The Impact of Feedback on Disk Galaxy Scaling Relations
We use a disk galaxy evolution model to investigate the impact of mass outflows (a.k.a. feedback) on disk galaxy scaling relations, mass fractions and spin parameters. Our model follows the accretion, cooling, star formation and ejection of baryonic mass inside growing dark matter haloes, with cosmologically motivated angular momentum distributions, and dark matter halo structure. Models without feedback produce disks that are too small, too gas poor, and which rotate too fast. Feedback reduces the galaxy formation efficiency, , (defined as the fraction of the universally available baryons that end up as stars and cold gas in a given galaxy), resulting in larger disks with higher gas fractions and lower rotation velocities. Models with feedback can reproduce the zero points of the scaling relations between rotation velocity, stellar mass and disk size, but only in the absence of adiabatic contraction. Our feedback mechanism is maximally efficient in expelling mass, but our successful models require 25% of the SN energy, or 100% of the SN momentum, to drive an outflow. It remains to be seen whether such high efficiencies are realistic or not. Our energy and momentum driven wind models result in different slopes of various scaling relations. Energy driven winds result in steeper slopes to the galaxy mass - halo mass, and stellar mass - halo mass relations, a shallower slope to the galaxy size - stellar mass relation at , and a steeper slope to the cold gas metallicity - stellar mass relation at . Observations favor the energy driven wind at stellar masses below , but the momentum driven wind model at high masses. The ratio between the specific angular momentum of the baryons to that of the halo, , is not unity in our models with inflow and outflow. Yet this is the standard assumption in models of disk formation. Above a halo mass of , cooling becomes increasingly inefficient, which results in decreasing with increasing halo mass. Below a halo mass of , feedback becomes increasingly efficient. Feedback preferentially ejects low angular momentum material because star formation is more efficient at smaller galactic radii, and at higher redshifts. This results in increasing with decreasing halo mass. This effect helps to resolve the discrepancy between the high spin parameters observed for dwarf galaxies with the low spin parameters predicted from .
keywords:galaxies: formation – galaxies fundamental parameters – galaxies: haloes – galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure
Understanding the origin and nature of galaxy scaling relations is a fundamental quest of any successful theory of galaxy formation. The success of a particular theory will be judged by its ability to reproduce the slope, scatter, and zero-point of any robust galaxy scaling relation. Of particular interest are the scaling relations between size (), luminosity (or stellar mass, ) (), and velocity (), as these parameters are related to each other via the virial theorem. For early type galaxies, these three parameters are indeed coupled, resulting in a two-dimensional plane known as the Fundamental Plane (FP: Dressler et al. 1987; Djorgovski & Davis 1987). However, for late-type galaxies, the relation between rotation velocity and luminosity, known as the Tully-Fisher relation (TF: Tully & Fisher 1977) is independent of galaxy size or surface brightness (e.g. Courteau & Rix 1999; Courteau et al. 2007; Pizagno et al. 2007). Understanding the origin of this surface brightness independence is likely the key to understanding the small scatter of the TF, and may even explain the origin of the zero point of the TF relation.
The slopes of the relations for disk galaxies can be broadly understood with galaxy formation models that include virial equilibrium after dissipation-less collapse of quasi-spherical cold dark matter (CDM) halos and angular momentum conservation (e.g. Mo, Mao, & White 1998; van den Bosch 1998, 2000; Navarro & Steinmetz 2000; Firmani & Avila-Reese 2000; Dutton et al. 2007).
However, reproducing these scaling relations in detail has been a problem for galaxy formation models. To date, no (self-consistent) CDM-based model of galaxy formation can simultaneously match the the zero points of the TF relation, galaxy sizes, and the luminosity (or stellar mass) function (e.g. Benson et al. 2003; Dutton et al. 2007). Models which match the zero point of the TF relation, do so by making the assumption that, , the observed rotation velocity is equal to, , the circular velocity at the virial radius (e.g. Somerville & Primack 1999), or, , the maximum circular velocity of the halo prior to galaxy formation, (e.g. Croton et al. 2006). For typical galaxy mass dark matter haloes (e.g. Bullock et al. 2001a), so these two assumptions are almost equivalent. Other more observational approaches also support the conclusion that (Eke et al. 2006; Blanton et al. 2008). The problem for galaxy formation theory is that both cosmological simulations and analytic models of disk galaxy formation that take into account the self-gravity of the baryons and the effect of halo contraction (Blumenthal et al. 1986) find that (Navarro & Steinmetz 2000; Abadi et al. 2003; Dutton et al. 2007; Governato et al. 2007).
Due to the almost one-to-one correlation between and , and the correlation between and baryonic mass (the baryonic TF relation e.g. McGaugh et al. 2000) an equivalent constraint to the ratio between and is the galaxy mass fraction: . Observations of halo masses using weak lensing (Hoekstra et al. 2005; Mandelbaum et al. 2006) find that for late type galaxies the maximum , where is the cosmic baryon mass fraction at redshift zero. Similarly low galaxy formation efficiencies are obtained by methods that match the stellar mass function with the dark halo mass function (e.g. Yang et al. 2007; Conroy & Wechsler 2008).
A partial explanation for the surface brightness independence of the TF relation, or equivalently the weak correlation between residuals of the and relations, is that observationally (e.g. McGaugh & de Blok 1997) gas fractions correlate with surface brightness, with higher gas fractions in lower surface brightness galaxies. Since lower surface density disks are expected to rotate more slowly at a given baryonic mass, the larger gas fractions shift these galaxies to lower stellar masses and hence lower luminosities (Firmani & Avila-Reese 2000; van den Bosch 2000). However, this solution is not very effective for the high surface brightness, low gas fraction galaxies. Dutton et al. (2007) showed that a reasonable amount of scatter in the stellar mass-to-light ratios helps to reduce the correlation between the residuals of the and relations further. Gnedin et al. (2007) proposed a correlation between disk mass fraction and disk surface density to explain the lack of a correlation between the and relations.
Thus understanding the physical mechanisms that determine galaxy mass fractions is fundamental to our understanding of the origin of the relations of disk galaxies. In the standard picture of disk galaxy formation, gas that enters the halo gets shock heated to the virial temperature, and then cools radiatively (Fall & Efstathiou 1980). Thus in order to produce low galaxy mass fractions, either a significant fraction of the gas has to be prevented from cooling, or a significant fraction must subsequently be ejected from the disk and halo. The latest hydrodynamical simulations indicate that rather than accreting gas via a cooling flow, below a critical halo mass of gas does not shock as it enters the halo, and instead accretes straight onto the galaxy in a cold flow (Birnboim & Dekel 2003; Keres et al. 2005). In this scenario, essentially all of the baryons accrete onto the galaxy, and the problem of stopping the baryons from cooling becomes one of stopping the cold flows from forming. Thus in the absence of some kind of pre-heating (e.g. Mo et al. 2005) which would shut down the cold flows, mass outflows are required in order to produce the low galaxy mass fractions observed in the Universe today.
