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For the last century astronomers have been modelling the structure of `nebulae', and here we focus on those external to the Milky Way. A key activity performed by many astronomers, past and present, is the catergorisation of these galaxies (Sandage 2005) and the quantification of their physical properties. How big are they? How bright are they? What characteristics distinguish or unite apparent subpopulations? Answers to such questions, and the establishment of "scaling relations" between two or more galactic properties provides valuable insight into the physical mechanisms that have shaped galaxies.

Understanding how galaxies form, increasingly through the use of simulations and semi-analytic modelling (e.g. Cole 1991; White & Frenk 1991; Kauffmann et al. 1993, 2004; Avila-Reese et al. 1998; Cole et al. 2000; de Lucia et al. 2006; Bower et al. 2006; Kauffmann et al. 2004; Di Matteo et al. 2005; Croton et al. 2006; Naab et al. 2006; Nipoti et al. 2006; Covington et al. 2011; Guo et al. 2011, and references therein), requires an accurate knowledge of galaxy properties and scaling laws, as elucidated by Driver (2011). Not surprising, our knowledge of galaxies is best in the nearby Universe - out to distances typically measured in megaparsecs rather than by redshift z - where galaxy structures can be reasonably well resolved. The properties of these galaxies provide the z = 0 benchmark used in the calibration of galactic evolutionary studies - both observed and simulated.

Popular scaling relations involving global galaxy parameters such as size, surface brightness, luminosity and concentration are reviewed here. As we shall see, many bivariate distributions, which are frequently assumed to be linear, are often only approximately so over a restricted luminosity range. For example, it may come as a surprise for many too learn that the useful Kormendy (1977b) relation is only the tangent to the bright arm of a continuous but curved effective radius-(surface brightness) relation which unifies dwarf and giant elliptical galaxies (section 3.2.4). Similarly, the Faber-Jackson (1976) relation with a slope of 4 represents the average slope over a restricted luminosity range to what is a curved or broken luminosity-(velocity dispersion) distribution, in which the slope is 2 rather than 4 at lower luminosities (section 3.3.3). Knowing these trends, the bulk of which cannot be established when assuming structural homology, i.e. using de Vaucouleurs' (1948) R1/4 model, is vital if one is to measure, model and make sense of galaxies.

This article has been structured into four main sections. Section 1 provides this general overview plus a further review and introduction to galaxies on the Hubble-Jeans sequence 1. Included are diagrams showing the location of dynamically hot stellar systems in the mass-size and mass-density plane, revealing that some high-z compact galaxies have properties equivalent to the bulges of local disc galaxies. Section 2 provides an historical account of how the radial distribution of stars in elliptical galaxies have been modelled, and the iterative steps leading to the development of the modern core-Sérsic model (section 2.2). Subsections cover the Sérsic model (section 2.1), its relation and applicability to dark matter halos (section 2.1.1), partially-depleted galaxy cores (section 2.2.1), excess nuclear light (section 2.3) and excess light at large radii in the form of halos or envelopes around giant elliptical galaxies (section 2.4). Section 3 presents and derives a number of elliptical galaxy scaling relations pertaining to the main body of the galaxy. From just two linear relations which unite the faint and bright elliptical galaxy population (section 3.1), a number of curved relations are derived (section 3.2). Several broken relations, at MB approx -20.5 mag, are additionally presented in section 3.3. For those interested in a broader or different overview of elliptical galaxies, some recent good reviews include Renzini (2006), Cecil & Rose (2007), Ciotti (2009) and Lisker (2009. Finally, the latter third of this paper is tied up in section 4 which contains a discussion of the light profiles of disc galaxies and their bulge-disc decomposition (4.1). Also included are subsections pertaining to dust (section 4.2), the difficulties with identifying pseudobulges (section 4.3), potential bulgeless galaxies (section 4.4) and methods to model bars (section 4.5). Throughout the article references to often overlooked discovery or pioneer papers are provided.

1.1. Early Beginnings

Looking out into the Milky Way arced across our night sky, the notion that we are residents within a pancake-shaped galaxy seems reasonable to embrace. Indeed, back in 1750 Thomas Wright also conjectured that we reside within a flat layer of stars which is gravitationally bound and rotating about some centre of mass. However, analogous to the rings of Saturn, he entertained the idea that the Milky Way is comprised of a large annulus of stars rotating about a distant centre, or that we are located in a large thin spherical shell rotating about some divine centre (one of the galactic poles). While he had the global geometry wrong, he was perhaps the first to speculate that faint, extended nebulae in the heavens are distant galaxies with their own (divine) centers.

