Inclination- and dust-corrected galaxy parameters: Bulge-to-disc ratios and size-luminosity relations
While galactic bulges may contain no significant dust of their own, the dust within galaxy discs can strongly attenuate the light from their embedded bulges. Furthermore, such dust inhibits the ability of observationally-determined inclination corrections to recover intrinsic (i.e. dust free) galaxy parameters. Using the sophisticated 3D radiative transfer model of Popescu et al. and Tuffs et al., together with Driver et al.’s recent determination of the average face-on opacity in nearby disc galaxies, we provide simple equations to correct (observed) disc central surface brightnesses and scalelengths for the effects of both inclination and dust in the and passband. We then collate and homogenise various literature data sets and determine the typical intrinsic scalelengths, central surface brightnesses and magnitudes of galaxy discs as a function of morphological type. All galaxies have been carefully modelled in their respective papers with a Sérsic bulge plus an exponential disc. Using the bulge magnitude corrections from Driver et al., we additionally derive the average, dust-corrected, bulge-to-disc flux ratio as a function of galaxy type. With values typically less than 1/3, this places somewhat uncomfortable constraints on some current semi-analytic simulations. Typical bulge sizes, profile shapes, surface brightnesses and deprojected densities are provided. Finally, given the two-component nature of disc galaxies, we present luminosity-size and (surface brightness)-size diagrams for discs and bulges. We also show that the distribution of elliptical galaxies in the luminosity-size diagram is not linear but strongly curved.
keywords:galaxies: fundamental parameters — galaxies: photometry — galaxies: spiral — galaxies: structure — ISM: Dust, Extinction — radiative transfer
Although the bulge-to-total (), or bulge-to-disc (), flux ratio was the third, not prime, galaxy-morphology criteria employed by Sandage (1961), its systematic behaviour along the Hubble sequence is well known (e.g. Boroson 1981; Kent 1985; Kodaira, Watanabe & Okamura 1986; Simien & de Vaucouleurs 1986). While one-third (two-thirds) of the stellar mass density in the Universe today is known to reside in bulges (discs) (e.g. Driver et al. 2007), the actual allocation of stars as a function of galaxy type is not securely known. This is however a quantity of interest. If (large) bulges, including elliptical galaxies, are the result of mergers, while discs formed via the gravitational collapse of a rotating proto-galactic cloud, or from the accretion of gas around a pre-existing galaxy, then the ratio reflects the dominance of a galaxy’s formation mechanisms (e.g. Navarro & Benz 1991; Steinmetz & Navarro 2002 and references therein). The reliable separation of bulge and disc stars is also important because of the connection between supermassive black hole mass and the physical properties of the host bulge (see the review in Ferrarese & Ford 2005).
For many years it was believed that the bulges of disc galaxies were universally described by de Vaucouleurs empirical model (de Vaucouleurs 1948, 1958, 1978). While deviations were occasionally noted for individual galaxies, such as the Milky Way (Kent, Dame & Fazio 1991), the above belief only began to change in earnest after Andredakis & Sanders (1994) showed that the exponential model provided a better description to the distribution of stellar light in the bulges of 34 late-type spiral galaxies; a result reaffirmed by de Jong (1996a) and Courteau, de Jong & Broeils (1996) using larger samples. However the fuller renaissance, if one can use such a term, started with Andredakis, Peletier & Balcells (1995) who showed that Sérsic’s (1963, 1968) model was particularly well suited to describing the light-profiles of bulges. This result was subsequently confirmed by other investigations (e.g. Iodice, D’Onofrio & Capaccioli 1997, 1999; Seigar & James 1998; Khosroshahi, Wadadekar & Kembhavi 2000) and Sérsic’s model has since become the standard approach for describing bulges in both lenticular and spiral galaxies (e.g. D’Onofrio 2001; Graham 2001a; Möllenhoff & Heidt 2001; Prieto et al. 2001; Castro-Rodríguez & Garzón 2003; MacArthur et al. 2003; Balcells et al. 2007; Carollo et al. 2007).
Today, bulges are observed to follow a roughly linear magnitude-Sérsic index relation, with most bulges having Sérsic indices (e.g. Graham 2001a; MacArthur et al. 2003). Indeed, Balcells et al. (2003, see their figure 1a) revealed that even lenticular galaxies typically have values of around 2, a result which has subsequently been echoed by others (e.g. Laurikainen et al. 2006). Application of the average ratios derived from studies which used bulge exponential disc decomposition introduces a bias into the separation of bulge and disc flux when a bulge does not have an profile (e.g. de Jong 1996b; Trujillo et al. 2001; Brown et al. 2003). Specifically, given that the bulges of most disc galaxies have Sérsic indices , application of the model (with its higher central concentration of stars and greater tail at large radii) assigns too much of the galaxy flux to the bulge. Figure 1 shows the average -band ratios, as a function of galaxy type, from the data in Simien & de Vaucouleurs (1986), whose bulge magnitudes were obtained using an model. Also shown in Figure 1 are the -band ratios obtained by Graham (2001a) using an bulge model. Figure 1 is not however the end of the story, as neither study addressed the issue of attenuation of the bulge flux due to dust in the disc of the galaxies.