In order to investigate how mass outflows (a.k.a. feedback) determines galaxy mass fractions, and the impact this has on disk galaxy scaling relations, we use an updated version of the disk galaxy evolution models presented in van den Bosch (2001; 2002a). In these models the input parameters are the concentration and spin of the dark matter halo. The galaxy spin parameter, galaxy mass fraction and gas to stellar mass ratio are collectively determined by the efficiencies of cooling, star formation, and feedback.
The main differences with respect to the van den Bosch (2002a) models are:
We use cosmologically motivated specific angular momentum distributions (AMDs) of the halo gas rather than shells in solid body rotation;
We consider a star formation recipe based on dense molecular gas, rather than on total gas with a Toomre star formation threshold;
We include scatter in halo concentration which we relate to the mass accretion history (MAH);
We explore two different feedback models: one based on kinetic energy conservation, the other based on momentum conservation.
An important aspect of this model is that we do not assume that the baryonic disk has an exponential density profile. In this model the surface density profile of the baryonic disk is determined by detailed conservation of angular momentum, starting from the AMDs of gas haloes as found in cosmological simulations. The surface density profile of the stars is then determined by the relative efficiencies of star-formation, outflows, and inflows as a function of radius. This allows us to self-consistently follow the evolution of the radial distributions of gas and stars. In future papers we use this galaxy formation model to investigate the origin of disk galaxy surface density profiles (Dutton 2008) and the evolution of disk galaxy scaling relations.
This paper is organized as follows: In §2 we describe the disk galaxy formation models; in §3 we discuss the effect of feedback on the relations; in §4 we discuss the impact of feedback on galaxy mass fractions and galaxy spin parameters; in §5 we discuss the effect of feedback on the mass-metallicity relation at ; in §6 we discuss how galaxies lose their gas; and in §7 we give a summary.
2 Disk Galaxy Formation Models
The main assumptions that characterize the framework of our models are the following:
Dark matter haloes around disk galaxies grow by the smooth accretion of mass;
The baryons acquire the same distribution of specific angular momentum as the dark matter;
Gas that enters the halo is shock heated to the virial temperature;
Gas cools conserving its specific angular momentum;
Star formation occurs according to a Schmidt type law on the dense molecular gas;
Supernova feedback re-heats some of the cooled gas, ejecting it from the halo;
The halo contracts and expands adiabatically to inflows and outflows, respectively;
Each model galaxy consists of five mass components: dark matter, hot halo gas, disk mass in stars, disk mass in cold gas, and ejected gas. The dark matter and the hot gas are assumed to be distributed in spherical shells, the disk mass is assumed to be in infinitesimally thin annuli. Throughout this paper we refer to as radius, as time (where is defined as the Big Bang) and as redshift.
For each galaxy we set up a radial grid with 200 bins quadratically sampled from between 0.001 to 1 times the redshift zero virial radius, and we follow the formation and evolution of the galaxy using 400 redshift steps quadratically sampled from to . For each time step we compute the changes in the various mass components in each radial bin. The prescriptions we use are described in detail below.
2.1 Limitations of the Model
Before we describe the details of the model, we first discuss the relevance of our model to understanding galaxy formation in a hierarchical universe.
The assumption of smooth mass accretion might seem inconsistent with the hierarchical merger picture of structure formation in a universe. However, major mergers of stellar rich galaxies are known to destroy disks (e.g. Barnes 1992; Cox et al. 2006), so disk galaxies are unlikely to form in haloes with recent major mergers. Minor mergers are more common than major mergers, and are likely to play and important role in the formation of galaxy bulges, either directly, or by triggering secular processes. Thus by not including mergers our model underestimates bulge fractions. However, one of the goals of these models is to determine how much of the structural properties of disk galaxies can be accounted for with the “zeroth order” scenario of disk formation (smooth accretion and quiescent star formation).
Our assumption about the way gas is accreted into galaxies (by a cooling flow of shock heated gas) is likely incorrect. Simulations suggest that disk galaxies accrete most of their mass though cold flows, and that in the absence of extra heating or outflows the baryon fraction of galaxies is close to the universal value (Keres et al. 2005). In our model the gas shock heats, but since for haloes with masses below the cooling time is short compared to the Hubble time, essentially all the gas that enters the halo accretes onto the disk in a free fall time. Thus although the physical mechanism by which galaxies accrete their gas in our model and cosmological simulations are different, we expect that the specific angular momentum distribution of cold flow baryons to be the same as that of the dark matter, though this needs to be verified using cosmological simulations.
Our assumption about the efficiency at which gas is converted into stars is necessarily over-simplified, but it is an improvement over the majority of star formation recipes (which may be physically or empirically motivated) used in semi-analytic models and hydro-dynamical simulations, which are based on the density (or mass) of the total cold gas (atomic plus molecular). We assume an empirical relation between the local star formation rate and the local density of molecular gas. We calculate the molecular gas fraction (as a function of radius in the disk) using the empirical relation between mid plane pressure and molecular gas fraction from Blitz & Rosolowski (2006). A more realistic treatment of star formation would model the formation and destruction of molecular gas in a physically motivated way (e.g. Pelupessy, Papadopoulos, & van der Werf 2006; Robertson & Kravtsov 2008)
In our feedback model we only consider winds that are able to escape the halo, and we assume that mass in such winds is lost forever. In reality mass that escapes the halo may fall back at later times. Furthermore, there may be winds that have enough energy/momentum to escape the disk but not the halo. The gas in these winds could then re-cool back onto the disk producing a galactic fountain. Since very little is known about how feedback works, and our main interest is to determine how much mass can be ejected from the disk and halo, rather than introducing additional free parameters to our wind model, we assume the maximal mass loss model.
