Issue 
A&A
Volume 516, JuneJuly 2010



Article Number  A7  
Number of page(s)  6  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/200913458  
Published online  16 June 2010 
Fitting functions for a diskgalaxy model with different CDMhalo profiles
(Research Note)
L. Darriba  J. M. Solanes
Departament d'Astronomia i Meteorologia and Institut de Ciències del Cosmos, Universitat de Barcelona, C/ Martí i Franquès, 1, 08028 Barcelona, Spain
Received 13 October 2009 / Accepted 7 April 2010
Abstract
Aims. We present an adaptation of the standard scenario of diskgalaxy formation to the concordant CDM
cosmology aimed to derive analytical expressions for the scale length
and rotation speed of presentday disks that form within four
different, cosmologically motivated protogalactic dark matter
halodensity profiles.
Methods. We invoke a standard galaxyformation model that
includes virial equilibrium of spherical dark halos, specific angular
momentum conservation during gas cooling, and adiabatic halo response
to the gas inflow. The mean massfraction and masstolight ratio of
the central stellar disk are treated as free parameters whose values
are tuned to match the zero points of the observed sizeluminosity and
circular speedluminosity relations of galaxies.
Results. We supply analytical formulas for the characteristic size and rotation speed of disks built inside Einasto r^{1/6},
Hernquist, Burkert, and NavarroFrenkWhite dark matter halos. These
expressions match simultaneously the observed zero points and slopes of
the different correlations that can be built in the
space of disk galaxies from plausible values of the galaxy and starformation efficiencies.
Key words: dark matter  galaxies: formation  galaxies: fundamental parameters  galaxies: spiral  galaxies: structure
1 Introduction
In the current hierarchical galaxyformation paradigm diskgalaxies are born out of the hot gasatmospheres associated with the potential well of virialized cold dark matter (CDM) halos. It is assumed that baryons have initially both the same density profile and specific angular momentum distribution as DM  the latter achieved, for instance, through tidal interactions with neighboring objects in the precollapse phase (e.g. Peebles 1969). As the gas radiates its energy it cools and starts to fall towards the center of the DM halo maintaining its specific angular momentum, where it settles into a rotationally supported disk. The assembly of a concentration of cold baryons at the bottom of the gravitational potential well on timescales longer than the freefall time produces the adiabatic contraction of the dark halo^{}. In this standard picture, the internal properties of disk galaxies are expected to be largely dictated by those of their host halos, and through the latter, by those of the background cosmology too.
Theoretical predictions for the distribution of disk galaxies in the space of disk scalelength (or size), fiducial (usually, maximum or asymptotic) rotational speed, and luminosity (or mass) based, partially or totally, on the scenario just outlined are abundant in the literature e.g., Mo et al. 1998, hereafter MMW; Pizagno et al. 2005; Dutton et al. 2007. They are widely used in semianalytic cosmological models, preprepared numerical simulations of galaxy groups and clusters, and studies of diskgalaxy scaling relations.
While nowadays there are extensive and comprehensive investigations of the correlations between diskgalaxy properties that deal with the scatter and covariances of the variables and allow for different modes of halo contraction (e.g. Dutton et al. 2007), it is not always feasible to implement such sophisticated treatments whenever one needs to estimate the scaling of the basic structural and kinematic parameters of galaxies. The simplest alternative is the use of scaling laws derived directly from fits to a given set of observations. However, because of their lack of theoretical foundation, these formulas cannot be extrapolated to explain the properties of galaxies other than those from which they are derived. Halfway between these two options is the possibility of using analytical expressions endowed with a physical basis that enables their application to a wide range of galactic and halo parameters. It is precisely with this aim that we here introduce a selfconsistent pure diskformation model that follows the wellknown approach by MMW adapted to the canonical CDM concordance cosmology and to four different massdensity distributions for the protogalactic dark halos. This updated scenario is capable of matching simultaneously with very good accuracy the zero points and slopes of the observed correlations in the space of disk galaxies from reasonably realistic values of its input parameters. Yet its most valuable characteristic is its ease of implementation, as we approximated the model predictions for the scale length and rotation speed of disks by analytical expressions. The supplied equations can come in handy for situations that require the generation of large numbers of galaxies with intrinsic attributes in good agreement with the mean observed trends, especially when the relative abundances of these objects are known in advance.