This brings us to our second, and often over-looked, concern. While bulges may contain little to no dust of their own — and have therefore in the past not had their flux corrected for dust — the galaxy discs which effectively cut them in half do contain dust. Consequently, particularly at optical wavelengths, one actually sees very little of the star light from the portion of a bulge on the far side of a disc. The extent to which the bulge and disc flux is dimmed depends on both the observed wavelength and the inclination of the (dusty) stellar disc. The reduction to the observed magnitude of the bulge and disc components can be as high as 2-3 mag in the -band (e.g. Driver et al. 2007). In spite of this, the overwhelming majority of published (optical) bulge magnitudes, and thus bulge-to-disc flux measurements, have not taken this into account and are therefore in considerable error.
On the other hand, for over the last decade most studies have corrected, at least in part, for the influence of dust on the discs. This has been accomplished by noting how parameters such as magnitude and surface brightness change with viewing angle and adjusting the observed values to those which would be observed with a face-on orientation (e.g. Valentijn 1990; de Vaucouleurs’ et al. 1991; Giovanelli et al. 1995; Tully et al. 1998; Graham 2001b; Masters et al. 2003). To obtain the intrinsic (dust-free) values of course requires an additional step: namely, correcting the face-on galaxy parameters for the influence of dust.
Sophisticated 3D dust/star galaxy models, which incorporate both clumpiness and explicit treatment of various grain compositions and sizes, now exist. The model of Popescu et al. (2000) and Tuffs et al. (2004) self-consistently explains the UV/optical/FIR/sub-mm emission from galaxies. Indeed, application of their dust model to 10,095 galaxies from the Millennium Galaxy Catalog (e.g. Allen et al. 2006) perfectly balances the amount of star light absorbed by dust in the Universe today with the amount re-radiated at infrared and sub-mm wavelengths (Driver et al. 2008). For almost a decade these models accounted for the fact that the dust is non-uniformly distributed, allowing for high extinction in the central regions, intermediate extinction in the spiral arms, and semi-transparent interarm regions which accounted for the detection of background galaxies. Their detailed model not only allows one to accurately correct for inclination-dependent extinction, but also for the additional extinction which is present when viewing galaxies with face-on orientations.
The one free parameter in their dust model which provides the calibration is the central, face-on, -band opacity . Fitting a range of simulated galaxies with varying integer-values of , Möllenhoff, Popescu & Tuffs (2006) provided tables to correct observed disc parameters to their face-on, dust-free values. Combining their tables with the statistically determined average opacity reported by Driver et al. (2007), Section 2 of this paper derives two new equations which can be used to easily correct the observed and -band disc scalelengths and central surface brightnesses for both inclination and the attenuating effect of dust, providing intrinsic, face-on, dust-free values. The above face-on, central -band opacity from Driver et al. (2007) was obtained by matching the observed inclination-attenuation relations for the Millennium Galaxy Catalog data with the dust models of Tuffs et al. (2004).
In Section 3.1 we introduce the galaxy data sets which shall be used in our analysis of disc galaxy structural parameters, while Section 3.2 describes the methodologies adopted to bring this data onto a uniform system. Section 4 provides tables of the mean intrinsic structural parameters as a function of galaxy type in various passbands. This encompasses disc scalelengths, central surface brightnesses and magnitudes. Average -band bulge effective radii, effective surface brightnesses, Sérsic indices and magnitudes are also listed. In addition, and also as a function of galaxy type, we provide the median bulge-to-disc size ratio in the -band and the median dust-corrected bulge-to-disc flux ratio in the near-infrared and various optical bands.
Finally, a selection of bivariate plots are shown in Section 5. In addition to (surface brightness)-size diagrams for discs and bulges, we provide new expressions for the size-luminosity relations of discs in early- to mid-type disc galaxies. We also present the size-luminosity relation for bulges and elliptical galaxies, emphasizing that it is not a linear relation.
While inclination corrections have been applied for many years, and are still necessary today (e.g. Bailin & Harris 2008; Maller et al. 2008; Unterborn & Ryden 2008), they only remove one of several systematic biases that cause the observed flux distribution to misrepresent the true (intrinsic) stellar distribution of galaxies. Allowances for non-homology through the use of Sérsic (bulge) models and separate dust corrections to both the disc and bulge are vital if we are to know the intrinsic physical properties of galaxies. It is hoped that the distributions and trends shown here, which have been acquired after dealing with the above three issues, as done in Driver et al. (2007), will provide valuable constraints for simulations of galaxy formation which are used to aid our understanding of galaxy evolution (e.g. Cole et al. 2006; Springel & Hernquist 2003; Almeida et al. 2008; Croft et al. 2008, and references therein).
A Hubble constant of 73 km s Mpc has been used when -independent distances were not available.