The assumption that the halo responds adiabatically to inflows and outflows may not be correct. When galaxies accrete their gas via a smooth cooling flow the gas radiates away its energy. However, when a galaxy acquires its gas via cold flows, clumps of cold gas can exchange energy with the halo via dynamical friction (e.g. El Zant, Shlosman, & Hoffman 2001; El-Zant et al. 2004; Jardel & Sellwood 2009), causing the halo to expand. In both scenarios the natural response of the halo to the deepening of the potential well due to the condensation of baryons is to contract, but in the latter scenario, the transfer of energy between baryons and dark matter will counter this effect. Processes internal to disks, such as bars, can also transfer energy to the halo via dynamical friction, causing it to expand (Weinberg & Katz 2002; Holley-Bockelmann, Weinberg & Katz 2005; Sellwood 2008). Mass outflows can also result in halo expansion, for example an adiabatic inflow followed by an instantaneous outflow can result in net halo expansion (Navarro, Eke, & Frenk 1996; Gnedin & Zhao 2002; Read & Gilmore 2005). Thus in order to asses how much impact the assumption of adiabatic inflow and outflow has on the structural properties of the resulting galaxies, we also consider models in which the halo does not respond to galaxy formation.
2.2 Dark Matter Haloes
The backbone of our galaxy formation model is the growth of the dark matter halo, which we model by a smooth mass accretion history (MAH). Van den Bosch (2002b) and Wechsler et al. (2002) have shown that the MAH is essentially a one parameter family. The MAH from Wechsler et al. (2002) is given by
where is the redshift zero mass and is related to the epoch of formation via
Here is defined as the scale factor when the logarithmic slope of the accretion rate falls below some specified value, . Following Wechsler et al. (2002) we adopt . Before we discuss how to compute we discuss how the structural properties of the halo depend on its mass.
In the standard spherical top-hat collapse model the virial radius, , of a halo of mass at a redshift, , is given by
where , and is the virial density, relative to the critical density for closure. We use the fitting formula of Bryan & Norman (1998)
with . The evolution of the matter density is given by
and the evolution of the Hubble parameter is given by
The relation between the virial velocity, , and virial radius, is given by
We assume that the density profile of the halo at each redshift is given by an NFW (Navarro, Frenk, & White 1997) profile
where is the radius where the slope of the density profile is -2, the so called scale radius, is the critical density of the universe, and is the characteristic overdensity of the halo. The concentration parameter of the halo is defined as and is related to via
Following Bullock et al. (2001a) we assume that the concentration parameter evolves as
with a minimum value of , corresponding to a constant concentration during the rapid accretion phase of dark halo growth (Zhao et al. 2003). Thus if we specify the redshift zero concentration we can compute the collapse epoch via , and hence the MAH through Eqs. (1) & (2).
To compute the mean concentration for a halo of a given mass at we use the Bullock et al. (2001a) model. This model requires us to specify the cosmology , as well as 2 free parameters, and . We assume the concentration is log-normally distributed with median from the Bullock et al. (2001a) model and scatter . Our adopted values for these parameters are given in §2.10.
2.3 Gas Cooling
At each time step a shell with mass virializes. A fraction of this mass is in baryons, and is heated to the halo’s virial temperature
where is the mass per particle, and is Boltzmann’s constant. The baryons dissipate energy radiatively, lose pressure support, and collapse until they reach centrifugal equilibrium. The time scale over which this occurs is given by . Here is the free fall time defined as
with the average halo density, and
is the cooling time. Here is the density of the hot gas, is the number of particles per electron, and is the normalized cooling function for a gas with metallicity . For we use times the density at the virial radius at the time the gas enters the halo. For we use the collisional ionization equilibrium cooling functions of Sutherland & Dopita (1993), assuming a helium mass abundance of 0.25.
For each time step, we compute the range of times between which gas that collapses onto the disk in the current time interval, entered the halo. We label these times as and .
2.4 Angular Momentum Distribution
The radius at which the cooled gas ends up depends on its specific angular momentum, . Van den Bosch (2001; 2002a) assumed the -distribution to be that of a shell in solid body rotation. The angular momentum of this shell can be computed assuming that the spin parameter, , is constant between time steps. The spin parameter is defined by
where , and are the mass, total angular momentum and energy of the halo, respectively. As shown by van den Bosch (2001) this results in density profiles that are more concentrated than exponential in the center, and also that truncate at shorter radii than is observed in some disk galaxies.
Building on the work of Bullock et al. (2001b), Sharma & Steinmetz (2005, hereafter SS05) used a series of non radiative N-body/SPH simulations in a CDM cosmology to study the growth of angular momentum in galaxy systems. They introduced a function that is able to describe, with a single parameter, , the specific angular momentum distribution of the gas and dark matter in their simulations, as well as that of exponential disks in NFW haloes:
where is the incomplete gamma function. SS05 found that the distribution of is log-normally distributed with mean and standard deviation in .
In summary the distribution of specific angular momentum of the dark matter halo and hot gas can be described by two parameters: a normalization () and a shape (). Both the normalization and shape parameters are log-normally distributed, with significant scatter. We assume that the spin and shape parameters are uncorrelated, although Bullock et al. (2001b) show that there may be a weak correlation between and . Furthermore we assume that, for a given halo, and are constant with redshift. These assumptions are made for simplicity, and need to be tested with cosmological simulations.
In order to compute the amount of mass, with a given specific angular momentum, , that has collapsed onto the disk in each time step, , we take the difference between the distributions of specific angular momentum of the halos at times and ( and are defined in §2.3 above);
2.5 Conservation of Angular Momentum and Halo Contraction
In order to compute the radius, , at which material with specific angular momentum, , ends up we assume specific angular momentum is conserved, i.e. one should solve
for . Here is the total circular velocity (from stars, cold gas, hot gas, and dark matter).
As the galaxy grows over time the circular velocity at a given radius increases. Thus to conserve specific angular momentum, material that settled at radius would need to drift to smaller radii over time. Given that the gas and stars effect the circular velocity, strictly conserving the specific angular momentum of the baryons over time is difficult to implement numerically. To get around this problem we use specific angular momentum, , rather than radius as our radial coordinate.
Under the simplifying assumption that , where is the total mass within a spherical radius, , the radius that corresponds to a given is given by
This has a number of desirable properties: 1) At each time step it is trivial to calculate how much cold gas is added to each bin in . 2) Over time, as the potential well changes, the specific angular momentum of the baryons is automatically conserved; 3) The response of the halo to the cooling of the baryons is automatically taken into account. 4) The resulting radial grid is adaptive, as the mapping between and depends on the amount of mass enclosed.
The price that we pay for these advantages is that the circular velocity due to the disk is not calculated correctly. Due to the disk geometry the true circular velocity deviates from that given by the spherically enclosed mass. For example the peak of a thick exponential disk is higher than that obtained using the enclosed mass, and at small radii the proper increases linearly with radius, whereas using the enclosed mass scales as .
However, given that computing the proper circular velocity of the disk is very time consuming, and that it is sensitive to gradients in the disk density profile, and that using it would remove the simplicity of the -grid approach we feel that it is a price worth paying.
