These systems have the advantage of being completely empirical, relying on no assumptions beyond our ability to calibrate the data properly. This is, in fact, not really trivial for bright galaxies, since we always see galaxies superimposed on foreground and background "sky" light which must be subtracted, and there is no clear place where we can guarantee that a galaxy's own light has dropped to absolutely zero. The best we can do is estimate the sky light from locations far enough away that the galaxy contribution is likely to fall below some small fraction of the likely sky error, while working to avoid systematic problems with the sky intensity from such causes as vignetting in the optical system, scattered light from bright stars, and actual changes in the sky brightness with position across large enough pieces of the sky.

These isophotal magnitudes have an obvious bias with galaxy surface brightness, missing an increasing fraction of the light for galaxies of lower surface brightness (a fraction as large as "all" for galaxies whose surface brightness nowhere exceeds the threshold level). In an effort to produce flux data which are more properly comparable for a wide range of galaxy properties, and less vulnerable to systematic bias from foreground absorption by Galactic dust or cosmological dimming, serious effort has gone into defining effective radii and magnitudes derived from flux within the effective radius. As defined by de Vaucouleurs, and used in the three successively larger Reference Catalogs, the effective radius is the radius within which half of the light from a galaxy originates (as projected in the sky; this differs from the projection of the three-dimensional radius enclosing a volume within which half of the light arises). This quantity will be directly comparable for all kinds of galaxies. The obvious problem, though, is that some theoretical input is needed to guide an extrapolation from the outermost observed piece of the galaxy to "total" flux. It is safe to assume that the surface brightness continues to decline monotonically outward, and the extrapolation is generally guided by the brightness profile where we can measure it. For example, in elliptical galaxies which follow an r1/4 profile closely, it is reasonable to assume that this continues outward. For spirals, some sort of combined profile, including an exponential distribution of light from the disk, is taken to derive an extrapolation rule depending on Hubble type, as was done in the Reference Catalogs by de Vaucouleurs and collaborators (1991). The upshot is that the effective radius can be well determined, since the allowed range of extrapolation of the brightness profile won't change it much. The luminosity within the effective radius is only half as uncertain as the total luminosity (though having the same relative uncertainty), and, again, this approach allows fair comparison across galaxies with very different surface brightnesses. For particular purposes, other measures of size have proven useful as well. One notable example is the Petrosian radius, defined as the radius within which the mean surface brightness is a specified multiple of the surface brightness at that radius. This has proven important in measuring Tolman dimming, a direct prediction of cosmological models in which space is expanding, as distinct from those in which some other process produces the redshifts of distant objects.

Using any of these measures, galaxies have a vast range of luminosity. The faintest dwarf galaxies in our neighborhood (such as the Draco dwarf) have blue-light luminosities only about 250,000 times that of the Sun, while the largest cD galaxies in the centers of some rich clusters (such as the extremely luminous central galaxy in the cluster Abell 1413) range as high as 2 x 1012 solar luminosities, a range of 10 million to one. Similarly, we can detect the starlight in the smallest dwarf systems to perhaps 1,000 parsecs from the center (as poorly defined as that point sometimes is) and in the largest galaxies as far as 2 megaparsecs from the nucleus. Galaxies thus span an enormous range in size and stellar content. The mass range of galaxies greatly exceeds the thousandfold range in mass of individual stars, though their brightnesses do not differ quite as much as the brightest and faintest individual stars because of stars' strong relation between mass and amount of light produced per unit mass. Even among spiral galaxies, which occupy a moderate span of size and luminosity among galaxies, similar-looking galaxies can in fact differ in size by a factor of 10.