2 Corrections for dust and inclination
2.1 Disc scalelengths and central surface brightnesses
The discs of disc galaxies contain dust, in particular various mixes of graphite and silicate. This reduces the amount of (ultraviolet, optical and near-infrared) light which escapes from such galaxies. The distribution of this dust is known not to be uniform, with the centres of discs containing more light-absorbing particles than their outskirts (Boissier et al. 2004; Popescu et al. 2005). Such dust not only decreases the observed central surface brightnesses, , of the discs, but the radial gradients in the opacity also make the observed disc scalelengths, , greater than their intrinsic values.
The sophisticated dust/star galaxy model of Popescu et al. (2000) and Tuffs et al. (2004) incorporates clumpiness and an explicit treatment of various grain compositions and sizes within a 3D distribution. Solving the radiative transfer equations they are able to measure the radial dependence of attenuation in various wavebands. Möllenhoff et al. (2006) has used this to provide figures and tables for the effects of varying opacity and inclination on the parameters and . This was done using a range of galaxy models with different integer values for the central, face-on -band opacity .
From a study of 10,095 nearby ()
galaxies in the Millennium Galaxy Catalogue, Driver et
al. (2007) have recently constrained this mean
Using the linear regression routine FITEXY from Press et al. (1992)
for some inclination , with degrees describing a face-on orientation. To assist with the understanding of Figure 2, we note that in the absence of dust, the line-of-sight depth through a disc increases as the disc is inclined, such that the pathlength changes by to give . The parameters , , and in equations 1 and 2 are a function of wavelength, and given in Table 1. These equations can be used to correct observed values to the intrinsic face-on (dust-free) values. However, we caution that because the data in Möllenhoff et al. only extends to inclinations where , the curves shown in Figure 2 at higher inclinations are an extrapolation of these empirical models and may therefore not be reliable for systems more inclined than 73 degrees.
2.2 Bulge and disc magnitudes
In addition to the above expressions, we shall also use the equations in Driver et al. (2008) to correct the observed disc and bulge magnitudes for the effects of inclination and dust. These are given by
where the coefficients in the above equations depend on wavelength and are provided in Driver et al. (2008) for various passbands, including the and bands. We have used their and coefficients for our and band data. This approximation will not introduce any noticeable bias, as inspection of Figure 2 from Driver et al. (2008) will reveal.
The effect of dust on bulges is far more severe than generally realised. Past studies of the ratio, while usually correcting the observed disc flux for inclination and dust (but typically overlooking any attenuation once adjusted to a face-on orientation), normally assume that the bulge is not affected by dust. This is wrong. Moreover, given that bulges and discs have their own inclination-attenuation correction, which is not surprising, there is no single galaxy inclination-attenuation correction. That is, one must perform a bulge/disc separation. Unfortunately this may severely hamper the ability of large-scale studies (which have fitted single Sérsic models to two-component disc galaxies) to recover intrinsic galaxy parameters.
In this study we use the above formula to provide dust-corrected bulge magnitudes and dust-corrected bugle-to-disc flux ratios. However the influence of dust on the individual Sérsic parameters of the bulge ( and , see equation 5) is not well known. It will of course depend not only on the wavelength of one’s observations but also on the precise star/dust geometry, with concentrated high- bulges likely experiencing a greater degree of attenuation. Moreover, the overwhelming majority of optical bulge parameters reported in the literature are dependent, to some degree, on the happenstance inclination of their galaxy’s disc. As yet unknown corrections to a dust-free configuration are required if we are to have an accurate set of physical parameters for the bulges. Nonetheless, for now, we have tabulated -band Sérsic bulge parameters as these will be the least affected by dust.
3 The Galaxy Data
In this study we primarily investigate the properties of ‘late-type galaxies’, and more generically ‘disc galaxies’. Today, the term ‘early-type galaxy’ is often used to refer to both elliptical galaxies and lenticular galaxies. While the average properties of elliptical galaxies are not considered to be a function of inclination — and so one should identify and exclude such systems from current inclination-dependent analyses of the effect of dust — this is not true for systems with large scale discs. We have therefore included lenticular disc galaxies in our analysis, rather than grouping/excluding them with the elliptical galaxies in an early-type galaxy bin.
However we note that elliptical galaxies are certainly not always devoid of dust (e.g. Ebneter & Balick 1985; Sadler & Gerhard 1985; Ebneter, Djorgovski & Davis 1988; Leeuw et al. 2008). Indeed, they also frequently possess small nuclear dust lanes and dusty nuclear discs (e.g. Ferrari et al. 1999; Rest et al. 2001) which probably originate from sources such as type Ia SNe and stellar mass loss from, for example, the winds of red giant stars (Ciotti et al. 1991; Calura et al. 2008). On a galactic scale, however, the dust is destroyed rather quickly by ion sputtering from the hot X-ray gas (e.g. Temi, Brighenti & Mathews 2007). In contrast, lenticular galaxies, such as the Sombrero galaxy, are commonly observed to possess considerable amounts of dust in their discs. In this regard lenticular galaxies are more like spiral galaxies. Interestingly, Bekki, Couch & Yasuhiro (2002) and Aragón-Salamanca, Bedregal & Merrifield (2006) have argued that lenticular galaxies may be spiral galaxies in which the star formation has been turned off and the spiral pattern dispersed. If such a morphological transformation occurs, one would simultaneously require the bulge-to-disc flux ratio to increase. This may proceed via the passive fading of the disc and, as noted by Driver et al. (2008), upon removal of centrally located dust.