As we show below (in §3), and as discussed in Dutton et al. (2007), models with halo contraction (and standard stellar IMF’s) are unable to reproduce the zero points of the relations as well as the low galaxy formation efficiency required to reconcile the halo mass function and galaxy stellar mass function. While there are processes such as dynamical friction and impulsive mass loss that can expand the halo, implementing these in a galaxy evolution model is a non-trivial task. Thus for simplicity we wish to consider a model in which the dark halo does not respond to galaxy formation.
Note that simply using the mapping between radius and based on the halo profile (at each time step) in the absence of galaxy formation would not conserve the specific angular momentum of the baryons. To calculate the mapping between and radius, for the case of no adiabatic contraction, we solve the equation
where is the mass (within a spherical radius ) of the dark matter plus hot gas halo in the absence of galaxy formation, and is the mass of the disk (gas plus stars) with specific angular momentum less than . This way the self-gravity of the disk is included but adiabatic contraction is ignored.
2.6 Star Formation
Observations have shown that the disk averaged star formation rates in nearby spiral galaxies are well fitted by a simple Schmidt (1959) law
Kennicutt (1998) used as the total gas density (molecular and atomic, but not including helium) and found , and (Kennicutt 1998)222 Including a helium correction of 1.36 results in .. This simple empirical law holds over many orders of magnitude in gas surface density, and even applies to circumnuclear starburst regions. However, when applied to local gas densities, the Schmidt law breaks down at low gas densities, corresponding to large radii, where star formation has been found to be abruptly suppressed. Kennicutt (1989) argued that this suppression is due to the gas density falling below the critical density for global disk stability as given by the Toomre criterion (Toomre 1964)
where is the velocity dispersion of the gas, is the epicyclic frequency, and is the Toomre Q parameter. Kennicutt (1989) found that and reproduces the observed star formation truncation radii. However other authors argue that this is just a coincidence, and that is not a threshold density (e.g. Schaye 2004).
The physical origin of the Schmidt-Kennicutt (SK) relation is also not clear. However, something that is well established is that stars form out of molecular gas, and predominantly in giant molecular clouds (GMC’s). This lead Wong & Blitz (2002) to argue for a Schmidt law based on the surface density of molecular gas. For high gas densities the molecular gas dominates, so the two prescriptions are identical. However, for low gas densities the molecular fraction is suppressed, resulting in a steep dependence of the star formation rate density on total gas density. This gives an alternative explanation for the suppression of star formation at low gas densities.
The fraction of gas that is molecular, , can be defined in terms of the mass ratio between molecular and atomic gas, by
Empirically Wong & Blitz (2002) and Blitz & Rosolowsky (2004; 2006) have argued that is determined to first order by the mid plane pressure, . The most recent relation from Blitz & Rosolowsky (2006) is
where is Boltzmann’s constant, and is in cgs units. For a gas plus stellar disk the mid plane pressure is given, to within 10% by (Elmegreen 1993)
where and are the velocity dispersions of the gas and stellar disk, respectively. For simplicity we will assume which is a reasonable assumption for the stellar dominated parts of disk galaxies. In the outer parts of disk galaxies this ratio is likely to be higher. However, in these regions gas dominates, and so the contribution of the stars to the mid plane pressure is negligible, regardless of the ratio between and . In the inner regions of galaxies is likely to be smaller than 0.1, but here the densities are high enough that .
Following Blitz & Rosolowski (2006) we assume that star formation takes place in dense molecular gas, traced by HCN, with a constant star formation efficiency
where (Gao & Solomon 2004, Wu et al. 2005). Expressing this equation in terms of the total gas content:
where is the ratio between the dense molecular gas (as traced by HCN) and the total molecular gas.
Based on the arguments and references in Blitz & Rosolowski (2006) we adopt the following relation for
In the low pressure, low molecular density regime, , and thus Eq. (25) asymptotes to
In the high pressure, high molecular density regime, , and eq.(25) asymptotes to the familiar SK relation
Furthermore, with , we recover the coefficient of of the standard SK relation.
Fig. 1 shows the relation between star formation rate density and gas density for our star formation model Eq. (25). Note that in order to compute one needs to know and . For illustrative purposes we have chosen a stellar and gas density profile representative of bright nearby disk galaxies (left panel of Fig. 1). At small radii, and high gas densities , and the star formation law follows the standard SK relation. At large radii the molecular fraction is very low, which results in a steeper slope to the star formation law.
We implement the star formation recipe given by Eq.(25) as follows. At each time step and annulus in the disk, we calculate the star formation rate. Then we use the following approximation (valid for times steps small compared to the star formation time scale) to calculate the mass of newly formed stars
with the area of the annulus, and the time step interval.
2.7 Supernova Feedback
When stars evolve they put energy back into the inter stellar medium. The effect of this on the star formation rate is partially taken into account by our empirically determined star formation recipe. What is not taken into account is a feedback driven outflow of gas from the disk. The physical mechanism responsible for driving outflows is a subject of debate (e.g. Finlator & Davé 2008), so in this paper we consider both energy and momentum driven winds.
Following van den Bosch (2001) we assume that the outflow moves at the local escape velocity of the disk-halo system. This choice is motivated by the fact that it maximizes the mass loss from the disk-halo system (lower velocity winds will not escape the halo, and higher velocity winds will carry less mass).
For our energy driven wind model following Dekel & Silk (1986) we assume that the kinetic energy of the wind is equal to a fraction, , of the kinetic energy produced by SN. However, contrary to Dekel & Silk (1986) we apply this energy condition locally in the disk as a function of radius, rather than globally to the whole galaxy. Thus the mass ejected from radius, , during a given time step is given by
Here is the mass in stars formed at radius, , erg is the energy produced by one SN, and is the number of SN per solar mass of stars formed (for a Chabrier IMF).
The local escape velocity is given by
where is the sum of the potentials due to the disk (stars plus gas) and halo (dark matter plus hot gas), and is computed assuming spherical symmetry.
For our momentum driven wind model we assume that the momentum of the wind is equal to a fraction, , of the momentum produced by SN, thus the mass ejected from radius, , during a given time step is given by
Here is the momentum produced by one SN, assuming that each SN produces of material moving at (Murray, Quataert & Thompson 2005). Note that this corresponds to a kinetic energy of erg.
We assume that the ejected mass is lost forever from the system: the ejected mass is not considered for later infall, and the corresponding metals are not used to enrich the infalling gas. This is clearly a dramatic oversimplification, but we make this choice to maximize the amount of gas that is lost from the system.