2 Model components
We recap here the key assumptions and associated equations of our selfconsistent CDMmodel of diskgalaxy formation:
 1.
 In the protogalactic state, the (hot) baryons and dark matter are well mixed within virialized spherical halos. Both components have the same distribution of specific angular momentum.
which, according to the results of Nbody simulations, follows a lognormal distribution with median lying in the range see, for instance, Shaw et al. 2006, and references therein, nearly independent of cosmology, halo environment, and redshift (e.g. Lemson & Kauffmann 1999). In Eq. (1), is the virial radius inside which the halo mean density, , is times the mean density of the universe at the redshift of observation, , and is a dimensionless function of the halo concentration (see below) that measures the deviation of the protogalactic halo's energy from that of a singular isothermal sphere with the same mass, see MMW. For the family of flat cosmogonies, (Bryan & Norman 1998).
The halo concentration parameter, c, characterizes the overall shape
of a halo density profile by measuring the ratio between its outer
radius and inner scalelength. Originally introduced for the
NavarroFrenkWhite function, its mean values are strongly correlated
with the halo mass given a cosmology (e.g. Navarro et al. 1997). We
approximate the mean concentrationmass relation at z=0 in the range
of halo masses of interest,
,
by the bestfitting powerlaw relation recently inferred by
Macciò et al. (2008) from relaxed halos simulated in the Wilkinson Microwave
Anisotropy Probe 5 years results (WMAP5) cosmology
where is defined here in a profileindependent form by adopting as the inner characteristic radius of the halo density profile the radius r_{2}, at which its effective logarithmic density slope equals 2. The WMAP5 cosmological parameters are , implying that times the critical density for closure at the current epoch.
 1.
 Disks form smoothly out of cooling flows preserving the specific angular momentum of the baryons. The cold gas settles in centrifugal equilibrium at the center of the halo's potential well following an exponential distribution.
where the values of this parameter, for which a plausible upper limit is the universal baryon fraction , do not seem to depend much on the halo mass or spin (Sales et al. 2009). Similarly, the angular momentum of the disk, expressed in units of that of its surrounding halo, can be written as
The common yet uncertain assumption that the specific angular momenta of the central disk galaxy and of the halo hosting it are equal, , is equivalent to setting .
On the other hand, a thin exponential mass distribution of total mass
,
surface density
,
and a rotation curve V(R), has a total angular momentum
where the factor is unity for a disk with a flat rotation curve at the level , and where the total circular speed is computed by summing in quadrature the contributions from the disk of cold baryons, , and from the dark halo, ,
with R the cylindrical radius. An expression for can be found in Binney & Tremaine (2008), p.101, Eq. (2.165).
Substituting Eqs. (1), (3), and (4) into
Eq. (5), one can then obtain the disk scalelength as a function
of the model parameters
with the effective spin of the disk.
 1.
 The halo contracts adiabatically and without shell crossing to gas inflow.
(8) 
where r_{i} and r are, respectively, the initial and final radius of the spherical shells, M_{i}(r) is the initial protogalactic halo mass profile, and comes from the replacement of the final thin exponential disk configuration by the spherical density profile that has the same enclosed mass.
The contribution to the total rotation curve (Eq. [6]) from
the dark matter (and the remaining hot baryons) is therefore
Taking into account that both halo and disk properties are directly proportional to their corresponding virial parameters, Eq. (6) allows one to express the amplitude of the total rotation curve at a given number of scalelengths and, in particular, its peak value, , in the compact form (cf. MMW)
with a dimensionless factor that, like , depends on the adopted halo density law and on the values of parameters , c, and .
3 Model predictions
We now proceed to tune the free parameters of our diskgalaxy formation model to match the scaling relations in space observed at . For a given halo virial mass, two are the free parameters in our modeling: the disk mass fraction, , and masstolight ratio, . This latter quantity is needed to convert the predicted disk masses into observed luminosities. We do not allow the average effective disk spin to vary freely however, but use the condition to set it equal to three representative values of : 0.03, 0.04, and 0.05^{}.