As elucidated by Hoskin (1970), it was Immanuel Kant (1755), aware of the elliptically-shaped nebulae observed by Maupertuis, and working from an incomplete summary of Wright (1750) that had been published in a Hamburg Journal 2, who effectively introduced the modern concept of disc-like galactic distributions of stars - mistakenly crediting Wright for the idea.

Using his 1.83 m "Leviathan of Parsonstown" metal reflector telescope in Ireland, Lord William Henry Parsons, the 3rd Earl of Rosse, discovered 226 New General Catalogue 3 (NGC: Dreyer 1888) and 7 Index Catalogue (IC: Dreyer 1895, 1908) objects (Parsons 1878). Important among these was his detection of spiral structure in many galaxies, such as M51 which affectionately became known as the whirlpool galaxy.

Further divisions into disc (spiral) and elliptical galaxy types followed (e.g. Wolf 1908; Knox Shaw 1915; Curtis 1918; Reynolds 1920; Hubble 1926) 4 and Shapley & Swope (1924) and Shapley (1928) successfully identified our own Galaxy's (gravitational) center towards the constellation Sagittarius (see also Seares 1928).

With the discovery that our Universe contains Doppler shifted 'nebulae' that are expanding away from us (de Sitter 1917; Slipher 1917; see also Friedmann 1922, Lundmark 1924 and the reviews by Kragh & Smith 2003 and Shaviv 2011), in accord with a redshift-distance relation (Lemaitre 1927, Robertson 1928, Humasson 1929, Hubble 1929) 5 - i.e. awareness that some of the "nebuale" are external objects to our galaxy - came increased efforts to catergorise and organise these different types of "galaxy". As noted by Sandage (2004, 2005), Sir James Jeans (1928) was the first to present the (tuning fork)-shaped diagram that encapsulated Hubble's (1926) early-to-late type galaxy sequence, a sequence which had been inspired in part by Jeans (1919) and later popularised by Hubble (1936a; see Block et al. 2004). Quantifying the physical properties of galaxies along this sequence, with increasing accuracy and level of detail, has occupied many astronomers since. Indeed, this review addresses aspects related to the radial concentration of stars in the elliptical and disc galaxies which effectively define the Hubble-Jeans sequence. Irregular galaxies are not discussed here.

1.2. The modern galaxy

For reasons that will become apparent, this review uses the galaxy notation of Alan Sandage and Bruno Binggeli, in which dwarf elliptical (dE) galaxies are the faint extension of ordinary and luminous elliptical (E) galaxies, and the dwarf spheroidal (dSph) galaxies - prevalent in our Local Group (Grebel 2001) - are found at magnitudes fainter than MB approx -13 to -14 mag (approx 108 Modot in stellar mass; see Figure 1a). Figure 1a reveals a second branch of elliptically-shaped object stretching from the bulges of disc galaxies and compact elliptical (cE) galaxies to ultra compact dwarf (UCD) objects (Hilker et al. 1999; Drinkwater et al. 2000; Norris & Kannappan 2011 and references therein). A possible connection is based upon the stripping of a disc galaxy's outer disc to form a cE galaxy (Nieto 1990; Bekki et al. 2001b; Graham 2002; Chilingarian et al. 2009) and through greater stripping of the bulge to form a UCD (Zinnecker et al. 1988; Freeman 1990; Bassino et al. 1994; Bekki 2001a). It is thought that nucleated dwarf elliptical galaxies may also experience this stripping process, giving rise to UCDs.