2.8 Stellar Populations and Chemical Enrichment
In order to convert stellar mass into luminosities we use the Bruzual & Charlot (2003) stellar population synthesis models. These models provide the luminosities of a single burst stellar population with a total mass of as a function of age, , and metallicity, , in various optical pass-bands. To compute the luminosities of our model stellar populations we convolve the star formation history of our galaxies with the single burst stellar population synthesis models.
In order to model the chemical enrichment of the ISM, we adopt the instantaneous recycling approximation. We assume that a fraction of the mass in stars formed is instantaneously returned to the cold gas phase with a yield (defined as the fraction of mass converted into stars that is returned to the ISM in the form of newly processed metals).
The equations for the change in the cold gas mass and metals are:
The metallicity of the cold gas is then given by . Note we assume that the ejected metals do not enrich the hot halo gas.
2.9 Book Keeping
We now briefly describe how we keep track of the evolution of the various mass components. Given a halo mass (dark matter plus hot gas) and concentration we compute the MAH of the halo using Eq. (1). The evolution of the virial radius and internal structure of the halo is then determined by Eqs. (3) & (10). We set up a grid in radius from 0.001 to 1 times the redshift zero virial radius. As described in § 2.5, for the purposes of conserving angular momentum it is more convenient to use a grid in specific angular momentum, . Thus we convert the grid in , to a grid in using .
For each time step () we compute the halo mass that is added to each radial bin so that the total mass follows the NFW profile for a halo of a given and . Thus
We assume that the baryons make up a fraction of this mass, so that
When we compute the circular velocity we assume that the hot gas follows the mass distribution of the dark matter. When computing the cooling time we assume is times the density at the virial radius at the time when the gas virialized.
At each time step we then compute, using the recipes in the previous sections, the amount of gas that cools, , the amount of stars formed , and the amount of ejected gas . For the stellar population modeling we keep track of the mass of stars formed at each time step and the metallicity of the gas from which the stars formed, .
2.10 Overview of Model Parameters
The input parameters of our models are as follows.
(1) Cosmology: . In this paper we adopt a flat CDM cosmology motivated by the 5th year WMAP results (Dunkley et al. 2009), with and .
(2) Halo structure: . We adopt the Bullock et al. (2001a) model with , , and . These parameters reproduce the distribution of halo concentrations of relaxed dark matter haloes in numerical simulations (Wechsler et al. 2002; Macciò et al. 2007, 2008).
(3) Angular momentum distribution: , , , . As fiducial values we adopt a median spin parameter with a scatter , corresponding to relaxed haloes (Macciò et al. 2007, 2008). For the angular momentum shape parameter we adopt a median , and scatter (Sharma & Steinmetz 2005).
(4) Star formation: . We use a star formation model based on dense molecular gas [Eq. (25)], and adopt .
(5) Feedback: , , , , . We adopt erg, , . We treat and as free parameters.
(6) Stellar populations and chemical enrichment: , and the choice of initial mass function (IMF). We adopt the Chabrier (2003) IMF, a return fraction , a stellar yield , and a metallicity of the hot gas of .
2.11 Overview of Output Parameters
The output parameters of our models, that we discuss in this paper, are as follows:
, total mass inside virial radius 
, circular velocity at the virial radius 
, maximum circular velocity of the halo without galaxy formation 
, galaxy mass (stars and cold gas) 
, stellar mass .
, cold gas mass .
, metallicity of cold gas.
, galaxy formation efficiency.
, galaxy mass fraction.
, stellar mass fraction.
, cold gas mass fraction.
, cold gas fraction.
, the circular velocity measured at 2.15 -band disk scale lengths .
, -band disk scale length .
, galaxy angular momentum fraction.
, stellar angular momentum fraction.
, cold gas angular momentum fraction.
, galaxy spin parameter.
, stellar spin parameter.
, cold gas spin parameter.
, the baryonic mass accretion rate 
, the mass cooling rate 
, the star formation rate 
, the mass outflow rate 
, the mass loading factor
, the (outflow) mass weighted mean outflow velocity , where the refers to the radial grid position.
The disk scale lengths are determined using the following procedure, which was developed to give robust disk scale lengths for the full range of surface brightness profiles produced by our model. We first compute the local disk scale length between the radii enclosing 50 and 99% of the stellar mass. The local disk scale length is computed at radial bin by using the surface densities and radii at radial bins and . We then determine the maximum of the local disk scale length, and the radius where this maximum occurs, . Finally we determine the scale length of the disk using a linear fit to the model data over the range .
2.12 Comparison with other Disk Galaxy Structure Models
In this section we place our model in the context of existing analytic and semi-analytic models of disk galaxy structure/formation in the literature. We classify these models into two general types: 1) models that conserve total specific angular momentum (i.e. the structural profile of the disk is assumed) and 2) models that conserve the distribution of specific angular momentum (i.e. the structural profile of the disk is derived). Both of these classes of models can be static or include evolution. The essential assumption in both classes of models is that the disk is in centrifugal equilibrium inside some potential (which may or may not include the self gravity of the disk).
2.12.1 Models that Conserve Total Specific Angular Momentum
In the simplest models of this class, the circular velocity is assumed to be constant, i.e. corresponding to an isothermal density profile, and the self gravity of the disk is ignored. This model has 3 parameters: the circular velocity, the spin parameter and the disk mass fraction. Such a model was discussed in Mo, Mao, & White (1998; MMW), and is widely used in Semi Analytic Models (e.g. Kauffmann, White, & Guiderdoni 1993; Cole et al. 1994; Somerville & Primack 1999; Hatton et al. 2003; Croton et al. 2006).
A more realistic version of this model includes the self gravity of the disk and adiabatic contraction of the halo (Blumenthal et al. 1986), which usually results in smaller sizes for a given spin parameter and disk mass fraction. In Mo, Mao & White (1998), the halo was assumed to be an NFW profile, and the disk was assumed to be exponential. This model has 4 parameters: the circular velocity of the halo, the concentration of the halo, the spin parameter and the disk mass fraction. This version of the MMW model is widely used in Semi Analytic Models (Cole et al. (2000); Benson et al. (2003); Somerville et al. 2008) and studies of disk galaxy scaling relations (e.g. Navarro & Steinmetz 2000; Pizagno et al. 2005; Dutton et al. 2007; Gnedin et al. 2007).