3.1 Literature catalogues
There has been a number of papers which have provided structural parameters
for nearby disc galaxies.
At the end of 2005,
when selecting which of these would be suitable for our
compilation of bulge and disc parameters,
we applied the following criteria:
(i) a Sérsic bulge model had been simultaneously fit with an
exponential disc model;
(ii) roughly a couple of dozen or more galaxies had been modelled;
(iii) morphological (Hubble or T)
Individual studies meeting the above criteria contained on the order of 20 to 100 galaxies. Previously, when these galaxies had been binned into their morphological type (e.g. Sab, Scd, etc.), this tended to result in less than ideal numbers per bin in each of these individual studies. As a result, past answers to questions of a statistical nature, such as the typical value of, and range in, some physical parameter for a given morphological type, were poorly defined. By combining here (see Section 3.2) the data from these studies, we hope to better answer such questions. Importantly, we also apply corrections to the magnitudes of both the discs and the bulges (see Section 2).
We stress that while the galaxies used here are typical representations of galaxies along the Hubble sequence, we have applied no sample selection criteria to provide a magnitude-limited sample. Given the inclination-dependent reduction to magnitudes due to dust, we note in passing that such a task is not as simple as one may first think: after correcting for dust, one’s original magnitude-limited sample will subsequently have a rather jagged boundary. However the desire to include as many galaxies as possible, rather than this issue, has been the driving ratioanle employed here.
We have started with the catalogues of Sérsic-bulge and exponential-disc parameters from the studies listed in Table 2. While the exponential model used to describe the radial stellar distribution in discs has been around for a long time (e.g. Patterson 1940; de Vaucouleurs 1957; Freeman 1970), Sérsic’s (1963) 3-parameter model has only become fashionable over the last decade. It can be written as
where is the surface brightness at the effective half-light radius , and is the Sérsic index quantifying the radial concentration of the stellar distribution. The quantity is not a parameter but instead a function of such that , where and are the complete and incomplete gamma functions as given in Graham & Driver (2005). For (Capaccioli 1989). The exponential model used to describe discs can be obtained by setting in Sérsic’s model, however it is more commonly expressed as
in which is the central () surface brightness and is the exponential disc scalelength.
The apparent magnitude of an exponential disc is such that
and that of a Sérsic bulge is given by the expression
with both and in arcseconds, and and in mag arcsec. The gamma function has the property . Conversion from apparent magnitudes to absoulte magnitudes proceeds via the standard expression . Although not done here, applying one’s preferred stellar mass-to-light ratio to the absolute luminosities yields the associated stellar masses.
In practice, because we use the observed disc magnitudes, due to the apparent ellipticity of these discs — as they are seen in projection – equation 7 had the term replaced with . The quantity is equal to the observed minor-to-major axis ratio of the disc. As we do not have information on the ellipticity/triaxiality of the bulges, this work has assumed they are spherical (but see Méndez-Abreu et al. 2008). Unless specified in their respective papers, the inclination of each galaxy has been estimated using , where is the observed outer axis ratio. Discs of course have a finite thickness, and so the applicability of this expression deteriorates when dealing with increasingly edge-on galaxies. In the past, discs have been modelled as transparent oblate spheroids (e.g. Holmberg 1946; Haynes & Giovanelli 1984; Guthrie 1992) giving rise to the relation , where is the intrinsic short-to-long axis ratio. Transparent edge on discs will have an ellipticity equal to Q rather than zero. Not surprisingly, the above relation breaks down in the presence of obscuring dust (Möllenhoff et al. 2006. their section 4.3 and Fig.12 which plots ).
The recovery of disc inclinations is not a major problem here because the
majority of our galaxies have observed axis ratios and therefore
inclinations less than 70 degrees, where the use of
remains a reasonable approximation. Only one galaxy (UGC 728, ) in
our optical data has smaller than 0.34. In our -band sample of
over 400 galaxies, where the
effect of dust on magnitudes and scalelengths are small, there are
three samples with (some) discs having . These include (1) Dong & De
Robertis (2006) where a mere six per cent of the galaxies have inclinations
greater than 75 degrees and (2) Möllenhoff & Heidt (2001) from which only
six galaxies have . As such, these few galaxies have no
statistically significant impact on the results. However, for the 19
early-type disc galaxies from (3) Balcells et al. (2003), the observed
ellipticities reach as high as 0.83. Unfortunately Möllenhoff et al. (2006) do not
provide corrections to for inclinations greater than 73 degrees,
and so we have adopted the -band value for which was used in the
dust model of Tuffs et al. (2004).