Table 1: Halo profiles.
We investigated the performance of our model for the
four functional forms of protogalactic DM halos listed in
Table 1. They are among the most representative functions
used in the literature to describe the equilibrium density profiles of
halos generated in CDM Nbody simulations. All of them are spherical
density distributions of the form
(11) 
where and are the characteristic density and scale radius of the profile respectively, and is a dimensionless function of the dimensionless radius .
With the aid of the
relation these expressions can be
reduced to uniparametric^{} density
laws in which the halo structure is fully determined from
.
It
can be shown that
(12) 
where the characteristic concentration is directly related to the profileindependent halo concentration parameter c defined in Eq. (2) (see Table 1).
In order to constrain our model predictions, we consider a subset of the SFI++ sample (Springob et al. 2007) consisting of 649 galaxies also included in the compilation of 1300 local field and cluster spiral galaxies by Courteau et al. (2007). The full SFI++ contains measures of intrinsic rotation velocity widths reduced to a homogeneous system based on the 21 cm spectral line, W, as well as absolute Iband magnitudes for near 5000 spiral galaxies, while the dataset by Courteau et al. provides inclinationcorrected estimates of disk scalelengths also in the Iband (below both observables and model parameters will refer to this nearIR band).
As stated by Catinella et al. (2007), for most intermediate and bright disks the width of the global H I profile provides a more reliable observational estimate of the peak rotation velocity than the widths of H rotation curves, at least for objects not affected by environmental interactions. This is probably because the latter are usually evaluated either at a radius where, on average, they are still rising (e.g., ), or on the asymptotic part of the optical disk. Accordingly, we adopt the approximation , where is the maximum width of our model total speed curve measured within .
3.1 Scaling laws
The distribution of R as a function of V provides the most effective way of determining the value of  which for bright galaxies represents to a good approximation the stellar mass fraction  that best fits the observations for each one of the values of under consideration. To allow for a more robust comparison between the model predictions and the data, the RV scaling law has been recast in the form of the tighter relation between the average specific angular momentum of disks computed from the fiducial rotation speed of the galaxies, , and . In a loglog scale this relationship is expected to follow a straight line with a slope near 2 and a zero point that is a sensitive function of .
In the upperleft panel of Fig. 1, we show the model relations that best fit the barycenter of the data cloud for the four halo profiles considered and the central value of (the best values of obtained for each one of the three values adopted for are listed in Col. 3 of Table 2). It can be seen from this plot that our disk models also reproduce the slope of the observed scaling law. We note in passing that on the basis of its location in this diagram, the angular momentum and disk scale of the Milky Way (MW) are unrepresentative of those of a typical spiral (see also Hammer et al. 2007).
Table 2: Model parameters.
Figure 1: Scale relations for nearby disks. Upperleft: Diskspecific angular momentum as a function of . The cross shows the barycenter of the data. Upperright: Iband TF relation. The dashed line shows the one derived by Masters et al. (2006) from SFI++ data (Springob et al. 2007). The cross shows the barycenter of the data. Lowerleft: Centraldisk surfacemass density vs. . Squares with error bars show the median observational values and the first and third quartiles in each velocity bin. In all these plots the solid lines show model predictions for pure disks with using the mean relation at z=0, while the dotted curves show from top to bottom the predictions resulting from adopting the 2.3th and 97.7th percentiles of the concentration distribution ( scatter) assuming (Macciò et al. 2008). The error boxes represent, for comparison, the values measured for the MW from: kpc, , and km s^{1} (Binney & Tremaine 2008), and (Portinari et al. 2007). The data clouds are build on the compilation of Iband absolute magnitudes and H I rotation widths by Springob et al. (2007) and on the Iband disk scalelength measurements by Courteau et al. (2007). 