2.12.2 Models that Conserve the Distribution of Specific Angular Momentum
The MMW type model is useful for understanding the origin of disk galaxy scaling relations, but it does not explain the origin of the density profiles of galaxy disks, or the relation between gas and stars in galaxy disks. In order to address these questions, one needs to start from some specific angular momentum distribution (AMD). This AMD may be that of a sphere in solid body rotation, or preferably that found in cosmological simulations (e.g. Bullock et al. 2001b). The radial density profile of the disk is then determined by detailed conservation of specific angular momentum.
As with the MMW type models, these models may ignore the self gravity of the disk by assuming the total density profile is isothermal (e.g. Ferguson & Clarke 2001), or include the self gravity of he disk inside a dark matter halo (e.g. Dalcanton, Spergel & Summers 1997). These models may also include evolution of the baryonic disk, by following the cooling of gas inside growing dark mater haloes, and evolution of the stellar disk, by following the star formation as a function of radius (e.g. Firmani & Avila Reese 2000; van den Bosch 2001, 2002; Stringer & Benson 2007).
Our model fall into this later category, being an evolution of the van den Bosch (2001; 2002) models. The strength of these models over the MMW type models is that they can be used to self-consistently study the origin and evolution of disk density profiles (stars, gas, star formation rates, inflows, outflows, metallicity, stellar ages) and rotation curve shapes.
3 The dependence of the Galaxy Velocity - Mass - Size relations on feedback
In this section we investigate the dependence of the scaling relations between rotation velocity, stellar mass and stellar disk scale size on the feedback model.
3.1 Observed Disk Galaxy Scaling Relations
Here we overview the main observed velocity, mass, size, scaling relations that we are going to compare our models to. We use the relations between rotation velocity at 2.2 -band disk scale lengths, , stellar mass, , and -band disk scale length, , from the data set of Courteau et al. (2007), as presented in Dutton et al. (2007). The stellar masses in Dutton et al. (2007) were derived from -band luminosities using the relations from Bell et al. (2003) corresponding to a diet Salpeter IMF. Here we adopt a Chabrier IMF, and so subtract 0.10 dex from the stellar masses.
The stellar mass TF relation is given by
the size-stellar mass relation is given by
and the corresponding size-velocity relation is given by
The intrinsic scatter in these relations is estimated to be , , and . The errors on the slopes of the , , and relations from fitting uncertainties are 0.01, 0.02, 0.12, respectively. However, systematic uncertainties are significantly larger, and harder to quantify. The most significant selection effect for the slope of the relation is surface brightness. The data set compiled by Courteau et al. (2007) is likely missing lower surface brightness galaxies, and thus over-estimates the slope of the size-mass relation at low masses. Such a conclusion is supported by Shen et al. (2003) who studied the half light radius-stellar mass relation () for a much larger sample of galaxies () from the SDSS. They find a log slope of 0.14 at low masses, increasing to 0.39 at high masses.
3.2 A Fiducial Model
To illustrate the effect that feedback has on observable properties of disk galaxies we consider a model with virial mass , and median concentration and angular momentum parameters: , and . The results of varying the energy and momentum feedback efficiency parameters from 0 to 1 are shown in Figs. 2 & 3, respectively. The main panels show the relations with the solid line showing the mean relations from observations and the dashed lines showing the 2 scatter. The upper right panels in these figures shows the more theoretical parameters and . These are not directly observable because they require knowledge of the halo mass, a quantity that cannot, at present, be reliably measured for individual galaxies.
We first focus on the models with energy feedback and adiabatic contraction (solid red points and lines in Fig. 2). A model with no feedback () results in a galaxy mass fraction of (where is the universal baryon fraction). The galaxy formation efficiency is less than 1 because cooling starts to become inefficient at late times. Since the highest specific angular momentum material is accreted last, and this does not have time to cool, the spin of the galaxy is slightly lower than that of the halo. The high galaxy formation efficiency results in a disk scale length a factor of too small and a circular velocity at 2.2 disk scale lengths a factor of too high.
When feedback is included, some of the cold baryons are ejected from the disc (and halo). This trivially results in lower (hence lower baryonic mass) but also, non-trivially, higher . The higher is due to the preferential ejection of low angular momentum material, which we discuss in more detail in §4. Both of these effects results in larger, lower surface density disks, which result in less efficient star formation, and hence higher gas fractions and lower stellar masses. The reduction in the amount and density of the baryons results in a lower rotation velocity, both because the baryons contribute less to the circular velocity at 2.2 disk scale lengths and because there is less halo contraction. With a high energy feedback efficiency of the galaxy formation efficiency drops to : of the baryons have not cooled, while of the baryons have been ejected from the disk and halo. This model also has a galaxy spin a factor of higher than the halo spin. The low galaxy mass fraction and higher galaxy spin parameter results in sizes that are more than a factor of too large. The feedback efficiency can be tuned so that the model reproduces the size-mass, or size-velocity relation. However, for all feedback efficiencies, the models rotate too fast at a given stellar mass. This is because higher feedback efficiency results in lower stellar masses as well as lower rotation velocities, with the net result that galaxies move almost parallel to the TF relation.
Fig. 3 shows the same results as Fig. 2, but for the momentum driven wind model. Even with 100 per cent efficiency this galaxy formation efficiency is , where of the baryons have been ejected from the disk and halo. This results in sizes that are in agreement with observations, but the models still rotate too fast. The reason that energy driven winds are more efficient at ejecting mass than momentum driven winds is shown in Fig. 4, which shows the mass loading factor, , versus the wind velocity, . The mass loading factor is defined as the ratio between the mass outflow rate and the star formation rate. For energy driven winds , whereas for momentum driven winds . Thus everything being equal, energy driven winds have higher mass loading factors than momentum driven winds for all typical wind velocities
3.3 The Tully-Fisher Zero Point Problem
A common problem to both feedback models, for all values of the feedback efficiency, is that they over predict the rotation velocities. This is a standard problem for galaxy formation models in CDM. As discussed in Dutton et al. (2007) and Gnedin et al. (2007), there are 3 solutions: 1) Lower the stellar mass-to-light ratio (i.e. for a given luminosity there is less stellar mass which shifts observed galaxies to the left and hence higher velocity in the plane)333Note that the stellar masses of the model galaxies are not completely independent of the IMF, as the return fraction is IMF dependent. However, changes in the return fraction are compensated for by changes in the star formation rate. . 2) Lower the initial halo concentration (which directly lowers ). 3) Reverse halo contraction (which directly lowers ).
A small change in the stellar mass-to-light ratio would be plausible due to systematic uncertainties (such as in the IMF or the stellar populations synthesis models) in the measurement of stellar mass-to-light ratios. However, the stellar mass-to-light ratios would have to be lowered by about a factor of 2 to match the TF zero point. Such a large change would require a top-heavy IMF. But all of the available constraints on stellar mass-to-light ratios point to IMF’s similar to Chabrier (e.g. de Jong & Bell 2007).