From the papers listed in Table 2 we took, when available or derivable, the five main structural parameters (). We also took the inclination of the disc, which, as noted above, usually came from the ellipticity of the outer isophotes such that ). Potential intrinsic disc ellipticities, i.e. non-circular shapes of face-on discs (Rix & Zaritsky 1995; Andersen et al. 2001; Barnes & Sellwood 2003; Padilla & Strauss 2008), and lopsidedness (e.g. Kornreich et al. 1998; Bournaud et al. 2006; Reichard et al. 2008), have not been measured for our galaxy sample and are consequently ignored. However given the intrinsic disc ellipticity has been estimated to have a mean value of only 0.05, the uncertainty this introduces to the corrected scalelengths via equation 2 is less than 1 percent, and the uncertainty on the central surface brightnesses obtained via equation 1 is less than 0.04 mag arcsec. Except in the -band, one can see from Figure 2 that the uncertainty in the face-on opacity will introduces a greater source of scatter.
For the distances to the galaxies we consulted Tonry et al. (2001) in the
case of the lenticular galaxies, and used the (Virgo GA Shapley)
distances given in NED
Inclination-corrections to the disc surface brightnesses which had been applied in some papers were undone according the prescription in each paper. The resultant ‘observed’ disc surface brightnesses were then corrected to a face-on, dust-free value using equation 1. Scalelengths — which had not been adjusted for inclination — were corrected using equation 2, and disc magnitudes were subsequently computed using equation 7.
Disc magnitudes were also derived using a second approach, in which the
observed (uncorrected) magnitude was corrected using equation 3.
Similarly, the observed bulge magnitudes have been corrected here using
Published Sérsic bulge parameters are not corrected for dust; expressions to
do so do not exist. We therefore report on only the -band Sérsic bulge
Galactic extinction corrections from Schlegel et al. (1998), as listed in NED, have been applied (if not already done so in the original paper). Cosmological redshift dimming also reduces the galaxy flux slightly, so we have applied an adjustment of to the magnitudes and surface brightnesses when not already done so. This is not a huge adjustment, for example, it amounts to 0.1 mag at a redshift corresponding to 7,000 km s. Given the proximity of the galaxy samples, we have not applied evolutionary nor -corrections.
In what follows we have grouped all S0 galaxies () into a single bin, denoted by in our Figures and Tables.
4 Structural parameters and their ratios
4.1 The near-infrared
We have computed the median (and width of the central 68 per cent) from the
distributions of various structural parameters as a function of galaxy type.
In Table 3 we report the face-on, dust-free, -band disc central
surface brightness and scalelength , obtained using
equations 1 and 2. These values of and have
been used to derive, via equation 7, the disc magnitudes given in
Table 3; they are also shown in Figure 3.
Spiral galaxies are commonly referred to as ‘early’ type or ‘late’ type
|mag arcsec||kpc||mag||mag arcsec||kpc||mag|
In Table 4 one can find the dust-corrected disc and bulge magnitudes obtained using equations 3 and 4, respectively. While equation 4 was applied to the bulge magnitudes in Table 3, equation 3 was not applied to the disc magnitudes from this table but the disc magnitudes obtained using the observed disc scalelengths and central surface brightnesses. Given that we now have two estimates of the corrected disc magnitude, one in Table 3 and the other in Table 4, these are compared in Figure 4 and shown to agree within a couple of tenths of a magnitude or better. A change of 0.2 mag for the disc magnitude corresponds to a change in of 0.08. We are also able to show such a comparison for the - and -band, as both sets of dust corrections are available in these bands. The slight difference in Figure 4 is thought to arise from the exponential function not providing a perfect description of dusty discs (see Möllenhoff et al. (2006, their Figure 2). Indeed, due to the prevalence of centrally located dust, the best-fitting exponential model can be seen in Möllenhoff et al. to overestimate the observed flux at small radii. Therefore, while the corrections between the measured and intrinsic central surface brightness and disc scalelength given in equations 1 and 2 are appropriate, application of equation 7, using from equation 1, and from equation 2, can lead to a slight over-estimate of the actual observed disc magnitude. As a result, the corrections to the observed disc magnitude using this approach are not quite as large as they should be (C.Popescu & R.Tuffs 2008, priv. comm.).
The bulge-to-disc size ratio
Using exponential bulge models, de Jong (1996b) suggested that the bulge-to-disc size ratio was independent of Hubble type. This unexpected result was reiterated by Courteau, de Jong & Broeils (1996) using an additional 250 Sb/Sc galaxies imaged in the -band. Adding to the mystery, Graham & Prieto (1999) pointed out that de Jong’s early-type spiral galaxies actually had an average ratio — derived using his exponential bulge models — that was smaller (at the level in the -band) than the average ratio from his late-type spiral galaxies.
Rather than only using the effective radii from de Jong’s exponential bulge models, Graham & Prieto explored use of the values from de Jong’s “best-fitting” bulge model; which was either an model, an model or an exponential model. Doing so, the mean ratio was shown to be slightly larger for the early-type disc galaxies than the late-type disc galaxies. Following up on this, Graham (2001a) fitted Sérsic bulge (plus exponential disc) models to all of the light profiles used by de Jong. Graham revealed little difference between the early- and late-type disc galaxy bulge-to-disc size ratios, and therefore ultimately reached the same conclusion as Courteau et al. (1996) but on somewhat different grounds. To explain this apparent discrepancy with what one sees when looking at spiral galaxies of different type, i.e. that early-type spiral galaxies appear to have larger bulge-to-disc size ratios than late-type spiral galaxies, Graham (2001a, his figure 21) subsequently presented the ice-berg’ model in which the intensity of the bulge varies (is effectively raised or lowered) relative to the intensity of the disc.