The distribution of galaxy luminosities is not only very deep, it is also strongly weighted toward dimmer systems, as expressed in the luminosity function This is the number density of galaxies (usually per cubic megaparsec) as a function of luminosity or absolute magnitude, strictly analogous to stellar luminosity functions. Summed over all galaxy types, the form of this function is well-determined except perhaps for the faintest dwarfs, and the shape is closely constant over a wide range of environments from the "field" to rich clusters. Its shape is well described by the Schechter function

in which L* is a fiducial luminosity and a is the asymptotic slope at the faint end in the (log number-absolute magnitude) plane. L* is often taken as the typical luminosity m G <U T3

En tuo o

Schechter luminosity function a = -0.7, M, = -22.0 Data: field galaxies, CfA survey

-24 -22 -20 -18 Absolute B magnitude

Figure 4.6. The luminosity function of galaxies, shown as the log of the space density (as expressed in Gpc-3) versus galaxy absolute magnitude. Dots represent the space density of galaxies in various luminosity bins, as retrieved from analysis of the Center for Astrophysics redshift survey for "field" galaxies outside rich clusters. The curve shows a fit of the Schechter function to these data. The free parameters are in the absolute normalization, slope of the faint end, and characteristic luminosity for the turnover at the bright end. The shape parameters are remarkably robust for large samples of galaxies, and provide useful measures of the characteristic mass and luminosity scales for galaxy formation.

for bright galaxies (Figure 4.6). Any viable theory of galaxy formation should be able to reproduce the luminosity function, as well as the global relations described below.


Star formation is an important facet of galaxy evolution caught "in the act". Much of the history of a galaxy is written through changes in its stellar population, inextricably linked through the processes of starbirth and stellar evolution. We can ask whether the current level of star formation is comparable with its past level, or has increased or decreased markedly. There are several disparate tracers from various wavelength regimes which give insight to the rate of star formation averaged over different timescales. In fact, the optical and near-infrared regimes are distinguished by being the only parts of the electromagnetic spectrum in which we can easily see any but the youngest stars in galaxies.

4.3.1 Emission lines: optical and infrared

The first tracer of star formation to be developed and extensively used involves optical emission lines. The recombination lines from hydrogen have a particularly straightforward relationship to the number of ionizing photons from the hottest stars, which yields a star-formation rate (SFR) when coupled with stellar models and an initial-mass function.

In equilibrium, the number of photoionizations over the whole volume of gas ionized by a star will equal the number of recombinations as the liberated electrons become bound to protons again. This generally happens only after many weaker Coulomb encounters, which have the helpful effect of making the velocity distribution of the electrons thermal (Maxwellian) even though it would have been quite different immediately upon their ejection from neutral atoms. During recombination, most electrons will be initially captured into an excited (high-n) state, and decay to the ground level by a cascade of photon emissions. The number of recombinations can thus be measured starting with the intensity of some emission line. This is usually one of the optical hydrogen lines (Ha, H$), although instrumental advances have started to allow significant surveys using the infrared lines from transitions between higher pairs of n-values. These lines are intrinsically weaker in both energy and photon flux, but have the enormous advantage of being much less sensitive to dust extinction.

Using the notation in Osterbrock and Ferland's treatment, a balance between recombination and ionization requires that the number NLC of ionizing photons per second satisfy

where v0 is the frequency at the Lyman limit of hydrogen and a is the recombination coefficient, calculated including effects of resonent scattering which can keep essentially all atoms in n = 2 in extensive nebulae. For comparison with observations, we use the calculations of what fraction of decay cascades leads to a given emission line, which may either be derived from a full cascade matrix incorporating all processes between the various levels, or using emissivities so derived for various lines at the relevant electron temperature. An effective recombination aeff rate can be defined, such that the emission rate of photons in the relevant emission line is simply npneaeff. (For example, at a typical electron temperature 104 K, each H^ photon stands in for 8.5 recombinations, while an Ha photon results on average from nearly half of all recombinations.) The line luminosity will be this quantity, integrated over the nebular volume, as diminished by distance (via 4-kD2 for small enough distances to be Euclidean):

fn„neaeff dV


A set of young stars will give off a number NLC of photons below the Lyman limit, and if the surrounding gas has sufficient column density, all these will be absorbed and eventually lead to recombinations (the ionization-bounded case). If we have a model of the stars' spectra in this region (which is unobservable because of this very absorption), we can calculate the number of ionizing phtons per second (also known as Q) as

Lv dv

lc hv in which the spectrum is converted to photons from the more usual energy units.