Lower halo concentrations would require less power on galaxy scales than in standard CDM. This would also reduce the amount of substructure, which could help solve the missing satellite problem (Klypin et al. 1999; Moore et al. 1999). However, the recent discovery of many satellite galaxies around the Milky Way has lessened the discrepancy between observations and standard CDM (e.g. Tollerud et al. 2008 and references therein). Furthermore using cosmological simulations with parameters from the latest WMAP results (Spergel et al. 2007; Dunkley et al. 2009), Macciò et al. (2008) have shown that the central densities of dark matter haloes are consistent (both in normalization and scatter) with those measured from dwarf and LSB galaxies (which typically have maximum rotation velocities of ). Thus there does not seem compelling evidence for a modification to CDM on small scales.
Given that reducing stellar mass-to-light ratios or the initial halo concentrations do not seem plausible, we consider the third possibility, that halos do not contract as expected. There are two processes that could cause the halo to expand. 1) Dynamical friction between baryons and the halo, e.g. by infalling baryonic clumps (e.g. El Zant et al. 2001; El-Zant et al. 2004; Tonini et al. 2006; Jardel & Sellwood 2009); or by galactic bars (Weinberg & Katz 2002; Holley-Bockelmann, Weinberg & Katz 2005; Sellwood 2008); 2) Rapid (i.e. non-adiabatic) mass loss from the galaxy, e.g. by SN driven winds (Gnedin & Zhao 2002; Read & Gilmore 2005). Both of these effects have been shown to be effective at expanding the halo, but the combined effect (which may be greater than the sum of its parts) has so far not been investigated. Note that our standard model with adiabatic contraction takes into account the adiabatic expansion of the halo due to outflows. However, since there is a net inflow of baryons into the centers of galaxies, the overall effect is halo contraction.
Furthermore halo contraction is based on the idea that galaxies form by cooling flows. The hot gas radiates away its binding energy, so when it falls to the center of the potential to form the galaxy the halo has to contract. However, recent simulations have indicated that the gas, in haloes that host disk galaxies, is accreted by cold flows (i.e. it does not shock heat when it enters the virial radius). This new scenario thus allows the possibility of the baryons transferring energy to the halo during galaxy formation.
The black circles in Figs. 2 & 3 show the results for two models without halo contraction. For both energy and momentum driven winds the galaxy mass fractions and spin parameters have the same dependence on the feedback efficiency in models with and without halo contraction. However, the models without halo contraction have significantly lower rotation velocities, allowing a match to the TF zero point for energy feedback efficiencies of , or momentum feedback efficiencies of . A model with also has , and a galaxy formation efficiency of , consistent with observations (see §1). However, this model has sizes that are too large. The solution to this is to lower the spin parameter of the baryons. This could occur if disk galaxies formed in haloes with lower than average halo spin, or by the baryons transferring some of their angular momentum to the halo during galaxy assembly, for example via dynamical friction. As we discussed above, such a process may be responsible for expanding the halo. As we show below, a model with a factor 1.4 lower halo spin reproduces both and relations as well as low galaxy formation efficiencies. The momentum feedback model with maximum feedback efficiency, , on the other hand, roughly reproduces the and zero points, but it has galaxy mass fractions and that are too high compared to observational and theoretical constraints.
|I: No Feedback:||0.0||0.0||0.035||0.35||N|
3.4 Models with Scatter
Having discussed the effects of energy vs. momentum driven winds, and halo contraction vs no halo contraction for a single halo mass, with the median concentration, and angular momentum parameters, we now turn our attention to models with the full range of halo masses relevant for disk galaxies, and with distributions of concentration, and angular momentum parameters.
We run a Monte Carlo simulations with halo masses ranging from , corresponding to virial velocities ranging from . In CDM there are many more low mass haloes than high mass haloes, however, since we are interested in the scaling relations between galaxies, rather than the number densities we sample the halo masses uniformly in log-space.
As discussed in Dutton et al. (2007) we also find that models with the expected scatter in halo spin parameter significantly over predict the amount of scatter in the and relations. This may signify that disk galaxies form in a special sub-set of haloes, or that the baryons acquire a different distribution of specific angular momentum than the dark matter. For the remainder of this paper we adopt , as this provides a reasonable agreement to the observed scatter in disk sizes.
To illustrate the effect of feedback on galaxy scaling relations we consider three models. Model I has no feedback, Model II has momentum feedback, and Model III has energy feedback and an average spin parameter a factor of 1.4 lower than models I and II. The parameters of these models are given in Table LABEL:tab:params. The parameters of Model II and III were chosen to match the zero points of the relations, and thus for reasons discussed above, they have no adiabatic contraction. For each model we run a Monte-Carlo simulation consisting of 2000 galaxies. For each galaxy we select the parameters , , and from log-normal distributions with means and scatters as determined by the parameters in § 2.10 and Table LABEL:tab:params.
Figs. 5-7 show the relations, as well as the correlation between the residuals of the and relations, for models I-III. Recall that is the circular velocity measured at -band disk scale lengths, is the stellar mass, and is the -band disk scale length.
The VMR relations in these figures are fit with two relations: a single power-law over the range where there is observational data, and a double power-law over the full range of masses. The parameters of the best-fit single power-law fits are indicated in the panels of Figs. 5-7. The double power-law is given by
Here is the slope at ; is the slope at ; is the transition scale; is the value of at . The best fit values of these parameters are given in Table LABEL:tab:power1.
The slope (as given by the single power-law fits) of the relation is only weakly dependent on the feedback model. This is expected since (as shown in Figs. 2 & 3) the offset of a galaxy from the relation is only weakly dependent on the galaxy mass fraction, which is determined by feedback (for haloes with ). By contrast the slope of the relation depends strongly on the feedback model. Again, this is expected given that the offset of a galaxy from the relation depends strongly on the galaxy mass fraction. The model without feedback (model I) has a slope of , the model with momentum driven feedback (model II) has a slope of , and the model with energy driven feedback (model III) has a slope of .