Using 400 disc galaxies, Table 4 and Figure 5, present the -band size ratio as a function of disc galaxy type. One of the nice features of this data set is in fact its heterogeneous nature. As a result, possible biases in any one paper’s modelling are minimised and/or effectively cancelled by those from the other papers. The results shown here are therefore something of a consensus from many papers. One can see that the median value is rather stable at around , with the only departure from this being (i) an increase to 0.31 for the Sa galaxies and (ii) a value of 0.12 for T-types less than zero. The first result arises from some of the shallow 2MASS data analysed by Dong & De Robertis (2006), which is responsible for the scatter to higher , and also higher , ratios than displayed by the other data. Excluding this data set, the remaining Sa galaxies have with a variance in this distribution of 0.04. The second result above is unusual: the size ratio is equal to or less than the lower 1 limit from every other galaxy type (not to mention that lenticular galaxies are commonly thought to have the largest bulge-to-disc ratios). This result is however again solely due to the data from one study. Modelling the bulge and bare (and disc) as separate components, Laurikainen et al. (2005) obtained these smaller ratios. In contrast, the barless galaxies from Balcells et al. (2003) with T-type less than 0 have . From the -band panel in Figure 2, one can see that uncertainties in the disc inclination for the Balcells et al. (2003) sample will not change this result by much.
We are unable to include the lenticular galaxies from Barway et al. (2007) as neither their fits nor bulge/disc parameters have yet been released. These authors claim that the ratio is not (roughly) constant for all disc galaxies and that luminous lenticular galaxies can have values as high as 5 to 10, implying the presence of small embedded discs whose light peters out well before the half-light radius of the bulge component. Such potentially new objects with bulge-to-disc size ratios 50 times greater than the median value of regular lenticular galaxies studied by others would surely be an important clue to the transition between disc and elliptical galaxies. If real, these discs might be more akin to the nuclear discs seen in many elliptical galaxies (e.g. Rest et al. 2001). However, we note that the sky-background should always be derived independently of one’s fitted models. Barway et al.’s (undesirable) treatment of the sky-background as a free parameter when fitting the bulge and disc models may be responsible for the extreme size ratios they report. Such an approach to modelling galaxies modifies the real surface brightness profile, and may result in the occurrence of item (vi) in Section 3.1.
The bulge-to-disc flux ratio
The inclination- and dust-corrected -band bulge-to-disc flux ratios, , are provided in Table 4 and shown in Figure 5 as a function of galaxy type. Unlike the size ratio, the flux ratio does indeed decrease with increasing galaxy type. The early-type spiral galaxies (Sa-Sb) have a median value of (i.e. , or equivalently a bulge-to-total flux ratio ), where and represent the luminosity (not magnitude) of the bulge and disc respectively. The late-type spiral galaxies (Scd-Sm) have a median value of ().
The tabulated -band flux ratios for the 12 lenticular galaxies in Andredakis et al. (1995. their table 4) give a mean ratio of 0.28. Similarly, as reported in Balcells et al. (2007), the mean (plus or minus the standard deviation) of the flux ratio for the lenticular galaxies analysed in Balcells et al. (2003) is 0.25 (). Laurikainen et al. (2005) also obtained a similar result from their -band data, reporting a mean value of 0.24 () from a sample of 14 S0 galaxies while the data in Gadotti (2008) yields an -band value of 0.28 from their 7 lenticular galaxies. Our result implies that (50+68/2=) 84 percent of lenticular galaxies have ratios smaller than 1/3. The remark in Kormendy (2008) that “almost no pseudobulges have ” could thus be expanded to read “most disc galaxies have ”.
The above ratios, obtained from the collective average of nine modern studies (see
Table 2), are smaller than those reported in the 1980s. Indeed, many
late-type spiral galaxies have bulge fluxes which are less than 4 per cent of
their galaxy’s total light.
While Shields et al. (2008) claim that the T=6 (Scd) galaxy NGC 1042 may be
a bulgeless galaxy, its optical light profile in Böker et al. (2003, their
Laurikainen et al. (2007, their figure 1) recently presented a set of - and -band flux ratios versus galaxy type for a sample of lenticular galaxies and the spiral galaxies from the Ohio State University Bright Galaxy Survey (Eskridge et al. 2002). Although that data has not been fully corrected for dust attenuation — which is not insignificant in the -band (e.g. Peletier & Willner 1992; Driver et al. 2008) — the trend which they show embodies the general behaviour observed here. In a separate study, Dong & De Robertis (2006) claim that their -band flux ratio versus galaxy type diagram (their fig.7) agrees with the original -band results given by Simien & de Vaucouleurs (1986). While roughly true, such an agreement should not occur given the differing bulge and disc stellar populations (and hence colours in the - and -band) for late-type disc galaxies. Aside from this problem, much of the agreement arises from the partial cancellation of two significant errors: Simien & de Vaucouleurs over-estimated their bulge flux due to the use of models while at the same time they neglected the dimming effects of dust. Working with -bulge models in the -band, Dong & De Robertis (2006) overcame these issues.