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With
fixed and given that the halo concentration is not allowed
to vary freely, the most sensitive tuning of the other free parameter
of the model,
,
is achieved by normalizing the model predictions
to the observed
relation. For the latter, which is fully
independent of surface brightness (Courteau & Rix 1999; Zwaan et al. 1995), we use the
calibration of the TullyFisher (TF) relationship corrected from
observational and sample biases calculated by Masters et al. (2006) using 807
cluster galaxies extracted from the SFI++ catalog, which we rewrite in
the form
to facilitate the comparison with our model predictions. In Eq. (13), mag has been adopted to transform model luminosities into absolute magnitudes. As in the former case, the upperright panel of Fig. 1 shows that the predicted relations (again we show only those inferred using the central value of ) closely match the slope of the empirical estimate. In this case, the best values of have been set by minimizing the residual between the model predictions and the observed TF relationship over the full available range of velocities. As could be expected, the agreement between the model predictions and our SFI++based comparison sample is fairly good too. This panel also illustrates the wellknown deficiency in luminosity of the MW with respect to the TF relation (Portinari et al. 2007).
Table 3: Coefficients of the approximations.
The excellent agreement between predictions and observations in the RV and planes is maintained for the joint distribution of the three variables. The lowerleft panel of Fig. 1 depicts, again for the central value of , the scatter diagram of central disk surface density, , and rotation speed. We have converted Springob et al.'s data on M_{I} into total disk luminosities, which in turn have been transformed into disk masses using the values of derived from the normalization of the relation. It can be seen that our model predictions are once more comfortably close to both the normalization and, in this case, curved mean trend delineated by the data.
3.2 Fitting functions for galaxy scaling parameters
By using the values quoted in Cols. 3 and 4 of Table 2
it is straightforward to calculate the average luminosity of a nearby
disk embedded in a halo of given
and
.
However, as shown
in Sect. 2, each of the remaining fundamental disk
properties, the characteristic scale and rotation speed, participates
in the calculation of the other. As a result, they can only be
computed by applying an iterative procedure that, despite its fast
convergence, remains cumbersome. For this reason, it is very
convenient to approximate the dimensionless factors
and
appearing in the calculation of
and
(Eqs. (7) and (10), respectively) by fitting
functions. Drawing inspiration from MMW, we propose the
following fitting formulas, which are valid for any of the
protogalactic halo mass density profiles explored:
The values of the coefficients corresponding to each profile, which are independent of the adopted relation, are listed in Table 3. Both approximations are accurate to within 8 for , , and . We note that attempts to fit these dimensionless factors using either solely polynomials or a linear combination of power laws of parameters , c, and have required a substantially larger number of terms to achieve a similarly satisfactory match. Given the countless number of real functions that can be implemented, it would be obviously possible to find other formulae also producing good fits, but most likely they will be more complicated than the above expressions.
4 Discussion and conclusions
We formulated a standard formation model of disk galaxies inside DM halos within the concordant CDM cosmology that simultaneously predicts with high accuracy the main trends of the observed fundamental scaling relations of nearby galaxies in space. This modeling has been developed with the sole aim of deriving physically sound analytical expressions for predicting the central properties that characterize the light profiles and rotation curves of typical spirals. We supply formulas for Einasto r^{1/6}, Hernquist, Burkert, and NavarroFrenkWhite protogalactic halo mass density distributions that provide a similarly good overall description of the data on disks for realistic enough values of the model free parameters. We find that, for a given , the predictions of the Einasto r^{1/6}, Hernquist, and NavarroFrenkWhite profiles are relatively similar, while the Hernquist profile  the only density law investigated that does not follow a behavior near  requires values of and that are lower by about a factor of 0.70 and 0.85, respectively.