The observed slope of the size-mass relation from (Courteau et al. 2007 and Dutton et al. 2007) is , which favors the momentum driven wind model. However, as discussed in § 3.1, at low stellar masses (), this slope is likely biased high by selection effects. Shen et al. (2003) find a slope of 0.14 at low masses for the half-light radius-stellar mass relation for a much more complete sample of late-type galaxies. Such a shallow slope is in much better agreement with our energy driven model. Thus the observed slope of the relation favors the energy driven wind model at low masses, and the momentum driven model at high masses. However, at high stellar masses () bulges are common is spiral galaxies (e.g. Weinzirl et al. 2008). Due to the correlation between the masses of bulges and black holes (Magorrian et al. 1998), AGN feedback may play a significant role in regulating galaxy formation efficiency in high mass spiral galaxies. Thus it is plausible that a model with energy driven SN feedback (which primarily effects galaxies in low mass haloes) and AGN feedback (which primarily effects galaxies in high mass haloes) could explain the slopes of the size-mass relation at low and high masses. However, since AGN feedback is not expected to be significant for galaxies in low mass haloes, it is unlikely that AGN feedback will be able to help the momentum driven wind model reproduce the shallow slope of the size-mass relation at low masses.
3.4.2 scatter and residual correlations
All models produce a relation with relatively small scatter, with smaller scatter in the models with feedback. The amount of scatter in the relation is directly related to the strength of the correlation between the and relations. The model without feedback has a very strong correlation (correlation coefficient, ; slope, ), which is caused by these galaxies being baryon dominated at 2.2 disk scale lengths. The models with feedback have weaker, but still significant, correlations (, ). These correlations are stronger than that observed by Courteau et al. (2007) for the -band and relations (, ), and by Gnedin et al. (2007) () and Avila-Reese et al. (2008) (, ) for the stellar mass and relations. However, as discussed in Dutton et al. (2007) scatter in the stellar mass-to-light ratio of dex, either from intrinsic variations or measurement uncertainties will weaken the correlation between the observed relations compared to the theoretical and relations. Thus we do not consider this failure of the model as a serious shortcoming.
3.5 The Baryonic Tully-Fisher Relation
The fundamental basis of the Tully-Fisher (linewidth-luminosity) relation is believed to be the relation between the asymptotic rotation velocity of a galaxy disk, , and the baryonic mass, , (the sum of stellar and cold gas mass). This relation is referred to as the Baryonic Tully-Fisher (BTF) relation. It was first studied by McGaugh et al. (2000), and subsequently by Bell & de Jong (2001), McGaugh (2005), Geha et al. (2006), Noordermeer & Verheijen (2007), and Avila-Reese et al. (2008).
The most significant source of uncertainty in the BTF is how one measures stellar mass. McGaugh (2005) measured the BTF for stellar masses calculated using different methods: stellar population synthesis models (e.g. Bell & de Jong 2001), the maximum disk hypothesis (van Albada & Sancisi 1986), the mass-discrepancy acceleration relation (i.e. Modified Newtonian Dynamics, or MOND, Milgrom 1983)). McGaugh (2005) found that the scatter in the BTF was minimized when the stellar masses were calculated with the mass-discrepancy acceleration relation. Under the assumption that the correct method would minimized the scatter in the BTF, this is evidence in favor of MOND over dark matter.
However, this is a circular argument because a relation between the asymptotic rotation velocity of a galactic disk, and the baryonic mass, , with zero scatter, is built into MOND. Thus if the stellar masses are chosen based on the MOND prescription, they will result in a BTF with scatter only due to measurement errors on and distance uncertainties. The scatter in the BTF (as defined as the relation between and ) thus cannot be used to discriminate between MOND and dark matter. However, MOND generally predicts higher stellar masses than stellar population synthesis models (based on a Kroupa IMF). Thus if stellar masses could be measured independently, this would provide a means of falsifying MOND.
Here we use the data from McGaugh (2005), using the stellar population synthesis stellar mass-to-light ratios, with an offset of -0.1 dex (corresponding to a Chabrier IMF). The majority of galaxies in this sample are in the UMa Cluster, for which the distance is somewhat uncertain. McGaugh (2005) adopted a distance of 15 Mpc. We adopt the HST Key Project distance of Mpc (Sakai et al. 2000), which is also the distance used by Bell & de Jong (2001).
The BTF data are plotted as green filled circles in Fig. 8. A linear fit to the data gives the following relation between the rotation velocity and baryonic mass
with a scatter of 0.053 dex in . This BTF is consistent with that from Bell & de Jong (2001) who report a slope of and a zero point of , and that from Avila-Reese et al. (2008) who report a slope of a zero point of and an intrinsic scatter of 0.051 dex in . The slightly steeper slope obtained by Avila-Reese et al. (2008) can be attributed (see Verheijen 2001) to their use of linewidths, compared to used by McGaugh (2005) and Bell & de Jong (2001). This good agreement is reassuring given that the data samples are largely based on the data set of Verheijen et al. (2001).
The BTF relations for our models are given by the grey dots in Fig. 8. For the rotation velocity we use , the circular velocity at a radius enclosing of the gas mass, , which usually corresponds to the flat part of the rotation curve (see Dutton 2008 rotation curves). Power-law fits to the models over baryonic masses between and are shown as red lines in the figure. The parameters of these fits are given in the top left of each panel. All three models result in BTF relations with similar slopes, zero points and scatter, and in reasonable agreement with observations. We note that for the model galaxies the slope of the BTF depends on galaxy mass, with slightly larger slopes for higher mass galaxies.
The BTF relation has been used to constrain the relation between baryonic mass and halo mass. By comparing the observed slope of the BTF () to the prediction from CDM (the slope of the relation for dark matter haloes is , Bullock et al. 2001a), Geha et al. (2006) argued that low mass galaxies have not preferentially lost baryons as would be predicted by feedback models (e.g. Dekel & Silk 1986). However, this is based on the incorrect assumption that the maximum observed rotation velocity is equal (or proportional) to the maximum circular velocity of the halo, independent of the baryon to halo mass ratio. As discussed by several authors (e.g. Navarro & Steinmetz 2000; Dutton et al. 2007; Avila-Reese et al. 2008), the maximum rotation velocity of a galaxy is, in general, not equal to the maximum circular velocity of the halo in the absence of galaxy formation, . As the baryon fraction increases, so to does the maximum circular velocity. This is because the baryons contribute a non-negligible amount of mass to the circular velocity. Thus for reasonable galaxy mass fractions, variation in galaxy mass fraction moves galaxies roughly parallel to the BTF.
Fig. 8 shows that the slope, zero point and scatter of the BTF are remarkably insensitive to the feedback model. Furthermore, as shown in §4.1, our energy feedback model results in substantial differential mass loss between haloes of mass and . Yet it has the same BTF slope as a model with no mass loss (and constant baryon to dark matter ratio within this range of halo masses). This provides a counter example to the claim by Geha et al. (2006) that models with preferential mass loss in dwarf galaxies cannot explain the slope of the BTF.