Recent claims that the luminosity of the bulge, rather than the ratio, is the driving force behind the trend seen in Figure 5 (e.g. Trujillo et al. 2002; Scannapieco & Tissera 2003; Balcells et al. 2007) reflect the results in Yoshizawa & Wakamatsu (1975, their figures 1 and 2) and echo the remarks in Ostriker (1977) and Meisels & Ostriker (1984) that the bulge luminosity, and ergo mass, may be a key parameter which distinguishes galaxies. In Figure 3 we show, separately, the bulge and disc magnitude as a function of galaxy T-type. One can see that the disc magnitude is roughly constant from T to T and then it falls by a couple of magnitude upon reaching T (Sdm galaxy). On the other hand, the bulge magnitude is roughly constant from T to T, but then falls five magnitude by T. We therefore confirm that the bulge magnitude is indeed predominantly responsible for the trend between the bulge-to-disc luminosity and disc galaxy type. In passing we note that Hernandez & Cervantes-Sodi (2006) have advocated that the spin parameter is the physical quantity which determines a disc galaxy’s Hubble Type (see also Foyle et al. 2008); implying that the spin parameter must therefore be connected with the magnitude of the bulge.
Simien & de Vaucouleurs (1986) presented a Figure showing the mean difference between the bulge and galaxy magnitude as a function of morphological T-type. Also shown in their Figure was the associated statistical uncertainty on their mean differences, which is of course much smaller than the actual scatter about each mean. They fitted a quadratic relation to this -T data, which has been popular in semi-analytical studies in which bulge-to-disc ratios are known but little information about the spiral arms is available. Although one could fit a relation to the median ratios as a function of T-type in Figure 5, we feel that this may be misleading. For example, if , the actual T-type could range from -1 (S0) to 6 (Scd), while an equation would give only one value. The range of values reminds us that the bulge-to-disc flux ratio was never used as the primary criteria for classifying disc galaxy morphology. However, given the prominace of a bulge reflects the relative dominance of certain construction processes, which may be different to those which generate spiral patterns, then it may at times be desirable to employ such quantitative measures without recourse to Hubble type.
Knowing the stellar mass-to-light ratio of the bulge and disc components allows one to transform the bulge-to-total flux ratios, , into stellar mass ratios. This aids comparison with cosmological simulations of galaxies. The stellar bulge-to-total mass ratio is given by , where is the ratio of the disc and bulge stellar mass-to-light ratios, and is of course the bulge-to-total flux ratio. If we use a slightly extreme (low) value of (see Bruzual & Charlot 2003), then for (0.33) we have a bulge-to-total stellar mass ratio of 0.4 (0.5).
To date, most simulations have a tendency to produce bulges rather than (pure) discs because of baryon angular momentum losses during merger events (Navarro & Benz 1991; Navarro & White 1994; van den Bosch 2001; D’Onghia et al. 2006). To avoid this issue, while still maintaining the hierarchical merger trees for CDM halos, potentially unrealistic amounts of feedback have been invoked in the past. Early feedback could blow much of the baryons out of high- galaxy discs for their late-time, and necessarily post major-merger, re-accretion into some preferred plane and thereby produce the disc-dominated galaxies observed in the local Universe (see Abadi et al. 2003a). From the 1000 spiral galaxies simulated by Croft et al. (2008), at only 3.5 per cent have a bulge-to-total stellar mass ratio smaller than 0.5. Given that 68 per cent of real lenticular galaxies have smaller than 1/3, these simulations are clearly inconsistent with what is observed in the Universe. The disc galaxy simulated by Abadi (2003b) has a stellar bulge-to-total mass ratio of 0.71, and as such it also does not represent a typical disc galaxy. While the high mass and force resolution CDM simulated spiral galaxy in Governato et al. (2004) has , subsequent high-resolution simulations (e.g. D’Onghia et al. 2006) reveal that resolution issues are not the key problem but rather some primary physics may still be missing. Current implementation of supernovae feedback, for example, has been shown to play a significant role in the evolution of simulated galaxies (Okamoto et al. 2005; Governato et al. 2007; Scannapieco et al. 2008 and references therein).
The current data may hopefully help determine how bulges should be assigned their mass in semi-analytical models. Such studies build galaxies by a trial and error process in which the model assumptions, parameters and processes are collectively tweaked until the simulations resemble real galaxies. For example, when a simulated disc becomes unstable (Mo et al. 1998; Cole et al. 2000), Croton et al. (2006, their equation 22) transfer enough disc mass to the bulge in order to maintain stability, while Bower et al. (2006, their equation 1) transform the entire system into an elliptical galaxy. In the case of minor mergers (mass ratio 0.3), some works assign all the accreted gas to the disc while transferring all the accreted stars to the bulge (de Lucia et al. 2006), while others add both the stars and cold gas to the disc of the primary galaxy (Kauffmann & Haehnelt 2000). The new bulge-to-disc flux ratio data may help to refine how these semi-analytical models operate.