A widely-used relation was derived by Kennicutt (1998), using models for the ionizing continuum of hot stars. Extrapolating to the entire mass in stars with a

Salpeter (1955) initial mass function of power-law form, it is:

SFR = L{Ha)/1.12 x 1041 erg/sec in solar masses per year, while counting only those stars with masses above 10 solar masses, which actually ionize surrounding hydrogen, it becomes:

4.3.2 Ultraviolet continuum from hot stars

Since young stellar populations are distinguished by hot, massive, and short-lived stars, it is natural to seek their direct signatures in the emitted ultraviolet range. Especially shortward of about 2000 A, only O and B stars are bright, so that a particular mix of stellar mass and age directly implies a star-formation rate averaged over less than 108 years. With an adequate spectrum, we can make progress in untangling the temperature mix of these stars, and estimate their abundances from the strengths of stellar-wind features. If the star-formation rate is constant over the lifetimes of the stars contributing in the far-UV (around 1500 A), the UV spectral shape is close to a power law with Fx <x A—2. This translates to flux per unit frequency Fv nearly constant with wavelength, so the star-formation rate is measured in a robust way using frequency-linked quantities:

again in solar masses per year.

The weakness of ultraviolet measurements lies in the necessary corrections for extinction and scattering by dust grains. Since star formation is aided by grains, and current star-forming regions are generally also dusty, the very galaxies which would otherwise be brightest in the UV are also the most susceptible to having their starlight absorbed by grains. Most of the stars' radiation is emitted in the wavelength range where absorption is most efficient. The abundance of grains depends on metallicity; empirically, there is a broad correlation between the slope of the observed UV spectrum and the metallicity as derived from emission lines. This fits with the dust/gas ratios found in nearby systems of various levels of chemical enrichment (particularly using the Magellanic Clouds for comparison), and has been used beginning with work by Daniela Calzetti and collaborators to estimate the effective extinction from the shape of the UV spectrum in star-forming galaxies. This approach has figured prominently in efforts to retrieve the overall history of star formation from galaxy statistics in, for example, the Hubble Deep Fields.

4.3.3 Far-infrared dust heating

The basic principle in retrieving star-formation rates from the far-IR is that whatever energy in starlight is absorbed by grains is reradiated at wavelengths corresponding to the equilibrium temperature of the grains. This statement needs some modification— small grains can be transiently heated by single UV photons so strongly as to give misleadingly high temperatures, not all the starlight is from young stars, grains are too small to be good blackbodies in the far-infrared—but it suffices to illustrate the basic energy balance. Let us consider a grain of albedo a and radius r. For incident starlight of intensity I in units of energy/(time x area), it will absorb ^r2I per unit time. In equilibrium, this will match the blackbody radiation from the surface

4nr aT where a = (2-k k )/(15c h ) is the Stefan-Boltzmann radiation constant.

The most important feature of this process is that, in equilibrium, luminosity is preserved (although, again, reality may introduce ugly complications along any given line of sight, due to structure in the stellar or dust distributions). The temperatures of the grains can be affected particularly by their location; active starbursts often have unusually hot grain populations, since the energy density can be high and the dust is often concentrated tightly around star-forming regions.

The relation between far-IR luminosity and star-formation rate is certainly affected by the detailed distribution of star and gas, as well as the initial-mass function of the stars. A relation such as

is representative, noting that there are assorted definitions of LFIR in use (as detailed in Kennicutt's 1998 review).

Taking extinction and dust heating into account, a better overall star-formation tracer has been proposed which combines the emerging UV intensity with the amount of starlight absorbed and re-emitted in the far-infrared. At this point, sensitivity and resolution limits prevent us from applying this approach to any but the most luminous galaxies at high redshifts. Comparison of UV- and infrared-selected samples suggests that more luminous galaxies are dustier, as intuition suggests for objects with long and intense histories of star formation, and therefore the ones most effectively hidden from short-wavelength surveys. This is especially relevant to color-selected samples of galaxies, in which very dusty systems may be completely missed.

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