Figure 1a Figure 1b

Figure 1. Left panel: The radius containing half of each object's light, R1/2 (as seen in projection on the sky), is plotted against each object's stellar mass. Open circles: dwarf elliptical (dE) and ordinary elliptical (E) galaxies from Binggeli & Jerjen (1998), Caon et al. (1993), D'Onofrio et al. (1994) and Forbes et al. (2008. Filled circles: Bulges of disc galaxies from Graham & Worley (2008). Shaded regions adapted from Binggeli et al. (1984, their figure 7), Dabringhausen et al. (2008, their figure 2), Forbes et al. (2008, their figure 7), Misgeld & Hilker (2011, their figure 1). The location of the so-called "compact elliptical" (cE) galaxies is shown by the rhombus overlapping with small bulges. The location of dense, compact, z = 1.5 galaxies, as indicated by Damjanov et al. (2009, their figure 5), is denoted by the dashed boundary overlapping with luminous bulges. Right panel: Stellar mass density within the volume containing half each object's light, rho1/2, versus stellar mass. The radius of this volume was taken to equal 4/3 × R1/2 (Ciotti 1991; Wolf et al. 2010).

While the identification of local spiral galaxies is relatively free from debate, the situation is not so clear in regard to elliptically-shaped galaxies. The discovery of UCDs, which have sizes and fluxes intermediate between those of galaxies and (i) the nuclear star clusters found at the centres of galaxies and (ii) globular clusters (GCs: e.g. Hasegan et al. 2005; Brodie & Strader 2006), led Forbes & Kroupa (2011) to try and provide a modern definition for what is a galaxy (see also Tollerud et al. 2011). Only a few years ago there was something of a divide between GCs and UCDs - all of which had sizes less than ~ 30 pc - and galaxies with sizes greater than 120 pc (Gilmore et al. 2007). However, as we have steadily increased our celestial inventory, objects of an intermediate nature have been found (e.g. Ma et al. 2007, their Table 3), raising the question asked by Forbes & Kroupa for which, perhaps not surprisingly, no clear answer has yet emerged. While those authors explored the notion of a division by, among other properties, size and luminosity, they did not discuss how the density varies. As an addendum of sorts to Forbes & Kroupa (2011), the density of elliptically-shaped objects is presented here in Figure 1b. This is also done to allow the author to wave the following flag.

Apparent in Figure 1b, but apparently not well recognised within the community, is that the bulges of disc galaxies can be much denser than elliptical galaxies. If the common idea of galaxy growth via the accretion of a disc, perhaps from cold-mode accretion streams, around a pre-existing spheroid is correct (e.g. Navarro & Benz 1991; Steinmetz & Navarro 2002; Birnboim & Dekel 2003; see also Conselice et al. 2011 and Pichon et al. 2011), then one should expect to find dense spheroids at high-z with 1010 - 1011 Modot of stellar material, possibly surrounded by a faint (exponential) disc which is under development. It is noted here that the dense, compact early-type galaxies recently found at redshifts of 1.4-2.5 (Daddi et al. 2005; Trujillo et al. 2006) display substantial overlap with the location of present day bulges in Figure 1a, and that the merger scenarios for converting these compact high-z galaxies into today's elliptical galaxies are not without problems (e.g. Nipoti et al. 2009; Nair et al. 2011). It is also noted that well-developed discs and disc galaxies are rare at the redshifts where these compact objects have been observed alongside normal-sized elliptical galaxies. Before trying to understand galaxy structure at high-redshift, and galaxy evolution - themes not detailed in this review - it is important to first appreciate galaxy structures at z = 0 where observations are easier and local benchmark scaling relations have been established.

1 This review does not encompass dwarf spheroidals, or any, galaxies fainter than MB approx -14 mag. These galaxies can not be observed (to date) at cosmologically interesting distances, and their increased scatter in the colour-magnitude relation may indicate a range of galaxy types (e.g. Penny & Conselice 2008, and references therein). Back.

2 Freye Urtheile, Achtes Jahr (Hamburg, 1751), translated by Hastie, op. cit., Appendix B. Back.

3 The NGC built upon the (Herschel family's) Catalog of Nebulae and Clusters of Stars (Herschel 1864). Back.

4 Reynolds (1927) called Hubble's attention to pre-existing and partly-similar galaxy classification schemes which were not cited. Back.

5 It is of interest to note that Hubble (1934, 1936b, 1937) was actually cautious to accept that the redshifts corresponded to real velocities and thus an expanding Universe as first suggested by others. He used the term "apparent velocity" to flag his skepticism. In point of fact, Hubble & Tolman (1935) wrote that the data is "not yet sufficient to permit a decision between recessional or other causes for the red-shift". Back.

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