Bulge profile shape
Although equations to correct the Sérsic parameters pertaining to the bulge are not yet available, we note that the influence of dust at wavelengths around 2 microns is noticeably less than in the optical bands (e.g. Whitford 1958; Cardelli, Clayton & Mathis 1989; Fitzpatrick 1999). The significant change to with inclination which is seen in Figure 2 is not because of dust but predominantly due to the greater line-of-sight depth through the (near-transparent) disc with increasing disc inclination. In the absence of dust, a spherical bulge will look the same from all angles, i.e. inclinations. We therefore do not expect the -band bulge parameters to change greatly (i.e. more than a few tenths of a magnitude, or 20 per cent in size) with inclination, and we report their observed values in Table 3.
As can be seen in Figure 3, Sc galaxies and earlier types typically
have values of and sometimes as high as 4. While Scd galaxies and
later types tend to have values of , some Sd galaxies can still have
values of n greater than 2. It may therefore be misleading to label all Sc
and later-type galaxies as possessing pseudobulges built from exponential
discs (Erwin et al. 2003; Kormendy & Kennicutt 2004, their section 4.2;
Athanassoula 2008). While a value of in Sérsic’s model describes an
exponential profile, it does necessitate the presence of a flattened,
rotating, exponential disc rather than a spheroidal, pressure supported bulge.
Dwarf elliptical galaxies also have values of (e.g. Caon et al. 1993; Young & Currie 1994; Jerjen et al. 2000) — they define the low-mass
end of a linear relation between magnitude and Sérsic index (e.g. Graham et al. 2006b, their figure 1; Nipoti,
Londrillo & Ciotti 2006). Dwarf ellipticals, however, are generally not
considered to have formed from the secular evolution of a disc.
Although not a new result, the preponderance of bulges with combined with the prevalence of late-type disc galaxies having bulges with necessitates that simulations of disc galaxies neither use nor create an -like bulge for every galaxy. While the projection of Hernquist’s (1990) useful model — employed in many numerical simulations (e.g. Springel et al. 2005b; Di Matteo et al. 2005) — reproduces an profile, it does not provide a particularly good representation of bulges, such as the Milky Way’s bulge, which have (e.g. Terzić & Graham 2005, their Figure 7; Terzić & Sprague 2007). Furthermore, efforts to simulate a range of disc galaxies with different morphological types by only varying the bulge-to-disc mass ratio, but using a Hernquist model for every bulge (e.g. Springel et al. 2005a), i.e. effectively assigning a de Vaucouleurs profile to every bulge, will not generate a realistic set of galaxies. In the case of bulges formed from disc instabilities, one requires the production of bulges that have exponential light profiles rather than Hernquist-like density profiles. In regard to simulations of minor mergers, in which bulges are built up within discs, the observed correlation between bulge profile shape and luminosity, and also supermassive black hole mass (Graham & Driver 2007), must be reproduced. The recovery of bulge parameters using -based models is known to introduce systematic biases with profile shape and thus also black hole mass (e.g. Trujillo et al. 2001, their Figures 4 and 5; Brown et al. 2003, their Figure 7). Therefore, once realistic bulges are generated, the computation of their sizes and binding energy by assuming a Hernquist model would be inappropriate and lead to spurious correlations when constructing, for example, a black hole fundamental plane (Younger et al. 2008).
Density models such as that from Einasto (1965) and Prugniel & Simien
(1997) contain a “shape parameter” that effectively captures the range of
structural shapes (Sérsic indices) which real bulges are observed to
By construction, the three-parameter () density model of Prugniel & Simien (1997; their equation B6) contains two identical parameters to Sérsic’s model and can be written as
where is the internal, i.e. not projected, radius (see Márquez et al. 2001; Terzić & Graham 2005, their equation 4). For clarity, the subscript has been dropped from the term . The new parameter is the internal density at and provides the normalisation such that the total mass from equation (9) equals that obtained from equation (5) after applying the appropriate stellar mass-to-luminosity ratio . Equation (10) has the same functional form as Sérsic’s model although multiplied by an aditional power-law term with exponent . We adopt Lima Neto et al.’s (1999) expression for , for which a high-quality match between deprojected Sérsic profiles and the above expression is obtained when .
After converting the observed -band bulge effective radii into units of parsecs, and the observed -band effective surface brightnesses into units of absolute solar luminosity per square parsec, we have used equation (10) to derive the mean values for bulges of each galaxy type and class. These luminosity densities are provided in Table 3. To do this we used an absolute -band magnitude for the Sun of 3.33 mag (Cox 2000). Multiplying by one’s preferred stellar ratio gives the stellar mass density of the bulge at . The mean , and values can be used to construct realistic bulges using the Prugniel-Simien model, and/or to test if simulated bulges match the typical density profiles of real galaxy bulges.
4.2 The optical
As with the -band data, we have collated literature data on the structural parameters of galaxies measured at optical wavelengths (see Table 2). After first removing any inclination corrections applied in these papers, we again employed equations 1 and 2 to determine the disc central surface brightnesses and scalelengths, and we computed the disc magnitudes using equation 7. The results are shown in Tables 5 to 8.