The reader may have noticed that our best models yield for , i.e. for the inverse of the average starformation efficiency, values somewhat lower than those inferred from population synthesis calculations (e.g. Pizagno et al. 2005). We stress however that the observational estimates of this parameter are affected by considerable uncertainties, our prediction that the average masstolight ratio of disks is , which is consistent with submaximal disks arguments (Courteau & Rix 1999; Kuzio de Naray et al. 2008), as well as relatively close to the values adopted as input in more sophisticated models of disk formation (Dutton et al. 2007). On the other hand, we find that the predicted values of are directly correlated with those adopted for . In particular we note that a value of , which coincides with the median of the distribution of the spin parameter for relaxed halos derived by Macciò et al. (2008), implies a small current average galaxyformation efficiency, . This agrees well with the predictions of galaxy evolution from halo occupation models (Zheng et al. 2007) and methods that match the stellar mass function to that of the halo (Conroy & Wechsler 2009). Further recent support for low (and , according to our model) comes for instance from weak lensing measurements (Mandelbaum et al. 2006) and from the roughly universal distributions of this parameter obtained by Sales et al. (2009) for various implementations of feedback in large cosmological Nbody/gasdynamical simulations. Notice also the fifth column in Table 2, where we list the ratio calculated for a MWmass halo, which increases with increasing and decreasing . As stated by Dutton et al. (2007), the relatively high values we predict for this ratio  a characteristic common to standard models  would likely hamper a simultaneous match to the galaxy LF that, according to semianalytical models of galaxy formation, requires the condition .
We made no attempt to explore the scatter of the observed scaling relations and the covariance that exists between model parameters, except for Fig. 1, where we carry out a naive comparison between the spread of the data and that resulting from taking into account the predicted scale of the probability distribution of the halo concentration. Including this and other sources of scatter, such as the variance of the halo spin parameter, or the dependence of the concentrationmass relation on the adopted cosmology (e.g. Macciò et al. 2008), would undoubtedly enrich the analysis. Yet, a thorough investigation of scatter requires dealing with the joint probability distribution of all the parameters entering the model and, in particular, with all their covariances (not just the variances), which ideally should be corrected from measurement errors. This far exceeds the scope of our present research. We note in addition that efforts in the direction just outlined will soon be much more effective when they can be applied to objective, homogeneous, and complete datasets free of nontrivial selection biases, as those build from the crosscorrelation of widearea spectroscopic optical and H I surveys (e.g., Toribio et al. 2010, in preparation).
Finally, we wish to comment on the possibility of extending our model
predictions to distant galaxies by adopting a
relationship
of the form
with given in Eq. (2) and g(z) the concentration growth factor, which can be taken proportional to H(z)^{2/3}, as found in a recent modification of the original Bullock et al. (2001) model for WMAP cosmologies by Macciò et al. (2008). Acknowledgements
We thank the anonymous referee for his/her thorough review and appreciate the comments and suggestions, which significantly helped to improving the manuscript. This work is supported by the Spanish Dirección General de Investigación Científica y Técnica, under contract AYA200760366.
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Footnotes
 ... halo^{}
 In modern literature, the mode and amount of halo contraction are actually a matter of debate (e.g. Dutton et al. 2007; Tissera et al. 2010). The outcome, however, remains unchanged: the properties of disk galaxies are linked to those of their host halos.
 ... 0.05^{}
 We ignore here a possible dependence of this parameter on halo mass (e.g. Berta et al. 2008).
 ... uniparametric^{}
 The Einasto r^{1/n} model has an additional parameter n controlling the curvature of the profile. In our modeling this parameter is kept fixed to n=6, a value representative of galaxysized halos (Merrit et al. 2005).
All Tables
Table 1: Halo profiles.
Table 2: Model parameters.
Table 3: Coefficients of the approximations.
All Figures
Figure 1: Scale relations for nearby disks. Upperleft: Diskspecific angular momentum as a function of . The cross shows the barycenter of the data. Upperright: Iband TF relation. The dashed line shows the one derived by Masters et al. (2006) from SFI++ data (Springob et al. 2007). The cross shows the barycenter of the data. Lowerleft: Centraldisk surfacemass density vs. . Squares with error bars show the median observational values and the first and third quartiles in each velocity bin. In all these plots the solid lines show model predictions for pure disks with using the mean relation at z=0, while the dotted curves show from top to bottom the predictions resulting from adopting the 2.3th and 97.7th percentiles of the concentration distribution ( scatter) assuming (Macciò et al. 2008). The error boxes represent, for comparison, the values measured for the MW from: kpc, , and km s^{1} (Binney & Tremaine 2008), and (Portinari et al. 2007). The data clouds are build on the compilation of Iband absolute magnitudes and H I rotation widths by Springob et al. (2007) and on the Iband disk scalelength measurements by Courteau et al. (2007). 

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